Russ Tedrake
Seminar on Computational Geometry and Robotics
Jan 18, 2023
Slides available live at https://slides.com/d/PbOeKDs/live
or later at https://slides.com/russtedrake/2023-cgr-seminar
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
Efficient algorithms for (approximate) global optimization-based planning for manipulators with dynamic constraints
image credit: James Kuffner
Trajectory optimization
Sample-based planning
AI-style logical planning
Combinatorial optimization
RRT* by Karaman et al
start
goal
start
goal
start
goal
fixed number of samples
collision-avoidance
(outside the \(L^1\) ball)
nonconvex
Default playback at .25x
3. Approximate convex decompositions of configuration space
Kinematic Trajectory Optimization
(for robot arms)
goal
start
disjunctive
constraints
"Convex relaxation" replaces this with:
"Mixed-integer convex" iff \(f\) and \(g\) are convex.
Convex relaxation is "tight" when the relaxed solution is a solution to the original problem.
Convex relaxations provide lower bounds
Feasible solutions provide upper bounds
convex
convex
convex
convex
convex
\(\Rightarrow\) 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?"
\(\varphi_{ij} = 1\) if the edge \((i,j)\) in shortest path, otherwise \(\varphi_{ij} = 0.\)
\(c_{ij} \) is the (constant) length of edge \((i,j).\)
"flow constraints"
binary relaxation
path length
Note: The blue regions are not obstacles.
Classic shortest path LP
now w/ Convex Sets
Non-negative scaling of a convex set is still convex (e.g. via "perspective functions")
Achieved orders of magnitude speedups.
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.
Previous best formulations | New formulation | |
---|---|---|
Lower Bound (from convex relaxation) |
7% of MICP | 80% of MICP |
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.
As Elon Rimon says "performance bounds are missing"
IRIS (Fast approximate convex segmentation). Deits and Tedrake, 2014
WAFR 2022;
Journal version out this month.
Solve using rational polynomial kinematics
The separating plane (green) is the non-collision certificate between the two highlighted polytopic collision geometries (red), with a distance of 7.3mm.
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!
with Tommy Cohn, Mark Petersen, and Max Simchowitz
with Tommy Cohn, Mark Petersen, and Max Simchowitz
GCS version (top down)
Prelimary results by Savva Morozov
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
http://manipulation.mit.edu
http://underactuated.mit.edu
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