Biped Stabilization by Linear Feedback of the Variable-Height - PowerPoint PPT Presentation

Biped Stabilization by Linear Feedback of the Variable-Height Inverted Pendulum Model Stéphane Caron May–August 2020 IEEE Virtual Conference on Robotics and Automation

1 height variation strategy F ext F ext Rest position Ankle strategy Height variation strategy

in this video Idea : • Divergent Component of Motion goes 4D Techniques : • Variation dynamics around reference trajectory • Least-squares pole placement 2

1 height variation strategy 1. Twan Koolen , Michael Posa et Russ Tedrake . « Balance control using center of mass 3 F ext F ext Rest position Ankle strategy Height variation strategy height variation : Limitations imposed by unilateral contact ». In : Humanoids 2016 .

pendulum models Linear Inverted Pendulum Variable-Height Inverted Pendulum 4 • Model : ¨ c = λ ( c − z ) + g • Model : ¨ c = ω 2 ( c − z ) + g • Inputs : u = z ∈ R 2 • Inputs : u = [ z λ ] ∈ R 3

balance control Linear Inverted Pendulum Variable-Height Inverted Pendulum • Control : nonlinear MPC [1] 5 • State : ξ = c + ˙ c /ω ∈ R 3 • State : [ c ˙ c ] ∈ R 6 • Control : z = − k ( ξ d − ξ )

divergent component of motion 3D DCM 2. Johannes Englsberger , Christian Ott et Alin Albu-Schäffer . « Three-dimensional bipe- 2 Viability Convergent dynamics c Divergent dynamics 6 ˙ ξ := c + ω ˙ ξ = ω ( ξ − z ) ˙ c = ω ( ξ − c ) Diverges ifg ξ / ∈ support ( z ) dal walking control based on divergent component of motion ». In : IEEE T-RO (2015).

pole placement Derive feedback et Fumio Kanehiro . « Balance control based on capture point error compensation for biped walking 3. Mitsuharu Morisawa , Nobuyuki Kita , Shin’ichiro Nakaoka , Kenji Kaneko , Shuuji Kajita 3 7 Desired error dynamics Error dynamics Tracking error : ∆ ξ := ξ − ξ d ∆ ˙ ξ = ω (∆ ξ − ∆ z ) ξ ∗ = − k p ∆ ξ ∆ ˙ ∆ z ∗ = arg min ∥ ∆ ξ ∗ − ∆ ξ ∥ ∆ z [ ] = − 1 + k p ∆ ξ ω on uneven terrain ». In : Humanoids 2012 .

time-varying dcm Model Time-varying DCM c 4 4. Michael A. Hopkins , Dennis W. Hong et Alexander Leonessa . « Humanoid locomotion on 8 ¨ c = λ ( c − z ) + g Pre-defjning c z ( t ) → λ ( t ) makes system LTV : ˙ ξ = c + ω ( t ) ω = ω 2 − λ . with the Riccati equation ˙ uneven terrain using the time-varying Divergent Component of Motion ». In : Humanoids 2014 .

dcm in space and time Viability 5. Stéphane Caron , Adrien Escande , Leonardo Lanari et Bastien Mallein . « Capturability- 5 Viability Convergent dynamics Divergent dynamics Cartesian space Phase space Input 9 Divergent dynamics Input z Convergent dynamics λ ˙ ω = ω 2 − λ ξ = ω ( ξ − z ) ˙ ˙ ζ = ω ( z − ζ ) γ = λ − γ 2 ˙ Diverges ifg ω 2 / Diverges ifg ξ / ∈ support ( z ) ∈ [ λ min , λ max ] based Pattern Generation for Walking with Variable Height ». In : IEEE T-RO (2020).

interpretation 10

4d dcm for the vhip 0 0 g z 0 0 Divergent component of motion 0 11 Divergent dynamics x = [ ξ ω ] ∈ R 4 : Cartesian 3D DCM ξ + natural frequency ω [ ˙ ] [ ] [ ] [ ] [ ] ξ λ I 3 λ I 3 ˙ x = = 1 x − 1 + 1 ω ω ω ω ˙ ω 2 ω λ

4d dcm for the vhip 0 0 1 0 1 Tracking error dynamics Take its linearized error dynamics ( a.k.a. variation dynamics) : 0 g Divergent component of motion z 0 0 11 0 Divergent dynamics x = [ ξ ω ] ∈ R 4 : Cartesian 3D DCM ξ + natural frequency ω [ ˙ ] [ ] [ ] [ ] [ ] ξ λ I 3 λ I 3 ˙ x = = 1 x − 1 + 1 ω ˙ ω ω 2 ω ω λ ω ( ξ d − z d ) [ ] [ ] [ ] λ d I 3 − ¨ c d /ω d λ d I 3 ∆ z ∆˙ x = ∆ x − 2 ( ω d ) 2 ω d ∆ λ ω d ω d Linear system : ∆˙ x = A ∆ x + B ∆ u .

least-squares pole placement Derive feedback Subject to : Error dynamics 12 Desired error dynamics ∆˙ x = A ∆ x + B ∆ u x ∗ = − k p ∆ x ∆˙ Minimize : ∥ ∆˙ x − ∆˙ x ∗ ∥ 2 • Linearized dynamics : ∆˙ x = A ∆ x + B ∆ u • ZMP support area : C ( z d + ∆ z ) ≤ d • Reaction force : λ min ≤ λ d + ∆ λ ≤ λ max • Kinematics : h min ≤ ξ d z + ∆ ξ z ≤ h max

behavior https://github.com/stephane-caron/pymanoid/blob/master/examples/vhip_stabilization.py 13

controller adjusts the dcm Previously, the DCM was a measured state : Now, the DCM is an output that can be adjusted : 14 – DCM Pole placement + Constrained – Pole Placement + The controller can vary ω (height) to maintain ξ ∈ support ( z ) .

force control https://github.com/stephane-caron/vhip_walking_controller 15 LIP tracking VHIP tracking

what have we seen ? Idea : • Divergent Component of Motion goes 4D Techniques : • Variation dynamics around reference trajectory • Least-squares pole placement To go further • Link with exponential dichotomies (Coppel, 1967) : https://scaron.info/talks/jrl-2019.html 16

thanks ! Thank you for your attention ! 17

references i [1] Shuuji Kajita et Fumio Kanehiro . « Balance control based on capture point error Mitsuharu Morisawa , Nobuyuki Kita , Shin’ichiro Nakaoka , Kenji Kaneko , [5] Twan Koolen , Michael Posa et Russ Tedrake . « Balance control using center of mass [4] 2014 . 18 Michael A. Hopkins , Dennis W. Hong et Alexander Leonessa . « Humanoid locomotion on [3] Johannes Englsberger , Christian Ott et Alin Albu-Schäffer . « Three-dimensional [2] T-RO (2020). Stéphane Caron , Adrien Escande , Leonardo Lanari et Bastien Mallein . « Capturability-based Pattern Generation for Walking with Variable Height ». In : IEEE bipedal walking control based on divergent component of motion ». In : IEEE T-RO (2015). uneven terrain using the time-varying Divergent Component of Motion ». In : Humanoids height variation : Limitations imposed by unilateral contact ». In : Humanoids 2016 . compensation for biped walking on uneven terrain ». In : Humanoids 2012 .