Port-based Modeling and Control for Efficient Bipedal Walking Machines

Transcription

1 Port-based Modeling and Control for Efficient Bipedal Walking Machines Vincent Duindam Control Laboratory, EE-Math-CS University of Twente, Netherlands Joint work with Stefano Stramigioli, Arjan van der Schaft Gijs van Oort, Edwin Dertien, Martijn Visser, Frank Groen CHESS UC Berkeley Oct 31 / Nov 7,

2 Introduction Twente University and Geoplex Efficient walking robots Models of Mechanisms in Contact Mechanism dynamics Compliant and rigid contact Efficient Model-Based Control of Walking Robots Control by mechanical optimization Zero-Energy Control of Walking Robots Dribbel Experimental walking robot Conclusions CHESS UC Berkeley Oct 31 / Nov 7,

3 Introduction CHESS UC Berkeley Oct 31 / Nov 7,

4 University of Twente (UT) Founded in 1961 Only Dutch university with a real campus Typical engineering depts, but also biomedical technology, environmental sciences, educational policy CHESS UC Berkeley Oct 31 / Nov 7,

5 Control UT UT Dept. EE-Math-CS Control Engineering 6 scientific staff, 10 PhD students Focus on mechatronics, robotics, and embedded systems Favorite sport: klootschieten CHESS UC Berkeley Oct 31 / Nov 7,

6 Geoplex Geometric Network Modeling and Control of Complex Physical Systems ( European FP5 project, Consortium: 9 universities + 1 company Goal: use port-hamiltonian systems theory to solve modeling and control problems across different domains My research: apply Geoplex philosophy to walking robots CHESS UC Berkeley Oct 31 / Nov 7,

7 Port-Hamiltonian Systems Developed by A.J. van der Schaft and B.M. Maschke (1992) Idea: study energy flow instead of information (signal) flow b m m b k k Collocated power variables effort (force) and flow (velocity) CHESS UC Berkeley Oct 31 / Nov 7,

8 Port-Hamiltonian Systems Developed by A.J. van der Schaft and B.M. Maschke (1992) Idea: study energy flow instead of information (signal) flow 1 m - - k b Collocated power variables effort (force) and flow (velocity) CHESS UC Berkeley Oct 31 / Nov 7,

9 Port-Hamiltonian Systems Developed by A.J. van der Schaft and B.M. Maschke (1992) Idea: study energy flow instead of information (signal) flow 1 m power - - power power k b Collocated power variables effort (force) and flow (velocity) CHESS UC Berkeley Oct 31 / Nov 7,

10 Port-Hamiltonian Systems Graphical language: bond graphs Mathematical language: structured differential equations d H x = (J(x) R(x)) dt x + G(x)u y = G T (x) H x with J(x) skew-symm, R(x) pos-def, H(x) energy function d dt H = y T u T H x R H x y T u CHESS UC Berkeley Oct 31 / Nov 7,

11 Walking Robots Active static walking: versatile, robust, use traditional control methods, but not very efficient Passive dynamic walking: efficient, natural, but generally not robust or versatile CHESS UC Berkeley Oct 31 / Nov 7,

12 Passive Dynamic Walking CHESS UC Berkeley Oct 31 / Nov 7,

13 Goal: explain and extend passive walkers Modeling goals: Build physical models of walking robots using port-hamiltonian techniques Analyze models and explain walking behavior Control goals: Use natural passive motions as much as possible Use adaptive passive elements to change natural motions Augment passive walking with active control if needed In this way, combine efficiency, versatility, and robustness. CHESS UC Berkeley Oct 31 / Nov 7,

14 Models of Mechanisms in Contact CHESS UC Berkeley Oct 31 / Nov 7,

15 Model setup Build up models as power-port interconnection of subsystems Mechanism + ground contact + actuation + other parts Continuous-time model with switching and state jumps Implemented in project software 20sim ( CHESS UC Berkeley Oct 31 / Nov 7,

16 Dynamics of general rigid mechanisms Standard result for dynamics of rigid mechanisms M(q) q + C(q, q) q + G(q) = τ with coordinates q(t) and velocities q(t) as dynamic states. Allows only joint spaces isomorphic to R n singularities Not suitable for more general joints, e.g. ball joints with configuration R(t) SO(3) and velocity ω(t) R 3 CHESS UC Berkeley Oct 31 / Nov 7,

17 Dynamics of general rigid mechanisms Proposed solution: 1 Parameterize joints with local and global coordinates 2 Write down Boltzmann-Hamel equations in local coordinates, parameterized by global coordinates 3 Evaluate results at origin of local coordinates 4 Result: similar equations with extra coupling on joint-level p 2 u 2 v 2 v 1 u 1 p 1 CHESS UC Berkeley Oct 31 / Nov 7,

18 Dynamics of general rigid mechanisms Generalized dynamics equations Works for all Lie groups (take exponential coordinates as local coordinates) Can be directly extended to nonholonomic constraints (e.g. rolling-but-no-slipping) Has been implemented into 20sim 3d mechanics toolbox Paper still to be written, so comments are very welcome! CHESS UC Berkeley Oct 31 / Nov 7,

19 Contact modeling Contact models consist of two parts Kinematics: what are the closest points between bodies? Dynamics: what forces are involved, how do bodies react? Compliant: finite compression and contact forces Rigid: limit-case compression and contact forces CHESS UC Berkeley Oct 31 / Nov 7,

20 Contact kinematics Extension and geometrification of David Montana s equations ([ ] [ ]) α f = M 1 ( f Kf + K ) 1 ωy o ω K vx o x ([ α o = Mo 1 ( R ψ Kf + K ) 1 ωy o ω x ψ = ω z + T f M f α f + T o M o α o 0 = v z ] v y + K f [ vx v y Limitations with this formulation Only suitable when bodies are in contact Not symmetric in two bodies Depends heavily on surface parameterization ]) CHESS UC Berkeley Oct 31 / Nov 7,

21 Contact kinematics f f 1 g D R 2 S E 3 S 2 E 3 CHESS UC Berkeley Oct 31 / Nov 7,

22 Contact kinematics p 2 p 1 Ψ 2 Ψ 1 Basic observations Contact points are separated by line along normal vectors p 1 + g 1 = H 1 2 p 2 Normal vectors are equal in direction but opposite g 1 = H 1 2 g 2 CHESS UC Berkeley Oct 31 / Nov 7,

23 Contact kinematics p 1 + g 1 = H 1 2 p 2 g 1 = H 1 2 g 2 Time-differentiation and merging gives contact kinematics ( ) ( g 1 + H2 1 g 2 H1 2 1,1 (I + g 1 ) ṗ 1 = T 2 g 1 + H2 1 g 2 g 2 ( ) ( g 2 + H1 2 g 1 H2 1 2,2 (I + g 2 ) ṗ 2 = T 1 g 2 + H1 2 g 1 g 1 Geometric equations, add surface coordinates separately 2,2 T 1 p 2 T 1,1 2 p 1 ) ) CHESS UC Berkeley Oct 31 / Nov 7,

24 Compliant contact dynamics First model of contact dynamics: Assumption: forces generated by spring and damper Implementation: Upon impact, connect spring/damper at contact points During contact, keep connected between these fixed points Release contact when distance > 0 Model only realistic for small deformations (reasonable for most walking robots) Simple submodel, can be used at different locations Stiff, fast contact Stiff differential equations CHESS UC Berkeley Oct 31 / Nov 7,

25 Compliant contact dynamics CHESS UC Berkeley Oct 31 / Nov 7,

26 Rigid contact dynamics Simpler rigid dynamics through limiting operation Assumptions: impact and deformation phase instantaneous, no slipping, impulsive contact forces Impact = discontinuity in velocity/momentum on impact For walking robots, results mostly in instantaneous release of rear leg on impact of front leg Interesting detail: release of contact not determined by force direction but by acceleration direction f 2 u f 1 CHESS UC Berkeley Oct 31 / Nov 7,

27 Efficient Model-Based Control of Walking Robots CHESS UC Berkeley Oct 31 / Nov 7,

28 Finding natural walking gaits How to find the most efficient walking gait? Purely passive: Search space of initial conditions that give stable cycle (Poincare mappings) Efficient (almost-passive): Search space of all trajectories? CHESS UC Berkeley Oct 31 / Nov 7,

29 Finding natural walking gaits Make problem solvable by polynomial approximation Parameterize joint trajectories over one step as q(t) = α 0 + α 1 t + α 2 t α n t n Force cyclic behavior by matching initial to final conditions. Minimize torque requirements T min τ(t) 2 dt α,t 0 with τ expressed in terms of α using model equations. Does not say anything about gait stability. CHESS UC Berkeley Oct 31 / Nov 7,

30 Finding natural walking gaits Gait search as minimization problem Recovers passive gait if available Otherwise, gives most efficient gait Can be extended with other parameters, e.g. for mechanical structure. x s k 0 x s x m m 0 m 0 x m CHESS UC Berkeley Oct 31 / Nov 7,

31 Constructing natural walking gaits Optimization of mechanical structure for variable speed walking Example: compass-gait walker with inter-leg spring and variable mass distribution J 10 1 m k v (m/s) v (m/s) 0 CHESS UC Berkeley Oct 31 / Nov 7,

32 Constructing natural walking gaits Optimization of mechanical structure for variable speed walking Example: compass-gait walker with inter-leg spring and variable mass distribution J 10 1 m k v (m/s) v (m/s) 0 CHESS UC Berkeley Oct 31 / Nov 7,

33 Power-continuous tracking Given efficient trajectory, how to follow it? Time-synchronization not important, only joint-synchronization tassume full actuation n degrees of freedom 1 degree of freedom for energy/speed control q (e.g. 2 (t) potential-energy shaping) n 1 degrees of freedom for curve-tracking (zero-energy, or, power-continuous control) Related q 1 (t) work q 3 (t) P.Y. Li and R. Horowitz ( ) PVFC: Passive Velocity-Field Control CHESS UC Berkeley Oct 31 / Nov 7,

34 Power-continuous tracking Given efficient trajectory, how to follow it? Time-synchronization not important, only joint-synchronization tassume full actuation n degrees of freedom 1 degree of freedom for energy/speed control q (e.g. 2 (t) q 2 potential-energy shaping) n 1 degrees of freedom for curve-tracking (zero-energy, or, power-continuous control) Related q 1 (t) work q 3 (t) q 1 P.Y. Li and R. Horowitz ( ) PVFC: Passive Velocity-Field Control q 3 CHESS UC Berkeley Oct 31 / Nov 7,

35 Power-continuous tracking Given efficient trajectory, how to follow it? Time-synchronization not important, only joint-synchronization Assume full actuation n degrees of freedom 1 degree of freedom for energy/speed control (e.g. potential-energy shaping) n 1 degrees of freedom for curve-tracking (zero-energy, or, power-continuous control) Related work P.Y. Li and R. Horowitz ( ) PVFC: Passive Velocity-Field Control CHESS UC Berkeley Oct 31 / Nov 7,

36 Step 1: encode desired behavior Encode the desired behavior 1 Extend desired curve to vector field 2 Split kinetic energy in M(q)-orthogonal good and bad directions 3 Transform into good/bad coordinates Result: kinetic energy is sum of two parts, one describing desired behavior, the other describing undesired behavior. q(t) q(t) q d (t) curve CHESS UC Berkeley Oct 31 / Nov 7,

37 Step 1: encode desired behavior Encode the desired behavior 1 Extend desired curve to vector field 2 Split kinetic energy in M(q)-orthogonal good and bad directions 3 Transform into good/bad coordinates Result: kinetic energy is sum of two parts, one describing desired behavior, the other describing undesired behavior. q(t) q(t) curve CHESS UC Berkeley Oct 31 / Nov 7,

38 Step 1: encode desired behavior Encode the desired behavior 1 Extend desired curve to vector field 2 Split kinetic energy in M(q)-orthogonal good and bad directions 3 Transform into good/bad coordinates Result: kinetic energy is sum of two parts, one describing desired behavior, the other describing undesired behavior. q(t) q(t) w curve CHESS UC Berkeley Oct 31 / Nov 7,

39 Step 1: encode desired behavior Encode the desired behavior 1 Extend desired curve to vector field 2 Split kinetic energy in M(q)-orthogonal good and bad directions 3 Transform into good/bad coordinates Result: kinetic energy is sum of two parts, one describing desired behavior, the other describing undesired behavior. q(t) q(t) w curve CHESS UC Berkeley Oct 31 / Nov 7,

40 Step 1: encode desired behavior Example (2d point mass in a circular field) Point mass dynamics q d q 2 dt p 1 = p m 1 0 p p Representation of desired vector field [ ] q2 w(q) = q 1 q 2 m [ u1 u 2 p/m ] q 1 CHESS UC Berkeley Oct 31 / Nov 7,

41 Step 1: encode desired behavior Example (2d point mass in a circular field) Coordinate transformation (only for momentum) [ ] p1 = 1 [ ] [ ] q2 q 1 α1 p 2 q q 1 q 2 α 2 Dynamics equations in transformed coordinates d dt 0 0 q 2 q 1 q q 0 q = q 2 q q q 2 q 1 0 α 1 0 q q q α 1 + q 1 q m 2 α 1 α 2 0 q q q m q 1 q 2 α 1 α q 1 q 2 q q 1 q q q 2 q and the energy equals H(q, α) = H(α) = 1 2m α m α2 2. [ u1 u 2 ] CHESS UC Berkeley Oct 31 / Nov 7,

42 Step 1: encode desired behavior Example (2d point mass in a circular field) undesired energy 1 2m α2 2 input G J desired energy 1 2m α2 1 CHESS UC Berkeley Oct 31 / Nov 7,

43 Step 2: design controller Step 2: derive the controller almost trivially 1 Keep energy in separate bins 2 Create uni-directional flow from bad to good 3 Take care of the potential energy 4 Design 1D speed/energy controller u y power- continuous interconnection undesired (bad) energy desired (good) energy CHESS UC Berkeley Oct 31 / Nov 7,

44 Step 2: design controller Step 2: derive the controller almost trivially 1 Keep energy in separate bins 2 Create uni-directional flow from bad to good 3 Take care of the potential energy 4 Design 1D speed/energy controller decoupling controller u y power- continuous interconnection undesired (bad) energy desired (good) energy CHESS UC Berkeley Oct 31 / Nov 7,

45 Step 2: design controller Step 2: derive the controller almost trivially 1 Keep energy in separate bins 2 Create uni-directional flow from bad to good 3 Take care of the potential energy 4 Design 1D speed/energy controller undesired (bad) energy desired (good) energy CHESS UC Berkeley Oct 31 / Nov 7,

46 Step 2: design controller Step 2: derive the controller almost trivially 1 Keep energy in separate bins 2 Create uni-directional flow from bad to good 3 Take care of the potential energy 4 Design 1D speed/energy controller asympt. controller undesired (bad) energy desired (good) energy CHESS UC Berkeley Oct 31 / Nov 7,

47 Step 2: design controller CHESS UC Berkeley Oct 31 / Nov 7,

48 Dribbel experimental walking robot Features First walking robot at the University of Twente Planar six link mechanism One actuator in hip joint Electro-magnetic knee lock Design and controller fully tested in simulation, setup worked on first try Very simple control algorithm Low energy consumption (6 W) Starting point and testbed for new research projects CHESS UC Berkeley Oct 31 / Nov 7,

49 Dribbel experimental walking robot CHESS UC Berkeley Oct 31 / Nov 7,

50 Dribbel experimental walking robot CHESS UC Berkeley Oct 31 / Nov 7,

51 Conclusions CHESS UC Berkeley Oct 31 / Nov 7,

52 Conclusions Physical models can help in rigorous design and analysis of walking robots, to go beyond physical intuition Benefit of port-based approach in mechanics usually a matter of taste, but sometimes gives just right level of abstraction Theoretical results still need to be tested in experiments! CHESS UC Berkeley Oct 31 / Nov 7,