Russ Tedrake
w/ Alexandre Megretski, Pablo Parrilo
Slides available live at https://slides.com/d/0jpKUsI/live
or later at https://slides.com/russtedrake/2022-multibody
Shortest Paths in Graphs of Convex Sets.
Tobia Marcucci, Jack Umenberger, Pablo Parrilo, Russ Tedrake.
Available at: https://arxiv.org/abs/2101.11565
Motion Planning around Obstacles with Convex Optimization.
Tobia Marcucci, Mark Petersen, David von Wrangel, Russ Tedrake.
Available at: https://arxiv.org/abs/2205.04422
Global optimization-based planning for vehicles and manipulators with dynamic constraints and certificates of optimality / correctness
image credit: James Kuffner
Trajectory optimization
Sample-based planning
AI-style logical planning
Combinatorial optimization
start
goal
Default playback at .25x
Kinematic Trajectory Optimization
(for robot arms)
goal
start
disjunctive
constraints
\(\Rightarrow\) Long solve times.
This is the convex relaxation
(it is very loose!).
"We know that the LP formulation of the shortest path problem is tight. Why exactly are your relaxations so loose?"
Note: The blue regions are not obstacles.
Conservation of flow
Spatial conservation of flow
(this was the missing constraint!)
start
goal
This is the convex relaxation
(it is tight!).
is the convex relaxation. (it's tight!)
Previous formulations were intractable; would have required \( 6.25 \times 10^6\) binaries.
Formulating motion planning with differential constraints as a Graph of Convex Sets (GCS)
+ time-rescaling
duration
path length
path "energy"
note: not just at samples
continuous derivatives
collision avoidance
velocity constraints
minimum distance
minimum time
Transcription to a mixed-integer convex program, but with a very tight convex relaxation.
IRIS (Fast approximate convex segmentation). Deits and Tedrake, 2014
The Probabilistic Roadmap (PRM)
from Choset, Howie M., et al. Principles of robot motion: theory, algorithms, and implementation. MIT press, 2005.
Graph of Convex Sets (GCS)
PRM
PRM w/ short-cutting
Preprocessor now makes easy optimizations fast!
I've focused today on Graphs of convex sets (GCS) for motion planning
GCS is a more general modeling framework
This is version 0.1 of a new framework.
There is much more to do, for example:
Give it a try:
pip install drake
sudo apt install drake
Trajectory optimization
Sample-based planning
AI-style logical planning
Combinatorial optimization
Shortest Paths in Graphs of Convex Sets.
Tobia Marcucci, Jack Umenberger, Pablo Parrilo, Russ Tedrake.
Available at: https://arxiv.org/abs/2101.11565
Motion Planning around Obstacles with Convex Optimization.
Tobia Marcucci, Mark Petersen, David von Wrangel, Russ Tedrake.
Available at: https://arxiv.org/abs/2205.04422