IROS2019国际学术会议论文集 0520

Stability and Gait Switching of Underactuated Biped Walkers Martin Fevre1 Hai Lin2 and James P Schmiedeler1 Abstract This paper introduces a new gait switching ap proach for underactuated biped walkers The switching condi tion relies on a reduced region of attraction that of the unactuated dynamics which is shown to be suffi cient to predict falls A gait transition is accordingly stable as in the robot does not fall if the biped s state is within the reduced region of attraction of the switched in gait when switching takes place Two and fi ve link biped models complete a sequence of random gait transitions using the switching logic The condition is also used to enlarge the full region of stability of the fi ve link model by embedding feedback stabilized trajectories from a gait library in a mapping The mapping stitches stable trajectories to the orbit of the desired gait based on each gait s reduced region of attraction This improves the robustness of the fi ve link model walking on uneven terrain without ground perception regardless of the size of the gait library I INTRODUCTION Fully actuated biped robots with powered ankles often rely on the zero moment point principle to guarantee that the robot does not fall 1 typically prioritizing robustness over energetic effi ciency In contrast underactuated bipeds exploit the same natural dynamics of human locomotion to offer the promise of effi ciency 2 but have limited ability to reject large disturbances in their unactuated degrees of freedom DoF One way to enlarge a dynamic walker s region of attraction RoA is to store a library of stable periodic gaits that span different regions of the state space and design a switching logic among them 3 The multiple Lyapunov function theorem MLFT provides theoretical tools for de signing such switching laws 4 5 giving robots the ability to transition among walking gaits just like humans switch gaits to robustly and effi ciently navigate uneven terrain While conventional dynamic programming is exponen tial in the DoF other motion planning approaches like probabilistic road maps 6 or rapidly exploring randomized trees RRT 7 can quickly search high dimensional spaces to design feasible trajectories RRT inspired the LQR tree algorithm 8 which generates a lookup table control policy consisting of feedback stabilized trajectories leading to a goal state The intent is analogous to MLFT except that branches trajectories are added to the tree goal state to maximize the set of stabilizable states i e enlarge the RoA The resulting set becomes the union of all branches which are stabilized using local controllers 9 10 This work was supported in part by the National Science Foundation under Grant IIS 1527393 Corresponding author mfevre nd edu 1Martin Fevre absolute stance angle q2is unactuated b Five link biped model with curved feet Relative hip q1 q2 and knee q3 q4 angles are actuated absolute torso angle q5is unactuated Parameters in Table I match ERNIE robot 15 TABLE I Parameters of the biped models a Two link SymbolParameter DescriptionUnitsValue Lleg lengthm1 0 LCleg CoM from proximal joint m0 4 mleg masskg2 5 mHhip masskg10 0 Ileg inertia about CoM kg m20 01 b Five link SymbolParameter DescriptionUnitsValue XFankle offsetm0 RFfoot radiusm0 201 LFfemur lengthm0 360 LTtibia length knee to foot bottom m0 378 LBtorso lengthm0 132 LFCfemur CoM from proximal joint m0 125 LTCtibia CoM from proximal joint m0 136 mFfemur masskg1 47 mTtibia masskg1 07 mBtorso masskg14 47 IFfemur inertia about CoM kg m20 0238 ITtibia inertia about CoM kg m20 0242 IBtorso inertia about CoM kg m20 1001 Im refl ected hip knee motor inertiakg m20 1731 used to orchestrate stable gait transitions on both models Section IV reports simulation studies to assess the validity of the switching logic For walkers underactuated by one DoF the reduced RoA can be represented by a 2D surface For 3D walkers with two degrees of underactuation the reduced RoA becomes a 4D surface which can be estimated offl ine via numerical computations Thus the approach is not limited to planar bipeds but is applicable to 3D walkers as well II DYNAMICMODEL OFWALKING The two and fi ve link planar biped models in Fig 2 consist of rigid links and are left right symmetric so one can search for periodic walking gaits using a single step Bipedal walk ing dynamics can be modeled as a series of fi nite time single support phases and infi nitesimally short ground impacts For a planar biped with n rigid links and m actuators generalized coordinates q q1 qn T Q can be used to derive the equations of motion during single support M q q C q q q G q B q u 1 where Q is an open subset of 0 2 nthat represents feasible confi gurations of the robot M q Rn nis the inertia matrix C q q Rn nis the Coriolis matrix G q Rn is the gravity vector B q Rn mis the input matrix and u U Rmis a vector of independent control inputs The hypersurface S is the guard set that defi nes the limit of the continuous dynamics of Eq 1 S n py Q R py h py 0 o 2 where pyand pyare the swing foot s vertical position and velocity and h is the step height The impacts assumed perfectly inelastic update the coordinates q before touch down to q after touchdown via the discrete map q qq q S q dq q q S 3 where qis the n n switching matrix and dqis an n n matrix relating pre and post impact velocities 15 The hybrid system may be written in state space form using coupled fi rst order differential equations x f x g x u x S x x x S 4 where X x q q q Q q Rn represents the state space and S X is the impact map III METHODOLOGY A Parameterization in the Unactuated Dynamics Many complex dynamical systems can be decomposed into a strongly actuated subsystem with nearly linear dynamics and a weakly actuated subsystem with highly nonlinear dynamics 16 For example the system of Eq 4 can be rewritten as x1 f1 x1 x2 x2 f2 x1 x2 u 5 where x1 Rn1and x2 Rn2represent the weakly and strongly actuated states respectively 14 The x1states can be used to parameterize x2 trajectories in the full order model For dynamic walkers with no ankle actuation a coherent choice for x1is a measure of gait progression Simulation and experimental studies alike have suggested that while dynamic walkers can reject large disturbances in their actuated DoF their dominant source of instability is their unactuated dynamics 17 18 Figure 1 shows an example of how most dynamic walkers typically fall The angle between the ground and the line from the ground contact to the hip quantifi es gait progression so observing its post disturbance behavior is suffi cient to predict a fall As ERNIE experiences an unseen terrain height change 6 of leg length in Fig 1 increases at the beginning of the gait cycle left image Later in stance though changes 2280 ait orbit 2 17 ait orbit ait orbit or21 DISTURBANCE DISTURBANCE DISTURBANCE Fig 3 Representation of realization t x0 of Eq 4 switching gaits when at boundary of desired gait s region of attraction At steady state system evolves on zero dynamics trajectory top After disturbance t x0 can 1 remain in bottom left or 2 exit bottom right in which case t x0 must transition to another gait until it reenters sign middle image and the x1 dynamics exit their RoA As a result the robot falls backward right image This highlights that one need not consider the full order dynamics x1 x2 to predict falls in dynamic walkers If denotes the reduced RoA of the x1states any realization t x0 of Eq 4 exiting the full RoA as a result of bounded disturbances must have exited in the fi rst place Because provable stability must rely on a full order representation of the dynamics this work aims to validate this hypothesis through simulation studies on different biped models In what follows to facilitate intuition as to their physical meanings and qaare used in lieu of x1and x2to label the unactuated and actuated coordinates respectively B Switching Logic While the key idea is to avoid falling by switching gaits the primary challenge lies in designing the switching logic so as to enlarge the RoA Figure 3 conceptually depicts an arbitrary realization t x0 of Eq 4 subject to bounded external disturbances The top illustration shows t x0 evolving at steady state on the zero dynamics trajectory of the desired gait After a disturbance two cases can arise 1 t x0 remains in bottom left fi gure and will eventually return to steady state or 2 the perturbation is large enough that t x0 exits bottom right fi gure When t x0 gets pushed out of and into 1 t x0 re starts in 1 By switching to the feedback stabilized trajectories of gait 1 instead of falling the robot converges to the zero dynamics of gait 1 and is stable in that sense 11 This logic allows switching at any time of the gait cycle simply based on the unactuated states Critically the approach leverages underactuation of dy namic walkers to focus on a low order representation of their dynamics their most unstable component s For walkers underactuated by one DoF is equivalent to a 2D surface For 3D walkers with two degrees of underactuation one must consider both the sagittal and frontal plane components of the unactuated dynamics f s f s and is a 4D surface In both cases the reduced RoA can be estimated rapidly by discretizing and simulating Eq 4 Thus the approach can be scaled to higher dimensional systems Estimating via simulation offers the following advantages It is applicable to a wide variety of dynamical systems i e any that can be simulated 8 Input state constraints that typically carry discontinu ities in optimizations are easily enforced in simulation It tends to be less conservative than sum of squares verifi cations and other Lyapunov techniques which is key for walkers with narrow stable funnels 9 16 In estimating for a given gait a set of initial conditions x0is sampled around the periodic orbit O and Eq 4 is simulated forward in time If the biped falls shrinks to exclude x0 If the dynamics converge to O x0is added to As the number of iterations increases grows to a less conservative fi t of the actual RoA and the process is repeated for every gait in the library The reduced RoA is n x0 R2n x0 q a q a lim t dist t x0 O 0 o 6 where the actuated states are assumed to take on their steady state values hence the notation C Autonomous Mapping Feedback stabilized trajectories from different gaits can be embedded in a mapping via the stable switching condition to patch together their RoA The approach divided into offl ine and online stages is conceptually shown in Fig 4 The offl ine step fi rst generates a library of gaits walking motions for the full order system The reduced RoA of each gait is then estimated numerically through simulation Lastly the feedback stabilized trajectories denoted are stored offl ine in individual mappings for each gait The online stage includes a motion planner that uses the switching logic to construct the autonomous state based mapping policy Given a desired walking speed with corresponding mapping the planner stitches trajectories from the library to to enlarge the overall stabilizable region The RoA of all the gaits in the library are concate nated to form the system s patched RoA Whereas is only defi ned in for the single gait case is defi ned in the union of all stable sets The mapping can be fed to the controller as a lookup table facilitating high bandwidth control Moreover the mapping is autonomous it only changes with desired walking speed More importantly the periodic orbit is now not only an attractor of but of N S i 1 i N T i 1 i 7 2281 Gait Optimization RoA Estimation Offline Gait library generation Simulate the RoA 1 step periodic gaits 2 Feedback Control Virtual constraints Feedback linearization High gain PD control N Gait Library Gait library i 1 i 1 x q q Online Fast time scale control system x f x u O2 2 2 2 2 O1 1 1 1 1 ON N N N N Motion Planning N 2 1 2 Fig 4 Methodology and structure of control system where N is the number of gaits in the library Thus if there exists a state x0that is feedback stabilizable by at least one gait in the library must capture that point In Fig 3 the autonomous mapping is defi ned for all points inside 1 1 Outside this region the mapping is not defi ned since there is no stable gait transition according to the switching logic and the robot will fall In Fig 3 the zero dynamics trajectory of gait 1 explicitly crosses the boundaries of meaning that t x0 will eventually reenter At that point the autonomous mapping automatically orchestrates the transition to the nominal gait Concretely for 1 t 0 rad s c 0 1 c 1 0 0 5 0 1 u N m 15 u 15 50 ui 50 Step length D m 0 30 D 0 700 40 D 0 50 Step duration T s 0 40 T 1 500 40 T 1 50 Friction coeffi cient s 0 6 s 0 4 Objective JTorque squaredCost of transport RT 0 u2 d P4 1 RT 0 ui qi d MtotalgD MethodOrthogonal collocationSingle shooting NLP solverIpoptFmincon points Direct transcription was used for the two link case with the trajectory parameterized by discrete collocation points consisting of the states and control inputs and the equations of motion satisfi ed only at the collocation points Both shooting and transcription methods create nonlinear programming problems NLP that can be solved by state of the art solvers MATLAB s Fmincon was used to generate stable closed loop gaits for the fi ve link model Ipopt was used with 100 discrete way points to generate open loop gaits for the two link model Within each fi nite element the states and control inputs were approximated with 5th order Lagrange polynomials B Feedback Control Collocation is common for trajectory optimization of dy namical systems governed by ordinary differential equations but unavoidable discretization errors require that open loop trajectories be feedback stabilized Hybrid zero dynamics HZD based control 15 was used to stabilize the two link gaits whereas the fi ve link gaits were optimized within the HZD framework and thus provably stable The monotonically increasing phase variable measures step progression and is used to parameterize a set of holo nomic constraints on the m actuated coordinates y hi qi i i 1 m 9 2282 where the functions i represent the biped s desired con fi guration during a step and the output functions are the tracking errors for each actuated DoF Feedback linearization is applied to obtain a new system that is input output linear The output equation must be differ entiated twice before the control input u appears explicitly y L2 fh LgLfh u 10 where Lfh is the Lie derivative of h with respect to f 20 Hence the input of the new system is u LgLfh 1 v L2 fh 11 where v is the double integrator y v For HZD based control high gain PD feedback is a common choice for v v KPy KD y 12 which collapses the dynamics onto the zero dynamics man ifold The PD gains were not optimized for this work but were selected to match those used in experiment in 17 on the ERNIE hardware KP 500 and KD 10 C Two Link Results The 8 gaits in the two link library were optimized for the integral of torque squared over walking speeds from 0 40 m s to 1 10 m s in 0 10 m s increments L m s 0 40 1 10 Table III summarizes the gait library details Via offl ine computation the reduced RoA of each gait was estimated numerically by discretizing using 1 000 evenly spaced intervals for and an adjustable step size search in The full RoA of the 1 00 m s gait was also computed numerically using a 4D grid in q1 q2 q1 q2 As expected simulation of the full RoA indicated that the biped can reject much larger disturbances in the actuated DoF the failure mode for large disturbances in q1and q1was friction constraint violation not actual falling The resulting 1 00 m s stable sets are shown in Fig 5 where the red surface represents the reduced 2D RoA and the enclosed blue regions are trimmed evenly spaced snapshots of the full RoA Since the q1 bounds on the full RoA are far from the periodic orbit the blue region was trimmed based on the size of the reduced RoA at each snapshot To test the switching logic s validity a sequence of 100 random gait transitions was simulated using MATLAB s ode45 default options The two link biped walked 25 steps with a given gait at which point a transition was enforced at a random time during the gait cycle The lone condition to check for switching is highlighted in Section III the reduced RoA of the switched in gait must contain the current state when switching takes place If this condition was not met another gait transition was randomly generated TABLE III Two link model gait library Gait no 12345678 Speed m s 0 400 500 600 700 800 90 1 001 10 Step length m 0 300 320 380 450 520 58 0 640 67 Objective N2m2s 0 30 82 35 913 426 9 52 7 120 6 Fig 5 Reduced red and full blue RoA of 1 00 m s gait Enclosed blue regions are trimmed evenly spaced snapshots of full RoA from simulation of 4D grid TABLE IV Ten gait transitions for two link walker Timing of switching reported as percentage of original gait cycle Switching no from m s to m s timing 10 401 1045 3 21 100 6028 1 30 600 909 6 40 901 0047 8 51 000 9078 3 60 900 6047 3 70 600 8033 6 80 800 9029 6 90 900 4063 7 100 401 1019 5 and so on In the end the two link model completed the full switching sequence and did not fall as a result of the transitions Ten of the 100 total transitions in the sequence are listed in Table IV and the step velocity data are in Fig 6 The video attached with the paper shows the two link biped transitioning from one gait to the next D Five Link Results The 50 gaits in the fi ve link library were optimized for cost of transport over walking speeds from 0 35 m s to 1 10 m s L m s 0 35 1 10 Cost of transport varied from 0 071 slowest gait to 0 259 fastest and step length from 0 44 m to 0 49 m The reduced RoA of each gait was estimated numerically as for the two link case except that was discretized using 400 evenly spaced intervals for and a 0 01 rad s incremental search in A sequence of 50 random gait transitions was generate