Alexander W. Winkler
Robotics researcher specialized in motion planning for legged systems.
Online Walking Motion and Foothold Optimization for Quadruped Locomotion
Alexander W. Winkler Farbod Farshidian Michael Neunert Diego Pardo Jonas Buchli
IEEE Internation Conference on Robotics and Automation , 2017
www.awinkler.me
The Footholds' Primary Function is to Help the CoM move
A. Herdt et al, “Online Walking Motion Generation with Automatic Foot Step Placement,” HAL archives-ouvertes, 2010
Full-body To
Legged Locomotion MOdules
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Optimization Variables
\mathbf{c}^{x,y}(t,\color{red}{\mathbf{a}_i})
cx,y(t,ai)
\mathbf{p}_1^{x,y}
p1x,y
\mathbf{p}_2^{x,y}
p2x,y
\mathbf{p}_3^{x,y}
p3x,y
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\mathbf{u}
u
Unilateral Contact Forces
CoP inside Support-Area
\Leftrightarrow
⇔
\mathbf{u}(\color{red}{\mathbf{a}})^T
\mathbf{n}_i(\color{red}{\mathbf{p}})
+ f(\color{red}{\mathbf{p}}) > 0
u(a)Tni(p)+f(p)>0
\mathbf{n}_2
n2
\mathbf{n}_1
n1
\mathbf{n}_3
n3
M. Kalakrishnan et al, “Learning, planning, and control for quadruped locomotion over challenging terrain,” IJRR, 2010
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\mathbf{u}
u
Unilateral Contact Forces
CoP inside Support-Area
\Leftrightarrow
⇔
\mathbf{u}(\color{red}{\mathbf{a}})^T
\mathbf{n}_i(\color{red}{\mathbf{p}})
+ f(\color{red}{\mathbf{p}}) > 0
u(a)Tni(p)+f(p)>0
\mathbf{n}_2
n2
\mathbf{n}_1
n1
\mathbf{n}_3
n3
M. Kalakrishnan et al, “Learning, planning, and control for quadruped locomotion over challenging terrain,” IJRR, 2010
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p_1^{RH}
p1RH
p_1^{LF}
p1LF
p_1^{LH}
p1LH
p_1^{RF}
p1RF
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\color{red}{\mathbf{u}}^T
\mathbf{n}_i(\color{black}{\mathbf{p}})
+ f(\color{black}{\mathbf{p}}) > 0
uTni(p)+f(p)>0
How to Satisfy
?
Thanks!
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Simultaneous Footstep Planning
Nonlinear Constraint
\mathbf{u}(\color{red}{\mathbf{a}})^T
\mathbf{n}_i(\color{red}{\mathbf{p}})
+ f(\color{red}{\mathbf{p}}) > 0
u(a)Tni(p)+f(p)>0
1. Modify CoM motion a so that it can be generated by a CoP inside the current base of support.
2. Modify base of support by changing the footholds p to accommodate the CoM acceleration
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LIP Model to estimate Center-of-Pressure u
\mathbf{u}
\approx
\begin{bmatrix}
\color{black}{u_x} \\
\color{black}{u_y}
\end{bmatrix}
=
\mathbf{c}(\color{red}{\mathbf{a}})
-
\frac{h}{g}
\ddot{\mathbf{c}}(\color{red}{\mathbf{a}})
u≈[uxuy]=c(a)−ghc¨(a)
S. Kajita et al, “Biped walking pattern generation by using preview control of zero-moment point,” IEEE International Conference on Robotics and Automation, 2003.
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\begin{bmatrix}
\mathbf{c}(t) \\
\dot{\mathbf{c}}(t)
\end{bmatrix}
=
\sum\limits_{i=1}^5
\begin{bmatrix}
t \\
i
\end{bmatrix}
\color{blue}{\mathbf{a}_i}
t^{i-1}
+
\begin{bmatrix}
\color{blue}{\mathbf{a}_0} \\
\mathbf{0}
\end{bmatrix}
[c(t)c˙(t)]=i=1∑5[ti]aiti−1+[a00]
CoM Parametrization
Represent by sequence of fifth-order polynomials
\ddot{\mathbf{c}}[t_k] =
\sum\limits_{i=2}^5
i (i-1) \color{blue}{\mathbf{a}_i} t^{i-2}
=
\frac{g}{h}(\mathbf{c}[t_k,\color{blue}{\mathbf{a}_i}] - \mathbf{u}[t_k])
c¨[tk]=i=2∑5i(i−1)aiti−2=hg(c[tk,ai]−u[tk])
Physical Feasiblity
An CoP at u in state c should cause a specific CoM acceleration.
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foothold
change
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p_1^{RH}
p1RH
p_1^{LF}
p1LF
p_1^{LH}
p1LH
p_1^{RF}
p1RF
foothold
change
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Range Of Motion COnstraint
-\mathbf{r}
< \mathbf{p}_i - \mathbf{c} - \color{black}{\mathbf{p}_{i,nom}}
< \color{black}{\mathbf{r}}
\quad \quad
\text{for } i=1,...,4
−r<pi−c−pi,nom<rfor i=1,...,4
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Tracking 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,” 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]:
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Walk
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Online Walking Motion and Foothold Optimization for Quadruped Locomotion Alexander W. Winkler Farbod Farshidian
Michael Neunert
Diego Pardo
Jonas Buchli IEEE Internation Conference on Robotics and Automation , 2017 www.awinkler.me
ICRA 2017
By Alexander W. Winkler
