Implicit Null-space Manifold Generation for Redundant Robotic Systems
Authors: Taiki Ishigaki, Teresa Vidal-Calleja, Ko Ayusawa, Eiichi Yoshida
Organizations: Tokyo University of Science, Japan · University of Technology Sydney, Australia · National Institute of Advanced Industrial Science and Technology, Japan
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.
Self-motion manifold (SMM) characterizes the geometric structure of the infinite inverse kinematic solutions set of a redundant manipulator at a fixed end-effector pose, and its efficient recovery underpins feasible and global optimal motion planning. Existing methods such as null-space continuation and learning-based methods are formulated around the assumption that an SMM is a curve, and do not extend to higher redundancy orders. We instead adopt a probabilistic view: SMMs are the support of the conditional posterior over configurations given a target pose, so that recovering it reduces to sampling from a learned distribution and separating its disjoint components by clustering. The formulation is independent of the manifold dimension and requires no architectural change as the redundancy order grows. In this work, we demonstrate that our method can approximate 1-D SMMs with performance comparable to the latest null-space continuation and learning-based approach, and that it is the first method capable of approximating highly redundant 4-D SMMs in a 7R manipulator for position tasks. Project website: \href{https://github.com/accuracy-maker/high-dimenstional-self-motion-manifold-approximation}{https://github.com/accuracy-maker/high-dimenstional-self-motion-manifold-approximation}
Learned manipulation policies increasingly predict motions for abstract "hands" and are attractive in practice because they rely on easily collected demonstrations and transfer across robot platforms. Executing these trajectories on multi-arm robots, however, is not trivial. Multi-hand policy outputs must be assigned to physical arms, each arm must realize a configuration-space motion that tracks its prescribed end-effector trajectory, and all arms must respect kinematic limits and avoid collisions. In the absence of algorithms that directly address this problem, practitioners typically extend single-arm inverse-kinematics (IK) pipelines in an ad hoc way, with no guarantees of feasibility or safety. In this work, we close this execution gap with a search-based framework that is theoretically complete for grounding policy-generated multi-hand trajectories onto physical multi-arm systems. Building on Conflict-Based Search, our method explicitly searches over both the discrete assignment of trajectories to arms and the continuous Jacobian null spaces of redundant manipulators, using redundancy to avoid inter-arm collisions while tracking the prescribed motions. This unified treatment of assignment and null-space motion yields a practically efficient planner that safely realizes coordinated manipulation-policy outputs on multi-arm robots. See omcbsa.github.io for more.
We propose a workspace-fibered decomposition framework for motion planning in nR planar redundant manipulators operating in cluttered environments. Rather than planning directly in the full n-dimensional configuration space, the method incrementally constructs obstacle-constrained reachable workspaces of lower-dimensional non-redundant sub-chains and recursively lifts them through redundant orientation fibers. This yields a sequence of reduced planning manifolds that preserve branch-consistent reachability structure while avoiding explicit construction of the full configuration-space obstacle geometry. We first establish that, for planar position-only manipulators, the obstacle-constrained reachable workspace induced by the minimal non-redundant sub-chain provides an exact characterization of feasibility with respect to the connected component of the start configuration, enabling early infeasibility detection prior to introducing redundant degrees of freedom (DOF). We then introduce an incremental fiber-lifting procedure that propagates reachable workspace structure through successive redundant links while enforcing local inverse-kinematic branch consistency using Jacobian determinant continuity constraints. The resulting representation admits efficient reduced-space planning directly on recursively-constructed workspace-fiber manifolds. Experimental results on redundant nR planar manipulators demonstrate that the proposed construction preserves collision-free connectivity structure across successive lifting stages while substantially reducing collision checking complexity relative to direct configuration space reasoning.