Robot Locomotion Group
ICCOPT 2025
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
https://mapsplatform.google.com/maps-products/routes/
Broadly speaking
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
What has changed?
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
https://github.com/ethz-asl/amr_visualisations
Task Space
Configuration Space
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Michigan Robotics 320, Lecture 15
Standard algorithms include PRM (Kavraki 1996), RRT (LaValle 1998), EST (Hsu 1999), and PRM* (Karaman 2011).
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Underactuated Robotics, Russ Tedrake
Control points of a trajectory are decision variables.
Use costs and constraints to enforce objectives, dynamics, and more.
Trajectory optimization approaches include STOMP (Kalakrishnan 2011), CHOMP (Zucker 2013), KOMO (Toussaint 2014), ALTRO (Howell 2019), cuRobo (Sundaralingam 2023).
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
+Global (probabilistic) completeness
-"Curse of Dimensionality"
-Non-smooth paths
-Nonconvex
+Scales with dimension
+Smooth paths, dynamics
Motion Planning, Wikipedia
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Reproduced from Slides for the Technion Robotics Seminar, Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Motion Planning around Convex Obstacles with Convex Optimization, Marcucci et. al.
Convex obstacle avoidance guarantees!
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
A directed graph \(G=(V,E)\)
Main Reference: Shortest Paths in Graphs of Convex Sets, Marcucci et. al.
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
\(\mathcal X_u\)
\(\mathcal X_v\)
\(\ell_{(u,v)}\)
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Trajectory Optimization on Manifolds: A Theoretically-Guaranteed Embedded Sequential Convex Programming Approach, Bonalli et. al.
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
A Riemannian Metric assigns an inner product \(\left<\cdot,\cdot\right>_g\) to each tangent space of a manifold \(\mathcal M\).
The "speed" of a curve \(\gamma:(a,b)\to\mathcal M\) at time \(t\) is \(\sqrt{\left<\dot\gamma(t),\dot\gamma(t)\right>_g}\)
The arc length of \(\gamma\) is \[ L(\gamma)=\int_a^b \sqrt{\left<\dot\gamma(t),\dot\gamma(t)\right>_{g}}\,\mathrm dt\]
Shortest path planning: want to minimize \(L(\gamma)\).
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
The arc length equation gives us a distance metric
\[d(p,q)=\inf\{L(\gamma)\,|\,\gamma\in\mathcal{C}^{1}_{\operatorname{pw}}([0,1],\mathcal{M}),\\\gamma(0)=p,\gamma(1)=q\}\]
Curves which minimize this function are called geodesics.
Without any obstacles, the solution to the shortest-path problem is a geodesic.
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
A set \(X\subseteq\mathbb{R}^n\) is convex if \(\forall p,q\in X\), the line connecting \(p\) and \(q\) is contained in \(X\)
A set \(X\subseteq\mathcal{M}\) is geodesically convex (or g-convex) if \(\forall p,q\in X\), there is a unique length-minimizing geodesic (w.r.t. \(\mathcal{M}\)) connecting \(p\) and \(q\), which is completely contained in \(X\).
Convex Set, Wikipedia
Textbook references: Convex Optimization (Boyd 2004) and An Introduction to Optimization on Manifolds (Boumal 2023).
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Sets can be convex in isometric local coordinates, but not geodesically-convex on the manifold
Radius of convexity: how large can a set be and still be convex?
Which Continent Is Situated In All Four Hemispheres?, Junior
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
A function \(f:X\to\mathbb{R}\) is convex if for any \(x,y\in X\) and \(\lambda\in[0,1]\), we have
\[f(\lambda x+(1-\lambda)y)\le\]
\[\lambda f(x)+(1-\lambda) f(y)\]
A function \(f:\mathcal{M}\to\mathbb{R}\) is geodesically convex if for any \(x,y\in \mathcal{M}\) and any geodesic \(\gamma:[0,1]\to\mathcal{M}\) such that \(\gamma(0)=x\) and \(\gamma(1)=y\), \(f\circ\gamma\) is convex
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
If \(\mathcal M\) has a geodesic loop, then every g-convex function is constant. Most robot configuration spaces have geodesic loops.
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Solution: In some cases, we can restrict the domain of the function to a domain on which it is g-convex. This is not always possible.
Example: \(\mathrm d:S^1\times S^1\to\mathbb R\), the Riemannian distance between two points on the unit circle, is g-convex if both points lie in the interior of a common semicircle.
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Configuration space \(\mathcal{Q}\) as a Riemannian manifold
\(\mathcal{M}\subseteq\mathcal{Q}\) is the set of collision free configurations, \(\bar{\mathcal{M}}\) its closure
The shortest path between \(p,q\in\bar{\mathcal{M}}\) is the solution to the optimization problem
\[\begin{array}{rl} \argmin & L(\gamma)\\ \textrm{subject to} & \gamma\in\mathcal{C}^{1}_{\operatorname{pw}}([0,1],\bar{\mathcal{M}})\\ & \gamma(0)=p\\ & \gamma(1)=q\end{array}\]
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
A directed graph \(G=(V,E)\)
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
For each \(v\in V\), let \(\psi_v:\mathcal{Y}_v\to\mathbb{R}^n\) be a coordinate chart
Define \(X_v=\psi_v(\mathcal{Y}_v)\)
For each \(e=(u,v)\in E\), new edge cost:
\[\ell_e(x_u,x_v)=\ell_e^\mathcal{Y}(\psi_u^{-1}(x_u),\psi_v^{-1}(x_v))\]
If all \(X_v\) and \(\ell_e\) are convex, we have a valid GCS problem
Mixed-Integer Riemannian Optimization? No existing literature.
Better option: transform into a GCS problem
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Have to consider the curvature of the manifold.
Positive curvature
E.g. \(S^n,\operatorname{SO}(3),\operatorname{SE}(3)\)
Negative curvature
E.g. Saddle, PSD Cone \(\vphantom{S^n}\)
Zero curvature
E.g. \(\mathbb R^n, \mathbb T^n\)
?
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Fetch Robot Configuration
Space: \(\operatorname{SE}(2)\times \mathbb{T}^3\times \mathbb{R}^7\)
Trajectory Control for Differential Drive Mobile Manipulators, Karunakaran and Subbaraj
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
A robot arm with 5 continuous revolute joints in the plane
Configuration Space: \(\mathbb{T}^5\)
(No self collisions)
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Black: Shortest path on PL approximation
Red: Shortest path on sphere
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Intuition: How much does the Jacobian of the chart vary?
Intuition: The amount we can "slip" along the boundary also depends on the Hausdorff distance \(\epsilon_H\) between the manifold and its triangulation, as well as the angle \(\alpha_{\operatorname{max}}\) between adjacent charts.
Intuition: The error grows additively with the number of sets traversed in the optimal path.
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
The piecewise-linear approximation strategy is probably
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Flatten the manifold, and pick a convex surrogate objective.
Learning the Metric of Task Constraint Manifolds for Constrained Motion Planning, Zha et. al.
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Constrained Bimanual Planning with Analytic Inverse Kinematics, Thomas Cohn, Seiji Shaw, Max Simchowitz, Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Planning Shorter Paths in Graphs of Convex Sets by Undistorting Parametrized Configuration Spaces, Shruti Garg, Thomas Cohn, and Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Non-Euclidean Motion Planning with Graphs of Geodesically-Convex Sets
Thomas Cohn, Mark Petersen, Max Simchowitz, and Russ Tedrake
Robot Locomotion Group
ICCOPT 2025