Problem
Research questions and friction points this paper is trying to address.
Riemannian Metrics
Motion Planning
Constraints
Submanifold
Configuration Space
Induced Riemannian Metrics for Motion Planning with Constraints
🤖 AI Summary
本文解决了约束运动规划问题，通过引入诱导黎曼度量来统一不同表示方法下的路径长度计算，从而在采样规划和轨迹优化中实现一致的几何处理。
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
induced Riemannian metrics
motion planning with constraints
submanifold representation independence
parameterization distortion
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