3D Bipedal Walking including COM height variations St´ ephane Caron CRI Group Seminar Series May 14, 2018
What do we want? COMANOID project – https://comanoid.cnrs.fr 2
What do we want? COMANOID project – Aircraft entry plan (2017) 3
What do we want? Hard part: dynamic stair climbing 4
Interest reaches farther than humanoids Duality between manipulation and walking Figures adapted from [Eng+11] (left) and [HRO16] (right) 5
How to walk a humanoid robot? Walking pattern generation (= planning) Walking stabilization (= tracking) 6
How to walk a humanoid robot? Walking pattern generation (= planning) Walking stabilization (= tracking) 7
Walking on a plane Newton equation: c = f + m � m ¨ g Force is grounded at CoP: f = ω 2 ( c − r ) Holonomic constraint: c z = 0 ⇒ ω 2 = g / h ¨ Newton equ. simplifies to: c xy = ω 2 ( c xy − r xy ) ¨ 8
Walking on a plane Linear time-invariant system: c = ω 2 ( c − r ) ¨ Plus, feasibility constraint: r ∈ S Question: how to stop? 9
Walking on a plane � � System: x = ˙ where c c c = ω 2 ( c − r ) ¨ Input: center of pressure r Balance: starting from � c 0 � x 0 = ˙ c 0 How to bring the sys. to a stop? With a stationary solution? 10
Instantaneous Capture Point c = ω 2 ( c − r ) Recall that ¨ Define the capture point : ξ = c + ˙ c ω First-order dynamics: ˙ ξ = ω ( ξ − r ) c = ω ( ξ − c ) ˙ Stopped by r = ξ (stationary) On this topic, go and read [Eng+11] 11
Towards 3D, take one Apply same equation but in 3D: c = ω 2 ( c − ν ) ¨ ν : Virtual Repellent Point Feasibility constraint becomes: r = ν + g ω 2 ∈ S Equation of motion is LTI but system nonlinear from feasibility constraint Related references: [EOA15; CK17] 12
Time-varying DCM Newton equation: ¨ c = λ ( c − r ) + g Divergent component of motion: ξ = ˙ c + ω c First-order dynamics: ˙ ξ = ωξ + g − λ r ω = ω 2 − λ ... under the Riccati equation: ˙ Discussed in [CM18; Car+18] 13
Boundedness condition Differential equation: ˙ ξ = ωξ + g − λ r 14
Boundedness condition Differential equation: ˙ ξ = ωξ + g − λ r Solution is: � t � � ξ ( t ) = e Ω( t ) e − Ω( τ ) ( λ ( τ ) r ( τ ) − g ) d τ ξ (0) + 0 15
Boundedness condition Differential equation: ˙ ξ = ωξ + g − λ r Solution is: � t � � ξ ( t ) = e Ω( t ) e − Ω( τ ) ( λ ( τ ) r ( τ ) − g ) d τ ξ (0) + 0 As t → ∞ , the DCM ξ should stay bounded 16
Boundedness condition Differential equation: ˙ ξ = ωξ + g − λ r Solution is: � t � � ξ ( t ) = e Ω( t ) e − Ω( τ ) ( λ ( τ ) r ( τ ) − g ) d τ ξ (0) + 0 As t → ∞ , the DCM ξ should stay bounded Therefore , � ∞ ( λ ( t ) r ( t ) − g ) e − Ω( t ) d t ξ (0) = 0 Constraint between current state (LHS) and all future inputs λ ( t ) , r ( t ) of the inverted pendulum (RHS) 17
Problem formulation Change of variable: s ( t ) = e − Ω( t ) Boundedness condition becomes: � 1 r xy ( s )( s ω ( s )) ′ d s = ˙ c xy + ω i c xy i i 0 � 1 1 c z i + ω i c z g ω ( s ) d s = ˙ i 0 Optimize over ϕ i = s 2 i ω ( s i ) 2 From ϕ ∗ , derive λ ( s ) , ω ( s ) , λ ( t ) , ω ( t ) , r ( t ) , c ( t ) , . . . 18
Optimization problem N − 1 � 2 � ϕ j +1 − ϕ j − ϕ j − ϕ j − 1 � minimize (1) ∆ j ∆ j − 1 ϕ 1 ,...,ϕ N j =1 N − 1 − c z c z ∆ j √ ϕ N = ˙ � i i √ ϕ j +1 + √ ϕ j subject to (2) g g j =0 ω 2 i , min ≤ ϕ N ≤ ω 2 (3) i , max ∀ j , λ min ∆ j ≤ ϕ j +1 − ϕ j ≤ λ max ∆ j (4) ϕ 1 = ∆ 0 g / z f (5) (1): min. height variations (2): boundedness (3): CoP polygon (4): pressure constraints (5): stationary height z f 19
Behavior of solutions Figure : CoM trajectories obtained by solving the resultant nonlinear optimization for different initial velocities. 20
Resulting walking patterns Code: https://github.com/stephane-caron/capture-walking 21
What did we see? Horizontal walking → LTI system With CoM height variations → nonlinear system Solve first the boundedness condition → LTV system Link with TOPP, nonlinear optimization... Outcome: dynamic stair-climbing walking patterns For details, see [CM18; Car+18] 22
Thanks! Thank you for your attention! 23
References I [Car+18] St´ ephane Caron, Adrien Escande, Leonardo Lanari, and Bastien Mallein. “Capturability-based Analysis, Optimization and Control of 3D Bipedal Walking”. In: Submitted. 2018. url : https://hal.archives-ouvertes.fr/hal- 01689331/document . [CK17] St´ ephane Caron and Abderrahmane Kheddar. “Dynamic Walking over Rough Terrains by Nonlinear Predictive Control of the Floating-base Inverted Pendulum”. In: Intelligent Robots and Systems (IROS), 2017 IEEE/RSJ International Conference on . Sept. 2017. [CM18] St´ ephane Caron and Bastien Mallein. “Balance control using both ZMP and COM height variations: A convex boundedness approach”. to be presented at ICRA 2018. May 2018. url : https://hal.archives-ouvertes.fr/hal- 01590509/document . 24
References II [Eng+11] Johannes Englsberger, Christian Ott, Maximo Roa, Alin Albu-Sch¨ affer, Gerhard Hirzinger, et al. “Bipedal walking control based on capture point dynamics”. In: Intelligent Robots and Systems (IROS), 2011 IEEE/RSJ International Conference on . IEEE, 2011, pp. 4420–4427. [EOA15] Johannes Englsberger, Christian Ott, and Alin Albu-Schaffer. “Three-dimensional bipedal walking control based on divergent component of motion”. In: IEEE Transactions on Robotics 31.2 (2015), pp. 355–368. [HRO16] Bernd Henze, M´ aximo A. Roa, and Christian Ott. “Passivity-based whole-body balancing for torque-controlled humanoid robots in multi-contact scenarios”. In: The International Journal of Robotics Research (July 12, 2016). doi : 10.1177/0278364916653815 . 25
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