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Dynamic Walking Over Rough Terrains by Nonlinear Predictive Control - - PowerPoint PPT Presentation
Dynamic Walking Over Rough Terrains by Nonlinear Predictive Control - - PowerPoint PPT Presentation
Dynamic Walking Over Rough Terrains by Nonlinear Predictive Control of the Floating-Base Inverted Pendulum . Stphane Caron & Abderrahmane Kheddar September 27, 2017 IROS 2017, Vancouver, Canada goal . 1 goal . 1 Quasi-static
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goal .
Quasi-static Dynamic
isometric 1
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linear inverted pendulum mode .
Equation of motion ¨ c = ω2(c − z) + ⃗ g Feasibility conditions
- Constant: ω =
√ g/h
- ZMP support area: z ∈ S
- Friction?
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inverted pendulum mode .
Linear Inverted Pendulum ¨ c = ω2(c − z) + ⃗ g Feasibility conditions
- Constant: ω =
√ g/h
- ZMP support area: z ∈ S
- Friction?
Inverted Pendulum ¨ c = λ(c − z) + ⃗ g Feasibility conditions
- Unilaterality: λ ≥ 0
- ZMP support area: z ∈ S
- Friction?
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inverted pendulum mode .
Equation of motion ¨ c = λ(c − z) + ⃗ g Feasibility conditions
- Unilaterality: λ ≥ 0
- ZMP support area: z ∈ S
- Friction?
c z
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inverted pendulum mode .
Equation of motion ¨ c = λ(c − z) + ⃗ g Feasibility conditions
- Unilaterality: λ ≥ 0
- ZMP support area: z ∈ S
- Friction?
c z
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inverted pendulum mode with friction .
Equation of motion ¨ c = λ(c − z) + ⃗ g Feasibility conditions
- Unilaterality: λ ≥ 0
- ZMP support area: z ∈ S
- Friction: c − z ∈ C
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nonlinear optimal control .
Equation of motion ¨ c = λ(c − z) + ⃗ g Forward integration1
- Direct multiple shooting2
- Discretization: # of sample
points, integration step
- Resolution of integrator?
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1See also Takasugi et al. (this session): "3D Walking and Skating..." 2Carpentier, Tonneau, Naveau, Stasse, and Mansard 2016.
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a model with exact integration .
Equation of motion ¨ c = ω2(c − z) + ⃗ g Virtual Repellent Points
- The ZMP/eCMP/VRP2 can
leave the contact area
- Fwd integration is exact:
c(t) = αeωt + βe−ωt + γ
- Feasibility conditions?
(Figure adapted from 2)
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3Englsberger, Ott, and Albu-Schaffer 2015.
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floating-base inverted pendulum .
Equation of motion ¨ c = ω2(c − z) + ⃗ g Floating-base pendulum
- Floating ZMP (eCMP)
- Exact forward integration
- New feasibility condition
Friction cone ZMP support cone
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floating-base inverted pendulum .
Equation of motion ¨ c = ω2(c − z) + ⃗ g Feasibility conditions
- Constant: ω > 0
- ZMP support cone: z ∈ Z
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goal .
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nonlinear model predictive control .
Nonlinear optimization...
- DMS over FIP model
- Adaptive step timings
- Runs at 30 Hz
... but significant failures
- Model is nonconvex
- Noise and delays in ZMP
control / COM estimation ⇒ jumps in PO map
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recovering from failures .
This communication: Constrained LQ regulator
- Linear EoM + linearized ZMP cones = Quadratic Program
- Runs at 300 Hz, recovers locally from failures
Next communication:3 Spoiler! A convexly-constrained model: one global optimum, 1000 Hz
https://scaron.info/research/3d-balance.html
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4Caron and Mallein 2017.
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recovering from failures .
This communication: Constrained LQ regulator
- Linear EoM + linearized ZMP cones = Quadratic Program
- Runs at 300 Hz, recovers locally from failures
Next communication:3 Spoiler! A convexly-constrained model: one global optimum, 1000 Hz
https://scaron.info/research/3d-balance.html
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4Caron and Mallein 2017.
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check it out! .
https://github.com/stephane-caron/dynamic-walking
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conclusion .
Floating-base Pendulum
- LTI model for 3D walking
- ZMP support area ⇒ cone
Nonlinear Predictive Control
- Can solve full problem
- Failures (nonconvexity)
- Recovery: constrained LQR
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thank you for your attention!
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