Simulation of Rover Locomotion on Sandy Terrain - Modeling, Verification and Validation Rainer Krenn, Gerd Hirzinger German Aeropace Center, DLR, Intitute of Robotic and Mechatronic Oberpfaffenhofen, D-82234 Weling, Germany rainer.krenn@dlr.de, gerd.hirzinger@dlr.de Abtract In thi paper a modeling technique called SCM (Soil Contact Model) for imulating the phyical interaction between a rover mobility ytem (wheel, leg, track etc.) and a planetary oil urface will be preented. SCM compute the contact force between the oil and the contact object baed on empirical terramechanic formulae. Within SCM the oil geometry i decribed a digital elevation model (DEM) while the contact object are decribed a general polygonal urface meh grid. Baed on thi method SCM i able to deal with arbitrarily haped contact object without any retriction in term of application cae. Moreover, SCM provide a module for platic oil deformation in order to take typical terramechanic phenomena like inkage, bulldozing and multi-pa effect into account, which trongly affect the entire contact dynamic. The preferred imulation engine for SCM i SIMPACK, a 3D multi-body imulation package. In the verification proce the correctne of the model ha been demontrated by bevameter tet propoed by. A firt tep in the validation proce wa made by applying SCM for imulation of ExoMar rover drawbar pull tet. It ha been demontrated that SCM create imulation reult with partly good correlation with real tet meaurement. Additional invetigation are required for low contact velocitie. INTRODUCTION In the recent year planetary exploration by mean of mobile robotic ytem became a more and more important iue within the pace robotic community. Currently, ESA ExoMar miion with it dedicated rover (Fig. ) i one of the mot powerful driver in thi field of reearch. But alo maller project like the Crawler of DLR Intitute of Robotic and Mechatronic (Fig. 2) provide important contribution to the detailed undertanding of rover locomotion on planetary terrain. A key cience regarding rover locomotion i terramechanic. It addree the interaction between wheel, leg or track of the rover and the oil urface in term of inkage and traction. ESA Fig. : Artit View of ExoMar Rover Fig. 2: DLR Crawler In the pat a large number of variou terramechanic baed modeling and imulation approache were preented. For the imulation of locomotion in andy oil mot of the author apply the empirically found oil tre-train law a introduced by, e.g. [] and extended by contribution of Wong [2]. An exemplary approach for the application of theory i preented by Bauer, Barfoot et al. in [3]. Here, a tire-oil-interaction model ha been validated veru experimental ingle-wheel tet uch that the overall rover locomotion performance could be plauibly predicted. Ihigami, Yohida et al. extended the ingle wheel etup to a full rover imulation with four teerable and controlled wheel in [4]. Herein, the complexity of the model wa increaed by taking bulldozing force and lateral force at the
wheel into account. The goal of our new approach wa the extenion of the current tate of the art in wheel-oil interaction modeling uch that the following requirement can be fulfilled:. Model compatibility with tandard multi-body imulation engine like Matlab/Simulink and SIMPACK. 2. Applicability for arbitrary contact object in order to be able to imulate the locomotion of ytem other than wheeled one like crawler and tracker, and to be able to validate the model veru reult of tandard oil penetration tet like bevameter experiment. 3. Computation of typical terramechanical phenomena like a. rolling reitance variation depending on inkage (bulldozing), b. lateral contraint force inide track depending on inkage and lateral bulldozing, c. drawbar pull variation a function of lippage, d. multi-pa effect of wheel running inline in a track. The according modeling activitie reulted in a module called Soil Contact Model (SCM) that will be introduced in the following chapter. Moreover, a number of SCM performance tet will be preented, which have been accomplihed for the verification and validation of the tool. IMPLEMENTATION OF THE SOIL CONTACT MODEL (SCM) The terramechanical contact problem oil contact body can be generally decribed a the contact between a platically deformable body and a rigid one. Thi implie that elatic deformation of the bodie, which penetrate the oil, are negligible in application uing SCM. Soil Surface Decription The oil urface inide the SCM algorithm i decribed a a digital elevation grid DEM. It provide height information z at dicrete horizontal coordinate x and y, which are defined by the grid node of a regular meh grid. The ditance between adjacent grid node are contant over the total urface. An example i given in Fig. 3. The contact counterpart i decribed a ordinary polygonal meh grid with vertice and face defining the body urface (Fig. 4). Z Soil X Y Fig. 3: Soil Surface Elevation Grid Fig. 4: Polygonal Surface of Contact Object (Wheel) Beide the geometrical information each grid node of the oil i aociated with a number of oil parameter and deformation tate. The mot important parameter of the oil grid node are collected in Tab. together with two example of planetary oil imulant. Tab. : Soil Attribute of Grid Node Parameter Soil Simulant DLR-A Soil Simulant DLR-D Exponent of inkage n [-].63.8 Coheive modulu k c [N/m n+ ] 63-6.675e5 Frictional modulu k ϕ [N/m n+2 ] 237.92e8 Coheion c Pa 88 3 Internal friction ϕ deg 24.8 3.4
Contact Detection For contact detection purpoe the contact body i treated a a cloud of urface vertice (Fig. 5, a). In the econd tep the horizontal co-ordinate of the vertice will be mapped into the grid of the oil DEM uch that the vertice will be arranged in column afterward (Fig. 5, b). Finally, the minimum vertice of each column (Fig. 5, c) will be elected for comparion with the vertical co-ordinate of the according oil grid node, repectively to detect contact and to define the footprint (Fig. 5, d). The footprint i the interection volume of wheel and oil. a) b) c) d) Fig. 5: Contact Detection and Footprint Computation Contact Force Computation Once the contact i detected we are able to calculate the penetration depth z for each individual node of the footprint grid. Together with the oil parameter of Tab. we have all pre-requiite for the computation of the local contact preure p according to the empirical formula () of. k c n p = + kϕ z b () Herein, the variable b i repreenting a meaure of the contact footprint hape. According to, b i the radiu of a circular contact area or the length of the horter ide of a rectangular contact area. However, thi definition i not applicable for a generally haped footprint contour that change during imulation. Therefore, SCM ubtitute the variable b by b *, which i a function of the total contact area ize A and the total contact contour length U according to (2). Both, A and U can be eaily tracked during imulation. * b = 2 A (2) U For circular contact plane b * i identical with b and for rectangular contact plane b * i a quite good approximation of b. If the inkage z i contant over the entire contact area A the preure p according to () could be interpreted a the mean preure acting in the contact zone. For the general cae in SCM with individual inkage of each grid node in the footprint the averaging approach i not valid anymore. Here, the preure ditribution inide the contact area ha to be taken into account in order to provide correct reult. In SCM the preure ditribution i derived from the centrality γ of contact grid node inide the contact area, which i a function of the ditance to all other contact grid node (3). mr γ = ; R = ; m = number of contact grid node m m T R (3) (( rj ri) ( rj ri) ) i= j= ; j i Example for the centrality ditribution are preented in Fig. 6. Taking the centrality (3) into account the original equation () ha to be lightly adapted for ue in SCM according to (4). * k c n = γ = γ + ϕ p p k z b (4)
.2..5.9...8.5.7 -.5 Centrality.9.8.6.5.4 -..7.3 -.5 -.2 -.2 -.5 -. -.5.5..5.2.6 -.2 -...2 -.2 -...2.2..5..4.3.9.2.8. -. -.2 -.3 -.4 Centrality..9.8.7.6.5 -.5.5.7.6.5.4.3.2. -.5 -.5 -.4 -.3 -.2 -...2.3.4.5.5 -.5 Fig. 6: Centrality Ditribution over Contact Area Expreed by Color Mapping Top: Circular contact area. Bottom: Grid like contact area) Left: Vertical view on contact area. Right: Centrality ditribution function In order to be able to finally compute the contact force, we have to take the oil urface normal vector n and the relative velocity vector v between oil urface and contact body urface into account. Baed on them we can calculate normal oil penetration vector and tangential liding vector t. Auming normalized vector and t, whoe length are equal to the individual contact area ize at the grid node, we can define the implemented component of the contact force in SCM:. Sinkage reitance force: * FSinkage = p (5) 2. Bulldozing reitance force: * FBulldozing = ( c+ p tanϕ ) (6) 3. Friction force a function of the friction coefficient μ between the oil and the contact body urface: * * * FFriction = μp t ; μp < c+ p tanϕ (7) Platic Soil Deformation The implementation of platically deformable oil, repectively the footprint and landfill computation in SCM wa motivated by typical rover imulation tak. During cruiing the front wheel are motly rolling through untouched oil
with a ignificant bulldozing reitance that i typically increaed by a hump in front of the wheel. On the other hand, the wheel following inline can drive in a pre-compreed rut at lower rolling reitance (multi-pa) but higher lateral guidance force. The imulation of thee phenomena require a platic deformation in the contact zone and an according depoition of the diplaced oil. And the ame requirement apply for imulating elf-carving of wheel into oil under advere condition. The algorithm for the platic oil deformation, which i currently a firt approach, i inpired by computer graphic algorithm for terrain generation (e.g. Olen [5]) and animation of footprint in oil (e.g. Sumner et al. [6]). It conit of three tep:. Soil diplacement from the contact area, 2. Temporary oil depoition at the border of the contact area and 3. Eroion of the diplaced oil in the vicinity of the contact zone. Soil Diplacement The total volume of the oil diplacement i equal to the volume of the footprint that ha been created by the contact body (Fig. 5, right). Thi implie that the total oil height reduction i equal to the wheel inkage z at the according grid node. The diplaced volume i divided into two part dz who depend on the normal oil penetration vector. The part are related to inkage (7) and bulldozing motion (8). T (8) dzinkage = z T dz bulldozing T = z T Temporary Soil Depoition In thi tep the diplaced oil volume will be temporary ditributed over the border grid node of the footprint. So each border node get a certain portion of each footprint node. The individual weighting factor w, which indicate how big the portion will actually be, depend on the ditance vector d from the footprint node to the border node for inkage (8) and the angle α between d and for bulldozing (9). w = () dd inkage T T ; coα d wbulldizing = m ; co α = ; m=,3,5 () co α ; coα > d An impreion for oil diplacement and temporary depoition i given in Fig. 7. Here a wheel wa rolling over the oil with a contant inkage rate. T (9) Fig. 7: Soil Shape Appearance After Diplacement and Frozen, Temporary Depoition without Eroion Fig. 8: Soil Shape Appearance After Soil Diplacement, Depoition and Eroion
Eroion of Soil When and i piled a maximum and hill lope angle can be achieved, which i equal to the internal friction angle ϕ of the and. Therefore, after temporary depoition of the diplaced oil volume SCM applie an eroion algorithm to the oil grid node in the vicinity of the contact area in order to meet thi phyical limitation. At oil grid reolution d the maximum oil height difference dz Limit to the adjacent node i limited a given in (2). dzlimit = d tanϕ (2) In the eroion algorithm (3) half of the height that exceed dz Limit will be removed and added to the adjacent node according to their individual fraction of the total eroion potential. max ( dz) dzlimit dz dzeroion = ; i =,, n 4 2 (3) dz In Fig. 8 the reult of the complete platic oil deformation proce i preented. VERIFICATION The verification proce of SCM we have to prove, that the model generate correct reult in term of the implemented terramechanic, repectively theory. In particular we have to prove that the computed contact force, which are a function of the parameter of Tab. and the relative kinematic between contact body and oil urface, match ufficiently the experimental reult, where theory i baed on. For the verification of SCM we apply the bevameter tet a propoed by. In thi oil penetration tet probe of different ize with circular or rectangular contact area will be preed into the oil with a well defined vertical force. The according maximum penetration depth of the probe will be recorded. By varying probe ize and penetration force we can identify the oil parameter of (), k c, k φ and n, which decribe the preure inkage function. Simulation Condition i= Tab. 2 Verification of SCM Parameter Identification i n [-] k c [N/m n+ ] k φ [N/m n+2 ] Applied Soil Parameter.63 237 63 Identified Soil Parameter Reolution Soil/Probe =.5 m/.5 m.6343 227.48 6262.6 Reolution Soil/Probe =.75 m/.75 m.633 2357.78 639.75 Reolution Soil/Probe =. m/. m.6323 235.8 6874.48 In Fig. 9 the reult of the bevameter tet imulation with cylindrical probe of two different diameter are preented. The verification reference for all imulation are the ideal function according to (). In order to analyze the enitivity of the imulation accuracy in term urface hape dicretization both, the oil grid reolution and contact body urface meh grid reolution ha been varied at each tet (oil reolution >= contact body reolution). Generally we can tate that SCM meet the ideal reult very well and the impact of reolution variation i abolutely acceptable. Thi tatement will be confirmed by the reult of the oil parameter identification baed the imulation reult mentioned above. In Tab. 2 the identified oil parameter are preented. All of them match the originally applied parameter quite well.
4 Probe Diameter.5, Probe Reolution.5 4 Probe Diameter.5, Probe Reolution.75 4 Probe Diameter.5, Probe Reolution. 35 35 35 3 3 3 25 25 25 2 2 2 5 5 5 5 Simulation - Soil Grid.5 Simulation - Soil Grid.75 Simulation - Soil Grid..2.4.6.8..2 5 Simulation - Soil Grid.75 Simulation - Soil Grid..2.4.6.8..2 5 Simulation - Soil Grid..2.4.6.8..2 4 Probe Diameter.3, Probe Reolution.5 4 Probe Diameter.3, Probe Reolution.75 4 Probe Diameter.3, Probe Reolution. 2 2 2 8 6 8 6 8 6 4 4 4 2 Simulation - Soil Grid.5 Simulation - Soil Grid.75 Simulation - Soil Grid..2.4.6.8..2 2 Simulation - Soil Grid.75 Simulation - Soil Grid..2.4.6.8..2 2 Simulation - Soil Grid..2.4.6.8..2 Fig. 9: Verification of SCM with Senitivity Analyi (Mar Soil Simulant DLR-A, ee Tab. ) VALIDATION In the validation proce we have to demontrate, that SCM i able to imulate general terramechanical problem. Thi implie the quetion if the theory of i applicable without further problem pecific adaptation. In the very firt tep of the SCM verification proce the drawbar pull reult of the ExoMar breadboard chai tet were taken a reference for according drawbar pull imulation with SCM. The tet were prepared and executed by ETH, Zurich at Oerlikon Space, Zurich. In Fig. and Fig. the cenario of the SCM imulation and the real breadboard chai inide a oil timulant tetbed are preented. Fig. : Senario of Rover Simulation Uing SCM and SIMPACK Fig. : ExoMar Breadboard Chai, Oerlikon Space, Zurich
In the drawbar pull tet the rover chai i tethered onto a wire that will be unrolled with controlled contant velocity v Tether. The angular velocity of the rover wheel ω Wheel i adjuted uch that the deired rover velocity v Rover,deired would be lightly fater then the tether deploying velocity. But due to the velocity limitation of the tether, the wheel are well defined lipping in the and. The relation between the lippage and the velocitie of tether and rover i the following: vtether vtether m = ; rwheel = wheel radiu; vtether.8 v = Rover, deired ωwheelr = (4) Wheel The applied pull force will be recorded by a enor mounted between the rover and the tether. In Fig. 2 the drawbar pull force of SCM imulation and the according meaurement made during the ExoMar breadboard chai tet are preented a function of lippage. In the diagram the force meaurement, in particular the plot of drawbar pull force veru experiment time (e.g. mall ub-diagram in Fig. 2), have been horizontally trongly compreed and placed cloe to the according lippage value in the diagram. By thi method both, the function pull force veru lippage and the range of the force bandwidth of the force can be preented in one diagram. 4 35 3 25 Drawbar Pull [N] 2 5 Drawbar Pull [N] 36 34 32 3 28 26 Slippage.7 5 24 Time SCM Simulation ExoMar Chai Tet -5..2.3.4.5.6.7.8.9 Slippage [-] Fig. 2: Drawbar Pull a Function of Slippage (Mar Soil Simulant DLR-D, ee Tab. ) The output of SCM imulation provide a claical function drawbar pull veru lippage with aymptotical approach to the maximum value. However, the real meaurement came out with an atypical, almot linear function. Therefore, the imulation matche the real value only at lippage > 4%, repectively at higher relative velocitie between wheel and oil, with ufficient accuracy. On the other hand, the bandwidth of the force, which relate with the effect of the wheel grouer, can be reproduced by SCM quite well. Hence, the future focu of the SCM validation proce will be the modeling of velocity dependent component of the oil contact dynamic. REFERENCES [] M.G., Introduction to Terrain-Vehicle Sytem, The Univerity of Michigan Pre, Ann Arbor, USA, 969 [2] J.Y. Wong. Theory of Ground Vehicle, Wiley, New York, 3 rd Edition, 2 [3] R. Bauer, W. Leung and T. Barfoot, Experimental and Simulation Reult of Wheel-Soil Interaction for Planetary Rover, IROS 25, Edmonton, Alberta, Canada, 2-6 Augut 25 [4] G. Ihigami, A. Miwa, K. Nagatani and K. Yohida, Terramechanic-Baed Model for Steering Maneuver of Planetary Exploration Rover on Looe Soil, Journal of Field Robotic 24(3), pp 233-25, Wiley InterScience, 27 [5] J. Olen, Realtime Procedural Terrain Generation, IMADA, Univerity of Southern Denmark, 3 October, 24 [6] R.W. Sumner, J.F. O Brien and J.K. Hodgin, Animation Sand, Mud and Snow, Computer Graphic Forum, Volume 8, No, 999