SLIDE 1
2
3D Bipedal Walking including COM height variations St ephane Caron - - PowerPoint PPT Presentation
3D Bipedal Walking including COM height variations St ephane Caron - - PowerPoint PPT Presentation
3D Bipedal Walking including COM height variations St ephane Caron CRI Group Seminar Series May 14, 2018 What do we want? COMANOID project https://comanoid.cnrs.fr 2 What do we want? COMANOID project Aircraft entry plan (2017) 3
SLIDE 2
SLIDE 3
3
What do we want?
COMANOID project – Aircraft entry plan (2017)
SLIDE 4
4
What do we want?
Hard part: dynamic stair climbing
SLIDE 5
5
Interest reaches farther than humanoids
Duality between manipulation and walking
Figures adapted from [Eng+11] (left) and [HRO16] (right)
SLIDE 6
6
How to walk a humanoid robot?
Walking pattern generation (= planning) Walking stabilization (= tracking)
SLIDE 7
7
How to walk a humanoid robot?
Walking pattern generation (= planning) Walking stabilization (= tracking)
SLIDE 8
8
Walking on a plane
Newton equation: m¨ c = f + m g Force is grounded at CoP: f = ω2(c − r) Holonomic constraint: ¨ cz = 0 ⇒ ω2 = g/h Newton equ. simplifies to: ¨ cxy = ω2(cxy − rxy)
SLIDE 9
9
Walking on a plane
Linear time-invariant system: ¨ c = ω2(c − r) Plus, feasibility constraint: r ∈ S Question: how to stop?
SLIDE 10
10
Walking on a plane
System: x =
- c
˙ c
- where
¨ c = ω2(c − r) Input: center of pressure r Balance: starting from x0 = c0 ˙ c0
- How to bring the sys. to a stop?
With a stationary solution?
SLIDE 11
11
Instantaneous Capture Point
Recall that ¨ c = ω2(c − r) Define the capture point: ξ = c + ˙ c ω First-order dynamics: ˙ ξ = ω(ξ − r) ˙ c = ω(ξ − c) Stopped by r = ξ (stationary)
On this topic, go and read [Eng+11]
SLIDE 12
12
Towards 3D, take one
Apply same equation but in 3D: ¨ c = ω2(c − ν) ν: Virtual Repellent Point Feasibility constraint becomes: r = ν + g ω2 ∈ S Equation of motion is LTI but system nonlinear from feasibility constraint
Related references: [EOA15; CK17]
SLIDE 13
13
Time-varying DCM
Newton equation: ¨ c = λ(c − r) + g Divergent component of motion: ξ = ˙ c + ωc First-order dynamics: ˙ ξ = ωξ + g − λr ... under the Riccati equation: ˙ ω = ω2 − λ
Discussed in [CM18; Car+18]
SLIDE 14
14
Boundedness condition
Differential equation: ˙ ξ = ωξ + g − λr
SLIDE 15
15
Boundedness condition
Differential equation: ˙ ξ = ωξ + g − λr Solution is: ξ(t) = eΩ(t)
- ξ(0) +
t e−Ω(τ)(λ(τ)r(τ) − g)dτ
SLIDE 16
16
Boundedness condition
Differential equation: ˙ ξ = ωξ + g − λr Solution is: ξ(t) = eΩ(t)
- ξ(0) +
t e−Ω(τ)(λ(τ)r(τ) − g)dτ
- As t → ∞, the DCM ξ should stay bounded
SLIDE 17
17
Boundedness condition
Differential equation: ˙ ξ = ωξ + g − λr Solution is: ξ(t) = eΩ(t)
- ξ(0) +
t e−Ω(τ)(λ(τ)r(τ) − g)dτ
- As t → ∞, the DCM ξ should stay bounded
Therefore, ξ(0) = ∞ (λ(t)r(t) − g)e−Ω(t)dt Constraint between current state (LHS) and all future inputs λ(t), r(t) of the inverted pendulum (RHS)
SLIDE 18
18
Problem formulation
Change of variable: s(t) = e−Ω(t) Boundedness condition becomes: 1 rxy(s)(sω(s))′ds = ˙ cxy
i
+ ωicxy
i
g 1 1 ω(s)ds = ˙ cz
i + ωicz i
Optimize over ϕi = s2
i ω(si)2
From ϕ∗, derive λ(s), ω(s), λ(t), ω(t), r(t), c(t), . . .
SLIDE 19
19
Optimization problem
minimize
ϕ1,...,ϕN N−1
- j=1
ϕj+1 − ϕj ∆j − ϕj − ϕj−1 ∆j−1 2 (1) subject to
N−1
- j=0
∆j √ϕj+1 + √ϕj − cz
i
g √ϕN = ˙ cz
i
g (2) ω2
i,min ≤ ϕN ≤ ω2 i,max
(3) ∀j, λmin∆j ≤ ϕj+1 − ϕj ≤ λmax∆j (4) ϕ1 = ∆0g/zf (5) (1): min. height variations (2): boundedness (3): CoP polygon (4): pressure constraints (5): stationary height zf
SLIDE 20
20
Behavior of solutions
Figure : CoM trajectories obtained by solving the resultant nonlinear
- ptimization for different initial velocities.
SLIDE 21
21
Resulting walking patterns
Code: https://github.com/stephane-caron/capture-walking
SLIDE 22
22
What did we see?
Horizontal walking → LTI system With CoM height variations → nonlinear system Solve first the boundedness condition → LTV system Link with TOPP, nonlinear optimization... Outcome: dynamic stair-climbing walking patterns
For details, see [CM18; Car+18]
SLIDE 23
23
Thanks!
Thank you for your attention!
SLIDE 24
24
References I
[Car+18] St´ ephane Caron, Adrien Escande, Leonardo Lanari, and Bastien Mallein. “Capturability-based Analysis, Optimization and Control of 3D Bipedal Walking”. In: Submitted. 2018. url: https://hal.archives-ouvertes.fr/hal- 01689331/document. [CK17] St´ ephane Caron and Abderrahmane Kheddar. “Dynamic Walking
- ver Rough Terrains by Nonlinear Predictive Control of the
Floating-base Inverted Pendulum”. In: Intelligent Robots and Systems (IROS), 2017 IEEE/RSJ International Conference on.
- Sept. 2017.
[CM18] St´ ephane Caron and Bastien Mallein. “Balance control using both ZMP and COM height variations: A convex boundedness approach”. to be presented at ICRA 2018. May 2018. url: https://hal.archives-ouvertes.fr/hal- 01590509/document.
SLIDE 25
25
References II
[Eng+11] Johannes Englsberger, Christian Ott, Maximo Roa, Alin Albu-Sch¨ affer, Gerhard Hirzinger, et al. “Bipedal walking control based on capture point dynamics”. In: Intelligent Robots and Systems (IROS), 2011 IEEE/RSJ International Conference
- n. IEEE, 2011, pp. 4420–4427.