Riemannian metrics improve robot motion planning with constraints
Induced Riemannian Metrics for Motion Planning with Constraints
Robotics
Summary
Planning how robots move can be tricky when their movement is limited by specific task rules or loops that must close perfectly. The authors show that by using a special way to measure distances called an induced Riemannian metric, planners can better account for these restrictions no matter how the robot’s possible moves are described. This approach helps planners find motion paths that truly reflect the robot's capabilities and energy costs. They tested this idea using two robotic arms performing tasks, comparing it to traditional methods.
What this means in practice
- •For robotic system engineers: Design motion planners that use induced metrics to better handle constraints in robot tasks involving multiple arms or closed-loop conditions.
- •For animation software developers: Improve realistic character motion by integrating constraint-aware distance metrics to optimize movement paths in virtual environments.
Authors
Phone Thiha Kyaw, Thomas Cohn, Miguel Angel Rogel Garcia, Jonathan Kelly
Abstract
In constrained motion planning problems, task and loop-closure constraints restrict a robot's motion to a curved, lower-dimensional submanifold of its configuration space. Planners measure path length with a metric, which sets the cost of moving in each direction. Under the Euclidean metric, this cost is the same everywhere, whereas under a general Riemannian metric, such as the kinetic-energy metric, the cost can vary with direction and configuration. Existing methods often describe the submanifold either implicitly, as a constraint level set, or explicitly, through a parameterization. The implicit representation is typically combined with the Euclidean metric of the configuration space, and the explicit representation with the parameter domain, so the path length that a planner minimizes depends on the representation. Instead, we measure path length with the induced metric, which the submanifold inherits from a Riemannian metric on the configuration space. The implicit and explicit representations yield the same induced metric, expressed in different coordinates, and hence the same geometry. This result holds for any Riemannian metric on the configuration space, not only the Euclidean one. The choice of metric is therefore independent of the choice of representation. Using this result, we extend planning under a Riemannian metric from unconstrained spaces to constraint submanifolds by applying the induced metric in both a sampling-based planner and a trajectory optimizer. For an explicit representation, the induced metric also accounts for the distortion that the parameterization introduces. In experiments on a bimanual manipulation setup with two Franka arms under end-effector task constraints, we compare the Euclidean and kinetic-energy metrics.