Walking and stair climbing stabilization for position-controlled biped robots . Stéphane Caron March 19, 2019 Presentation given at Wandercraft, Paris
context . COMANOID project - https://comanoid.cnrs.fr 1
final demonstrator . 2
stair climbing part . https://www.youtube.com/watch?v=vFCFKAunsYM 3
hrp-4 humanoid robot in short . Stiff position control on all joints Mechanical flexibility at the ankles 1 1. Kenji Kaneko, Fumio Kanehiro, Mitsuharu Morisawa, Kazuhiko Akachi, Gou Miyamori, Atsushi Hayashi et Noriyuki Kanehira. « Humanoid robot HRP-4 - Humanoid Robotics Platform with Light- 4 weight and Slim Body ». In : IEEE/RSJ International Conf. on Intelligent Robots and Systems . 2011.
system overview . 5 Whole-body Walking Whole-body Kinematic Pattern Admittance Generation Control Control DCM DCM Control Observer
walking pattern generation . 6 Desired Kinematic Footstep Whole-body Walking Whole-body Targets Locations Kinematic Pattern Admittance Generation Control Control DCM DCM Control Observer
linear inverted pendulum mode m IEEE/RSJ International Conference on Intelligent Robots and Systems . 2001. 2. Shuuji Kajita, Fumio Kanehiro, Kenji Kaneko, Kazuhito Yokoi et Hirohisa Hirukawa. « The 3D 2 . 7 • Equation of motion : • Floating base dynamics : M ¨ q + N = S T τ + J T f ¨ c = 1 ∑ i f i ˙ L c = ∑ i ( p i − c ) × f i • Angular momentum ˙ L c = 0 : c = ω 2 ( c − z ) ¨ with ω 2 = g / h and z the ZMP Linear Inverted Pendulum Mode : A simple modeling for a biped walking pattern generation ». In :
divergent component of motion . rence on Intelligent Robots and Systems . 2011. 3. Johannes Englsberger, Christian Ott, Maximo Roa, Alin Albu-Schäffer, Gerhard Hirzinger et 3 • Unstable dynamics : 8 c • Divergent Component of Motion : • LIPM equation of motion : c = ω 2 ( c − z ) ¨ ξ = c + ˙ ω ˙ ξ = ω ( ξ − z ) al. « Bipedal walking control based on capture point dynamics ». In : IEEE/RSJ International Confe-
walking pattern generation . Generate a CoM-ZMP trajectory that is : Consistent Feasible • ZMP belongs to support area • Contact force within friction cone Viable Not falling. For this system, same as Figure adapted from [Gri+17]. 9 c ( t ) = ω 2 ( c ( t ) − z ( t )) ∀ t > 0 , ¨ bounded : ∃ M > 0 , ∀ t > 0 , ∥ c ( t ) ∥ ≤ M
walking pattern generation . So far we have tested three methods : • Linear Model Predictive Control [Wie06] • Foot-guided Agile Control through ZMP Manipulation [SY17] • Capturability of Variable-Height Inverted Pendulum [Car+18] 10
linear model predictive control • Viability : terminal DCM 2006. 4. Pierre-Brice Wieber. « Trajectory free linear model predictive control for stable walking in the 4 • Variable step timings : [BW17] • Variable CoM height : [Bra+15] Allows a number of extensions, including : • Feasibility : ZMP in support area . • Consistency : state equation Constraints • Minimize CoM jerk • Track desired CoM velocity • Track desired ZMP reference Cost function Formulate preview control [Kaj+03] as a Quadratic Program (QP) : 11 presence of strong perturbations ». In : IEEE-RAS International Conference on Humanoid Robots .
foot-guided agile control through zmp manipulation . tems . 2017. 5. Tomomichi Sugihara et Takanobu Yamamoto. « Foot-guided Agile Control of a Biped Robot 5 • Call many times to adapt step timings • Analytical solution : • Finite horizon, continuous time dynamics 12 subject to Predictive control with ZMP as input minimize ∫ T ( z ( t ) − z d ) 2 d t ⇒ Feasibility (best effort) z ( t ) 0 c = ω 2 ( c − z ) ¨ ⇒ Consistency c ( T ) + ˙ c ( T ) = ξ d ⇒ Viability ω z ∗ (0) = z d + 2( ξ (0) − z d ) − ( ξ d − z d ) e − ω T 1 − e − 2 ω T through ZMP Manipulation ». In : IEEE/RSJ International Conference on Intelligent Robots and Sys-
variable-height inverted pendulum . ouvertes.fr/hal-01689331/document . 6. Stéphane Caron, Adrien Escande, Leonardo Lanari et Bastien Mallein. « Capturability-based 6 • Call many times to adapt step timings 13 • New input λ > 0 for height variations : ¨ c = λ ( c − z ) + g • Viability ⇒ boundedness condition : ∫ ∞ ( λ ( t ) r ( t ) − g ) e − Ω( t ) d t ξ (0) = 0 • Solve : tailored optimization (30-50 µ s) Analysis, Optimization and Control of 3D Bipedal Walking ». 2018. url : https://hal.archives-
visualization . Visualization of stair climbing pattern in mc_rtc 14
walking stabilization . 15 Desired Kinematic Footstep Whole-body Walking Whole-body Targets Locations Kinematic Pattern Admittance Generation Control Control DCM DCM Control Observer
role of stabilization . Actuated joints converge but unactuated floating base diverges : In walking pattern By robot without stabilization Figure adapted from [Tak+09]. 16
floating base facts . Let's review the facts : • Floating base translation is unactuated • Its dynamics are reduced to : • Only way to control it is via indirect force control of the ZMP z 17 c = ω 2 ( c − z ) ¨
indirect force control . ... but our robot is position-controlled ? Split control into two components : Admittance control Allow position changes to improve force tracking Floating-base control Assuming force control, select reaction force to control the floating base 18
visualization . Standing stabilization under external forces 19
admittance control . 20 Commanded Desired Kinematic Kinematic Commanded Footstep Whole-body Walking Whole-body Targets Targets Joint Angles Locations Kinematic Pattern Admittance Generation Control Control Distributed Measured Foot Wrenches Foot Wrenches DCM DCM Control Observer
strategies . Different strategies for different components of the net contact wrench : • CoP at each contact [Kaj+01b] • Pressure distribution [Kaj+10] • CoM admittance control [Nag99] This stabilizer implements : • Ankle strategy : yes • Hip strategy : translation, no rotation • Stepping strategy : no 21
center-of-pressure control . • Rotate end-effector to move its CoP • Assumes compliance at contact : • Apply damping control : Figure adapted from [Kaj+01b] 7 7. Shuuji Kajita, Kazuhito Yokoi, Muneharu Saigo et Kazuo Tanie. « Balancing a Humanoid Robot Using Backdrive Concerned Torque Control and Direct Angular Momentum Feedback ». In : IEEE International Conference on Robotics and Automation . 2001. 22 τ = K e ( θ − θ e ) ˙ θ = A cop ( τ d − τ ) • Closed-loop behavior has τ → τ d
pressure distribution control . Kaneko, Fumio Kanehiro et Kazuhito Yokoi. « Biped walking stabilization based on linear inverted 8. Shuuji Kajita, Mitsuharu Morisawa, Kanako Miura, Shin'ichiro Nakaoka, Kensuke Harada, Kenji 8 Figure adapted from [Kaj+10] 23 • Apply damping control : • Push down with foot that needs more • Net vertical force compensates pressure, lift the other one gravity ⇒ only need to control : ∆ f z = f Rz − f Lz * d z p * d ctrl p Lz Rz ˙ z ctrl = A z (∆ f zd − ∆ f z ) f f Rz Lz pendulum tracking ». In : IEEE/RSJ International Conference on Intelligent Robots and Systems . 2010.
com admittance control • Closed-loop behavior : analysis is 9. Discussions with Pr T. Sugihara. 10 . only possible with delay or 24 • Accelerate CoM against ZMP error : c = ω 2 ( c − z ) • Recall that ¨ ¨ c = A c ( z − z d ) disturbance observer 9 measured desired 10. Ken'ichiro Nagasaka. « Whole-body Motion Generation for a Humanoid Robot by Dynamics Filters ». In : PhD thesis (1999). The University of Tokyo, in Japanese.
choice of strategies . Which ones to choose ? End-effector strategies • CoP at each contact [Kaj+01b] • Pressure distribution [Kaj+10] ... are sufficient to control the net wrench, yet : CoM admittance control [Nag99] • uses other joints, e.g. hips 11. The effect is similar to Model ZMP Control [Tak+09]. 25 • helps recover from ZMP saturation 11
com admittance in stair climbing . 26 Measured ZMP 0.3 Measured DCM 0.2 Desired DCM 0.1 Sagittal Coordinate (m) 0.0 -0.1 Desized ZMP 0.3 Maximum ZMP 0.2 Minimum ZMP 0.1 0.0 -0.1 Time (s) Figure 1 : Top : no CoM admittance control. Bottom : with A c = 20 [Hz 2 ].
dcm control . 27 Commanded Desired Kinematic Kinematic Commanded Footstep Whole-body Walking Whole-body Targets Targets Joint Angles Locations Kinematic Pattern Admittance Generation Control Control Distributed Measured Foot Wrenches Foot Wrenches Estimated Desired DCM Measured Joint Angles DCM DCM DCM Desired CoM & Contacts Measured IMU Orientation Control Observer
net wrench • Apply feedback control to it : mation . 2009. 12. Tomomichi Sugihara. « Standing stabilizability and stepping maneuver in planar bipedalism 12 This gives us the net wrench . . 28 yields best CoM-ZMP regulator [Sug09] requires less control input [Tak+09] • Assume control of z in ¨ c = ω 2 ( c − z ) • Controlling only the DCM ˙ ξ = ω ( ξ − z ) : ξ = ˙ ˙ ξ d + k p ( ξ d − ξ ) [ ] z = z d − 1 + k p ( ξ d − ξ ) ω based on the best COM-ZMP regulator ». In : IEEE International Conference on Robotics and Auto-
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