AI summaryⓘ
The authors study how robots that have more ways to move than needed can perform the same task using many different positions, which form shapes called manifolds in their movement space. Instead of just finding single solutions, they create a method to understand the whole shape of these solutions by representing it as a special kind of field where points on the field at zero represent valid robot positions. They use a technique guided by robot movement math (the Jacobian) to sample points around this shape, building a continuous map that shows how close any position is to a valid solution. Their tests on different robots show this approach helps model the full range of possible solutions, even as tasks change smoothly.
redundant degrees of freedomconfiguration spacesolution manifoldJacobianimplicit scalar fieldzero-level setdistance fieldrobot manipulatortask-defining mapscontinuous variation
Authors
Taiki Ishigaki, Teresa Vidal-Calleja, Ko Ayusawa, Eiichi Yoshida
Abstract
Robotic systems with redundant degrees of freedom can achieve the same task outcome using multiple configurations, resulting in solution sets that form manifolds in the configuration space. Existing approaches typically exploit such redundancy locally through Jacobian-based techniques to compute individual solutions or trajectories. While effective for solution computation, these methods do not retain a representation of the geometry of the solution set itself. In this work, we adopt a representation-centric approach to estimate the geometric structure of the solution space. We consider solution manifolds induced by general task-defining maps and construct an implicit scalar field over the configuration space, whose zero-level set corresponds to the solution manifold. To this end, we generate samples in the neighborhood of the solution manifold using a Jacobian-guided exploration strategy, which efficiently captures its local and global structure. The resulting implicit representation is defined over the configuration space and naturally induces a continuous, distance field that encodes proximity to the solution manifold. Experiments on a planar three-link robot and a seven-degree-of-freedom Franka manipulator demonstrate the effectiveness of the proposed representation. Furthermore, the framework enables consistent modeling of solution spaces across families of tasks with continuous variation.