Tobia Marcucci*, Mark Petersen*, David von Wrangel, Russ Tedrake*. In preparation.
Tobia Marcucci, Jack Umenberger, Pablo Parrilo, Russ Tedrake. Shortest Paths in Graphs of Convex Sets. On arxiv. Submitting to TAC (hopefully this week).
is the convex relaxation. (it's tight!)
Previous best formulations | New formulation | |
---|---|---|
Lower Bound (from convex relaxation) |
7% of MICP | 80% of MICP |
Collision-free planning with dynamic constraints
IRIS (Fast approximate convex segmentation)
Contact forces are
\( \Rightarrow \) Piecewise-affine or mixed logical-dynamical systems (MLDS)
Shortest path, \(P:\) \[ \min_P \min_{(x_i)_{i \in P}} \sum_{(i,j) \in P} \ell(x_i,x_j).\]
\(\varphi_{ij} = 1\) if the edge \((i,j)\) in shortest path, otherwise \(\varphi_{ij} = 0.\)
\(c_{ij} \) is the (constant) length of edge \((i,j).\)
Use convex hull reformulation + perspective functions to rewrite this as mixed-integer convex.
Finding the shortest path from A to B while avoiding polygonal obstacles (“Euclidean shortest path”):
Our approach:
Going forward...
The Probabilistic Roadmap (PRM)
from Choset, Howie M., et al. Principles of robot motion: theory, algorithms, and implementation. MIT press, 2005.
Still need