Model Uncertainty and Robustness for Interactive Robots ... · These more complex dynamics introduce additional sources of model uncertainty, and are typically accompanied by hierarchical

Model Uncertainty and Robustness for Interactive Robots with JointFlexibility

by

Kevin Robert Haninger

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Engineering - Mechanical Engineering

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor Masayoshi Tomizuka, ChairProfessor Karl Hedrick

Associate Professor Pieter Abbeel

Fall 2016

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Abstract

Model Uncertainty and Robustness for Interactive Robots with Joint Flexibility

by

Kevin Robert Haninger

Doctor of Philosophy in Engineering - Mechanical Engineering

University of California, Berkeley

Professor Masayoshi Tomizuka, Chair

Transforming robots from laborers to collaborators promises to significantly broaden theirsocietal impact, but is presently limited by, among other factors, technical feasibility. In-teractive robots seek to achieve safe and productive behavior in non-deterministic settingsby realizing reactive behavior, which can enable direct human-robot interaction, promisingnew modalities of power and information flow between human and robots. This can enableassistive or rehabilitative robots which restore or augment human’s physical capabilities, aswell as expand robotic roles in manufacturing contexts.

However, several novel considerations must be made in analysis of robots which seekto achieve interactive behavior. For physical interaction, guarantees of safety become bothmore important and harder to demonstrate. Often, complex hardware is introduced tomeet interactive design criteria (e.g. joint torque sensors which introduce joint flexibility).These more complex dynamics introduce additional sources of model uncertainty, and aretypically accompanied by hierarchical controllers which further obfuscate both safety andperformance. Additionally, interactive robots will couple with unmodeled environments,changing the effective dynamics of the robot and presenting further analytical challenges.

This dissertation examines the safety and performance of uncertain, interactive systemsfrom several perspectives. Limitations to achievable model accuracy and the effects of thismodel uncertainty on performance and safety are examined analytically and experimentallyon series-elastic actuated systems.

First, the objectives of interactive robots and constraints introduced by their hardwareare introduced. A model for interactive flexible joint robots is motivated which explicitlyconsiders backdriveability of the motor and load-side dynamics. Conditions for passivity offlexible-joint robots which render a load-side impedance are developed, then extended tohold over an uncertain motor model. This robust passivity condition is shown to inducesensible constraints on inner-loop torque controllers. Rigorous means of relaxing this passiv-ity condition are introduced, and the relaxed condition shown to explain interactions knownin literature to be unsafe in practice. A model and corresponding uncertainty bound of anexperimental setup is characterized through bilateral system identification (i.e. both motor

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and environment driven) and the results used to validate the robust passivity condition asa practical design tool. This analytical methodology assists hierarchical controller synthesisby generating practical constraints on controller parameters.

Performance of a linear interactive system is then defined, and shown to be limited bymodel uncertainty. However, by incorporating direct measurement of interactive variables(force and motion), this performance can be improved and the robust rendering of desired dy-namics can be achieved. A novel controller structure, derived from the disturbance observer,is proposed and analyzed. Again, conditions for passivity are developed, then extended tohold over an uncertain model. For a fixed-structure controller, these conditions are thenpropagated back onto parameter constraints to inform controller design. The performanceof this approach is then validated experimentally.

When direct measurement of the interactive force is not feasible, performance improve-ments can be made by improving model accuracy. Here, a data-driven modeling techniqueis used to describe non-idealized dynamics. However, interactive systems will couple withunknown environments, making their effective dynamics multimodal and potentially intro-ducing unknown input which confounds identification. A modeling approach suited to thesechallenges is introduced which allows for the identification of inverse dynamics which aremultimodal and subject to intermittent unobserved external disturbances. The passivityof the overall resulting controller policy is shown, and the performance and passivity arevalidated experimentally.

Uncertainty in the environment motivates the need for interactive control, but realizationof this control is in turn limited by uncertainty in the robot model. Explicitly exploring therelationship between model uncertainty, safety, and performance can allow relevant limita-tions to be improved when possible, and respected when not.

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To all the cats, dolphins, and hummingbirds.

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Contents

Contents ii

List of Figures iv

1 Introduction 11.1 Development of Interactive Control . . . . . . . . . . . . . . . . . . . . . . . 21.2 Challenges to Interactive Control . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Model Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Summary of Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Modeling for Interactive Control 102.1 Admittance and Impedance Control . . . . . . . . . . . . . . . . . . . . . . . 102.2 Flexible Joint Robot Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Passivity and Safety 283.1 Interactive Systems Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Passivity in Interactive Control . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Robust Passivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Passivity Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Load Side Uncertainty 484.1 Position-Based Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Robust Impedance Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3 Series-Elastic Actuated System . . . . . . . . . . . . . . . . . . . . . . . . . 584.4 General Dynamic Error Feedback . . . . . . . . . . . . . . . . . . . . . . . . 624.5 Design Within Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.6 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Load-Side Dynamics Learning 695.1 Impedance Controlled Interactive Robots . . . . . . . . . . . . . . . . . . . . 715.2 Learning of Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3 Disturbance Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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5.4 Passivity in Feedforward Dynamics Compensation . . . . . . . . . . . . . . . 785.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6 Conclusion 866.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.2 The Future of Interactive Robots . . . . . . . . . . . . . . . . . . . . . . . . 88

Bibliography 90

iv

List of Figures

1.1 Impedance Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Interactive System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Control Diagram for Impedance Control . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Desired and Real Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Impedance and Admittance Representations . . . . . . . . . . . . . . . . . . . . 112.3 Integrated SEA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 One DOF Test Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Dynamics for Serial Manipulator with Joint Flexibility . . . . . . . . . . . . . . 162.6 Five DOF Upper-Limb Exoskeleton . . . . . . . . . . . . . . . . . . . . . . . . 172.7 Integrated SEA to high-DOF Exoskeleton . . . . . . . . . . . . . . . . . . . . . 182.8 Bowden-Cable Transmission Schematic . . . . . . . . . . . . . . . . . . . . . . 182.9 Ground Mounted Motor Supports for Exoskeleton . . . . . . . . . . . . . . . . 192.10 Bowden-Cable Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.11 Impedance Model for Flexible Joint . . . . . . . . . . . . . . . . . . . . . . . . 202.12 Effect of Backdriveability on τ d → τ . . . . . . . . . . . . . . . . . . . . . . . . 202.13 Effect of Backdriveability on τ d → τ with High Gain Controller . . . . . . . . . 212.14 Motor Model Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.15 Desired and Real Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.16 Bowden-Cable Friction Characteristics, Experimental Results . . . . . . . . . . 242.17 Time Series of Static Friction Break Tests . . . . . . . . . . . . . . . . . . . . . 252.18 Torque Comparison and Fit Model . . . . . . . . . . . . . . . . . . . . . . . . . 252.19 Stiction τfr with fit β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.20 Effect of Load Dynamics on τ d → τ . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Generalized Environmental Interaction . . . . . . . . . . . . . . . . . . . . . . . 293.3 Flexible-Joint Robot with Inner-loop Torque Control and Impedance Control . . 323.4 Collocated Passivity Model for Flexible Joint . . . . . . . . . . . . . . . . . . . 333.5 Full Passivity Model for Flexible Joint . . . . . . . . . . . . . . . . . . . . . . . 333.6 Uncertainty Bounds as Impedance Increases . . . . . . . . . . . . . . . . . . . . 373.7 Uncertainty Bounds as Controller Gain Increases . . . . . . . . . . . . . . . . . 383.8 Minimum Uncertainty as Bandwidth Ratio Changes . . . . . . . . . . . . . . . 38

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3.9 Rendered Impedance θ to τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.10 Experimentally Observed and Model Response for R1 . . . . . . . . . . . . . . 413.11 Experimentally Observed and Model Response for R2 . . . . . . . . . . . . . . 423.12 Experimentally Observed and Model Response for R3 . . . . . . . . . . . . . . 423.13 Experimental Model Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 433.14 Controller Parameters Satisfying Robust Passivity and Stability . . . . . . . . . 443.15 Uncertainty Bounds for Controllers . . . . . . . . . . . . . . . . . . . . . . . . . 453.16 Phase Response of Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.17 State Space Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.18 Interaction Port Power Flow and Total Energy Transfer . . . . . . . . . . . . . 47

4.1 Impedance Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2 Inverse dynamic compensation for impedance control . . . . . . . . . . . . . . . 524.3 Disturbance observer based impedance control . . . . . . . . . . . . . . . . . . . 544.4 Inner loop compensation DOB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.5 Allowable Uncertainty and α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.6 Allowable Uncertainty and ωc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.7 Allowable Uncertainty and Gimp . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.8 Allowable Uncertainty and Bimp . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.9 Controller Parameters which Satisfy Safety Conditions Bimp . . . . . . . . . . . 594.10 Series-elastic actuator integrated to driven inertia . . . . . . . . . . . . . . . . . 604.11 Inner-loop torque-mode control . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.12 Integrated inner loop control with DOB . . . . . . . . . . . . . . . . . . . . . . 614.13 Model and Uncertainty for G(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.14 Experimental Response from τ d and τ to θ . . . . . . . . . . . . . . . . . . . . . 644.15 Experimentally Found Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 654.16 Stiffness rendering comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.17 Variety of stiffnesses rendered with DOB approach . . . . . . . . . . . . . . . . 664.18 DOB vs Traditional in impedance rendering . . . . . . . . . . . . . . . . . . . . 674.19 Two different impedances rendered on SEA . . . . . . . . . . . . . . . . . . . . 68

5.1 Interactive System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3 Rendering of Zero Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4 Regressed Model, Nominal State x =[θ, θ, θ

]. . . . . . . . . . . . . . . . . . . 82

5.5 Regressed Model, Augmented State x =[θ, θ, θ, sgn(θ)

]. . . . . . . . . . . . . 82

5.6 Rendering of Zero Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.7 Rendering of Pure Stiffness Kimp = 3.5 . . . . . . . . . . . . . . . . . . . . . . . 845.8 Power and Energy into Actuator During Environmental Impact . . . . . . . . . 85

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Acknowledgments

For the support, guidance, advice and insight of those around me, I am deeply thankful.To my advisor, Professor Tomizuka, whose advice I have found not only unfailingly true,but with layers of insight and meaning, I am deeply grateful. For the patience and wisdomwith which you have encouraged my intellectual development, I am thankful. To Junkai,in our collaboration, discussion and comments I have found a helpful colleague and friend,and for this I am thankful. To Suraj, I am thankful for the diligent, and often amusing,collaboration we’ve had. To Wenjie, for the insight, honest advice and guidance whichhelped my early development. To Sebastian, Aaron, Robert, Daisuke, Cong, my labmatesand student cohort, I appreciate the opportunity to engage honestly and discuss ideas. Ourdiscussions have helped develop technical ideas, and your passions and personalities havekept me inspired. To Professors Hedrick and Abbeel, I am grateful for the insight you havegiven on my work, and the opportunity to learn from your perspectives and methodologies.

To my family and Cassie, I am thankful for the support and encouragement (and pa-tience) in fostering my curiosity, creativity and compassion. To Francisco and Drew, for theopportunity to develop a vague notion into a rewarding execution. To Arqui, Ann, Fayad,and Max for keeping me balanced.

To John and Janet McMurty, I gratefully acknowledge the financial support which hasgiven me flexibility to pursue my interests. To the National Science Foundation (and thetaxpayers which fund it) I appreciate the opportunity to engage in an exciting, fulfillingproject.

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Chapter 1

Introduction

Robots have long been envisioned as autonomous entities, providing human assistancewhile seamlessly embedded in society. Media representations reinforce this public perceptionwith books and films where the robot’s intelligence is presumed, often as a plot device.Though perhaps compelling literature, this belies the nature of real-world robots. Robots,especially early ones, are closer to a literal translation of ‘robotnik’ from Czech - ‘laborer’.In the deterministic settings of factories and assembly lines, industrial robots were the firstto be widely deployed, capable of performing high-precision, repetitive tasks. In doing so,they have been largely isolated from humans: out of the public eye, and physically separatedfrom their human coworkers.

As robot perception, actuation and control continue to develop, these barriers betweenrobot and human can be relaxed. Robots can move towards direct interaction with humansand the realization of safe, productive behavior in less-structured settings. However, severalnovel technical considerations must be made to realize this transition.

Deterministic environments admit accurate models or simplifying assumptions whichallow a priori predictions of how the robot and environment will behave. This allows thebehavior of the total system (robot and environment) to be predicted, and robot behavior tobe designed to achieve desired output. This approach allows the designer to leverage a prioriknowledge on the environment to simplify many stages of decision-making and control, andwas a natural domain for early robots.

Though successful in deterministic environments, not all of these established analysis andsynthesis techniques can be directly transferred to nondeterministic environments, wheredictation of total system behavior is not only difficult but often dangerous. When the robotinteracts with systems which are unknown or unmodeled, coupled system behavior cannotbe predicted, thus coupled system behavior cannot be completely designed a priori as isdone in determinstic settings. What cannot be controlled, must be accommodated - if thesystem-level objectives can be encoded in reactive robot behavior, the environment need notbe modeled, just an appropriate characterization measured.

Extracting a quantifiable, relevant state of the environment to inform the robot’s reac-tion has been one of the major challenges for interactive robots. Devices such as cameras,

CHAPTER 1. INTRODUCTION 2

force sensors, tactile sensors and microphones (many bearing resemblance to biological sen-sory systems) enable new modalities of interaction. Ever-growing perception and inferencetechniques process these raw sensations into compact and relevant representations, or evendirectly determine robot response. These sensors have the potential to enable robots to inferand react as humans do. This goal of human-like interaction, although largely implicit inmany areas of robotics research, is not unreasonable. Not only is there precedent for thesemodes of interaction being successful on intelligent agents (humans), but human-like robotscan use established infrastructure, tooling, transportation and production, allowing a morenatural transfer of responsibilities and objectives.

Perception, inference and decision making are important and necessary components ofall types of autonomous agents. For robots, agents which are embodied to enable interac-tion with the material world, physical interaction is also important. To varying degrees,these interactions can be realized with well-established techniques - position control, forcecontrol or a combination thereof. However, to approach the physical dexterity and skill ofhumans, challenges remain. In particular, interacting with uncertain manipulanda, realizinghighly dynamic manipulation, and presenting a large dynamic range in proprioception andpower are still ongoing areas of research. Physical interaction underpins other, higher-leveldecision-making processes, thus making its accurate and safe realization crucial for advancinginteractive robots.

Physical interaction is sometimes defined as ‘power exchange between systems’. In thisdefinition industrial robots manipulating payloads are admitted by their changing the kineticand potential energy of their payload. However, the properties and behavior of the payloaddo not impact the robot’s behavior. In fact, a standard objective for these robots is to makethe robot behavior invariant to, for example, payload mass. This is a physical manifestationof prescriptive systems, where the system behavior is dictated a priori and external effectsminimized. Interactive robots are to be responsive, where the properties or state of theenvironment inform the behavior of the robot. The definition can be extended: ‘power andinformation exchange between systems’.

1.1 Development of Interactive Control

The historical development of interactive control is rooted in classical robot control.Position control and subsequently force control were the first operational modes of robots,arising from traditional regulation of a directly sensed quantity. The control objective (aposition or force reference) is pursued while taking other perturbations as disturbances tobe rejected. For position controlled robots, this manifest as high-stiffness drivetrains, underhigh-gain control. These systems maintain positional accuracy under high external forces,which is exactly what makes them dangerous for interactive tasks. If the external force arisesfrom unexpected contact with the environment, damage to the robot or environment canoccur.

Early works on interactive robots arose from a manufacturing context, such as robotic


drilling and deburring, which cannot be easily achieved by position or force control alone [124,69]. Stiffness control, damping control, and other interactive control modes were generalizedto impedance control in [49], presenting a framework where the controller objective is topresent desired dynamics at an interaction port.

Figure 1.1: Impedance Control Objective

A schematic for this objective can be seen in Figure 1.1, where Mimp, Bimp and Kimp

are the mass, damping and stiffness of the desired dynamics, chosen to meet higher-levelsystem goals. These parameters define a desired relationship between interactive force fintand position x. Impedance control has become a typical name for many types of interactivecontrollers, although in rigorous definition an impedance takes an across variable as input(here, generalized motion x) and produces a through variable response (here, generalizedforce fint). A physical admittance is complementary, taking a force input and producing amotion response.

These port variables are chosen such that their product is the energy flow into the system.By assuming a coupling is power continuous, the power out of one system is the power flowinto its complement. The terminology, as well as some of the underlying methodology forsafety, is inherited from network and large-scale systems [74, 122]. For physical systems,an impedance can be intuitively viewed as a generalization of stiffness - a ‘high impedance’gives a large magnitude force response for an equivalent displacement. Similarly, admittancecan be intuitively viewed as a generalized flexibility, where ’high admittance’ systems offera large position response for a small force input.

For a number of practical and theoretical reasons this coalesced an ideological shift to-wards interactive systems, and became the standard framework. Theoretically, impedancecontrol generalizes the objectives of position and force control. By taking the impedanceparameters Kimp, Bimp, and Mimp towards infinity (respectively zero), and letting x (fint)


be the deviation from the reference, motion (force) control objectives are recovered. Theimpedance dynamics in Figure 1.1 are simple enough to be algebraically convenient and in-tuitive, but also match the order of the differential equations which govern inertial systems,making design of realizable controllers more feasible. Additionally, this framework allowedthe adoption of the theoretically powerful and intuitive concept of passivity [121]. Briefly;if the dynamics shown in Figure 1.1 are achieved, the system can be stably coupled to anypassive environment under some very mild constraints on impedance parameters.

Subsequent to the formalization of impedance control, two major application domainsdeveloped, with slightly different objectives and constraints. Interactive robots began tobe used directly with humans, with objectives including rehabilitation, augmentation andcommunication. Here, hardware which is typically custom designed for the task coupleseither through the end-effector (manipulandum) or joint-space (exoskeleton) to allow therobot to reshape the human’s dynamics or achieve other interactive goals. Haptic interfaces[70], exoskeletons (upper-limb [90] and lower-limb [59]) and even planar manipulator designs[62] all explored human-robot interaction using the same framework of impedance control.

Another variant, often with manufacturing applications, is the modification of more tra-ditional serial-manipulators. With the addition of force sensing, these robots can presenta variety of dynamics for complex environmental manipulation. Such robots are capableof polishing, peg-in-hole insertion, grinding and other (traditionally human) tasks [4, 33,63]. Interactive control with appropriately chosen impedance parameters allows these robotsto adapt to small deviations in geometry and contact properties while still maintainingtask-relevant behavior. Serial manipulator designs are standard for commercial interactiverobots, with several now offering impedance control as-sold [33, 63, 100, 102]. Of course,these divisions are not rigid and many applications look at human-robot collaboration formanufacturing tasks [91] or exoskeletons for human laborers [55].

Across all these applications, the selection of desired dynamics is often done by an en-gineer with domain knowledge of the task and the robot, and no generalized methodologyfor choosing impedance parameters is accepted in literature. However, these parameters areconstrained by the robot - many parameter values cannot be safely realized, and others maynot be realized accurately. Furthermore, the environmental dynamics impact both safetyand performance, making these parameter constraints environment-dependent. To properlycompartmentalize, such that end-users can abstract the robot to a mass-spring-damper sys-tem, explicit bounds on the realizable impedance, as well as characterization of performanceis important.

1.2 Challenges to Interactive Control

Several key challenges in hardware, modeling and control have motivated much of thework in physical interaction. Two major challenges, indirect sensing and safety, underlymany studies in literature and are also key aspects of this dissertation.


Indirect Sensing

Indirect sensing refers to the inference of a variable which isn’t directly sensed through useof directly measured variables and a model. Interactive force is often not directly measuredat the interaction port due to constraints in form-factor, force characteristics (force sensorscan rarely tolerate shock loads) or cost. Position sensing is rarely problematic in this way, asit can often be sensed through an approximately rigid kinematic chain to encoders. Force,as a through variable, cannot be as easily remotely sensed, as any transmission which carriesthe force to the sensor introduces dynamics, thereby modifying the force characteristics.Often, these transmission elements have design constraints imposed by the application (e.g.sufficient strength to avoid yield) and thus may unavoidably present significant dynamics.

If these transmission elements can be modeled as function of measured variables, this canbe used to infer the force of interest. However, in some cases the model is indeterminate inmeasured state (e.g. static friction at zero velocity), and in general this approach is limitedby model accuracy. This has manifest in two major challenges for interactive control.

Bilateral Actuation

Original implementations of impedance control directly controlled the motor torque ofDC motors through current control [49, 70]. In some cases this is still used today [105, 1],but is limited to applications where the robot can be actuated with a direct or low gear ratiotransmission. When a gear reduction is introduced, nonlinear friction is injected betweenthe output of the gearbox and the motor, causing a mismatch between the motor torque andgearbox output torque. Critically, a torque on the gearbox output below the breakaway valueof the static friction causes no change in measurable state when motor position and currentare measured. In interactive control, the system should be capable of being driven from boththe motor and external input, referred to in the design community as backdriveability.

Backdriveability is a critical consideration in the design of interactive devices - leadingsome researchers to pursue novel gear reductions, e.g. cable and capstan [34], to reduce fric-tion. However, standard in industry and research is still traditional gearboxes, either strain-wave based (e.g. harmonic drive) or planetary. Such gearboxes present a high impedanceto the output, but backdriveability can be improved through the intentional introductionof compliant elements after the gearbox [96]. In these series-elastic actuators (SEAs), theelastic element allows the interactive side of the robot to be moved without overcoming thestatic friction of the gearbox, reducing apparent impedance. Measuring the relative displace-ment on the spring allows inference to the torque it carries, thus a closed-loop controller canbe introduced to regulate this torque, further improving responsiveness.

A related approach is the introduction of a torque sensor after the gearbox output [4]. Atorque sensor, which measures the mechanical strain on an element of known stiffness, bearssimilarities to the series elastic actuator in measuring the displacement and using knownproperties to infer force, albeit with a much higher stiffness.

As these systems present elasticity in the joint, they will be treated in a unified framework


here, referred to as flexible-joint robots. Flexibility is inherent to force/torque measurement- from an instrumentation perspective, displacement (and strain) are measurable, whereasforce is not. Even recent developments in force sensing technology measure displacementwith alternative means, such as light interferometry [84] or piezoelectric voltage.

Load Dynamics

The issues of indirect sensing remain even when torque measurements are taken after thegear reduction. Rigid-body robot dynamics, with limited model accuracy, are still betweenthe torque measurement and interaction port. These will be referred to as the load dynamicshere. Although theoretically easy to treat with an inverse dynamics model, typical measure-ments are of only position, making velocity and acceleration terms noisy. This, coupled withmodel inaccuracy, can limit the performance of this approach.

Current interactive applications are typically quasi-static, where low velocity and accel-eration reduces the effects of these errors. However, even for these quasi-static tasks, errorsin the position-dependent dynamics (e.g. gravitational terms) can be significant.

One more unusual configuration bears mentioning here - the integration of a force/torquesensor at the base of the robot, allowing traditional industrial manipulators to become in-teractive without modifying the robot [85]. Measurements of the ground reaction forces,coupled with a dynamic model of the robot, allows inference of the interactive force at theend effector. However, this base force/torque sensor must safely carry the significant forcesof the robot, and this high load capacity typically comes at the cost of sensitivity. Theuncertainty in the robot model further limits the accuracy of the end-effector force inference,although this may find useful application when high sensitivity is not necessary.

Safety in Complex Systems

For interactive systems, safety is critical, with humans or unknown environments poten-tially at risk. At the same time, guarantees of safety are more difficult, as a priori predictionson coupled system behavior are not available. Stability of the robot in isolation (e.g. bytraditional Lyapunov stability) is no longer sufficient, as when a robot is coupled with theenvironment, the effective dynamics of the robot change, potentially compromising stability.One manifestation of this is a classical problem for industrial robots - interaction with ahigh-stiffness environment (e.g. unintended collision) causing instability. The high stiffnessrobot, under high gain control, has smaller effective robustness margins, making them moresensitive to the change in dynamics arising from environmental contact [54].

The trade-off between performance and robustness is well-established in control theory,implicit in gain and phase margin in textbooks and notably discussed in [114]. Some treat-ments of robustness and performance specific to interactive control been explored [118, 53,107], although there are no canonical results or best practices.

Specific demonstrations of coupled system stability can be made if the environment is as-sumed to belong to a class of systems (e.g. inertial) through well-established robust stability


techniques [60, 14]. To show the stability when interacting with a general environment, oneapproach is to show the passivity of the controlled system [48]. By showing the passivityof the controlled system, it can be safely coupled with any passive environment (passivityis considered a mild condition for most environments). Human limbs are often regarded aspassive, as well as any kinematic constraints or strictly mechanical environments. However,rigorously showing passivity of the robot can be difficult given the complexity of both thedynamics and hierarchical controller.

One further challenge for this approach is that passivity loses sufficiency for coupledstability in several applications, notably in high-stiffness environments, especially when ren-dering a high impedance. This has been well-documented in literature [115, 30], and reflectsthe inherent challenge of two high impedance systems interacting. To explore the limita-tions of passivity, some authors make considerations of unmodeled practicalities, such assampled-data effects [23] or quantization [25]. However, these works analyze haptic systems,where simpler, direct-drive dynamics allow closed-form derivations of passivity conditions.Although the intuition gained from these works transfers to flexible-joint robots (namely,the importance of physical damping), direct application is not possible, and additional tech-niques may need to be developed.

1.3 Model Uncertainty

Interactive control is, in a sense, deeply rooted in model uncertainty. If the environmentor human was well-modeled, it could be brought into the system model, and analysis ofcoupled behavior performed. Interactive control is pursued when the feasibility of modelingthese environments is limited. Although on a philosophical level the environment may bedeterministic, from a practical engineering perspective, models will always be limited by(among other reasons) instrumentation, observability and complexity.

Model uncertainty motivates the need for interactive control, but also manifests withinthe controlled system. When there is model uncertainty in the model for actuator or robotdynamics, it implicitly and explicitly limits the performance of the controlled system. Takethe schematic shown in Figure 1.2, and let the port fint, x be the desired interaction port.The effects of the model uncertainty in the actuator and load dynamics will be introducedbelow.

Figure 1.2: Interactive System Model


Actuator Uncertainty

Uncertainty in the actuator dynamics sets performance constraints by directly limit-ing the accuracy of the torque control, as well as constraining the controllers (torque andimpedance) which can be safely realized. Difficult-to-model phenomena, such as nonlinear ortransient friction and drivetrain backdriveability directly affect the performance of the innertorque-control loop, seen in Figure 1.3. As safety must be rigorously guaranteed, in prac-tice the inner-loop controllers are conservatively designed in an ad-hoc or iterative manner,further limiting the torque-tracking performance that can be achieved.

Figure 1.3: Control Diagram for Impedance Control

With the typical hierarchical structure of interactive flexible-joint robots, stability andpassivity analysis is complex. Often the design heuristic of letting an inner loop have 5 to10 times the bandwidth of the outer loop is used to simplify analysis for other hierarchicalcontrollers. However, the highest inner-loop torque bandwidth documented in literature is60Hz [86], so this approach would put severe limitations on the achievable impedance. Topush the outer-loop bandwidth higher requires consideration of both loops concurrently.

In addition to the hierarchical controller structure, some systems which are in theorypassive do not exhibit stability in demanding applications (e.g. very stiff impedances or ina high-stiffness environment). With the complex dynamics of flexible-joint robots, modeluncertainty may play a role. To respect system limitations and inform design, it is useful todevelop conditions which predict the real safety of the system.

In Chapter 3, conditions for passivity of a flexible-joint robot under impedance controlare developed, with consideration of actuation dynamics. This passivity condition is thenextended to hold over an uncertain model. It is shown that this analysis method givespredictive power over what impedances are safely realizable, as well as capturing the effectsof inner-loop torque control on overall system passivity. With an experimentally derivedmodel uncertainty bound, constraints on controller design which respect robust passivitycan be explicitly derived.


Load Uncertainty

The load side dynamics also affect the rendering of an impedance at the interaction port,as motivated in Section 1.2. Depending on hardware and application, this can significantlylimit the system’s performance. To improve overall system performance with implicit sensing,two approaches are developed here; direct sensing, and improving the load side model.

In Chapter 4 direct measurement of both interaction quantities is shown to allow for therobust regulation to the desired dynamics. This is a reflection of fundamental motivations forfeedback control - sensing and feedback can reduce the effects of an uncertain model. First,fundamental limitations to interactive performance of an uncertain system are derived, thenshown to be improved with direct sensing. A controller architecture similar to a disturbanceobserver is used to achieve the robust rendering of desired dynamics, and it’s performanceanalyzed. The passivity of the resulting controller is considered, then extended to hold overan uncertain system. The effects of actuation dynamics are taken, and the conditions forpassivity extended to hold while considering actuation dynamics.

When sensing is not possible, improving the accuracy of the model can improve perfor-mance. Although certain aspects of the load-side dynamics are difficult to model, such asnon-idealized gravitational terms or dynamics arising from non-traditional actuation, theyare still fundamentally functions of the robot state. As such, Chapter 5 explores the mod-eling of these effects with a data-driven method. The objectives of inverse dynamic mod-eling in interactive tasks is introduced, and conditions for model regression developed. Amultimodal Gaussian process regression is used to improve inverse dynamic modeling asneeded for interactive systems. Intermittent unmeasured disturbance during identificationcan be accommodated, and multimodal dynamic behavior clustered and identified in a uni-fied framework. The passivity of the resulting control policy is shown, and the performanceand passivity is validated experimentally.

1.4 Summary of Contribution

A brief summary of the contributions in this dissertation are as follows:

• Analytically derived and experimentally validated model for flexible-joint robot, withconsiderations for inverse gearbox transmission and load-side dynamics [39].

• Development of a passivity condition for a flexible-joint robot which renders impedanceon the load-side motion, then extension to hold over uncertain robot model [42, 44].

• Proposal of novel control architecture for robust rendering of desired impedance on atorque-controlled robot and experimental validation [41].

• Development of inverse dynamics learning approach suited for interactive robots whichrejects intermittent disturbance and captures multimodal behavior [43].

10

Chapter 2

Modeling for Interactive Control

Modeling of interactive robots begins with separation of the robot from the environment,formalized by placement of the interaction port and assignment of the causality for thetwo systems (i.e. what is input/output for each). Though seemingly simple, this requiresjoint consideration of the objective, hardware limitations and safety. No standard model isaccepted in literature due to the variety of tasks, hardware, design criteria and objectives.This chapter introduces these modeling questions in general, then analytically motivates andexperimentally validates models appropriate to the hardware used in this dissertation.

The contribution in this chapter is the development and validation of a model for series-elastic driven robots which explicitly considers gearbox backdriveability and load-side dy-namics. Prior works investigating flexible-joint robot control consider either non-backdriveable[104, 50] or perfectly backdriveable [86, 59, 4] gearboxes. This chapter introduces a novelmotivation for an analytical model of a planetary gearbox under bilateral actuation (i.e.driven by both motor and load). This model is then validated on the experimental setupunder varying output load. Typical models for torque controller design on series-elastic ac-tuators (e.g. [96, 86, 59]) design and validate a controller with the load-side position fixed.Here, the effects of load-side dynamics on inner-loop torque tracking performance are alsoinvestigated [39].

2.1 Admittance and Impedance Control

Let the interactive port variables be denoted fint, x, the interactive force and velocityrespectively. Inherited from circuit analysis; these variables are chosen such that their prod-uct gives the instantaneous power flow into a system [126, 74]. The power flow representationallows a rigorous and intuitive means of stability analysis of these coupled systems, to bediscussed in Chapter 3.

Although velocity offers a theoretically satisfying choice of port variable, many tasks aremore naturally represented with position. Position offers the advantage of being directlysensed not only by encoders on robot joints, but also cameras or other task-relevant sensors

CHAPTER 2. MODELING FOR INTERACTIVE CONTROL 11

which can inform robot behavior. Position and velocity representations are both common inliterature, often with position used to analyze performance and velocity to examine passivity.

The desired relation between fint and x is typically that of a mass-spring-damper system,as shown in Figure 2.1a. By selecting the impedance parameters Mimp, Bimp and Kimp

(inertia, damping and stiffness respectively), a system response consistent with system-levelgoals can be realized.

A real system, such as the one shown in Figure 2.1b driven by an actuator fact, canrealize a control policy for fact which allows the presentation of a variety of desired interactivedynamics. Rendering a desired impedance or admittance through a control policy is calledimpedance or admittance control respectively, with the objective to present the desireddynamics at the interaction port (performance to be defined rigorously in Chapter 4).

(a) Desired Interactive Dynamics (b) Real System Dynamics

Figure 2.1: Desired and Real Dynamics

Figure 2.2: Impedance and Admittance Representations

Admittance Control

An admittance controlled system takes as input the interactive force, and achieves aposition output. Some systems which can be easily moved by external force, such as hapticdevices, do not require measurement of interactive forces [26, 23, 70]. Several exoskeleton


implementations with low gear ratios also adopt a direct admittance control approach [34,79].

Other systems which cannot be easily backdriven (driven from the output), require directmeasurement of interactive force to achieve acceptable performance [14, 90, 17, 2, 72, 127,85, 128, 120]. This is a common choice for adapting a traditional industrial manipulator foruse in interactive tasks - a force/torque sensor at the end-effector allows measurement of fint,and the well-established position control of the robot can realize the appropriate positionresponse [35]. When an existing (typically manufacturer provided) motion controller is used,performance is typically acceptable within the bandwidth of the position controller, butrigorous demonstrations of passivity are made impossible from the unknown, typically high-gain nature of the motion controller. The high intrinsic stiffness of these manipulators makescontact transition with a stiff environment difficult [115], as the mutually high stiffness cancause skipping, especially when the contact must arrest a large momentum. This limitsdirect force sensing approach, and is more commonly seen in more compliant environments(e.g. in human-robot interaction).

Impedance Control

Impedance control takes the environmental motion as input to the system and producesa force response. For impedance control to be realized, issues of co-location and implicitsensing become important, as introduced in Section 1.2.

Series Elastic Actuators

In the work of [96], the output impedance of a geared motor was significantly reducedby the introduction of physical elasticity - a spring between the output of the gearbox andthe environment. Measurement of the relative displacement on the elastic element allowsinference to the torque it carries, allowing a closed-loop control to regulate this torque,further reducing the apparent admittance of the geared motor. As position sensors are oftenless expensive than traditional force sensors, this can be a cost-effective approach.

Furthermore, as the compliance is physical there is no bandwidth limitation to the in-creased admittance, reducing the effects of the gearbox and the high reflected motor inertiaeven at high frequencies. This makes these systems particularly well suited when complianceis needed at high frequencies. This offers major advantages in collision, which introduceshigh-frequency excitation. This is especially relevant in high-stiffness environments, e.g.ground impact of bipedal robots [101]. Several other studies have found intrinsic elasticityoffering safety benefits in human-robot interaction [38, 3].

Related to the SEA are variable stiffness or variable impedance actuators. These actua-tors are designed such that the joint stiffness can be adjusted on-line, theoretically allowingthe advantages of high and low stiffnesses to be leveraged [93, 3]. Here, these are groupedwith series elastic actuators as they use measurement of bulk displacement to infer torque.


Joint Torque Measurements

A similar approach is to integrate torque sensors after the gearbox [4]. These torquesensors, by measuring the strain on a structural element of known properties, allow inferenceof the torque. This similarly allows closed-loop regulation of the torque, allowing a reductionof geared motor impedance within the bandwidth of the torque controller.

Although these robots with joint flexibility are structurally identical and treated jointlyin this text, several practical considerations distinguish the torque sensor approach from theseries-elastic actuator. The higher stiffness of torque sensors gives lower admittance beyondthe bandwidth of the torque controller. However, this higher stiffness also places resonancemodes at higher frequencies when integrated to an inertial load. The effect of differentintrinsic stiffness has been well-studied in literature [38, 97].

Another difference between torque sensing and elastic element approaches arise fromthe instrumentation. Strain gauges, the strain measurement device in torque sensors, aresensitive to electrical noise as well as heat-expansion and other sources of strain [58]. Thesenoise characteristics differ from the quantization in encoders, but differentiation of eithersignal will introduce noise: broad spectrum noise for strain gauges, and impulses from thequantization of optical or magnetic encoders.

2.2 Flexible Joint Robot Models

As all major types of impedance-controlled robots exhibit joint flexibility, a unified modelflexible-joint model willl be used for both a linear one DOF case and a high-DOF serialmanipulator.

Joint flexibility has considerable history in robotics. For most industrial robots, harmonicdrive gearboxes are used for a variety of favorable design characteristics, including minimalbacklash, compact size, and co-linear input/output shafts. However, these gearboxes exhibita non-negligible stiffness [117], introducing dynamics at the joints which can give rise tooscillation in load side behavior, especially in highly dynamic applications (high-accelerationor significant load). Much prior work has been done on modeling and control which allowscompensation of these joint dynamics. Some of these approaches are model based, mostlyderived from the flexible joint model of [113]. Some used this model in a singular perturbationframework to treat simplified dynamics [110, 116]. Joint flexibility motivated some adaptivecontrol techniques [66, 51] to handle the relatively difficult-to-identify model parameters. Thedeviations in load-side motion induced by joint flexibility has also been a large motivationfor other advanced approaches such as Iterative Learning Control [21]. In general, theobjective of these approaches is to improve load-side position tracking performance. This isan important distinction from interactive robots, where deviation from the desired trajectoryis expected, but this deviation should be controlled.

An additional distinction for interactive flexible-joint robots is that the joint flexibilitycomes with additional measurement (either torque or position). This addresses some of


Figure 2.3: Integrated SEA Model

the major issues with model-based joint flexibility, where although in theory the additionalstates induced by motor/joint mismatch may be observable, reliable estimation of them maybe difficult. The additional measurement afforded in interactive robots allows the directdeployment of control architectures which can use full state information. In fact, some ofthe proposed controllers for interactive flexible-joint robots [4] are viewed as extensions ofprior flexible joint position control [116] to the full-state information case. The prior workin position control for flexible joint robots also established passivity as a useful means ofstability analysis on these systems [116].

Linear

To treat one-DOF systems as well as gain intuition to inner-loop torque control behavior,a linear SEA model will be considered here. In this case, the output of the elastic elementis coupled to a load inertia, with damping as seen in Figure 2.3.

The system dynamics can then be found as:

Jφ+Bφ = −K (φ− θ) + τm (2.1)

Mθ + V θ = K (φ− θ) + τint (2.2)

where τint and τm are the interactive and motor torque respectively, φ and θ the positions,J and M the inertias, B and V the damping, of the motor and load side respectively. Let

τ = K (φ− θ) (2.3)

where τ is the torque on the elastic element which couples the two systems. The dynamicscan be re-written to directly express the torque dynamics:

K−1τ +M−1V τ +(J−1 +M−1

)τ +

(J−1B −M−1V

)φ = J−1τm +M−1τint (2.4)


Figure 2.4: One DOF Test Actuator

Although (2.4) requires an assumption of the load side dynamics, which may change as thesystem is interacting with (and coupled to) new environments, this formulation will be usedin later sections to gain intuition to system performance.

The hardware setup for this linear model can be seen in Figure 2.4.

Serial Manipulator

When a flexible joint is integrated to a serial manipulator, the equations of motion becomemuch more complex. Following the development in [112], most flexible-joint robots [93, 4,43] are modeled as

M (θ) θ + C(θ, θ)θ + g (θ) = JTint (θ)Fint + τ (2.5)

Jφ+Bφ = τm − τ (2.6)

(2.7)

where again τ = K (φ− θ), but now τ ∈ RN , if this is an N -degree of freedom (DOF)manipulator. Once more, the open-loop torque dynamics with these load-side dynamics canbe written

JK−1τ +(BK−1 +Kd

)τ +Kpτ + Jθ +Bθ = τm (2.8)


Figure 2.5: Dynamics for Serial Manipulator with Joint Flexibility

The experimental setup for the high-DOF flexible joint robot can be seen in Figure 2.6,with a detail of one of the joints in Figure 2.7.

Bowden Cable Drives

For both the single-DOF test actuator and the three distal joints on the exoskeleton,actuation is realized through bowden cables. A sheath of high axial rigidity, but which offerstorsional flexibility allows a cable to be pulled through it, provide remote transmission ofpower. This allows the geared motors to be placed at a ground-mounted location, signif-icantly reducing the sprung inertia on the exoskeleton. A schematic of the bowden cableintegration can be seen in Figure 2.8. In this work, it will be assumed that the cables areaxially rigid (no spring effects), and thus φm = φp = φ. The load-side pulleys that the cablesattach to can be seen in Figures 2.6 and 2.4. The motor side mount can be seen in Figure2.9.

Although this transmission allows significant reduction in the sprung inertia, it comes atthe cost of dynamic behavior. Seen in Figure 2.10 is a cutaway view of a cable under load.The cables introduce friction which is a function of their geometry and the tensile load, asthese jointly determine the normal force between cable and sheath as well as contact area[9]. Furthermore, as the load-side of the exoskeleton moves, the sheaths exert a spring-likeforce which resists their bending, adding elastic terms to the load-side dynamics. The effectsof bowden cables, manifest in both the motor-side and load-side are major sources of modeluncertainty, motivating the emphasis on these effects in this dissertation.

Backdriveability

One of the motivations for introducing joint flexibility is to improve backdriveability, boththrough inner-loop torque control and (for SEAs) on a direct, mechanical level. However,gearbox backdriveability still affects modeling and performance, as it modifies how the elasticelement torque τ induces motion in the motor. In the standard flexible joint equations of


Figure 2.6: Five DOF Upper-Limb Exoskeleton

motion (2.2),(2.6) it was assumed that a torque at the flexible element is directly appliedto the motor. Depending on the gear reduction ratio and technology, friction within thegearbox and other losses mean that the torque induced on the motor differs from the torqueat the gear reduction output (here, τ).

Shown in Figure 2.7, a series-elastic joint on a high-DOF robot is shown, where a brush-less, DC Maxon motor is geared through a 1:92 planetary gearbox. Figure 2.11 shows amodel for a general flexible joint under a closed-loop torque controller of

τm = C(s)(τ d − τ

). (2.9)

A variety of inverse gearbox dynamics have been considered in literature, with both back-driveable [86, 59] and non-backdriveable [104, 50] models used for SEAs. Let the motor sidedynamics be represented from output equivalent parameters (e.g. reflected motor inertia anddamping). Then, when inverse gearbox dynamics are taken as 1 (i.e. fully backdriveable),the motor side inertia couples with the load side dynamics through the elastic element, giving


Figure 2.7: Integrated SEA to high-DOF Exoskeleton

Figure 2.8: Bowden-Cable Transmission Schematic


Figure 2.9: Ground Mounted Motor Supports for Exoskeleton

Figure 2.10: Bowden-Cable Model


Figure 2.11: Impedance Model for Flexible Joint

higher order dynamics. If the inverse gearbox dynamics are 0 (i.e. non-backdriveable), thetorque τ does not induce motion in the motor, giving a simpler system, though increasingthe uncompensated low-frequency impedance of the system.

With a damped inertia model for the motor dynamics, G(s) = (Js2 + Bs)−1, and aPD controller for inner-loop torque control, τm = (Kp + Kds)(τ

d − τ), the torque trackingperformance of these two models, τ d to τ , can be seen in Figure 2.12. The motor parametersare fit to the hardware in Figure 2.7, and PD parameters of Kp = 3.5, Kd = .5.

Figure 2.12: Effect of Backdriveability on τ d → τ

The backdriveable system exhibits lower DC gain and increased resonance. However, asthe gain of the PD controller increases, the performance of the two models converge, as seenin Figure 2.13 (where Kp ≈ 150, Kd ≈ 25). The convergence of these two limiting cases


Figure 2.13: Effect of Backdriveability on τ d → τ with High Gain Controller

suggests that the effects of inverse gearbox dynamics will be reduced under high-gain control,but may still be significant for more realistic control gains.

To improve the low-frequency torque tracking performance, a backdriveable system caninclude an additional term in the controller to compensate τ ’s effect on the motor

τm = C(s)(τ d − τ) + βτ (2.10)

where β is an estimated parameter to capture the effects of the load torque on motor.Uncertainty in the spring stiffness and inverse gearbox dynamics will cause a mismatchbetween the τ induced by the elastic element and the compensating motor term, effectivelygiving a modified backdriveability.

Stiction and Backdriveability Modeling

For the purposes of improving inner-loop control, it is desired to more clearly motivatea model for the inverse gearbox dynamics. In all other sections of this text, the motorcharacteristics (motor torque, position, damping and inertia) are treated as viewed from thegearbox output, but this section will consider motor-side quantities as well. Let the nominalmotor model be as seen in Figure 2.14, with gear ratio N , motor torque τmm, motor positionφm, with corresponding output torque and position of τ and φ. In idealized gearboxes, thefollowing relations hold:

φm = Nφ (2.11)

τmm = N−1τ (2.12)


Figure 2.14: Motor Model Variables

Here, it will be assumed that the gearbox is rigid (i.e. the position relations hold as ideal),but the torque transmission will be considered more carefully.

Let the friction on the motor side, τfr,m and friction between the gear reduction andload output τfr,l. A reasonably accurate friction model typically includes the Stribek effectas well as Coulomb and viscous friction, as seen in Figure 2.15a [8]. Although even morecomplex models have been developed (e.g. LuGre), they are not considered here due to thedifficulty in fitting the model parameters, complexity in implementation, and requirementof steady-state velocity. The limited range of motion for these robot joints, and the oftenintermittent nature of the tasks means steady-state friction conditions may not be reached.Note that the gearbox friction τfr,l includes friction introduced between the gear reductionand the torque τ , for example the friction introduced by the Bowden cables which drive thetest actuator seen in Figure 2.4.

Here, the friction characteristics are taken as load dependent [89, 47, 8]. Although thegear tooth profile will be involute, such that there is no sliding contact between the teeth andthus no direct sliding friction between teeth, as the load increases, the supporting bearingreaction forces increase, giving corresponding increases in normal surface forces and therebyincreasing friction. Furthermore, for the three distal joints on the exoskeleton, as well asthe test actuator, the bowden cable transmission introduces a load-dependent friction. Thiscan be seen in Figure 2.10, where increasing tensile load increases the normal force betweencable and sheath, changing the friction characteristics.


(a) Stribek Friction Model (b) Static and Viscous Friction Model

Figure 2.15: Desired and Real Dynamics

The exact relationship between normal force and friction, as well as the friction char-acteristics themselves may be too complex to be tractable for controller compensation. Towrite the equations with full dependencies:

Jmφm + τmm − τfr,m(τmm, φm

)= N−1

(τ − τfr,l

(τ, φm

))(2.13)

To develop a tractable model for analysis and control, it will be assumed that the frictioncan be decoupled into two terms, a speed-dependent term which is independent of load,and a load-dependent term which represents the effects of coulomb friction. This separationis motivated by experiments on the Bowden cable which show significant variation in thecoulomb coefficient as bend radius changes, but not much change in the damping, as seen inFigure 2.16. Bend radius and tensile load are closely related for bowden-cables, as decreasingbend radius increases the preload on the cables. Further investigations into bend radius canbe seen in [9]. Under this assumption, τfr = τfr,c (τ) +Bφm.

Rearranging the terms of (2.13) to be viewed from the output (J = N2Jm, φ = N−1φm),and taking a linear approximation to τfr,c gives a modified backdriveability as:

Jφ+Bφ = τm − τ + τfr,c (τ) + τc (2.14)

Jφ+Bφ ≈ τm − βτ + τc (2.15)

where τc is the offset in coulomb friction - the coulomb present even under no load.This model can be identified by increasing the output load on the gearbox and testing

the slip condition. For series-elastic actuators this can be easily done by fixing the output ofthe SEA, such that as the motor angle increases, the output load increases proportionally.Motor torque is measured from motor current, and output torque is measured by the SEA.


Figure 2.16: Bowden-Cable Friction Characteristics, Experimental Results

By repeatedly increasing τm to the slip condition, the value of τc and β can be fit. Note thatbelow the slip condition, the static friction is indeterminate as it will match the motive forceto prevent motion. By fitting the model at the slip condition, φ and φ are minimal, so allthe dynamic terms in (2.15) are neglected, leaving τc and β to be directly identified.

Several typical time series for these tests can be seen in Figure 2.17. The motor torqueincreases until the breakaway friction is exceeded, then the increased τ brings the systemback to rest. A comparison of the torques can be seen in Figure 2.18, and it can be seen thatthe relationship is approximately linear within the tested range of torques. Thus, a linearfit will be used to find experimental values for β and τfr as minimize

minβ∈[0,1],τc∈R+

‖τm − βτ − τc‖2 (2.16)

These found values of β = .79, τfr = .32 are then used for compensation in the control law.The effective static friction magnitude after this model fit can be seen in Figure 2.19, andalthough variance increases with load, the major dependencies on load have been removed.

τm = C(s)(τ d − τ) + βτ + τfrsgn(τ d − τ

)(2.17)

where the sgn function, which matches the sign of it’s argument, is used to resolve the staticfriction model according to the direction of desired motion, where the direction of desiredmotion is inferred by the error in torque.


Figure 2.17: Time Series of Static Friction Break Tests

Figure 2.18: Torque Comparison and Fit Model

Load Dynamics Model

Another modeling question is the treatment of environmental input. For torque controllerdesign, commonly the load side motion is taken as exogenous input, and the torque τ asoutput [96, 86, 87, 59, 67, 119]. However, as the load side of the elastic element is usuallyfixed to additional kinematic structure which has an inertial component, this raises questionsabout performance of the integrated system [73, 40].

When load side motion is taken as input, performance is typically experimentally vali-


Figure 2.19: Stiction τfr with fit β

dated with output θ fixed (i.e. high load impedance). This may be a good approximationwhen the system will be interacting with a high-stiffness environment (e.g. bipedal robots),but a lower impedance load may change performance. Comparing the torque tracking per-formance (τ d to τ) of the same controller with a high impedance load versus an inertialload can be seen in Figure 2.20. Performance of the two models differs significantly, withthe inertial output having lower DC gain and increased resonance. However, as the gain ofthe inner-loop controller increases, these two models again converge, showing that a loaddynamics model becomes less important as the gain of the torque controller increases.

Practically, it is important to render the torque control performance invariant of loaddynamics as the impedance controller relies on satisfactory performance of the torque controlloop. As the load dynamics change by, for example coupling to new environments, if inner-loop performance varies, it may require situational adjustment to the impedance controlparameters.

A high-gain controller reduces the uncertainty in backdriveability, the effects of motorfriction and renders the torque control invariant of the load dynamics. All these factors areimportant for the performance of interactive robots, but pursuit of high-performance inner-loop control cannot compromise safety. As such, the next section will motivate constraintson the controller which guarantee stability and passivity.


Figure 2.20: Effect of Load Dynamics on τ d → τ

28

Chapter 3

Passivity and Safety

One of the challenges for interactive control is to provide meaningful guarantees of safety,even when the robot is coupled with an unknown environment. If two interacting systems arefully modeled, established techniques derived from Lyapunov theory or direct linear analysiscan be used to show stability. However, for interactive control the environment is unmodeled,and it is often desired to make as few assumptions about it as possible. Internal states ofthe environment, and dynamics which govern their evolution may be difficult to motivate.However, two major approaches in robust control theory allow conclusions about coupledsystem stability based off only input-output behavior of the two systems.

This chapter introduces these techniques in the context of interactive control. Exist-ing approaches for showing safety of interactive flexible joint robots are reviewed, then anew condition introduced, extended for uncertain systems, then relaxed. The real-worlddescriptive power of this condition is then investigated.

The main contribution in this chapter is a novel passivity condition for flexible joint robotswhich render an impedance on the load-side motion. Existing work shows passivity for eitherquasi-equivalent motor-side position impedance control [4, 93] or only when rendering a zeroimpedance [119]. This passivity condition is then extended to hold over an uncertain systemmodel to understand and inform real-world limitations to passivity. Others have investigatedlimitations of interactive control under discretization [23] and quantization [25], and only oneknown work investigates model uncertainty [115]. Unlike the work in [115], which focuseson impedance parameter constraints for contact transitions with a stiff environment, here ageneral condition is developed which can inform constraints on both inner-loop torque andouter-loop impedance parameters.

3.1 Interactive Systems Stability

Let two systems be as shown in Figure 3.1, with controlled system H and environmentE. In the most general form for the following analysis, these need only be square, causalsystems. A more thorough introduction to passivity can be found in e.g. [121, 74].

CHAPTER 3. PASSIVITY AND SAFETY 29

Figure 3.1: Generalized Environmental Interaction

Let G be a general operator which maps G : Lq (U) → Lq(Y ), where U and Y arelinear spaces for the input and output respectively, and where Lq (X) denotes the set ofall measurable functions f : R+ → X where

∫∞0|f(t)|q dt < ∞. Let the norm operator

‖f‖q,T =(∫ T

0|f(t)|q dt

) 1q

Small-Gain

The map G is said to have finite Lq gain if there exists finite γ and b such that:

‖G (u)‖q,T ≤ γ ‖u‖q,T + b ∀u ∈ Lq (U) , ∀T > 0 (3.1)

Let γH , γE denote the values attained by controlled system and environment respectivelyfor some q. Then the overall system will have finite Lq gain (i.e. be bounded-input, bounded-output stable) with respect to input additively injected if:

γH · γE < 1 (3.2)

Equation (3.2) is referred to as the small-gain condition.

Passivity

Let G be an operator mapping a finite-dimensional inner-product space U to a dual spaceU∗ = Y (i.e of the same dimension) and let a dual product 〈y, u〉, y ∈ Y, u ∈ U be definedbetween these two spaces which is linear in u, and satisfies the conditions of an inner-product

〈y, u〉T =

∫ T

0

〈y(t), u(t)〉dt (3.3)

G is passive if, and only if

〈G(u), u(t)〉T ≥ −α ∀u ∈ L1(u),∀T ≥ 0 (3.4)

for some α > 0. Passivity is most usefully understood when 〈y, u〉 is the instantaneous powerflow from a system. Then 〈y, u〉T becomes the power extracted over time T under input u.


The passivity definition (3.4) can then be interpreted as a lower bound on the energy whichcan be extracted from a system.

Several extensions to this definition can be made, where an operator G is strictly outputpassive if:

〈G(u), u(t)〉T ≥ ε ‖G(u)‖22,T − α ∀u ∈ L2(U), ∀T ≥ 0 (3.5)

for some α, ε ≥ 0.One passivity theorem is that if two interconnected systems H and E are passive, and one

is strictly output passive, then the coupled system has bounded L2 gain (i.e. bounded-inputbounded-output stable) with respect to any additive input and either system output.

This statement of passivity is defined from only inputs and outputs (independent of staterepresentation), which is useful for analysis of the environment where a state representationmay be difficult to motivate. An alternate statement of passivity can be made which usesa state representation, which will be useful for demonstrating passivity of the controlledsystem.

Suppose a storage function S : X → R+, where X is the state space and S is bothcontinuous and differentiable. Then a system is passive if

S(x(t1))− S(x(t0)) ≤∫ t1

t0

s(y, u)dt (3.6)

where s(y, u) = 〈y, u〉. Equivalently; for a dynamic system of the form

x = f(x, u) (3.7)

y = h(x, u) (3.8)

condition (3.6) can be written as

Sx(x)f (x, u) ≤ s(u, h(x, u))∀x, u (3.9)

where Sx (x) is the partial derivative of S with respect to x. This formulation is actually astatement of the more general dissapivity, where passivity is recovered if the storage functions(y, u) = 〈y, u〉, but finite-gain, strictly-input passive and strictly-output passive conditionscan all be derived with different definitions of s [125].

Cayley Transform

These two results have a fundamental relation which allows a passivity condition to berestated as a small-gain condition, and vice versa [5]. The Cayley Transform states that

G ∈ P+ ⇐⇒∥∥∥∥G− IG+ I

∥∥∥∥ < 1 (3.10)


where G is a causal, square system and P+ is the set of strictly-output passive systems.If G is linear, time-invariant and SISO, the H∞ norm can be used to simplify the right-

hand side as ∥∥∥∥G− IG+ I

∥∥∥∥ < 1 ⇐⇒∣∣∣∣G(jω)− IG(jω) + I

∣∣∣∣ < 1 ∀ω ∈ [−∞,∞] (3.11)

Linear Systems

Intuition for these two robust stability conditions can be most easily gained from con-sidering H and E as linear systems H(s) and E(s). In this case, the system response to anadditive input to either τ or θ will have the denominator of

Den(s) = 1 +H(s)E(s) (3.12)

To guarantee the denominator does not go to zero, the norm |H(s)E(s)| < 1 or |∠H(s)E(s)| <180 deg can be enforced. These can be conservatively bounded as:

‖H(s)‖ ‖E(s)‖ < 1 ⇒ ‖H(s)E(s)‖ < 1 (3.13)

|∠H(s)| < 90 deg∧ |∠E(s)| < 90 deg ⇒ |∠H(s)E(s)| < 180 deg (3.14)

This can be seen visually as the constraints that these impose for both systems in thes−plane, as seen in Figures 3.2a and 3.2b

(a) Passivity Constraint (b) Small-Gain Constraint

With a linear model, a positive-real condition can be used to show the passivity of atransfer function G, as positive realness and passivity are equivalent for linear systems [61].Necessary and sufficient conditions for a positive real system G are

Re(G(jω)) ≥ 0 ∀ω ∈ [−∞,∞] (3.15)

G∗(jω) +G(jω) ≥ 0 ∀ω ∈ [−∞,∞] (3.16)


It is useful to note a necessary condition for a system to be positive-real is that it have at mostrelative degree one (e.g. one pure integrator). This can be seen from the phase constraint|∠G| < 90 deg. This is another reason that the port variables for interactive systems aretypically θ, as the interactive systems typically are inertial, and will have relative degree twobetween force and position.

For linear systems, it may also be useful to note that the Cayley transform (3.10) bearsstructural similarities to the Tustin Transform, used to check stability in discretization ofcontinuous time systems. This is because the transform maps the imaginary axis to the unitcircle, transforming the constraint on pole locations between the two domains.

3.2 Passivity in Interactive Control

Passivity is widely accepted as a standard assumption about typical environments forinteractive control. However, to use the passivity theorem to conclude stability, the complexdynamics of a flexible-joint robot must be considered. Seen in Figure 3.3 is a model for aflexible joint robot, with motor side position φ, load side position θ and flexible element K.

The impedance controller determines the desired interactive torque according to theimpedance parameters and load side position. The inner-loop controller tracks the desiredtorque, to realize it on an elastic element. The general hierarchical controller structure(outer-loop impedance with inner-loop torque control) is standard not only within SEAliterature [60, 87] but also in work for torque-sensing robots [93, 4].

Figure 3.3: Flexible-Joint Robot with Inner-loop Torque Control and Impedance Control

As the passivity of systems in series cannot be directly concluded from the passivity ofeach sub-system individually, the impedance controller in series with the closed-loop torqueresponse complicates the passivity argument. Some authors have investigated passivity ofsystems in series [111, 7], however these conditions are not met by standard models usedfor flexible-joint impedance control. Namely, the impedance controller I(s) which includesa spring term τ d = Kimpθ is not strictly-output passive as required for this series passivityargument.


Figure 3.4: Collocated Passivity Model for Flexible Joint

Figure 3.5: Full Passivity Model for Flexible Joint

One approach, popularized by [4], is to realize impedance control with a motor side refer-ence position which is, in steady-state, equivalent to the desired load side position, as shownin Figure 3.4. As each coupling is a power continuous inter-connection, the passivity of theseinterconnected systems can be directly concluded from the passivity of each individually.This is very similar to passivity arguments for position control of flexible joint robots whichused motor-side position measurements for control [116]. However, for systems with lowerjoint stiffness (e.g. SEAs) the accuracy of the rendered impedance in this approach is re-duced, and although modifications have been proposed for variable-stiffness actuators [93] itremains a fundamental limitation of this approach.

Total Response Passivity

Work examining the passivity of a system where the impedance is derived from the load-side motion is limited. In [107], the passivity of a velocity-sourced SEA is shown, althoughneglecting the effects of the inner-loop velocity controller. Others, e.g. [119] show thepassivity in the direct response of a torque-controlled SEA, with the impedance controllerimplicitly assumed to be 0. However, the realization of an outer-loop impedance has atheoretical and practical effect on the passivity of the overall system. If the desired impedancefrom θ to τ d is denoted I(s), with controller C(s) and motor side dynamics ofG(s), the overallsystem response and resulting passivity constraint is

τ

θ=I (s)KG (s)C (s) +K

1 +KG (s)C (s)s−1 ∈ P (3.17)


where P denotes the set of positive real systems. This condition can be decoupled into twosystems in parallel, and showing the passivity for each path individually is sufficient (butnot necessary) for overall passivity.

For any known model and controller, these conditions can be easily checked (there’s evena Matlab command, isPassive()). However, a point-wise passivity check does not muchhelp inform intuition or controller design. Furthermore, once condition (3.17) is met, therendered impedance or controller gain can typically be scaled arbitrarily while maintainingpassivity. However, every interactive system has an in-practice limit to the impedance orstiffness which can be safely rendered. Thus, the real-world difficulty in realizing a highimpedance (or a high-gain inner loop controller) is not captured by this condition.

3.3 Robust Passivity

Passivity suffers from several limitations as a real-world condition. Many authors [14,64, 73, 107] have noted the conservatism of passivity, that non-passive systems can exhibitacceptable behavior in practice. Others [22, 14, 20, 115] have noted the in-practice instabilityof coupling two passive systems. Although passivity in theory provides reasonable constraintsand guarantees, the sufficiency of it for interactive stability can be compromised and althoughpassivity was never a necessary condition (i.e. is inherently conservative), the conservatismcan be restrictive.

To explain this discrepancy, several real-world effects have been considered, most notablydiscretization [24] and quantization [25, 27]. These analyses are taken on haptic systems,which have direct-drive actuation, so considerations of inner-loop torque control are notmade. With the uncertainty of the inner-loop torque control loop (from, e.g. the backdriv-ability and friction), this section will consider model uncertainty in the passivity argument.

In the work [115], the small-gain theorem is considered to additionally restrict theimpedance parameters to improve stability in contact transitions (i.e. coming into con-tact with high-stiffness environment). However, here the concept of robust passivity will bedeveloped, where condition (3.17) should hold not only for a nominal G(s) (here referredto as nominal passivity), but for all possible systems in a bounded set G(s) ∈ G (robustpassivity).

Let the set of systems G be parameterized by a nominal model G, and a perturbation ofunknown phase but bounded magnitude ∆ as:

G =G(s) : G(s) (1 + ∆(s)) ‖∆(jω)‖ < ∆(jω)

(3.18)

The ultimate goal of this condition is to assist in the synthesis of the controller designfor interactive robots. By combining this condition with a rigorous means of relaxation,the necessity and sufficiency of these conditions can come closer to that needed in practice,allowing explicit and meaningful safety constraints.

These constraints can then be incorporated with other means of controller synthesisfor interactive robots, such as algorithmic [95] or computational [14] optimization. For


optimization-based synthesis methodologies especially, explicit constraints induced by prac-tical stability and passivity are important. In general, performance comes at a cost of ro-bustness and as these techniques, by pushing towards high-performance controllers can oftenreduce safety margins by generating controllers which are high-gain (or high-order). Explicitbounds, such as H∞ constraints, or parameter bounds for a fixed-structure controller canallow these optimization based techniques to be used with less ad-hoc modification to theobjective or resulting controller to ensure safety.

Robust Passivity of Inner-Loop Torque Control

Using the Cayley Transform (3.10), the condition for total system passivity (3.17) canbe shown to hold over all G ∈ G, as parameterized in (3.18). Let the total objective be tofind constraints on I and C such that:

I (s)KG (s)C (s) +K

1 +KG (s)C (s)s−1 ∈ P ∀G ∈ G (3.19)

By directly substituting the construction of G and rearranging, the following conditioncan be shown to be necessary and sufficient for (3.19):∥∥∥∥A∆ +N

B∆ +M

∥∥∥∥ < 1 (3.20)

where A = G(s)C(s)(I − s), B = G(s)C(s)(I + s), N = A + (1 − K−1s)) and M =B + (1 +K−1s). Using the H∞ norm, a conservative condition can be written as

|∆(jω)| < |M | − |N ||A|+ |B|

⇒ |A∆(jω)+N | < |B∆(jω)+M | (3.21)

where N , M , A, and B are implicitly functions of jω. Alternatively, a computationalapproach without conservatism can be taken to find a magnitude bound ∆(jω) over a set offrequencies Ω as seen in Algorithm 1.

Algorithm 1 Uncertainty Bound Fitting

1: for ω ∈ Ω do2: while ∃θ s.t. ‖A∆(ω)e(θi)+N‖ > ‖B∆(ω)e(θi)+M‖ do3: ∆(ω)← α∆(ω), α < 14: end while5: end for


Impedance Bound

For any impedance controller and system which meets condition (3.19), the impedancecan be reduced without compromising the passivity under some mild conditions.

Proposition 1 For systems A(u) and B(u)

A+B ∈ P (3.22)

A ∈ P (3.23)

⇒ A+ ηB ∈ P ∀ ∈ [0, 1] (3.24)

Proof By conditions (3.22) and (3.23)∫ T

0

〈A(u), u〉+ 〈B(u), u〉 > −α0 ∀T > 0 (3.25)∫ T

0

〈A(u), u〉 > −αa ∀T > 0 (3.26)

Necessary and sufficient for (3.24) is∫ T

0

〈A(u), u〉+ η〈B(u), u〉 > −αη αη ≥ 0,∀T > 0 (3.27)

By direct substitution, αη ≥ ηα0 + (1− η)αa. From the positivity of α0 and αa, αη > 0 willhold for η ∈ [0, 1].

This argument can be directly extended if A and B are not a specific system but only knownto belong to sets A and B respectively, provided α0 and αa can be found which bound allsystems in the respective sets.

In the application here, (3.19) can be split into two transfer functions over the summationin the numerator. If (3.19) holds for an impedance I(s), and

Ks−1

1 +KG(s)C(s)∈ P ∀G ∈ G (3.28)

then the impedance can be scaled as I(s) = ηI(s), η ∈ [0, 1] while maintaining robustpassivity. In practice, this is important as the structure of impedance I(s) is often fixed,and changing the stiffnesses rendered is done by scaling the entire controller such that thedamping characteristics are not significantly affected.


Simulation Results

If C(s) is a PD compensator of the form C(s) = (Kp + Kds), nominal plant is G(s) =(Ms2 + Bs)−1, and desired impedance is I(s) = Bimps + Kimp, the effects of impedance onthis robustness condition can be examined. Using parameters identified in Section 3.5,themaximum allowable ∆(jω) which satisfies condition (3.17) is shown in Figure 3.6. Increasingrendered impedance reduces the robustness of the system, which can explain the implemen-tation difficulties with rendering a high impedance.

Figure 3.6: Uncertainty Bounds as Impedance Increases

Similarly, the effect of increasing the controller gain can be seen in Figure 3.7. Thesystem tolerance of uncertainty decreases as the controller gain increases, demonstrating onetrade-off between performance and safety for the torque controller.

In this framework, the design heuristic for heirarchical control to keep the outer-loopbandwidth 5-10 times lower than the inner-loop bandwidth can be revisited. The gain forthe inner-loop controller is varied to adjust the ratio of bandwidth. Seen in Figure 3.8 is theminimum uncertainty allowed (over all frequencies) as the inner-loop bandwidth increases.When the bandwidth of inner and outer loop are very close, the allowable uncertainty isnegligible (−50dB), but as the bandwidth of the inner-loop increases relative to the outer-loop, the minimum allowable uncertainty increases. This may be one of the motivatingfactors for this design heuristic.

The achieved uncertainty bounds under the robust passivity condition are most sensitiveto the magnitude of the damping term. Parallel to the findings in [23], damping has a hugeimpact on the passivity of the system. Damping is the term which dissipates energy, andis thus fundamental to achieving passivity. However, the damping which can be realized by


Figure 3.7: Uncertainty Bounds as Controller Gain Increases

Figure 3.8: Minimum Uncertainty as Bandwidth Ratio Changes

controllers (Bimp and Kd) is limited by the quantization error induced by the encoders [27],especially as the nested controller gives effectively second order differentiation of the positionmeasurement. For this reason, several researchers and companies [88, 6] are investigatingthe use of viscoelastic actuators, which instead of a pure stiffness, also presents dampingcharacteristics.


3.4 Passivity Relaxation

Passivity is noted for being restrictive in certain applications [115, 107], and this sectionwill discuss relaxation of passivity. When information about the interactive environmentis known a priori, this can be used to make conclusions about coupled system behaviorwithout using passivity [14, 60, 16]. In general, if the environment is known to belong toa class of dynamic systems, techniques from robust control can be used to demonstratecoupled stability. However, for systems which interact with arbitrary environments, thetypical assumption is that the environment will be passive.

Frequency-Domain Relaxation

Another approach, if the environment can be treated linearly is a mixed passivity andsmall-gain condition, as proposed in [37]. In this approach, the frequency domain is parti-tioned into discrete frequency bands, and either passivity or a small-gain condition shown tohold within each band. This result has also found application in the synthesis of controllerswhich can accommodate passivity violations of the controlled system [32].

For flexible joint realizations in impedance control, the total response of θ → τ , denotedH(s), will roll off at high frequencies. A typical H(s) with PD inner-loop control gives thefrequency response shown in Figure 3.9.

Figure 3.9: Rendered Impedance θ to τ

Denote the environmental dynamics as E(s), and let ω0 be such that

ω0 = min |ω| (3.29)

s.t. ∀ω ≥ ω |H(jω)| < γ ∧ |E(jω)| < γ−1 (3.30)


where γ is a positive scalar. Then, by Theorem 1 of [37], H(s) and E(s) need only be passivein the range [−ω0, ω0]. This can be used to relax conditions such as (3.17), which instead ofholding ∀ω ∈ [−∞,∞] now need only hold ∀ |ω| < ω0.

One advantage to this approach over other means of incorporating environment informa-tion is that less assumptions about the environment need to be made, as long as a conservativebound on the cutoff frequency ω0 can be made. If a component of the environment is inertial(e.g. the kinematic drivetrain after the flexible element), E(s) will roll off above a cutofffrequency, and finding an ω0 which satisfies (3.30) can be easily shown.

This relaxation can also explain some situations where torque control and impedancecontrol struggle. Typically there is more model uncertainty at higher frequencies, and ac-tuator or sensor dynamics can also add phase at high frequencies, making high-frequencypassivity violations a common occurrence [31]. A high stiffness environment increases the ω0

imposed by the environment, requiring passivity to hold over higher frequencies. Increasingthe stiffness rendered by the robot increases the ω0 constraint imposed by the controlled sys-tem. Both of these can cause the passivity violations at higher frequencies to meaningfullyimpact coupled stability.

3.5 Experimental Results

To validate the approach introduced here, a nominal model and uncertainty are experi-mentally fit on an SEA system, as seen in Figure 2.4. A Maxon BLDC motor, geared througha planetary gearbox at 132:1 actuates the system through bowden cables. Additional detailscan be found in [67].

System identification is performed in multiple configurations to capture uncertainty intro-duced by the inverse gearbox dynamics. In the first identification mode, the elastic elementis removed, and the system is identified without significant load on the output of the gearbox.Then, the spring is re-introduced, with the load-side fixed, which gives a reactive load torqueas the motor side is displaced. These identifications are done by exciting the input at a fixedfrequency, then fitting the steady-state response. These sinusoidal excitations are done ata range of frequencies and amplitudes to capture uncertainty which may be encountered inuse. A final identification is done through manual excitation of the output, which tests thepure backdriveability of the system. As the human input is bandwidth limited, it is onlyused for validation of the other two model identifications.

Nominal Model Fitting

For the model fitting, the model is used as shown in Figure 2.3, with the inverse gearboxdynamics taken as a scalar β, and all uncertainty grouped in with G. The three transferfunctions induced by the experimental conditoins are:


R1(s) = G(s) (3.31)

R2(s) =G(s)

1 + βKG(s)(3.32)

R3(s) =βKG(s)

1 + βKG(s)(3.33)

The identification is done over a range of input amplitudes and offsets, and the experi-mental responses can be used to motivate the model uncertainty. First, a nominal model isfit to R1, giving

G(s) =1

(7.5e−4)s2 + (4.5e−3)s(3.34)

Then, the nominal value of spring stiffness used K = 3.39, and inverse gearbox dynamicsβ per the identified value in Section 2.2. The resulting fit magnitude responses can be seenin Figures 3.10, 3.11, 3.12.

Figure 3.10: Experimentally Observed and Model Response for R1

Model Uncertainty

The error between the response predicted by the model and the experimentally observederror is attributed to ∆ under the assumption of multiplicative uncertainty as G(s) =G(s)(1 + ∆(s)). Expressions for ∆ can be found from model error of each experimentalconfiguration, as a function of the actual observed system response Ri(jωi). The result for




the three sets of experiments are shown in Figure 3.13.

∆1(jωi) =R1(jωi)

G(jωi)− 1 (3.35)

∆2(jωi) =R2(jωi)(1 +KG(jωi))

1−R2(jωi)KG(jωi)(3.36)

∆3(jωi) =

R3(jωi)

((βKG(jωi)

)−1

-1

)-1

1 +R3(jωi)(3.37)

The resulting model uncertainty can be seen in Figure 3.13. Higher frequencies have


Figure 3.13: Experimental Model Uncertainty

higher mangitude of model uncertainty, and the identifications which involve the inversegearbox dynamics (∆2 and ∆3) present more uncertainty than ∆1.

Explicit Constraints For Fixed-Structure Controllers

Given that all controllers documented for SEA are fixed structure: PD, PI or Lead-lag,the design parameter space for these systems is low enough to be discretized and visualized.

Another major constraint can also be incorporated at this stage - the limitations inderivative terms due to quantization or discretization. In practice, increasing these gainssignificantly impacts noise in the control signal, with consequences of backlash chatter, wearand saturation. Letting these parameters (Bimp and Kd) take their maximum realizablevalue; the effect of Kp on the maximum realizable Kimp which respects the passivity condition(3.20) can be found. The maximum stiffness is found, and as suggested by Proposition (1),a lower stiffness will also respect this constraint. The additional constraint of closed-loopstability for (3.19) can also be written with the same uncertainty parameterization by thesmall gain condition:

|∆i(jωi)| ≤∣∣∣KG(jωi)C(jωi)

∣∣∣−1

∀i (3.38)

Applying the passivity and stability robustness constraint to the G and ∆i which wereidentified above produces the parameter bounds seen in in Figure 3.14. Note that the robustpassivity condition makes no consideration of performance, so selecting a controller for veryhigh Kimp, but low Kp, would result in a system unable to render this stiffness accurately.


Figure 3.14: Controller Parameters Satisfying Robust Passivity and Stability

High Damping Low Damping

Kimp 8 8Bimp .3 .1Kp 15 15Kd 1.5 .3

Table 3.1: Controller Parameters for Comparison

Validation of Passivity

The explanatory power of robust passivity is examined here by comparing the responsesof two systems. The controller settings shown in Table 3.1 have significantly different damp-ing for both torque and impedance controllers. Both systems are nominally passive (i.e. arepassive on the model G), but these controllers have significantly different passivity robust-ness, as seen in Figure 3.15. Note that the stiffness rendered here is higher than the stiffnessof the spring, yet nominal passivity holds for both systems.

Frequency Domain Passivity

The system under controllers shown in Table 3.1 were then excited with manual excitation(i.e. by hand manipulating SEA output), then with impulses delivered by a rubber mallet toensure a broad range of frequency excitation. The resulting input joint angle θ and outputSEA torque τ are transformed to frequency domain; giving the phase response shown inFigure 3.16. It can be seen that the low damping system comes close to violating passivity


Figure 3.15: Uncertainty Bounds for Controllers

(i.e. approaches −90 deg phase). To verify the passivity violation, the system was excitednear the violation frequency, and the time domain response examined.

Figure 3.16: Phase Response of Controllers


Time Domain Passivity

Recall the state-space, storage function representation of passivity (3.6). The direct time-domain representation of passivity, shown in (3.4) cannot be proven to hold experimentally,as realizing all inputs over arbitrarily long time period is not feasible. However, the state-space condition can be shown to be violated through a finite-time trajectory.

If there exists a closed trajectory in the state space which extracts power, condition(3.6) will be violated (right-hand side will evaluate to zero). Exciting the system near thesuspected passivity violation frequency found in Figure 3.16, a portion of the state-spacetrajectory response is shown in Figure 3.17. This trajectory is closed, and in Figure 3.18,the power flow at the port and total energy extracted over the same time period can beseen. As power can be extracted from the low-damping system over a closed trajectory inthe state space, it is not passive.

Figure 3.17: State Space Trajectory


Figure 3.18: Interaction Port Power Flow and Total Energy Transfer

48

Chapter 4

Load Side Uncertainty

Even with perfect inner-loop torque control, impedance control performance will still belimited by model uncertainty in the load dynamics. Here, load dynamics refers to dynamicelements which are between the controlled torque source (e.g. spring, torque sensor) and thedesired interactive port.

Many of the early mechanical devices for interactive control were designed to be lightweightand low friction [70, 62]. In such systems, the load dynamics can be neglected and the desiredimpedance directly realized on the actuator. These systems can demonstrate passivity moresimply, as the control policy (sometimes called virtual environment) is in feedback with thereal system and environment. This allows the total system passivity to be concluded fromthe passivity of the mechanism and control policy. Advanced techniques such as the passivityobserver [45] allow guarantees on controller passivity even with discretization, quantizationor other practical effects.

However, more demanding applications, necessitating high DOF or reinforced linkages,may require a mechanical system which has significant and difficult to model dynamics.These systems sometimes include a force/torque sensor at the interaction port and use thismeasurement to inform the control strategy. One common form of this approach is to usethe force/torque measurements with the desired admittance to define desired motion for aninner-loop motion controller. This approach has been taken in exoskeleton control [17, 2, 28,76, 72, 56], and traditional robot manipulators [106, 115]. These high-DOF systems, oftenwith significant load side dynamics, benefit from the robustness offered by the inner-loopmotion controller where well-established techniques can be used to achieve high-performancetracking of the desired trajectory. However, there are two major limitations in using aninner-loop motion controller. Above the bandwidth of the position controller, the system’s(typically high impedance) dynamics cannot be modified. Furthermore, a demonstration ofpassivity with a position control inner-loop is difficult to show rigorously.

In general, direct sensing of the interaction force allows the application of much morecomplex techniques to regulate to the desired dynamics. In [19], the use of H∞ designtechniques to shape the closed-loop system towards the desired admittance are explored.Others have also explored H∞ techniques, combined with other controller structures [57].

CHAPTER 4. LOAD SIDE UNCERTAINTY 49

Similarly, in [15, 95] fixed-structure controllers are optimized over to match the desiredapparent dynamics. One challenge to these optimization based design techniques is balancingperformance and robustness, typically achieved by penalizing control action in the objectivefunction.

Natural admittance control [80] considers the design of a velocity and force feedbackcontroller with the objective of matching the desired impedance. Although no rigorouscontroller synthesis approach is presented, this has found practical application in many high-stiffness drivetrains. However, as it assumes only motor-side joint information is known, thishas not been incorporated on systems with joint-space torque control, where additionalinformation may improve performance. However, the explicit design objective of enforcingthe desired impedance bears similarity to the approach here.

The Disturbance Observer (DOB) [83] is a classic approach in precision motion control,and has been applied in several ways to interactive control. First, it was used to estimatethe interactive force indirectly by using a nominal model and motor torque measurements[75, 103]. This estimation of interactive force is then used in an admittance control scheme.Others have used the DOB to enforce nominal dynamics of the robot in a force or impedancecontrol task [82, 12]. In this chapter, the structure of the DOB is used with direct measure-ment of the interactive force to improve interactive control.

The main contribution in this chapter is the development and analysis of a DOB-basedcontroller architecture which regulates the interactive closed-loop dynamics to that of thedesired impedance. Just as traditional feedback control must directly measure an output torobustly regulate it, this approach uses direct measurement of both interactive force and mo-tion to robustly achieve the desired relationship between them (i.e. the desired impedance).The control structure is most similar to that used in [53, 57], where the ‘dynamic error’ (i.e.τint−Gimpθ, where Gimp is desired dynamics and τint, θ desired port variables to regulate)is corrected with feedback. Here, a DOB-like structure is used to regulate this dynamic error,allowing rigorous treatment of multiplicative uncertainty and a simple tuning methodology[44]. Again, passivity conditions are developed to constrain the controller parameters, thenextended to consider the actuation dynamics of the system. Model uncertainty is foundexperimentally and used to design a controller which is then validated experimentally.

4.1 Position-Based Controller Design

This section introduces an analysis of classical impedance control in the presence ofuncertainty. Let the system in Figure 4.1 be described by

Mθ +Bθ = τint + τact + d (4.1)

where M and B are the true inertia and damping of the load-side system. The system isdriven by actuator and interactive torques τact and τint, and achieves position θ. A distur-bance d also affects the system, representing additional forces applied to the system whichare not the desired interactive forces. The transfer function G denote the dynamics of the


Figure 4.1: Impedance Control System

load-side system such that

θ = G(s) (τint − τact + d) (4.2)

Let a model for the load-side system be denoted G, and the desired impedance Gimp. Letthe following relations be introduced:

G (s) = G (s) (1 + ∆ (s)) (4.3)

Gimp (s) = G−1 (s) (1 + Ω (s)) (4.4)

where ∆ (s) characterizes the model uncertainty, and has a bounded, known magnitude.The quantity Ω (s) characterizes the difference between the nominal model and the desiredimpedance, and is known.

Performance Limits of Impedance Control

Let a control policy as a function of θ be introduced, τact = C(s)θ, which renders theapparent dynamics of

θ

τint=

G

1 + CG

Let the system performance be characterized by the following cost function

Vp(C) =

∫ ∞0

ln

∣∣∣∣G−1imp(jω)− G(jω)

1 +G(jω)C(jω)

∣∣∣∣ dω (4.5)

This cost function Vp(C) characterizes the tracking of the desired impedance in the frequencydomain, approaching −∞ when perfect tracking of the desired dynamics is achieved. Letthe controller design be viewed in this framework by the optimization problem


minCVp(C) =

∫ ∞0

ln∣∣G−1

imp (1 +GC)−G∣∣+ ln

∣∣(1 +GC)−1∣∣ dω. (4.6)

Note that the term ln |(1 +GC)−1| can be evaluated by the bode integral [13, 114]. Let Pdenote the set of right-half plane zeros of GC

ln∣∣(1 +GC)−1

∣∣ =∑p∈P

Re(p) (4.7)

Without loss of generality, re-parameterize the control policy C(s) = Gimp + GC. This thengives ∫ ∞

0

ln∣∣∣G−1

imp

(1 + (1 + ∆) C

)∣∣∣ dω +∑p∈P

Re(p). (4.8)

By canceling known terms, the following minima is reached

V ∗p =

∫ ∞0

ln∣∣G−1

imp∆∣∣ dω +

∑p∈P

Re(p) (4.9)

arg minVp(C) = Gimp − G. (4.10)

To evaluate the bode integral term by finding P , note that the inertial systems used forinteractive control do not have unstable zeros or poles. If Gimp − G has only left-half planezeros, this term will be zero. This is satisfied for standard impedance control tasks if thedesired impedance damping is higher than the system’s intrinsic damping.

This leaves only the first term in V ∗p , which is a function of both the desired impedanceand the model uncertainty. Increasing model uncertainty decreases the performance of thesystem. As the desired impedance decreases (i.e. rendering a less stiff system), this costfunction again increases, as an equivalent model uncertainty will cause further deviation inthe position response.

Inverse Dynamic Compensation

A block diagram realization of the optimal position feedback (4.10) is shown in Figure4.2, generalized to a linear plant G(s) = (Ms2 + Bs)−1, corresponding nominal model G(s)and desired impedance Gimp(s) = Iimps

2 +Bimps+Kimp. The inputs τint, τact and d are theinteractive, actuator and disturbance torques respectively.

This minimizer shown in (4.10) gives the canonical statement of position based impedancecontrol. The controller expression for this approach is

τact =(Mimp-M

)θ +

(Bimp-B

)θ +Kimpθ. (4.11)


Figure 4.2: Inverse dynamic compensation for impedance control

Any disturbance terms affect the system response as an interactive force does, as thetwo are not distinguished in this approach. Note one further restriction - an arbitrarycompensator C is not guaranteed to be realizable. For the form given in (4.11), typicallyθ can be estimated from differentiation of encoder signals, and the impedance mass will betaken to be the model system mass. However, some have explored changing the apparentmass of a system [25, 107] on haptic devices.

Stability of Inverse Dynamic Compensation

The stability of the block diagram shown in Figure 4.2 will be investigated in this sec-tion under the assumption that relations between the true dynamics, model, and desiredimpedance are as shown in (4.3), (4.4).

This gives a closed-loop transfer function from τint to θ of

Gidc = G−1imp

1 + ∆

1 + ∆Ω (1 + Ω)−1 . (4.12)

For |∆ (s) | 6= 0, (4.12) diverges from the desired transfer function G−1imp.

The stability of (4.12) can be shown by using the small gain theorem. Sufficiency forstability is G be stable, (1 + Ω) be minimum-phase stable and∣∣∣∣∆ (jω) Ω (jω)

1 + Ω (jω)

∣∣∣∣ < 1 ∀ω . (4.13)

Typically this sufficient condition will not be difficult to meet, but other aspects of animplementation may compromise stability.


4.2 Robust Impedance Control

In the controllers above, performance was limited by model accuracy and the presenceof disturbances. This section will present an approach which can use direct measurement ofθ, τint to enforce the desired impedance.

Force and Position Sensing

When the interaction port variables are directly measured, measurements of both positionand force can be used to choose τact, which may improve performance. Let the control lawτact = Cτ (s)τint +Cp(s)θ, then the performance objective from (4.5) can be written with thenew dynamics from τint → θ.

V (C,Cτ ) =

∫ ∞0

ln

∣∣∣∣G−1imp −

G (1 + Cτ )

1 +GCp

∣∣∣∣ (4.14)

(4.15)

Again, without loss of generalization let the controller be reparametrized as Cp = Gimp−G−1 + G−1C, gives reduction to:

V(C, Cτ

)=

∫ ∞0

ln∣∣−G−1

imp∆ +G (GimpC∗ − Cτ )

∣∣− ln∣∣∣1 + Ω + Ω∆ +GC

∣∣∣ (4.16)

By fixing Cτ = GimpC, the performance metric becomes:

V(C)

=

∫ ∞0

ln∣∣−G−1

imp∆∣∣− ln

∣∣∣1 +(C + Ω

)(1 + ∆)

∣∣∣ (4.17)

As∣∣∣C∣∣∣ increases and dominates the other terms, this gives a reduction in the overall cost

function compared to the original formulation (4.9). By introducing a torque sensor (andincorporating the feedback from it), the performance of the system, as defined by (4.5) canbe improved.

Note that in this simplification it was assumed that Cτ = GimpC, and that then increasingthe gain on these two controllers improves performance. This can provide motivation for thecontroller proposed in the next section.

Disturbance Observer

The disturbance observer is a technique to improve performance in feedback controlsystems [83]. In a traditional DOB, an inverse model of the plant and observed outputare used to estimate disturbances and enforce that the closed-loop system has the modeldynamics.


If in the feedback path of the DOB, the transfer function for the desired impedancedynamics are used instead of the inverse plant dynamics, this system will enforce the desiredimpedance dynamics. The proposed architecture is shown in Figure 4.3, where Q(s) is afilter introduced to allow tuning and make Gimp(s)Q(s) realizable.

Figure 4.3: Disturbance observer based impedance control

In this architecture, the expression for θ is

θ(s) =G

1 +QGGimp-Qτint(s)+

G (1−Q)

1 +QGGimp-QD(s) (4.18)

When Q (s) = 1, the transfer function from τint to θ approaches the desired transfer functionG−1imp. Also, the transfer function from D to θ goes to 0, giving ideal disturbance rejection.

The expression for this control law is

τact =Q

1−Q(τint −Gimpθ)

This term reflects the properties of a high-performance controller which achieves (4.17).As Q→ 1, the effective gain on this error term increases, improving performance.

Stability of DOB Impedance Control

As stability of a DOB is limited by the difference between the inverse model and theactual system dynamics, the block diagram in Figure 4.4 is introduced. The Gimp − G−1

term compensates known discrepancies between the model and desired impedance, and isdirectly analogous to the traditional implementation of impedance control. An outer DOBloop robustly enforces the desired impedance.

This block diagram structure gives a closed-loop transfer function from τint to θ of

Gdob =G

1−Q+GimpG− (1−Q)GG−1. (4.19)


Figure 4.4: Inner loop compensation DOB

Again, using the multiplicative uncertainty models in (4.3) and (4.4) allows stability analysis.A sufficient condition for stability requires G should be stable, (1+Ω) minimum-phase stableand ∣∣∣∣∆ (jω) (Ω (jω) +Q (jω))

1 + Ω (jω)

∣∣∣∣ < 1 ∀ω. (4.20)

The disturbance rejection properties are preserved with the inner-loop compensation, andcan be shown with a similar approach to that shown in (4.18).

In the block diagram proposed in Figure 4.4, the control action takes the form

τact =Q

1-Q(τint−Gimpθ) -

(Gimp-G

−1)θ. (4.21)

The first term is again the high-gain term which provides the high-performance regulationof the desired dynamics. The second terms (Gimp − G−1) are the same as those used inimpedance control, reflecting the compensation of known differences between the systemand desired dynamics.

Passivity of DOB Impedance Control

Passivity of the controlled system allows stability to be concluded when the system iscoupled with any passive environment [49]. In the proposed approach, if the model uncer-tainty goes to zero (∆→ 0), passivity can be immediately found from passivity of Gimp(s).However, ∆ is a magnitude-bounded uncertainty (i.e. unknown phase) and passivity (forlinear systems, positive realness) is a phase condition. The Cayley Transform can be used,


where if T is positive real, ‖(T − 1)(T + 1)−1‖ < 1 [5]. Applying this transform here tosGdob(s) with the H∞ norm gives∣∣∣∣∣ G (s+Gimp) + ∆

G (s−Gimp)− ∆

∣∣∣∣∣ < 1 ∀ω ∈ [−∞,∞] , (4.22)

where ∆ = (1−Q)∆(1 + ∆)−1. This can be rearranged to put in the same form as (3.20):∣∣∣∣A∆ +N

B∆ +M

∣∣∣∣ < 1 ∀ω ∈ [−∞,∞] (4.23)

where A = Ω +Q− Gs, B = Ω +Q+ Gs, N = 1 + Ω− Gs and M = 1 + Ω + Gs.

Effect of Design Parameters on Passivity and Stability

The Q filter and the desired impedance Gimp are the two design parameters to this system.A Q filter which approaches |Q| = 1 over as large of frequency as possible is desirable, aswhen Q = 1 the system response will approach the desired impedance. The typical form fora Q filter is a low-pass filter, to keep the controller expression (4.21) realizable.

Design Parameters and Stability

The condition (4.20) can be quite easily rearranged to get

|∆| ≤∣∣∣∣ 1 + Ω

Q+ Ω

∣∣∣∣ ∀ω ∈ [−∞,∞] (4.24)

If Q and Ω are parameterized, this robust stability condition can be used to motivatebounds on these parameters. Let Q a low-pass filter parameterized by (α, ωc) of the form:

Q (s) =αω2

c

s2 + 1.415ωcs+ ω2c

(4.25)

Suppose a nominal system (identification to be done in a later section), and an impedanceGimp(s) = Kimpθ+Bimpθ, where here Kimp = 2 and Bimp = 1. By increasing α, the DC gainof the Q filter, the allowable uncertainty magnitude is shown in Figure 4.5. This directlyshows a tradeoff between performance and robustness, as Q → 1, the allowable systemrobustness decreases. Similarly, by increasing the bandwidth of the Q filter, the stabilityrobustness is impacted as seen in Figure 4.6. In practice, model uncertainty increases athigher frequencies, so this condition will be harder to meet as ωc, the roll-off frequency, getslarger. Finally, by jointly scaling the impedance parameters (e.g. a scalar gain in frontof Gimp), it can be seen in Figure 4.7 that increasing the rendered impedance increasessensitivity to model uncertainty. This matches the in-practice limit to rendered stiffnesswhich has been noted by many practitioners.


Figure 4.5: Allowable Uncertainty and α

Figure 4.6: Allowable Uncertainty and ωc


Figure 4.7: Allowable Uncertainty and Gimp

Design Parameters and Passivity

Next, the effect of design parameters on the passivity condition is explored. The passivitycondition is almost invariant to the DOB design parameters α and ωc, but is significantlyaffected by the virtual damping Bimp. Varying this parameter is shown in Figure 4.8, andas the virtual damping increases, passivity robustness increases, tolerating more uncertaintyat higher frequencies.

Incorporation of Design Constraints

With the DOB formulation, and an impedance controller of the form Gimp = Kimpθ +Bimpθ, there are 4 parameters which impact the evaluation of the robust passivity andstability constraints. As increasing Bimp improves robustness, but limited by the dampingvalue which can be satisfactorily realized in hardware, suppose that Bimp is fixed at it’smaximum value, found experimentally. For this system, this value is approx Bimp = .35. Atthis value, at a given stiffness (here, Kimp = 3), the bounds on controller parameters ωc andα can be found which satisfy the passivity and stability constraints for the experimentallyfound uncertainty. The bounds can be seen in Figure 4.9.

4.3 Series-Elastic Actuated System

Here, the effect of actuation dynamics on the DOB controller will be investigated on aseries-elastic actuated system [87, 67, 59]. Let the SEA, integrated to an inertial system, be


Figure 4.8: Allowable Uncertainty and Bimp

Figure 4.9: Controller Parameters which Satisfy Safety Conditions Bimp


Figure 4.10: Series-elastic actuator integrated to driven inertia

Figure 4.11: Inner-loop torque-mode control

as shown in Figure 4.10, where J is the motor inertia viewed from the output of the gearbox,M is the load-side inertia, φ and θ are their respective positions, and Ksp is the stiffness ofthe spring which couples them. Motor torque and interactive torque, τm and τint, drive thesystem. Under the proposed model,

K−1sp τsea = -

(M-1 + J-1

)τsea+J

-1τm-M-1τint (4.26)

where τsea = Ksp (φ− θ).

Torque-mode Control

This model under closed-loop torque control with control law τm = Ct (s)(τ dsea − τsea

)is

shown in Figure 4.11, where A (s) is defined as

A (s) =1

K-1sps

2 + (J-1 +M-1). (4.27)

The closed-loop dynamics can be found as

Acl (s) =τseaτ dsea

=Ct (s)A (s)

M + Ct (s)A (s)(4.28)

Aint (s) =τseaτint

=MA (s)

JM + JCt (s)A (s). (4.29)


Figure 4.12: Integrated inner loop control with DOB

DOB with Actuator Dynamics

The system with actuator dynamics under DOB control can be seen in Figure 4.12. TheDOB outer-loop is realized as in (4.21), with τact = τ dsea. Writing the expression from τint toθ under the controller shown in Figure 4.12

θ

τint=

GAclQ+ (1-Q)G (1 + Aint)

1−Q+GAclGimp − (1−Q)GAclG−1. (4.30)

When Q (s) = 1, the desired dynamics are still recovered, so performance robustness withan uncertain G can be preserved even with actuator dynamics. In practice, the additionof actuator dynamics will reduce the maximum |Q| which can be safely realized, reducingperformance of the DOB approach compared with an ideal force source.

Stability with Actuation Dynamics

Looking at the denominator of (4.30), and taking the same multiplicative uncertaintymodel shown in (4.3) and (4.4), a sufficient stability criteria can be found. If G, Acl and Aintare stable, the following condition must also be met:

|Acl|∣∣−QA−1

cl + (Ω +Q) (1 + ∆)∣∣ < 1 ∀ω ∈ [−∞,∞] (4.31)

Typically, Acl will have unity gain up to some bandwidth (i.e. |Acl(jω)| ≈ 1, ω ≤ ωc) androlling off above that frequency. To keep the term QA−1

cl small in the stability criteria, Qcould be a low-pass filter with cutoff frequency below that of the actuator.


4.4 General Dynamic Error Feedback

In some cases, the high-gain control inherent to the DOB is to be avoided. In this case,it is still possible to use direct measurement of interactive force to improve the accuracyof the rendered impedance. In the DOB impedance control, high-gain control corrects thedeviation from the desired closed-loop dynamics as

τdob =Q

1−Q(τint −Gimpθ) (4.32)

where τdob is the component of τact which can be attributed to the disturbance observer. Theexpression τdyn = τint−Gimpθ can be considered as the deviation from the desired impedancedynamics expressed as a force. A position representation of this error is θdyn = G−1

impτint− θ,which is the basis of admittance control. However, for a system with force as the primitiveinput (i.e. with direct control over the actuator, instead of an inner-loop position control),this force representation of the dynamic error can be more natural.

This dynamic error term can be compensated with a general compensator Cimp (s) togive the following control law:

τact = Cimp (s) (τint −Gimpθ) +(G−1 −Gimp

)θ (4.33)

This then gives the following:

θ(s)

τint(s)= G−1

imp

CimpN(s)

1 + CimpN(s)(4.34)

where

N(s) =GGimp

1−G(G−1 −Gimp

) (4.35)

This now resembles the classical feedback architecture of a compensator Cimp(s) in series witha plant N(s) under negative feedback. Traditional approaches to stability and performancecan inform the controller design (e.g. root locus). The design objective of letting τint/θapproach G−1

imp can be achieved by letting CimpN(1 +CimpN)−1 → 1. This feedback can alsobe shown to improve the disturbance rejection.

4.5 Design Within Constraints

To meaningfully apply the conditions developed for both passivity and stability ((4.23),(4.20), and (4.31)), several considerations must be made. First, a form for ∆ must bemotivated or found experimentally. In this section, frequency domain identification andmodel fitting is used to motivate a model and model uncertainty.

Once this has been found, the relationship between the design parameters Gimp, α andωc and the necessary passivity and stability conditions is not clear. For a given Gimp, a set


Figure 4.13: Model and Uncertainty for G(s)

of α and ωc can be found which meet the conditions. Similarly, for a fixed α and ωc, a rangeof impedances Gimp can be found which meet the conditions. The usefulness of either ofthese approaches will be dependent on the application (whether one impedance or a rangeof impedances is to be rendered). Although there is not a closed form relation betweenthese design parameters and these conditions, the dimension of the parameter space is smallenough that it can be easily discretized and computational approaches used to establishthese relationships.

Model and Model Uncertainty Identification

Identification of the response from torque τ to position θ can be done in a variety ofmethods - chirp excitation, frequency sweep or random excitation. In this identification,a range of frequencies and amplitudes was used to excite the system, by being passed asreference to the inner-loop torque controller (i.e. τ d = Aj sin(ωjt)). Identification at a rangeof amplitudes allows characterization of the effects of nonlinearities such as static friction.A least-squares sinusoid fit onto the input and output allows characterization of magnitudeand phase at a variety of frequencies, which can then be passed to a model-fitting program inMATLAB. The result is shown in Figure 4.13, where the experimentally observed responsesand fit model response are shown.

Note that the response from τ d to θ can also be extracted from the same data. Overlayingthe responses of θ

τdand θ

τ, the effects of the inner-loop torque control can be seen. In Figure

4.14, it can be that although in general the differences are not significant, the magnituderesponse from τ d is smaller at the lowest and highest frequencies experimentally validated.


Figure 4.14: Experimental Response from τ d and τ to θ

At low frequencies, static friction causes the realized torque to be less than the commandedtorque, and at high frequencies, τ d → τ begins to roll off, so the response is reduced.

For a given response Gj(ωj), a corresponding ∆j(ωj) such that Gjωj = G(ωj)(1 +∆j) exp(−iφ) can be found, where φ is a phase adjustment to ∆j.

∆j(ωj) =

∣∣∣∣∣ Gj

G(ωj)

∣∣∣∣∣ (4.36)

The resulting uncertainty can then be plotted, as seen in Figure 4.15. Just as with themotor model, uncertainty increases at higher frequencies. The resulting model is

G (s) =1

.233s2 + 1.43s(4.37)

4.6 Experimental Validation

The discussed controllers were implemented on a cable-driven series-elastic actuated sys-tem for validation. Additional details on the experimental setup can be found in [67]. AnATI Mini45 force/torque sensor was integrated to allow direct measurement of the interactivetorque. A picture of the experimental setup can be seen in Figure 2.4.


Figure 4.15: Experimentally Found Uncertainty

The DOB implementation used a Q filter of the form seen in (4.25), with values of α = .8and ωc = 50 (rad/sec) chosen to respect robust passivity and stability as seen in Figure 4.9.This Q allows compensator realizability (i.e. relative degree of total compensator expression).The dynamic feedback in (4.33) was realized with Cimp(s) = 2.5, a lowpass filter on the τintwith cutoff frequency 100 (rad/sec), and first-order differentiation of θ.

First, a quasi-static, manually applied interactive force was slowly increased, then de-creased to test the rendering of an almost pure stiffness. The comparison between a tradi-tional controller (4.11), DOB controller (4.21), and dynamic feedback (4.33) can be seen inFigure 4.16. The effects of coulomb friction on the load side are apparent for all implementa-tions, but greatly reduced under both DOB and dynamic feedback, where dynamic feedbackand the DOB have almost identical performance.

Next, a variety of stiffnesses were realized on the DOB system with the same inner-loopand DOB gains. These results are seen in Figure 4.17. Stiffnesses up to 10 N-m/rad could berealized, which is greater than the stiffness of the SEA spring (4.8 N-m/rad). The low stiffness(.2 N-m/rad) has oscillation, possibly from the stick-slip of the bowden cables, evident inthe response. At lower impedances (here, lower stiffnesses), the coulomb friction, which wasnot fully compensated, is more noticeable in the position response. The second test wasperformed by exciting the system with a variety of interactive torques for an extended period.Significant frequency content in the interactive torque was in the range 0-50 rad/sec. Thefrequency domain representation of τint and θ were used to find the experimental transferfunction, shown in Figure 4.18. The DOB provides a much more accurate rendering ofthe desired impedance, and the dynamic feedback controller has moderate performance atlower frequencies, but does not match the desired impedance as well at higher frequencies.


Figure 4.16: Stiffness rendering comparison

Figure 4.17: Variety of stiffnesses rendered with DOB approach


Figure 4.18: DOB vs Traditional in impedance rendering

The performance loss of the traditional controller can be partially attributed to the ratherlarge coulomb friction on the load side of this setup. In Figure 4.19, the same process wasused to render two different impedances on the DOB system (Kimp = 1.25, Bimp = .35and Kimp = 3 Bimp = .25 respectively). The DOB system was able to render these twoimpedances accurately.


Figure 4.19: Two different impedances rendered on SEA

69

Chapter 5

Load-Side Dynamics Learning

It has been seen in previous chapters how model uncertainty restricts interactive controlsafety and performance. Learning a more accurate dynamic model offers a means of directlyimprove performance and, by decreasing the effective uncertainty (e.g. ∆ in (4.9)), allow alarger set of controllers to meet stability and passivity robustness conditions.

A general schematic for an interactive robot is shown in Figure 5.1. A control policy,realized at the actuator with τ and θ measured is to render desired dynamics at τint, θ.To relocate the interaction port like this requires compensation of the load dynamics. Assuch, the performance of this approach is fundamentally limited by the accuracy of the loaddynamics model.

Even in cases where dynamics are not to be compensated, learning the load or envi-ronmental dynamics can be useful. This representation may encode information about theobjectives or intentions of the environment, which can be useful in determining robot behav-ior. For example, if a system-level objective is known a priori, an environmental model canallow the selection of robot responsive behavior which, when coupled with this environment,is desirable. The use of nominal human models are now being explored, notably [91], wherethe stiffness presented by a human determines a complementary stiffness on the robot.

Figure 5.1: Interactive System Model

The learning of load or environmental dynamics can be useful, but fundamental properties

CHAPTER 5. LOAD-SIDE DYNAMICS LEARNING 70

of interactive systems can make this difficult. Many of these systems present unconventionaldynamics which may make traditional modeling difficult. For example, consider the ex-oskeleton shown in Figure 2.6, where bowden cables exert an elastic force which resists theirbending, and a subject wearing the exoskeleton adds additional dynamics. This limits theapplication of traditional serial-revolute manipulator identification techniques.

In general, interactive robots are intended to be coupled with additional systems - tools,payloads, collaborators, or other manipulanda with dynamics unknown a priori. In somecases, it is desired to bring these additional dynamics into the system model, such that theycan be compensated and additional input reacted to. For example, a robot compensatingthe weight of a workpiece allows a human collaborator to manipulate it more freely.

All modeling techniques, which, for the argument here includes model-free data-driventechniques, rely on knowing all inputs and output of a system. Currently, this is done withphysical separation such that all input to the system of interest arises from a controlledsource. However, as robots move from laboratory environments towards well-connected,real-world environments, isolation of the system of interest may be difficult. In some ap-plications, isolation may not even be possible - here, a subject wearing an exoskeleton willexert additional muscular torques, and the passive dynamics of the subject’s limb cannot becoupled without introducing this additional input. In other cases, it may simply be moreconvenient to have additional input - for example to manually lead the robot through a partof the state space during model identification.

To varying degrees, model identification relies on a roboticist applying a priori knowledgethrough design of both experiment and environment to isolate relevant parts of the model.However, current practices are time consuming - each set of interactive dynamics requiringits own set of experiments and analysis. To address these challenges, here these dynamicsof interest are treated as multimodal, having distinct dynamic modes which are transitionedbetween. There is evidence that multimodal model identification and switching also informshuman motor control [108], suggesting this approach can help these robots achieve moregeneral, human-like interaction.

Model Learning

Model learning has a rich history in the controls and robotics communities, and severaltechniques are now well-established for traditional industrial manipulators: adaptive control,iterative learning control, and nonparametric learning.

Adaptive control uses input and output data of the system to do either online identifi-cation of model parameters or direct tuning of the controller [78]. By identifying a model(function of the state), this can offer improved performance even on novel trajectories. How-ever, adaptive techniques require a structured model or controller, making nonlinear frictionand non-idealized gravitational terms difficult to compensate.

One model-free learning approach is Iterative Learning Control, which uses historicalinformation on error to improve future performance [21]. This has been very successful inindustrial robots, but faces challenges for use with interactive robots. Non-repetitive pertur-


bations may be introduced by the environment, which should not be learned or compensatedin the future. Although some have explored iterative learning control with non-repetitivedisturbances [65], this is still a fundamental challenge in this field. Furthermore, by con-struction, this approach does not easily generalize to new trajectories as it uses a time anditeration domain representation of error, instead of being state-based.

Nonparametric modeling techniques have found application to inverse dynamics learning[81]. By being state-based, these models can generalize from historical data to new trajecto-ries, although the degree to which it can generalize is application and parameter dependent.Being constructed from historical data, they can capture more difficult non-linear effects anddon’t require a priori knowledge of model structure.

Multimodal identification has also been treated in some of these frameworks. Adaptivecontrol has treated multimodal models [78, 77], but again requires a priori knowledge onmodel structure for each mode. Nonparametric models can be extended to multimodalmodels [109, 46, 71, 99] or even continuously varying models [94]. However, the extension ofthese to learning multimodal inverse dynamics is sparse [52, 94].

There is also some prior work on classification of external perturbations for interactiverobots, such as under cyclic system motion [10] or in collision [36]. Here, perturbation is notidentified with an existing model, but occurs during model identification and is separatedfrom the underlying model within the multimodal framework.

The contribution in this chapter is a modeling approach which addresses both multi-modal inverse dynamics and intermittent external perturbations under a unified framework.The aims of model learning in interactive control are introduced, then necessary conditions(on robot inverse dynamics and observed states) for achieving this objective with regres-sion techniques are developed. An algorithm for the batch processing of unlabeled multi-modal data with intermittent external perturbation is introduced, based on the stochasticexpectation-maximization algorithm. The passivity of the resulting overall control policyis shown analytically. Finally, experiments validate both the model identification, and thepassivity of the resulting control policy.

5.1 Impedance Controlled Interactive Robots

A general schematic for an impedance controlled robot interacting with the environmentis shown in Figure 1.2.

For now ignoring actuation dynamics, and taking τ as the controlled input to the system,the robot’s inverse dynamics can be written as

M (θ) θ + C(θ, θ)θ + g (θ) = JTint (θ)Fint + τ. (5.1)

The interaction force Fint, modulated by the Jacobian Jint(θ), is the environmental input to


the system. The objective is to realize desired Cartesian space or joint space dynamics of

M jimpθ +Bj

impθ +Kjimpθ = JT (θ)Fint (5.2)

JT (θ)(M c

impx+Bcimpx+Kc

impx)

= JT (θ)Fint. (5.3)

where [·]jimp represents the joint space impedance parameters, [·]cimp the cartesian spaceimpedance parameters and x represents the end-effector position achieved by the manip-ulator. These equations represent joint (5.2) and Cartesian space impedance control (5.5)respectively, and the impedance parameters are chosen with respect to system-level goals.In trajectory tracking, θ or x can be regarded as the deviation from the desired position.

In early realizations of impedance control the entire robot dynamics were directly com-pensated with a feedforward term (e.g. τ d = −M(θ)θ...) [49]. Although in theory thisallowed assignment of the apparent inertia, damping and stiffness of the manipulator, itrequired direct measurement of interactive force and measurement of acceleration, makingit difficult to realize. Letting the inertia be the manipulator inertia, Mimp = M (θ) improvesthe practical realizability, and currently most implementations simply compensate gravita-tional terms [4], leaving the relatively minor Coriolis terms uncompensated. If the gravitymodel is g (θ), with the true gravitational terms g (θ), the cartesian impedance control lawis expressed as:

JT (θ)(Kcimpx (θ) +Bc

impx (θ))

+ g (θ) = τ (5.4)

Under this control law, the equilibrium equations of (5.5) become:

JT (θ)Kcimpx (θ) + g (θ) = JT (θ)Fint. (5.5)

where g is the error in the gravitational model g = g− g. This error directly shows up in theequilibrium conditions, and affects the system just as an external force. Especially as thestiffness Kc

imp decreases, or more generally a low impedance is rendered, a large deviationfrom desired position can occur. This is the manifestation of the fundamental limitation ofimpedance control on an uncertain system as in (4.9). This sets in-practice limitations tothe lowest stiffness which can be rendered on a robot, and due to the position dependencyof the error g, this can give unexpected performance in different regions of the joint space.

5.2 Learning of Dynamics

To learn a more accurate compensation for the inverse dynamics, nonparametric statis-tical learning [92, 52] among many other data-driven approaches have been explored. Forinteractive robots, this dynamic compensation must be state-based, such that even when thesystem is perturbed by unmeasured external input, acceptable compensation is achieved.In general, such techniques regress a function τ = f(θ, θ, θ), where τ is the motor torque


(trained from previous values of τ) and the arguments θ, θ, θ in theory allow complete evalu-ation (5.1) when there is no external forces, giving conditional independence to a time-seriesof samples τ1, τ2, . . . .

Note that if this regression is successful (i.e. τ(θ, θ, θ) = M(θ)θ + C(θ, θ)θ + g(θ)), acontrol policy of τ = τ(θ, θ, θ)− τ gives:

τ = JTintFint (5.6)

The additional input τ can be designed to realize, e.g. the desired impedance. Ignoringquestions of realizability, reducing the dynamic equations of (5.1) to an algebraic one (5.6)conceals the dynamic behavior of the system. Errors in the inverse dynamic compensationwill generate dynamic terms f(θ, θ, θ), a dynamic system with unknown characteristics andunverifiable stability. Even analytical techniques which seek to significantly change dynamicsystem behavior, e.g. feedback linearization, does not seek to change the order of the plant.

To more generally discuss learning dynamic compensation, let the system dynamics beexpressed as

ξ = f(ξ) + g(ξ)u. (5.7)

Let a system model be identifiable if

∃u = u(ξ,D) + u s.t. ξ = u. (5.8)

where the feedforward compensation term u is a function of the current evaluation of ξ, as

well as the historical data of D =u1, ξ1, ξ1, . . . , uT , ξT , ξT

. Robustness issues, similar to

those from the reduction to an algebraic relation in (5.6) can arise, but by using notationwhich explicitly shows the dynamics, a u(ξ) can be designed to address these issues.

Although in general the convergence of a regression u(ξ,D) to achieve (5.8) will rely on the

properties of the regression technique used, it is also necessary that the mappingξ, ξ→ u

is unique. Suppose that different inputs u1 6= u2, realized at the same state ξ1 = ξ2, achievethe same instantaneous motion ξ1 = ξ2. Then, by direct consequence f(ξ1) 6= f(ξ2). Thatis, f is not a function, and therefore cannot be exactly canceled by a regression function and(5.8) cannot be guaranteed.

Several factors can cause the loss of uniqueness for the system dynamics. One is theexistence of unobserved states. If these states take different values at the same value of theobserved state, the apparent dynamics may change. If these hidden states are slowly varying,this may be well suited for adaptive or online techniques which capture the dynamics nearthe current value of the hidden state. Additionally, loss of uniqueness can also occur whenunknown input to the system occurs.

Even if the system is isolated (no additional input) and well-characterized (no hiddenstates), another source of non-uniqueness in real robots is coulomb or static friction, suchas in Figure 2.15a. By being discontinuous at θ = 0, such systems do not admit uniquetrajectories (i.e. are not Lipschitz continuous). Any torque below the breakaway value willinduce no motion. Techniques will be discussed later in the text to address this, but itremains a theoretical and practical limitation of truly model-free dynamics learning.


Multimodal Dynamics Learning

For a multimodal system, the dynamics are assumed to switch between multiple dynamicmodels; i.e. τ = fk (x) where k ∈ [1, . . . K] indexes the possible dynamic modes. Given a data

collection D = τ1, x1, . . . , τT , xT, where here xt =[θt, θt, θt

], assume that these samples

are independently and identically drawn as

τt ∼ p (τ |xt, wt,Θ) (5.9)

where wt ∈ [1, . . . , K] is a latent indicator variable showing the membership of sample t, andΘ = Θ1, . . . ,ΘK, where Θk parameterizes the kth distribution.

Here, Gaussian processes are used to model each mode’s inverse dynamics [98]. Otherwork has offered certain improvements for modeling inverse dynamics by using LocallyWeighted Projection Regression [123], Incremental Support Vector Machine [68] and Infi-nite Mixture of Linear Experts [29], but the simplicity of Gaussian processes allows thefocus on phenomena which are largely invariant to specific regression technique.

Under this assumption, each data point takes the following distribution:

p (τt|xt, wt = k,Θ) = GP (xt,Dk,Θk) (5.10)

= N (µt,Σt) . (5.11)

where GP denotes the posterior distribution of the Gaussian Process, based off data Dk =τt, xt : wt = k, the subset of data labeled to this dynamic mode. This evaluates toa normal distribution, where µt and Σt are implicitly functions of the mode data Dk andparameters Θk.

Under a slight abuse of notation (as the distribution will also be conditioned on the state)the clustering and parameter fitting can be framed as a maximum likelihood problem,

maxΘ

p (D|Θ) =∑w∈W

p (D|Θ, w) p (w) (5.12)

where W is the set of all admissible combinations of clusterings. The expected completelikelihood and membership probabilities can be written in a relatively straight-forward ap-plication of the Expectation-Maximization algorithm:

q (w|D,Θ) ∝∏k

p(Dk|w,Θk)p(w) (5.13)

〈l (Θ;D, w)〉q =∑w∈W

q(w|D,Θ) log p (D, w|Θ) (5.14)

where Dk is the partition of data belonging to the kth mode. Note that neither step of theEM algorithm is computationally feasible. The set of possible clusterings W is of size KT ,


which will be prohibitively large for even modest data sets. This differs from classical appli-cations of EM to clustering as in a nonparametric model, the likelihood of a sample’s latentmode depends on the classification of other data, so mode membership is not independentconditioned on the parameters. To address this issue for multimodal nonparametrics, others(e.g. [109]) use Markov-Chain Monte Carlo techniques, such as Gibbs sampling.

As the full distribution q(w|D,Θ) is prohibitively large, some MCMC variants of EMsample from the distribution q(w|D,Θ) to approximately marginalize the expected log like-lihood (5.14). The limiting case of pulling a single sample from q(n), and using this has beenformalized as the SEM algorithm [18], where demonstrations of convergence (in probabil-ity) can be found therein. Under this approach, let w(n) denote a sample from the currentparameter estimates Θ(n), and the sample likelihood written as:

lw(n)(D|Θ) =∑k

∑t∈Tk

log(p(yt|Dk,Θk, w

(n)t

))p(w(n)|D,Θ(n)) (5.15)

where Tk is the set of time indices which are labeled to the kth mode.

Sampling Latent Class Membership w

Direct sampling from (5.13) is again not computationally feasible. Generating thesesamples can be done with Gibb’s sampling, alternatively viewed as a leave-one-out clusteringapproach. This gives conditional distributions of

p(wt|D, w−t,Θ) ∝ p(wt, yt|D, w−t,Θ) (5.16)

= p(yt|D, w,Θ)p(wt|w−t) (5.17)

where w−t denotes the set of all w excluding wt.The evaluation of p(yt|D, w,Θ) = GP(xt,Dk,Θk) is a straightforward evaluation of the

distribution returned by the gaussian process. If the mode wt is assumed to be indepen-dent and identically distributed, the conditional distribution p(wt|w−t) = p(wt) is trivial toevaluate. However, different priors for wt can be applied to characterize the mode switchingbehavior. If additional sensing gives inference to the mode, this can be incorporated bymaking this conditional p(w|Dnew). Here, a simple time correlation will be assumed as:

p (wt|wt−1) =

π wt = wt−1

1− π wt 6= wt−1

(5.18)

This gives overall probability

p (w) = πc0 (1− π)T−c0 (5.19)

c0 =∑

I (wt = wt−1) . (5.20)

Where I is an indicator function of one when it’s argument is true and zero otherwise. Thiscan be easily normalized to find p(wt|w−t) as needed for (5.17).


Parameter Identification

Straightforward maximum likelihood estimation of parameters is again not tractable dueto the coupling of the latent variables. Given a sample of the classification w(n), parameterscan be estimated as follows:

〈l(D|Θ)〉w(n) =∑k

l (Dk|Θk) p(w(n)|D,Θ(n)) (5.21)

where the data partions Dk are induced by w(n). Although the total marginal likelihood ofa set of data drawn from a Gaussian process can be easily computed (e.g. [98]), here thehold-one-out likelihood is used; as in practice this has given better performance:

l (Dk|Θk) =∑t∈Tk

− (τt − µt)T Σ−1t (τt − µt)T (5.22)

where µt and Σt the posterior mean and covariance returned by evaluating Gaussian process

GP(Dk,Θ(n)

k

). As the parameters Θk are all scalar, they can be searched over with a

numerical gradient descent in each iteration of the algorithm.

5.3 Disturbance Identification

This framework can also be used to rigorously motivate a means of separating intermittentdisturbance from model uncertainty. This is motivated by an exoskeleton application, wherea subject couples their limb to the exoskeleton, as in Figure 2.6. The subject’s limb introducesadditional gravitational terms to the load-side dynamics of the exoskeleton, which should belearned and compensated to improve the system performance. However, additional musculartorques intermittently introduced by the subject may limit the learning of these dynamics,especially as the subject is a non-human primate and muscular relaxation cannot be enforcedor verified. The gravitational terms are functions of the measured state (load-side position),and are therefore viewed as model uncertainty. Additional forces, arising from the muscularexertions of the subject will be taken as exogenous input.

Let the general dynamic system from (5.7) be extended to include unknown input d. Thesystem dynamics become:

ξ = f(ξ) + g(ξ) (u+ d) (5.23)

The exogenous input d is not measured, but by being additive with u will affect thesystem evolution. As this input is unmeasured (but it is presumed that u, ξ, and ξ are),this will cause an instantaneous deviation in ξ, leading to an evaluation of the regression

function equivalent toξd, ξ

→ u, where ξd = ξ − g(ξ)d. If the regression function is

unique in the first argument, i.e. ξ1 6= ξ2 ⇒ τ(ξ1, ξ

)6= τ

(ξ2, ξ

), this perturbation will


cause an evaluation which differs from the measured input, i.e. τ(ξd, ξ

)= ud 6= u. Typical

regression techniques assume some distribution on samples ut to accommodate noise andregression error, however if ud significantly diverges from the u, it is more likely the resultof an external perturbation.

Note that if there is a state dependency of d, i.e. d = f(ξ), ideologically it could be moreaccurately viewed as model uncertainty. When the disturbance has state dependency, it willthen be captured with the regression technique, and ud → u as this disturbance’s effect islearned by the regression technique. This provides the ideological and practical statementthat dk ⊥ ξk, the time series of disturbances should be statistically independent of the state.

To formalize the distinguishing of u and ud, let two dynamic modes be nominal inversedynamics, then with an added disturbance. Under assumptions of the exogenous input’stime-series characteristics, these two modes can be clustered, allowing the inverse dynamicsto be separated and compensated. Assume the dynamic behavior in these two modes can betaken as

τt =

f (xt) wt = 1

f (xt) + dt wt = 2(5.24)

where f (xt) is the inverse dynamics and disturbance dt. If f is fit with a Gaussian process,each evaluation of f(xt) will yield a conditional distribution of τt. Let p (τt|xt, wt = 1,Θ) ∼N (µt,Σt), where µt and Σt are the mean and covariance returned by the Gaussian Process.

If the disturbance is assumed to be independent and identically distributed with dt ∼N (0,Σd), this gives the distribution

p (τt|xt, wt = 2,Θ) ∼ N (µt,Σt + Σd) (5.25)

When the system is perturbed, it will deviate from the nominal dynamics, here characterizedby the additional covariance in it’s distribution.

Disturbance Parameter Identification

An appropriate value of Σd is key to the performance of sampling and parameter updatingas in (5.17). Building off the SEM framework used in (5.21), let the identification be doneby maximizing the following expected likelihood:

〈l (D2|Σd)〉w(n) =∑t∈T2

−p(w(n)t |D,Θ(n)) τt

T (Σt + Σd)−1 τt (5.26)

where τt = τt − µt, µt and Σt are the posterior mean and covariance of Gaussian process

GP(D1,Θ

(n)1

)evaluated at xt. Again, if Σd is low-dimensional (e.g. Σd ∝ I), this can be

easily searched over numerically.


5.4 Passivity in Feedforward Dynamics Compensation

For interactive robots, the safety must be guaranteed when the robot is coupled to un-known environments, not just stability in isolation (e.g. Lyapunov approaches). The mostcommon approach in interactive control is to show the passivity of the robot [49, 4]. Ifthe robot and controller are passive, they can be coupled to an arbitrary environment -payload, working surface, without compromising coupled system stability. The intuitive in-terpretation of a passive system is one which has a lower bound on the energy which can beextracted through the interaction port. However, feedforward techniques can be problematicfor stability, much less passivity.

In particular, friction compensation can be difficult. Viscous friction compensation, e.ga term of the form τ = vθ is directly injecting power into the system (τ · θ > 0), and ifv exceeds the real viscous friction, system energy may grow without bound. Furthermore,physical damping is key to providing robustness in interactive control when discretizationor model uncertainty is considered [23, 44], so it’s compensation may be detrimental toreal-world performance

Demonstrating total system passivity is a major challenge to model-free or nonparametriclearning approaches which, by not having a model, limit the a priori analyses which can beundertaken. Here, let the robot dynamics be as shown in (5.1), with a control policy of:

τ = τff (θ) +Kimpθ +Bimpθ (5.27)

A system with state x, dynamics x = f(x, u) and output y = g(x), can be shown to bepassive with respect to y, u if ∃ a storage function S(x) : X → R which is bounded frombelow, and Sx(x) · f(x, u) < 0, where Sx denotes the partial derivative of S(x) with respectto x.

When τff (θ) is chosen from a Gaussian process using a square exponential kernel,

k(xi, xj) = σy exp(−‖x1 − x2‖2

2l−2)

+ σnδij (5.28)

where δij = 1 ⇐⇒ i = j and a zero mean function, the feedforward torque can be writtenas:

τff (θt) = KTt K

−1D y (5.29)

Kt =

σyexp

(− (θt−θ1)T (θt−θ1) l−2

)...

σyexp(− (θt−θT )T (θt−θT ) l−2

) (5.30)

where σy and l are parameters of the GPR, and KD denotes the kernel induced by data setD, and y is the previously observed values in D.


Let the following function be defined:

E (x) =

√

2πσ−1y l2erf

(− (x−x1)T (x−x1)

l2√

2

)...

√2πσ−1

y l2erf(− (x−xT )T (x−xT )

l2√

2

) (5.31)

where erf is the Gauss error function. By construction ∂E(x)∂x

= Kt. As the error function isbounded (above and below), any finite-valued training data means that a storage functionSf (θ) = E(θ)KDy is bounded from below.

Note that the passivity of two systems in parallel can be directly concluded from thepassivity of each. As the impedance controller terms (Kimpθ + Bimpθ) are by constructionpassive, only the passivity of the feedforward term will be shown below. Let the followingstorage function be defined:

S(θ, θ) =1

2θTM (θ) θ + E(θ)KDy + Vg(θ) (5.32)

S(θ, θ)

= −θT(M (θ)− 2C(θ, θ)

)θ + θTJT (θ)Fint (5.33)

where Vg (θ) is the gravitational potential energy of the system, such that ∂Vg(θ)

∂θ= g(θ).

As(M (θ)− 2C(θ, θ)

)is skew-symmetric, this quadratic term is zero. Thus we achieve

S ≤ θT τint, and the robot is passive.Note that there are some limitations to this passivity approach. It shows passivity for

only regressions of the joint angles, although this is often sufficient for many interactiverobot applications it may limit the higher-frequency performance. However, to regress onlyon position breaks the conditional independence of τt. Fortunately, the overall regressiontechnique can still include higher-order terms as θ and θ, it is just Kt(θ), the evaluationdependent on current state, which can only depend on θ. In the implementation here, thisreduces to the simple evaluation of the regression at τ(0, 0, θ).

A further limitation of this passivity argument is that it is only valid when an offlinedataset D is used for inference, with no obvious extensions to a general online learningstrategy.

5.5 Experimental Results

The efficacy of this approach will be tested by validation on a test actuator, seen inFigure 5.2. An integrated force sensor allows validation of the mode classification, and theinertia of .73 kg acts as the load side dynamics. Data collection is done with a high-gain PDposition control in quasi-static exploration of the range-of-motion. Data was downsampledto 20Hz for the model fitting.


Figure 5.2: Experimental Setup

Classification

To test the ability to identify with perturbations, the system was manually perturbedduring the identification process. The results can be seen in Figure 5.3. Using the forcesensor data to determine when the system was unperturbed, 192 of these 235 data pointswere correctly identified as being unperturbed. Of data that had significant external force,14 of the 92 samples were incorrectly identified as being from the unperturbed system. Theseresults were obtained with the final parameters seen in Table 5.1.

Parameter Value

l .42σy .05σn .14Σd .07

Table 5.1: Final GPR Parameters

Coulomb Friction

To account for the significant coulomb friction in this setup, an additional feature was

added to the state representation: sgn(θ)

. Note this violates the conditions for passivity,

but as this feature is constant in θ, it’s ability to inject energy is limited. Seen in Figures 5.5and 5.4, the discontinuity of coulomb friction can be hard for Gaussian processes to regress(Gaussian processes are continuous and smooth). Introducing this additional feature helpswith the separation of these two data sets.


Figure 5.3: Rendering of Zero Impedance

Also note the load-dependent coulomb friction. This is just one of many position depen-dent effects which can make analytical models difficult to implement.


Figure 5.4: Regressed Model, Nominal State x =[θ, θ, θ

]

Figure 5.5: Regressed Model, Augmented State x =[θ, θ, θ, sgn(θ)

]Impedance Rendering

The identified model is then used for online compensation. In Figure 5.6, the resultof rendering a zero impedance can be seen. This system seeks to present no resistance to


constant velocity input (the uncompensated inertia means it will resist acceleration). TheGPR model, which captures the position-dependent static friction term is better able tocompensate for this term.

Figure 5.6: Rendering of Zero Impedance

In Figure 5.7, the rendering of a pure stiffness can be seen. Again, the GPR compensatedsystem provides improved performance. However, the perfect compensation of static frictionis not achieved.

Passivity Validation

Although a rigorous demonstration of passivity experimentally is not feasible (cannotrealize all admissable inputs over arbitrary time period), by demonstrating the passivity ofthe system in a demanding application, passivity can be suggested. Here, broad-spectrumfrequency content is introduced by delivering impulses with a rubber mallet to the load side.Instantaneous power flowing into the load side from the actuator can be found from τ , whichis directly measured and θ, which can be found through differentiation and filtering. Theinstantaneous power flow and total energy into the actuator can be seen in Figure 5.8. Theactuator absorbs energy through the impact even as it realizes an impedance and feedforwardcompensation, showing the overall control policy is passive.

5.6 Conclusion

As always, sensing and actuation accuracy limit the performance which can be achievedby feedforward compensation, however there is a theoretical and practical limit involved for


Figure 5.7: Rendering of Pure Stiffness Kimp = 3.5

learning inverse dynamics.As developed in Section 5.2, static friction limits the ability to model the inverse dy-

namics. Of course, there are several ways to work around this. Static friction has longbeen separately modeled (or assumed to be well-compensated) in robot control, then sub-sequently removed from the (inverse) dynamics. When the direction of desired motion isknown, a breakaway torque can be applied to induce motion. When the desired trajectory isknown a priori, this can be easily inferred, however in interactive tasks the desired directionof motion is not always known.

Again, there are several approaches which can work around this, such adding high-frequency excitation (‘dither’) to the input torque so a smaller external force is neededto reach slip conditions [11]. Alternatively, additional sensing (e.g. a force sensor) can beintroduced to improve responsiveness. However, in general the compensation techniquesrely on the a priori knowledge of the designer, so claims that this learning approach is trulymodel-free are only partially true.


Figure 5.8: Power and Energy into Actuator During Environmental Impact

86

Chapter 6

Conclusion

Model uncertainty sets fundamental limitations for the performance of all control systems,and manifests in several significant ways for interactive robots. The complex robot dynamicsintroduced to realize interactive behavior (e.g. joint flexibility) introduce additional sourcesof model uncertainty, and bilateral actuation exposes the uncertainty in inverse gearboxdynamics. Not only is model uncertainty arising from more sources, it must not compromiseguarantees of safety. Interactive performance is also directly affected by model uncertainty;as interactive systems become more sensitive to environmental input, sensitivity to modeluncertainty is also increased. These factors make the characterization and analysis of modeluncertainty especially important to the realization of interactive systems.

Model Limitations

The accuracy of models, or tractability of a system for modeling has limitations whichare problem-specific, but most can be characterized as follows.

Hidden States

Hidden states imply that the complete dynamic behavior of the system has not beencaptured. Classical examples of hidden states which often appear inH∞ control is the higher-order vibrational modes of a complex structure. In traditional position-controlled flexiblejoint robots, hidden states have been the load-side motion (only motor-side measurementsavailable). Actuation and sensing dynamics, especially those with lower bandwidth (e.g.SEAs) can also give rise to hidden states which are significant. Other traditional sourcesof model uncertainty such as hysteresis, time-varying parameters and more complex frictioncharacteristics can also be viewed as arising from hidden states, as the state by definitionmust render current system behavior independent of history.

Sometimes states which are observable are neglected to make the model tractable foranalysis or controller synthesis. Complex models can limit the intuitive insight which thedesigner uses to inform the higher-level design choices in controller synthesis. In other cases,

CHAPTER 6. CONCLUSION 87

it is simply not easy to observe these states. The hidden state may be only loosely correlatedto observable output, or sensor noise and other practical effects can limit the accuracy ofinference.

External Input

Classical model identification seeks to isolate the system in question, with the inputs andoutputs well-measured. For some interactive systems, it might not be possible to isolate thesystem of interest, for example a subject wearing an exoskeleton which is realizing musculartorques. As interactive robots move out of the relatively controlled environment of labora-tories, it may be increasingly difficult to isolate systems for identification. This additional,unobserved input to the system is a fundamental challenge to relating inputs to outputs.

6.1 Contributions

This dissertation has examined the sources and implications of model uncertainty on theperformance and safety of interactive robots. Broadly speaking, these considerations aremade to make the performance and safety limits explicit and allow safety constraints to beprojected back to controller parameter bounds, to assist in design of the inner and outer-loopcontrollers.

Actuation Robustness

To provide guarantees of safety in interaction with a general environment, the passivityof the controlled system should be shown. A novel demonstration of flexible-joint passivitywith impedance rendered from the load-side behavior is developed.

In general, passivity is both conservative and not sufficient in several real-world appli-cations. Understanding why coupled stability is lost is key to developing systems whichcan overcome these limitations if possible, or avoid them if not. Towards this, the passivityof a system which has model uncertainty was examined. The above condition for passiv-ity extended to hold over a class of systems when parameterized by a bounded magnitudemultiplicative uncertainty. A result showing that the impedance can be reduced while main-taining robust passivity is developed, simplifying the application of this condition. Theexplanatory power of this robust passivity condition is validated in simulation and experi-ment, and explicit constraints on a general compensator and parameter bounds for a fixedstructure controller developed.

Load-Side Robustness

Even with passive and accurate actuation of a system, model uncertainty sets funda-mental limitations on the accuracy of the presented dynamics. A fundamental bound to


system performance under position-based impedance control is developed, then it is shownhow introducing direct force sensing allows improvement beyond this limitation.

A controller architecture capable of achieving robust regulation of an uncertain systemto the desired dynamics is presented. As performance often comes at the cost of stabilityrobustness (and, as is shown, passivity robustness), explicit conditions on the controllerparameters are developed which respect these constraints.

Load-Side Modeling

Model uncertainty manifests in both direct and indirect limitations to performance. Toimprove the performance of interactive systems, improving model accuracy can help, butseveral limitations arising from hardware and application are considered. The use of anonparametric statistical learning technique based on state-space regression is developed.In particular, the developed approach allows the separation of intermittent perturbationsas well as the clustering and identification of separate dynamic modes in a unified frame-work. Passivity of the resulting feedforward control law is demonstrated, and experimentallyvalidated. The additional properties of this technique are then validated (separation of per-turbation, multimodal classification), providing a means of improving models for interactivesystems.

6.2 The Future of Interactive Robots

Discussed in this dissertation is only a small portion of the problems currently facingthe realization of physically interactive robots. Interactive robots will continue to be re-fined, growing more capable of realizing acceptable performance in rendering a wider rangeof dynamics in more general environments. Some applications will likely have design re-quirements which necessitate the development of application-specific hardware and control,however, considering the (relatively) limited dynamic range and proprioceptive accuracyof humans, it is likely that a single hardware platform will be able to achieve human-likedynamic behavior over a significant range of tasks.

Once such a platform is realized, selection of proper interactive dynamics to achievesystem-level goals becomes the next major challenge. One approach for this is the incor-poration of a nominal environment model, such that desired coupled system behavior canbe predicted by appropriate selection of robot behavior. Modern, data-driven modeling ap-proaches (many of which are ‘model-free’) promise to greatly expand the ability to describecomplex systems, and may help develop environmental models in more challenging appli-cations. However, these approaches may need to be embedded in frameworks which allowrigorous guarantees on safety. Furthermore, to form compact and relevant models for de-sired systems in the well-connected real world, these modeling techniques must be capableof identifying systems which are driven by additional (unknown) input.


Many questions remain in realizing systems which are less reliant on the a priori knowl-edge of their creator and the careful arrangement of the robot’s environment. The technicalaspects of these questions are being enthusiastically explored by sectors of academia, indus-try and government. How these technological advances are integrated to society, and theensuing social impact remains to be seen.

90

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