Control and Planning with Asymptotically Stable Gait Primitives: 3D Dynamic Walking to Locomotor Rehabilitation?

Transcription

1 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg 1 1 Control and Planning with Asymptotically Stable Gait Primitives: 3D Dynamic Walking to Locomotor Rehabilitation? Robert D. Gregg * and Mark W. Spong Coordinated Science Laboratory Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Department of Electrical Engineering Erik Jonsson School of Engineering and Computer Science University of Texas at Dallas Special thanks to T. Bretl, A. Tilton, E. Hsiao-Wecksler Research supported by NSF grants CMS and CMMI

2 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg 2 2 Some Challenges in Biped Control Motion planning for 3D dynamic walkers? Shih, Grizzle, Chevallereau. Robotica 2010 (submitted) Gregg, Bretl, Spong. ICRA 2010 Control/planning of robotic locomotor therapies? ASIMO: ZMP walker Cornell biped: Dynamic walker Lokomat: Gait-training exoskeleton

3 Talk Outline I. 3D Dynamic Walking Background Gait primitives Building gaits via controlled reduction Planning with primitives II. Locomotor Rehabilitation Control design for robotic orthosis Underactuated potential shaping Sequential therapies July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg 3 3

4 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg 4 4 Reduction for 3D Walking Most 3D bipeds do not have passively stable walking gaits. Stable limit cycles may exist in the sagittal plane. periodic limit cycle

5 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg 5 5 Reduction for 3D Walking Most 3D bipeds do not have passively stable walking gaits. Stable limit cycles may exist in the sagittal plane. Reduction-based control: exploit symmetries to separate sagittal, frontal, and transverse control problems.

6 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg 6 6 Motion Primitives for Planning Goal: Derive set of 3D dynamic walking primitives and rules for stable sequential composition.

7 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg 7 7 Hybrid Dynamics A biped is a hybrid control system HC : ½ ẋ = f(x) + g(x)u x + = (x ) x D\G x G H cl In closed loop, this is a hybrid system : x G ẋ = f cl (x) (x) Poincare map Discrete system P : G G x j+1 = P (x j ) D eig{δp (x )} < 1 = local exp. stability about fixed-point x

8 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg 8 8 Gait Primitives Strategies to walk straight-ahead and turn: Definition: An asymptotically stable gait primitive is a pair G = (H cl, x ), where H cl is a closed-loop hybrid system and x is an asymptotically stable fixed-point (modulo yaw) of H cl. Invariant w.r.t. time, heading, and global position. Energy shaping: no reference trajectories! arcs on walking surface:

9 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg 9 9 Lagrangian Mechanics Given config. space Q, a mechanical system has state (q, q) T Q and Lagrangian function L(q, q) = K(q, q) V (q) = 1 2 qt M(q) q V (q), Then, EL q {L} := d dt L q L q = u control input M(q) q + C(q, q) q + q V (q) = u inertia matrix Coriolis matrix potential vector

10 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Controlled Reduction A mechanical system with C-space has a cyclic variable q 1 G : L q 1 = 0 energy-shaping control: Q = G S L λ = L + L aug T Q Invariant surface Z : phase space p 1 := L λ q 1 = λ(q 1 ) mod G T S : reduced phase space [Ames, Gregg, and Spong, CDC 2007]

11 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg General Framework Open kinematic chains have recursively cyclic inertia matrices [Gregg & Spong, IJRR 10]: M(q 2,..., q n ) = Related to slope invariance [Spong & Bullo, TAC 05] Allows controlled reduction by stages: k constrained coordinates q c = (q 1,..., q k ) T momenta p c := J c (q2 n ) q = K(q c q c ) invariant surface m 1 (q2 n) m 12 (q2 n ) m 12(q2 n) m 2 (q3 n ) M 13(q2 n) M 23 (q3 n ) M13(q T 2 n ) M23(q T 3 n ) M 3 (q4 n ) Z T Q (q 2,..., q n ) (q 3,..., q n ) (q 4,..., q n )

12 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg D Compass-Gait Biped Example 5-DOF 3D biped (no dynamic gaits) reduction-based control p 1 = K 1 (ψ ψ) 5-DOF 3D biped with dynamic gaits yaw 4-DOF 3D biped with dynamic gaits p 2 = K 2 ϕ lean ψ ϕ 3-DOF planar biped with dynamic gaits

13 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Straight-Ahead Gait G st x st = P 2 st(x st ) LES 2-periodic limit cycle along heading : ψ x st Phase portrait Joint trajectories

14 Constant-Curvature Steering Constant steering angle s = ψ induces exp. stable periodic turning gaits modulo yaw: [IROS 09, CDC 09] July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg 14 14

15 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg CW-Turning Gait G tu(s) x tu(s) + s 0 = P 2 tu(s) (x tu(s) ) LES 2-periodic limit cycle (mod yaw) for : s = π/14 x tu(s) Phase portrait Joint trajectories

16 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Planning with Primitives A 3D biped is a discrete-time switched system x k+2 = P 2 σ(k) (x k). Switching signal σ(k) {s, 0, s} chooses a primitive every step cycle to control path: Must constrain σ( ) for stability! constant-curvature walking arcs

17 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Stably Switching Gaits Primitive set {G st, G tu(s), G tu( s) }, G tu( s ) G st = G tu(0) as s 0. where Lyapunov funneling [Burridge, Rizzi, Koditschek, 99] Rules for stable path planning: CCW: Upper bound on curvature Lower bound on dwell time ST: CW: G tu(s) G tu( s) G tu(0) x 0 x tu(-s) x tu(0) x tu(s)

18 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Planner Formulation Dynamically stable planning reduces to tree search! Candidate paths: black Suboptimal path: red Goal region Plan output: sequence of primitives (s,s,s,s,s,s, 0,0,0,0, ) Start New work with A. Tilton, S. Candido, and T. Bretl

19 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Walking Path Simulation Traces path (planned a priori) with minor drift: Candidate paths: black Suboptimal path: red Simulated steps: blue Goal region Plan output: sequence of primitives (s,s,s,s,s,s, 0,0,0,0, ) Start New work with A. Tilton, S. Candido, and T. Bretl

20 Significance to Robot Walking Applicable to any biped with asymp. stable gaits *[Collins & Ruina, 05] Mechanical Cost of Transport Comparison Straight Turning DOF Sarcos humanoid? [submitted to Humanoids 2010 with Righetti, Buchli, Schaal] DOF hipless DOF hipped Human* Cornell biped* Honda ASIMO* July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg 20 20

21 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Related Challenges in Rehabilitation Robotic orthosis provides underactuated control with unknown human input. What can we prove? u Illinois Portable Powered Ankle-Foot Orthosis (PPAFO): new work with E. Hsiao- Wecksler, A. Shorter, T. Bretl Unknown! M(θ) θ + C(θ, θ) θ + N(θ) = Su + v n m torque map S (non-invertible)

22 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Underactuated Potential Shaping V Potential energy fundamental to walking. Idea: shape into ev with therapeutic properties. I.e., closed-loop single-support dynamics M(θ) θ + C(θ, θ) θ + e N(θ) = v, e N = q e V. Exactly attainable when the energy matching condition is satisfied: S ³N(θ) N(θ) e = 0, θ Q where S S = 0.

23 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Underactuated Potential Shaping V Potential energy fundamental to walking. Idea: shape into ev with therapeutic properties. I.e., closed-loop single-support dynamics M(θ) θ + C(θ, θ) θ + e N(θ) = v, e N = q e V. Exactly attainable with underactuated control u = (S T S) 1 S ³N(θ) T N(θ) e. left pseudo-inverse

24 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Designing Locomotor Therapies Assistive energy-shaping paradigm: alter meaningful traits of gait enable stable dynamic walking systematic tuning to patient Progressive therapy by funneling: plan sequence of intermediate training gaits for safe composition. Goal: drive stable fixed-point (healthy gait). x k x G k G 2 G 1 x k x 2 x 0 x 1 [To appear in 2010 IEEE Conf. on Decision & Control]

25 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Future Work Dynamic walking with the Sarcos humanoid? Robotic locomotor therapy: Underactuated energy shaping as a tool for meaningful control strategy design? Rapid simulations of personalized strategies to help plan/prescribe sequential therapies? Region of attraction estimation is essential! Clinical research with Kevin Lynch and Yasin Dhaher at NU and RIC (starting Fall 2010).

26 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Select Publications A Control Theoretic Approach to Robot-Assisted Locomotor Therapy. Gregg, Bretl, and Spong. Submitted to 2010 CDC, Atlanta, GA. Acceleration-Based Inverse Dynamics for Controlled Reduction: Sagittal-Plane Decoupling in Biped Locomotion. Gregg, Righetti, Buchli, Schaal. Submitted to 2010 Humanoids, Nashville, TN. Asymptotically Stable Gait Primitives for Planning Dynamic Bipedal Locomotion in Three Dimensions. Gregg, Bretl, and Spong. In 2010 ICRA, Anchorage, AK. Passivity and Symmetry-Breaking in the Controlled Geometric Reduction of Mechanical Systems. Gregg and Spong. Submitted to IEEE TAC, Reduction-Based Control of 3D Bipedal Walking Robots. Gregg and Spong. Int. J. of Robotics Research, Reduction-Based Control of Branched Chains: Application to Three-Dimensional Bipedal Torso Robots. Gregg and Spong. In 2009 CDC, Shanghai, China. Bringing the Compass-Gait Bipedal Walker to Three Dimensions. Gregg and Spong. In 2009 IROS, St. Louis, MO. A Geometric Approach to 3D Hipped Bipedal Robotic Walking. Ames, Gregg, and Spong. In 2007 CDC, New Orleans, LA.

27 July 8, 2010 Dynamic Walking, Cambridge, MA R. Gregg Reduction-Based Control Inner loop: u = M(q 2 ) q d + C(q 2, q) q + N(q 2 ) Outer loop: s.t. Z invariant + globally attractive, and Or, enforce constraints via nullspace projection: k x k upper-triangular q d = M λ (q) 1 ( C λ (q, q) q N λ (q) + v) EL θ {L λ (q, q)} Z = EL θ {L θ (θ, θ)}. q d = µ M 1 c ( J c q + K q c + M θd cθ ) θ d + v Passive feedback u 1 Z = K 1 q 1 + v 1.