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

### Quick glimpse of the results...

Default playback at .25x

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.

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

The Probabilistic Roadmap (PRM)
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

## Not just for motion planning

• Tight formulations for many mixed discrete (on graphs) + continuous optimizations
• e.g. orders of magnitude faster for traveling salesman problems (TSP) with neighborhoods

• Task and motion planning
• Make Chris' "joint solution" approach tractable?

## Graphs of Convex Sets

• The vertices in the graph are symbols.

• Adding continuous variables (convex sets) allows symbol grounding.

​...and they are now deeply connected in the optimization.