Bringing the Compass-Gait Bipedal Walker to Three Dimensions

Transcription

1 October 14, 2009 IROS 2009, St. Louis, MO 1 1 Bringing the Compass-Gait Bipedal Walker to Three Dimensions Robert D. Gregg* and Mark W. Spong Coordinated Science Laboratory University of Illinois at Urbana-Champaign

2 October 14, 2009 IROS 2009, St. Louis, MO 2 2 Motivation Passive dynamic walking mechanism by Collins, Wisse, and Ruina, Mechanically demonstrates efficient, natural bipedal walking no actuation!

3 October 14, 2009 IROS 2009, St. Louis, MO 3 3 Powered Passive Dynamics ~7.5 W/kg 0.85 W/kg [Collins et al.]

4 October 14, 2009 IROS 2009, St. Louis, MO 4 4 Planar Compass-Gait Biped Many theoretical results in dynamic walking consider a biped constrained to the sagittal plane. Passive walking gaits (stable limit cycles) down shallow slopes [McGeer 1990]. Mapped to flat ground using potential shaping [Spong & Bullo].

5 October 14, 2009 IROS 2009, St. Louis, MO 5 5 Planar Compass-Gait Biped Many theoretical results in dynamic walking consider a biped constrained to the sagittal plane. [Goswami et al. 1999]

6 October 14, 2009 IROS 2009, St. Louis, MO 6 6 Decomposing Complex Motion Most 3-D bipeds do not naturally have stable limit cycles for walking. Stable limit cycles may exist in the sagittal plane. Propose: Exploit symmetries to extend these gaits to 3-D with reduction-based control, separating sagittal, lateral, and axial control problems.

7 October 14, 2009 IROS 2009, St. Louis, MO 7 7 Building 3-D Dynamic Gaits [ACC 2008, IJRR 2009] How does steering effect dynamic walking stability when navigating 3-D space?

8 October 14, 2009 IROS 2009, St. Louis, MO 8 8 Ultimate Goal Extend the compass-gait biped into 3-D space for fast and efficient walking: Straight-ahead gaits Turning gaits Period-doubling bifurcations

9 October 14, 2009 IROS 2009, St. Louis, MO 9 9 Ultimate Goal Extend the compass-gait biped into 3-D space for fast and efficient walking:

10 October 14, 2009 IROS 2009, St. Louis, MO Symmetries and Reduction 1. Dynamic Walking Background 2. Symmetries and Reduction 3. Controlled Reduction of Robots 4. 3-D Bipedal Walking 5. Turning Gaits 6. Closing Remarks

11 October 14, 2009 IROS 2009, St. Louis, MO Lagrangian Mechanics A mechanical system with config. space Q is described by (q, q) T Q and Lagrangian L(q, q) = K(q, q) V (q) satisfying = 1 2 qt M(q) q V (q), d dt L q L q = u Inertia matrix! M(q) q + C(q, q) q + N(q) = u

12 October 14, 2009 IROS 2009, St. Louis, MO Example of Symmetry q 1 Variable is said to be cyclic if Generalized momentum p 1 = J 1 (q, q) := L q 1. L q 1 = 0. Uncontrolled E-L equations show d dt L q 1 L Conservation law: 0 q 1 = d dt p 1 = 0. p 1 is constant.

13 October 14, 2009 IROS 2009, St. Louis, MO Geometric Reduction Based on Lagrangian symmetries. In Routhian reduction, a system with config. space Q = G S has cyclic variables : q i G i L q i = 0 T Q : phase space J 1 (μ) : momentum map surface p i = J i (q, q) = μ i mod G T S : reduced phase space

14 October 14, 2009 IROS 2009, St. Louis, MO What about the divided variables? Likely to be unstable, e.g., yaw and lean for a biped

15 October 14, 2009 IROS 2009, St. Louis, MO Controlled Reduction Decoupled and stabilized subsystem Symmetry-Breaking Geometric Reduction Stabilized cyclic coordinates J i (q, q) = λ i (q i ) Reduction-based (energy-shaping) control law.

16 October 14, 2009 IROS 2009, St. Louis, MO Lagrangian Shaping Lagrangian L with cyclic coordinate : L(q n 2, q) = K(q n 2, q) V (q n 2 ) = 1 2 qt M(q n 2 ) q V (q n 2 ) Closed-loop system corresponds to an almost-cyclic Lagrangian: L λ (q, q) = K(q n 2, q) V (q n 2 ) + K aug λ (q, q 2 n ) V aug λ (q 1, q2 n ), where K aug λ, V aug λ are special energy shaping terms based on momentum function λ(q 1 ). q 1

17 October 14, 2009 IROS 2009, St. Louis, MO Reduction Revisited J 1 (λ) T Q: phase space p 1 = J 1 (q, q) = λ(q 1 ) mod G T S : reduced phase space L λ (q, q) L red (q n 2, qn 2 )

18 October 14, 2009 IROS 2009, St. Louis, MO Application to Robots 1. Dynamic Walking Background 2. Symmetries and Reduction 3. Controlled Reduction of Robots 4. 3-D Bipedal Walking 5. Turning Gaits 6. Closing Remarks

19 October 14, 2009 IROS 2009, St. Louis, MO General Control Framework Need symmetries in mechanical systems for symmetry-based methods! Geometric property of inertia matrices.

20 October 14, 2009 IROS 2009, St. Louis, MO Extensive Symmetries Definition 1: An n n matrix M is recursively cyclic if it has the form: M(q n 2 ) = µ mq1 (q n 2 ) M q1,q n 2 (qn 2 ) M T q 1,q n 2 (qn 2 ) M q n 2 (q n 3 ) = m q1 (q n 2 ) m q1,q 2 (q n 2 ) M q1,q n 3 (qn 2 ) m q1,q 2 (q n 2 ) m q2 (q n 3 ) M q2,q n 3 (qn 3 ) M T q 1,q n 3 (qn 2 ) M T q 2,q n 3 (qn 3 ) M q n 3 (q n 4 ) (q 2,..., q n ) (q 3,..., q n ) (q 4,..., q n ) where q n j = (q j, q j+1,..., q n ) T [IJRR 2009]

21 October 14, 2009 IROS 2009, St. Louis, MO Extensive Symmetries Definition 1: An n n matrix M is recursively cyclic if it has the form: M(q n 2 ) = = µ mq1 (q n 2 ) M q1,q n 2 (qn 2 ) M T q 1,q n 2 (qn 2 ) M q n 2 (q n 3 ) m q1 (q2 n ) M q1,q2 n(qn 2 ).... Mq T 1,q2 n(qn 2 ) m qj (qj+1 n ) M q j,qj+1 n (qn j+1 ) Mq T j,qj+1 n (qn j+1 ) M qj+1 n (qn j+2 ) where q n j M q n n (q n n+1 ) = m q n R = (q j, q j+1,..., q n ) T and with base case and qn+1 n =. [IJRR 2009]

22 October 14, 2009 IROS 2009, St. Louis, MO Property of Serial Chains [IJRR09] Theorem: The inertia matrix of any n-dof serial kinematic chain is recursively cyclic. z x y Branched chains considered in [CDC09].

23 October 14, 2009 IROS 2009, St. Louis, MO Controlled Reduction by Stages An open kinematic chain is controlledreducible down to a lower-dof subrobot. L λ k 1 z x y

24 October 14, 2009 IROS 2009, St. Louis, MO Controlled Reduction by Stages An open kinematic chain is controlledreducible down to a lower-dof subrobot. L λ k 2 z x y Reduction Stage-1 J i (q, q) = λ i (q i ), i = 1

25 October 14, 2009 IROS 2009, St. Louis, MO Controlled Reduction by Stages An open kinematic chain is controlledreducible down to a lower-dof subrobot. L λ k 3 z x y Reduction Stage-2 J i (q, q) = λ i (q i ), i = 1, 2

26 October 14, 2009 IROS 2009, St. Louis, MO Controlled Reduction by Stages An open kinematic chain is controlledreducible down to a lower-dof subrobot. L λ k 4 z x y Reduction Stage-3 J i (q, q) = λ i (q i ), i = 1, 2, 3

27 October 14, 2009 IROS 2009, St. Louis, MO Controlled Reduction by Stages An open kinematic chain is controlledreducible down to a lower-dof subrobot. L λ k 5 z x y Reduction Stage-4 J i (q, q) = λ i (q i ), i = 1, 2, 3, 4

28 October 14, 2009 IROS 2009, St. Louis, MO D Bipedal Walking 1. Dynamic Walking Background 2. Symmetries and Reduction 3. Controlled Reduction of Robots 4. 3-D Bipedal Walking 5. Turning Gaits 6. Closing Remarks

29 October 14, 2009 IROS 2009, St. Louis, MO Compass-Gait Biped Model Ankle is ZYX-Euler joint Yaw about Roll about Pitch about z : ψ y : ϕ x : θ s 4-DOF configuration q = (ψ, ϕ, θ T ) T, with sagittal config. θ = (θ s, θ ns ) T.

30 October 14, 2009 IROS 2009, St. Louis, MO Reduction-Based Control energy shaping 4-DOF 3D biped (no dynamic gaits) 4-DOF 3D biped with dynamic gaits λ 1 (ψ) = α 1 (ψ ψ) 3-DOF 3D biped with dynamic gaits λ 2 (ϕ) = α 2 ϕ 2-DOF planar biped with dynamic gaits

31 October 14, 2009 IROS 2009, St. Louis, MO Straight-Ahead Gait Exponentially stable limit cycle:

32 October 14, 2009 IROS 2009, St. Louis, MO Compass-Gait-with-Torso Model 5-DOF configuration q = (ψ, ϕ, θ T ) T, with sagittal config. θ = (θ s, θ t, θ ns ) T.

33 October 14, 2009 IROS 2009, St. Louis, MO Turning Gaits 1. Dynamic Walking Background 2. Symmetries and Reduction 3. Controlled Reduction of Robots 4. 3-D Bipedal Walking 5. Turning Gaits 6. Closing Remarks

34 October 14, 2009 IROS 2009, St. Louis, MO Steering Motion Introduces subsystem perturbations. If steering angle s = ψ is constant over several steps, what happens to the gait?

35 October 14, 2009 IROS 2009, St. Louis, MO Full Circle Turning If s = 2π/h, converge to stable 360 -turning maneuvers, periodic over h steps. 4-DOF 5-DOF

36 October 14, 2009 IROS 2009, St. Louis, MO Periodic Turning Gaits Sufficiently small s = ψ induces 1-step periodic turning gaits modulo heading change.

37 October 14, 2009 IROS 2009, St. Louis, MO CW-Turning Gait s = 2π/13 = Exponentially stable limit cycle:

38 October 14, 2009 IROS 2009, St. Louis, MO Turning Gait Bifurcations Large steering angles induces perioddoubling behavior:

39 October 14, 2009 IROS 2009, St. Louis, MO Remarks and Future Work 1. Dynamic Walking Background 2. Symmetries and Reduction 3. Controlled Reduction of Robots 4. 3-D Bipedal Walking 5. Turning Gaits 6. Closing Remarks

40 October 14, 2009 IROS 2009, St. Louis, MO Locomotion Energetics Specific energetic cost of transport: c et = E/(m d g) Straight Turning DOF biped 5-DOF biped Cornell biped* Honda ASIMO* *Collins & Ruina 2005

41 October 14, 2009 IROS 2009, St. Louis, MO Path Planning by Primitives Motivated by motion primitives based on ZMP equilibrium constraints. Dynamic walkers have asymptotically stable gait primitives [submitted ICRA10]: What about varying curvature (clothoids)?

42 October 14, 2009 IROS 2009, St. Louis, MO Future Work: Motion Planning for Dynamic Walkers Send comments to Special thanks to Mark Spong, Tim Bretl, and Seth Hutchinson

43 Backup slides October 14, 2009 IROS 2009, St. Louis, MO 43 43

44 October 14, 2009 IROS 2009, St. Louis, MO Reduced Lagrangian Given almost-cyclic Lagrangian L λ (q, q) = 1 2 ³ q 1 q nt 2 µ m q1 (q n 2 ) M q n 2 (q n 2 ) µ q1 +K aug λ q n 2 V (q n 2 ) (q, qn 2 ) V aug λ (q) Lower-dimensional Lagrangian J(q, q) = λ(q 1 ) L red (q n 2, q n 2 ) = 1 2 qnt 2 M q n 2 (q n 2 ) q n 2 V (q n 2 ).

45 October 14, 2009 IROS 2009, St. Louis, MO Relating Solutions If J(q(t 0 ), q(t 0 )) = λ(q 1 (t 0 )), then L λ L red (q 1 (t), q 1 (t), q n 2 (t), q n 2 (t)) (q n 2 (t), q n 2 (t)), where J(q(t), q(t)) = λ(q 1 (t)), t > t 0. Conservation law allows us to uniquely reconstruct (q 1 (t), q 1 (t)). cf. Theorem 1

46 October 14, 2009 IROS 2009, St. Louis, MO Hybrid Systems A hybrid control system has the form HC : ½ ẋ = f(x) + g(x)u x + = (x ) x D\G x G A hybrid system H has no explicit input (e.g., uncontrolled or closed-loop systems) x G ẋ = f(x) D (x) µ x = q q D T Q

47 October 14, 2009 IROS 2009, St. Louis, MO Solutions to Hybrid Systems A hybrid flow x(t) is a solution curve to H. Called h-periodic if x(t) = x(t + P h where T i is the fixed time-to-impact. i=1 T i), An h-periodic hybrid orbit O = {x(t) t 0} D is locally exponentially stable (LES) if there exist k, α, δ > 0 such that d(x(0), O) < δ = d(x(t), O) ke αt d(x(0), O), t 0.

48 October 14, 2009 IROS 2009, St. Louis, MO Poincaré Analysis Stability is determined by studying the Poincaré map P : G G. The h-composition of P sends state x j G ahead h impact events by the discrete system x j+h = P h (x j ). Any h-periodic orbit has an h-fixed-point x G O such that x = P h (x ). G O

49 October 14, 2009 IROS 2009, St. Louis, MO Stability of Orbits Numerically analyze orbit stability by the map s linearization δp h. Locally exponentially stable if and only if max{ eig(δp h ) } < 1. x 0 x, Then, for sufficiently close to lim P hz (x 0 ) = x. z

50 October 14, 2009 IROS 2009, St. Louis, MO Biped Subsystems 4-DOF biped s Lagrangian is L 4D (q, q) = 1 2 qt M 4D (q) q V 4D (ϕ, θ), with recursively cyclic inertia matrix : m ψ (ϕ, θ) M ψ,ϕ,θ (ϕ, θ) M 4D (q) = m ϕ (θ) M ϕ,θ (θ) Mψ,ϕ,θ T (ϕ, θ) M ϕ,θ T (θ) M θ(θ) Planar Biped 3-DOF Biped M 4D Control heading to angle ψ, lean to vertical.

51 October 14, 2009 IROS 2009, St. Louis, MO Select Publications Asymptotically Stable Gait Primitives for Planning Dynamic Bipedal Locomotion in Three Dimensions. Gregg, Bretl, and Spong. Submitted to 2010 ICRA, Anchorage, AK. Bringing the Compass-Gait Bipedal Walker to Three Dimensions. Gregg and Spong. In 2009 IROS, St. Louis, MO. Reduction-Based Control of Branched Chains: Application to 3-D Bipedal Torso Robots. Gregg and Spong. To appear in 2009 CDC, Shanghai, China. Reduction-Based Control of 3-D Bipedal Walking Robots. Gregg and Spong. Int. J. of Robotics Research, Pre-print, Reduction-based Control with Application to 3-D Bipedal Walking Robots. Gregg and Spong. In 2008 ACC, Seattle, WA. A Geometric Approach to 3-D Hipped Bipedal Robotic Walking. Ames, Gregg, and Spong. In 2007 CDC, New Orleans, LA.