IROS2019國際學(xué)術(shù)會(huì)議論文集 1919

Energy Effi cient Locomotion Strategies and Performance Benchmarks using Point Mass Tensegrity Dynamics Brian M Cera Anthony A Thompson and Alice M Agogino Abstract This work introduces a novel 12 motor paired cable actuation scheme to achieve rolling locomotion with a spherical tensegrity structure Using a new point mass tensegrity dynamic formulation which we present we utilize Model Predictive Control to generate optimal state action trajectories for benchmark evaluation In particular locomotive performance is assessed based on the practical criteria of rolling speed energy effi ciency and directional trajectory tracking accuracy Through simulation of 6 motor 12 motor paired cable and 24 motor fully actuated policies we demonstrate that the 12 motor schema is superior to the 6 motor policy in all benchmark categories comparable to the 24 motor policy in rolling speed and is over fi ve times more energy effi cient than the fully actuated 24 motor confi guration I INTRODUCTION Spherical tensegrity robots tensegrities are lightweight soft robots that are comprised of an elastic tension net work that suspends and connects isolated rigid rods A six bar spherical tensegrity shown in Figure 1 has six rigid rods held together by 24 series elastic cables Notably the structural properties of compliant and low weight tensegrities have proven to be advantageous in applications that involve high impact loads and co robotic cooperation with humans potential applications for these tensegrity robots include space surface exploration rovers 1 and disaster response robotics However the performance and energy effi ciencies of spherical tensegrity robots has yet to be evaluated for practical use cases in realistic scenarios Motion planning and optimal control for rolling locomo tion has been a major driving force for tensegrity research in recent years As a result innovative approaches utilizing evolutionary algorithms data driven methods and model based optimal control have all been developed to control these novel complex robots In particular great emphasis has been placed on optimal performance with respect to rolling speed under non ideal conditions and rough terrain but less consideration has been made for practical implementation challenges such as energy effi ciency controllability and directional trajectory tracking accuracy The goals of this paper are to introduce and evaluate a novel 12 motor paired cable actuation scheme for tensegrity locomotion presenting new tools and benchmarks to ade quately assess the performance of a tensegrity s mobility with respect to energy effi ciency rolling speed and di rectional trajectory tracking Understandably the values for these benchmarks heavily depend on the hardware design the number of actuated cables the actuators used and which specifi c cables are controlled Nevertheless we believe that this preliminary exploration into these quantifi able metrics Fig 1 Example of a spherical tensegrity robot which locomotes through shape shifting by controlling individual cable tensions Each rod is 60 cm in length Photo courtesy of Squishy Robotics Inc elucidates a greater understanding of practical tensegrity hardware and control policy design for future mobile tenseg rity robots In this paper we fi rst explore the new point mass tensegrity dynamics in great detail to demonstrate how equations of motion for the tensegrity dynamics can be easily constructed in a rigorous and procedural manner Next we outline our approach for motion planning through the use of Model Predictive Control MPC in conjunction with the new point mass formulation Lastly we demonstrate the approach s versatility by generating optimal state action trajectories for different tensegrity actuation confi gurations and evaluating their locomotive performance using relevant benchmarks II PRIORRESEARCH Various rolling locomotion control policies and actuation confi gurations for spherical tensegrities have been explored Developments in single cable actuation i e where only one cable is actuated at a time have chiefl y relied on the design and analysis of hand engineered control policies 2 3 In contrast due to the nonlinear coupled dynamics of spherical tensegrity structures multi cable actuation i e simultaneous actuation of all 24 cables has proven to be a signifi cantly more challenging task Methods to explore multi cable actu ation primarily consist of Monte Carlo simulations and data driven machine learning methods In recent years research in multi cable actuation for tensegrities 4 5 has found that locomotion by rolling can be achieved by shape shifting to a desirable quasi static 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 IEEE4678 geometry that positions the center of mass outside of the support polygon and recent advancements in continuous rolling locomotion for tensegrities have utilized deep rein forced learning 6 7 Finally many tensegrity topologies and actuation con fi gurations have been explored Numerous designs of even just spherical tensegrities have demonstrated variability in the number of actuated cables degrees of freedom overall compliance weight distribution etc Continuing on the in novations of these explorative hardware designs this paper presents a novel 12 motor paired cable actuation scheme and compares its performance to other schemes using standard benchmarks of energy effi ciency rolling speed and direc tional trajectory tracking III THESIX BARSPHERICALTENSEGRITYROBOT AND POINTMASSDYNAMICS In this paper we focus on the Class 1 spherical tenseg rity topology Specifi cally Class 1 tensegrities are special tensegrity structures constructed with compressive bodies which bear no rigid joints and which are interconnected solely through series elastic tensile elements For this reason Class 1 tensegrity dynamic equations of motion are well structured and can be procedurally obtained when given a fi xed set of parameters In prior work 8 a minimal representation of the 3D rigid body dynamics for tensegrity systems is presented While the reduced state dimension of this minimal representation is advantageous this particular formulation is susceptible to dynamic singularities which can make robust and reliable motion planning and optimal control diffi cult In this section we present a new simplifi ed point mass tensegrity dynamics representation which can be easily formulated for any Class 1 tensegrity structure For this point mass formulation rather than representing true rigid body dynamics we assume that the entirety of each rod s mass can be distributed between two point masses located at the ends of the rod hereafter also refered to as nodes Notably this assumption s validity is largely depen dent on the actual hardware design of the tensegrity robot as an example consider the tensegrity SUPERball 1 designed by the Intelligent Robotics Group at NASA Ames which carries most of its mass closer to the ends of the rod where heavier motor assemblies and electronics are housed Thus this point mass assumption can often be relatively accurate and greatly simplifi es the formulation of tensegrity dynamic equations of motion enabling rapid design and prototyping of new innovative topologies in simulation With these simplifying assumptions we now consider only the positions velocities and accelerations of each point mass We defi ne vectors p and p R3Ncontaining the individual xyz positions and velocities of the N nodes as p x1 y1 z1 xN yN zN T 1 p x1 y1 z1 xN yN zN T 2 Next we assume that forces are imparted on each node purely through idealized two force members i e the rods and cables in pure compression tension or from the external environment e g contact forces with the ground For the re mainder of this section we discuss the dynamic formulation of the cable and rod forces intrinsic to tensegrity structures A Series Elastic Cable Forces Forces which act on the nodes due to the spring cables are calculated simply using Hookean approximations with special consideration that no compressive forces can be applied through the cables Fcables j max 0 kj Sj Lj 3 Here kjis the stiffness of the series elastic cable j Sjis the separation distance between the two end nodes attached to cable j and Ljis the spring cable assembly rest length see Fig 2 Given a cable connectivity matrix C RJ N see 8 for details with rows Cjthat encode cable interconnections between pairs of nodes we represent elements in the vector of cables forces RJ as j 2k j softplus j zj kzjk2 j 1 J 4 where variables zjand j are defi ned as follows zj CT jCj I3 p j Sj Lj s zT jzj 2 Lj zj RJis a sparse vector that contains the directional vector lying along the direction of the cable j The softplus function above is a smooth approximation to the non differentiable rectifi er function used in Eq 3 approximating max 0 i with tunable smoothness parameter softplus j q 2 j 2 j 2 0 5 In practice this Lipschitz smooth approximation demon strates better numerical stability in simulation and its contin uously differentiable property is well suited for calculating the locally linearized dynamic models used in the receding horizon control methods discussed in later sections B Rigid Body Constraint Forces The other set of essential forces in tensegrity structures are the rigid body constraints which constrain the nodal positions relative to each other and the environment Rather than model the rods using a linear elastic model as with cable forces in the prior section we instead adopt a constrained dynamics approach The motivation behind Fig 2 Hookean linear elastic model between two point masses 4679 this is that penalty or energy barrier methods which rely on restorative forces to maintain rigid connections necessi tate large stiffness parameters and lead to stiff differential equations Instead the constraint forces we describe here neatly cancel out the components of the applied forces that violate rigid constraints at each timestep creating accurate and numerically tractable dynamic simulations In our work we adopt a similar approach to 9 and defi ne constraint vectors G p and G p RM to represent the implicit constraint functions and their time derivatives where M is the number of active dynamic constraints Each scalar element Gi p is a single implicit constraint function that is satisfi ed when equal to zero If we assume that initial positions and velocities of the system satisfy dynamic constraints i e G p 0 and G p 0 then any forces which maintain legal accelerations i e G p 0 will be valid forces which satisfy all dynamic constraints We decompose the vector of legal forces F0which are ultimately applied to the particle masses into two compo nents F the total forces originally applied to the particle and F which are resultant constraint forces that cancel out any illegal accelerations We also introduce the inverse mass matrix W which contains the reciprocal of each particle s mass as elements along the diagonal Thus the legal acceleration condition can be written as G p G p p p G p p p 6 G p p p G p p W F F 0 7 Simplifying notation of G p p and G p p as matrices J p and J p respectively and dropping the matrices explicit dependencies on p we rewrite JW F J p JWF 8 JW JT J p JWF 9 where Eq 9 is a result of Eq 7 in combination with the principle of virtual work which restricts constraint forces F to lie in the subspace spanned by the constraint gradient vectors i e the rows of G p p The vector of Lagrange multipliers determines how much of each constraint gradient is applied providing a measure proportional to the reaction force applied due to the corresponding constraint To prevent the accumulation of numerical drift corrective stiffness and damping terms are appended to Eq 9 JW JT J p JWF ksG kd G 10 As a concrete example consider the constraint forces imposed by the rigid body connection between two endpoint nodes of a rod Given nodal positions and velocities rod length Lrod q and a rod connectivity matrix R RQ N with rows Rqthat encode rod interconnections we write an implicit rod constraint function constraining relative distance between nodal positions pAand pB the constraint function s respective time derivative and their associated Jacobian matrices as Gi p kpB pAk2 2 L 2 rod q 11 1 2p T RT qRq I 3 2p L2rod q 12 Gi p pT RT qRq I 3 2 p 13 Ji p pT RT qRq I 3 2 14 Ji p pT RT qRq I 3 2 15 Given these implicit constraint functions which are obtained for each rod we combine these results with the formulas in Section III A to guarantee that nodal accelerations are realis tically and stably simulated with no pair relative acceleration components lying along the axis of the rigid rods IV MOTIONPLANNING USINGMODELPREDICTIVE CONTROL Tensegrity motion planning and control can be overwhelm ingly complex due to the high dimensional highly coupled nonlinear dynamics inherent to tensegrity robots Generating optimal state action trajectories i e the control and time evolution of actuated cable rest lengths and the resulting dynamic states can be a diffi cult task when considering the entirety of the 72 dimensional state space and up to 24 dimensional action space Fortunately we are able to leverage the well structured dynamics of Class 1 tensegrities by importing the dynamic equations of motion derived in the Section III as optimization constraints for model based receding horizon control such as Model Predictive Control MPC In short MPC is a control schema which iteratively solves a constrained optimization problem and implements only the fi rst control input at the each timestep 10 11 The primary benefi t of this control scheme is the ability to leverage dynamic models to optimize future behavior over fi nite time horizons while also complying with state and input constraints such as those defi ned by realistic safety and actuator limitations Additionally because MPC is an iterative algorithm the approach is inherently robust to unforeseen disturbances In this work we utilize MPC to automatically design and evaluate tensegrity locomotion actuation policies i e how to optimally actuate cable rest lengths The continuous dynamics of the robot are linearized about the robot s current state and discretized using a trapezoidal approximation p k 1 pk 1 p k pk dT 2 pk pk 1 2 p0 p x xk p x xk 1 16 where dT is the simulation timestep x R96is a con catenated vector of cable lengths and nodal position velocity states xkis the deviation about the linearization point x0 and p0 R3Nis the current state acceleration at x x0 4680 where p is calculated as follows p W JT JWJT 1 J p ksG kd G I JT JWJT 1JW J X i i Fext 17 Eq 17 is obtained by combining the results of Sections III A and III B Note Fextare the total forces applied to the tensegrity robot which are external to the system e g ground contact reaction forces and are calculated using damped linear elastic collisions These formulas are similar to those discussed in Section III A and thus a formal discussion of these external force calculations is excluded for the sake of brevity Using the derived linearized and discretized dynamics as optimization constraints we minimize the following cost function T X k 1 k 1 1 N X i 1 pT iD 2k Lkk1 3k Lkk1 18 Here D R3is the desired direction of travel T is the MPC fi nite time horizon and 1 is a discount factor placing less weight on later states to account for linearization errors Finally 1 2 3are weighting parameters and Lk and Lk RJcontain deviations of the kth step cable rest lengths about the neutral pretensioned lengths and initial lengths used for linearization respectively Combined these cost terms reward rolling velocity in a desired direction while simultaneously penalizing cable rest length deviations from both initial pretensioned lengths and current rest lengths i e k 1 respectively preventing the robot from excessive deformations and generating sparse motor actuation The convex cost function above and linear equality and inequality constraints from the dynamics state actuator lim its and initial conditions thus form a linear program which is easily minimized using any convex optimization solver For this work Gurobi Optimizer and YALMIP 12 were used in MATLAB to solve the optimization problem at each timestep iteration Combined these tools enable us to rapidly evaluate the novel 12 motor tensegrity actuation policy and its relative performance V 12 MOTOR PAIRED CABLEACTUATION Similar to the 24 motor actuation scheme the 12 motor paired cable actuation scheme controls all 24 cables in a spherical tensegrity however for the 12 motor scheme two cables are coupled by a single motor For this actuation scheme a pair of cables meet at a single node the retraction of one cable means the extension of the other cable in that pair Thus while all 24 cables are actuated only 12 degrees of freedom exist in the system Interestingly this new paired cable schema has some practical advantages over its 24 motor schema counterpart The most immediate advantages are that fewer parts are necessary so the robot is less prone to mechanical failure and that the tensegrity robot will weigh signifi cantly less VI COMPARISONS OFROLLINGLOCOMOTION STRATEGIES In this fi nal section we discuss tensegrity rolling locomo tion in detail and compare three cable actuation schemes each with varying degrees of control authority 6 motor underactuated 12 motor paired actuation and 24 motor full actuation schema In particular these actuation schema vary the number of cables that are driven by motor actuators and consequently which cables remain as passive tensile elements As a result we demonstrate that greater control authority can provide improved performance at the cost of additional hardware and controller complexity In the results that follow we utilize MPC with the dynamic constraints introduced earlier to generate optimal state action trajectories for evaluation Nota