Alexander W. Winkler
Robotics researcher specialized in motion planning for legged systems.
Trajectory Optimization for Legged Robots
Alexander W. Winkler, Dario Bellicoso, Marco Hutter, Jonas Buchli
Paper published in IEEE Robotic and Automation Letters (RA-L 2018) \( \cdot \) DOI: 10.1109/LRA.2018.2798285
Why legged machines?
\( \bullet \) traverse rubble in earthquake \( \bullet \) reach trapped humans \( \bullet \) climb stairs \( \bullet \)...
Agility ...vs rolling
Strength ...vs flying
\( \bullet \) carry heavy payload \( \bullet \) open heavy doors \( \bullet \) rescue humans \( \bullet \) ...
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 \( \cdot \) position \( \cdot \) velocity \( \cdot \) duration \( \cdot \)
Robot \( \cdot \) kinematic \( \cdot \) dynamic
Environment \( \cdot \) terrain \( \cdot \) friction \( \cdot \) ...
Desired Motion-Plan
Actuator Commands
force \( \cdot \) torque
Tracking
Controller
\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 Method
Collocation
?
\mathbf{x}(t), \mathbf{u}(t)
Task
Optimization-based
Motion Planning
Gait and Trajectory Optimization for Legged Systems through Phase-based End-Effector Parameterization
IEEE Robotic and Automation Letters (RA-L) \( \cdot \) 2018
A. W. Winkler, D. Bellicoso, M. Hutter, J. Buchli
\min\limits_{\color{blue}{\mathbf{w}}} 0 \quad \text{subject to} \quad \color{blue}{\mathbf{b}(\mathbf{w})} = \mathbf{0}, \quad
\color{blue}{\mathbf{c}(\mathbf{w})} \ge \mathbf{0}
- Contact schedule
- CoM height (no jumps)
- Body orientation (horizontal)
- Foothold height (flat ground)
Mathematical Optimization Problem
predefined / "factorized":
Why integrated motion-planning?
restrict search space
all motion-plans \( \{ \mathbf{x}(t), \mathbf{u}(t) \} \)
fullfills all contraints
\text{find} \quad
\mathbf{r}(t) \in \mathbb{R}^3
\quad \text{(CoM)}
\mathbf{\theta}(t) \in \mathbb{R}^3
\quad
\text{(Base orientation)}
\text{for every foot } i \in \{1,\ldots,n_{ee}\}:
\color{darkblue}{\mathbf{p}_i}(t) \in \mathbb{R}^3
\quad
\text{(Foot position)}
\color{red}{\mathbf{f}_i}(t) \in \mathbb{R}^3
\quad
\text{(Foot force)}
\mathbf{p}_1
\mathbf{p}_2
\mathbf{p}_3
\mathbf{p}_4
\mathbf{f}_1
\mathbf{f}_2
\mathbf{r},
\theta
Towards integrated motion-planning
keeping search-space as open as possible
m \, \mathbf{\ddot{r}}
\quad \quad \quad \quad \quad =
\sum_{i=1}^{4}
{\color{red}\mathbf{f}_i}
- m \mathbf{g}
\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 \( \cdot \) Newton-Euler Equations
\mathbf{p}_1
\mathbf{p}_2
\mathbf{p}_3
\mathbf{p}_4
\mathbf{f}_1
\mathbf{f}_2
\mathbf{I}, m
\begin{bmatrix}
\mathbf{\ddot{r}} \\
\mathbf{\dot{\omega}}
\end{bmatrix}
Kinematic Model
{\color{#1c4587}\mathbf{p}_i} \in {\color{#1c4587}\mathcal{R}_i}(\mathbf{r},\theta)
\mathbf{p}_1
\mathbf{p}_2
\mathbf{r}, \mathbf{\theta}
\mathcal{R}_2
\mathcal{R}_1
Range-of-Motion Box \(\approx\) Joint limits
Gait Optimization
R | 2 | L | R | 2
R | 0 | R | 2 | R | 2
.... gait defined by continuous phase-durations \(\Delta T_i\)
\Delta T_{R,1}
\Delta T_{R,2}
\Delta T_{R,3}
\Delta T_{L,1}
\Delta T_{L,2}
\Delta T_{L,3}
\Delta T_{L,4}
without Integer Programming
Sequence:
swing
stance
individual foot always alternates between and
Phase-Based End-Effector Parameterization
Know if polynomial belongs to swing or stance phase
-
Foot \( \mathbf{p}_i(t)\) cannot move while
\color{red}{\mathbf{f}_i}
(t\notin\mathcal{C}_i)
=
\mathbf{0}
\color{blue}{\dot{\mathbf{p}}_i}
(t\in \mathcal{C})
=
\mathbf{0}
Physical Restrictions
- Forces \(\mathbf{f}_i(t)\) cannot exist while
standing
swinging
Terrain constraints
{\color{blue}p_{i,s}^z} = h({\color{blue}p_{i,s}^x}, {\color{blue}p_{i,s}^y})
{\color{red}\mathbf{f}_i(t)} \cdot
\mathbf{n}({\color{blue}p_{i,s}^x}, {\color{blue}p_{i,s}^y}) \ge 0
\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 \( h(x,y) \in \mathbb{R}\)
- normals \( \mathbf{n}(x,y) \in \mathbb{R}^3 \)
- tangents \( \mathbf{t}(x,y) \in \mathbb{R}^3 \)
t \in \mathcal{C}
Given:
open-sourced software
Summary
Computation Time 100 ms
1s-horizon, 4-footstep motion for a quadruped
$ sudo apt-get install ros-kinetic-xppThese slides, papers and more at
J. Buchli
M. Hutter
D. Bellicoso
$ sudo apt-get install ros-kinetic-towr_ros$ sudo apt-get install ros-kinetic-ifopt
A. Winkler
Gait and Trajectory Optimization for Legged Systems
By Alexander W. Winkler
Gait and Trajectory Optimization for Legged Systems
Paper: https://ieeexplore.ieee.org/document/8283570/ Recorded talk: https://youtu.be/KhWuLvb934g
