Why to Use an Articulated Vehicle in Underground Mineing Operations

8/3/2019 Why to Use an Articulated Vehicle in Underground Mineing Operations

1/6

Proceedings of the 1999 IEEEInternational Conference on Robotics & AutomationDetroit, Michigan May 1999

Why to use an Articulated Vehicle in Underground MiningOperat ons?Claudio Altafini *

Optimization and Systems TheoryRoyal Institute of Technology, Stockholm, Sweden

[email protected]

AbstractEngineers i n the underground mining in dustry haveknown for a long time that an articulated vehicle ispreferable to a normal truck for the navigation in thenarroui e nvironme nts of an underground mine becauseof its higher maneuverability. The two advantages ofthe articulated configuration can be explained i n term sof degrees of freedom in the selection of the directionsneeded to span the whole tangent space and of thereduced gap between the trajectories followed by thewheels of the truck when steering. Both characteris-tics can be used in a constructive way for path planning

and navigation purposes.

1 IntroductionThe action of transporting the material from the

stope to the dumping point of an underground mine isperformed by a truck called LHD (Load-Haul-Dump).The LHD is an articulated vehicle composed of twobodies connected by a kingpin hitch. Each body has asingle axle and the wheels are all non-steerable. Thesteering action is performed on the joint, changing theangle between the front and rear part by means ofhydraulic actuator s. Both the shape and the steer-ing mechanism are intended to improve the maneu-verability of the vehicle. The effect of the actuatedarticulation is twofold: the truck can steer on placei.e. the orientation of the vehicle changes varying thesteering angle alone and the width spanned by thevehicle when turning is reduced with respect to, forexample, a car-like vehicle. Both properties descenddirectly from the geometry of the articulation and canbe explained considering the kinematic model of thevehicle.

'This work was supported by the Swedish Foundation forStrategic Research through the Center for Autonomous Systemsat KTH

The articulated truck is an underactuated drift-freenonlinear system with two inputs, which was provento be controllable in [6]. The steering on place can bejustified looking at the vector fields associated withthe two inputs. In particular, when comparing with acar-like vehicle, it turns out that the articulated con-figuration has as consequence a richer range of possi-ble maneuvers that can be explained in terms of inde-pendence of the higher order Lie brackets of the vec-tor fields i.e. with the presence of an extra degreeof freedom in the selection of the directions needed tocomplete the tangent space of the configuration space.This degree of freedom can be used to optimize a costfunction in a path planning problem. Here we providea simple example where the cost to minimize is thenumber of elementary maneuvers required to reach agiven point in the configuration space.

Also the second characteristic can be explained con-sidering the geometry of the vehicle. In general, in amultiaxis wheeled vehicle, a nonnull steering angle im-plies that the wheels (or better the midpoints of theaxles) follow different trajectories. This fac t becomesa relevant problem when the free space in which thevehicle is allowed to move is limited, like in a narrowroad or in an underground tunnel. It is intuitivelyeasy to understand that a vehicle without articulation,say a car-like vehicle, would be more cumbersome i.e.would span a larger area than the LHD when turn-ing. For the articulated vehicle case, the off-trackingbetween the trajectories can be easily calculated insome situations (see also [4]). The main problem forthe autonomous navigation of the mining truck is to beable to follow the tunnel keeping a safety margin fromboth walls. Navigation of the truck requires properinteraction with the environment: in this case the en-vironment (the tunnel) can be modeled as a path tofollow and the proper criterion (keeping the middle ofthe tunnel) can be reformulated as reducing the off-tracking of the whole vehicle from the path. This idea,

0-7803-5180-0-5199$10.000 1999 IEEE 3020


2/6

formulated in [3],finds here its most suitable applica-tion because of the limited width of the workspace ofthe mining truck. All the several approaches proposedin the literature t o solve the pa th following problem forwheeled vehicles are essentially based on the selectionof a single point of the vehicle and on the definitionof a tracking criterion for this guidepoint. The taskof the controller, then, is to have the correspondingtracking error converging to zero. Here the proposedsolution consists in redefining the tracking error of thepath following problem not based only on one singledistance but on the sum of the signed distances ofthe midpoints of both axles of the vehicle from theirorthogonal projections on the path. Stability can beproven locally for paths of constant curvature.

Finally, these being the two advantages of havingan articulated configuration, it is also easy to see adrawback: as consequence of the central joint there isno direction of motion in which the open-loop systemhas a stable equilibrium point: in a sense that willbe clarified below the system always behaves as a carmoving backwards. Therefore such a configuration isuseful only for applications in which the speed rangeis quite low, like mining, earth moving, forest industryor similar.

2 Steering on placeA typical configuration for a mining truck is the

one shown in Fig. 1, where (xi, j), i = 0 , 1 are the

. . . . .L , Lo 9., pI__,,,....._.!......0E k@......

Figure 1: Two-unit articulated vehicle.

Cartesian coordinates and 6'i the orientation angles ofthe midpoints PO and PI of the axles of the vehicle.At kinematic level, the inputs of the system can betaken to be the speed vector v of the point Pi and thesteering speed U = B where /3 = 6'0 - 6'1.

A set of variables that describes the configurationspace of the truck is given by q = [ X I , 1, '1, /3] with

A

A

the equations:0

= Sl(S)V+ g2(q)u (1)The system (1) is well-defined in the domainD = R x R x S1 x ] arccos (-2) arccos (-$) }{If L O> L1 the system presents no singularity.

The system (1) was proven to be controllable in [SIusing tools from differential geometry, like the rankof the Control Lie Algebra generated by the vectorfields associated with the inputs. For the definition ofnonlinear controllability, as well as Lie bracket, filtra-tion, distribution etc. refer to a standard textbook onnonlinear control systems like [5].

When Lo = 0, we obtain a car-like vehicle with thewell-known system:

cos el 04 = [ yIv+[I u c

= g1,(q)v + g2,(q)'1Lc (2 )where U , stands for the steering input of the car. Thedomain of definition is:

D,= {R x R x s1 I- }The difference between the two models (1) and (2 )

is that in the articulated truck the steering input isentering also into the equation for the orientation an-gle 6'1. This allows to change 6'1 by means of thesteering actuator alone, whereas, in the car, 6'1 canbe varied only through a sequence of the two inputs.It is equivalent to say that 6'1 is locally accessible byU without need of Lie bracketing the two inputs. Thesteerability on place can be checked using lineariza-tion of the equations around the origin: for the sys-tem (1) the two vector fields are gl(0) = [l 0 0 0IT,92(0) = [o 0 - Lo& I]T , hereas for the car-like (2 )g1,(0) = [l 0 0 0IT and g2,(0) = [0 0 0 1IT.

We can use the original nonlinear system to analyzemore in depth this difference. As in all nonholonomicsystems, g1 and g2 (or for the car 91, and 92,) are non-commuting vectors and the distribution they generatespan {gl, 92) is not involutive: the combination of the

3021


3/6


4/6

(a) Car-like vehicle#. ,... ,., I, ' H(b) Articulated vehicle

teristic of multiaxis vehicles that during a bend themidpoints of the axles tend to follow different trajec-tories. The difference between these trajectories canbe taken as a measure of how much cumbersome a ve-hicle is. Comparing again the mining truck with thecar-like vehicle, it is intuitively clear that the articula-tion helps reducing the gap between the trajectories ofthe midpoints of the two axles. Detailed calculationsfor the articulated vehicle are reported in [4]. n syn-thesis, in the two cases, the distance between the twotrajectories can be easily computed for a motion withconstant steering angle p # 0. In fact, for v # 0, thetwo midpoints follow concentric circles whose radiusTO and r-1 (respectively for the front axle and the rearaxle) can be calculated using the geometry of the ve-hicle. For the car-like vehicle, the off-tracking is (seeFig. 3):

(a) Car-like vehi- (b) Articulatedcle vehicle

Figure 3: Off-tracking margins.

1- cospTo - 1 = L- sin pFigure 2: Change of orientation. whereas, for the articulated vehicle:

To - '1 = L1 p) - ( 2 L 1P )os ( 2 L ldesired endpoint, i.e. the arbitrariness in the choiceof the fourth vector needed to span the whole tan-gent space reflects in the larger range of combinationsof input commands and implies here the possibilityof finding simpler combinations tha t generate a pathbetween two given points. 3.1 A path tracking criterion

When the two units are symmetric, then L~ = , andthe off-tracking is zero i.e. the two trajectories overlay.

The rest of the paper is dedicated to the formu-lation of an algorithm for the navigation of the ar-ticulated vehicle, aiming at reducing the off-trackingof the whole vehicle from a given path. The idea isadapted from [3].

The underground tunnel in which the truck is nav-igating is usually represented in terms of a curvature

3 Off-trackingThe typical work environment for a mining truck is

an underground tunnel of limited width. Navigatingin such an environment implies an high risk of crashingagainst the walls of the gallery. It is common charac-

3023


5/6

function associated with the length of a trajectory rep-resenting for example the middle of the tunnel. Trans-lating this into the Cartesian coordinates of an inertialframe is not possible analytically because of the ab-sence, except for trivial cases, of a closed form in theline integral expressing the length of the pa th covered.Therefore, a particularly convenient local representa-tion is given by a frame moving on the path to follow(see [7, 81). Under the assumption that the path is atleast C1 and that the curvature has an upper bound(see [9] for the details), a Frenet frame can be used tolocally describe the motion of the point with respectto the reference path y of known curvature ny . Thecontinuity of the curvature function is not requiredand so also simple paths, composed of straight linesand arcs of circle, can be considered. The set in whichthe local coordinates are well defined is essentially a"tube" around the path .

In our case, we consider two Frenet frames movingon the curve to follow, corresponding to the projec-tions on y of the two points PO and PI of our vehi-cle. In our frames, we assume to have chosen a base

I;=

+

Y

- L I S L O C O S -sin 81LO+LI cos; sin 00

-sin P - (L I+Lo COS P ) K - , (ST" ) COS ioLo+L1 c05P ( L ~ + L ~osP)(l-ny(s,,,)zo) '

sinp - KT (s , , ) cose lLO+Ll COSP l - K , ( s , l ) z l0

LoLl sin P sin B OLO+Ll COSP- 0L O L L s i n ~ n - , ( ~ , ~ ) c o s i o U1 cosLO+Ll c0p.P + (LO+LI c o s ~ ) ( 1 - - n 7 ( s y o ) z o ~L o cospLa+L1 cos0- 1

vo. "

Figure 4: Frenet frames associated with POand P I .with the conventions of Fig. 4. Each of the curvilin-ear frames is represented by two coordinates (s?,, -,,)where s-,, is the curvilinear abscissa and e-,, is theorientation of the frame with respect to the inertialframe. In the Frenet frame, the point P; is repre-sented by the signed distance zi between the pointitself and its projection and by the relative orienta-tion angle 0; = 0; - 0,;. The equations describing thedynamics of the point P; in the local frame can befound for example in [7].

For this system, the main issue of the autonomousnavigation is by far to keep the middle of the tunnel,i.e. t o keep control of the lateral dynamics. The mainadvantage of the Frenet frame is that it provides anatural way to describe the lateral dispiacement of apoint from the path , since -7 i represents the signed dis-tance of P; from its orthogonal projection on y. Thisproperty has been used by several authors as tracking

- A

3024


6/6

i.e. they reduce the configuration space of the systemdown to the original number of variables. In what fol-lows we will simply drop them and continue workingwith the overparameterized model.Stability analysis For a car-like vehicle, the pathfollowing with positive speed implies that the openloop equilibrium point is naturally stable whereasbackward motion implies that the same equilibriumis open loop unstable. For a mining truck, the pathfollowing problem has always an unstable equilibriumpoint due to the steering action performed on the ar-ticulation joint .

We use Lyapunov linearization method to showthat the system can be locally asymptotically stabi-lized to a path of constant curvature. The fact tha tlinearization does not provide global results is not alimitation in our case since the mining truck has tonavigate into a tunnel of reduced width and also thelocal frames are isomorphically defined only in a regionaround the path.

For a path of constant curvature K ? , the equilibriumpoint pe can be calculated from the geometry of theproblem. The Jacobian matrix calculated at pe A =2 4 is a Hurwitz ma trix i.e. the closed loop stat ematrix A - K B with B = L?(p,) can be made stableby choosing an appropriate gain I ( . Therefore, thesystem (4) can be locally asymptotically stabilized toa circular pa th by means of a linear st ate feedback.

a p P=Pe

Simulations In Fig. 5, the path to track is an arcof circle. It can be seen tha t convergence is achievedfrom a generic admissible initial condition. A similarbehavior is obtained with a negative speed.

T

. .. .- _ - _ .. .s; ,.. -.. .. . .. .I... ... \ . . .-3 . , ,, . ... -.-.,.

. . ..

.. ._ _ _ _ .. .s; ,.. -.. .. . .. .I. . , .

-3.... . , ,, . ... -.-.,.. .\ . . . .. . ..

Figure 5: Following an arc of circle of curvature K? =-0.2 (dash-dotted line) from a wrong initial posture.

4 ConclusionIn this paper we have given some mathematical in-

sight into the higher maneuverability concept thatmakes an articulated vehicle more suitable than a car-like vehicle in an environment characterized by a lim-ited free space like it can be an underground mine.How to explain and exploit this difference is the sub-ject of this paper.

References[l]C. Altafini. The general n-trailer problem: conver-

sion into chained form. In Proc. of the Conferenceon Decision and Control, Tampa, FL, December1998.

[2] C. Altafini. A path tracking criterion for anLHD articulated vehicle. International Journal ofRobotics Research, Summer 1999.

[3] C. Altafini and P. Gu tman . Pa th following withreduced off-tracking for the n-trailer system. InProc. of the Conference on Decision and Control,Tam pa, FL, December 1998.

[4] L. Bushnell, B. Mirtich, A. Sahai, and M. Secor.Off-tracking bounds for a car pulling trailers withkingpin hitching. In Proc. of the Conferenceon Decision and Control, pages 2944-2949, LakeBuena Vista, FL, 1994.

[5] A. Isidori. Nonlinear Control System. Springer-Verlag, 3rd edition, 1995.

[6] J. -P. Laumond. Controllability of a multibody mo-bile robot. IEE E Trans. on Robotics and Automa -t ion, 9:755-763, 1993.

[7] A. Micaelli and C. Samson. Trajectory trackingfor unicycle-type and two-steering-wheels mobilerobots. Technical Report 2097, INRIA, November1993.

[8] M. Sampei , T. Tamura, T. Itoh, andM. Nakamichi. Path tracking control of trailer-likemobile robot. In Proc . IEEE /RSJ In ter . Workshopon Intelligent Robots and Systems, pages 193-198,Osaka, Japan , November 1991.

[9] C . Samson. Control of chained systems: applica-tion to path-following and time-varying point sta-bilization of mobile robots. IEEE Trans . on Auto-matic Control, 40:64-77, 1995.

3025

Documents

Why to Use an Articulated Vehicle in Underground Mineing Operations