IROS2019国际学术会议论文集0754

Modeling and Force Control of a Terramechanical Wheel-Soil Contact for a Robotic Manipulator Used in the Planetary Rover Design Process* Jan Wachter1, Ralf Mikut2and Fabian Buse3 AbstractThe German Aerospace Center (DLR) has devel- oped the Terramechanics Robotics Locomotion Lab (TROLL) to provide a feasible testing facility for developing planetary ex- ploration rovers, as well as validating terramechanical models. A robotic manipulator is used to provide the required degrees of freedom to the mounted wheel or subsystem, making it necessary to actively control the interaction force of the wheel- soil contact. This paper is concerned with the development of a feasible force control strategy for the testbench wheel-soil contact during single-wheel experiments. For this purpose, a terramechanical model has been developed to accurately map the dynamic processes relevant for the force control design, which is later used in a testbench simulation framework to predict and evaluate the performance of control strategies. The Adaptive Admittance Control (AAC) scheme developed is adapting the gain based on the current control deviation, the rotational velocity of the wheel and an estimated soil stiffness during the experiment. The AAC is evaluated using a benchmark single- wheel experiment and shows superior performance compared to standard admittance control. I. INTRODUCTION Mobility and therefore a robust and performant locomotion system are essential for the success of planetary explo- ration missions. Rough, sandy terrain in particular poses an immense risk to exploration rovers. Well-known examples include the NASA Mars rovers SPIRIT, which became ir- recoverably embedded in a crater fi lled with loose sand 1, and the CURIOSITY rover which faced severe diffi culties during sand ripples, due to prior wheel damage 2. In order to improve rover locomotion systems for future missions, the German Aerospace Center (DLR) is conducting research into deeper understanding of the wheel-soil interaction of planetary rovers through terramechanical modeling, as well as the development and control of locomotion systems. In order to provide a feasible and replicable testing envi- ronment for developing rover locomotion subsystems and validating terramechanical models, the DLR has developed the Terramechanics Robotics Locomotion Lab (TROLL) 3, as shown in Fig. 1. The TROLL uses a six-axis robotic ma- nipulator (KUKA KR210 R3100-ultra) to achieve the desired wheel trajectory, whereas other state of the art single-wheel testbenches deploy trolley systems that confi ne the wheel to a predetermined path 4, 5. Using the TROLL setup *This work was not supported by any organization 1Institute for Automation and Applied Informatics (IAI), Karlsruhe In- stitute of Technology, Karlsruhe, G 2Institute for Automation and Applied Informatics (IAI), Karlsruhe Institute of Technology, Karlsruhe, G 3Institute of System Dynamics and Control, German Aerospace Center (DLR), Oberpfaffenhofen, Germanyfabian.busedlr.de Fig. 1.Terramechanics Robotics Locomotion Lab (TROLL) Setup with Attached Single-Wheel-Tool enables arbitrary wheel paths within the robot workspace, which vastly increases freedom during experiment design. This comes at a cost: The vertical axis of the testbench, responsible for simulating gravitational forces of the rover, cannot be passively loaded with weight-plates as in other test setups, but has to be actively force-controlled. This work is concerned with the modeling and development of a feasible force control strategy including terramechanical wheel-soil interaction, in order to provide a robust testing environment. To use the control interface provided (RSI- KUKA) the output of the force control scheme needs to be a position correction based on the force feedback delivered by the six-axis force/torque sensor mounted at the connection of the robot hand and the wheel system. This position correction is enforced by the position control of the robot, which is considered a black box in this work. Taking these restrictions into account, the required control scheme can be classifi ed as a position-based explicit force control scheme 6. Further requirements for the force control system stem from the complexity of the wheel-soil interaction: The stiffness of the wheel-soil contact changes over multiple magnitudes, comparing loose and compressed states of the used soil simulant; dynamic effects such as slip sinkage depending on the imposed wheel slip as well as the used wheel grousers1 further infl uence the stiffness. In addition, the stiffness of the wheel-soil contact is back-coupled with the previous wheel- soil contact due to the plastic deformation introduced by the interaction itself. These requirements distinguish the system 1Grousers are paddle-like structures positioned on the outer surface of a wheel to improve locomotion on sandy terrain 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Macau, China, November 4-8, 2019 978-1-7281-4003-2/19/$31.00 2019 IEEE560 signifi cantly from conventional industrial force control tasks. In standard literature, no feasible control schemes tailored to these requirements can be found. As a result, this work is concerned with the development and evaluation of a suitable control approach as well as the investigation of the wheel- soil interaction itself. The model for wheel-soil interaction is described fi rst, focus- ing on the developed soil element and contact model (Section II). This is followed by the Adaptive Admittance Control and the benchmark standard admittance control (Section III). Next, the benchmark single-wheel experiment (Section IV) is presented together with the results (Section V) and discussed (Section VI). Some parts of the sections below are based on the fi rst authors masters thesis 7. II. SOILMODEL FOR THESIMULATIONFRAMEWORK In order to predict and evaluate the performance of the control schemes developed, a simulation framework is re- quired. State of the art terramechanical models used for rover development that are computationally fast 4 focus on the steady state behavior. This is suffi cient for the relatively slow driving speeds of current generation rovers. High-tier models 8, 9, 10 use discrete element approaches to map dynamic effects, which makes them too computation- ally expensive for this application. An extensive review of terramechanical models is given in 11. For use in a control approach, dynamic effects of the force response of the wheel- soil interaction must be mapped with a fast computation time, and a new model has therefore been developed for this specifi c task. The resulting reaction force in the vertical direction Fzof the wheel-soil interaction can be generally expressed as: Fz= Z Acon z(,)dAcon,(1) with Aconbeing the contact area of the wheel-soil interaction, z(,) the stress depending on the sinkage and the contact angle . The modeling process is divided into two sub-problems: (A) a soil element which gives a sinkage stress relationship and (B) a contact model which provides the contact area depending on the sinkage, as well as the stress distribution over the contact area. A. Soil Element The soil element is based on basic elasto-plastic behavior, which has already been successfully used in a non-wheel- soil interaction application 12, but has been extended in order to provide better performance with respect to wheel- soil interaction. The main effects covered are: Elasto-plastic behavior of the wheel-soil contact is used as the basic behavior. Increasing stiffness of deeper soil layers, due to gravi- tational pre-compaction as well as the previous loading. Elastic unloading characteristics: during the unloading process, elastic behavior with non-zero force response is modeled. Creep behavior for the soil is implemented. To account for the aforementioned main effects, the soil element shown in Fig. 2 is used. The resulting stress in the c1 Off set c2(tot) damp d zg w zw zg0 ptot Fig. 2.Block Diagram of the Developed Soil Element vertical direction zis calculated as follows: z= c1w0 damp 0 damp, (6) with the damping coeffi cient d, the damping threshold damp and the creep velocity of the ground zg. The creep velocity is integrated and superimposed with the plastic deformation to account for the changing surface height of the soil. 561 B. Contact Model Contact Area: As an estimation for the contact area in this work the circumference of the embedded part of the wheel is used, as shown in Fig. 3. The blue contact area Acir w r Fig. 3.Contact Area of the Wheel-Soil Interaction in Fig. 3 can be calculated as: Acir= br ,(7) using the contact angle , the width of the wheel b and the radius of the wheel r. The contact angle can be calculated using the sinkage of the wheel wwith the following relation: = 2arccos ?r w r ? , for 0 w 2r.(8) Stress Distribution: Since not all parts of the wheel have the same sinkage, the stress over the contact area is not constant and depends on the contact angle . The real stress distribution is unknown, a linear distribution in scdirection over the contact length lcis assumed, similar to the proposed linearization in 13. Along the width of the wheel (yc direction) the stress is assumed to be constant. This leads to the equation for the stress distribution over the contact area z(sc): z(sc) = max z ? 1 |sc| lc ? , sc lc,lc,(9) with max z calculated by (2), the arc coordinate scin the contact coordinate system and the contact length lcas shown in Fig. 4. The contact length lcis calculated using (8) according to: lc= rarccos ?r w r ? , for 0 w 2r .(10) Resulting Reaction Force: Inserting (9) for the stress distribution and (7) for the contact area into (1), the resulting force can be calculated as Fz= Z Acir z(sc)dAcir,(11) which can be expressed using the symmetry of the stress distribution, and (9): Fz= 2 Z b 0 Z lc 0 max z ? 1 |sc| lc ? dscdyc.(12) The integral in (12) can be solved using the contact length expression in (10) and expressed as a closed-form solution: Fz= brmax z arccos ?r w r ? .(13) w z sc r lc yc zc sc lclc Fig. 4.Linearized Stress Distribution in the Contact Area This equation can be rewritten for better interpretability by using (7) and (8), which results in: Fz= Acir 2 max z .(14) To conclude, this model can be implemented as computa- tionally effi cient in a closed-form solution. C. Parameter Identifi cation Process The data used for parameter estimation is obtained by means of an experiment with alternating loading and un- loading phases, whereby the penetration depth is increased with each loading cycle and contact is never lost during the unloading phases. The parameters d, c1, c20and c2pare fi tted using the SIMULINKParameter Estimation Toolbox. The parameter dampis chosen such that numerical stability is ensured. Fig. 5 shows that the developed wheel-soil interaction model is able to reproduce the qualitative force response observed during experiments. The model is integrated in the TROLL 0246810121416 Fig. 5. Resulting fi t of the developed wheel-soil interaction model with tuned parameters. simulation framework and used to develop the AAC control scheme. III. FORCECONTROLSYSTEMDESIGN As mentioned in the introduction, the required force control system can be classifi ed as a position-based explicit force control scheme. A standard approach is the admittance control (AC), which is used as a benchmark during the 562 practical trials. Based on the admittance control approach, an adaption scheme is developed to account for the specifi c demand of the described use case. A. Admittance Control (AC) Mechanical admittance A is the ratio of the velocity x to the applied force f: A = x f .(15) Accordingly, the admittance control scheme relates the con- trol deviation to a velocity perturbation of the end effector, which can be integrated to obtain the required position correction xf14. If formulated as a transfer function, the controller can be expressed as: K(s) = 1 s L(vf)(s) L(f)(s) = 1 s A(s),(16) where L is the Laplace transformation. Choosing zero, fi rst or second order functions for A(s) leads to I, PI and PID control actions, respectively. In this paper, a zero order function has been chosen for the admittance A(s) = A = const. in order to avoid a proportional control action to increase robustness, required due to noisy force feedback. In combination with an inner position control loop, this results in the block diagram shown in Fig. 6. + - fdesfmeaxtcp A Integrator vfxf Contact with Environment f Position Controlled Robot Fig. 6.Block Diagram of Admittance Control Using an Inner Position Loop B. Adaptive Admittance Control (AAC) This control scheme is based on the zero order admittance control scheme presented in the previous section. The AAC scheme uses the concept of gain adaption of the integral control action to maintain feasible performance as well as stable behavior during the experiment. Adaptive admittance control concepts using higher order admittance approaches are described in 15. The AAC gain Againis calculated using a correction factor kgainand the default gain A0 gain according to: A(s) = Again= kgainA0 gain. (17) The correction factor kgainis calculated in three subsystems that are described below. NormalizedForceError(NFE): This subsystem is designed to lessen the gain for small control deviations. The corresponding correction factor kNFEis calculated based on the Normalized Force Error (NFE) fNFE which is defi ned as: fNFE= |f| fdes ,(18) with the control deviation f and the force setpoint fdes. The resulting algorithm for calculating of kNFEdivides the fNFE input into three regions (acceptable (A), transitional (B) and unacceptable (C), see Fig. 7) according to the parameters: ac- ceptable threshold g0, lower limit of the correction factor v0, unacceptable threshold g1and upper limit of the correction factor v1. As a transition function in the transitional region, ft(fNFE), a square-root function is used. The correction factor kNFEcan be calculated as follows: kNFE= v0, fNFE g0 ft(fNFE),g0 fNFE g1 v1,g1 fNFE. (19) Fig. 7 shows the resulting characteristic line of the in- put/output behavior of the NFE subsystem. fNFE kNFE 0 g0g1 v0 v1 A B C Fig. 7.Characteristic Line of the Input/Output Behavior of the NFE Subsystem Wheel Movement Compensation (WMC): This subsystem is designed to account for the infl uence of the turning wheel and to assure a smooth transition during the start of wheel rotation. The rotational velocity of the wheel is ramped up from zero to the desired steady state value. The correction factor kWMCis calculated based on the ratio rWMCbetween the current setpoint curand the steady state setpoint of the rotational velocity set: rWMC= ? ? ? ? cur set ? ? ? ?. (20) The correction factor kWMCis calculated according to: kWMC= ( vmrWMC,0 rWMC 1 vm,1 rWMC, (21) with the parameter vmrepresenting the WMC gain. Normalized Stiffness (NST): This subsystem is designed to adapt the admittance gain to the changing stiffness of the soil. The corresponding correction factor kNSTis calculated based on the normalized stiffness cNST , which is defi ned as: cNST= cest c0 ,(22) where c0is the default stiffness and cestis the stiffness which is constantly estimated during the experiment. The algorithm for calculating kNSTdivides the cNSTinput into four regions (soft saturation (A), soft transition (B), 563 hard transition (C), hard saturation (D), see Fig. 8) which are characterized by the parameters: soft soil threshold gs, soft soil correction value vs, hard soil threshold gh, hard soil correction value vhand the default stiffness c0. The cNST kNST 0 1 1 gsgh vh vs A B C D Fig. 8.Characteristic Line of the Input/Output Behavior of the NST Subsystem correction factor kNSTis calculated according to: kNST= vs,cNST gs fts(cNST),gs cNST 1 fth(cNST),1 cNST gh vh,gh cNST, (23) where the transition functions in the soft transition zone fts(cNST) and hard transition zone fth(cNST) are chosen as linear functions. The stiffness cestis estimated using a linear spring model: Fmea= cest(rwhrsur),(24) where Fmeais the measured force in contact direction, cest the estimated soil stiffness, rwhthe wheel position and rsur the surface height. The surface height is determined during the experiment, where the measured force is larger than the force threshold of 10 N for the fi rst time during the setting phase. This value is not changed during the experiment. The stiffness is subsequently estimated using the recursive least square method implemented in SIMULINK. Calculating the Resulting Gain Multiplier: Combining the correction factors kNFE, kWMCand kNSTdescribed above, the gain multiplier kgainis calculated as follows: kgain= (kNFE+kWMC)kNST,(25) the correction factors for the NFE subsystem and the WMC subsystem are added because their effects complement each other. The correction factor for the NST subsystem is mul- tiplied, because different soil stiffnesses call for the same qualitative behavior, but the gain needs to be scaled accord- ing to the changed stiffness. Using the correction factor kgain the admittance control Matrix Againis calculated according to (17). IV. BENCHMARKEXPERIMENT The fi xed slip single-wheel experiment has been chosen for experimental evaluation of the force control schemes presented above. The benchmark experiment can be de- scribe