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Pendular models for walking over rough terrains . Stphane Caron June 20, 2017 Journes Nationales de la Robotique Humanode, Montpellier, France goal . 1 Standard model reduction . multi-body systems . Equation of motion c F

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Pendular models for walking over rough terrains

.

Stéphane Caron June 20, 2017

Journées Nationales de la Robotique Humanoïde, Montpellier, France

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goal .

1

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Standard model reduction .

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multi-body systems .

Equation of motion M¨ q + h(q, ˙ q) = STτ + JT

cF

Constraints

  • τ ∈ {feasible torques}
  • F ∈ {feasible contact forces}

Assumption

  • (Rigid bodies)

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newton-euler dynamics .

Equations of motion ¨ c =

1 m

∑

i fi + ⃗

g ˙ Lc = ∑

i(pi − c) × fi

Constraints

  • Friction cones: ∀i, fi ∈ Ci

Assumption

  • Infinite torques

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forward integration .

Equations of motion ¨ c =

1 m

∑

i fi + ⃗

g ˙ Lc = ∑

i(pi − c) × fi

Forward integration approximated by iterative methods (e.g. RK4)

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pendular mode .

Pendular mode ˙ Lc = 0 Conserve the angular momentum at the center-of-mass

  • Pro: enables exact forward

integration

  • Con: assumes ˙

Lc = 0 feasible regardless of joint state

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From 2D to 3D locomotion .

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lipm and cart-table .

LIPM [Kaj+01]

  • Control: z ∈ S
  • Output: ¨

c CART-table [Kaj+03]

  • Control: ¨

c ∈ ω2(c − S) + ⃗ g

  • Output: z

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linear inverted pendulum mode .

Equation of motion ¨ c = ω2(c − z) + ⃗ g Constraints

  • ZMP support area: z ∈ S

Assumptions

  • Infinite torques
  • Pendular mode
  • COM lies in a plane: cz = h
  • Infinite friction
  • Contacts are coplanar

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without infinite friction .

Figure 1: ZMP support area with friction [CPN17]

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without coplanar contacts .

Figure 2: ZMP support area with non-coplanar contacts [CPN17]

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linear pendulum mode .

Equation of motion ¨ c = ±ω2(c − z) + ⃗ g Constraints

  • ZMP support area: z ∈ S

Assumptions

  • Infinite torques
  • Pendular mode
  • COM lies in a virtual plane

chosen via ±ω2 = g/h

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  • bservation

.

ZMP support area S changes with COM position:

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lipm and cart-table .

2D LIPM

  • Control: z ∈ S
  • Output: ¨

c 2D CART-table

  • Control: ¨

c ∈ ω2(c − S) + ⃗ g

  • Output: z

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3D CART-table .

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com acceleration cone .

Algorithm [CK16] Compute the 3D cone C of COM accelerations

Figure 3: ZMP support areas for different values of ±ω2 Figure 4: COM acceleration cone for the same stance

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  • bservation

.

The cone C still depends on the COM position c:

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predictive control .

For predictive control, intersect cones C over all c ∈ preview:

Preview COM locations Preview COM accelerations

Walking patterns not very dynamic, but works surprisingly well!

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check it out! .

https://github.com/stephane-caron/3d-com-mpc

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3D Pendulum Mode .

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lipm and cart-table .

2D LIPM

  • Control: z ∈ S
  • Output: ¨

c 3D COM-accel [CK16]

  • Control: ¨

c ∈ C(c)

  • Output: z

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inverted pendulum mode .

Linear Inverted Pendulum ¨ c = ω2(c − z) + ⃗ g Plane assumption: ω = √

g h

↓ Remove this assumption: Inverted Pendulum ¨ c = λ(c − z) + ⃗ g

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inverted pendulum mode .

Equation of motion ¨ c = λ(c − z) + ⃗ g Constraints

  • Unilaterality λ ≥ 0
  • ZMP support area: z ∈ S

Assumptions

  • Infinite torques
  • Infinite friction
  • Pendular mode

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inverted pendulum mode with friction .

Equation of motion ¨ c = λ(c − z) + ⃗ g Constraints

  • Unilaterality λ ≥ 0
  • ZMP support area: z ∈ S
  • Friction: c − z ∈ C

Assumptions

  • Infinite torques
  • Pendular mode

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inverted pendulum mode: question .

Equation of motion ¨ c = λ(c − z) + ⃗ g

  • Product bwn control and state
  • Forward integration: how to

make it exact?

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reformulation .

Floating-base inverted pendulum (FIP) Allow the ZMP to leave the contact area.1

Figure 5: Friction constraint Figure 6: ZMP constraint

1At heart, it is used to locate the central axis of the contact wrench [SB04]

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floating-base inverted pendulum .

Equation of motion ¨ c = ω2(c − z) + ⃗ g Constraints [CK17]

  • Friction: c − z ∈ C
  • ZMP support cone:

∀i, ei · (vi − c) × (z − vi) ≤ 0 Assumptions

  • Infinite torques
  • Pendular mode

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properties of fip model .

Equation of motion ¨ c = ω2(c − z) + ⃗ g

  • Forward integration is exact:

c(t) = α0eωt + β0e−ωt + γ0

  • Capture Point is defined:

ξ = c + ˙ c ω + ⃗ g ω2

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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

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check it out! .

https://github.com/stephane-caron/dynamic-walking

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conclusion .

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conclusion .

2D LIPM

  • Control: z ∈ S
  • Output: ¨

c 2D CART-table

  • Control: ¨

c ∈ ω2(c − S) + ⃗ g

  • Output: z

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conclusion .

3D FIP [CK17]

  • Control: z ∈ S(c)
  • Output: ¨

c 3D COM-accel [CK16]

  • Control: ¨

c ∈ C(c)

  • Output: z

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thanks for listening!

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references i .

[CK16] Stéphane Caron and Abderrahmane Kheddar. “Multi-contact Walking Pattern Generation based on Model Preview Control

  • f 3D COM Accelerations”. In: Humanoid Robots, 2016

IEEE-RAS International Conference on. Nov. 2016. [CK17] Stéphane Caron and Abderrahmane Kheddar. “Dynamic Walking over Rough Terrains by Nonlinear Predictive Control

  • f the Floating-base Inverted Pendulum”. In: Intelligent

Robots and Systems (IROS), 2017 IEEE/RSJ International Conference on. to be presented. Sept. 2017. [CPN17] Stéphane Caron, Quang-Cuong Pham, and Yoshihiko Nakamura. “ZMP Support Areas for Multi-contact Mobility Under Frictional Constraints”. In: IEEE Transactions

  • n Robotics 33.1 (Feb. 2017), pp. 67–80.

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references ii .

[Kaj+01] Shuuji Kajita, Fumio Kanehiro, Kenji Kaneko, Kazuhito Yokoi, and Hirohisa Hirukawa. “The 3D Linear Inverted Pendulum Mode: A simple modeling for a biped walking pattern generation”. In: Intelligent Robots and Systems, 2001. Vol. 1.

  • IEEE. 2001, pp. 239–246.

[Kaj+03] Shuuji Kajita, Fumio Kanehiro, Kenji Kaneko, Kiyoshi Fujiwara, Kensuke Harada, Kazuhito Yokoi, and Hirohisa Hirukawa. “Biped walking pattern generation by using preview control

  • f zero-moment point”. In: IEEE International Conference on

Robotics and Automation. Vol. 2. IEEE. 2003, pp. 1620–1626. [SB04]

  • P. Sardain and G. Bessonnet. “Forces acting on a biped robot.

center of pressure-zero moment point”. In: IEEE Transactions

  • n Systems, Man and Cybernetics, Part A: Systems and

Humans 34.5 (2004), pp. 630–637.

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