ENHANCED PATH PLANNING FOR ARTICULATED AND SKID STEERING MINING VEHICLES. Vladimir Polotski

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1 Copyrigt IFC 15t Triennial World Congress, Barcelona, Spain ENHNCED PTH PLNNING FOR RTICULTED ND SKID STEERING MINING VEHICLES Vladimir Polotski NSERC Industrial Cair in Mining utomation, Ecole Polytecnique de Montreal Pone ext 4175, FX (514) bstract: utonomous guidance of te mining veicle is based on te pat tracking algoritm tat use pre-calculated desired pats. Tese pats are often calculated witout taking into account te nonolonomic nature of te veicle motion. s a result te calculated pat migt be not feasible and te pat tracking controller will be required to correct te situation, deteriorating at te same time te overall performance of te guiding algoritm. In tis paper we describe pat-planning algoritms for articulated and skid steering veicles used in mining applications. We propose numerically stable procedure for computing off-line te skidding and articulation angles. Te results of tese computations complement te conventional pat-planing procedures and improve significantly te pat-tracking precision, and overall guidance performance. Copyrigt c IFC Keywords: numerical stability, autonomous guidance, pat-planning. 1. INTRODUCTION Te azardous environment of a mine demands an increasing degree of autonomy of te mobile mining equipment in order to keep te uman presence on te site minimal. Tis researc targets autonomous guidance of slow moving eavy underground and surface mining macines as Laud-Haul-Dumping macines (LHD) and blast-ole drills. n underground LHD, equipped wit a bucket, as to move back and fort along te narrow drifts between loading and dumping sites were it stops for executing loading/dumping ore. surface drill, equipped wit a very tall drilling rig and two actuated tracks, as to move slowly between predefined locations were it stops for drilling oles tat are subsequently filled wit explosives. Developing autonomously guided LHD s and drills will allow te remote supervised control of te macine and migt evolve to te multi-unit supervision by a single remote operator, resulting in iger efficiency, and greater safety. Some degree of autonomy is acieved in te control of underground macines. Muc more as to be done in surface macine automation. However, in bot cases tere are still many callenges in transferring te advanced results obtained from planning and control of autonomous robots to industrial implementations. Some recent steps towards converting real mining macines to autonomous units wit on-board navigation and pat following capabilities are described in (DeSantis, 1997; Polotski et al., ) for underground mining and (Peck and Hendrics, 1997; madi et al., ) for open pit mining. Te veicle guidance problem is conventionally devised into te pat-planning and pat-tracking pase. First pase is mostly addressed on teoretical level. second pase is widely analyzed from practical point of view, owever te pat to be followed is most often taken in a simplified form as a set of te straigt segments and circular arcs. Tis approac being acceptable from te geometrical point of view neglects te computation of some parameters useful for control algoritms, tus letting to te pat tracking controllers to correct te situation. Tis work demonstrates ow te off-line computation of

2 C K C K? additional parameters suc as articulation angle for articulated veicles and skidding angle for skid-steering veicles may improve te overall performance of te veicle guidance witout increasing te complexity and te cost of te system. Te paper is organized as follows. Section and 3 are devoted to articulated veicles. pat-tracking control for articulated veicles as been te subject of numerous works (DeSantis, 1997; l. Busnell et al., 1994; Polotski, ) We analyze ere te pat planning computations tat must precede te pat-tracking, but ave not received substantial attention of te researcers. We calculate te articulation angle and demonstrate tat for backward maneuver our algoritm gives te correct result despite te numerical instability tat deteriorate conventional computations. In section 4 and 5 we study skid steering veicles. pat-tracking controller for tis type of veicles as been recently proposed in (madi et al., ). We extend te planning procedure and obtain, more accurately, te veicle eading. Taking into account te sape of te pat and te required veicle speed, we calculate te corresponding lateral slippage, and deduce te slip-angle and veicle eading for subsequent use in pat-tracking controller.. MODELING OF N RTICULTED VEHICLE two-part four-weel veicle articulated in te middle is under consideration. Te steering action is performed by canging articulation angle. Let N l P ϕ V p l 1 M Figure 1: Sceme of articulated veicle and be te middle points of two axles wit te junction point (see Figure 1). Let,, and be te velocities of points respectively. Te sign of is positive if and negative oterwise. Let denote by te articulation angle between and (positive for anti-clockwise case). Let us mark a point as front point. Let be Cartesian its coordinates and eading angle of wit respect to te world coordinates, and let denote by te angle of velocity vector at point wit te front part of te veicle. We will consider tis angle as te generalized control input. For low speed veicles like tose usually met in mining applications te kinematics level of description seems to be appropriate. Te kinematic equations ave te following form ( and are denoted by and respectively): "# %$'&)(* (1) "+# %(-,./ () "# : (3) ";# =< 1 (-,./ > 3657./< > $'&)(@ (4) : Te term generalized input means tat tis model may describe bot actuated articulation and tractor-trailer sceme. If only one part of te veicle is steered te model describes te motion of a car wit trailer (bot forward and backward cases). One can see tat equations (1) (3) describe te unicycle motion and equation (4) corresponds to te internal degree of freedom. noter approac based on te new reference point serving as linearizing or flattening point is discussed in (Polotski, ) 3. RTICULTION PLNNING smoot trajectory of te point directly specifies te trajectory of te point as anoter point of te same rigid body (tricycles ). We will first study te beavior of te tricycle wit respect to te first one. Let us denote te generalized input by B and introduce te lengt s(t) along te pat of point and an angular D variablec # < " 89. We ave C # D 89 #FE E " 89 " Canging te variable in (4) from to C and using notation G # <: 1 : 8<H7: we obtain # sign< < 1 (-,. > BI<:C <:$'&)(J >? > H G - (5) Tis equation as te type studied in (Kamke, 19833) (Eq. 1.79). and is equivalent to te Riccati equation. Canging we get variable from to K # 3657.I<:L87H # sign< Te factor sign< N < 1 K > BI<:C <? > G 1 BI<:C G)M (6) is positive for forward motion and negative oterwise. Te rigt-and side of ( 5),cange te sign wit tis factor becoming te stable for forward motion and unstable oterwise Let us first discuss te case G O # tat corresponds to te symmetrical macine P #. Riccati equation 5 is reduced in tis case to te linear one: < 1 K > BI<:C - (7) C # sign< Tis equation as unique exponentially stable attractive solution for forward motion and an unstable one oterwise. For backward motion, te growing solutions (see Figure, rigt bottom plot) escape from one repulsive solution and converge to te singular solution #RQ of ( 5). Tese solutions do not correspond to any feasible motion of te veicle.

3 y pase picture x inverse time calculation 1 x,y time istory 1 3 time articulation 4 were only circular and straigt line trajectories were considered. On te oter and, our results are motivated by te controlled joint sceme, in suc a case te unstable beavior as to be taken into account, te articulation angle along te pat (including backward maneuvers) as to be calculated and subsequently used by an appropriate pat tracking controller time Figure : Backward motion time Te case B # const corresponds to a desired pat of circular sape or straigt lines and was analyzed in (l. Busnell et al., 1994). Te Riccati equation as constant coefficients and can be analytically solved using reduction to te linear system. For industrial application bot assumptions of small G and slowly varyingb usually old. G For # O and B # const two stationary solutions are TS # HU57V6$W3657.X< B YTS #RQ. Using small parameter approximation, two solutions engendered by S and S can be found as follows S # HU57V6$W3657.X< BI<? >? G > H'B - H S # HU57V6$W3657.X< H B G 1 G BI<? > B H In te general case Riccati equation could be reduced to te second order system of linear differential equations. Essential property consists in te existence of two solutions, one of wic is always stable, but anoter one unstable. Unstable solutions can be made stable by reversing te time and te direction of motion (sign< ). Tus equations (1) (4) can be used to obtain te desired articulation angle along te backward maneuver. Using (1) and () to compute, te generalized input can be subsequently determined from (3), and ten (4) is used for stable computations of an articulation angle. Figure illustrates te backward maneuver of following a pat consisting of successively executed turns and straigt line parts (left top plot). Tis is an example of essentially non-stationary pat. Direct calculations of te articulation angle give an unnatural stable solution closed to Q for sligtly asymmetrical macine (rigt bottom plot). If tis solution is used for control, te veicle will fold up instead of following te required pat. Te time inversion allows to obtain te correct result via numerically stable calculations (left bottom plot). Te results discussed above are closely related to te beavior of a trailers convoy pulled by a car. (l. Busnell et al., 1994). Te description of te motion as using te Riccati equation generalizes tose of (l. Busnell et al., 1994) 4. MODELING OF TRCKED VEHICLE veicle model wit tree degrees of freedom is adopted (madi et al., ). Te veicle is supposed to undergo a motion in te orizontal plane, tat is usually te case for a drilling rig in open pit mines te targeted application of our researc Te traction and longitudinal resistance forces, as well as te distributed lateral friction forces are included in modeling. Most of existing models ave some intrinsic limitations, e.g., only forward maneuvers are considered or only circular and straigt line pats are allowed, etc. more complete model, capable of simulating a large variety of maneuvers of te veicle is presented ere. Te following assumptions and notations are used: te weigt of te veicle is equally divided on bot tracks; (,, ) represent te coordinates of te center of gravity (point ) in te fixed frame and te veicle eading. second coordinate system Z\[ is attaced to te veicle wit its Z -axis coinciding to forward looking longitudinal axe of te veicle, and [ -axis being left ortogonal to Z. Te desired pat is defined by te smoot curve (]< D ^< D ) in global coordinates wit te radii of curvature _ < D at eac point; te veicle moves wit te velocity `. Figure 3 illustrates te force distribution: acb, _db, ate, _fe are te traction and resistance forces exerted to te veicle troug te left and rigt tracks respectively. X b F r F l Y f l Figure 3: Tracked veicle s force model. We denote by g te signed desired velocity of te veicle (positive if te forward motion is required and negative for a backward motion), and by C an angle between te direction of motion gj` and Z. It sould be empasized tat te direction of backward motion is opposite to te `, and due to te veicle skidding, its direction is not tangent to te pat. s a result, te angular speed is not equal to M D y V R α r R l ψ x P

4 v # te rotation rate of te velocity vector i # E g E 89_. For te same reason, te instantaneous center of rotation does not coincide wit te center of te pat curvature and te corresponding radii are different. In tis modeling, te location of te vanising point j of te lateral speed defined by kj #%l # [\8 #1 gm(-,.@cc8n d (8) is very important and will be used later. Basic equations of motion in Z [ coordinates, (see (Siller et al., 1994; madi et al., )), are as follows: oqp Z # <:aib > ate 1 _db 1 _fe orp [ #srt (9) u p # <:aib 1 _fe -v 1 <:atw 1 _fw -v 1 H t < r 1 were o is te mass of te veicle,a and are introduced below. Under assumption of te uniform distribution of te weigt along te track, te sear force density t is proportional to te pressure per unit of te track lengt x # o^y 87H7 : t/#rz x #Rz o{y 8<H7 To compute te lateral forces, we introduce te saturation function c}j~< ZI-ZI -Z # o/ƒ ZX ZX m ou P ZI-Z ˆˆ and denote: # c}j~< l 1 P87H P87H (1) Ten we obtain using Figure 3: aib Š6 #qr DŒ y < gci t (11) Te traction forces atb and ate are related to te left and rigt track slips b and e via te commonly used exponential expression (Bekker, 1969)? 1 w 6 9 \ kš sign< ai e # 1 E EI (1) were #œ 89, #, - is te track area, and and are te soil parameters (Bekker, 1969). Te left and rigt track slips are defined as b # žbp8< Ÿ itb e # \e 8< Ÿ i e W žb # Z 1 1 Ÿ itb \e # Z > v 1 Ÿ i e were Ÿ and itb i e are te radius and te angular velocities of te sprockets, and \b and \e are te slip velocities for eac track. Detailed description of te modeling in (madi et al., ) and is not included for te sake of brevity. We focus ere on te computational aspects of te orientation planning for tracked veicles. 5. NONHOLONOMIC PLNNING FOR TRCKED VEHICLES Trajectory planning for conventional veicles consists of two separate procedures, namely: velocity planning (or sceduling) and pat planning. Due to te inevitable slippage of tracks and te dependence of traction forces on track slippage, tose two problems cannot be solved separately. Te sceduling problem and te sape definition ave to be solved first. Ten te orientation of te veicle along te pat can be calculated. Te pat (]< D ) ž^< D is supposed to be known as well as te linear speeds gn< D along te pat as it is described above. For a targeted application tis is a natural assumption as a drill rig is usually required to move along te prescribed pat at te constant speed between two pre-specified locations and its acceleration/deceleration can be easily accounted for. NGLE [rad] LPH*(t) TIME [sec] Figure 4: Structure of solutions In order to take te slippage into account, let us evaluate te balance of forces in te lateral direction [. From (1)- (4) we obtain: DŒ y < gli H z y Using te relationsip between c}j~< 1 g DŒ P C 1 P87H P87H # gliq$'&)(žc > g^(-,.@c (13) C,, and i : C > # ij (14) and introducing te natural frequency : #F H z y 89 (15) # te equation (13) can be rewritten in terms of C for before-saturation case ( l ) as C # i > DŒ y ig^(-,.jc ilg^$'&)(žc > (16) g^(-,.jc To illustrate te use of tis equation, one can calculate te stationary solution of (16) for te case wen g # O. For C # O we get 3657.JC #1 i E i E, ence Cc #1 ƒ " ƒ) < i E i E 87 W (17)

5 " 1 C Tis is a purely analytical result tat cannot be obtained as a numerical solution of (16)due to numerical instability. Namely, equation(9) as a closed form solution of te form <: " # < 3657.JC # E i E il > f < i E i E 57V6$W3657.@ >«ª. > i E i E >? fc -plane is il- Te global structure of solutions in te < lustrated in Figure 4..5 were (18) Y[m] NGLE[rad] CONVERGENCE LYER LPH*(t) TIME[sec] Figure 5: Time-varying balanced solution Te stationary solution (1) is a unique (up to te periodicity over C ) solution tat can be extended to te unbounded time domain, we will call it balanced solution. ll oter solutions as illustrated in figure (3) diverge from te balanced solution and escape to infinity (in C ) in finite time. Terefore any direct numerical solution of (9) diverges from te balanced solution (1) and cannot be used to compute te balanced solution. For te time invariant case, tis numerical instability seems not to be very important since te balanced solution is known. For te time-varying case, an analytical solution may not exist, wic makes impossible te direct computation of te balanced type solutions in te presence of suc instabilities. By torougly analyzing te timevarying before-saturation case, we ave demonstrated tat te solutions of (16) still as te similar structure. We ave sown tat under some tecnical assumptions tere exist a unique balanced-type solution. It is repulsive in te direct time, but it is attractive in te inverse time and can be obtained by backward integration from almost any terminal conditions. start- Namely, if we fix te terminal time ~ and introduce # ~ 1 ", and consider two solutions, C < and C < ing from C, S C at S # O, teir difference # C < C < decays monotonically to zero wit an explicit upper bound. Te demonstration is rater involved and is not presented ere. Due to tat monotonicity, all solutions converge to a unique limiting solution - balanced solution, wic is de X[m] Figure 6: Circular pat execution noted below byc < " inverse te time again # ~ 1 and use C < ". Once tis solution is computed, we as te desired value of te skidding angle along te pat. Figure 5 illustrates te results of te proposed computational metod weni andg O? vary in time ( D X@ 1 id< " O H D X o 8 D g? o 8 D ). ll solutions converge very fast to te balanced solution C < ", wic can not be found nor analytically neiter by te direct numerical integration. # In te above discussion we ave limited ourselves by te non-saturated case l. To include te saturation effects, E g^(-,.jcc i 1 as to be verified for te balanced solution. E.g. for te following parameters # o? O o -g # 8 D _ # 1 7O o z# O? ±, te balanced solution is C # O? and it does not satisfy to (1). s a result, te stationary turn wit tese parameters will not occur. If te te value ofz increases up to.5, we get C # O O H, satisfying to (1). In general, te following condition guarantees tat te balanced solution C is acceptable. E i E ² H E gm(-,.jc E E H (19) () Under an assumption tat P8<H E _ E is small (te lengt of te track wit respect to te diameter of turn) we may rewrite (13) as z y _ ² g (1) We ave found tis condition very important for planning te maneuver of tracked veicles, i.e., wen turning at a given radius_, iger speeds are allowed on a terrain wit iger coefficient of te lateral resistance. For te given speed tere is a lower bound for te allowable radius given a particular value of z. nd vice versa, tere is a lower bound forz if te maneuver wit a particular_ is required.

6 and skid-steering veicles is described in tis paper. dditional parameters specifying veicle eading, namely articulation and skidding angles are computed along te planning pase. and used for constructing feed-forward terms along te execution pase. Proposed metodology is of particular importance for te pats wit a varying curvature and backward maneuvers. Using computations in backward time, our proposed procedure overcomes te numerical instability reported in (l. Busnell et al., 1994) for articulated veicles and in (Siller et al., 1994) for skid steering veicles.. rticulated veicle: mini-lhd By incorporating our enanced planning procedure into te control loop we make te orientation error decaying faster and also improve te overall controller performance. Tis procedure constitutes te low cost improvements since no additional sensing capabilities or more actuators are needed but only some additional off-line processing and complementary feed-forward terms in controllers. CKNOWLEDGMENTS Te financial support of te NSERC Industrial Cair in Mining utomation is gratefully acknowledged. B. Tracked veicle: MR5 Figure 7: Experimental veicles Figure 6 illustrates te execution of a turn at _ #³1 7O o wit required speed g #? O o 8 D on a terrain wit O? ± z{# ( - corresponds to an anti-clockwise turn). Condition (1) is not met, ence, for te execution of tis maneuver te desired speed was reduced to 9.4 m/sec tat corresponds to te boundary of (1). Ten te control inputs corresponding to tis motion are calculated and te open loop run is executed. Te system follows exactly te required circular pat (Figure 6). If g #? O o 8 D is conserved and corresponding C # C is used, te smaller circle (_ #qr) o is realized. If te condition (1) is not taken into account at all andc # C is used, te spiral-sape trajectory wit fast growing C is observed (figure 6). 6. IMPLEMENTTION ISSUES Te described approac to te articulated and skid-steering veicle guidance as been implemented on te laboratory size platforms mini-lhd and MR5 illustrated in figures 7 () and 7 (B) respectively. On-board control algoritms for pat-following utilize feed-forward terms based on offline computations of articulation and eading described above. Ongoing projects address outdoor testing in more realistic environments and implementation on real mining equipment. 7. CONCLUSIONS n enanced pat-planning metodology for articulated References madi, M., V. Polotski and R. Hurteau (). Pat tracking control of tracked veicles. In: IEEE Int. Conf. on Robotic and utomation. Los ngeles, C. pp Bekker, M. G. (1969). Introduction to Terrain-Veicle Systems. University of Micigan Press. nn rbor. DeSantis, R. (1997). Modeling and pat-tracking for a load-aul-dump mining veicle. SME J. Dynamic Systems, Measurement and Control pp Kamke, E. (19833). DifferentialGleicungenk. Wiley.. Stuttgart. l. Busnell, B. Mirtic,. Saai and M. Secor (1994). Off tracking bounds for a car pulling trailers wit kingpin itcing. In: Int. Conf. on Decision and Control. Peck, J. and P. Hendrics (1997). Gps-based navigation systems on mobile mining equipment for open-pit mine. CIM Bulletin pp Polotski, V. (). New reference point for guiding an articulated veicle. In: Int. Conf. on Control pplications. Polotski, V., J. N. Bakambu, W. Wu and P. Coen (). Navigation system for underground mining veicles: rcitecture and experimental setup. In: 31st Int. Symp. on Robotics. Siller, Z., M. Serate and M. Hua (1994). Trajectory planning of tracked veicles. In: IEEE Int.Conf. on Robotic and utomation.