IROS2019国际学术会议论文集 2336

Effects of Limb Morphology on Transient Locomotion in Quadruped Robots Leanne Raw1 Student Member IEEE Callen Fisher1 Student Member IEEE and Amir Patel1 Member IEEE Abstract The natural world hosts a few mammals that display both a high degree of agility and speed However these animals have complex leg morphologies Robot designers are thus faced with the dilemma of which morphology to employ when designing the next generation of agile legged robots Thus this letter presents a novel investigation into the effects of limb morphology of quadrupeds during rapid transient maneuvers such as acceleration and deceleration Three leg confi gurations inspired by nature O X Type and All Ankle as well as All Knee confi gurations are compared Extensive large scale Monte Carlo simulations utilizing contact implicit trajectory optimization methods were employed on 100 randomly generated robots of varying sizes to determine the optimal confi guration for the task of rapid transient locomotion After extensive analysis the investigation revealed that an X Type leg confi guration outperformed all other confi gurations Ultimately these results will provide insight for the mechanical design of future agile quadruped robots I INTRODUCTION The animal kingdom is brimming with variations on a common limb design that facilitates specialized performance to enhance that animals ability to survive in its environment and within its niche in the food chain The quarter horse Equus Caballus greyhound Canis Familiaris and chee tah Acinonyx Jubatus are three examples of the complex morphological solution found in nature that service the task of attaining rapid transient motion An intriguing question we pose is What is the optimal leg bend direction for attaining rapid transient motions acceleration and deceleration The complex morphology found in nature has been broken down into three simplifi ed versions often utilized in robotics and an additional morphol ogy only found in robots has been added to give the four leg morphologies analyzed in this paper depicted in Fig 1 A brief description of each follows 1 O Type Joints face in opposite directions away from the center of the robot 2 X Type Joints face in opposite directions both point ing towards the center of the robot 3 All Ankles Joints both face towards the rear of the robot 4 All Knees Joints both face towards the front of the robot This work is based on the research supported in part by the National Research Foundation of South Africa for the Grant No 99380 and 101245 1 Authors are with the Faculty of Electrical Engineering University of Cape Town 7700 Rondebosch South Africa leanneraw uct ac za Fig 1 An example of the complex limb morphology common in nature and the four joint confi gurations they inspired that were simulated in this study The colored lines illustrate the joint confi gurations O Type green between the shoulder and hip joints X Type purple between the elbows and knees all backward facing at the ankles and wrists All Ankles blue The fourth confi guration that is pictured to the side of the image is the All Knee salmon layout which is not inspired by nature and can be found on physical robotic platforms Animal image adapted from 1 Many studies exist in both the biological sciences 2 4 as well as in the realm of engineering 5 7 that have delved into the investigation of certain aspects of limb design or performance Each of the above mentioned joint confi gurations have been implemented on a quadrupedal robot or simulation thereof allowing researchers to draw conclusions from their fi ndings which have been summarized in the info graphic in Fig 2 The following paragraphs will delve briefl y into some of these studies from the robotics front When observing the motion of the robot Thumper Jones et al studied the effect of a rear versus forward facing joint on impact forces and energy effi ciency on this monopedal robot 7 Results indicated that when equipped with a rear facing joint their robot displayed greater stability 7 The work reported that an All Ankle confi guration produced optimal performance due to its impact rejection and energy effi ciency resulting from gains at the point of touchdown These gains were associated with rotation of the tibia versus the translation of this strut experienced with an All Knee type confi guration Four other studies also utilized simulated platforms to test and evaluate the All Ankle leg confi guration 12 15 Conclusions were in favor of this confi guration over the All Knee option citing energy economy as the 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 IEEE3349 Fig 2 Summary of conclusions from existing research into leg confi guration for quadrupedal robots for the four joint orientations investigated in this work Images 8 11 driving factor The biped RAMone 17 showed that the All Ankle confi guration had an energy requirement of 60 less than that of the All Knee confi guration The reasons cited were due to the initiation of the swing phase where the motors on the All Knee robot had to reverse direction requiring a larger hip velocity This fi nding is supported by Monte Carlo simulations performed by Haberland 16 Smit Anseeuw et al 17 also concluded that the All Ankle confi guration benefi ts from propulsive forces generated by the extension of the joint during the toe off phase while the push off velocity of an All Knee confi guration was facilitated through extension of the hip joint Lee et al found that a knee type joint offered better propulsion while an ankle type joint yielded stronger braking forces 13 thus their fi nal choice for optimal leg confi guration was the X Type that combined these benefi ts A 2005 paper published by Zhang et al 14 detailing their study of their robot Biosbot found the All Ankle layout struggled to lift its hind limbs clear of the ground Both this confi guration and the All Knee confi gurations suffered from lack of grip and required complex controllers This lead to the conclusion that their preferred leg confi gurations were the O and X Type The robot on which these simulations were based had a cable attached to the rear of the torso which offset the center of gravity COG of their robot toward the rear thus affecting the results The researchers revealed that the O and X Type joint confi gurations resulted in those robots displaying a stronger propensity towards maintaining grip as well as decreasing the pitching motion of the torso 12 This decrease in the pitch of the torso during motion and inhibition of slip are two characteristics of the both the O and X Type confi gurations that have been repeated fi ndings for researchers over the years 13 14 This inherent stability makes for simpler control requirements along with robust motion of the platform 13 A common denominator among these papers has however been that each of these studies centered around periodic motion and as of writing this no investigation has been carried out into the realm of maneuverability No studies have considered the factors affecting the design and performance of quadrupedal limbs for transient motion In this letter we seek to answer the central question that has remained unanswered What leg confi guration is optimal for rapid transient motion This will be carried out via a comparison of the four leg confi gurations by performing large scale Monte Carlo simulations Section II describes the methodology utilized with a breakdown of the formulation of the equations of motion as well as how the trajectory optimization problem was solved in Section III The results of the optimization are presented in Section IV as well as analyzed and discussed in Section V The paper closes with concluding remarks and future work in Section VI II METHODOLOGY There exists a plethora of robots designed for a diverse range of problems These tasks include energy effi cient locomotion 18 steady state running 19 jumping 20 and locomotion on rough terrain 21 However our focus is on transient acceleration and deceleration maneuvers and particularly how leg bend direction effects the performance of the robot To investigate these transient motions and to form general insight into the effects of leg bend direction 3350 trajectory optimization methods were employed The task of interest is a long time horizon task start and end at rest while travelling a fi xed distance of 30 body lengths with an acceleration steady state and deceleration phase Current platforms vary in size from tiny micro quadrupeds to much larger robots As such theoretically there are an infi nite possible combination of parameters for a quadruped robot making it impossible to analyze them all Here based on a methodology developed by Haberland 16 we utilized large scale Monte Carlo trajectory optimization methods on 100 randomly generated robots to make the study feasible Parameters are selected from a carefully chosen design space Table I which limited the search space in order to remove infeasible robots Parameters were scaled according to the mass of the robot which was linked to the spine length The design space included the length and mass of the spine and leg limbs with their respective CoM positions set to the middle of the link The available torques max and allowed limb relative velocities max were scaled according to the mass of the robot on a non linear scale To further remove infeasible robots the parameters were related to each other to remove for example the possibility of having an extremely heavy robot that is physically small The design space was however large enough to encompass high speed quadruped animals and all the current quadruped robots 22 25 To date only a few large scale Monte Carlo optimization studies have been successfully completed 16 26 27 TABLE I PARAMETERBOUNDS ParameterLower boundUpper bound lspine0 2 m 1 5 m lspine COM50 lspine m mspine1 kg 30 kg for min spine lengthfor max spine length lfemur25 lspine m 75 lspine m ltibia25 lspine m 75 lspine m lfemur COM50 lfemur m ltibia COM50 ltibia m mfemur5 mspine kg 15 mspine kg mtibia5 mspine kg 15 mspine kg max 5mrobot Nm 5mrobot Nm max 3mspine rad s 3mspine rad s For this experiment four leg confi gurations were modelled using rigid body kinematics and are shown in Fig 1 Each leg consisted of two rigid links connected with a torque actuator The leg was connected to the rigid link spine through another torque actuator For each randomly gener ated parameter set four robots were generated each with a different leg confi guration All parameters were kept constant except for the leg confi guration allowing the performance to be compared Compliance was not added to the model as choosing the desired springs and dampers will effect the results and allowing the optimizer to select them will make the problem intractable Instead the optimizer was tasked with simulating the desired effect through optimal control For each parameter set the equations of motion EOM for the four leg confi gurations were generated using Euler Lagrange dynamics 28 The EOM are generated using the manipulator equation as follows M q q C q q q G q B A 1 where q are the generalized coordinates using absolute angles with being the applied actuator torques Ground reaction forces are represented by To ensure there is no biasing due to controller design or enforcing foot contact order optimal control with contact implicit methods were implemented This means that the optimizer selects the optimal applied torques and foot contact order to achieve the trajectory1 long time horizon task while minimizing a user defi ned cost function More detail can be found in Section III To improve the chances of fi nding an optimal solution 15 seed points were optimized for each parameter set and each leg confi guration The optimal solution was taken as the best result of the 15 seeds Once all the results had been gathered statistical methods in the form of bootstrapping 29 were applied to gain general insight into the optimal leg confi guration for these transient motions III TRAJECTORYOPTIMIZATION As mentioned previously to avoid the biasing effect of designing a controller optimal control methods are im plemented in the form of trajectory optimization This is achieved by defi ning several constraints and bounds that the optimizer must satisfy by varying the decision variables while minimizing a cost function A detailed tutorial on trajectory optimization can be found here 30 The relevant constraints and bounds are described in detail below A Constraints The optimizer needs to be guided to a feasible and realizable trajectory which is done through constraints These must be satisfi ed before the solution is considered feasible These constraints are as follows 1 Direct Collocation For this research the trajectories were discretized into N N 300 fi nite elements time periods using polynomials Each state trajectory is repre sented using a Runge Kutta bases with 3 collocation points 28 The differential equations 1 were solved using 3 point Radau polynomials with an accuracy of h2K 1 where K 3 since we are utilizing 3 point collocation 28 at the selected time points 28 31 The time step between each fi nite element is governed by the following constraint 0 5hM hi 2hM 2 where hiis the time period for the ithnode and hMis set to T N where T is used to scale the fi nal time see Section III C 2 Contact Implicit Optimization Methods The optimizer is tasked with fi nding the optimal trajectory and foot contact order which is achieved by implementing contact implicit optimization methods 32 This method has been shown to be a vital tool in studying locomotion 33 35 and have been implemented with increasing accuracy 28 The 1Trajectory of all the generalized coordinates of the robot 3351 drawback of this method is that it requires a large number of complimentary constraints 32 equation 8 to 16 to ensure that ground reaction forces are only applied when the foot is on the ground All contacts are modelled as inelastic collisions 32 36 and can have one of two modes slipping or sticking Slipping is modelled using a coulomb friction model2 32 Due to the complexity involved in solving these con straints regularization methods 37 are employed as fol lows 0i 0 i 0i K X j 0 ij 0 i K X j 0 ij 3 where ijand ijare the two parts of the complementary constraint for the ithnode and jthcollocation point To simplify the problem these are summed across collocation points and only evaluated at mesh points Once the com plimentary constraint has been solved it will equal zero ij ij 0 A relaxation method was implemented to solve these constraints where starts off large and tends towards zero further details in Section III C 3 Joint Angles Constraints on the joint angles are imple mented in order to enforce the desired leg bend confi guration Similar constraints are applied to the velocity of the limbs 4 Motor Model To improve the validity of the results a simple motor power model was implemented that limited the available torque according to the relative angular rate of the link as follows 16 max max max max max max 4 5 Terminal Condition In order to enforce the desired task start and end at rest while traveling a fi xed distance the following constraints were enforced qN q1 q N q1 5 where q1 0 to enforce zero velocity at the end and q1 is the rest confi guration This holds true except for the fi nal horizontal position which was constrained to 30 body lengths The initial conditions were enforced through bounds B Bounds To fi nd a feasible solution the optimizer varies the decision variables between the bounds to satisfy all the constraints while minimizing the cost function The decision variables include the following decV ar q q q h slack 6 where q along with its derivatives are the generalized coordinates of the robot with being the applied torques and representing the ground reaction forces The time between node points is represented by h and the complimentary constraints are solved using several slack variables 2 Coeffi cient of friction was set to 1 The initial conditions are bounded to the rest position of the robot with zero velocity The remaining variables are bounded suffi ciently high as to not bias the solution but to restrict the solve space C Solver Set up and Seed Loops Due to the complexity and non linearity of the problem there is no guarantee that the solver will fi nd the global optimal solution Therefore a number of optimization seeds points are executed for each robot and each leg confi gura tion 15 optimizations each Each seed point is randomly generated to properly explore the search space Once all optimizations had fi nished the best result for each parameter set and each leg confi guration was selected as the optimal solution For this study we optimize the trajectory for a minimum time solution As relaxation techniques were employed 37 was set to 1000 and the problem was solved iteratively 8 times with being divided by 10 after each successful solve If the optimizer could not fi nd a solution the seed was discarded and the next seed was generated After 8 successful solves the complimentary constraints were considered solved and the solution was saved The IPOPT 38 solver was used in GAMS General Algebraic Modelling System 39 The fi rst 5 seeds were solved with T set to 7 seconds the next 5 were solved with T set to 6 seconds and the last 5 were solved with T set to 5 seconds This was found to drastically improve the convergence rate across the randomly generated robots due to their varying sizes IV RESULTS A total of 6000 seeds were optimized with 1067 successful results that satisfi ed all constraints mentioned in the method section This gives a total convergence ratio of 17 8 which is comparable to other large scale trajectory optimization studies 16 26 This experiment was performed on a cluster of 15 machines with 3 25GHz Intel i5 proce