# Motion planning around obstacles with convex optimization

Russ Tedrake

Mark Petersen

David von Wrangel

Bernhard Græsdal

Boyuan Chen

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​

start

goal

## Planning as a nonconvex optimization

### A preview of the results...

Default playback at .25x

Graph of Convex Sets (GCS)

PRM

PRM w/ short-cutting

Preprocessor now makes easy optimizations fast!

## Two key ingredients

1. The linear programming formulation of the shortest path problem on a discrete graph.

2. Convex formulations of continuous motion planning (without obstacle navigation), for example:

Kinematic Trajectory Optimization

(for robot arms)

## Graphs of Convex Sets

• For each $$i \in V:$$
• Compact convex set $$X_i \subset \R^d$$
• A point $$x_i \in X_i$$
• Edge length given by a convex function $\ell(x_i, x_j)$

Note: The blue regions are not obstacles.

## Motion planning with Graph of Convex Sets (GCS)

\ell_{i,j}(x_i, x_j) = |x_i - x_j|_2

start

goal

## Motion planning with Graph of Convex Sets (GCS)

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

minimum distance

minimum time

Transcription to a mixed-integer convex program, but with a very tight convex relaxation.

• Solve to global optimality w/ branch & bound orders of magnitude faster than previous work
• Solving only the convex optimization (+rounding) is almost always sufficient to obtain the globally optimal solution.

## Sampling-based motion planning

from Choset, Howie M., et al.
Principles of robot motion: theory, algorithms, and implementation. MIT press, 2005.

• Guaranteed collision-free along piecewise polynomial trajectories
• Complete/globally optimal within convex decomposition

## Discussion points

JBT has said "people like RRT as being cognitively relevant".  The connection to graph search seems more natural to me?

LPK + TLP -- for TAMP, this motion planner says "yes" or "no" (compared to "yes" or "maybe, if you draw more samples")

By russtedrake

• 461