Pendular models for walking over rough terrains . Stéphane Caron June 20, 2017 Journées Nationales de la Robotique Humanoïde, Montpellier, France
goal . 1
Standard model reduction .
multi-body systems . Equation of motion c F Constraints Assumption • (Rigid bodies) 2 M ¨ q + h ( q , ˙ q ) = S T τ + J T • τ ∈ {feasible torques} • F ∈ {feasible contact forces}
newton-euler dynamics . • Infinite torques Assumption Constraints L c g m c Equations of motion 3 ¨ 1 ∑ i f i + ⃗ = ˙ = ∑ i ( p i − c ) × f i • Friction cones: ∀ i , f i ∈ C i
forward integration . by iterative methods (e.g. RK4) Forward integration approximated L c g 4 m c Equations of motion 1 ¨ ∑ i f i + ⃗ = ˙ = ∑ i ( p i − c ) × f i
pendular mode . Pendular mode Conserve the angular momentum at the center-of-mass • Pro: enables exact forward integration regardless of joint state 5 ˙ L c = 0 • Con: assumes ˙ L c = 0 feasible
From 2D to 3D locomotion .
lipm and cart-table . LIPM [Kaj+01] c CART-table [Kaj+03] g • Output: z 6 • Control: ¨ c ∈ ω 2 ( c − S ) + ⃗ • Control: z ∈ S • Output: ¨
linear inverted pendulum mode . Equation of motion g Constraints Assumptions • Infinite torques • Pendular mode • Infinite friction • Contacts are coplanar 7 c = ω 2 ( c − z ) + ⃗ ¨ • ZMP support area: z ∈ S • COM lies in a plane: c z = h
without infinite friction . Figure 1: ZMP support area with friction [CPN17] 8
without coplanar contacts . Figure 2: ZMP support area with non-coplanar contacts [CPN17] 9
linear pendulum mode . Equation of motion g Constraints Assumptions • Infinite torques • Pendular mode • COM lies in a virtual plane 10 c = ± ω 2 ( c − z ) + ⃗ ¨ • ZMP support area: z ∈ S chosen via ± ω 2 = g / h
observation . 11 ZMP support area S changes with COM position:
lipm and cart-table . 2D LIPM c 2D CART-table g • Output: z 12 • Control: ¨ c ∈ ω 2 ( c − S ) + ⃗ • Control: z ∈ S • Output: ¨
3D CART-table .
com acceleration cone . Algorithm [CK16] Figure 3: ZMP support areas for Figure 4: COM acceleration cone for the same stance 13 Compute the 3D cone C of COM accelerations different values of ± ω 2
observation . 14 The cone C still depends on the COM position c :
predictive control . Walking patterns not very dynamic , but works surprisingly well! 15 For predictive control, intersect cones C over all c ∈ preview: Preview COM accelerations Preview COM locations
check it out! . https://github.com/stephane-caron/3d-com-mpc 16
3D Pendulum Mode .
lipm and cart-table . 2D LIPM c 3D COM-accel [CK16] • Output: z 17 • Control: ¨ c ∈ C ( c ) • Control: z ∈ S • Output: ¨
inverted pendulum mode . g Inverted Pendulum Remove this assumption: h g Linear Inverted Pendulum g 18 ¨ c = ω 2 ( c − z ) + ⃗ √ Plane assumption: ω = ↓ ¨ c = λ ( c − z ) + ⃗
inverted pendulum mode . Equation of motion g Constraints Assumptions • Infinite torques • Infinite friction • Pendular mode 19 ¨ c = λ ( c − z ) + ⃗ • Unilaterality λ ≥ 0 • ZMP support area: z ∈ S
inverted pendulum mode with friction . Equation of motion g Constraints Assumptions • Infinite torques • Pendular mode 20 ¨ c = λ ( c − z ) + ⃗ • Unilaterality λ ≥ 0 • ZMP support area: z ∈ S • Friction: c − z ∈ C
inverted pendulum mode: question . Equation of motion g • Product bwn control and state • Forward integration: how to make it exact ? 21 ¨ c = λ ( c − z ) + ⃗
reformulation . Floating-base inverted pendulum (FIP) Allow the ZMP to leave the contact area. 1 Figure 5: Friction constraint Figure 6: ZMP constraint 1 At heart, it is used to locate the central axis of the contact wrench [SB04] 22
floating-base inverted pendulum . Equation of motion g Constraints [CK17] • ZMP support cone: Assumptions • Infinite torques • Pendular mode 23 ¨ c = ω 2 ( c − z ) + ⃗ • Friction: c − z ∈ C ∀ i , e i · ( v i − c ) × ( z − v i ) ≤ 0
properties of fip model . Equation of motion g • Forward integration is exact : • Capture Point is defined: c g 24 c = ω 2 ( c − z ) + ⃗ ¨ c ( t ) = α 0 e ω t + β 0 e − ω t + γ 0 ξ = c + ˙ ω + ⃗ ω 2
model predictive control . NMPC Optimization • Runs at 30 Hz • Adapts step timings • FIP for forward integration • Sometimes fails... Linear-Quadratic Regulator • Runs at 300 Hz • Takes over when NMPC fails 25
check it out! . https://github.com/stephane-caron/dynamic-walking 26
conclusion .
conclusion . 2D LIPM c 2D CART-table g • Output: z 27 • Control: ¨ c ∈ ω 2 ( c − S ) + ⃗ • Control: z ∈ S • Output: ¨
conclusion . 3D FIP [CK17] c 3D COM-accel [CK16] • Output: z 27 • Control: ¨ • Control: z ∈ S ( c ) c ∈ C ( c ) • Output: ¨
thanks for listening! 27
references i Robots and Systems (IROS), 2017 IEEE/RSJ International on Robotics 33.1 (Feb. 2017), pp. 67–80. Mobility Under Frictional Constraints”. In: IEEE Transactions Yoshihiko Nakamura. “ZMP Support Areas for Multi-contact Stéphane Caron, Quang-Cuong Pham, and [CPN17] Conference on . to be presented. Sept. 2017. Walking over Rough Terrains by Nonlinear Predictive Control . Stéphane Caron and Abderrahmane Kheddar. “Dynamic [CK17] IEEE-RAS International Conference on . Nov. 2016. Walking Pattern Generation based on Model Preview Control Stéphane Caron and Abderrahmane Kheddar. “Multi-contact [CK16] 28 of 3D COM Accelerations”. In: Humanoid Robots, 2016 of the Floating-base Inverted Pendulum”. In: Intelligent
references ii . Humans 34.5 (2004), pp. 630–637. on Systems, Man and Cybernetics, Part A: Systems and center of pressure-zero moment point”. In: IEEE Transactions P. Sardain and G. Bessonnet. “Forces acting on a biped robot. [SB04] Robotics and Automation . Vol. 2. IEEE. 2003, pp. 1620–1626. “Biped walking pattern generation by using preview control Kensuke Harada, Kazuhito Yokoi, and Hirohisa Hirukawa. Shuuji Kajita, Fumio Kanehiro, Kenji Kaneko, Kiyoshi Fujiwara, [Kaj+03] IEEE. 2001, pp. 239–246. Mode: A simple modeling for a biped walking pattern and Hirohisa Hirukawa. “The 3D Linear Inverted Pendulum Shuuji Kajita, Fumio Kanehiro, Kenji Kaneko, Kazuhito Yokoi, [Kaj+01] 29 generation”. In: Intelligent Robots and Systems, 2001. Vol. 1. of zero-moment point”. In: IEEE International Conference on
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