Alexander W. Winkler
Robotics researcher specialized in motion planning for legged systems.
Fast Trajectory Optimization for Legged Robots using Vertex-Based ZMP Constraints
Alexander W. Winkler Farbod Farshidian Diego Pardo Michael Neunert Jonas Buchli
Robot and Automation Letters (RA-L), 2017
(www.awinkler.me)
(ZMP)
(Capture Point)
Trotting
Push recovery
Walking
\mathbf{v}_0
v0
Pacing
Bounding
(Capture Point)
Single Formulation/Planner
Motivation
Alexander W. Winkler et al.
M. Kalakrishnan et al, “Learning, planning, and control for quadruped locomotion over challenging terrain,” IJRR, 2010
J. Pratt et al, “Capture point: A step toward humanoid push recovery,” in Humanoids, 2006.
Single Formulation/Planner
Alexander W. Winkler et al.
M. Kalakrishnan et al, “Learning, planning, and control for quadruped locomotion over challenging terrain,” IJRR, 2010
J. Pratt et al, “Capture point: A step toward humanoid push recovery,” in Humanoids, 2006.
Youtube: "animal gaits for animators"
Trajectory Optimization
\text{find} \quad
\color{black}{x(t)}, \color{black}{u(t)} ,
\quad \text{for } t \in [0,T]
findx(t),u(t),for t∈[0,T]
\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0
subject tox(0)−x0=0
\dot{x} - f(x(t), u(t)) = 0
x˙−f(x(t),u(t))=0
h(x(t), u(t)) \leq 0
h(x(t),u(t))≤0
x(T) - x_T = 0
x(T)−xT=0
x(t), u(t) = \text{arg min } J(x,u)
x(t),u(t)=arg min J(x,u)
given initial state
dynamic Model
Path Constraints
Desired Final State
Objective
\text{find} \quad
\color{red}{x(t)}, \color{red}{u(t)} ,
\quad \text{for } t \in [0,T]
findx(t),u(t),for t∈[0,T]
Alexander W. Winkler et al.
\mathbf{c}(t)
c(t)
\mathbf{u}(t)
u(t)
\mathbf{p}_{3}
p3
\mathbf{p}_{4}
p4
\mathbf{p}_{2}
p2
\mathbf{p}_{1}
p1
OPtimization Variables
Alexander W. Winkler et al.
Dynamic Consistency
\text{find} \quad x(t), u(t) , \quad \text{for } t \in [0,T]
findx(t),u(t),for t∈[0,T]
\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0
subject tox(0)−x0=0
\dot{x} - f(x(t), u(t)) = 0
x˙−f(x(t),u(t))=0
h(x(t), u(t)) \leq 0
h(x(t),u(t))≤0
x(T) - x_T = 0
x(T)−xT=0
x(t), u(t) = \text{arg min } J(x,u)
x(t),u(t)=arg min J(x,u)
Linear Inverted Pendulum Model
S. Kajita et al, “Biped walking pattern generation by using preview control of zero-moment point,” IEEE International Conference on Robotics and Automation, 2003.
\ddot{\mathbf{c}}(t)
=
\frac{g}{h}(\mathbf{c}(t)- \color{red}{\mathbf{u}}(t))
c¨(t)=hg(c(t)−u(t))
Alexander W. Winkler et al.
foothold
change
Alexander W. Winkler et al.
Simultaneous Base and Feet Optimization
Unilateral Contact Forces
\text{find} \quad x(t), u(t) , \quad \text{for } t \in [0,T]
findx(t),u(t),for t∈[0,T]
\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0
subject tox(0)−x0=0
\dot{x} - f(x(t), u(t)) = 0
x˙−f(x(t),u(t))=0
h(x(t), u(t)) \leq 0
h(x(t),u(t))≤0
x(T) - x_T = 0
x(T)−xT=0
x(t), u(t) = \text{arg min } J(x,u)
x(t),u(t)=arg min J(x,u)
\color{green}{\mathbf{c}} =
\begin{bmatrix}
1 \\ 1 \\ 1 \\ 0
\end{bmatrix}
c=1110
\color{green}{\mathbf{c}} =
\begin{bmatrix}
0 \\ 1 \\ 1 \\ 0
\end{bmatrix}
c=0110
u
u
u
u
\color{red}{\mathbf{u}}^T
\mathbf{n}_i(\mathbf{p})
+ \text{offset}(\mathbf{p}) > 0
uTni(p)+offset(p)>0
Unilateral Contact Forces CoP inside Support-Area
\Leftrightarrow
⇔
M. Kalakrishnan et al, “Learning, planning, and control for quadruped locomotion over challenging terrain,” IJRR, 2010
Alexander W. Winkler et al.
- Cannot represent single point-contacts or lines
- Heuristic expansion of points or lines into areas
- Ordering of contact points necessary.
\color{red}{\mathbf{u}} = \sum\limits_{i=1}^4
\lambda_i
\mathbf{p}_i
u=i=1∑4λipi
\sum\limits_{i=1}^{4} \lambda_i = 1
i=1∑4λi=1
0 < \lambda_i
0<λi
\color{green}{\mathbf{c}} =
\begin{bmatrix}
1 \\ 1 \\ 1 \\ 0
\end{bmatrix}
c=1110
\color{green}{\mathbf{c}} =
\begin{bmatrix}
0 \\ 1 \\ 1 \\ 0
\end{bmatrix}
c=0110
< c_i
<ci
u
u
u
u
Vertex-Based Zmp-Constraint Formulation
Alexander W. Winkler et al.
Future Work
(under review for ICRA/RAL 2018)
F. Farshidian
D. Pardo
M. Neunert
J. Buchli
A. Winkler
Paper, Video and Presentation: www.awinkler.me
Reduce heuristics by simultaneously optimizing over body motion and footholds
Vertex-based ZMP-constraint formulation allows to uniformly handle point-, line- and area-contacts
Take-Away MSGs
Walk
Trot
Bound
Pace
Alexander W. Winkler et al.
Kinematic Range
\text{find} \quad x(t), u(t) , \quad \text{for } t \in [0,T]
findx(t),u(t),for t∈[0,T]
\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0
subject tox(0)−x0=0
\dot{x} - f(x(t), u(t)) = 0
x˙−f(x(t),u(t))=0
h(x(t), u(t)) \leq 0
h(x(t),u(t))≤0
x(T) - x_T = 0
x(T)−xT=0
x(t), u(t) = \text{arg min } J(x,u)
x(t),u(t)=arg min J(x,u)
kinematic Range:
\mathbf{p}_{i=1..4} \in \mathcal{R}(\mathbf{c})
pi=1..4∈R(c)
\mathbf{p}_{LF}
pLF
\mathcal{R}_{LF}
RLF
\mathcal{R}_{RF}
RRF
\mathcal{R}_{RH}
RRH
\mathcal{R}_{LH}
RLH
\mathbf{c}
c
Alexander W. Winkler et al.
Robust Motions
\text{find} \quad x(t), u(t) , \quad \text{for } t \in [0,T]
findx(t),u(t),for t∈[0,T]
\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0
subject tox(0)−x0=0
\dot{x} - f(x(t), u(t)) = 0
x˙−f(x(t),u(t))=0
h(x(t), u(t)) \leq 0
h(x(t),u(t))≤0
x(T) - x_T = 0
x(T)−xT=0
x(t), u(t) = \text{arg min } J(x,u)
x(t),u(t)=arg min J(x,u)
\mathbf{u}
u
\mathbf{\lambda}
=
\begin{bmatrix}
0 \\ 0.5 \\ 0.5 \\ 0
\end{bmatrix}
λ=00.50.50
\mathbf{\lambda}^*
=
\begin{bmatrix}
0.33 \\ 0.33 \\ 0.33 \\ 0
\end{bmatrix}
λ∗=0.330.330.330
J = \sum\limits_{k=1}^{n}
|\!|
\mathbf{\lambda}_k - \mathbf{\lambda}_k^*
|\!|_2^2
J=k=1∑n∣∣λk−λk∗∣∣22
\mathbf{u}^*
u∗
"If possible, Avoid The Extremes"
Alexander W. Winkler et al.
\mathbf{u} \in \mathcal{P}(\mathbf{p},c)
u∈P(p,c)
\color{red}{\mathbf{u}} = \sum\limits_{i=1}^8 \lambda_i \mathbf{v}_i(\mathbf{p},\alpha)
u=i=1∑8λivi(p,α)
\sum\limits_{i=1}^{8} \lambda_i = 1
i=1∑8λi=1
\Leftrightarrow
⇔
0 < \lambda_i
0<λi
Biped with non-point-feet:
\mathbf{c} =
\begin{bmatrix}
1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1
\end{bmatrix}
c=11111111
< c_i
<ci
Generating Torques to Track the Optimized Motion
\Leftrightarrow
\ddot{\mathbf{q}}_{j,ref}
=
\mathbf{J}_j^+
(
\color{red}{\ddot{\mathbf{p}}} - \dot{\mathbf{J}}\dot{\mathbf{q}} - \mathbf{J}_b \color{red}{\ddot{\mathbf{q}}_{b,ref}}
)
⇔q¨j,ref=Jj+(p¨−J˙q˙−Jbq¨b,ref)
\ddot{\mathbf{p}}
=
\mathbf{J} \ddot{\mathbf{q}}
+
\dot{\mathbf{J}} \dot{\mathbf{q}}
p¨=Jq¨+J˙q˙
\mathbf{P} = \mathbf{I} - \mathbf{J}_c^+
\mathbf{J}_c
P=I−Jc+Jc
\mathbf{\tau}
= (\mathbf{P} \mathbf{S}^T)^+
\mathbf{P}
(
\mathbf{M}\ddot{\mathbf{q}}_{ref} + \mathbf{C}
)
τ=(PST)+P(Mq¨ref+C)
[1] F. Aghili, “A unified approach for inverse and direct dynamics of constrained multibody systems based on linear projection operator: Applications to control and simulation,” IEEE T-RO, 2005.
[2] M. Mistry, J. Buchli, and S. Schaal, “Inverse dynamics control of floating base systems using orthogonal decomposition,” IEEE ICRA, 2010
Cartesian → Joint:
Joint+Contacts → Torque [1,2]:
Fast Trajectory Optimization for Legged Robots using Vertex-Based ZMP Constraints Alexander W. Winkler Farbod Farshidian
Diego Pardo
Michael Neunert
Jonas Buchli Robot and Automation Letters (RA-L) , 2017 (www.awinkler.me)
RA-L 2017: "Vertex-based ZMP constraints"
By Alexander W. Winkler
RA-L 2017: "Vertex-based ZMP constraints"
Paper: https://ieeexplore.ieee.org/document/7970144/
- 4,356
