Implicit Null-space Manifold Generation for Redundant Robotic Systems
For roboticists dealing with redundant systems, this provides a novel representation-centric approach to model entire solution manifolds rather than individual solutions, though it is incremental over existing Jacobian-based methods.
This work introduces an implicit scalar field representation whose zero-level set captures the solution manifold for redundant robotic systems, enabling continuous distance fields that encode proximity to the solution space. Experiments on planar and 7-DOF manipulators show effective manifold modeling and consistent representation across task families.
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.