Alexander W. Winkler
Robotics researcher specialized in motion planning for legged systems.
Optimization-based Motion Planning for Legged Robots
Alexander W. Winkler
May 14, 2018 ⋅ PhD Defense
Why legged machines?
∙ traverse rubble in earthquake ∙ reach trapped humans ∙ climb stairs ∙...
Agility ...vs rolling
Strength ...vs flying
∙ carry heavy payload ∙ open heavy doors ∙ rescue humans ∙ ...
vs
Source:
ANYbotics, Anymal bear, "Image: https://www.anybotics.com/anymal", 2018; Boston Dynamics, Atlas, "Image: https://www.bostondynamics.com/atlas", 2016; Italian Institute of Technology, HyQ2Max "Image: https://dls.iit.it/robots/hyq2max, 2018; Alphabet Waymo, Firefly car, "Image: https://waymo.com", 2016, DJI, Phantom 2 drone, "Image: https://www.dji.com/phantom-2", 2016
Source: https://www.youtube.com/watch?v=NX7QNWEGcNIa
Source: https://www.youtube.com/watch?v=arCOVKxGy9E
Goal ⋅ position ⋅ velocity ⋅ duration ⋅
Robot ⋅ kinematic ⋅ dynamic
Environment ⋅ terrain ⋅ friction ⋅ ...
Outline
Desired Motion-Plan
Actuator Commands
force ⋅ torque
Tracking
Controller
wmina(w)subject tob(w)=0,c(w)≥0
\min\limits_{\mathbf{w}} a(\mathbf{w}) \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad
\mathbf{c}(\mathbf{w})\ge \mathbf{0}
off-the-shelf
NLP Solver
Mathematical Optimization Problem
Direct Methods
e.g. Collocation
?
Paper I
"Fast Trajectory Optimization for legged Systems using Vertex-based ZMP Constraints"
Paper 2
"Gait and Trajectory Optimization for Legged Systems through Phase-based End-Effector Parameterization"
x(t),u(t)
\mathbf{x}(t), \mathbf{u}(t)
Task
Generating dynamic motions
Dynamic Model
r¨(t)=hg(r(t)−pc(t))
\ddot{\mathbf{r}}(t)
=
\frac{g}{h}(\mathbf{r}(t)- \color{red}{\mathbf{p}_c}(t))
pc
\color{red}{\mathbf{p}_c}
pc
\color{red}{\mathbf{p}_c}
Linear Inverted Pendulum
pc
\mathbf{p}_c
pc
\mathbf{p}_c
pcTni(p)+offset(p)>0
\color{red}{\mathbf{p}_c}^T
\mathbf{n}_i(\color{blue}{\mathbf{p}})
+ \text{offset}(\color{blue}{\mathbf{p}}) > 0
Unilateral Contact Forces ⇔ CoP inside Support-Area
Difficult for single point-contacts or lines
p1
\color{blue}{\mathbf{p}_1}
p2
\color{blue}{\mathbf{p}_2}
p3
\color{blue}{\mathbf{p}_3}
Ordering of contact points
1.pc=i=1∑4λipi
1. \quad\color{red}{\mathbf{p}_c} = \sum\limits_{i=1}^4
\color{red}{\lambda_i}
\color{blue}{\mathbf{p}_i}
2.i=1∑4λi=1
2. \quad \sum\limits_{i=1}^{4} \color{red}{\lambda_i} = 1
3.0≤λi
3. \quad 0 \le\color{red}{\lambda_i}
c=1110
\mathbf{c} =
\begin{bmatrix}
1 \\ 1 \\ 1 \\ 0
\end{bmatrix}
c=0110
\mathbf{c} =
\begin{bmatrix}
0 \\ 1 \\ 1 \\ 0
\end{bmatrix}
≤ci
\le c_i
pc
\mathbf{p}_c
pc
\mathbf{p}_c
Vertex-Based Zmp-Constraint Formulation
p1,λ1,c1
\color{blue}{\mathbf{p}_1}, \color{red}{\mathbf{\lambda}_1}, \color{#7f6000}{c_1}
Fast Trajectory Optimization for Legged Robots using Vertex-based ZMP Constraints
IEEE Robotic and Automation Letters (RA-L) ⋅ 2017
A. W. Winkler, F. Farshidian, D. Pardo, M. Neunert, J. Buchli
foothold
change
Simultaneous Foothold and CoM Optimization
Fast Trajectory Optimization for Legged Robots using Vertex-based ZMP Constraints
IEEE Robotic and Automation Letters (RA-L) ⋅ 2017
A. W. Winkler, F. Farshidian, D. Pardo, M. Neunert, J. Buchli
- Contact schedule
- CoM height (no jumps)
- Body orientation (horizontal)
- Foothold height (flat ground)
Mathematical Optimization Problem
predefined:
Motion-Plan Search Space
restrict search space
all motion-plans {x(t),u(t)}
fullfills all contraints
\text{find} \quad
\mathbf{r}(t) \in \mathbb{R}^\color{#07d507}{3}
\quad \text{(CoM)}
θ(t)∈R3(Base orientation)
\mathbf{\theta}(t) \in \color{#07d507}{\mathbb{R}^3}
\quad
\text{(Base orientation)}
for every foot i∈{1,…,nee}:
\text{for every foot } i \in \{1,\ldots,n_{ee}\}:
\color{darkblue}{\mathbf{p}_i}(t) \in \mathbb{R}^\color{#07d507}{3}
\quad
\text{(Foot position)}
fi(t)∈R3(Foot force)
\color{red}{\mathbf{f}_i}(t) \in \color{#07d507}{\mathbb{R}^3}
\quad
\text{(Foot force)}
p1
\mathbf{p}_1
p2
\mathbf{p}_2
p3
\mathbf{p}_3
p4
\mathbf{p}_4
f1
\mathbf{f}_1
f2
\mathbf{f}_2
r,
\mathbf{r},
θ
\theta
Gait and Trajectory Optimization for Legged Systems through Phase-based End-Effector Parameterization
IEEE Robotic and Automation Letters (RA-L) ⋅ 2018
A. W. Winkler, D. Bellicoso, M. Hutter, J. Buchli
Towards integrated motion-planning
mr¨=∑i=14fi−mg
m \, \mathbf{\ddot{r}}
\quad \quad \quad \quad \quad =
\sum_{i=1}^{4}
\color{red}{\mathbf{f}_i}
- m \mathbf{g}
I(θ)ω˙+ω×I(θ)ω=∑i=14fi×(r−pi)
\mathbf{I}(\theta) \, \dot{\omega} + \omega\!\times\!\mathbf{I}(\theta) \omega
= \sum_{i=1}^{4} \color{red}{\mathbf{f}_i}\!\times\!(\mathbf{r}-\color{#1c4587}{\mathbf{p}_i})
Dynamic Model
Single Rigid Body ⋅ Newton-Euler Equations
p1
\mathbf{p}_1
p2
\mathbf{p}_2
p3
\mathbf{p}_3
p4
\mathbf{p}_4
f1
\mathbf{f}_1
f2
\mathbf{f}_2
I,m
\mathbf{I}, m
[r¨ω˙]
\begin{bmatrix}
\mathbf{\ddot{r}} \\
\mathbf{\dot{\omega}}
\end{bmatrix}
Kinematic Model
pi∈Ri(r,θ)
\color{#1c4587}{\mathbf{p}_i} \in \color{#1c4587}{\mathcal{R}_i}(\mathbf{r},\theta)
p1
\mathbf{p}_1
p2
\mathbf{p}_2
r,θ
\mathbf{r}, \mathbf{\theta}
R2
\mathcal{R}_2
R1
\mathcal{R}_1
Range-of-Motion Box ≈ Joint limits
Gait Optimization
R | 2 | L | R | 2
R | 0 | R | 2 | R | 2
.... gait defined by continuous phase-durations ΔTi
ΔTR,1
\Delta T_{R,1}
ΔTR,2
\Delta T_{R,2}
ΔTR,3
\Delta T_{R,3}
ΔTL,1
\Delta T_{L,1}
ΔTL,2
\Delta T_{L,2}
ΔTL,3
\Delta T_{L,3}
ΔTL,4
\Delta T_{L,4}
without Integer Programming
Gait and Trajectory Optimization for Legged Systems through Phase-based End-Effector Parameterization
IEEE Robotic and Automation Letters (RA-L) ⋅ 2018
A. W. Winkler, D. Bellicoso, M. Hutter, J. Buchli
Sequence:
swing
stance
individual foot always alternates between and
Phase-Based End-Effector Parameterization
Know if polynomial belongs to swing or stance phase
-
Foot pi(t) cannot move while
fi(t∈/Ci)=0
\color{red}{\mathbf{f}_i}
(t\notin\mathcal{C}_i)
=
\mathbf{0}
p˙i(t∈C)=0
\color{blue}{\dot{\mathbf{p}}_i}
(t\in \mathcal{C})
=
\mathbf{0}
Physical Restrictions
- Forces fi(t) cannot exist while
standing
swinging
Terrain constraints
pi,sz=h(pi,sx,pi,sy)
\color{blue}{p_{i,s}^z} = h(\color{blue}{p_{i,s}^x}, \color{blue}{p_{i,s}^y})
fi(t)⋅n(pi,sx,pi,sy)≥0
\color{red}{\mathbf{f}_i(t)} \cdot
\mathbf{n}(\color{blue}{p_{i,s}^x}, \color{blue}{p_{i,s}^y}) \ge 0
∣fi(t)⋅t(pi,sx,pi,sy)∣<μfi(t)⋅n(pi,sx,pi,sy)
\lvert \color{red}{\mathbf{f}_i(t)}\cdot
\mathbf{t}(\color{blue}{p_{i,s}^x}, \color{blue}{p_{i,s}^y}) \rvert
< \mu \color{red}{\mathbf{f}_i(t)} \cdot
\mathbf{n}(\color{blue}{p_{i,s}^x}, \color{blue}{p_{i,s}^y})
Foot can only stand on terrain
Forces can only push
Forces inside friction pyramid
- height map h(x,y)∈R
- normals n(x,y)∈R3
- tangents t(x,y)∈R3
t∈C
t \in \mathcal{C}
Given:
4
open-sourced software
Summary
Computation Time 100 ms
1s-horizon, 4-footstep motion for a quadruped
Software
Publications
+ co-authored various others with F. Farshidian, D. Pardo, M. Neunert, ...
1st author
open-sourced
Additional Material:
mr¨=∑i=14fi−mg
m \ddot{\mathbf{r}}
=
\sum_{i=1}^{4}
\color{red}{\mathbf{f}_i}
- m \mathbf{g}
I(θ)ω˙+ω×I(θ)ω=∑i=14fi×(r−pi)
\mathbf{I}(\theta)\dot{\omega} + \omega\!\times\!\mathbf{I}(\theta) \omega
= \sum_{i=1}^{4} \color{red}{\mathbf{f}_i}\!\times\!(\mathbf{r}-\color{#1c4587}{\mathbf{p}_i})
Centroidal Dynamics ⇒ Single Rigid Body Dynamics
Newton-Euler Equations
+ Assumption A2: Momentum produced by the joint velocities is negligible.
+ Assumption A3: Full-body inertia remains similar to the one in nominal configuration.
A(q)q¨+A˙(q,q˙)q˙=[∑i=14fi−mg∑i=1nifi×(r(q)−pi(q))]
\mathbf{A}(\mathbf{q}) \mathbf{\ddot{q}} + \mathbf{\dot{A}}(\mathbf{q},\mathbf{\dot{q}}) \mathbf{\dot{q}} =
\begin{bmatrix}
\sum_{i=1}^{4} \color{red}{\mathbf{f}_i} - m \mathbf{g} \\
\sum_{i=1}^{n_i} \color{red}{\mathbf{f}_i}\!\times\!(\mathbf{r}(\mathbf{q})-\color{#1c4587}{\mathbf{p}_i}(\mathbf{q}))
\end{bmatrix}
| (pos) | Assumptions | ||
|---|---|---|---|
| Rigid Body Dynamics (RBD) | A1 | ||
| Centroidal Dynamics (CD) | A1 | ||
| Single Rigid Body Dynamics (SRBD) | A1, A2, A3 | ||
| Linear Inverted Pendulum (LIP) | A1, A2, A3, A4, A5, A6 |
qb,qj
\mathbf{q}_b, \mathbf{q}_j
qb,qj
\mathbf{q}_b, \mathbf{q}_j
r,θ,
\mathbf{r}, \mathbf{\theta},
rx,ry
r_x, r_y
τ,fi
\tau, \mathbf{f}_i
fi
\mathbf{f}_i
fi
\mathbf{f}_i
pc
\mathbf{p}_c
pi
\mathbf{p}_i
x
\mathbf{x}
x˙=F(x(t),u(t))
\mathbf{\dot{x}} = \mathbf{F}(\mathbf{x}(t), \color{red}{\mathbf{u}(t)})
u
\mathbf{u}
x(t)=a0+a1t+a2t2+a3t3
x(t) = a_0 + a_1t + a_2t^2 + a_3t^3
{
\{
x0
\color{black}{x_0}
x˙0
\color{black}{\dot{x}_0}
{
\{
{
\{
{
\{
−ΔT1−2[3(x0−x1)+ΔT1(2x˙0+x˙1)]
-\color{#7f6000}{\Delta T_1}^{-2} [ 3(\color{black}{x_0} - \color{black}{x_1}) + \color{#7f6000}{\Delta T_1}(2\color{black}{\dot{x}_0} + \color{black}{\dot{x}_1}) ]
ΔT1−3[2(x0−x1)+ΔT1(x˙0+x˙1)]
\color{#7f6000}{\Delta T_1}^{-3} [ 2(\color{black}{x_0} - \color{black}{x_1}) + \color{#7f6000}{\Delta T_1}( \color{black}{\dot{x}_0} + \color{black}{\dot{x}_1}) ]
wj={x0,x˙0,ΔT1,x1,x˙1,ΔT2,x2,x˙2,ΔT3,xT,x˙T}
\color{black}{\mathbf{w}_j} =
\{
\color{black}{x_0},
\color{black}{\dot{x}_0},
\color{#7f6000}{\Delta T_1},
\color{black}{x_1},
\color{black}{\dot{x}_1},
\color{#7f6000}{\Delta T_2},
\color{black}{x_2},
\color{black}{\dot{x}_2},
\color{#7f6000}{\Delta T_3},
\color{black}{x_T},
\color{black}{\dot{x}_T}
\}
Cubic-Hermite Spline for f{x,y,z}(t),p{x,y,z}(t)
Optimization-based Motion Planning for Legged Robots Alexander W. Winkler www.awinkler.me May 14, 2018 ⋅ PhD Defense
PhD Defense 2018: "Optimization-based motion planning for legged robots"
By Alexander W. Winkler
PhD Defense 2018: "Optimization-based motion planning for legged robots"
Pdf: https://www.research-collection.ethz.ch/handle/20.500.11850/272432
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