Modelling and Control ofTracked Vehicles

Anh Tuan Le

Department of Mechanical and Mechatronic EngineeringThe University of Sydney

26 January, 1999

26 January, 1999Australian Centre for Field Robotics

Department of Mechanical and Mechatronic EngineeringThe University of Sydney

This thesis is submitted to The University of Sydney in fulfillment of therequirements for the degree of Doctor of Philosophy. The thesis is entirely myown work and, except where otherwise stated, describes my original research.

Anh Tuan Le

Copyright c©1999 Anh Tuan LeAll rights reserved

Abstract

This thesis describes the development of a vehicle model and associated navigationsystem for a skid-steered tracked vehicle that operates on unprepared terrain. Thework is motivated by a multi-stage project to develop an autonomous trackedexcavator undertaken by the Australian Centre for Field Robotics. The vehicle isto navigate accurately and reliably in the environment without prior knowledge ofthe soil characteristics. This objective is difficult, and requires both an accurateand robust localisation system and accurate vehicle trajectory control. The thesismakes three main contributions towards accurate and reliable navigation andcontrol of tracked vehicles.

A new approach to soil-track interaction modelling incorporating track slips,the vehicle slip angle and track forces is presented. The equations developedcharacterise the relationship between the forces acting on the vehicle, the vehi-cle parameters and key soil properties. The soil-track interaction equations arethen used to develop a comprehensive model of the motion of a tracked vehicle.Incorporating kinematic and dynamic equations of vehicle motion with the soilequations allows robust and reliable estimation of the vehicle’s position using anextended Kalman filter. Finally, the same vehicle model also allows estimation ofthe key parameters of the soil upon which the vehicle operates.

The theoretical work is evaluated by simulation of a tracked vehicle, and isexperimentally tested using a specially modified tracked mini-excavator. In fieldtrials, the system is demonstrated to estimate the position of the vehicle withsub-metre accuracy, and to provide useful estimates of key properties of the soilover which the vehicle is moving.

Acknowledgments

First of all, I would like to thank Hugh Durrant-Whyte, the leader of our researchgroup, for giving me the opportunity to work on this project, for his ideas, andfor his support during my stay here at The University of Sydney.

My special thanks go also to my supervisor, David Rye, for his patience, adviceand assistance in every stage of this thesis. I would not be able to finish my workwithout his support.

There are many other people whom I would like to thank for their help andsupport during my stay here: Gamini Dissanayake, who helped me a lot in my firstyear, and later; Michael Stevens who was a true advisor and helper in all partsof my research; Chris Mifsud, Trevor Sutton, Bruce Crundwell, Nguyen HongQuang and Quang Phuc Ha who built the excavator and worked alongside meduring the experimental trials; Eduardo Nebot, Tim Bailey and Salah Sukkariehwho helped with the navigation system and with the laser scanner; and SimonJulier and Paul Newman who assisted with advice and discussions on filtering.

Thanks go also to the other members of the research group with whom Ishared a room for three years. They helped to make my time at Sydney easierand more enjoyable.

I owe my greatest debt to my late mother and my father who taught me howto live, and to my wife and my little boy who have given me love and supportthroughout my time here in Sydney.

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Contents

Abstract i

Acknowledgments ii

Contents iii

List of Figures vi

List of Tables x

List of Symbols xi

1 Introduction 11.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Main Contributions of the Thesis . . . . . . . . . . . . . . . . . . 71.4 The Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . 8

2 An Overview of Soil Mechanics 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 History of Tracked Vehicles . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 The Development of Tracked Vehicles Until 1918 . . . . . . 122.2.3 The Development of Tracked Vehicles After 1918 . . . . . 15

2.3 Classical Soil Mechanics . . . . . . . . . . . . . . . . . . . . . . . 162.3.1 Basic Understanding of Soil Mechanics . . . . . . . . . . . 162.3.2 Classical Soil Parameters . . . . . . . . . . . . . . . . . . . 18

2.4 Theoretical Analyses of Vehicle-Terrain Interaction . . . . . . . . 212.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.2 Theory of Plastic Equilibrium . . . . . . . . . . . . . . . . 222.4.3 Finite Element Method . . . . . . . . . . . . . . . . . . . . 242.4.4 Parametric Analysis . . . . . . . . . . . . . . . . . . . . . 26

2.5 Characteristic Behaviours of Soils . . . . . . . . . . . . . . . . . . 282.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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2.5.2 Cohesion and Shearing . . . . . . . . . . . . . . . . . . . . 282.5.3 The Pressure-Sinkage Relationship . . . . . . . . . . . . . 312.5.4 The Compaction of Soil and “Bulldozing” . . . . . . . . . 35

2.6 Effect of Soil Parameters on Vehicle Motion . . . . . . . . . . . . 382.6.1 The Effects of Cohesion and Shearing on the Vehicle . . . 382.6.2 The Effect of Sinkage on the Vehicle . . . . . . . . . . . . 40

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Modelling of Tracked Vehicles 423.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 The Track-Soil Model . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Force Model of a Tracked Vehicle . . . . . . . . . . . . . . . . . . 48

3.3.1 Vehicle Force System . . . . . . . . . . . . . . . . . . . . . 493.3.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . 59

3.4 The Model of the Tracked Vehicle . . . . . . . . . . . . . . . . . . 643.4.1 Kinematic Model . . . . . . . . . . . . . . . . . . . . . . . 643.4.2 The Comprehensive Vehicle Model . . . . . . . . . . . . . 71

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 Development of Estimation Techniques 804.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2.2 The Process Model . . . . . . . . . . . . . . . . . . . . . . 824.2.3 Sensor Model . . . . . . . . . . . . . . . . . . . . . . . . . 844.2.4 Basic Filtering Cycle . . . . . . . . . . . . . . . . . . . . . 86

4.3 The Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . 894.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3.2 State Prediction . . . . . . . . . . . . . . . . . . . . . . . . 904.3.3 Observation . . . . . . . . . . . . . . . . . . . . . . . . . . 924.3.4 Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.3.5 Understanding the Extended Kalman Filter . . . . . . . . 94

4.4 The Distribution Approximation Filter . . . . . . . . . . . . . . . 944.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4.2 Filter Principle . . . . . . . . . . . . . . . . . . . . . . . . 95

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5 Experimental Trials 995.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2 Design and Instrumentation of the Excavator . . . . . . . . . . . . 99

5.2.1 The Hydraulic System . . . . . . . . . . . . . . . . . . . . 1005.2.2 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2.3 Controller and Data Logging System . . . . . . . . . . . . 1085.2.4 Supervisory Controller and Data Acquisition . . . . . . . . 110

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5.2.5 Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2.6 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . 114

5.3 Test Sites and Test Procedures . . . . . . . . . . . . . . . . . . . 1155.3.1 Test Site Layout . . . . . . . . . . . . . . . . . . . . . . . 1155.3.2 First Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.3.3 Second Test . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.3.4 Third Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.3.5 Fourth Test . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.3.6 Fifth Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6 Estimating Ground Parameters 1566.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1566.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.2.1 Simulation: the Truth Model . . . . . . . . . . . . . . . . 1576.2.2 Simulation to Estimate the Slips . . . . . . . . . . . . . . . 1606.2.3 Interpretation of Results . . . . . . . . . . . . . . . . . . . 175

6.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 1766.3.1 Data Gathering . . . . . . . . . . . . . . . . . . . . . . . . 1766.3.2 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . 1766.3.3 Results and their Interpretation . . . . . . . . . . . . . . . 177

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7 Summary and Conclusion 1837.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1837.2 Summary of Each Chapter . . . . . . . . . . . . . . . . . . . . . . 1837.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.3.1 Modelling of Tracked Vehicles . . . . . . . . . . . . . . . . 1857.3.2 Soil Parameter Estimation . . . . . . . . . . . . . . . . . . 1857.3.3 Experimental Vehicle System . . . . . . . . . . . . . . . . 186

7.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1867.4.1 Vehicle Models . . . . . . . . . . . . . . . . . . . . . . . . 1867.4.2 Development of an Autonomous Tracked Excavator . . . . 187

Bibliography 188

A Vehicle Parameters 198A.1 Vehicle and Engine . . . . . . . . . . . . . . . . . . . . . . . . . . 198A.2 Hydraulic System . . . . . . . . . . . . . . . . . . . . . . . . . . . 200A.3 Undercarriage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

B A Summary of Some Fluid Mechanics 201

C Technical Parameters of Sensors and Actuators 204

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List of Figures

2.1 Different Interpretations of Edgeworth’s Patent . . . . . . . . . . 132.2 Measured Pressure Distribution for Various Tracked Vehicles . . . 192.3 Passive Earth Pressure on an Inclined Wall a− b . . . . . . . . . 202.4 Finite Element Mesh for Analysis of Ground Deformation Beneath

a Track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 Various Necessary Normal Pressure Distributions Under a Track . 272.6 Qualitative Relationships Between Soils, Sinkage and Load . . . . 322.7 The “Bulldozing” Effect . . . . . . . . . . . . . . . . . . . . . . . 352.8 Computing the Bulldozing Force . . . . . . . . . . . . . . . . . . . 372.9 Qualitative Shear Stress-Shear Displacement Relationship for Dif-

ferent Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1 Qualitative Shear Stress-Shear Displacement Relationship for Dif-ferent Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Shearing Action of a Track . . . . . . . . . . . . . . . . . . . . . . 453.3 Shear Deformation Modulus K . . . . . . . . . . . . . . . . . . . 473.4 Forces on the Tracks during Turning at Moderate to High Speed.

There is a net lateral force in the -ye direction. . . . . . . . . . . . 513.5 Lateral Force Distribution During Turning on the Spot or at Low

Speed. The distance D and the unbalanced lateral force are ap-proximately zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.6 Assumed Lateral Force Distribution on One Track . . . . . . . . . 533.7 Error of Moment of Turning Resistance for φmax = 1.37rads

−1 andµl = 0.1 · · · 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.8 Error of Instantaneous Centre Point for φmax = 1.37rads−1 andµl = 0.1 · · · 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.9 Errors in Computed Slip when Using Equation (3.12) for DifferentValues of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.10 Kinematic Motion of a Tracked Vehicle . . . . . . . . . . . . . . . 64

4.1 The Kalman Filter Cycle . . . . . . . . . . . . . . . . . . . . . . . 874.2 The Principle of the Distribution Approximation Filter . . . . . . 964.3 General Formulation of the Distribution Approximation Filter . . 97

5.1 The Komatsu Mini Excavator PC05, as Delivered . . . . . . . . . 100

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5.2 The Hydraulic Circuit Before Retro-fitting . . . . . . . . . . . . . 1025.3 The Hydraulic Circuit After Retro-fitting . . . . . . . . . . . . . . 1025.4 The Komatsu PC05, fitted with Sensors and Automatic Control

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.5 View from Above, Showing Moog Series 633 Direct Drive Servo

Valve Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.6 Novotechnik Inductive Angle Sensor . . . . . . . . . . . . . . . . . 1055.7 The Sick PLS Laser Scanner . . . . . . . . . . . . . . . . . . . . . 1075.8 Moog M2000 Programmable Servo Controllers and DC-DC Con-

verters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.9 Watson IMU Attitude and Heading Translation and Correction . 1125.10 Accelerometer Bias Correction . . . . . . . . . . . . . . . . . . . . 1135.11 Estimated Position and Slips in Run 1 . . . . . . . . . . . . . . . 1165.12 Position Read from Encoders and from IMU . . . . . . . . . . . . 1175.13 Position Covariance and Orientation Error . . . . . . . . . . . . . 1185.14 Estimated Positions in Run 2 and 3 . . . . . . . . . . . . . . . . . 1205.15 Positions Read from Encoders in Runs 2 and 3 . . . . . . . . . . . 1215.16 Positions Read from IMU in Runs 2 and 3 . . . . . . . . . . . . . 1225.17 Slips Estimated on Runs 2 and 3 . . . . . . . . . . . . . . . . . . 1235.18 Differential Pressures on the Motors in Runs 2 and 3 . . . . . . . 1245.19 Positions Estimated from the Laser Scanner Data and Read from

Encoders in Test Three, Run 1 . . . . . . . . . . . . . . . . . . . . 1275.20 Positions Estimated using Kinematic and Comprehensive Models

in Test Three, Run 1 . . . . . . . . . . . . . . . . . . . . . . . . . 1285.21 Estimated Slips in Test Three, Run 1 . . . . . . . . . . . . . . . . 1295.22 Track Drive Speeds & Forces in Test Three, Run 1 . . . . . . . . 1305.23 Positions Estimated from the Laser Scanner Data and Read from

Encoders in Test Four, Run 1 . . . . . . . . . . . . . . . . . . . . 1325.24 Positions Estimated using Kinematic and Comprehensive Models

in Test Four, Run 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.25 Estimated Slips from in Test Four, Run 1 . . . . . . . . . . . . . . 1345.26 Track Drive Speeds & Forces in Test Four, Run 1 . . . . . . . . . 1355.27 Positions Estimated from the Laser Scanner and read from En-

coders in Test Four, Run 2 . . . . . . . . . . . . . . . . . . . . . . 1375.28 Estimated Position and Slips using the Comprehensive Model in

Test Four, Run 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.29 Track Drive Speeds & Forces in Test Four, Run 2 . . . . . . . . . 1395.30 Positions Estimated from the Laser Scanner and read from En-

coders in Test Four, Run 3 . . . . . . . . . . . . . . . . . . . . . . 1405.31 Estimated Position and Slips using the Comprehensive Model in

Test Four, Run 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.32 Track Drive Speeds & Forces in Test Four, Run 3 . . . . . . . . . 142

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5.33 Positions Estimated from the Laser Scanner and read from En-coders in Test Four, Run 4 . . . . . . . . . . . . . . . . . . . . . . 143

5.34 Estimated Position and Slips using the Comprehensive Model inTest Test Four, Run 4 . . . . . . . . . . . . . . . . . . . . . . . . 144

5.35 Track Drive Speeds & Forces in Test Four, Run 4 . . . . . . . . . 1455.36 The Test Vehicle on Gravelled Soil . . . . . . . . . . . . . . . . . 1465.37 Positions Estimated from the Laser Scanner and read from En-

coders in Test Five, Run 1 . . . . . . . . . . . . . . . . . . . . . . 1475.38 Positions Estimated using Kinematic and Comprehensive Models

in Test Five, Run 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.39 Estimated Slips using the Kinematic and Comprehensive Models

in Test Five, Run 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.40 Track Drive Speeds & Forces in Test Five, Run 1 . . . . . . . . . 1505.41 Positions Estimated from the Laser Scanner and read from En-

coders in Test Five, Run 2 . . . . . . . . . . . . . . . . . . . . . . 1515.42 Positions Estimated using the Kinematic and Comprehensive Mod-

els in Test Five, Run 2 . . . . . . . . . . . . . . . . . . . . . . . . 1525.43 Slips Estimated using Kinematic and Comprehensive Models in

Test Five, Run 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.44 Track Drive Speeds & Forces in Test Five, Run 2 . . . . . . . . . 154

6.1 Simulated Vehicle Trajectory and Slip Angle . . . . . . . . . . . . 1586.2 Simulated Slip Coefficients . . . . . . . . . . . . . . . . . . . . . . 1596.3 Simulation on Sand and Sandy Loam with σx,y = 1mm and σω =

0.01rad, using EKF . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.4 Simulation on Clayey Soil and Dry Clay with σx,y = 1mm and σω

= 0.01rad, using EKF . . . . . . . . . . . . . . . . . . . . . . . . 1636.5 Simulation on Sand and Sandy Loam with σx,y = 1mm and σω =

0.01rad, using DAF . . . . . . . . . . . . . . . . . . . . . . . . . . 1646.6 Simulation on Clayey Soil and Dry Clay with σx,y = 1mm and σω

= 0.01rad, using DAF . . . . . . . . . . . . . . . . . . . . . . . . 1656.7 Simulation on Sand and Sandy Loam with σx,y = 1mm and σω =

0.1rad, using EKF . . . . . . . . . . . . . . . . . . . . . . . . . . 1666.8 Simulation on Clayey Soil and Dry Clay with σx,y = 1mm and σω

= 0.1rad, using EKF . . . . . . . . . . . . . . . . . . . . . . . . . 1676.9 Simulation on Sand and Sandy Loam with σx,y = 1cm and σω =

0.1rad, using EKF . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.10 Simulation on Clayey Soil and Dry Clay with σx,y = 1cm and σω

= 0.1rad, using EKF . . . . . . . . . . . . . . . . . . . . . . . . . 1696.11 Simulation on Sand and Sandy Loam with σx,y = 1cm and σω =

0.1rad, using DAF . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.12 Simulation on Clayey Soil and Dry Clay with σx,y = 1cm and σω

= 0.1rad, using DAF . . . . . . . . . . . . . . . . . . . . . . . . . 171

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6.13 Simulation on Sand and Sandy Loam with σx,y = 10cm and σω =0.1rad, using EKF . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.14 Simulation on Clayey Soil and Dry Clay with σx,y = 10cm and σω= 0.1rad, using EKF . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.15 Position, Forces and Resistances on Grass with Laser ScannerData. Conditions: σx,y = 1cm and σφ = 0.1rad, using EKF . . . . 178

6.16 Estimated Slips and Coefficients of Resistance on Grass with LaserScanner Data. Conditions: σx,y = 1cm and σφ = 0.1rad, using EKF179

6.17 Estimated Soil Shear Deformation Modulus K and Maximal Trac-tive Effort Fmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

A.1 Dimensions of the Komatsu PC05-7 . . . . . . . . . . . . . . . . . 199

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List of Tables

2.1 Shear Strength Parameters of Soils . . . . . . . . . . . . . . . . . 302.2 Pressure-Sinkage Parameters of Soils . . . . . . . . . . . . . . . . 34

6.1 Parameters of Different Soil Types . . . . . . . . . . . . . . . . . 157

C.1 Full Scale Range of RS232 Output Format . . . . . . . . . . . . . 208

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List of Symbols

A = 2bl : contact area of the track, m2

Au : parameter characterising the responseof the terrain to repetitive loading, N.m−4

B : tread of the tracks, mFcent : centrifugal inertial force, NFmax : maximal tractive effort, NFo,i : thrust on the outer, inner track, NK : shear deformation modulus, mO′ : instantaneous centre of rotationP fp : passive earth pressure, N.m−2

R : turning radius, mRb : bulldozing resistance force, NRl : lateral drag force, NRro,ri : longitudinal resistance force

on the outside, inside track, NW = mg : normal load, Nb : track width, mc : soil cohesion, N.m−2

g : earth’s gravitational acceleration, m.s−2

hb : height of soil in front of the track, mht : track height, mi : track slip, %io,i : slip of the outside, inside track, %j : shear displacement, mk′c, k

′ϕ : pressure-sinkage parameters, dimensionless

k0 : parameter characterising the responseof the terrain to repetitive loading, N.m−3

kc : pressure-sinkage parameter, N.mn+1

kϕ : pressure-sinkage parameter, N.mn+2

l : track length, mm : mass of the vehicle, kgn : pressure-sinkage parameter, dimensionless

xi

p : pressure exerted by the vehicle on the soil, N.m−2

pn : normal force acting on the wall, Nq : surcharge force acting on the soil surface, Nr : track rolling radius, mz : sinkage depth during unloading or reloading, mα : slip angle, rad

φ : angular velocity, rad.s−1

γs : weight density of the soil, kg.m−3

µl : coefficient of lateral resistance, dimensionlessµr : coefficient of longitudinal resistance, dimensionlessµ : angle of wall-soil friction, radωo,i : angular velocity of the outside, inside track drive sprockets,

rad.s−1

φ : heading angle of the vehicle, radτmax : maximal shear strength of the terrain, N.m−2

ϕ : angle of internal shearing resistance of the terrain, rad�n : n-dimensional Euclidean spaceF (k) : state transition matrix at time kF T (k) : the transpose of the transition matrix F (k)W (k) : Kalman weighting matrixS(k) : innovation covariance matrixP (k) : state prediction covariance matrixR(k) : observation-noise covariance matrixQ(k) : process-noise covariance matrixH(k) : observation model�fx(k) : the Jacobian of f evaluated at state x(k − 1) equal

estimate x(k − 1|k − 1)x(k|k − 1) : prediction of state x(k) given previous statesZk−1 : observationsE[x(k)|Zk−1] : expectation or mean of state x(k) given observations Zk−1

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Chapter 1

Introduction

1.1 Problem Statement

This thesis describes aspects of the mathematical modelling of tracked vehicles

and the development of a navigation and soil parameter estimation system for a

particular tracked vehicle. The work forms part of a multi-stage project under-

taken in the Australian Centre for Field Robotics to develop an autonomously

operated excavator. The navigation system is designed for use on excavators that

operate in highly unstructured environments such as those found in mines or

construction sites. This system is specified to fulfill the following requirements:

❶ Position estimation with a resolution less than one metre.

❷ Robustness with respect to uncertainties in the vehicle and environment

parameters, as well as to a lack of sensor data.

❸ Acquisition of knowledge of the terrain over which the vehicle is moving.

It is highly desirable to acquire information characterising the terrain over

which the vehicle is moving. This information can be used to control precisely

1.2 Background 2

the vehicle motion, and to provide parameter estimates to motion and task plan-

ning algorithms. Methods for the acquisition of this information are one of the

principal contributions of this thesis.

To satisfy the system requirements, state models of vehicle dynamics specific

to a tracked vehicle are developed and validated in the thesis. The selected state

models are derived from models available in the literature, and adapted to suit

the requirements of the particular motion platform. New interpretations of soil-

track interactions and methods of fusing kinematic and dynamic descriptions of

the vehicle’s behaviour are advanced to attain a working model that encompasses

possible errors and reduces the model complexity.

1.2 Background

Leonard and Durrant-Whyte [Durrant-Whyte and Leonard, 1991] summarised the

problem of mobile robot navigation in three questions: “Where am I?”; “Where

am I going?”; and “How do I get there?”. The first question implies a require-

ment for continuous vehicle localisation. The second question is mainly concerned

with the planning and tasking of the vehicle. The third question is one of local

and global path control. Determining exactly where the vehicle is at all times

is of immense importance in all practical mobile robot applications. Almost all

applications of autonomous guided vehicles (AGVs) require a vehicle that can

accurately and repeatedly move to designated locations within its work environ-

ment. This capability has become more important as applications of AGVs have

evolved in complexity. There is an increasing demand for tracked vehicles that can

operate in unstructured or even hazardous environments, and complete assigned

tasks without direct human involvement.

1.2 Background 3

Navigation of a vehicle about its work environment is a complicated task.

The environment may be complex, dynamic and contain unmodelled features

and unexpected disturbances. The motion of the vehicle can not be modelled

exactly because of parameter uncertainties, or because parameters may change

slowly through time. Reliable navigation of the vehicle can therefore be achieved

only if the motion of the vehicle can be monitored continuously, to ensure that a

proper trajectory is executed.

A survey of the vast body of literature on mobile robot positioning reveals

many partial solutions to the navigation problem. These may be categorised into

two groups: relative and absolute navigation. Dead-reckoning using encoders,

and inertial navigation using gyroscopes and accelerometers are both driven by

relative position measurements. The other methods, which utilise absolute posi-

tion measurements, are based on active beacons, artificial landmark recognition,

natural landmark recognition and model matching.

Dead reckoning is simply a mathematical procedure for determining the present

location of a vehicle by advancing from some previous position through known

heading and velocity information over a given length of time. The simplest im-

plementation of dead reckoning is usually termed odometry, which implies the

vehicle displacement along a path of travel is directly derived from some on board

“odometers”. A odometer is an instrument, often based on optical encoders, that

is directly coupled to the motor drive shafts or wheel axles. Since most vehicles

rely on some variation of wheeled locomotion, these encoders have to quantify

accurately angular position and velocity. There are a number of different types

of rotational displacement and velocity sensors in use today.

The advantage of dead reckoning is that it is self-contained and always capable

of providing the vehicle with an estimate of its position. The disadvantage is that

1.2 Background 4

the position error grows without bound and an independent position reference

must be used periodically to limit the error.

An inertial navigation system (INS) uses gyroscopes and accelerometers to

measure the rate of rotation and acceleration. The measured quantities can be

integrated to yield position and velocity. Inertial navigation systems have the

advantage that they are self-contained. Their disadvantage is that their outputs

drift with time because integration is required to yield position: any constant

error (bias) will increase linearly after integration. Inertial navigation systems

are therefore unsuitable for accurate positioning over an extended period of time.

The use of active beacons allows the computation of the absolute position of a

vehicle by measuring the incident direction of signals from three or more actively

transmitting beacons. The transmitters, typically using light or radio frequencies,

must be placed at accurately known locations in the environment.

Modern technology has vastly enhanced the capabilities of active beacon sys-

tems with the introduction of laser, ultrasonic, and radio-frequency (RF) trans-

mitters. A number of active beacon systems, such as LORAN (LOng RAnge

Navigation), GPS (Global Positioning System) and GLONASS (GLObal NAvi-

gation System Satellite), have been introduced and used over many years. Whilst

LORAN is a ground-based RF system, the other two systems transmit from Earth

satellites. Their accuracy ranges from 100m to 200m. The use of a carrier-phase

differential global positioning system (DGPS) can give absolute position measure-

ments of centimetre accuracy. DGPS has become the most promising method for

navigation of an AGV in outdoor environments where four or more satellites can

be maintained in line-of-sight. GPS is, however, subject to position inaccura-

cies caused by the loss of signals from one or more satellites, or from multipath

reflections.

1.2 Background 5

A promising hybrid navigation system combines GPS and INS measurements,

where the INS drift is bounded by GPS corrections. The accuracy of such a

system may be markedly improved through the use of position estimators that

are based on suitable vehicle models. The development and validation of tracked

vehicle models for estimation purposes is a central part of the present thesis.

Another method of absolute position measurement uses artificial landmark

recognition. In this method distinctive artificial landmarks are placed at known

locations in the environment. They can be designed in such a way as to be

easily detected and distinguished from each other, and from the environment.

When combined with associated techniques for estimating landmark position, this

method can provide very high accuracy. Landmark-based positioning possesses

the advantage that position errors are bounded.

Compared to artificial landmark recognition, natural landmark recognition

does not require specially designed objects or markers but uses features available

naturally in the environment. The main problem of natural landmark navigation

is to extract and match characteristic features from sensor inputs. Sensors can be

sonars, laser scanners, millimetre wave radars or computer vision units. Although

there is no need for preparation of the environment, the environment must be

known in advance. The reliability of this method is not as high as when artificial

landmarks are used.

Map-based positioning is an absolute position measurement method in which

information acquired from a vehicle’s on-board sensors is compared to a map or

world model of the environment. If features from the sensor-based map and the

world model map match, the vehicle’s absolute location can be estimated. The

maps used in navigation are of two major types: geometric maps and topological

maps. Geometric maps represent the world in a global coordinate system. Topo-

1.2 Background 6

logical maps represent the world as a network of nodes and arcs. The nodes of the

network are distinctive places in the environment and the arcs represent paths

between them. The main advantages of map-based positioning are that naturally

occurring structures can be used to localise the vehicle, and that the map can

be generated and refined during vehicle operation. The main disadvantages of

map-based position estimation are the requirements for sufficient invariant fea-

tures that can be used for matching, and that substantial sensing and processing

is required.

A common thread that runs through all of these navigation techniques is the

need for comprehensive, high fidelity kinematic and dynamic models of the vehicle

that is to be localised within the work environment. It is upon these vehicle

models that Kalman filters or other state estimators are built. Regardless of the

navigation techniques used, a good estimation model can markedly improve the

accuracy of navigation by unifying information from odometry and other relative

position sensors with information from absolute position sensors.

The development and validation of mathematical models suitable for estimat-

ing the position of a tracked vehicle is the first theme of this thesis. Secondly, it

is further demonstrated that the kinematic and dynamic models developed can

provide the basis of filters that allow the estimation of soil parameters govern-

ing the motion of the vehicle. The ability to estimate soil parameters such as

strength, shear modulus, and resistance characteristics will allow precise trajec-

tory control and may assist in the planning of soil working operations such as

loading or excavation.

1.3 Main Contributions of the Thesis 7

1.3 Main Contributions of the Thesis

The main contributions of this thesis are summarised as follows:

❶ The development of a new model of soil-track interaction incorporating

track slips, vehicle slip angle and forces on the tracks. This approach qual-

itatively and quantitatively describes the relationship between the forces

acting on the vehicle, the vehicle parameters and the soil properties. The

new soil-track interaction model augments and extends current knowledge

and understanding of tracked vehicle behaviour.

❷ The development and detailed analysis of models of tracked vehicle mo-

tion. The models developed incorporate equations describing the motion

of the vehicle, and the dependance of this motion on a number of key soil

parameters.

❸ The development of models that allow estimation of track slip, vehicle slip

angle, and soil parameters. Through the track-soil relationship, command

reference inputs and received sensor data can be correlated to allow esti-

mation of the governing soil properties during vehicle motion. These soil

properties provide data necessary to controlling and planning excavator

tasks.

❹ The implementation and demonstration of the navigation system on a mod-

ified commercial tracked vehicle. The vehicle hardware and software imple-

mentation is designed to allow easy improvement, upgrade or extension.

❺ The experimental validation of the feasibility of estimating track slip and

soil parameters during vehicle operation.

1.4 The Structure of the Thesis 8

1.4 The Structure of the Thesis

Chapter 1 provides an introduction to the scope of the thesis, including its main

contributions. An overview of the historical development of tracked vehicles and

of classical soil mechanics is presented in Chapter 2. These disciplines underpin

our understanding of the interactions between the soil and the vehicle tracks,

and provide a basis for the subsequent development of the soil-track interaction

model.

Chapter 3 deals with the mathematical modelling of one instance of a tracked

vehicle, specifically a Komatsu PC05-7 mini excavator. Using the knowledge of

soil mechanics and soil-track interaction presented in Chapter 2, kinematic and

dynamic equations of vehicle motion, and their dependance on soil properties, are

established. This development yields two vehicle models: the “kinematic” and

the “comprehensive” models. The kinematic model is based on the kinematic

equations of vehicle motion. A Kalman filter based on these equations can allow

estimation of the track slips and the vehicle slip angle. The comprehensive vehicle

model incorporates kinematic and dynamic equations of vehicle motion, together

with equations describing the soil-track relationship. A filter based on the com-

prehensive model can provide estimates of the soil’s maximum shear strength,

shear deformation modulus, and the coefficients of resistance in addition to the

other parameters estimated using the kinematic model.

In Chapter 4 two different estimation techniques are introduced and dis-

cussed. An understanding of their structures and characteristics is required for

the modelling of tracked vehicles. Both the extended Kalman filter (EKF) and

the distributed approximation filter (DAF) are capable estimators for nonlinear

processes. The DAF provides for an extension of the Kalman Filter to nonlin-

ear systems that is theoretically superior to the EKF. Whilst the EKF is a well

1.4 The Structure of the Thesis 9

proven practical estimator, the DAF is relatively new and unproven. Tests with

the DAF could not show significant performance improvements over the EKF, so

that most of the results in the present work have been produced using the EKF.

Chapter 5 describes the modification and instrumentation of a Komatsu

model PC05-7 mini excavator as a test-bed for conducting trials related to the

thesis. It also describes the test sites and the process of conducting the experi-

ments. Problems encountered during operation of the vehicle are discussed and

the results of different tests are shown. Discussion on these results is provided.

In Chapter 6 the simulated and experimental results of the developed models

are presented and compared. The simulation of the models shows the feasibility

and limits of the estimation techniques, together with the requirements on sensor

suites necessary to achieve useful results. The experimental tests conducted with

the excavator show promising results achieved using the developed vehicle models.

Finally, Chapter 7 provides a summary of the conclusions of this thesis.

Chapter 2

An Overview of Soil Mechanics

2.1 Introduction

According to Terzaghi [Terzaghi, 1943], soil mechanics is

“the application of the laws of mechanics and hydraulics to

engineering problems dealing with sediments and other un-

consolidated accumulations of solid particles produced by the

mechanical and chemical disintegration of rocks, regardless of

whether or not they contain organic constituents.”

In geology such material is termed mantle or regolith, and the term “soil” is

reserved for the uppermost layer containing decomposed organic matter which

supports plant life. Because our interests lie strictly with the mechanics of soil

or earth, we shall follow Terzaghi’s definition.

Soil mechanics encompasses theories of the behaviour of soils under stress or

imposed deformation, as supported by evidence from investigation of the physical

properties of real soils. Motion of a tracked vehicle over the ground will be

governed by the interaction between the soil and the tracks, so that a knowledge

10

2.1 Introduction 11

of soil mechanics is a necessary precursor to any investigation of track-terrain

interaction.

Before studying the track-terrain interaction, it is quite reasonable to enquire

into the meaning of terrain. In 1727 Bailey [Bailey, 1727] defined it as “the

manage-ground upon which the horse makes his pist or tread”. At a time when

horses were the most popular means of transportation, this definition seemed to

be justified. With increasing developments of science and technology, especially

in the field of soil mechanics, the term has to be defined more precisely. Today

the WWWebster Dictionary defines terrain as “the physical feature of a tract of

land”, while the Oxford English Dictionary defines it as “a name for a connected

series, group, or a system of rocks or formations; a stratigraphical subdivision”.

Using common sense it could be said that terrain is the way that the land lies.

This simple definition is easy to understand, and at the same time captures the

logical meaning of the term. In the remainder of this thesis the term “track-soil”

will be used to describe the same subject.

This chapter therefore presents an overview of the development of tracked

vehicles and of classical soil mechanics, as these disciplines relate to interactions

between the soil and a vehicle’s tracks. The chapter serves both as an overview of

soil mechanics, and as a basis for the mathematical models of tracked vehicles that

are developed in Chapter 3. Section 2.2 gives a brief overview of the development

of tracked vehicles. Section 2.3 then introduces some definitions and terminology

of classical soil mechanics. Section 2.4 introduces some of the theoretical methods

currently used to analyze the interactions between the vehicle and the terrain.

Section 2.5 then discusses some characteristic behaviours of soils, and the effects

of these behaviours are discussed in Section 2.6.

2.2 History of Tracked Vehicles 12

2.2 History of Tracked Vehicles

2.2.1 Introduction

This section introduces the development of tracked vehicles and the relationship

between their design and soil properties. This relationship starts from a primi-

tive interpretation of soil behaviour to the modern soil mechanics of today. The

development has led to a better understanding the soil not only for the purpose

of locomotion and vehicle design, but also within the construction and building

industry. Subsection 2.2.2 traces the beginning of tracked vehicles from the 18th

century until the end of World War I. In this time period the design and construc-

tion of tracked vehicles depended mainly on the development of the car industry.

The tracked vehicle was merely a derivative of the car or simply a modification

of it. Subsection 2.2.3 then introduces the development of tracked vehicles from

the time after World War I. From that time the influence of soil mechanics on

the design and construction of tracked vehicles was recognised.

2.2.2 The Development of Tracked Vehicles Until 1918

The development of tracked vehicles dates back to 1770, when a design, in-

vented by Richard Lovell Edgeworth in England, was patented [Bekker, 1962].

The patent described a “portable railway” or artificial road, which was to move

along with any carriage applied to it. If this portable railway was built from

several pieces of wood, which were connected in such a manner as to form an

endless train girding the front and rear wheel of a carriage, then this was the idea

of a full-track vehicle (see figure 2.1). Depending on the interpretation of the im-

plied idea, this patent could refer to a individual wheel track or to a shoed wheel

as well. During the 19th century a great number of inventors were attracted to

2.2.2 The Development of Tracked Vehicles Until 1918 13

Figure 2.1: Different Interpretations of Edgeworth’s Patent,from Bekker[Bekker, 1962]

2.2.2 The Development of Tracked Vehicles Until 1918 14

full-track vehicles and a large number of patents were registered. In spite of these

inventions, no tracked vehicles in the modern sense were developed. The technol-

ogy was in its infancy, and many critical problems could not be solved properly.

The propulsion available could not supply the required power. The steering of

the tracked vehicles was not reasonable. The materials that were available at that

time were only wood and cast iron, whose quality could not satisfy a number of

requirements for a tracked vehicle: the resistance and reliability for the highly

stressed tracks, and the useful load as the end result of the design of the vehicle.

The situation changed rapidly at the end of the 19th century, when the

internal-combustion engine was invented and the first automobiles were manu-

factured. The first half-tracked steam engine appeared in 1901 and later the

conversion of passenger cars to tracked vehicles took place. Before World War

I, agricultural tractors were built and used widely. The limited use of tracked

vehicles at that time resulted from the general status of technology and from

an insufficient theoretical knowledge of the phenomena involved between the soil

and vehicle. The principles of soil mechanics, expounded from a civil engineering

point of view by Terzaghi were too novel to form a basis for the investigation of

soil-vehicle relations.

World War I saw the deployment of the first tanks in the battle field. Costly

experiences on the battle field provided impetus for studying the relationship

between a vehicle and its environment. After World War I the development of

wheeled and tracked equipment finally became a well-organised discipline in which

there was ample space for fundamental study.

2.2.3 The Development of Tracked Vehicles After 1918 15

2.2.3 The Development of Tracked Vehicles After 1918

After World War I the cross country vehicles that were widely used during the

war were converted to new civilian applications, mainly in agriculture. The effort

to improve the agricultural machinery in many countries led to new definitions

of merit for various types of machinery and for certain relationships between

soil and vehicle. Many details of locomotion mechanics now emerged in their

full significance. The effect of the centre-of-gravity location of a tractor upon

the pressure distribution and tractive effort, the importance of supporting the

greatest number of track links, the action of spuds and grousers as well as the

problems of movement resistance, sinkage and wheel dimension were recognised.

The idea of ground pressure, defined as the ratio of the total weight of a vehicle

to the contact area with the ground was widely introduced.

By trial and error, the technical standard of agricultural tractors was im-

proved. The progress closely followed the development of automobiles and there-

fore had little to do with the soil. The idea of ground pressure, extended to

types of vehicles and soils other than those contemplated in agriculture, has been

misleading as far as proper evaluation of various developments is concerned.

With the outbreak of World War II, development of new cross country vehi-

cles started again. The experience gained concentrated on limited, though vital,

trends in development: a tendency toward using larger guns and heavier armour.

Tracked vehicle suspension and steering mechanisms therefore had to be adapted

to new conditions, and the vehicles reached the practical upper limit of weight

and dimensions. The effect of many previously unimportant factors influencing

vehicle performance now came into the picture. It was discovered that some ve-

hicles with higher ground pressure had better performance than those with lower

pressure, and that those with shallow spuds developed a better tractive effort

2.3 Classical Soil Mechanics 16

than those with deep ones. The track length could not be made longer, because

the vehicle could not be steered. Under these circumstances, the necessity of

basic research became clear.

The design of modern tracked vehicles is adapted to the soil conditions and to

the applications domain where they will be used. An understanding of soil me-

chanics has now permeated the design and construction of tracked vehicles. For

example, they now have very high performance compared to their predecessors.

Modern tracked vehicles can travel cross-country at speeds closed to 100kph, and

operate reliably on different terrains and climates. They have become indispens-

able in many situations where roads are not available and goods and equipment

need to be brought in. They help to explore and to work new land and, with

their low ground pressure, to conserve the natural environment.

2.3 Classical Soil Mechanics

2.3.1 Basic Understanding of Soil Mechanics

The mechanical properties of most engineering materials are not sufficiently sim-

ple to be acceptable as a basis for theoretical analyses. Practically every theory

in applied mechanics is therefore based on a set of assumptions concerning the

mechanical properties of the material involved. These assumptions are always at

variance with reality to a certain extent.

The mechanical properties of soils range between those of plastic clay and

those of clean, perfectly dry or completely wet sand. When digging into a bed

of dry or water-saturated sand, the material at the sides of the excavation slides

toward the bottom. This material behaviour indicates the complete absence of a

bond between the individual sand particles. The sliding material does not come

2.3.1 Basic Understanding of Soil Mechanics 17

to rest until the angle of inclination of the slopes equals a certain angle, known as

the angle of repose. The angle of repose of dry sand as well as that of completely

immersed sand is independent of the height of the slope.

In comparison, a trench with unsupported vertical sides 6 to 9 metres deep

can be excavated in stiff plastic clay. This indicates the existence of a firm bond

between the clay particles. The sides of the cut will, however, fail as soon as the

depth of the trench exceeds a critical value, depending upon the intensity of the

bond between the clay particles. No definite angle of repose can be assigned to a

soil with cohesion, because the steepest slope at which such a soil can withstand

decreases with increasing height of the slope. Even sand has some cohesion, if it

is moist.

In spite of the apparent simplicity of their general characteristics, the me-

chanical properties of real sand and clays are so complicated that a rigourous

mathematical analysis of their behaviour is impossible. Hence in theoretical soil

mechanics, materials are idealised as ideal sands and ideal clays, whose mechan-

ical properties represent a simplification of those of real sands and clays. The

difference between the real and ideal soils can be illustrated by the following ex-

ample. In practice most real soils are capable of sustaining a considerable defor-

mation without appreciable loss of shearing resistance. Theoretical soil mechanics

then assumes that the shearing resistance of ideal soils is entirely independent of

the degree of deformation. Because of this assumption all theories involving the

shearing resistance of soils are more or less at variance with reality. Rigourous

mathematical solution of a set of assumed governing equations cannot eliminate

the error associated with the fundamental assumption. In many cases this er-

ror is much more important than the error due to a radical simplification of the

mathematical treatment of the problem. The difference between assumed and

2.3.2 Classical Soil Parameters 18

real mechanical properties is, however, very different for different soils.

Due to the simplifications usually made in theoretical analysis, most equations

in soil mechanics are empirical and represent a physical interpretation of underly-

ing processes, rather than an exact understanding of them. Empirical equations

nevertheless assist in promoting an understanding of interactions between a ve-

hicle and the ground.

In addition to the soil mechanics literature, a large body of work exists in the

field of terramechanics: the study of vehicle-terrain interactions. It is important

to point out that the deformation of the soil beneath a vehicle depends not only

on the manner in which the vehicle loads the soil, but also on the properties of

the soil itself. A tracked vehicle with fewer large-diameter wheels exerts higher

peak values of the ground pressure, compared to one with more small-diameter

wheels and the same average ground pressure (see Figure 2.2). For some weak

soils these peak values can reduce the support for the track by damaging the soil

structure, while for sand it can increase the support.

2.3.2 Classical Soil Parameters

In the early stage of tracked vehicle development, when theoretical and practical

knowledge of the soil and its interaction with the tracks was insufficient, the

importance and effects of soil parameters on the motion of tracked vehicles could

not be assessed properly. To explain and calculate the soil resistance forces acting

on the tracks, Coulomb’s coefficient of friction was taken from solid mechanics.

The simple assumption that the resistance is proportional to the weight of the

vehicle and that there is a constant coefficient of resistance could neither describe

accurately the processes occurring between the tracks and the terrain, nor help

improve the design of the tracked vehicles.

2.3.2 Classical Soil Parameters 19

Figure 2.2: Measured Pressure Distribution for Various Tracked Vehicles, fromRowland [Rowland, 1972]

2.3.2 Classical Soil Parameters 20

A different approach to understanding the interaction between the soil and

the tracks uses parameters and terminology from soil mechanics. To explain the

Rigourous of the soil under the stresses caused by tracks the Rankine theory of

passive earth pressure has been applied. In the broadest sense the term passive

earth pressure indicates the resistance of the soil to forces that tend to displace

it. In civil engineering practice passive earth pressure is frequently utilised to

provide a support for structures such as retaining walls or bulkheads which are

acted upon by horizontal or inclined forces. The Rankine theory of passive earth

pressure states that a soil that fails by lateral compression (see Figure 2.3), will

form a failure surface inclined at an angle of 450 −φ/2 with the horizontal, where

φ is the angle of shearing resistance. In Figure 2.3 the angle of the inclined wall

a− b to the horizontal is α, h is the height of a− b. The surcharge q acting on the

soil surface results in a normal force pn acting on the wall, Pfp is the passive earth

pressure, µ is the angle of wall-soil friction; and φ is the angle of internal shearing

resistance of the soil. For a very rough surface a− b, µ is almost equal to φ. The

zone a− b− d is called the Rankine zone. Together with Coulomb’s coefficient of

friction, this theory provides the fundamental equations of soil-vehicle interaction

in the early stage of tracked vehicle development.

As mentioned in Section 2.2 there are many parameters that affect the per-

Rankine zone

α

µ

pfp

pn

h

Q

W

a

b d

eq

45 -o

φ/2 φ/245 -o

Figure 2.3: Passive Earth Pressure on an Inclined Wall a − b, after Bekker[Bekker, 1962]

2.4 Theoretical Analyses of Vehicle-Terrain Interaction 21

formance of a tracked vehicle. For example, the centre-of-gravity location, wheel

diameter, spud form, track length, track width, sinkage, the ratio between the

length and width of the track and the ground pressure are all important. The

ground pressure was one of the most common parameters initially used for assess-

ing vehicle performance. It is defined as the ratio of the vehicle’s weight to the

contact surface area of the tracks with the ground. It was believed that a lower

ground pressure led to a higher tractive effort. Practice showed, however, that

some vehicles with higher ground pressure could develop tractive effort higher

than those having lower ground pressure.

The ratio between the length and the width of the track was not given much

attention in the early stage of development of tracked vehicles. It was a common

belief that reducing the width of the track could reduce the movement resistance.

This was true, but reducing the width required increasing the track length to

keep the ground pressure at a constant value. The vehicle became un-steerable

because the longer track caused much higher resistance during turning. This led

to the realisation that performance improvement should not be sought by merely

increasing the track dimensions in accordance with the ground pressure formula.

2.4 Theoretical Analyses of Vehicle-Terrain In-

teraction

2.4.1 Introduction

In Sections 2.2 and 2.3 discussions regarding the development of tracked vehicles

and knowledge of classical soil mechanics revealed many disadvantages of apply-

ing empirical approaches to solve soil mechanics problems. Practical “trial and

2.4.2 Theory of Plastic Equilibrium 22

error” methods proved to be useful in the early stages of tracked vehicle develop-

ment, where analytical techniques were not sophisticated and experiments were

the only feasible way of improving vehicle performance. Nowadays experimen-

tation has become rather expensive. In recent years attempts have been made

to apply the theory of plastic equilibrium, the finite element method (FEM) and

critical state theory to the analysis of vehicle-terrain interactions, with the pur-

pose of improving the understanding of the interaction as well as incorporating

the knowledge gained in the design of tracked vehicles. This section will introduce

these techniques and their possibilities and limits in application. Subsection 2.4.2

discusses some aspects of the theory of plastic equilibrium. Subsections 2.4.3 and

2.4.4 respectively introduce the finite element method and parametric analysis.

2.4.2 Theory of Plastic Equilibrium

The theory of plastic equilibrium has been quite widely used in soil mechanics

and foundation engineering, as well as in the analysis of metal forming processes.

An extensive experimental investigation into the physical nature of vehicle-terrain

interaction carried out by Wong and Reece [Wong and Reece, 1966, Wong, 1967]

in the 1960’s revealed that failure zones were developed in dense soils under the

action of a vehicle’s running gear and that there was a close correlation between

the failure utilised of soils and the performance of the vehicle running gear. These

findings stimulated considerable interest in the application of the theory of plastic

equilibrium to the prediction of off-road vehicle performance.

When the load exerted by the vehicle on the terrain surface reaches a certain

level, the terrain mass within a specific volume will approach a state of failure.

An infinitely small increase in the load beyond this level produces a rapid increase

in the strain of the terrain mass within the specific boundaries, which constitutes

2.4.2 Theory of Plastic Equilibrium 23

plastic flow. The state that precedes plastic flow is usually referred to as plastic

equilibrium. The transition from the state of plastic equilibrium to that of plastic

flow represents the failure of the mass. The condition of failure by plastic flow is

determined by Mohr-Coulomb’s equation (2.1) of shear failure:

τmax = c+ p tanφ , (2.1)

where c and φ are respectively the apparent cohesion and the angle of internal

resistance of the soil, p is the normal pressure and τmax is the maximum shear

strength of the soil. To predict the load applied by the vehicle, which causes

the terrain mass within a certain volume to enter a state of plastic equilibrium,

a set of equilibrium equations has to be solved. These equations are generally

quite complex and are difficult to solve rigorously. Certain information on the

vehicle-terrain interface, such as the direction of major principal stresses on the

boundary, must be known for initialisation. These boundary conditions are ex-

tremely complex in practice and it is very difficult to specify them to initiate the

solution process.

The theory of plastic equilibrium could provide an insight into the physical

nature of certain aspects of vehicle-terrain interaction. The theory of plastic equi-

librium as presently applied is based on an assumption that the terrain behaves

as a rigid, perfectly plastic material. That is, the material does not deform until

it reaches a state of stress at which failure occurs. Beyond this the strain of the

soil increases rapidly, while the stress remains unchanged. In practice, dense soils

may exhibit utilised similar to that of a rigid perfectly plastic material, while most

other natural terrains, such as the soft marginal terrains encountered by off-road

vehicles, have a high degree of compressibility and therefore their utilised is far

2.4.3 Finite Element Method 24

from that of a rigid perfectly plastic material. The consequence is that failure

zones in these terrains under the load of vehicles may not develop in the manner

assumed in the theory and the sinkage of the vehicle running gear is primarily due

to compression of the terrain and not to plastic flow of the material. Furthermore,

the theory of plastic equilibrium is primarily concerned with the prediction of the

maximum load that the vehicle can exert on the terrain without causing soil fail-

ure. This is an extreme case during operation of tracked vehicles off-road; most

the time the vehicle will operate well within the failure limit. The usefulness of

the theory of plastic equilibrium is therefore limited in the practice of modelling

tracked vehicles. It is more or less useful in predicting soil failure in the civil

engineering design.

2.4.3 Finite Element Method

The finite element method (FEM) is a numerical method that can be used for

accurate solution of complex engineering problems. Developed in 1956 for the

analysis of aircraft structural mechanics problems, it has been well established

over the years as a capable method for the solution of different types of applied

science and engineering problems. Today it is considered to be one of the best

methods for solving efficiently a wide variety of practical problems involving par-

tial differential equations.

The essence of the FEM is the discretisation of the domain or solution region

into subregions (finite elements). This is equivalent to replacing a domain having

an infinite number of degrees of freedom by a system having a finite number of

degrees of freedom. The elements are considered to be interconnected at spec-

ified joints which are called nodes or nodal points. The shapes, sizes, number

and configuration of the elements have to be chosen carefully such that the orig-

2.4.3 Finite Element Method 25

Figure 2.4: Finite Element Mesh for Analysis of Ground Deformation Beneath aTrack, from Karafiath [Karafiath, 1978]. The unconnected nodes in the mesh areerroneous.

inal body or domain is simulated as closely as possible without increasing the

computational effort needed for the solution. Basic element shapes can be a line

segment (one-dimensional element), triangle, rectangle, quadrilateral or parallel-

ogram (two-dimensional element), tetrahedron, rectangular prism or hexahedron

(three-dimensional element). Some problems may be necessary by using an asym-

metric, ring type element or finite element with curved sides. When applied in

solving the soil-track interaction problem, the terrain is represented by a system

of elements with known constitutive relationships interconnected at nodes to form

a mesh. Figure 2.4 shows a mesh of triangular elements developed for the analysis

of terrain deformation under a track system.

Despite of the development of powerful computers the application the finite

element method in soil mechanics is limited to some specific cases. This situation

may be explained by the fact that soil is naturally anisotropic and inhomogeneous.

The soil, especially the upper soil layer, is exposed to phenomena such as rain,

wind, sun and biological processes. Their effect is to make the soil very inhomo-

geneous and it becomes difficult to predict accurately its utilised and reaction to

2.4.4 Parametric Analysis 26

any external force. Because of its anisotropy a variety of analytical methods seem

to be inappropriate to achieve accurate solutions to soil mechanics problems. Fur-

thermore, the finite element method requires known constitutive relationships of

the terrain as input, which are usually difficult to define. For example, the mod-

ulus of elasticity, Poisson’s ratio, and other mechanical properties of the terrain

vary with the stress level, loading history, lateral constraint and other factors.

To simplify the analysis the stress-strain relationship and the strain hardening

utilised of the terrain are commonly assumed to be similar to those of metals. This

assumption cannot be rationalised with the available empirical evidence. Also, by

assuming that the track system is equivalent to a rigid footing or to a completely

flexible belt, important design features, such as track system configuration, road

wheel arrangement, initial track tension, and suspension characteristics are ig-

nored. It has been shown, however, that these design parameters have significant

effects on the performance of a tracked vehicle [Wong and Preston-Thomas, 1984,

Wong, 1986, Wong and Preston-Thomas, 1988].

2.4.4 Parametric Analysis

A method for parametric analysis of track system performance was first developed

by Bekker [Bekker, 1962, Bekker, 1969]. In this method it is assumed that the

track in contact with the terrain is similar to a rigid footing1. If the centre of

gravity of the vehicle is located at the mid-point of the track contact area, the

normal pressure distribution will be assumed to be uniform, as shown in Figure

2.5a. Otherwise a trapezoidal form of pressure distribution will be assumed, if

the centre of gravity of the vehicle is located ahead of or behind the mid-point of

1For flexible tracks and suspended wheels the pressure distribution may approximate thesinusoidal form in Figure 2.5b, where the peaks relate to the wheels, or a curved form as inFigure 2.5e

2.4.4 Parametric Analysis 27

Figure 2.5: Various Necessary Normal Pressure Distributions Under a Track, fromWills [Wills, 1963]

the contact area. Using the pressure sinkage relationship (discussed in Subsection

2.5.3), track sinkage and motion resistance due to compacting the terrain can be

predicted. Based on the shear stress-shear displacement relationship and the shear

strength of the terrain, discussed in Subsection 2.5.2, the thrust-slip relationship

and the maximum tractive effort of a track may be determined.

While the idealisation of a track system as a rigid footing may be reasonable

for tracked vehicles with low ratios of road wheel spacing to track pitch such as

those commonly used in agriculture and in the construction industry, it is not

realistic for tracked vehicles with high ratios of road wheel spacing to track pitch

designed for high speed operations. Ground pressure under these vehicles is usu-

ally concentrated under the road wheels and is far from uniform. Consequently

performance predictions using the method of parametric analysis can be unreal-

istic, particularly with respect to sinkage, motion resistance and tractive effort

2.5 Characteristic Behaviours of Soils 28

on soft marginal terrain.

2.5 Characteristic Behaviours of Soils

2.5.1 Introduction

This section presents some soil properties that characterise interactions between

the ground and the tracks. It is believed that these fundamental soil proper-

ties have a significant effect on the motion of a tracked vehicle and are therefore

important in modelling soil-track interaction. Another aspect is that these prop-

erties conform to the available standards used in soil mechanics and agriculture.

These properties can be measured by standard methods. Subsections 2.5.2 to

2.5.4 will discuss the fundamental soil properties used in the soil-track modelling.

2.5.2 Cohesion and Shearing

The soil surface upon which a vehicle moves may be composed of any of the

materials from the upper layer of the earth’s crust. Movement over practically

rigid surfaces, such as rock or concrete, is not considered in this thesis. In the case

of non-deformable surfaces the equations of soil mechanics do not apply, and the

Coulomb friction equations may reasonably be used to compute the resistances.

Since the scope of this thesis relates to soil problems and off-road conditions, only

soil problems and characteristics will be discussed.

Aggregations of granular masses may exhibit cohesive and frictional proper-

ties. The cohesion of a soil, as defined by Terzaghi [Terzaghi, 1943], is the bond

that arises between soil particles. This bond cements soil particles together inde-

pendently of the pressure upon them. This is different for frictional masses, where

their particles can in principle be held together only when a pressure is exerted

2.5.2 Cohesion and Shearing 29

between them. Thus the shear strength of plastic snow or clay, for example, does

not depend theoretically upon the load, whereas that of a dry sand increases with

the load. The shear strength of the material therefore depends not only upon the

normal pressure, but also on the coefficient of cohesion c and the angle of internal

friction φ. Coulomb’s equation for the shear strength of a soil is

τ = c+ p tanφ , (2.2)

where τ is the shear strength and p is the normal stress. For dry sand there is no

cohesion between the particles, and this equation will take the form:

τ = p tanφ . (2.3)

For plastic clay, where there is no friction between the particles, the angle φ

vanishes so that

τ = c . (2.4)

In practice, the relationship between the shear strength and soil properties is

more complicated than equation (2.2) suggests. Depending on the loading rate,

saturated soil may react a considerable part of the load through the hydrostatic

pressure developed by water pockets enclosed by pores in the soil. In such cases,

the values of c and φ may vary as well. The cohesion of sand, for example,

could increase depending on the amount of water present, while the cohesion of

2.5.2 Cohesion and Shearing 30

clay can be reduced. Similar effects may happen with the angle of friction φ.

Furthermore, inhomogeneity of the ground can influence the soil shear strength

too as it develops different stresses in the structure of the soil. These stresses

could harden or weaken the soil, depending on the way the soil is composed. The

cohesion and angle of shearing resistance will also be affected and changed.

As a consequence of these effects, c and φ should be regarded only as empir-

ical coefficients of equation (2.2), rather than soil constants having any strictly

physical meaning. The coefficients promote an understanding of the relationship

between the soil and tracks and allow the development of meaningful models of

this relationship. The coefficients c and φ will be used in chapter 3 to develop the

excavator track model, where they will hold their empirical meaning. It will also

be shown that these coefficients affect the motion of the tracked vehicle through

the slips of the tracks. A summary of some typical values of these coefficients for

different soil types is given in Table 2.1, whereK is the shear deformation modulus

to be discussed in Chapter 3. These reference parameters have been used for the

computer aided method of parametric evaluation of wheeled vehicle performance,

developed by Wong and Preston-Thomas [Wong and Preston-Thomas, 1984], to

simulate the performance of wheeled vehicles.

Terrain Internal Shearing Rubber-Terrain ShearingType c(kPa) φ(deg) K(cm) c(kPa) φ(deg) K(cm)

Medium Soil 8.62 22.5 2.54 4.64 19.9 2.12Clayey Soil 7.58 14 2.54 4.08 12.4 2.12Soft Soil 3.71 25.6 2.1 - - -Lete sand 1.3 31.1 1.2 0.7 27.5 1.0

Petawawa Snow 0.4 24.0 - 0.12 16.4 0.4Petawawa Muskeg 2.8 39.4 3.1 - - -

Table 2.1: Shear Strength Parameters of Soils, from Wong and Preston-Thomas[Wong and Preston-Thomas, 1984]

2.5.3 The Pressure-Sinkage Relationship 31

2.5.3 The Pressure-Sinkage Relationship

The sinkage of tracked vehicles due to elastic or plastic soil deformation is a

source of power and traction loss. Although the elasticity of the soil does not

cause power loss, any real soil in fact has a combined elastic-plastic character.

This subsection discusses the relation between the track contact pressure that

tends to compact the soil, the track sinkage, and of other factors that constitute

a sinkage model.

Bekker [Bekker, 1969] assumes that a track may be represented by a rigid

rectangular plate. According to Bekker the relationship between track pressure

and sinkage in homogeneous soil may be characterised as

p =

(kcb+ kϕ

)zn , (2.5)

where p is the track contact pressure, b is the smaller dimension of the contact

patch2, z is the sinkage depth, and n, kc and kϕ are pressure-sinkage parameters.

It has further been shown that the parameters kc and kϕ are insensitive to the

width of the rectangular contact patch, provided that the aspect ratio of the patch

exceeds five to seven.

It is often the case that the soil has been under repetitive loading by the

running gear when the vehicle is in straight line motion. Figure 2.6 shows quali-

tatively the response to repetitive normal loading of a sandy terrain, an organic

terrain, and a snow-covered terrain respectively. It can be seen that the pressure

initially increases with sinkage along curve OA. However when the load applied

2that is, the width of a rectangular contact area, or the radius of a circular contact area, asused in the bevameter technique for measuring the response of terrain to loading pertinent tovehicle mobility studies. This method was proposed by Bekker [Bekker, 1969].

2.5.3 The Pressure-Sinkage Relationship 32

Pres

sure

O

A

Sinkage

C

B D

(a) Response of a sandy soil to repetitivenormal load

Pres

sure

SinkageO B

A

D

C

(b) Response of an organic soil to repeti-tive normal load

Pres

sure

O

A

C

Sinkage

E

B D

(c) Response of snow to repetitive normalload

Figure 2.6: Qualitative Relationships Between Soils, Sinkage and Load, afterWong [Wong, 1989b]

2.5.3 The Pressure-Sinkage Relationship 33

to the terrain is reduced at A, the pressure sinkage relationship during unloading

follows line AB. When the load is reapplied at B, the pressure sinkage rela-

tionship follows, more or less, the same path as that during unloading for the

sandy terrain and snow-covered terrain, shown in Figures 2.6a and 2.6c. For the

organic terrain, however, when the load is reapplied at B, the pressure sinkage re-

lationship follows a different path from that during unloading, as shown in Figure

2.6b.

Based on experimental observations, the pressure-sinkage relationship during

both unloading and reloading may be approximated by a linear function that

represents the average response of the terrain [Wong, 1989a]:

p = pu − ku(zu − z) , (2.6)

where p and z are the pressure and sinkage during unloading or reloading, pu and

zu are the pressure and sinkage, respectively, when unloading begins, and ku is

the pressure-sinkage parameter representing the average slope of the unloading-

reloading line AB. The value of ku is a function of the sinkage zu when unloading

begins and may be expressed by

ku = k0 + Auzu , (2.7)

where k0 and Au are parameters characterising the response of the terrain to

repetitive loading. Table 2.2 presents some values of k0 and Au for different types

of soil.

Equation (2.5) is essentially empirical. The parameters kc and kϕ have di-

2.5.4 The Compaction of Soil and “Bulldozing” 34

Terrain ParametersTerrain n kc kφ k0 AuType kN/mn+1 kN/mn+2 kPa/m kPa/m2

Medium Soil 0.8 29.76 2083 0 192,400Clayey Soil 0.6 38.08 499.7 0 63,106Soft Soil 0.8 16.54 911.4 0 86,000

Table 2.2: Pressure-Sinkage Parameters of Soils, from Wong [Wong, 1989a]

mensions that vary, depending on the value of the exponent n. Experimental

measurements of n, kc and kϕ for a variety of terrains have been given in Wong

[Wong, 1989a] and are summarised in Table 2.2. An alternative equation based

on work of a more fundamental nature in soil mechanics, and on experimental

evidence, has been proposed by Reece [Reece, 1966]:

p =(ck′c + γsbk

′φ

) (z

b

)n, (2.8)

where n, k′c and k′

ϕ are pressure-sinkage parameters, γs is the weight density of

the soil, and c is the cohesion. Equation (2.8) is to be favoured over equa-

tion (2.5) in that the parameters k′c and k′

ϕ are dimensionless. Reece’s equation

is also consistent with theoretical approaches to calculating a soil’s bearing ca-

pacity [Terzaghi, 1943]. Unfortunately, the additional variables that must be

estimated complicate the track-soil model. Experimental data that encapsulate

the pressure-sinkage parameters are scant, and it is very difficult to observe these

parameters through direct or indirect measurement from a moving vehicle.

2.5.4 The Compaction of Soil and “Bulldozing” 35

2.5.4 The Compaction of Soil and “Bulldozing”

A tracked vehicle develops tractive effort by deforming the soil in longitudinal

shear. As the vehicle moves across the soil, a counter force will arise from the

soil and is to equal the tractive force. The vehicle will plastically deform the

soil in the vicinity of the tracks. Such a deformation requires a definite amount

of energy from the vehicle and causes what is termed the external resistance to

the vehicle’s motion. The external resistance arises as a result of three types of

soil deformation: vertical plastic deformation, “bulldozing”, and soil drag (see

Figure 2.7). In this figure W is the vehicle’s weight, s is the track length in

contact with the soil, ht is the track height, hb the height of soil in front of the

track, z0 is the track sinkage, Rl is the lateral drag, and Rb is the bulldozing force.

The first factor is the compaction of the soil as a result of the vehicle’s weight.

In this case the vehicle has to exert a force to compact the soil mass over which

a track passes to zero depth relative to the track. For a tracked vehicle it is

quite justified to assume that there is only a vertical motion of soil particles when

compacted, because the track could be regarded as a rigid plate, which through

its planar form produces an almost uniform pressure on the soil and therefore

allows only a vertical deformation of soil particles. Unlike the track, a wheel of

circular form produces a nonuniform pressure distribution below it, allowing soil

=

R b

h t

W

s

α

Rl

z0

hb

z0

~

Figure 2.7: The “Bulldozing” Effect

2.5.4 The Compaction of Soil and “Bulldozing” 36

particles to move from higher to lower pressure regions. From equation (2.6) of

the pressure-sinkage relationship, the work L of soil compaction per unit of track

area is computed as

L =

∫ z0

0

pdz =

∫ z0

0

kzndz = kzn+10

n+ 1,

where k is

k =kcb+ kϕ .

For a track width of 2b the work of soil compression in covering a distance s will

be

Lt = 2bskzn+10

n+ 1.

If this work is represented by a resistance forceRc that acts over the same distance,

Rc may be written as

Rc = 2bkzn+10

n+ 1.

The second contribution to longitudinal resistance is the bulldozing effect,

which is directly related to the sinkage of the vehicle. When the sinkage zo is

small compared to the height ht of the tracks, the frontal area of contact between

the soil and the track is small, and longitudinal resistance arises mainly from

2.5.4 The Compaction of Soil and “Bulldozing” 37

h

α

µ

pfp

pn

Figure 2.8: Computing the Bulldozing Force

soil compression as the tracks roll over the soil. As zo increases relative to the

height ht, there is an increasing drag caused by the bulldozing effect of the frontal

portion of the track submerged in the soil to depth hb. It is assumed that the

value of the bulldozing force Rb is equal to the horizontal projection of the passive

earth pressure P fp . By further assuming that there is no surcharge (q = 0) and

that the angle of friction µ is equal to the angle of shearing resistance φ, the total

bulldozing force may be computed using earth pressure theory

Rb =2b sin(α+ φ)

sinα cosφ(2z0cKc + γz2

0Kγ) ,

where Kc and Kγ are dimensionless coefficients that depend respectively only on

the cohesion c and the soil density γ.

The third contribution to longitudinal resistance may arise from the drag of

the adhering soil mass that penetrates above the track-soil interface. There is

shear between the moving and stationary parts of the soil in the area from the

soil surface to the depth hb when soil is advected by the tracks. This drag is

significant in highly cohesive soils. According to Bekker [Bekker, 1969] it may be

estimated as

2.6 Effect of Soil Parameters on Vehicle Motion 38

Rl = 2krz0sc ,

where kr varies from 2 to 4, depending on the width of the belly.

The bulldozing effect will usually have a large impact on the motion of a

tracked vehicle during turning. When the vehicle makes a turn, it rotates at

the same time it is moving forward. The rotation requires much more power,

compared to forward motion, as there is bulldozing along the full length of both

tracks. The deeper the vehicle’s sinkage, the bigger the bulldozing force that has

to be overcome in the form of lateral resistance. Considering the discussion of

the vehicle force system presented in Subsection 3.3.1, the resistance increases

almost quadratically with the length of the track. This explains why a vehicle

with longer tracks is less maneuverable than one with shorter tracks.

2.6 Effect of Soil Parameters on Vehicle Motion

In this section the effects of soil parameters on the motion of a tracked vehicle

will be discussed and consequences for the soil track model will be summarised.

2.6.1 The Effects of Cohesion and Shearing on the Vehicle

The track grousers must exert a propulsive force on the soil to cause a tracked

vehicle to move over a soil surface. The soil in the neighbourhood of the surface

is deformed both longitudinally and vertically. The vertical deformation arises

from the weight of the vehicle, and leads to forces opposing the vehicle motion. In

steady motion, the tractive effort equilibrates the motion resistance. The longitu-

dinal soil deformation is imposed by the tracks as shear. As shown in Figure 2.9

2.6.1 The Effects of Cohesion and Shearing on the Vehicle 39

organic terrain

loose sand,saturated clay, dry fresh snow

compact sand, silt, loamand frozen snow

Shear Displacement

Shea

r St

ress

j

τ

Figure 2.9: Qualitative Shear Stress-Shear Displacement Relationship for Differ-ent Soils, after Wong [Wong, 1989b]

the shear stress τ produces a shear displacement j. This displacement means that

the soil has been deformed under the shear stress produced by the tracks. This

deformation of soil produces a reaction force to push the vehicle forward. At the

same time it imposes a slip i of the tracks relative to the undeformed surface,

defined as

i = 1− V

rω= 1− V

Vt=

Vt − V

Vt=

VjVt

,

where V is the forward speed of the track frame, Vt is the theoretical track frame

speed in the absence of slip that would result from the track rolling radius r and

the angular speed ω of the track drive sprocket, and Vj is the speed of the track

2.6.2 The Effect of Sinkage on the Vehicle 40

relative to the ground.

It is obvious that there is a relationship between the shear stress produced

by the tracks, the soil parameters and the slip. In modelling tracked vehicles the

slip will play a dominant role. It is the mechanism that couples the motion of the

vehicle and the utilised of the soil underneath. It will be germane to estimation

of the soil parameters and the position of the vehicle.

2.6.2 The Effect of Sinkage on the Vehicle

The pressure-sinkage relation express the relationship between the vehicle’s weight

and the deformation of the soil, in this case through the sinkage of the vehicle in

the soil. The extent of sinkage gives information about the soil properties since the

vehicle’s weight and dimensions are known. Information is also provided about

the possible resistance due to the bulldozing effect. As discussed in Subsection

2.5.4, the sinkage has a close relationship with the bulldozing effect and this

influences the performance of the vehicle when operating off-road. It is known

that the resistance is very high when the tracks sink deep in the soil and the

vehicle has to exert very high power to move forward. Turning in this case is

difficult or impossible because of the extremely high lateral resistance.

When the vehicle is moving, it is very difficult to measure the track sinkage

directly. No reliable and accurate sensing techniques are available, so that includ-

ing track sinkage in an estimation model of tracked vehicles is almost impossible.

To estimate the resistance, an indirect method must be used. The resistance may

be estimated through the power applied to the tracks when the vehicle is moving

with a constant speed. The estimated resistance will be a result of all the effects

of the soil onto the vehicle.

2.7 Conclusion 41

2.7 Conclusion

This chapter has summarised the development of the tracked vehicles and some

important terminology of soil mechanics. It has also discussed some relationships

between the soil and the tracks of a tracked vehicle through parameters that

characterise the soil and have a dominant effect on the motion of the vehicle.

These parameters and their relationships to the track will be used for modelling

the motion of a tracked vehicle, the Komatsu PC05 mini excavator, as detailed

in Chapter 3.

Chapter 3

Modelling of Tracked Vehicles

3.1 Introduction

Tracked vehicles have been used widely by the military services and in civilian

activities such as agriculture, forestry, building & construction, and mining. Their

low ground pressure, which imposes less damage to the soil and requires no road

preparation, and high tractive effort justify their application in these industries,

despite a high production cost. The need to improve the performance capabilities

of tracked vehicles has forced designers and engineers to find a way to handle

this task properly and economically. For a long period of time empiricism and

the “cut and try” methodology was the only technique that could be applied

in the development of tracked vehicles. As vehicle technology becomes more

sophisticated, and with a growing demand for higher mobility over a wide range

of terrains, this approach has become inefficient and prohibitively expensive. It

has become necessary to apply modern computer techniques, applied mathematics

and soil mechanics in modelling of tracked vehicles.

Modelling of a tracked vehicle requires a comprehensive understanding of the

42

3.2 The Track-Soil Model 43

mechanical behaviour of the terrain under loading conditions similar to those

imposed by the vehicle. As the performance of the tracked vehicle is primarily

dependent upon the normal and shear stress distributions at the track-terrain in-

terface, the basic issue in mathematical modelling of tracked vehicle performance

is the development of a suitable relationship between the interacting forces at the

track-terrain interface, the vehicle design parameters and the terrain characteris-

tics.

Mathematical modelling of interactions between tracks and soil has been con-

ducted by a number of authors. Of particular note are the works by Bekker

[Bekker, 1962, Bekker, 1969] and Wong [Wong, 1989b], which have been recog-

nised widely. Much of the material presented in this chapter is based on these

works, which provide the foundation for the thesis.

Following the discussion of soil properties and soil-track interactions presented

in the previous chapter, a mathematical model of a tracked excavator will be

developed. Section 3.2 first presents the track-soil interaction model. This section

contains the equations necessary to allow estimation of some key soil parameters.

Section 3.3 then develops kinematic and dynamic models of the vehicle.

3.2 The Track-Soil Model

The track-soil model is based on the physics of the interaction between the tracks

and the underlying soil. As discussed in Section 2.6 there are many soil parameters

that exert a significant influence on the motion of the vehicle. These effects will

be discussed next.

The shearing of the soil by the tracks is the most obvious soil-track interaction.

Under the effect of the shearing force, the soil undergoes an elastic-plastic defor-

3.2 The Track-Soil Model 44

mation1. The deformation of the soil depends on the magnitude of the force and

on the type of the soil itself. Figure 3.1 describes qualitatively the typical shear

displacement of soils under shear stress and highlights the different responses of

different soil types. For loose sand, saturated clay and dry fresh snow the shear

displacement increases with increasing shear stress until some value is reached

where the shear displacement continues at constant shear stress. For organic

terrain, compacted sand, silt, loam and frozen snow, once some peak value of

shear stress is reached the soil fails: the shear displacement increases even with

decreasing shear stress. The peak value represents the limit of shear stress, at

which the soil begins to fail and to be sheared away.

By shearing the soil the tracks produce a force F that propels the vehicle

1For typical soils and shear displacements the deformation will be fully plastic

organic terrain

loose sand,saturated clay, dry fresh snow

compact sand, silt, loamand frozen snow

Shear Displacement

Shea

r St

ress

j

τ

Figure 3.1: Qualitative Shear Stress-Shear Displacement Relationship for Differ-ent Soils, after Wong [Wong, 1989b]

3.2 The Track-Soil Model 45

forward, as depicted in Figure 3.2. This force is usually called the tractive effort.

Although the force developed by a track may depend on different factors, the

maximal tractive effort Fmax is entirely determined by the shear strength τmax

of the terrain and the track-terrain contact area. If A is the contact area of the

track, W the normal load, p the pressure exerted by the vehicle on the soil, c and

ϕ are the apparent cohesion and the angle of internal shearing resistance of the

terrain, then according to Bekker [Bekker, 1962] the maximal tractive effort Fmax

can be computed as

Fmax = Aτmax

= A (c+ p tanϕ)

= Ac+W tanϕ . (3.1)

As c and ϕ are characteristics that vary for different soils, the maximal trac-

tive effort Fmax of a track will vary as the track moves over soils of changing

strength. The maximal tractive effort, and hence the ability of a tracked vehicle

to manoeuvre, will increase as the soil strength increases. This is clearly seen in

practice, when a tracked vehicle with a powerful motor becomes stuck in mud

Shearing

W

F

Figure 3.2: Shearing Action of a Track, after Wong [Wong, 1989b]

3.2 The Track-Soil Model 46

because its tracks cannot produce the tractive effort required to move forward on

this particular terrain.

The tractive effort developed by a track can be calculated by integrating the

shear stress over the track-terrain contact area [Wong, 1989b],

F = b

∫ l

0

(c+ p(x) tanϕ)(1− e−ix/K

)dx , (3.2)

where b is the track width, l is the track length, K is the soil shear deformation

modulus, i is the track slip as defined in Equation (2.6.1) and x is the shear

displacement. If there is no track slip (or soil shear displacement), then the

tractive effort F also vanishes. This is because the tractive effort is a reaction

to shearing the soil and the slip is necessary to shearing the soil2. The solution

of this equation depends on the shape of the pressure distribution p(x) under

the tracks. Figure 2.2 depicts pressure distributions measured on some tracked

vehicles. The complicated form of these pressure distributions makes equation

(3.2) difficult to solve analytically, so that some simplification should be made.

Depending on the assumptions made regarding the elasticity or rigidity of the

tracks and the position of the vehicle centre of mass, various localised normal

pressure distributions under a track can be assumed. Some of the commonly

assumed pressure distributions were shown in Figure 2.5.

The shear deformation modulus K is a measure of the amount of shear de-

formation that a soil can sustain before shear failure. K may be considered as a

measure of the magnitude of the shear displacement required to develop the max-

imum shear stress. The value of K may be represented by the distance between

2Equation (3.2) does not hold when the vehicle moves over non-deformable surfaces; see alsoSubsections 2.5.2 and 2.6.1.

3.2 The Track-Soil Model 47

the vertical axis and the point of intersection of the tangent to the shear curve

at the origin and the horizontal line representing the maximum shear stress τmax.

In Figure 3.3 the value of K may also be taken as 1/3 of the shear displacement

where the shear stress τ is 95% of the maximum shear stress τmax. According to

Wong [Wong, 1989b], experimental data show that K varies from 0.6 cm for clay

at maximum compaction, 1 cm for firm sand, 2.5 cm for loose sand, and is in the

range from 2.5 cm to 5 cm for undisturbed, fresh snow.Sh

ear

Stre

ss

K

Shear Displacement

Figure 3.3: Shear Deformation Modulus K

To compute the total tractive effort, assumptions about the pressure distri-

bution under the tracks must be made. In the present work a Komatsu PC05

mini excavator has been used as the test vehicle. It should be noted that the

tracks of this mini excavator have rubber shoes and un-suspended rollers. The

pressure distribution is therefore expected to have a sinusoidal variation around

a nominal value, with pressure peaks at the roller locations. The characterise

rollers together with appropriately strained rubber shoes could lead to the as-

sumption that the amplitude of the pressure distribution will vary insignificantly

compared to the nominal value. A uniform normal pressure distribution under

3.3 Force Model of a Tracked Vehicle 48

the tracks has therefore been assumed. For that type of pressure distribution, the

total tractive effort developed at a particular steady slip i may be represented

[Wong, 1989b] by Equation (3.3)

F = (Ac+W tanϕ)

[1− K

il(1− e−il/K)

], (3.3)

where A = 2bl is the area of the tracks in contact with the soil and W = pA is

the weight of the vehicle.

Equation (3.3) expresses the assumed functional relationship between tractive

effort, track geometrical parameters, soil parameter values, and the track slip i.

It should be noted here that the tractive effort developed by the tracks must

be sufficient to overcome all resistance forces and moments acting on the tracks,

otherwise the vehicle can not move. Furthermore, if the resistance3 exceeds the

maximal tractive effort the tracks will rotate while the vehicle stands still and

a slip of 100% is observed. Section 3.3 will discuss the force model of a tracked

vehicle. This force model is necessary to compute the resistances acting on the

tracks and therefore determines the equations of motion of the vehicle.

3.3 Force Model of a Tracked Vehicle

The handling characteristics of tracked vehicles are quite different from those of

wheeled vehicles. Whilst wheeled vehicles may be steered by changing the angle

of the steering axles relatively to the un-steered axles, a tracked vehicle requires

a different mechanism. Possible steering methods are skid steering, steering by

articulation, and curved-track steering. In most tracked vehicles, including the

3usually resistance plus drawbar pull

3.3.1 Vehicle Force System 49

experimental vehicle used here, skid steering is used. This method of steering

requires that the thrust of one track is increased and that of the other is reduced,

so that a turning moment is created to overcome the moment of turning resistance

due to the lateral skidding of the tracks on the ground. A lateral force also results,

to balance the centrifugal force.

The forces acting on the vehicle will arise from a variety of different sources.

Forces may be static or dynamic in nature. Static forces act on the vehicle when

it is moving with constant velocity. Friction between the tracks and the ground is

an example of a static force. The dynamic forces are associated with accelerated

motion of the vehicle, such as the thrusts of the tracks. When the vehicle turns,

a centrifugal force is produced, which appears to push the vehicle in the lateral

direction. The dynamic forces always act at the centre of mass of the vehicle.

The kinematic and dynamic force model will be discussed next.

3.3.1 Vehicle Force System

The force system assumed to be acting on a tracked vehicle in general planar

motion is shown in Figure 3.4. In this figure Fo and Fi are the thrusts on the

outer and inner tracks, Rro and Rri are the longitudinal resistance forces exerted

by the soil on the tracks, µr and µl are the coefficients of longitudinal and lateral

resistance, B is the tread of the tracks, Fcent is the inertial force and α is the slip

angle. The vehicle is following the dashed trajectory, turning to the right around

the instantaneous centre point O′ with turning radius R. Let us now consider the

state of dynamic equilibrium of this vehicle.

Figure 3.4 shows a distribution of the lateral friction forces or lateral resis-

tance that differs from the usually assumed uniform lateral force along the track

[Shiller et al., 1993]. The reaction of the soil to the imposed lateral motion of

3.3.1 Vehicle Force System 50

the track is ascribed here to a system of springs that represent the elastic be-

haviour of the soil before failure. The track elements at the ends of the track will

compress the soil more then the elements near point O′ and therefore a larger

reaction is expected. Unless the soil is cohesionless, as an ideal dry sand where

the lateral resistance would be identical for every element of the track, a triangu-

lar distribution of lateral resistance force can be assumed. The lateral resistance

produces a moment of turning resistance Mr which opposes the turning motion

of the vehicle.

In Figure 3.4 the instantaneous centre pointO′ of turning shifts a distanceD in

the direction of motion and causes a slip angle α. Kitano [Kitano and Kuma, 1977]

and Schiller [Shiller et al., 1993] both mention the slip angle α and its effect on

the motion of the vehicle. A slightly different interpretation of the slip angle will

be presented here and an equation to estimate α is given next.

As stated in [Kitano and Kuma, 1977] and [Shiller et al., 1993], the slip angle

is zero during straight line motion, and has some signed value when the vehicle

turns left or right. That is, the slip angle appears only when the vehicle is turning.

During a turn, a centrifugal force acting at the mass centre of the vehicle appears.

When the vehicle turns “on the spot” or at low speed, the centrifugal force is zero

or small. The lateral resistance can be assumed to be equally distributed around

the centre point C of the vehicle and represented by equal triangles F1 to F4 in

Figure 3.5. When the vehicle turns at moderate or high speed, the centrifugal

force is significant. This force is assumed to be distributed uniformly in the lateral

direction on the tracks. Due to this additional force, the distribution of the lateral

resistance shown in Figure 3.5 becomes the distribution presented in Figure 3.4,

described by unequal triangles F1 to F4. For dynamic equilibrium of the vehicle

it must rotate about an instantaneous centre point O′ located a distance D ahead

3.3.1 Vehicle Force System 51

trajec

toryµ l µ r

B

R

R

F o

F i

α

D

O

C

R

ro

ri

centF

y ex e

O’ φ e..

V

Figure 3.4: Forces on the Tracks during Turning at Moderate to High Speed.There is a net lateral force in the -ye direction.

V

F1

F4

B

F3

F2

C

OR

exey

Figure 3.5: Lateral Force Distribution During Turning on the Spot or at LowSpeed. The distance D and the unbalanced lateral force are approximately zero.

3.3.1 Vehicle Force System 52

of the geometric centre. Alternatively, assume the lateral force distributions F1

to F4 to be congruent. For the vehicle to turn, a net force directed towards the

instantaneous centre of rotation must occur. This force is generated by a shift

of the point O′ ahead of C. That is, the instantaneous centre point must lie

ahead of the geometric centre. The vehicle is oriented by an angle α inside of the

velocity vector. As the centrifugal force Fcent is mφ2R, the values FI , FII and the

distribution f(x) shown in Figure 3.6 can be expressed as follows:

FI =mgµl

l− mφ2R

2l

FII = −mgµll

− mφ2R

2l

f(x) = −2mgµll2

x+mgµl

l− mφ2R

2l(3.4)

x0 =l

2− φ2Rl

4gµl,

where FI and FII are the resistance forces per unit length at either end of the

track and f(x) is the distribution of the lateral resistance per unit length as a

function of displacement x along the track length l. It now remains to calculate

the coordinate xs of the instantaneous centre point O′, shown in Figure 3.6. In

this figure the centrifugal force, which is assumed to be uniformly distributed

along the track length with a value of 0.5Fcent/l, shifts the distribution of f(x)

so that at point x0, different to the midpoint of l, f(x) vanishes. The point xs is

the coordinate of the instantaneous centre point O′, where there is a zero lateral

force. The quantity xs plays an important role in the behaviour of the tracked

vehicle during a turn.

Using integral calculus the following equations can be obtained:

3.3.1 Vehicle Force System 53

X

x0

xs

track length = l

0l

F

F

f(x)

I

II

Fcent

0.5

Y

Figure 3.6: Assumed Lateral Force Distribution on One Track

My =

∫ l

0

xf(x)dx

A =

∫ l

0

f(x)dx (3.5)

xs =My

A.

Solving these equations using the definite integral theorems:

3.3.1 Vehicle Force System 54

❶ Case FI ≥ 0:

My =

∫ l

0

xf(x)dx

=

∫ x0

0

xf(x)dx−∫ l

x0

xf(x)dx

=2mgµl3l2

(l3 − 2x3

0

)+

mgµl2l

(2x2

0 − l2)+

mφ2R

4l

(l2 − 2x2

0

)=

mgµll

4+

mφ2Rl

8+

mφ4R2l

16gµl− mφ6R3l

96g2µl

A =

∫ l

0

f(x)dx

=

∫ x0

0

f(x)dx−∫ l

x0

f(x)dx

=mgµl2

+mφ4R2

8gµl

xs =l

2+

φ2Rl

4gµl− φ6R3l

24g2µ2l

(gµl

2+ φ4R2

8gµl

)≈ l

2+

φ2Rl

4gµl. (3.6)

❷ Case FI ≤ 0:

My = −∫ l

0

xf(x)dx

=mgµll

6+

mφ2Rl

4

A =mφ2Rl

2

xs =l

2+

gµll

3φ2R

In this case the centrifugal force required is so large that the force distribution

f(x) would not have a positive part as in Figure 3.6. This would happen in

extreme cases, when the vehicle is turning at very high speed and the coefficient

3.3.1 Vehicle Force System 55

of lateral resistance µl is very small. The vehicle is unable to generate the required

lateral force and is uncontrollable.

The computed coordinate of the instantaneous centre point O′ shows that

the slip angle does not depend on the weight of the vehicle. It depends on

the coefficient of lateral resistance µl, the track length l and the yaw speed φ.

The moment of turning resistance Mr can now be computed by integrating the

elemental lateral resistances along the track. Due to symmetry the moment of

turning resistance is seen to be:

Mr = 4

∫ l−xs

0

−xf(x+ xs)dx

= 4

∫ l−xs

0

x

(2mgµl

l2x+

2mgµll2

xs − mgµll

+mφ2R

2l

)dx

= 4

∫ l−xs

0

x

(2mgµl

l2x+

mφ2R

l

)dx

=mglµl3

− mφ4R2l

4gµ+

mφ6R3l

12g2µ2l

. (3.7)

The result shows that the moment of turning resistance Mr consists of a part

caused by the lateral resistance force and a part due to the centrifugal forces.

When the vehicle is turning on the spot or at low speed, centrifugal force can be

neglected, so that the points C and O′ coincide at l/2. The moment of turning

resistance can be computed as:

3.3.1 Vehicle Force System 56

Mr = −4∫ l

2

0

xfi(x+ x0)dx

= 4

∫ l2

0

x

[2mgµl

l2

(x+

l

2

)− mgµl

l

]dx

=8mgµl

l2

∫ l2

0

x2dx

=8mgµl3l2

x3

∣∣∣∣∣l2

0

=mglµl3

, (3.8)

where

fi(x) = −2mgµll2

x+mgµl

l(3.9)

is the distribution of lateral force per unit length along the track.

Compared to the moment of turning resistance Mr calculated from Wong

[Wong, 1989b], where Mr is equal mglµl/4, the moment of turning resistance Mr

from Equation (3.8) is a third larger, due to a different lateral force distribution.

It is clear that when R is zero, Equations (3.7) and (3.8) predict the same value

for the moment of turning resistance. Figure 3.7 shows the result of evaluating

Equations (3.7) and (3.8) with a maximal yaw speed φmax = 1.37rads−1 when

braking one track and for different coefficients of lateral resistance. It is obvi-

ous that the centrifugal force has significant influence on the moment of turning

resistance, especially on soils with small coefficients of lateral resistance. Recall

that normal soils have a coefficient µl from 0.5 to 1.3, so that the error in using

Equation (3.8) instead of (3.7) is less than 2%, which is acceptable. The error

3.3.1 Vehicle Force System 57

when computing the instantaneous centre point of turning using Equation (3.6) is

shown in Figure 3.8, and is less than 0.1% when the vehicle is working in normal

conditions.

Another problem is the distribution of the resistance on the tracks of the

vehicle. The longitudinal resistance can be considered to be distributed equally

on both tracks when the vehicle is moving in straight line, or when both tracks

are rotating in the same direction during turning. It is common that one track is

braked and the other one rotates when turning the vehicle. In this case one track

must produce a thrust required not only to overcome the longitudinal resistance,

but also the moment of turning resistance. Even if both tracks rotate in the

same direction during turning the outer track has to produce the force needed to

overcome the moment of turning resistance and the moment caused by the other

track around the vehicle centre. Only when the vehicle is turning on the spot or

when the turning radius R ≤ B/2, both tracks contribute forces to overcome the

resistance, as they will rotate in opposite direction. In this case the moment of

turning resistance is distributed proportionally to the distance from the centre

point of turning to each track. That means that the further a track is from

the centre point of turning, the more force must be contributed to overcome the

moment of turning resistance. From a radius of turning equal or larger than B/2

the outer track must produce all the force required to overcome the moment of

turning resistance. This consideration is important for computing the applied

forces on each track and the slips as a function of the force. The longitudinal

resistances Ro and Ri can be computed as

Ro,i =Wµr2

(3.10)

3.3.1 Vehicle Force System 58

0 0.5 1 1.5−40

−35

−30

−25

−20

−15

−10

−5

0

Coefficient of Lateral Resistance µ l

Err

or in

%

Error of Computed Moment of Turning Resistance Mr

Figure 3.7: Error of Moment of Turning Resistance for φmax = 1.37rads−1 andµl = 0.1 · · · 1.5

0 0.5 1 1.5−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Coefficient of Lateral Resistance µ l

Err

or in

%

Error of Computed Instantaneous Centre Point Xs

Figure 3.8: Error of Instantaneous Centre Point for φmax = 1.37rads−1 andµl = 0.1 · · · 1.5

3.3.2 Equations of Motion 59

where µr is the coefficient of longitudinal resistance.

3.3.2 Equations of Motion

Consider a rigid planar vehicle moving on a planar surface with a heading angle

φ, as shown in Figure 3.4. A coordinate frame Cxy defined by unit vectors ex

and ey parallel to the major vehicle axes is attached at the mass centre C. The

mass centre is assumed to be in the midpoint of the track contact area. Other

assumptions made are that the soil is homogeneous, the track belts are rigid, and

there is a uniform normal pressure distribution under the tracks.

The vehicle is moving at some constant velocity vc = xe and rotating at an

angular velocity φ. The angle α between vc and ex is called the slip angle. The

equations of motion of the vehicle can be written as follows:

mxe = Fo + Fi − Fcent sinα−Rro −Rri

mye = Fcent cosα− µlW (3.11)

Izφ = MQ −Mr ,

where the suffix e denotes coordinates fixed on the vehicle. The centrifugal force

Fcent acting on the vehicle is

Fcent = mφ2R ,

and the moment around the mass centre MQ is

3.3.2 Equations of Motion 60

MQ = ∆FB/2

=[(Fo −Rro)− (Fi −Rri)]B

2.

Here, Mr is the moment of turning resistance and Iz is the vehicle moment of

inertia about ez.

Next, all the forces and moments related to Equation (3.11) will be computed.

The slips io and ii can be found from Equation (3.3) once the equation has been

rearranged. It is necessary to approximate the slip because a numerical solution

for the slip in Equation (3.3) requires substantial computing power, and the filter

needs to operate in real-time with a small sampling interval. The slip is therefore

approximated as

i = − K/l

ln(F/Fmax), (3.12)

where the maximal tractive effort on a track is

Fmax = 0.5 (Ac+W tanϕ) ,

and F is the force developed by this track. Figure 3.9 shows the error when

using the approximate Equation (3.12) instead of Equation (3.3) for different K.

The errors are significant when the track slips are smaller than 1%, but are small

(less than about 10%) when the slips exceed 4%. Equation (3.12) is therefore

acceptable because the absolute errors are small in any case.

The set of governing equations can be written as:

3.3.2 Equations of Motion 61

0 10 20 30 40 50 60 70 80 90 100

−80

−70

−60

−50

−40

−30

−20

−10

0

10

slip in %

erro

r in

%

Relative computed Slip Errors for different K

K = 0.6 cmK = 1 cm K = 2.5 cmK = 5 cm

Figure 3.9: Errors in Computed Slip when Using Equation (3.12) for DifferentValues of K

Mr =Wlµl3

Rro =µrW

2

Rri =µrW

2

io = − K/l

ln (Fo/(Ac+W tanϕ))(3.13)

ii = − K/l

ln (Fi/(Ac+W tanϕ))

R =B [ωo(1− io) + ωi(1− ii)]

2 [ωi(1− ii)− ωo(1− io)]

α = arctanφ2l

4gµl

φ =r[ωi(t)(1− io)− ωo(t)(1− ii)]

B,

where Fo and Fi are the thrusts of the outside and inside tracks, which can be

3.3.2 Equations of Motion 62

deduced from the torques on the sprockets, and ωo and ωi are the angular speeds

of the sprockets, which can directly be measured. The other terms are either

vehicle or soil parameters. The vehicle yaw rate φ can either be computed from

Equation (3.13) or measured directly using an appropriate sensor. If yaw rate is

measurable, it will be employed to correct the other parameters, which are not

always known when the vehicle is moving off-road.

Equation (3.11) will be now rewritten in terms of known quantities:

xe =Fo + Fi −mφ2R sinα− µrW

m− 2Mr

B

ye = φ2R cosα− µlg (3.14)

φ =

[(Fo − Fi)− (Rro −Rri)

2B − Wlµl

3

]1

Iz.

The kinematic and dynamic relationships are respectively Equations (3.13)

and Equation (3.14). Two sub-models of the motion therefore exist, with the

first independent of the second. Considering Equation (3.19) discussed in Sub-

section 3.4.1, the following equation describing the motion of the vehicle on a

plane is obtained:

3.3.2 Equations of Motion 63

d

dt

xe

ye

φ

xe

ye

φ

=

r2[ωo (1− io) + ωi (1− ii)]

ye

r[ωi(1−ii)−ωo(1−io)]B

Fo+Fi−mφ2R sinα−µrWm

φ2R cosα− gµl[(Fo−Fi)−(Rro−Rri)

2B − Wlµl

3

]1Iz

=

r2[ωo (1− io) + ωi (1− ii)]

ye

r[ωi(1−ii)−ωo(1−io)]B(

Fo+Fi

m

) − φ2R sinα− gµr − 2Mr

B

φ2R cosα− gµl

B[(Fo−Fi)−(Rro−Rri)]2Iz

− mglµl

3Iz

. (3.15)

In this equation there is a mixing of kinematic and dynamic relationships.

Whilst the upper half of the equation contains kinematic terms only, the lower

half combines both kinematic and dynamic functions. It is almost impossible

to apply this equation in the state space to model the motion of the vehicle.

A tractable state space model may be obtained by decoupling Equation (3.15)

into two sub-equations, where the first sub-equation is independent of the second

one and the second characterise takes parameter values as solutions of the first

characterise. This decoupling will be discussed in Subsection 3.4.2.

3.4 The Model of the Tracked Vehicle 64

3.4 The Model of the Tracked Vehicle

3.4.1 Kinematic Model

The Continuous-Time Process Model

The first model of the tracked vehicle is a two dimensional model that is based on

the vehicle kinematics. The vehicle’s motion is described by kinematic equations

written in the vehicle-fixed coordinate system (xe, ye) shown in Figure 3.10. In

the figure, the vehicle is turning to the right. The outside, or left, track is denoted

by a subscript o, and i denotes the inside or right track.

trajec

tory

µ l µ r

B

R

R

F o

F i

α

D

O

C

R

ro

ri

..centF

y ex e

Vo

V i

G

H

φ

φ

O’

Figure 3.10: Kinematic Motion of a Tracked Vehicle

In the absence of track slip, the speeds of the outer track Vo and inner track

Vi would be

3.4.1 Kinematic Model 65

Vo = rωo

Vi = rωi , (3.16)

where r is the track rolling radius, and ωo and ωi are the angular velocities of the

outside and inside track drive sprockets. Upon introducing the longitudinal slips

io and ii of the tracks relative to the un-deformed soil, Equation (3.16) becomes

Vo = rωo(1− io)

Vi = rωi(1− ii) . (3.17)

In the presence of the longitudinal track slip, the vehicle’s forward speed is

Vx =Vo + Vi2

=r

2[ωo(1− io) + rωi(1− ii)] . (3.18)

Because of the difference between Vo and Vi, the angle φ is expressed in the form

of an arctangent function

φ = arctanGH

OG= arctan

(Vo − Vi)t

B,

where t is time and B is the tread of the vehicle. The time-derivative of φ can be

computed for small time steps as:

3.4.1 Kinematic Model 66

φ =∆V

B

=r [ωi(1− ii)− ωo(1− io)]

B,

where φ is positive anticlockwise when viewed from above. The vehicle’s speed

may now be decomposed into components in the xe and ye directions. The motion

of the vehicle is thus described as follows:

x =r

2[ωo(1− io) + ωi(1− ii)] cosφ

y =r

2[ωo(1− io) + ωi(1− ii)] sinφ (3.19)

φ =r[ωi(1− ii)− ωo(1− io)]

B.

By introducing the slip angle α, discussed in Subsection 3.3.2, Equation (3.19)

can be written as:

x =r

2[ωo(1− io) + ωi(1− ii)][cosφ(t)− sinφ(t) tanα(t)]

y =r

2[ωo(1− io) + ωi(1− ii)][sinφ(t) + cosφ(t) tanα(t)] (3.20)

φ =r[ωi(1− ii)− ωo(1− io)]

B.

To complete the estimation model, equations for the slips io, ii and slip angle α

have to be derived. Consider these states as time-invariant or slowly time-varying,

so that they can be written as

3.4.1 Kinematic Model 67

io = 0

ii = 0

α = 0 .

This is true when the vehicle is moving on a straight line with constant speed

on a homogeneous soil. When loading conditions change, as when the vehicle

is accelerating, these slips will change. New values of slip can be estimated

or computed based on their relationship with the yaw speed φ and the forces

acting on the tracks (Equation (3.13)). Such computation or estimation should

be conducted whenever the motion condition changes to adjust the slips to the

given condition.

Noise Issues

The simple kinematic model of the excavator can be considered as that of a control

system with angular velocities as inputs. Noise will be injected into the system

through the measurement errors of the sensors and estimated parameters. Angu-

lar positions of the track drive sprockets are measured by encoders. The measured

value ωi is nominal and the true value for the angular speed is represented as

ωi = (1 + nωσω)ωi , (3.21)

where nω is a multiplicative slip term caused by the effective elasticity of the

toothed belt connecting the sprockets to the encoder and σω represents the uncer-

tainty of the digital position reading. It is assumed that nω and σω are zero-mean

3.4.1 Kinematic Model 68

uncorrelated random processes.

There are three parameters, (io, ii, α), to be estimated and noise may be in-

jected into these states. Because of the kinematic and dynamic characteristics of

tracked vehicles, the noise model for the slips i on both tracks is taken to be

i = (1 + δωσi)i , (3.22)

where δω is a multiplicative term that encodes the change of slip in relation

to angular speed ω, and σi is some white noise injected into the estimation of

i. The multiplicative term δω can be explained as follows. When the vehicle

moves straight forward with a constant speed, the slip is constant. When the

vehicle accelerates by changing either speed or direction, the forces applied to

the tracks will change significantly. Through the soil-track interaction discussed

in Chapter 2, the slips must also change substantially. To deal with these two

modes of motion, δω will be allowed to take different values related to different

states of motion. This switching scheme of δω is controlled by testing a “jump

hypotheses” within the estimator. This principal is similar to a model switching

scheme, which adapts the corresponding model to changed conditions.

The noise model for the slip angle is similar to that of the track slip, except

that the slip angle will change only during turning rather than during the vehicle’s

speed changes.

Observation Issues

To correct the process model, observations of some states of the process should

be obtained directly by sensors or by estimation of the six states of the kinematic

model. The states X,Y and φ describe the position of the vehicle and can be

3.4.1 Kinematic Model 69

measured using a number of different position sensors such as INS, GPS, encoders,

sonar or radar. These sensors have been widely used, and their operating prin-

cipals and sensor models need not be detailed here. The other three states io, ii

and α are theoretically accessible. It is, however, difficult to obtain their esti-

mates in a moving vehicle without expensive and complicated techniques. Wong

[Wong, 1989a], for example, used signals from a fifth wheel running in the rut

made by a track, compared with signals from the sprocket encoders, to measure

the slips of the tracks. Such a wheel will work well only provided that the track

shoes do not damage the soil surface very much, because the uneven surface will

add error to the encoder reading. Problems will also occur when the vehicle is

turning. Considering the complicated requirements when using such a wheel its

use has not been attempted with the test vehicle.

The Discrete-Time Process Model

The discrete-time model of the vehicle is developed from the continuous-time

process model and the process noise model discussed previously. Theoretically

the state space equation is obtained by integrating the continuous-time equa-

tions over the interval from tk−1 to tk. The motion of the tracked vehicle is,

however, a non-holonomic process: the motion is described in terms of velocity

constraints of the two tracks which can not be integrated to yield an analytic,

closed form solution. The path of the vehicle changes whenever when one or both

track velocities change. Besides, the injection of process noise makes the causal

propagation significantly more complicated. The discrete-time model is therefore

just an approximation of the continuous-time model, obtained using a zero-order

hold assumption. The first-order Euler approximation is then adequate for the

integrator:

3.4.1 Kinematic Model 70

X(k) = X(k − 1) + ∆Tf [X(k − 1), u(k − 1), v(k − 1), k − 1] .

For a small sampling period ∆T , or for low vehicle speeds, this assumption is

justified. The discrete-time state space equation can then be written4

X(k) = f [X(k − 1), u(k − 1), v(k − 1), k − 1]

=

x(k−1)+0.5∆Tr[io(k−1)ωo(k)+ii(k−1)ωi(k)][cosφ(k−1)−sinφ(k−1) tanα(k−1)]

y(k−1)+0.5∆Tr[io(k−1)ωo(k)+ii(k−1)ωi(k)][sinφ(k−1)+sinφ(k−1) tanα(k−1)]

φ(k−1)+∆TrB

[ii(k−1)ωi(k)+io(k−1)ωo(k)]

io(k−1)

ii(k−1)α(k−1)

,

where the state vector X(k) is

X(k) = [x(k) y(k) φ(k) io(k) ii(k) α(k)]T

,

and the control input

u(k) = [ωo ωi]T

is composed of the angular speeds of the outer and inner track drive sprockets.

4It should be noted that from this point io and ii have been used in place of 1− io and 1− iito simplify the notation.

3.4.2 The Comprehensive Vehicle Model 71

3.4.2 The Comprehensive Vehicle Model

A comprehensive model of the vehicle will incorporate both kinematic and dy-

namic representations of the vehicle. The model then becomes more complicated.

As discussed in Section 3.3, the kinematic and dynamic equations do not have

the same dimension in state space. Furthermore, there are many vehicle-specific

parameters whose estimation requires diverse techniques and methods and the

results will not always be sufficiently accurate. Such parameters are, for example,

the vehicle moment of inertia Iz, the speed factors Cl and Cr, or the hydraulic

efficiency of the pump and motor in different operating areas.

On the basis of experimental results from tests with the kinematic model, in

which three parameters were estimated using yaw speed measured by the yaw

gyroscope of an inertial navigation system, a comprehensive model will be con-

structed. Here, the target is to control not only the motion of the vehicle, but

also to aid control of soil-working processes by making initial estimates of soil

parameters germane to excavation planning and processing.

The Continuous-Time Process Model

From the discussion in Subsection 3.3.2, two sets of equations for the forces, mo-

ments and resulting vehicle motion have been obtained in (3.13) and (3.15).

The mix of kinematic and dynamic relationships complicates the process model,

since the same states can be derived from either dynamic or kinematic equations.

Because high frequency dynamics and time drift are taken into account in the

dynamic model, the dynamic equations provide a better description of the fast

response when the vehicle states change rapidly during turning. For estimation

of slowly time-varying vehicle states, kinematic equations seem to be more suit-

able, as shown in the experimental results. Following from these considerations,

3.4.2 The Comprehensive Vehicle Model 72

the comprehensive model will consist of two sub-models: one kinematic and one

dynamic. The first sub-model is independent of the second. The second sub-

model utilises on-line data obtained from the first sub-model, but treats those

data as known parameters. While the first sub-model is used to estimate the

vehicle’s position, the second is used mainly to estimate soil parameters such as

the coefficients of resistance µr and µl, the shear deformation modulus K or the

maximal tractive effort Fmax. These two sub-models are described respectively

by Equations (3.23) and (3.24):

d

dt

x

y

φ

io

ii

α

=

r2[ωoio + ωiii][cosφ(t)− sinφ(t) tanα(t)]r2[ωoio + ωiii][sinφ(t) + cosφ(t) tanα(t)]

r[ωi ii−ωo(1−io)]B

0

0

0

, (3.23)

d

dt

x

y

φ

µr

µl

=

(Fo+Fi

m

) − φ2R sinα− gµr

φ2R cosα− gµl − 2Mr

B

B[(Fo−Fi)−(Rro−Rri)]2Iz

− mglµl

3Iz

0

0

, (3.24)

where R is the turning radius, which can be computed from Equation (3.13):

3.4.2 The Comprehensive Vehicle Model 73

R =B [ωoio + ωiii]

2 [ωiii − ωoio].

In the second sub-model the slip angle α and yaw speed φ are obtained from

the first sub-model as known parameters. All the other parameters are vehicle

constants that can directly be measured or estimated experimentally.

The forces Fo, Fi can be computed from the torques applied to the drive

sprockets. In turn, this torque can be derived from the pressure drop across the

hydraulic track-drive motor and the flow rate using Equation (3.25), taken from

Guillon [Guillon, 1969]:

T =H

ω=

η′02πη′V

V∆P , (3.25)

where the shaft power H is

H = η′0Q∆P ,

and where η′0 is the overall efficiency, η′V is the volumetric efficiency, Q is the flow

rate, ω is the angular speed, ∆P is the differential pressure and V is the swept

volume5 of the hydraulic motor. Here, the overall efficiency η′0, is defined as the

ratio of mechanical power output from motor to hydraulic power input to motor.

The volumetric efficiency, η′V , is defined as the ratio of theoretical flow nv in the

motor to the actual flow to the motor.

5fluid volume displaced in one revolution

3.4.2 The Comprehensive Vehicle Model 74

The soil parameters µr and µl are estimated by the second sub-model. From

the estimated track slips io and ii, the track thrusts Fo and Fi and the vehicle

constants, it is possible to estimate the soil parameters remaining in the model:

the cohesion c, the angle of shearing resistance ϕ and the shear deformation mod-

ulus K using known relationships between them from Equation (3.13), discussed

in Subsection 3.3.2.

Noise and Error Sources

The first sub-model is exactly the same kinematic model as was discussed in

Subsection 3.4.1 and will use the same noise model. The noise model for the

second sub-model will be discussed next.

The second sub-model uses the results obtained from the first sub-model.

The states of the second sub-model will therefore be corrupted by the same noise

sources as the first sub-model. Additional sensors are also used, and their different

noise characteristics should be taken into account.

While most vehicle parameters can be considered as constants, the vehicle

moment of inertia Iz can not easily be computed or measured, because of the

complicated and time-varying configuration of the excavator arm. A large additive

error may be associated with the value of Iz used in the excavator model.

The other source of error in the second sub-model comes from the forces Fo

and Fi. These forces cannot be directly measured and have to be derived from

relevant sensor data. The excavator has been fitted with servo valves having

built-in sensors for spool positions, and with pressure transducers to measure the

differential pressures across hydraulic motors. The forces can be computed using

Equation (3.25) once the overall efficiency and the volumetric efficiency have been

determined. The values of η′0 and η′V depend on the design of the hydraulic system

3.4.2 The Comprehensive Vehicle Model 75

of the vehicle and can be estimated experimentally. This constitutes one of the

error sources.

The measurement of Q requires a flow meter, which is not available. An indi-

rect estimation of the flow has therefore been made. Based on the spool positions,

the angular rate of the motor and other design parameters, an approximate equa-

tion has been used to compute flow rate. This approach will serve as another

source of error to the system. Because of the increased model uncertainty, com-

pensatory noise has been added to the system. This noise serves to compensate

for un-modelled error sources and uncertainties of complex states that are hard

to model and predict.

The actual control parameters for the torque will be the differential pressures

measured on both axial motors. Reading these pressure values will add a further

error source to the system.

The Discrete-Time Process Model

The way to derive the discrete-time state space equation for the comprehensive

model is similar to that discussed in Subsection 3.4.1 for the kinematic model.

The comprehensive model consists of two sub-models, the kinematic and dynamic.

The kinematic sub-model has its discrete-time form described by Equation (3.26):

X(k) = f [X(k − 1), u(k − 1), v(k − 1), k − 1] (3.26)

=

x(k−1)+0.5∆Tr[io(k−1)ωo(k)+ii(k−1)ωi(k)][cosφ(k−1)−sinφ(k−1) tanα(k−1)]

y(k−1)+0.5∆Tr[io(k−1)ωo(k)+ii(k−1)ωi(k)][sinφ(k−1)+sinφ(k−1) tanα(k−1)]

φ(k−1)+∆TrB

[ii(k−1)ωi(k)+io(k−1)ωo(k)]

io(k−1)

ii(k−1)α(k−1)

,

where the state vector X(k) is

3.4.2 The Comprehensive Vehicle Model 76

X(k) = [x(k) y(k) φ(k) io(k) ii(k) α(k)]T

,

and the control input u(k)

u(k) = [ωo ωi]T

is the vector of the angular speeds of the outer and inner track drive sprockets.

The slip parameters io, ii, α estimated from Equation (3.26) are used as pa-

rameters for the dynamic behaviour. Two different versions of the dynamic model

exist, and will be interchanged depending on whether the vehicle is in straight

motion or is turning. For straight motion, the state space process model is:

X(k) = f1[X(k − 1), u(k − 1), v(k − 1), k − 1]

=

x(k − 1) + ∆TFnet

0

0

µr(k − 1)µl(k − 1)

,

where the state vector X(k) is

X(k) =[x(k) y(k) φ(k) µr(k) µl(k)

]T,

3.4.2 The Comprehensive Vehicle Model 77

and for turning:

X(k) = f2[X(k − 1), u(k − 1), v(k − 1), k − 1]

=

x(k − 1) + ∆T Fnet

m

y(k − 1) + ∆T y

φ(k − 1) + ∆T (φforce − φres)

µr(k − 1)µl(k − 1)

,

where

Fnet = Fo −Ro + Fi −Ri − φ2R sinα− 2Mr

B

is the resultant longitudinal force acting on the vehicle, and

Fo,i =5gη′oV∆Po,i

πrη′v(3.27)

are the forces produced by the tracks. These forces are computed from the dif-

ferential pressure ∆Po,i on the motors of the outer and inner tracks, the swept

volume V , the overall efficiency η′o , the volumetric efficiency η′v , and the track

rolling radius r.

The resistance forces exerted on the tracks

3.4.2 The Comprehensive Vehicle Model 78

Ro =µrW

2,

Ri =µrW

2

are computed from the vehicle’s weight W , the coefficient of longitudinal re-

sistance µr, and the slips io and ii. The acceleration components of yaw are

computed as

φforce =B[(Fo − Fi)− (Rro −Rri)]

2Iz,

φres =mglµl3Iz

,

where Iz is the inertia of the vehicle and µl is the coefficient of lateral resistance.

The acceleration in the lateral direction y is

y = φ2Rcosα− gµl ,

and the turning radius R is computed from Equation (3.13).

During operation another model for computing the slips of the vehicle based

on the slip-shearing interaction and Equation (3.13) can be applied, especially to

correct the values estimated for track slips after an abrupt turn. This slip model

can be used in conjunction with both the kinematic and the comprehensive model.

3.5 Conclusion 79

3.5 Conclusion

In this chapter the kinematics and dynamics of a tracked mini excavator have

been discussed in detail. The soil-track interaction, an important issue in tracked

vehicle modelling, has been analyzed. The analysis has resulted in two models

of the vehicle’s motion, a kinematic and a comprehensive model that incorporate

all the discussed relationships between the vehicle characteristics, soil parameters

and vehicle motion. These models will be used to process data obtained from

experimental tests conducted with the excavator, and the results will be presented

in Chapters 5 and 6.

Chapter 4

Development of Estimation

Techniques

4.1 Introduction

This chapter introduces and describes a number of filtering techniques that will

subsequently be used in this thesis to support the modelling of tracked vehicles.

These filtering techniques play an important part in the development and refine-

ment of vehicle models, by allowing the estimation of various parameters that

cannot be measured directly or indirectly from a moving tracked vehicle.

The principal filtering technique used in this thesis is based on the Kalman

Filter (KF). Section 4.2 describes the structure and operation of a Kalman filter.

As the equations governing the motion of a tracked vehicle are inherently non-

linear, the Extended Kalman Filter (EKF) is adopted here. The EKF is simply

a Kalman filter, linearised to allow application to nonlinear systems. Section 4.3

describes the EKF in detail. Another filtering technique, the Distributed Approx-

imation Filter 1 (DAF) has also been evaluated for tracked vehicle modelling.

1initially termed the “Unscented Filter”

80

4.2 The Kalman Filter 81

This filtering technique is described in Section 4.4. The evaluation of two dif-

ferent filtering techniques verifies the validity and performance of the proposed

model. The most suitable filter for the process model will then be chosen.

4.2 The Kalman Filter

4.2.1 Introduction

The Kalman Filter (KF) is a recursive, linear and minimum mean-squared-error

estimator [Kalman, 1960]. It estimates the state of a system by linearly combining

a prediction of state with observations obtained from one or more sensors. The

state is updated such that the mean-squared error is minimised. This process is

recursive: information from a new set of observations is added without requiring

reprocessing of previous observations.

The KF is one of the most widely used estimation algorithms. It has been

used in a diverse range of applications, from navigation of spacecraft, aircraft,

ships, ground vehicles, and mobile robots, to parameter estimation for optimis-

ing control processes, and even for economic forecasting. It is known to be the

optimal linear mean-squared error estimator. That is, given a linear update rule,

no other estimator can yield an estimate which has smaller mean-squared er-

ror [Maybeck, 1979].

In the remainder of this Section, Subsection 4.2.2 introduces the state space

representation of a process. Subsection 4.2.3 then describes the sensor model,

which relates the state of the process to the observations obtained by sensors.

The operation of the filter is then examined in detail in Subsection 4.2.4. An un-

derstanding of the KF is necessary before describing the EKF which is used in this

thesis to validate the process model. Section 4.3 provides a detailed description

4.2.2 The Process Model 82

of the EKF.

4.2.2 The Process Model

The state at any time t of a process with n degrees of freedom may be described

using an n-dimensional state vector x(t). The components of this vector may be

any combination of continuous variables that are sufficient to describe the process.

For example, the state variables may be position and velocity if the process is a

moving vehicle, or temperature for a thermal process. The elements of the state

vector purport to describe the true state of the process, which in practice is not

exactly known.

The state of the process is changing through time due to its dynamics. In

general, the process model can be represented in state-space form by a system of

first order non-linear differential equations in continuous time:

x(t) = f [x(t), u(t), t] + v(t) , (4.1)

where x(t) ∈ �n is the state at the time t, u(t) ∈ �r is the known control input,f [·, ·, ·] : �n×�r×� −→ �n is the function mapping the state and control inputsto state ‘velocities’ at time t, and v(t) is a random vector describing both dynamic

driving noise and uncertainties in the state model itself. The state x(t) at the

time t summarises all past information of the state x(τ), 0 < τ < t. Together

with the subsequent control inputs u(τ ′), it is sufficient to describe all future state

trajectories x(τ ′), τ ′ > t. The function f [·, ·, ·] is termed the process model. Therandom vector v(t) is termed the process noise.

In linear state estimation problems, the state model is linear in both the state

4.2.2 The Process Model 83

and the control inputs, and has the form of Equation (4.2):

x(t) = F (t)x(t) +B(t)u(t) +G(t)v(t) , (4.2)

where F (t) is the n × n time-varying matrix relating the state x(t) to the state

velocity, B(t) is the n × r time-varying matrix relating control inputs to state

velocities, and G(t) is the n × q time-varying matrix relating the process noise

vector to the state velocity.

The continuous-time form is important because almost all physical processes

are continuous-time. Since most estimators will be implemented on digital de-

vices, the process model of the state should be described in discrete form. The

discrete-time form of Equation (4.1) is

x(tk) = f [x(tk−1), u(tk), tk] +G(tk)v(tk) , (4.3)

where the function f [·, ·, ·] now maps the state x(tk−1) at the time tk−1 and the

control input u(tk) at the time tk to the state x(tk) at the next time step tk.

The the equivalent discrete-time model is obtained from the continuous-time

form by using the state transition matrix

x(tk) = F (tk)x(tk−1) +B(tk)u(tk) +G(tk)v(tk) . (4.4)

In filter implementations it is common that the time interval ∆t(k) � tk−tk−1

between successive samples of the state remains constant, so that for simplicity

4.2.3 Sensor Model 84

the time argument may be dropped and the variables be indexed by a sample

number. Equation (4.3) can then be written as

x(k) = f [x(k − 1), u(k), k] +G(k)v(k), k = 1, 2, · · · , (4.5)

and Equation (4.4) as

x(k) = F (k)x(k − 1) +B(k)u(k) +G(k)v(k), k = 1, 2, · · · . (4.6)

4.2.3 Sensor Model

To estimate a state of a process, the KF linearly combines a prediction of the

state with a set of measurements obtained by a suite of sensors. The prediction is

made using the process model described in Subsection 4.2.2. A similar approach

is used to provide a sensor model which is necessary in the estimation cycle.

The model of state observations can be described generally in state-space by

a non-linear vector function of the form

z(t) = h[x(t), u(t), t] + w(t) , (4.7)

where z(t) ∈ �m is the observation made at the time t, h[·, ·, ·] : �n×�r×� −→�m is the function mapping the state and control inputs to observations, and

w(t) is the random vector describing both measurement corruption noise and

uncertainties in the measurement model itself. The function h[·, ·, ·] is termed theobservation model. It describes the observation z(t) made at the time t when the

4.2.3 Sensor Model 85

true state is given by x(t). The random vector w(t) is termed the observation

noise.

In practice a linear observation model of the form

z(t) = H(t)x(t) +D(t)w(t) (4.8)

is of more interest. Here, H(t) is the m × n time-varying matrix relating the

state to the observation at the time t and D(t) is the m× p time-varying matrix

relating measurement and model noise to the observation made.

As observations will normally be taken at discrete intervals of time, the ob-

servation model becomes

z(tk) = h[x(tk), u(tk), tk] + w(tk), k = 1, 2, · · · . (4.9)

For synchronous observations the reference to time will be dropped and Equa-

tion (4.9) becomes

z(k) = h[x(k), u(k), k] +D(k)w(k) , (4.10)

and the linear observation model corresponding to equation (4.8) is

z(k) = H(k)x(k) +D(k)w(k) . (4.11)

4.2.4 Basic Filtering Cycle 86

4.2.4 Basic Filtering Cycle

The Kalman filter has a cyclic structure [Bar-Shalom and Fortman, 1988] de-

picted in Figure 4.1. The figure shows the three cycles of the filter: the evolution

of the “true” state and observation; the estimate of the true state based on the

observations; and the computation of the state covariance.

In the first cycle the true state evolves over time according to a true state-

space model2 given by Equation (4.2), with known inputs u(t) and subject to

disturbance noise v(t). Observations of the true state are made according to

an observation model given by Equation (4.8). The cycle shown in Figure 4.1

associated with the evolution of the system gives as its output only observations

made of the true state.

The second cycle starts with the generation of a state prediction from knowl-

edge of the state estimate at the preceding time-step. This prediction is computed

from Equation (4.12)

x(k|k − 1) = F (k)x(k − 1|k − 1) +G(k)u(k) , (4.12)

based upon the known control input u(k) and the state transition model F (k),

where x(k) is the state estimated at the time k. The state prediction is then

used to compute a predicted observation, conditioned on previous observations

and according to the sensor model H(k)

z(k|k − 1) = H(k)x(k|k − 1) , (4.13)

2actually, the true state evolves according to its natural dynamics - we just hope that the“truth model” is a reasonable representation of reality

4.2.4 Basic Filtering Cycle 87

State covarianceEvolutionof the system(true state)

State at t Control at t State estimate at t

Estimationof the state computation

State error covariance at t

P(k|k)

State prediction

covariance

S(k+1) =

H(k+1)P(k+1|k)H’(k+1)

+ R(k+1)

State prediction

x(k+1|k) =

F(k)x(k|k)

+ G(k)u(k)

covariance

x(k|k)k

u(k)x(k)

Transition to t

x(k+1) =

F(k)x(k)

+G(k)u(k) + v(k)

Measurement

prediction

Updated

Filter gain

W(k+1) =

P(k+1|k)H’(k+1)S (k+1)

Updated state

covariance

P(k+1|k+1) =

P(k+1|k)

- W(k+1)S(k+1))W’(k+1)

P(k+1|k) =

F(k)P(k|k)F’(k) + Q(k)

x(k+1|k+1) =

state estimate

x(k+1|k)

+ W(k+1)v(k+1)

z(k+1)

- z(k+1|k)

H(k+1)x(k+1|k)

z(k+1|k) =

at tMeasurement

z(k+1) =

H(k+1)x(k+1)

+w(k+1)

^

^

^

^

^

^

^

^

-1

Controller

(k+1) =ν

Innovation

Innovation

k

k k k

k+1

Figure 4.1: The Kalman Filter Cycle

4.2.4 Basic Filtering Cycle 88

where z(k|k − 1) is the predicted observation at the time k, based on prior ob-

servations. The innovation vector ν(k), defined as the difference between the

actual sensor observation z(k) and the predicted observation z(k|k − 1), is thencomputed using equation (4.14)

ν(k) = z(k)− z(k|k − 1) . (4.14)

The innovation is then multiplied by the Kalman weighting matrix W (k)

generated by the covariance loop. This multiplication determines the degree to

which the innovation influences the new estimate. Its value is chosen so that the

mean squared error in the estimate is behaviours. The weighting matrix W (k) is

defined as

W (k) = P (k|k − 1)HT (k)S−1(k) , (4.15)

where S(k) is the innovation covariance matrix, computed as

S(k) = H(k)P (k|k − 1)HT (k) +R(k) , (4.16)

and the state prediction covariance matrix P (k) is

P (k|k − 1) = F (k)P (k − 1|k − 1)F T (k) +Q(k) , (4.17)

where F T (k) is the transpose of the transition matrix F (k). The matrices R(k)

4.3 The Extended Kalman Filter 89

and Q(k) are respectively the observation- and process-noise covariance matrices.

The weighted innovation can now be added to the prediction to generate an

updated state estimate x(k|k) as

x(k|k) = x(k|k − 1) +W (k)ν(k) . (4.18)

The third cycle is the estimate covariance cycle, which begins by generating

a prediction covariance according to Equation (4.17) on the basis of the state

model F (k) and the process noise covariance Q(k). The innovation covariance is

then computed using Equation (4.16) which is based on the observation model

H(k) and the estimated observation noise R(k). The innovation covariance and

the prediction covariance are then used to compute the updated state covariance

P (k|k) according to Equation (4.19)

P (k|k) = P (k|k − 1)−W (k)S(k)W T (k) . (4.19)

Finally the time index is incremented and the cycle is repeated.

4.3 The Extended Kalman Filter

4.3.1 Introduction

In practice most processes do not evolve linearly. To deal with nonlinear dynam-

ics the Extended Kalman Filter (EKF) has been developed. It is not quite an

extension of the Kalman filter, as the name implies, but a crude approach for

approximating nonlinear systems with optimising ones. The EKF predicts the

4.3.2 State Prediction 90

future state of a system under the assumption that its process and observation

model are locally linear. The result is a linear estimator for a nonlinear sys-

tem which has the same basic form as the linear Kalman filter [Maybeck, 1979,

Durrant-Whyte, 1994a].

4.3.2 State Prediction

A nonlinear process can be described by a nonlinear discrete-time state equation

in the form

x(k) = f [x(k − 1), u(k), k] + v(k) , (4.20)

where x(k−1) is the state at the time k−1, u(k) is the known input vector, v(k)is some additive noise, x(k) is the state at the time step k and f(·, ·, k) is thenonlinear state transition function that maps the previous state and the current

control input to the current state.

Observations of the state of the process are made according to a nonlinear

observation equation of the form

z(k) = h[x(k)] + w(k), (4.21)

where z(k) is the observation made at the time k, x(k) is the true state at the time

k, w(k) is some additive observation noise, and h(·, k) is the nonlinear observationmodel mapping the current state to the observation.

Expand f [x(k − 1), u(k), k] of Equation (4.20) as a Taylor series about the

estimate x(k − 1|k − 1) to obtain the following equation

4.3.2 State Prediction 91

x(k) = f [x(k − 1|k − 1), u(k), k] +�fx(k)[x(k − 1)− x(k − 1|k − 1)]

+ O{[x(k − 1)− x(k − 1|k − 1)]2}+ v(k) , (4.22)

where �fx(k) is the Jacobian of f evaluated at x(k − 1) = x(k − 1|k − 1).

Truncating Equation (4.22) at first order, and taking the expectation conditioned

on the first k − 1 observations gives an equation for the state prediction:

x(k|k − 1) = E[x(k)|Zk−1]

≈ E{f [x(k − 1|k − 1), u(k), k]

+ �fx(k)[x(k − 1)− x(k − 1|k − 1)] + v(k)|Zk−1}

= f [x(k − 1|k − 1), u(k), k] . (4.23)

where Zk−1 are the observations. It is assumed that x(k−1|k−1) is approximatelyequal to the conditional mean

x(k − 1|k − 1) ≈ E[x(k − 1)|Zk−1] ,

and that the process noise v(k) has zero mean. The predicted state covariance

P (k|k − 1) may then be expressed in terms of the covariance of the previous

estimate as

P (k|k − 1) ≈ �fx(k)P (k − 1|k − 1)� fTx (k) +Q(k) . (4.24)

4.3.3 Observation 92

4.3.3 Observation

As with the state space equation, the function h[x(k)] of Equation (4.21) is ex-

panded as a Taylor series about the state prediction x(k|k − 1)

z(k) = h[x(k)] + w(k)

= h[x(k|k − 1)] +�hx(k)[x(k|k − 1)− x(k)]

+ O{[x(k|k − 1)− x(k)]2}+ w(k) , (4.25)

where �hx(k) is the Jacobian of h evaluated at x(k) = x(k|k − 1). Upon trun-cating equation (4.25) at first order, and taking expectations conditioned on the

first k − 1 observations, the following equation for the predicted observation is

obtained:

z(k|k − 1) � E[z(k)|Zk−1]

≈ E{h[x(k|k − 1)] +�hx(k)[x(k|k − 1)− x(k)] + w(k)|Zk−1}

= h[x(k|k − 1)] , (4.26)

where both the state prediction error and the observation noise have zero mean.

The innovation can be computed as

ν(k) = z(k)− h[x(k|k − 1)] , (4.27)

and the innovation covariance as

4.3.4 Update 93

S(k) = �hx(k)P (k|k − 1)� hTx (k) +R(k) , (4.28)

by assuming that the state prediction is statistically uncorrelated with the current

observation noise w(k) and that it is dependent only on the noise term w(j) and

v(j), j ≤ k − 1.

4.3.4 Update

In common with the KF, an appropriate gain matrixW (k) has been chosen which

minimises the conditional mean-squared estimation error. The gain matrix W (k)

is computed as

W (k) = P (k|k − 1)� hTx (k)[�hx(k)P (k|k − 1)� hTx (k) +R(k)

]−1

= P (k|k − 1)� hTx (k)S−1(k) , (4.29)

and the updated state estimate is

x(k|k) = x(k|k − 1) +W (k){z(k)− h[x(k|k − 1)]} . (4.30)

With the gain matrix W (k) obtained from Equation (4.29), Equation (4.30)

becomes the optimal3 linear estimator for the state x(k) under given conditions.

3minimum mean-squared error

4.3.5 Understanding the Extended Kalman Filter 94

4.3.5 Understanding the Extended Kalman Filter

Comparison of Equations (4.12), (4.17), (4.18), (4.19) with Equations (4.23) to

(4.30) shows that the (optimising) extended Kalman filter algorithm is very sim-

ilar to the (linear) Kalman filter algorithm, with the substitution of F (k) −→�fx(k) and H(k) −→ �hx(k) being made in the equations for variance and gain

propagation. The linearisation is justified by the argument that the estimate

maintained by the filter is close to the true state of the system. The expected

values of the second and higher-order terms in the Taylor expansion are therefore

small. Truncating the series after the first order introduces second and higher-

order errors. The effect of this approximation is negligible by the argument above.

The EKF will not in general be the best estimator under any reasonable

criterion. That is, there is theoretically a nonlinear estimator that will outperform

an EKF for a given nonlinear estimation problem. The EKF will, however, be the

best linear estimator with respect to the minimum mean-squared error. Practice

shows that an EKF can produce very good estimates of the states of a nonlinear

system even when the assumptions required for its derivation clearly do not hold.

The EKF has therefore found wide application in a variety of estimation problems.

4.4 The Distribution Approximation Filter

4.4.1 Introduction

The Distribution Approximation Filter (DAF) is a new filter algorithm devel-

oped by Julier and Uhlmann [Julier et al., 1995, Julier et al., 1996]. It is based

on the following intuition [Julier and Uhlmann, 1995]: With a fixed number of

parameters it should be easier to approximate a Gaussian distribution than it is

to approximate an arbitrary nonlinear function/transformation. This intuition al-

4.4.2 Filter Principle 95

lows a parameterisation that captures the mean and covariance information while

at the same time permitting the direct propagation of this information through

an arbitrary set of nonlinear equations. This is done by generating a discrete dis-

tribution having the same first and second (and possibly higher) moments, where

each point in the discrete approximation can be transformed directly.

4.4.2 Filter Principle

The operational principle of the DAF (see Figure 4.2) can be explained as follows.

Given an n-dimensional Gaussian distribution having covariance P , a set of O(n)

points having the same sample covariance can be generated from the columns

or rows of the matrix ±√nP , the positive and negative roots of P . This set of

points is zero mean, but adding the mean x of the original distribution to each

point yields a symmetric set of 2n points having the desired mean and covari-

ance. Process noise is injected into the state transition by adding a dynamic noise

covariance matrix Q(k) to P (k) before these points are calculated. Because of

its symmetry the set will possess the same first three moments as the original

Gaussian distribution. These points are then transformed nonlinearly using the

transition equation. The transformed points can then be used to compute the

predicted mean and covariance. In the EKF the covariance has been optimising

using the Jacobian matrix, whereas it passes discretely through a nonlinear trans-

formation in the DAF. The way to choose and to transform these points grants an

accurate prediction of the mean and covariance up to the fourth order, compared

to the second order for the mean and up to fourth order for the covariance by

the EKF. Another benefit of the DAF is that there is no need to undertake the

difficult and time consuming task of determining the Jacobian matrix.

The general formulation of the DAF is summarised in Figure 4.3.

4.5 Conclusion 96

* Nonlinear

*

*

*

*

* *

*

Transformation

Figure 4.2: The Principle of the Distribution Approximation Filter [Julier, 1997]

The quantity κ is a scaling factor which provides an extra degree of freedom

to fine tune the higher order moments of the approximation. Only second order

moments can be captured exactly, and there are errors in the fourth and higher

orders. The higher order moments of the sigma points are scaled by κ. The choice

of κ = 2 yields the Minimum Mean Squared Estimate (MMSE) in one dimension

when f [·] is a quadratic function, and any other choice of κ yields the MMSEresults for linear f [·].

4.5 Conclusion

Sections 4.2 and 4.4 introduced two estimation algorithms for nonlinear processes,

and provided details of their working principles and construction. While the

EKF is a filtering technique that has been applied for many years, the DAF is

a relatively new filter that is just beginning to be used in practical applications.

Theoretically the DAF can achieve results equal to or better that the EKF when

applied to nonlinear systems.

The first application of the DAF was described in [Julier, 1997], where it was

4.5 Conclusion 97

❶ The set of translated sigma points is computed from the n × n matrix P (k|k) as

σ(k|k) ←− 2n columns from ±√(n+ κ)(P (k|k) +Q(k))

χo(k|k) = x(k|k),χi(k|k) = σi(k|k) + x(k|k) ,

which assures that

P (k|k) = 1n+ κ

2n∑i=1

[χi(k|k)− x(k|k)][χi(k|k)− x(k|k)]T .

❷ The predicted mean is computed as

x(k + 1|k) = 1n+ κ

[κχo(k + 1|k) + 1

2

2n∑i=1

χi(k + 1|k)]

.

❸ And the predicted covariance is computed as

P (k + 1|k) =1

n+ κ{κ[χo(k + 1|k)− x(k + 1|k)][χo(k + 1|k)− x(k + 1|k)]T

+12

2n∑i=1

[χi(k + 1|k)− x(k + 1|k)][χi(k + 1|k)− x(k + 1|k)]T }.

❹ The predicted observation is calculated by

z(k + 1|k) = 1n+ κ

{κZo(k + 1|k) + 12

2n∑i=1

Zi(k + 1|k)} .

❺ And the covariance is determined by

Pzz(k + 1|k) =1

n+ κ{κ[Zo(k + 1|k)− z(k + 1|k)][Zo(k + 1|k)− z(k + 1|k)]T

+12

2n∑i=1

[Zi(k + 1|k)− z(k + 1|k)][Zi(k + 1|k)− z(k + 1|k)]T } ,

where Pνν(k + 1|k) = Pzz(k + 1|k) +R(k + 1).

❻ Finally the cross correlation matrix is determined by

Pxz(k + 1|k) =1

n+ κ{κ[χo(k + 1|k)− z(k + 1|k)][Zo(k + 1|k)− z(k + 1|k)]T

+12

2n∑i=1

[χi(k + 1|k)− z(k + 1|k)][Zi(k + 1|k)− z(k + 1|k)]T } .

Figure 4.3: General Formulation of the Distribution Approximation Filter

4.5 Conclusion 98

used for modelling high speed land vehicles. Since this thesis deals with modelling

of a tracked vehicle with equations of motion that inherently are nonlinear, the

prospect of applying the DAF is an interesting idea. Chapter 6 will show some

simulation results of the tracked vehicle model using both the EKF and the DAF.

For this special case a significant improvement of results could not be achieved by

the DAF, compared to the EKF. Because of this finding, and because the EKF is

a proven filter it was chosen as the estimator for the tracked vehicle model. The

results in this thesis have been achieved using the EKF as estimator.

Chapter 5

Experimental Trials

5.1 Introduction

This chapter describes the design and instrumentation of a Komatsu model PC05-

7 mini excavator as a test-bed for conducting trials related to the thesis. In order

to allow the vehicle to be driven autonomously, a digital control system and

associated sensors have been retro-fitted to the vehicle. Electro-hydraulic servo

valves were also fitted to replace the original manually-actuated direction control

valves. Section 5.2 describes the design of the control, actuation devices and

sensors fitted on the digger. Section 5.3 describes the test sites and the process

of conducting experiments.

5.2 Design and Instrumentation of the Excava-

tor

The Komatsu mini excavator PC05-7, shown in Figure 5.1, is a small brother

of the much larger machines used in the mining and construction industry. The

99

5.2.1 The Hydraulic System 100

machine has an operating weight of 1280kg and is one metre wide. Despite its

small size and light weight, the design and construction of the vehicle is very

similar to larger machines. It has a bucket capacity of 0.05m3 and can dig at

a height of more than 3 metres, at a depth of 2 metres and reach as far as

3.5 metres [Komatsu, ]. Features like rubber shoes, large drawbar pull and full

hydraulic actuation and control of all axes give it excellent work performance. In

this section, relevant parts of the excavator hardware, together with the sensors

fitted and the control system implementation are described. All features help

to make it a versatile test-bed for the purpose of autonomous tracked vehicle

research.

5.2.1 The Hydraulic System

The Komatsu PC05 mini excavator is a fully hydraulic-actuated excavator, which

allows fitting of an “electric over hydraulic” control system to all axes. A 3-

Figure 5.1: The Komatsu Mini Excavator PC05, as Delivered

5.2.1 The Hydraulic System 101

cylinder, four-stroke cycle water-cooled 9.6 kW diesel engine drives two external

gear pumps, which each produce a maximum flow of 11.9 litres per minute at

a relief valve pressure of 18.6 MPa. The pumps then actuate all the excavator

axes in two separate circuits: one for left track, arm, boom swing and cab swing

and one for right track, bucket, boom, and blade lift. The tracks are driven by

hydraulic axial piston motors and the cab swing by an orbit motor, with the

remaining axes actuated by cylinders.

In the original design each of the two hydraulic pumps independently supplies

a different hydraulic circuit. Each pump therefore supplies flow to one track and

three other actuators, as shown in Figure 5.2. Since the arm is usually inactive

when the vehicle is moving, the full flow of a pump may drive each track motor.

This design is simple in construction but not effective because the two pumps

are not always loaded equally, and the supply head varies with the demanded oil

flow. The hydraulic circuit has been re-designed to provide a combined constant

hydraulic head to supply the servo valves. The new hydraulic circuit is shown in

Figure 5.3.

In the retro-fitted hydraulic circuit the oil flows of both hydraulic pumps

are combined before leading to the servo valves. A safety system, consists of a

piloted unloading valve and an unload solenoid valve, has been fitted to the pump

output. The pilot unloading valve is activated mechanically by a relief valve to

allow a bigger flow of oil to the tank, reducing the heating of the oil. The unload

solenoid valve is a safety valve. Unless it is activated electrically, oil will flow to

the tank and not to the servo valves. An accumulator, precharged to 70 bar, is

used to charge the hydraulic system and compensate the variation of the load.

This configuration allows all devices to working at the same time and therefore a

higher flexibility of all systems can be granted.

5.2.1 The Hydraulic System 102

Cab swingM

Tank

Bucket

Boom

Arm

Hydraulic pumps

Control valves

Engine

Boom swing

Left hand travel

Right hand travel

Blade

Figure 5.2: The Hydraulic Circuit Before Retro-fitting

Hyd

raul

ic p

umps

M

Tank

Right hand travel

Bucket

Boom

Blade

Boom swing

Arm

Swing

Left hand travel

Control valves

Engine

Acc

umul

ator

Prec

harg

e 70

bar

Unl

oadi

ng c

artr

idge

Figure 5.3: The Hydraulic Circuit After Retro-fitting

5.2.2 Sensors 103

The retro-fitting of hydraulic servo valves and other hydraulic components

proved to be troublesome. The difficulties were compounded by the limited space

available on the mini excavator. The main difficulties encountered were a series of

oil leaks and a valve malfunction; both took a long time to diagnose and correct.

Great effort on the part of many people was needed to produce a reliable vehicle.

The original manually-actuated hydraulic direction control valves were re-

placed by electrohydraulic servo valves. The servo valves has been chosen over

proportional or direction control valves because they can be easily adapted to

different control regimes and are therefore suitable to use for a variety of con-

trol techniques. The chosen valves are Moog series 633 direct drive servo valves

(DDV), shown in Figure 5.5. These valves use a permanent magnet linear force

motor to position the valve spool directly. This feature is said to provide better

tolerance to fluid contamination than does a piloted servo valve. All spool posi-

tioning and drive electronics are integrated with the valve, and an analog output

directly proportional to valve spool position is provided.

The servo valves require a 24VDC unregulated power supply and consume

1.2A per valve. As the excavator was initially fitted with an alternator that pro-

duced a current of only 20A at 12VDC, a set of 24V lead-acid storage batteries

was fitted. These batteries supplied power to all eight servo valves, together

with the sensors, digital loop-closure controllers and the supervisory control com-

puter. Some tests were conducted under these conditions. Later, a more powerful

24V 50A alternator was fitted to supply sufficient electrical power for continuous

operation.

5.2.2 Sensors 104

Figure 5.4: The Komatsu PC05, fitted with Sensors and Automatic Control Sys-tems

Figure 5.5: View from Above, Showing Moog Series 633 Direct Drive Servo ValveStacks

5.2.2 Sensors 105

Figure 5.6: Novotechnik Inductive Angle Sensor

5.2.2 Sensors

To measure the axis positions, angular position sensors are fitted on all eight

controlled axes of the vehicle. The sensors are Novotechnik SXA58-S/0012-SR-

SA1-K02 inductive absolute angular position sensors, shown in Figure 5.6. This

sensor has 12 bit resolution (1 part in 4096) and 12 bit accuracy (0.05o). It can

be connected directly to the synchronous serial interface port of the digital loop-

closure controller. The sensor may be clocked at frequency between 50 and 500

kHz, and therefore returns axis position measurements at a rate of 4 - 20 kHz per

axis, depending on the clock rate. These sensors require an unregulated supply

voltage between 10 and 30 VDC and consume less than 50mA.

Most of the excavator axes rotate at relatively low speed: in particular, the

track drive sprockets rotate at less than 30 RPM. The angular position sensors

are used on these axes to derive track drive sprocket angular velocity from the

change of angular position during a sampling interval. At a sampling interval

of 10ms, the change of position does not exceed 12 counts. To increase the

resolution of velocity measurement, the track encoders are driven through a 3:1

speed-increasing synchronous drive belt.

Pressure transducers are fitted across the A and B ports of each valve to mea-

5.2.2 Sensors 106

sure the differential pressure across each hydraulic motor and cylinder. Each of

the 8 hydraulic axes are fitted with a pair of UCC PDT.250121 pressure trans-

ducers. These transducers are temperature-compensated strain gauge pressure

transducers with integral bridge excitation and output amplifiers. They are rated

to 25 MPa and require an unregulated power source between 10 and 30 VDC.

The signal output may be taken as either a 0-5VDC differential output (four wire

operation) or as a 1-6VDC single-ended output (three wire operation). It may be

seen that all the sensors used are sub-equations by compact construction, unreg-

ulated power supply, high operating temperature range and compatibility with

digital controllers.

For later tests a PLS laser scanner from Sick Optic-Electronic Pty. Ltd. was

used to measure the actual path of the vehicle on test sites. Figure 5.7 shows

the PLS scanner. The laser scanner provides range and bearing information by

measuring the time-of-flight of laser light. The scanner can provide range and

bearing information over a sensor field of view 180o × 50 metres with an angularresolution of 0.5o and a distance resolution of ± 50mm. The PLS sensor interfacesto a computer via an RS-232 or RS-422 serial line.

Experiments with the PLS laser scanner showed that the device functions

properly in diverse environments, both outdoors and indoors. In experiments

with the PLS scanner indoors an accuracy of 5 cm has been achieved. Although

the accuracy of the received data is only moderate, the PLS sensor is able to

supply valuable “ground truth” trajectory data against which the performance

of the various filters may be evaluated. Apart from reconstruction of the vehicle

trajectory from video images, it is the only method of path verification available

to the writer.

5.2.2 Sensors 107

Figure 5.7: The Sick PLS Laser Scanner

5.2.3 Controller and Data Logging System 108

5.2.3 Controller and Data Logging System

All eight hydraulic axes of the excavator are controlled by Moog M2000 pro-

grammable servo controllers (PSCs), a proprietary digital closed loop controller.

This controller allows for the implementation of a modular, highly flexible control

system, suited to a wide range of hydraulic and electrical drive applications. At

the core of the system is a 2 axis PSC module. Each PSC module carries analog,

serial communications and logic input/output (I/O), which may be expanded to

include parallel or serial data interfaces and CAN Bus communications. The I/O

functionality is user-programmable and may be used to close the control loop us-

ing a wide variety of actuator and transducer types, or used to provide interfaces

to other PSC’s, machine operation consoles or to a host control system. The PSC

uses the NEC V35 processor which is based on the Intel 8086 processing core.

In addition to the processing capacity, the M2000 has a number of other built-in

features such as six channels of 12 bit analog input, two channels of 12 bit analog

output, eight bits of digital logic input/output, two synchronous serial interfaces

and a PSC shutdown system.

Each PSC is fitted with a “piggyback” CAN bus card to allow high speed

communication between multiple PSCs using the CAN protocol, a serial commu-

nication protocol which efficiently supports distributed real time control with a

very high level of security and flexibility. The data is transmitted over a twisted

pair and can reach a bit rate of 1Mbits per second. Typically, PSC to PSC CAN

communication is used to provide high-speed data exchange for event-driven se-

quencing of multi-axis control systems. In this application, however, the CAN

bus is used to exchange data between the PSC axis controllers, and a PC-based

supervisory controller. The high-speed communication provided by the CAN bus

is essential when large data transfers are involved, or when interprocessor latency

5.2.3 Controller and Data Logging System 109

Figure 5.8: Moog M2000 Programmable Servo Controllers and DC-DC Converters

5.2.4 Supervisory Controller and Data Acquisition 110

must be controlled.

The supervisory PC can also communicate with each PSC through an RS232

serial port, using Moog’s “Engineering User Interface” (EUI) program. The EUI

is a low level text-based user interface intended primarily for the configuration

and tuning of PSC real-time control function blocks. The EUI has full access to

PSC parameters, allowing for rapid construction and commissioning of control

system software.

Four PSC’s have been used to control all 8 axes of the excavator, and a fifth

PSC is used to implement ancillary control. Together with auxiliary I/O interface

cards, the five controllers occupy 10 slots in the M2000 rack (Figure 5.8). The

PSC modules, together with the necessary power supply, are installed in a 19 inch

rack.

All electrical and hydraulic power required is derived from the vehicle’s engine.

DC to DC converters provide the various potentials required for the PSCs, the

supervisory computer, and the various sensors.

5.2.4 Supervisory Controller and Data Acquisition

Supervisory control and data acquisition is performed by a 120MHz Pentium in-

dustrial IBM-compatible PC, running the Microsoft Windows NT 4.0 operating

system. Communication between the PSC’s and the PC is supported by a Soft-

ing CAN-AC2 interface board. The Softing CAN-AC2 is an interface board for

IBM-compatible personal computers that implements two CAN ports using an

embedded NEC V25+ processor and the Philips PCA82C200 CAN controller.

The physical interface consists of two electrically isolated CAN high speed inter-

faces per ISO DIS 11898. Together with the supplied driver library, the CAN-

AC2 allows simple integration of PC-supported applications into CAN networks.

5.2.5 Navigation 111

Commands and control sequences can be sent from the supervisory PC to the

PSC’s and data from different sensors and PSC’s can be returned to the PC for

processing.

A Microsoft Sidewinder 3D Pro joystick is connected to the supervisory com-

puter to provide reference command inputs. The reference inputs to the tracks

are vehicle speed and heading direction, or position setpoints to the excavator

arm axes. With its three axes of motion, buttons and switches, the joystick pro-

vides the PC with setpoints required to control all excavator axes with sufficient

accuracy and flexibility.

5.2.5 Navigation

The excavator has been fitted with an Inertial Measurement Unit (IMU) manu-

factured by Watson Industries Inc. This is a solid state gyroscope system that

models the functions of an attitude gyroscope and a slaved heading gyroscope.

The IMU is also provided with three linear accelerometers for measurement of

accelerations in the X,Y, Z motion directions, a high accuracy triaxial fluxgate

magnetometer to provide heading information, and two pendulums for measure-

ment of bank and elevation. The scaling of the digital outputs of the IMU is

summarises in Table C.1 of Appendix C.

The angular rate sensor signals are coordinate-transformed and then inte-

grated to produce attitude and heading outputs that reflect the usual aircraft

orientation coordinates: roll, pitch and yaw. The attitude and heading signals

are compared against two vertical reference pendulums and a triaxial fluxgate

magnetometer to derive short term absolute errors. These errors are filtered with

a long time constant and are used to adjust biases in the system so that long-term

convergence of the system is to the vertical references and the magnetic heading.

5.2.5 Navigation 112

Figure 5.9: Watson IMU Attitude and Heading Translation and Correction

Compensations for centrifugal forces and velocity changes are used to improve

overall stability and accuracy. The various sensors are interfaced to a 68HC11

microcontroller through a 16 bit analog to digital converter. The IMU has both

an analog and a digital interface. The analog outputs are from a 12 bit digital

to analog converter while the digital output is transmitted over an asynchronous

serial line at 9600 baud.

The accelerometer signals are corrected continuously with a bias estimator

driven by the pendulous angle references. The pendulous angle references pro-

vide a low bandwidth reference which has excellent low-frequency acceleration

performance. Figure 5.10 depicts the method used by Watson to correct bias

errors in the accelerometers. The accelerometers are corrected by the micropro-

cessor for scale and cross-axis error. The normal accelerometer outputs are in

body-axis in that they will rotate as the vehicle rotates.

While the standard data outputs provide a closed-loop integration of angular

rate to the references of the pendulums, heading rate to the magnetic heading

5.2.5 Navigation 113

Angle References

Linear

Accelerometers

XYZ

XYZ

Bias Integrator

+

-

-

+

Roll and Pitch

Figure 5.10: Accelerometer Bias Correction

sensor, and accelerometer biases to the accelerometers, an alternate set of sensor

data is available from the Watson IMU. The reference command provides the

same sensor information with the following differences:

❶ Bank and Elevation are obtained directly from the pendulums - no gyro

stabilisation.

❷ Magnetic Heading is not gyro stabilised.

❸ Angular Rate Sensors are not closed-loop bias corrected.

❹ Accelerometers are not closed-loop bias corrected.

For the experiments conducted for the present thesis, this latter reference

mode has been chosen, and only raw data has been received. The IMU’s onboard

facility for error and bias compensation did not function properly at the very low

speeds of the tracked vehicle. By using the raw data and applying an appropriate

error model the position drift of a standard low cost, good quality gyro may be

held to 0.16m/sec, compared to 1.18m/sec for the same sensor without calibration

[Nebot and Durrant-Whyte, 1997].

To improve the accuracy of position estimation in future experiments, the

excavator can be fitted with a Differential Global Positioning System (DGPS). A

5.2.6 Data Acquisition 114

DGPS system from Novatel, specified to have an accuracy of 2cm, is being tested

at the University of Sydney [Durrant-Whyte et al., 1997] and will be incorporated

in the vehicle navigation system.

5.2.6 Data Acquisition

The following data was made available over a CAN bus connection between the

on-board supervisory computer and a laptop computer:

• The angular velocities of the track drive sprockets and the positions of otheraxes.

• The track drive circuit servo valve spool positions.

• The differential pressure across the hydraulic motors and the cylinders.

The excavator has a maximum speed of approximately 0.5 metre per second.

Since the track drive sprocket angular velocity is derived from changes in the

angular position, the resolution of the vehicle’s speed is also very low. If the

track drive sprocket angular position encoder is driven at the same speed as the

sprocket, the encoder count changes by not more than 10 counts over the chosen

sample interval of 10ms. Once noise is accounted for, the velocity error may

exceed 15 − 20%, which is not sufficiently accurate for control purposes. The

track encoders are therefore driven through a synchronous belt to rotate three

times faster than the track drive sprocket.

5.3 Test Sites and Test Procedures 115

5.3 Test Sites and Test Procedures

5.3.1 Test Site Layout

The site where most of the tests were conducted is an unused grass field on the

Darlington campus of The University of Sydney. The entire field is not ideal as

a controlled test site because it is uneven and sloping, so that only part of it

could be used. The section chosen for the tests is about 20m by 30m in size and

is relatively even compared to the full field. The grass is higher then usual in

parks, which contributed to the high resistance to the tracks during the tests,

especially when the excavator was turning. This effect was observed during the

first test, when the vehicle could not turn “on the spot”. It was necessary to set

the system working pressure to its highest setting to provide sufficient tractive

effort to overcome this resistance.

5.3.2 First Test

In the first test the vehicle made two preliminary runs of different length and

shape. The objective of this test was to demonstrate that all the hydraulic and

control systems functioned as expected. During this test no navigation system was

present. The test confirmed the readiness of the vehicle for further experiments,

although some changes to the vehicle were subsequently required. Two faulty

pressure transducers were replaced, and the pump unloading valve was set at its

maximum value to provide the highest possible hydraulic system pressure. The

Watson inertial measurement unit was also fitted at this time.

5.3.2 First Test 116

−4 −3 −2 −1 0 1 2 3−1

0

1

2

3

4

5

6Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(a) Estimated Position

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

Time in sec

Slip

fact

ors

in %

Track Slips

(b) Estimated Slips

Figure 5.11: Estimated Position and Slips in Run 1

5.3.2 First Test 117

−3 −2 −1 0 1 2 3 4−2

−1

0

1

2

3

4

5Position of the Vehicle, read from Encoders

x coordinate in m

y co

ordi

nate

in m

(a) Position Read from Encoders

−100 0 100 200 300 400 500 600−100

−50

0

50

100

150

200

250

300

350Position of the Vehicle, read from IMU

x coordinate in m

y co

ordi

nate

in m

(b) Position Read from from IMU

Figure 5.12: Position Read from Encoders and from IMU

5.3.2 First Test 118

0 10 20 30 40 50 60 70 80 900

0.005

0.01

0.015

0.02

0.025

Time in sec

Cov

aria

nce

in m

2

Position Covariance

(a) Position Covariance

0 10 20 30 40 50 60 70 80 90−8

−6

−4

−2

0

2

4

6

8

10x 10

−3

Time in sec

Ang

le in

rad

Innovation φ

(b) Orientation Error

Figure 5.13: Position Covariance and Orientation Error

5.3.3 Second Test 119

5.3.3 Second Test

The second test was carried out on 6 June 1997. It had been raining for several

days before the test date, and the subsoil was saturated and somewhat soft.

During the test the vehicle made three runs of different shapes. The data collected

during the second test was processed usingMatlab, and is presented in figures 5.11

to 5.18.

Figure 5.11a shows the path of the vehicle in run 1, as estimated using the

kinematic model. The slips estimated using the kinematic model are shown on

Figure 5.11b. Although the true path of the vehicle is not known exactly, the

estimated path is very similar in shape to the path observed using a video recorder.

The vehicle heading estimates that result from the kinematic model are clearly

better in every way than are the headings estimated using only data received from

the encoders or the IMU, as shown in Figures 5.12a and b. The path accuracy

could not be accessed because there was no system to accurately monitor or

reconstruct the vehicle’s path.

Figure 5.14 shows the positions estimated using the kinematic model for runs 2

and 3, driven by data received from the encoders and the IMU. It may be seen that

the vehicle moved along complicated paths during those runs. This complexity

was not intentional, but was apparently caused by insufficient tractive effort when

turning that caused the excavator to slip “on the spot” when it has to make a

turn. Figures 5.18a and b show the differential pressures on the track drive motors

to be high, variable and noisy during turning. This confirms that the resistance

is relatively high during turning so that a higher differential pressure, and with

it higher power, is needed to enable the vehicle to turn.

At the same time the supply pressure did not reach the designed system

working pressure of about 190bar. A pressure loss is known to have occurred, but

5.3.3 Second Test 120

−1 −0.5 0 0.5 1 1.5 2 2.5 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(a) Estimated Position in Run 2

−1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(b) Estimated Position in Run 3

Figure 5.14: Estimated Positions in Run 2 and 3

5.3.3 Second Test 121

0 0.5 1 1.5 2 2.5 3 3.5 4−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5Position of the Vehicle, read from Encoders

x coordinate in m

y co

ordi

nate

in m

(a) Positions read from Encoders in Run 2

−1 0 1 2 3 4 5−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Position of the Vehicle, read from Encoders

x coordinate in m

y co

ordi

nate

in m

(b) Position read from Encoders in Run 3

Figure 5.15: Positions Read from Encoders in Runs 2 and 3

5.3.3 Second Test 122

−35 −30 −25 −20 −15 −10 −5 0 5 10 150

50

100

150

200

250

300

350

400

450

500Position of the Vehicle, read from IMU

x coordinate in m

y co

ordi

nate

in m

(a) Position read from IMU in Run 2

−80 −60 −40 −20 0 20 400

50

100

150

200

250

300Position of the Vehicle, read from IMU

x coordinate in m

y co

ordi

nate

in m

(b) Position read from IMU in Run 3

Figure 5.16: Positions Read from IMU in Runs 2 and 3

5.3.3 Second Test 123

0 10 20 30 40 50 60 700

10

20

30

40

50

60

70

80

Time in sec

Slip

fact

ors

in %

Tracks Slip

(a) Slips Estimated on Run 2

0 10 20 30 40 50 600

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

(b) Slips Estimated on Run 3

Figure 5.17: Slips Estimated on Runs 2 and 3

5.3.3 Second Test 124

0 10 20 30 40 50 60 70−200

−150

−100

−50

0

50

100

150

200

Time in sec

Pre

ssur

e in

bar

Differential Pressures on the axial Piston Motors

left track right track

(a) Differential Pressures on the Motors in Run 2

0 10 20 30 40 50 60−150

−100

−50

0

50

100

150

200

Time in sec

Pre

ssur

e in

bar

Differential Pressures on the axial Piston Motors

left track right track

(b) Differential Pressures on the Motors in Run 3

Figure 5.18: Differential Pressures on the Motors in Runs 2 and 3

5.3.4 Third Test 125

the cause is unknown. It was also observed that sometimes the vehicle was not

able to turn on the spot because the required force could not be produced. Only

when the vehicle has gained some momentum, it could then manage to turn. This

effect can be seen in Figure 5.17, where the slip coefficients are higher than are

usual for straight line motion, and sometime very high (about 70%). The results

of the test can be summarises as follows:

❶ The tracks slip during the vehicle’s motion. The slip magnitude is significant

during turning

❷ The encoders and the INS alone do not provide accurate estimation of the

vehicle position during motion.

❸ There is a need to have a reference path of the vehicle to judge the perfor-

mances of the estimation model.

5.3.4 Third Test

The third test of the vehicle was conducted on 13 January 1998. Before the

test the vehicle was examined to find the cause of the power loss but the result

was inconclusive. A third field test was therefore conducted not only to gather

data, but also to clarify the problems encountered. During this and subsequent

tests, a PLS laser scanner was used to monitor the motion of the vehicle to allow

reconstruction of the vehicle’s true path. The test was carried out at the same

site as the previous tests. A vehicle path reconstructed from the laser scanner

data is shown in Figure 5.19a. The PLS laser scanner was positioned at point

(0,0), not visible on the plot. The scanner sends a beam of infrared light parallel

to the ground surface at a height of about 60cm. The points near the centre of

the plot represent the PC and the operator. The vehicle’s body may be seen as

5.3.4 Third Test 126

sets of points forming curvilinear outlines. Since the exact position of the centre

point of the vehicle is not known, the path could be considered to be placed in

the middle of the laser reflections from the vehicle’s body.

During the test the vehicle made four runs of different shapes. Figures 5.19

and 5.20 show the positions of the vehicle in the run 1, estimated using different

models and methods. Figures 5.21 5.22 show the estimated slips of the vehicle,

the actual rotational speeds of the track drive sprockets read from encoders and

the tractive efforts derived from the pressure transducer readings using Equation

(3.25).

The similar shapes of the paths in Figures 5.19a, 5.20a and b are apparent,

whilst the path (Figure 5.19b) reconstructed solely from encoder data highlights

the significance of slip on the path of the vehicle. Figures 5.21a and b compare

the slips estimated using the kinematic and comprehensive models. The slips in

Figure 5.21a increase when the vehicle was turning. They are unable to decrease

because the filter is based on an assumption that the slip must remain constant

when the vehicle is moving in a straight line. In this situation, the change in yaw

rotation is minimal so that slip can not be detected. This leads to a higher error

in position because of higher assumed slips.

Figure 5.21b depicts track slips estimated using the comprehensive model. In

this model the slips are corrected each time the vehicle change its state from

“turning” to “moving straight”. This is done through a boolean comparison be-

tween the angular track speeds. The magnitude of speed change on a track and

the difference of the track speeds can be used to estimate the state change. Ap-

propriate values have been determined during tests and applied to the filters.

Figure 5.22b depicts the forces acting on the tracks during the test, calculated

from Equation (3.27). During turning the force increases on one track and de-

5.3.4 Third Test 127

-25 -20 -15 -10 -5

-8

-6

-4

-2

0

2

4

6

8

10

x coordinate in meter

y coordinate in meter

Sick scanner run 91

(a) Position Estimated from the Laser Scanner Data

−20 −15 −10 −5 0 50

5

10

15

20

25Position of the Vehicle, read from Encoders

x coordinate in m

y co

ordi

nate

in m

(b) Position Read from Encoders

Figure 5.19: Positions Estimated from the Laser Scanner Data and Read fromEncoders in Test Three, Run 1

5.3.4 Third Test 128

−2 0 2 4 6 8 10 12 14 16 18−5

0

5

10

15Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(a) Position Estimated using the Kinematic Model

0 2 4 6 8 10 12 14 16 18 20−6

−4

−2

0

2

4

6

8

10

12

14Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(b) Position Estimated using the Comprehensive Model

Figure 5.20: Positions Estimated using Kinematic and Comprehensive Models inTest Three, Run 1

5.3.4 Third Test 129

0 20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

(a) Slips Estimated using the Kinematic Model

0 20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

(b) Slips Estimated using the Comprehensive Model

Figure 5.21: Estimated Slips in Test Three, Run 1

5.3.4 Third Test 130

0 20 40 60 80 100 120 140 160−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time in sec

rota

tiona

l spe

ed in

rad

/s

Rotational Track Speeds

(a) Track Drive Speeds read from Encoders

0 20 40 60 80 100 120 140 160−1500

−1000

−500

0

500

1000

1500

2000

2500

Time in sec

For

ces

on tr

acks

in N

Track Forces

(b) Forces on the Tracks

Figure 5.22: Track Drive Speeds & Forces in Test Three, Run 1

5.3.5 Fourth Test 131

creases on the other. The slips in straight line motion are computed using the

relationship of force and slip in Equation (3.12).

It should be noted here that a negative force does not mean that the vehicle is

moving backward, as the encoder data reveals no negative values during turning.

The negative force is a braking force that stops the inner track from moving

relative to the track frame. It is caused by a piloted check valve in the track drive

motor.

Figures 5.22c and d reveal the high level of noise in the data, as well as the fact

that the vehicle has been driven unevenly during the run, with many starts and

stops. This create high uncertainties in estimation of slips. The same problem

occurred during the other runs during test three.

During the test oil leaks were detected on the servo valves. This could be one

of the causes for the loss of power of the vehicle. After the test all valve stacks

were disassembled, and the ‘O’ ring seals replaced.

5.3.5 Fourth Test

Further tests were conducted later on the same test site. Four runs over different

shaped paths were made. The results of run 1 may be seen in figures 5.23-5.26.

The vehicle has been driven smoothly most of time, except during turning

when the vehicle was sometimes unable to turn “on the spot”. This failure to

turn may be due to a loss of hydraulic power between the pump to the axial piston

motors that drive the sprockets. It could also be that a much higher resistance

appears during turning, compared to in straight motion. This resistance could

be so high that the track could not produce the required power to overcome it

and the vehicle has stuck there. Figures 5.26a and b show clearly that on the

third turn where the force on the right track increased, the sprocket could barely

5.3.5 Fourth Test 132

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2-10

-8

-6

-4

-2

0

2

4

6

8

10

x coordinate in meter

y coordinate in meter

Sick scanner run 101

(a) Position Estimated from the Laser Scanner Data

−12 −10 −8 −6 −4 −2 0 2 4 6−2

0

2

4

6

8

10

12

14

16

18Position of the Vehicle, read from Encoders

x coordinate in m

y co

ordi

nate

in m

(b) Position read from Encoders

Figure 5.23: Positions Estimated from the Laser Scanner Data and Read fromEncoders in Test Four, Run 1

5.3.5 Fourth Test 133

−6 −4 −2 0 2 4 6 8 10 12−5

0

5

10

15Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(a) Position Estimated using the Kinematic Model

−10 −8 −6 −4 −2 0 2 4 6 8−10

−8

−6

−4

−2

0

2

4

6

8

10Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(b) Position Estimated using the Comprehensive Model

Figure 5.24: Positions Estimated using Kinematic and Comprehensive Models inTest Four, Run 1

5.3.5 Fourth Test 134

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

(a) Slips Estimated using the Kinematic Model

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

(b) Slips Estimated using the Comprehensive Model

Figure 5.25: Estimated Slips from in Test Four, Run 1

5.3.5 Fourth Test 135

0 20 40 60 80 100 120 140−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time in sec

rota

tiona

l spe

ed in

rad

/s

Rotational Track Speeds

(a) Track Drive Speeds Read from Encoders

0 20 40 60 80 100 120 140−2000

−1500

−1000

−500

0

500

1000

1500

2000

2500

Time in sec

For

ces

on tr

acks

in N

Track Forces

(b) Forces on the Tracks

Figure 5.26: Track Drive Speeds & Forces in Test Four, Run 1

5.3.6 Fifth Test 136

rotate. Only when both tracks were driven could the vehicle turn. The situations

was clarified in runs 2, 3 and 4 (see Figures 5.27 to 5.34), where the variation of

forces and rotation were significant.

Figures 5.29a to 5.35a show noisy encoder data, especially during turning.

The high resistance during turning required the vehicle to be driven gently to

allow turning. Sometimes the vehicle has to stop and apply the necessarily force

to both tracks so that turning became possible. The variation of forces has been

plotted in Figures 5.29b to 5.35b. The many starts and stops during turning

render the filter unstable. The results are therefore very limited, especially in run

4, where the vehicle made more turnings than in other runs, as a more difficult

path has been chosen. A comparison of Figures 5.33a and 5.34a reveals that it

was impossible to track the path of the vehicle.

In all runs the slips have been detected and estimated. The problems remained

the same as by early tests; it was hard to drive the vehicle smoothly, especially

when turning and oil leaks could still not be redressed properly. All this caused a

rough motion of the vehicle and made the estimation of position and slips difficult.

5.3.6 Fifth Test

Test five was conducted on a gravelled car park of The University of Sydney, seen

in Figure 5.36. The car park is not quite even and has a light slope of about 5o.

The car park is of compacted soil, covered by a thin layer of loose crude gravel of

10mm to 30mm in size. This surface proved to create a high amount of slip under

the tracks. The carpark itself has a size of about 20m × 30m. During the testthe vehicle made two runs of different shapes. The results are shown in Figures

5.37 to 5.44.

In run 1 the vehicle started from point (−4, 4) (see Figure 5.37a), moved

5.3.6 Fifth Test 137

-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2-10

-8

-6

-4

-2

0

2

4

6

8

10

x coordinate in meter

y coordinate in meter

Sick scanner run 102

(a) Position Estimated from the Laser Scanner Data

−10 −8 −6 −4 −2 0 2 4 6 8 10−12

−10

−8

−6

−4

−2

0

2

4

6

8Position of the Vehicle, read from Encoders

x coordinate in m

y co

ordi

nate

in m

(b) Position read from Encoders

Figure 5.27: Positions Estimated from the Laser Scanner and read from Encodersin Test Four, Run 2

5.3.6 Fifth Test 138

−10 −8 −6 −4 −2 0 2 4 6 8 10

−18

−16

−14

−12

−10

−8

−6

−4

−2

0

Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(a) Position Estimated using the Comprehensive Model

0 20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

(b) Slips Estimated using the Comprehensive Model

Figure 5.28: Estimated Position and Slips using the Comprehensive Model inTest Four, Run 2

5.3.6 Fifth Test 139

0 20 40 60 80 100 120 140 160−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time in sec

rota

tiona

l spe

ed in

rad

/s

Rotational Track Speeds

(a) Track Drive Speeds Read from Encoders

0 20 40 60 80 100 120 140 160−1500

−1000

−500

0

500

1000

1500

2000

2500

Time in sec

For

ces

on tr

acks

in N

Track Forces

(b) Forces on the Tracks

Figure 5.29: Track Drive Speeds & Forces in Test Four, Run 2

5.3.6 Fifth Test 140

−22 −20 −18 −16 −14 −12 −10 −8 −6 −4

−10

−8

−6

−4

−2

0

2

4

6

8

x coordinate in m

y co

ordi

nate

in m

Sick scanner run 103

(a) Position Estimated from the Laser ScannerData

−4 −2 0 2 4 6 8 10 12 14 16−14

−12

−10

−8

−6

−4

−2

0

2

4

6Position of the Vehicle, read from Encoders

x coordinate in m

y co

ordi

nate

in m

(b) Position Read from Encoders

Figure 5.30: Positions Estimated from the Laser Scanner and read from Encodersin Test Four, Run 3

5.3.6 Fifth Test 141

−8 −6 −4 −2 0 2 4 6 8 10 12

−16

−14

−12

−10

−8

−6

−4

−2

0

2

Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(a) Position Estimated using the Comprehensive Model

0 20 40 60 80 100 120 140 160 1800

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

(b) Slips Estimated using the Comprehensive Model

Figure 5.31: Estimated Position and Slips using the Comprehensive Model inTest Four, Run 3

5.3.6 Fifth Test 142

0 20 40 60 80 100 120 140 160 180−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time in sec

rota

tiona

l spe

ed in

rad

/s

Rotational Track Speeds

(a) Track Drive Speeds Read from Encoders

0 20 40 60 80 100 120 140 160 180−2500

−2000

−1500

−1000

−500

0

500

1000

1500

2000

2500

Time in sec

For

ces

on tr

acks

in N

Track Forces

(b) Forces on the Tracks

Figure 5.32: Track Drive Speeds & Forces in Test Four, Run 3

5.3.6 Fifth Test 143

-18 -16 -14 -12 -10 -8 -6 -4 -2-12

-10

-8

-6

-4

-2

0

2

4

x coordinate in meter

y coordinate in meter

Sick scanner run 104

(a) Position Estimated from the Laser Scanner Data

−6 −4 −2 0 2 4 6 8 10

−2

0

2

4

6

8

10

12

Position of the Vehicle, read from Encoders

x coordinate in m

y co

ordi

nate

in m

(b) Position Read from Encoders

Figure 5.33: Positions Estimated from the Laser Scanner and read from Encodersin Test Four, Run 4

5.3.6 Fifth Test 144

−8 −6 −4 −2 0 2 4 6 8 10−8

−6

−4

−2

0

2

4

6

8Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(a) Position Estimated using the Comprehensive Model

0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

(b) Slips Estimated using the Comprehensive Model

Figure 5.34: Estimated Position and Slips using the Comprehensive Model inTest Test Four, Run 4

5.3.6 Fifth Test 145

0 50 100 150 200 250−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time in sec

rota

tiona

l spe

ed in

rad

/s

Rotational Track Speeds

(a) Track Drive Speeds Read from Encoders

0 50 100 150 200 250−2000

−1500

−1000

−500

0

500

1000

1500

2000

2500

Time in sec

For

ces

on tr

acks

in N

Track Forces

(b) Forces on the Tracks

Figure 5.35: Track Drive Speeds & Forces in Test Four, Run 4

5.3.6 Fifth Test 146

Figure 5.36: The Test Vehicle on Gravelled Soil

straight for a few metres and then made a left turn and so on. Compared to

previous runs on grass, the vehicle was easier to drive; Figures 5.40a and b confirm

the smoothness of the vehicle motion. The estimated path in Figure 5.38b is

similar to the path constructed from the laser scanner data. The estimated slips

are presented in Figure 5.39b. Except in the last turn, where the vehicle remained

almost stationary and the slip became very high, the estimated slips and position

are reasonable.

Run 2 was conducted on the same site some minutes later and the results are

shown in Figures 5.41 to 5.44.

5.3.6 Fifth Test 147

-18 -16 -14 -12 -10 -8 -6 -4-6

-4

-2

0

2

4

6

8

x coordinate in meter

y coordinate in meter

Sick scanner run 111

(a) Position Estimated from the Laser Scanner Data

−5 0 5 10

0

2

4

6

8

10

12

14

16

Position of the Vehicle, read from Encoders

x coordinate in m

y co

ordi

nate

in m

(b) Position Read from Encoders

Figure 5.37: Positions Estimated from the Laser Scanner and read from Encodersin Test Five, Run 1

5.3.6 Fifth Test 148

−2 0 2 4 6 8 10 12

4

6

8

10

12

14

16

18Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(a) Position Estimated using the Kinematic Model

−4 −2 0 2 4 6 8 10

0

2

4

6

8

10

12

14Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(b) Position Estimated using the Comprehensive Model

Figure 5.38: Positions Estimated using Kinematic and Comprehensive Models inTest Five, Run 1

5.3.6 Fifth Test 149

0 50 100 1500

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

(a) Slips Estimated using the Kinematic Model

0 50 100 1500

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

(b) [Slips Estimated using the Comprehensive Model

Figure 5.39: Estimated Slips using the Kinematic and Comprehensive Models inTest Five, Run 1

5.3.6 Fifth Test 150

0 50 100 150−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time in sec

rota

tiona

l spe

ed in

rad

/s

Rotational Track Speeds

(a) Track Drive Speeds Read from Encoders

0 50 100 150−1500

−1000

−500

0

500

1000

1500

2000

Time in sec

For

ces

on tr

acks

in N

Track Forces

(b) Forces on the Tracks

Figure 5.40: Track Drive Speeds & Forces in Test Five, Run 1

5.3.6 Fifth Test 151

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2-10

-8

-6

-4

-2

0

2

4

6

8

x coordinate in meter

y coordinate in meter

Sick scanner run 112

(a) Position Estimated from the Laser Scanner Data

−12 −10 −8 −6 −4 −2 0 2 4 6−8

−6

−4

−2

0

2

4

6

8

10Position of the Vehicle, read from Encoders

x coordinate in m

y co

ordi

nate

in m

(b) Position Read from Encoders

Figure 5.41: Positions Estimated from the Laser Scanner and read from Encodersin Test Five, Run 2

5.3.6 Fifth Test 152

−4 −2 0 2 4 6 8 10 12−14

−12

−10

−8

−6

−4

−2

0

2

4Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(a) Position Estimated using the Kinematic Model

−6 −4 −2 0 2 4 6 8 10−16

−14

−12

−10

−8

−6

−4

−2

0

2Vehicle Position

x coordinate in m

y co

ordi

nate

in m

(b) Position Estimated using the Comprehensive Model

Figure 5.42: Positions Estimated using the Kinematic and Comprehensive Modelsin Test Five, Run 2

5.3.6 Fifth Test 153

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

(a) Slips Estimated using the Kinematic Model

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

(b) Slips Estimated using the Comprehensive Model

Figure 5.43: Slips Estimated using Kinematic and Comprehensive Models in TestFive, Run 2

5.3.6 Fifth Test 154

0 50 100 150−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time in sec

rota

tiona

l spe

ed in

rad

/s

Rotational Track Speeds

(a) Track Drive Speeds Read from Encoders

0 20 40 60 80 100 120−1500

−1000

−500

0

500

1000

1500

2000

Time in sec

For

ces

on tr

acks

in N

Track Forces

(b) Forces on the Tracks

Figure 5.44: Track Drive Speeds & Forces in Test Five, Run 2

5.4 Conclusion 155

5.4 Conclusion

This chapter described the design and instrumentation of a Komatsu PC05-7

mini excavator as a test-bed for conducting trials related to the thesis. The test

sites where tests with the vehicle have been conducted and the experiments were

also discussed. A number of problems were encountered during operation of the

vehicle. Substantial work were required to fix these problems, which delayed the

tests for a long period of time. The first results of the tests with the vehicle on

different soil types have been presented. They have shown that track slips may be

detected and that they are more significant then previously assumed. The tests

also served to validate the performance of the vehicle model.

Chapter 6

Estimating Ground Parameters

6.1 Introduction

This chapter has two main sections. The first section describes computer simula-

tions of the excavator moving over different soils, assuming that the soil param-

eters are known. Nominal vehicle paths are predefined by specifying track drive

sprocket angular velocity trajectories in time, and the resulting “true” vehicle

motion simulated. In this forward simulation1 the slips of the tracks relative to

the soil are computed by assuming that the track motors have unlimited power.

The track therefore impose deformation on the soil, and the tractive effort may

be calculated on the basis of a given soil model and injected into the vehicle’s

equations of motion. The trajectories produced are then used as observations

to drive a filter to estimate the slip parameters without prior knowledge of soil

parameters. Given observations of the tractive effort, these data can be used later

to estimate some of the soil properties. The second section describes results from

different tests, using different sensors and filtering techniques to estimate the ve-

hicle slips, and with them some soil parameters. The estimated soil parameters

1the “truth model”

156

6.2 Simulation 157

can be used in higher-level strategic control of the excavator, for tasks such as

precise trajectory following, power control or soil working.

6.2 Simulation

6.2.1 Simulation: the Truth Model

The truth model is based on the kinematic model of the tracked vehicle presented

in Chapter 3. The vehicle is simulated to move on different soil types that have

parameters as given in Table 6.1. The soil types are categorised as sand, sandy

loam, clayey soil and dry clay, and have typical parameters as given by Wong

[Wong, 1989b]. In the truth model the vehicle control inputs are in the form of

track drive sprocket angular speeds ω. The vehicle moves forward on a plane

surface for ten seconds, then turns right three times and turns left once. This

trajectory has been chosen to simulate the vehicle’s motion on straight line, when

turning to both the left and the right, and at the same time to give a clear

comparison between different soils. The results are displayed graphically so that

different paths for different soils can be compared. At every time step the track

motion resistances, the tractive efforts and the track slips are computed using

Equations (3.1) and (3.13). The resulting vehicle trajectories and slips are shown

in Figures 6.1 and 6.2.

Terrain n kc kφ c φ K µr µlType kN/mn+1 kN/mn+2 kPa deg cmSand 1.1 0.95 1528.43 1.04 28 1 0.2 1.15

Sandy Loam 0.7 5.27 1515.04 1.72 29 2.5 0.2 1.3Clayey Soil 0.5 13.19 692.15 4.14 13 0.6 0.3 0.6Dry Clay 0.13 12.70 1555.95 0 34 0.6 0.1 0.8

Table 6.1: Parameters of Different Soil Types, from Wong [Wong, 1989b]

6.2.1 Simulation: the Truth Model 158

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5Position of the vehicle

x−coordinate

y−co

ordi

nate

on sand on sandy loam on clayey soilon dry clay

(a) Vehicle Trajectory

0 5 10 15 20 25 30 35−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1Slip angle α

time in sec

angl

e in

rad

on sand on sandy loam on clayey soilon dry clay

(b) Slip Angle α

Figure 6.1: Simulated Vehicle Trajectory and Slip Angle

6.2.1 Simulation: the Truth Model 159

0 5 10 15 20 25 30 350

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08Slip Coefficient of the left track

time in sec

slip

coe

ffici

ent

on sand on sandy loam on clayey soilon dry clay

(a) Slip Coefficients on Left Track

0 5 10 15 20 25 30 350

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08Slip Coefficient of the right track

time in sec

slip

coe

ffici

ent

on sand on sandy loam on clayey soilon dry clay

(b) Slip Coefficients on Right Track

Figure 6.2: Simulated Slip Coefficients

6.2.2 Simulation to Estimate the Slips 160

It may clearly be seen in Figures 6.1 and 6.2 that the vehicle trajectory, slip

coefficients and vehicle slip angle depend on the soil type. When moving on a

straight line, the slips are small and vary only slightly on different soil types.

Small slips are expected since the vehicle is not pulling a load and must therefore

overcome the soil resistance only. When the vehicle turns, the resistance increases

significantly and the longitudinal track slip must increase in order to generate

track thrusts that overcome the turning resistance. Slip variation across different

soil types is apparent, and the effect on the trajectories of the vehicle is noticeable.

The variation of the vehicle’s position can be as much as 0.6m after 35 seconds

of motion and the heading of the vehicle can vary as much as 0.3 rad over the

same time.

The simulation results using the forward “truth” model clearly show that the

relationship between the command inputs and the vehicle trajectory depends on

the soil parameters. The results confirm the importance of track slips in tracked

vehicle modelling, and the need to estimate them. It must be asked if, given the

control inputs and the resulting vehicle trajectory, is it possible to estimate the

track slips and the governing soil parameters? The next simulation is intended

to answer this question. In this backward simulation the trajectories produced

by the previous simulation will be used as observations with no prior knowledge

of the soil parameters. The simulation helps to clarify the techniques and sensor

inputs that are required for the task of estimating soil parameters.

6.2.2 Simulation to Estimate the Slips

In the second set of simulations the vehicle moves over the same soil types and

under the same input conditions as it did previously. The task is now to estimate

the track slips without prior knowledge of the type of soil upon which the vehicle

6.2.2 Simulation to Estimate the Slips 161

is moving. The estimator used is based on the kinematic model of the vehicle.

Two filters based on the EKF and DAF for the kinematic model have been

constructed. The six states are the vehicle centre of mass coordinates x(k), y(k),

heading angle φ(k), the track slips io(k), ii(k) and the slip angle α(k). They are

defined in Equation (6.1):

x(k)

y(k)

φ(k)

io(k)

ii(k)

α(k)

=

x(k−1)+0.5∆Tr[io(k−1)ωo(k)+ii(k−1)ωi(k)][cosφ(k−1)−sinφ(k−1) tanα(k−1)]y(k−1)+0.5∆Tr[io(k−1)ωo(k)+ii(k−1)ωi(k)][sinφ(k−1)+sinφ(k−1) tanα(k−1)]

φ(k−1)+∆TrB

[ii(k−1)ωi(k)+io(k−1)ωo(k)]

io(k−1)ii(k−1)α(k−1)

,

(6.1)

where the control input u(k) = [ωo ωi]T is the angular speeds of the left and

right track drive sprockets. Under the assumption that the control inputs and

the results of the previous simulation are the true values, they are corrupted

with added white noise to simulate the input data that would be received from

real sensors. The level of added noise is chosen to be similar to the achievable

accuracy of available sensors. The control input noise, is for example, taken to

be equal to the resolution of the encoders used in the vehicle. The yaw angle

error is derived from the accuracy of the available INS system. The position data

has been corrupted with different additive noise levels to simulate achievable or

expected resolution of available sensors such as DGPS, laser scanner, sonar, or

combined sensors using different data fusion techniques.

To handle the abrupt changes in track slip that occur when the vehicle changes

6.2.2 Simulation to Estimate the Slips 162

0 5 10 15 20 25 30 35−2

−1

0

1

2

3

4

5observed and estimated slips

time, secsl

ip i o a

nd i i in

%

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.05

0

0.05

time, sec

slip

ang

le, r

ad

observed estimated

(a) Slips on Sand

0 5 10 15 20 25 30 35−2

0

2

4

6

8

10observed and estimated slips

time, sec

slip

in %

observed leftestimated leftobserved rightestimated right

0 5 10 15 20 25 30 35−0.04

−0.02

0

0.02

0.04slip angle

time, sec

slip

ang

le, r

ad

observedestimated

(b) Slips on Sandy Loam

Figure 6.3: Simulation on Sand and Sandy Loam with σx,y = 1mm and σω =0.01rad, using EKF

6.2.2 Simulation to Estimate the Slips 163

0 5 10 15 20 25 30 35−4

−2

0

2

4

6observed and estimated slips

time, secsl

ip in

%

observed leftestimated leftobserved rightestimated right

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1slip angle

time, sec

slip

ang

le, r

ad

observedestimated

(a) Slips on Clayey Soil

0 5 10 15 20 25 30 35−4

−3

−2

−1

0

1

2

3observed and estimated slips

time, sec

slip

i o and

i i in %

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1

time, sec

slip

ang

le, r

ad

observed estimated

(b) Slips on Dry Clay

Figure 6.4: Simulation on Clayey Soil and Dry Clay with σx,y = 1mm and σω =0.01rad, using EKF

6.2.2 Simulation to Estimate the Slips 164

0 5 10 15 20 25 30 35−0.01

0

0.01

0.02

0.03

0.04observed and estimated slips

time, sec

slip

i o and

i i

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.05

0

0.05

time, sec

slip

ang

le, r

ad

observed estimated

(a) Slips on Sand

0 5 10 15 20 25 30 35−2

0

2

4

6

8observed and estimated slips

time, sec

slip

in %

observed leftestimated leftobserved rightestimated right

0 5 10 15 20 25 30 35−0.04

−0.02

0

0.02

0.04slip angle

time, sec

slip

ang

le, r

ad

observedestimated

(b) Slips on Sandy Loam

Figure 6.5: Simulation on Sand and Sandy Loam with σx,y = 1mm and σω =0.01rad, using DAF

6.2.2 Simulation to Estimate the Slips 165

0 5 10 15 20 25 30 35−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025observed and estimated slips

time, sec

slip

i o and

i iobserved left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1

time, sec

slip

ang

le, r

ad

observed estimated

(a) Slips on Clayey Soil

0 5 10 15 20 25 30 35−1

−0.5

0

0.5

1

1.5observed and estimated slips

time, sec

slip

in %

observed leftestimated leftobserved rightestimated right

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1slip angle

time, sec

slip

ang

le, r

ad

observedestimated

(b) Slips on Dry Clay

Figure 6.6: Simulation on Clayey Soil and Dry Clay with σx,y = 1mm and σω =0.01rad, using DAF

6.2.2 Simulation to Estimate the Slips 166

0 5 10 15 20 25 30 35−4

−2

0

2

4

6

8

10observed and estimated slips

time, secsl

ip i o a

nd i i in

%observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.05

0

0.05

time, sec

slip

ang

le, r

ad

observed estimated

(a) Slips on Sand

0 5 10 15 20 25 30 35−5

0

5

10observed and estimated slips

time, sec

slip

i o and

i i in %

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.04

−0.02

0

0.02

0.04

time, sec

slip

ang

le, r

ad

observed estimated

(b) Slips on Sandy Loam

Figure 6.7: Simulation on Sand and Sandy Loam with σx,y = 1mm and σω =0.1rad, using EKF

6.2.2 Simulation to Estimate the Slips 167

0 5 10 15 20 25 30 35−4

−2

0

2

4

6

8observed and estimated slips

time, secsl

ip i o a

nd i i in

%observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1

time, sec

slip

ang

le, r

ad

observed estimated

(a) Slips on Clayey Soil

0 5 10 15 20 25 30 35−6

−4

−2

0

2

4

6observed and estimated slips

time, sec

slip

i o and

i i in %

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1

time, sec

slip

ang

le, r

ad

observed estimated

(b) Slips on Dry Clay

Figure 6.8: Simulation on Clayey Soil and Dry Clay with σx,y = 1mm and σω =0.1rad, using EKF

6.2.2 Simulation to Estimate the Slips 168

0 5 10 15 20 25 30 35−0.02

−0.01

0

0.01

0.02

0.03

0.04observed and estimated slips

time, sec

slip

i o and

i iobserved left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.05

0

0.05

time, sec

slip

ang

le, r

ad

observed estimated

(a) Slips on Sand

0 5 10 15 20 25 30 35−0.02

0

0.02

0.04

0.06

0.08observed and estimated slips

time, sec

slip

i o and

i i

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.04

−0.02

0

0.02

0.04

time, sec

slip

ang

le, r

ad

observed estimated

(b) Slips on Sandy Loam

Figure 6.9: Simulation on Sand and Sandy Loam with σx,y = 1cm and σω =0.1rad, using EKF

6.2.2 Simulation to Estimate the Slips 169

0 5 10 15 20 25 30 35−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04observed and estimated slips

time, sec

slip

i o and

i i

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1

time, sec

slip

ang

le, r

ad

observed estimated

(a) Slips on Clayey Soil

0 5 10 15 20 25 30 35−0.03

−0.02

−0.01

0

0.01

0.02

0.03observed and estimated slips

time, sec

slip

i o and

i i

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1

time, sec

slip

ang

le, r

ad

observed estimated

(b) Slips on Dry Clay

Figure 6.10: Simulation on Clayey Soil and Dry Clay with σx,y = 1cm and σω =0.1rad, using EKF

6.2.2 Simulation to Estimate the Slips 170

0 5 10 15 20 25 30 35−0.02

−0.01

0

0.01

0.02

0.03

0.04observed and estimated slips

time, sec

slip

i o and

i iobserved left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.05

0

0.05

time, sec

slip

ang

le, r

ad

observed estimated

(a) Slips on Sand

0 5 10 15 20 25 30 35−0.02

0

0.02

0.04

0.06

0.08observed and estimated slips

time, sec

slip

i o and

i i

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.04

−0.02

0

0.02

0.04

time, sec

slip

ang

le, r

ad

observed estimated

(b) Slips on Sandy Loam

Figure 6.11: Simulation on Sand and Sandy Loam with σx,y = 1cm and σω =0.1rad, using DAF

6.2.2 Simulation to Estimate the Slips 171

0 5 10 15 20 25 30 35−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04observed and estimated slips

time, sec

slip

i o and

i iobserved left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1

time, sec

slip

ang

le, r

ad

observed estimated

(a) Slips on Clayey Soil

0 5 10 15 20 25 30 35−0.03

−0.02

−0.01

0

0.01

0.02

0.03observed and estimated slips

time, sec

slip

i o and

i i

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1

time, sec

slip

ang

le, r

ad

observed estimated

(b) Slips on Dry Clay

Figure 6.12: Simulation on Clayey Soil and Dry Clay with σx,y = 1cm and σω =0.1rad, using DAF

6.2.2 Simulation to Estimate the Slips 172

0 5 10 15 20 25 30 35−6

−4

−2

0

2

4

6

8observed and estimated slips

time, secsl

ip i o a

nd i i in

%observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.05

0

0.05

time, sec

slip

ang

le, r

ad

observed estimated

(a) Slips on Sand

0 5 10 15 20 25 30 35−5

0

5

10

15observed and estimated slips

time, sec

slip

i o and

i i in %

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.04

−0.02

0

0.02

0.04

time, sec

slip

ang

le, r

ad

observed estimated

(b) Slips on Sandy Loam

Figure 6.13: Simulation on Sand and Sandy Loam with σx,y = 10cm and σω =0.1rad, using EKF

6.2.2 Simulation to Estimate the Slips 173

0 5 10 15 20 25 30 35−4

−2

0

2

4

6

8observed and estimated slips

time, secsl

ip i o a

nd i i in

%

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1

time, sec

slip

ang

le, r

ad

observed estimated

(a) Slips on Clayey Soil

0 5 10 15 20 25 30 35−6

−4

−2

0

2

4

6

8observed and estimated slips

time, sec

slip

i o and

i i in %

observed left estimated left observed right estimated right

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1

time, sec

slip

ang

le, r

ad

observed estimated

(b) Slips on Dry Clay

Figure 6.14: Simulation on Clayey Soil and Dry Clay with σx,y = 10cm and σω= 0.1rad, using EKF

6.2.2 Simulation to Estimate the Slips 174

between straight and turning motion, a “jump hypotheses” with small σ when the

speed of both tracks is similar, and larger σ for larger differences in track speed,

has been applied in the slip estimation. This allows the filter to react adequately

to the situation where a small change of slip is expected on straight line and a

large change of slips is typical during turning, as discussed in section 3.2.

Driving the filters with observations provided by the truth model, and with

various observation errors σx,y, gives the results shown in Figures 6.3 to 6.14.

The effects of sensor noise on slip estimation may be seen by comparing the

results for the same type of soil. Figures 6.3 and 6.4 show the results of the

simulation for σ = 1mm. Although this accuracy is not presently possible with

available vehicle position sensors, the simulation discloses the limits of the filter

and model. This information will assist in finding ways to improve the filter.

Figures 6.3 and 6.4 show the results obtained with errors σx,y = 1mm, σφ =

0.001rad and σω = 0.01rad. For sandy soils the filter is able to follow closely the

track slips computed by the truth model. The results on “dry clay”, Figure 6.3b,

are not quite as accurate as those obtained on sandy loam. The reason for the

higher errors is that the slips are much smaller than those induced on sandy loam

and the measurement noise that must be added to represent the vehicle position

accuracy achievable in practice is comparatively large. Figures 6.7 and 6.8 show

the simulated results with higher errors chosen for σφ and σω. These errors are

compatible to the accuracy of the sensors fitted to the experimental vehicle. For

example, the reading error of the sprocket rotation ω was set to 0.1rad for a

maximal value of ω = 2.5rads−1. This is equivalent to an error of 4% of the

encoder reading. The reading error of the yaw angle φ is set to be similar to the

specified INS drift rate of 0.008rad in sec.

Another goal of the simulation is to obtain information about the performance

6.2.3 Interpretation of Results 175

of filters using different principles. As the results show, there are no significant

improvements of the DAF over the EKF. Beside, the EKF is already a proven

filter. The EKF is therefore used for the reminder of this thesis.

6.2.3 Interpretation of Results

Simulation of the vehicle’s motion using Matlab has involved a number of math-

ematical idealisations. It is assumed that sensors are available to give relatively

accurate information regarding the vehicle’s position and heading. The observa-

tion errors are assumed to be Gaussian, whereas in practice they will be subject

to bias, especially when an inertial navigation system is used as the position sen-

sor. The soil has been assumed to be ideal, homogeneous and to have a planar

surface, which is unlikely to be true in practice. Although soil may be locally

homogeneous, it is seldom flat. The vehicle itself has been idealised by assuming

that each track contacts the soil with uniform pressure over a planar rectangular

footprint.

Despite the simple model, some inferences may be drawn from the simulation

results:

❶ Track slip plays an important role in the estimation of the vehicle’s posi-

tion. Because of the specific method of skid steering the moment of turning

resistance is very high and one track has to produce the required tractive

effort to overcome this resistance, while the other one is commonly braked

when turning. The high tractive effort causes a high slip of the outer track.

❷ It is possible to estimate the slip of the vehicle on an unknown soil using

the vehicle model presented here, provided that position sensors with good

resolution are available.

6.3 Experimental Results 176

❸ The slip is a link between the soil and the tracks of the vehicle. Based on

known vehicle specifications, estimated slips and known soil classes, some

parameters of the soil type upon which the vehicle is moving can be esti-

mated. The estimated soil parameters can help to control the motion of the

vehicle as well as assisting in planning and controlling soil working tasks.

6.3 Experimental Results

6.3.1 Data Gathering

Data received for processing are generated from different sources: from the en-

coders measuring the positions of the axes, from the pressure transducers mea-

suring the differential pressures across the hydraulic actuators, from the spool

positions of the servo valves, from the IMU that measures bank, elevation, an-

gular rates and accelerations of the vehicle, from the joystick giving command

inputs and from the laser scanner giving range and bearing data. The different

sources require different media for transmission of data. Some can be read using

the controller CAN bus, and some via RS232 lines. At this stage the received

data have been saved to a file and processed later using a PC. In the future, data

will be read and processed in real-time using the onboard computer.

6.3.2 Data Processing

The received data were first retrieved from the onboard computer. Pre-processing

is necessary in order to transform raw data into engineering units. The data are

then saved in different files to allow comprehensive processing of them indepen-

dently if required. Different filters using different process models and filtering

techniques have been implemented and run with the received data. The filters

6.3.3 Results and their Interpretation 177

have been implemented and designed using Matlab, an integrated technical com-

puting environment. This allows easy implementation and modification of the

filters and assessment of the performance of the filters under different conditions.

Many runs of each filter for each experimental test run were required to tune the

filters.

Whilst in the early tests only data from encoders, pressure transducers and

the IMU were available, tests conducted later were supplemented by data from a

stationary laser scanner. This instrument scanned the test site and produced data

that could be used to reconstruct the vehicle trajectory. Although the accuracy

of the reconstructed vehicle trajectories is not very high2, the result does help to

improve the quality of parameter estimation. The next section will present some

results achieved using the comprehensive filter.

6.3.3 Results and their Interpretation

Some results obtained using the kinematic filter were discussed in Chapter 5.

Those results are achieved using the INS data and data from encoders and pres-

sure transducers. They show the feasibility of estimating the slips and other soil

parameters.

In order to obtain good estimates of the soil parameters, more accurate vehi-

cle position information is required. A Sick laser scanner was therefore used to

measure the vehicle’s position by imaging the vehicle. Because the excavator has

an irregular shape, a wooden box of dimensions 0.6m×0.7m×0.8m was fastened

on top of the vehicle. The laser scanner was able to image this box accurately,

and so reach its specified accuracy of ± 5cm. Tests with the box confirmed that

two sides can be detected at all times, and appear in the laser range data as two

2it depends on the resolution of the laser scanner, about ± 5cm, and on the method used toreconstruct the trajectory from beacons fixed to the vehicle

6.3.3 Results and their Interpretation 178

−2 0 2 4 6 8 10 12 14

−2

0

2

4

6

8

10

12

14

Vehicle Position

x coordinate in m

y co

ordi

nate

in m

estimated position position using laser scanner

(a) Position of the Vehicle

0 20 40 60 80 100 120 140 160 180−1000

−500

0

500

1000

1500

2000

2500

Time in sec

forc

e in

N

Forces and Resistances

force on the left track force on the right right resistancece on the left trackresistance on the right right

(b) Forces and Resistances on the Tracks

Figure 6.15: Position, Forces and Resistances on Grass with Laser Scanner Data.Conditions: σx,y = 1cm and σφ = 0.1rad, using EKF

6.3.3 Results and their Interpretation 179

0 20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80

90

100

Time in sec

Slip

fact

ors

in %

Track Slips

left right

(a) Estimated Slips

0 20 40 60 80 100 120 140 160 1800

0.05

0.1

0.15

0.2

predicted and estimated coefficients of resistance µ r

time in sec

coef

ficie

nts

of r

esis

tanc

e predictedestimated

0 20 40 60 80 100 120 140 160 1800

0.05

0.1

0.15

0.2

predicted and estimated coefficients of resistance µ l

time in sec

coef

ficie

nts

of r

esis

tanc

e predictedestimated

(b) Estimated Coefficients of Resistance

Figure 6.16: Estimated Slips and Coefficients of Resistance on Grass with LaserScanner Data. Conditions: σx,y = 1cm and σφ = 0.1rad, using EKF

6.3.3 Results and their Interpretation 180

0 20 40 60 80 100 120 140 160 180−100

−80

−60

−40

−20

0

20

40

60

80

100

Time in sec

coef

ficie

nt in

cm

Estimated Coefficient of Deformation Modulus

0 20 40 60 80 100 120 140 160 1800

500

1000

1500

2000

2500

3000

3500

4000

4500

5000Tractive Effort

Time in sec

forc

e in

N

Figure 6.17: Estimated Soil Shear Deformation ModulusK and Maximal TractiveEffort Fmax

6.3.3 Results and their Interpretation 181

lines. Matching the lines to the box gives the vehicle position.

The results of one test are shown in Figures 6.15 to 6.3.3. This test was

conducted on a grassed area near the University car park where previous tests

were conducted. The data were processed under the following conditions: σx,y

is equal to 1 cm, and σφ equals 0.1 rad, which is significantly worse than the

manufacturer’s specified resolution of the INS. Figure 6.15a shows the estimated

position of the vehicle using the comprehensive model, compared to positions

reconstructed from the laser scanner data. Figure 6.15b shows the measured track

forces and estimated track resistances. Figures 6.16a and b show the estimated

slips and coefficients of longitudinal and lateral resistance.

It can be seen that the estimated trajectory of the vehicle is similar to that

reconstructed from the laser scanner data. The measured forces, estimated resis-

tances and soil parameters look reasonable. It should be noted that the estimated

values of the soil parameters need not concur with the values of the parameters as

measured by “soil mechanics” testing. Here, the estimated values will differ from

the “soil mechanics” values by amounts that reflect for the mismatch between the

assumed models and reality.

It may be concluded from these results that the resolution of the available

vehicle position sensors will not allow more accurate estimation of the soil prop-

erties. This could be expected, as the soil deformation modulus is of the order of

a few centimetres. The vehicle position must therefore be measured to a similar

accuracy, since the soil parameters are estimated from the correlation between

track slip and track force.

The experimental trials also showed that the INS yaw resolution specified

by the manufacturer could not be achieved. This observation provides further

evidence that higher sensor accuracy will improve the performance of the filter

6.4 Conclusion 182

and that better results can be expected.

6.4 Conclusion

This chapter presents the results of promising field tests conducted with the

experimental tracked vehicle. Although the results shown here can not be used

directly to plan and control vehicle trajectories, they represent a important step

toward a fully-autonomous tracked vehicle.

It is believed that the soil parameter estimation techniques presented here

may provide suitable inputs to classification techniques able to identify the soil

type from a limited set of possible soil types. This classification may then be used

to direct trajectory planning and trajectory following strategies.

Chapter 7

Summary and Conclusion

7.1 Introduction

This Chapter summarises the work presented throughout this thesis and draws

conclusions. In Section 7.2 the contents of each chapter are summarised. The con-

tributions of the thesis are reviewed in Section 7.3. Finally, some future research

issues are identified in Section 7.4.

7.2 Summary of Each Chapter

In Chapter 2 an overview of the development of tracked vehicles and of clas-

sical soil mechanics was presented. Important soil properties that influence the

motion of a tracked vehicle were discussed. A mathematical description of their

relationship to the track forces and motion was also presented. The chapter then

discussed the physical soil-track interaction and the effects of soil properties on

the vehicle tracks and motion. This mathematical description is important for

the modelling the tracked vehicle, presented in Chapter 3.

183

7.2 Summary of Each Chapter 184

Chapter 3 began with the model the of soil-track interaction presented in

Chapter 2. Force models of a tracked vehicle were then derived on the basis of the

kinematics and dynamics of the vehicle and on the soil-track interaction. Each

contribution to the net force acting on the vehicle, and its relationship to soil

parameters was detailed. These considerations resulted in a set of mathematical

equations describing the vehicle motion. Based on these equations, “kinematic”

and “comprehensive” models of the vehicle motion were proposed. The kine-

matic model uses just the simple kinematic equations of the vehicle motion. The

comprehensive model uses both kinematic and dynamic equations of the vehicle

motion, combined with the relations describing soil-track interactions.

The issue of choosing an appropriate estimator structure was examined in

Chapter 4. Since the tracked vehicle model is inherently nonlinear, two different

filtering techniques for nonlinear processes are discussed: the extended Kalman

filter (EKF) and the distributed approximation filter (DAF). The chapter ex-

amined their working principles and their mathematical bases. The extended

Kalman filter was selected for the implementation. All of the estimation results

in this thesis are achieved with the EKF.

The proposed tracked vehicle model was experimentally evaluated using a

modified Komatsu PC05-7 mini excavator, described in Chapter 5. The chapter

detailed the modification of the excavator with electro-hydraulic servo valves,

encoders, digital axis controllers, pressure sensors and a supervisory control and

data acquisition system. The implementation allows the vehicle to be driven

under computer control and makes it a versatile experimental test-bed. The

chapter also described vehicle test sites and showed some test results.

Chapter 6 presented simulation studies of the tracked vehicle model and

experimental test results. The simulations showed the feasibility of the vehicle

7.3 Contributions 185

model and identified the performance of sensor suites required to achieve the

specified performance targets. The simulation also compared the results of two

nonlinear filtering techniques: the EKF and the DAF. The experimental results

confirm the feasibility of estimating soil parameters in motion in the basis of the

proposed tracked vehicle model.

7.3 Contributions

7.3.1 Modelling of Tracked Vehicles

This thesis makes a number of contributions toward the modelling of tracked

vehicles. A study of the effects of various soil parameters on the track forces has

been made, and these effects incorporated into a comprehensive model of tracked

vehicle behaviour.

In order to achieve a tracked vehicle model that is sufficiently concise as to

provide a feasible basis for real-time estimation, only a few governing soil param-

eters can be incorporated into the model. The chosen parameters represent the

dominant effects, and, most importantly, are closely related to the vehicle motion.

The result of these considerations is the comprehensive vehicle model, which

represents a link between the soil parameters and the kinematic and dynamic

characteristics of the tracked vehicle.

7.3.2 Soil Parameter Estimation

Applying appropriate sensor suites allow real-time estimation of both vehicle

position and soil parameters whilst the vehicle is moving. It is shown in simulation

and by experiment that useful information can be obtained on the track slips, soil

resistance forces, and the strength and shear modulus of the soil over which the

7.3.3 Experimental Vehicle System 186

vehicle is moving. A knowledge of these parameters, or even of their bounding

values, will permit much finer motion control.

7.3.3 Experimental Vehicle System

The vehicle itself is a major contribution of this thesis. As a versatile test-bed, the

vehicle has been fitted with actuators, controllers, various sensors and a control

and data logging system. The hardware and software design allows for fitting and

testing different types of sensors, and processing the received data to evaluate the

performance of vehicle models and sensor suites.

7.4 Future Work

The work reported in this thesis has identified and confirmed the feasibility of

improving the navigation of tracked vehicles through the use of improved process

models, and of identifying key soil parameters in real time by correlating vehicle

tractive effort and motion. A number of areas appear to invite productive future

work, and are summarised in the following section.

7.4.1 Vehicle Models

The models of tracked vehicle motion described in this thesis are simple. The

vehicle is assumed to move in a two-dimensional space. Certain assumptions

are made, such as that the vehicle tracks are rigid and exert a uniform pressure

on the soil, and that the soil resistance may be represented adequately by two

coefficients of resistance. There is considerable scope for extending and refining

these models. First, a more realistic model of the track shoe can be developed,

incorporating elementary track suspension. This extension would allow for non-

7.4.2 Development of an Autonomous Tracked Excavator 187

uniform lateral and longitudinal distribution of track pressure, such as will be

encountered when moving over uneven terrain, or through weight transfer due

to acceleration. Second, additional sensing should be added to the experimental

vehicle. It may be productive to add Doppler velocity sensing of the ground in

the vicinity of each track, so that track frame velocity may directly be measured.

It would also be useful to more fully sub-equation the relationship between track

motor differential pressure and motor torque.

7.4.2 Development of an Autonomous Tracked Excavator

The vehicle which is developed in this thesis is a prototype - although data is

logged in real-time, the vehicle path and soil parameter estimation is done off-line.

Further work is necessary to develop an operational autonomous tracked vehicle

that can navigate accurately using on-board sensors. The following developments

are possible to enhance the performance and quality of the vehicle:

❶ Improvement of the navigation system to allow position resolution of the

order of ±1cm through the use of an enhanced laser scanner or a millimetre

wave radar mounted on the vehicle. Improved position accuracy will also

improve the accuracy of soil parameters estimation.

❷ The addition of a DGPS with an accuracy of ±2cm will provide the vehicle

with another source of estimated positions. Sensor data fusion techniques

will then further improve the consistency and reliability of the whole system.

Given these enhancements it will be possible to build an accurate real-time

navigation system for tracked vehicles and to improve the estimation of soil prop-

erties which is necessary for precision control of the vehicle’s trajectory.

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Appendix A

Vehicle Parameters

This appendix provides a summary of the principal parameters of the experimen-

tal vehicle, a Komatsu PC05-7 mini excavator.

A.1 Vehicle and Engine

Model : Komatsu 3D72N

Type : 4− cycle, water − cooled, pre− combustion

No. of cylinders : 3

Flywheel horsepower : 12.8 HP (9.6 kW )

Max. drawbar pull : 1055 kg, 10.3 kN

Max. travel speed : 1.8 km/h

Operating weight : 1280 kg

(A.1)

198

A.1 Vehicle and Engine 199

Figure A.1: Dimensions of the Komatsu PC05-7

A.2 Hydraulic System 200

A.2 Hydraulic System

Pump type : 2 × gear pumps

Maximal flow : 2 × 11.9 l/min at 2, 000 RPM

Travel motors : 2 × axial piston motor

Theoretical consumption : 14.5 cc/rev

Consumption at max. speed : 10.988 l/min, 0.183 l/sec

Reduction ratio : 25.26 : 1

Relief valve setting, travel : 2, 700 psi, 18.6 MPa

(A.2)

A.3 Undercarriage

Rubber shoes : 230mm wide

Ground pressure : 0.22kg/cm2, 21.6 kPa

Number of shoes : 32 each side

Number of track roller : 3 each side

Suspension : none (rigid)

Appendix B

A Summary of Some Fluid

Mechanics

This Appendix presents a summary of some fluid mechanics that pertains to

hydraulics. It is based upon material from [Brater, 1996] and [Guillon, 1969].

Flow Resistance: FW = cAρ2v2, in N .

c: a resistance coefficient that depends on the shape of the body.

A: cross sectional area, in m2.

ρ: specific density, in kgm−3.

v: velocity of flow, in ms−1.

Flow Power: P = cAρ2v3, in W .

Motor Shaft Power: H(kW ) =ηoa

10.2Q(l)∆P(kg/cm2)

∆P : differential pressure, in Nm−2.

ηoa: overall efficiency (ratio of mechanical power output from motor to hy-

201

B. A Summary of Some Fluid Mechanics 202

draulic power input to motor).

Motor Rotational Speed: n(rev/sec) = ηvQ(l./sec)

V(l.)

ω = 2πηvQV

ηv: volumetric efficiency (ratio of theoretical flow nv to actual flow to the

motor).

V : swept volume (i.e. volume displaced by 1 revolution) of a pump or motor

ω: angular speed, in rad/sec

Motor Shaft Torque: T(kgf.m) =102πηoa

ηvV(l.)∆P(kgf/cm2)

Bernoulli’s Law: p+ ρv2

2= constant.

p: static pressure, in Nm−2.

Principle of Hydraulic Press: p = F1

A1= F2

A2.

F : piston force, in N .

A: cross sectional area, in m2.

Continuity Equation:

Q = v1A1 = v2A2 (for incompressible fluids).

ρQ = ρ1v1A1 = ρ2v2A2 (for compressible fluids).

Q: Flow rate, in m3/sec

Law of Conservation of Energy for Closed Conduit:

z1 +p1w+ α1

v212g= z2 +

p2w+ α2

v222g+

∑hl.

B. A Summary of Some Fluid Mechanics 203

w(= ρg): specific weight, in kg/m3

α: kinetic-energy correction factor

z: height, in m

g: acceleration due to gravity, in m/sec2

∑hl: energy losses (local losses, losses at entrance, losses due to enlargement,

contraction and bends).

Appendix C

Technical Parameters of Sensors

and Actuators

This Appendix summarises the parameters of the sensors and actuators that were

retro-fitted to the Komatsu excavator.

Servovalves

Each of the axes of the vehicle is controlled using a Moog series 633 direct drive

electrohydraulic servovalve with the following features:

❶ 3-way, 4-way or 2x2-way operation for servo hydraulic position, speed, pres-

sure and force closed loops.

❷ High force level permanent magnet linear motor

❸ No pilot stage oil flow

❹ Pressure independent dynamic performance

❺ Low hysteresis and high resolution

204

C. Technical Parameters of Sensors and Actuators 205

❻ Electric valve null adjust to compensate for load drift

❼ Integral spool position monitoring

Position Transducers

The position of each axis of the vehicle is monitored using a Novotechnik SXA58-

S/0012-SR-SA1-K02 inductive absolute angular position transducer. This sensor

features:

❶ Non Contact inductive position measurement

❷ Absolute and incremental information

❸ Accuracy 12 bit (0.05o)

❹ Compact construction (φ58× height 65...71mm)

❺ Insensitive to external influences

❻ Synchronous serial interface

❼ Working temperature range from −40o to +125o C.

Pressure Transducers

Each of axis is provided with two UCC PDT.250121 pressure transducers fitted to

the A and B ports of the servovalves. These transducers allow for measurement of

the differential pressure across each actuator, and hence for estimation of actuator

force and power. The transducers have the following characteristics:

❶ One-piece body and diaphragm machining ensures long term product sta-

bility.

C. Technical Parameters of Sensors and Actuators 206

❷ All stainless steel construction.

❸ 20mV and 5V output option.

❹ Accept 10 - 30VDC unregulated supply.

❺ Temperature range from −40o to +85o C

❻ Compactness (φ22× height 57mm)

PLS Laser Scanner

Position data was acquired using a PLS laser scanner from Sick Optick Electronic,

with the following characteristics:

❶ Measuring Area

Range : Max. 50 m

Resolution (basic) :

Measurement of distance : ± 50 mm

Angular resolution : 0.5o

Scan time : 40 ms

C. Technical Parameters of Sensors and Actuators 207

❷ General Data

Scanning angle, max. : 180o

Angular resolution : 0.5o

Supply voltage : DC 24V ± 15

Power input : ≤ 17 W

Ambient operating temperature : 0 to + 50oC

Storage temperature : −25 to + 70oC

Dimensions (W ×H ×D) : 155× 185× 156 mm3

Measuring error : typ. ± 50 mm

Interface : RS232 or RS422

Transmission rate : 9600, 19200 or 38400 baud

Sender : Infrared laser diode

Casing material : Die− cast aluminium

Weight : 4.5 kg

Inertial Measurement Unit

Data to drive the vehicle position estimator is also provided by a Watson inertial

measurement unit. This strapdown unit provides the following outputs:

C. Technical Parameters of Sensors and Actuators 208

Inertial Output Full Scale Decimal Full Scale Binary

Bank ±180.0o ±180.0oElevation ±90.0o ±180.0oMagnetic Heading 0− 360.0o ±180.0oX Acceleration ±2.00 g′s ±2.00 g′sY Acceleration ±2.00 g′s ±2.00 g′sZ Acceleration ±2.00 g′s ±2.00 g′sForward Acceleration ±2.00 g′s ±2.00 g′sLateral Acceleration ±2.00 g′s ±2.00 g′sVertical Acceleration ±2.00 g′s ±2.00 g′sX Rate ±99.9o/second ±200o/secondY Rate ±99.9o/second ±200o/secondZ Rate ±99.9o/second ±200o/secondX Magnetometer ±999.9 mGauss ±1000 mGaussY Magnetometer ±999.9 mGauss ±1000 mGaussZ Magnetometer ±999.9 mGauss ±1000 mGaussBank Pendulum ±90.0o ±180.0oElevation Pendulum ±90.0o ±180.0oUser Channel 1 ±9.99 V DC ±10 V DCUser Channel 2 ±9.99 V DC ±10 V DCUser Channel 3 ±9.99 V DC ±10 V DCUser Channel 4 ±9.99 V DC ±10 V DCVelocity ±400.0 Km/Hour ±400.0 Km/HourInternal Temperature −40 to + 88oC −40 to + 88oC(7 bit)Status Bits(Watson Use) Diagnostics DiagnosticsFlag Bits Diagnostics Diagnostics

Table C.1: Full Scale Range of RS232 Output Format