CSSDF-Net: Safe Motion Planning Based on Neural Implicit Representations of Configuration Space Distance Field

Robot manipulators are monitored by constraint-specific indicators whose units and gradient scales are not comparable, so they do not provide a common measure of the configuration-space motion remaining before violation. We define the feasibility distance field (FDF) as the distance, under a fixed positive-definite joint-space metric, to the union of infeasible configuration sets. Classical distance-to-set theory gives 1-Lipschitz continuity, almost-everywhere differentiability, and unit dual-gradient norm wherever the nearest projection is unique. The robotics contribution is an admissibility analysis showing when practical constraints define non-empty closed sets. We derive admissible formulations for external and self-collision, joint limits, dexterity, Cartesian and task-projected compliance, joint torque under payload, and dynamic manipulability. Since every field uses the same metric, heterogeneous constraints compose by a pointwise minimum, conditioned constraints retain a fixed distance space, and multi-robot constraints produce block-sparse gradients that identify which robots must react. We generate projection-based labels and train neural approximations with a distance loss and an Eikonal penalty. Simulations on a UR5e and a dual-arm cell evaluate seven fields using value, projection, sign, gradient, composition, and moving-obstacle diagnostics. Across 8,000 configurations, the largest feasible-side secant ratio is 0.920, mean learned gradient norms range from 0.994 to 0.998, and projection residuals range from 0.011 to 0.034 rad. Across 24 random obstacle paths, the external and composed collision fields achieve 90.4% and 91.6% success within 3 cm, with sign-error rates below 2%. The results support a common configuration-space margin and identify approximation errors near medial axes and sparsely sampled boundaries.

Robust motion planning and control for manipulators is the problem of guaranteeing safe execution and collision-free motion under disturbances. The Kinodynamic Motion Planning via Funnel Control (KDF) is a recently proposed planner that comes with robustness guarantees and is particularly attractive for collision-free control as it guarantees closed-loop evolution inside a funnel, i.e., a collision-free region around a nominal trajectory. Despite its attractive theoretical properties, the current approach for computing funnels is computationally demanding, which limits its expressiveness and performance. We address its current limitations by proposing a novel funnel planning and control solution. Our key tool to enable this is a neural Signed Configuration Distance Function (nSCDF) which allows us to estimate collision-free balls and gradients in a computationally efficient way. We use the nSCDF in our planner to compute a nominal trajectory, where path length and clearance are balanced. To track the trajectory with collision-avoidance guarantees, we design a funnel controller that exploits the geometric information from the nSCDF. From numerical experiments, we observe that our planner produces nominal paths $2$ times faster, while at the same time obtaining $20 \%$ shorter path length and $5$ times the clearance. Finally, we also observe improved motion performance and robustness properties when using the nSCDF for funnel control.

Mainstream plan-then-track approaches to multifingered grasp execution entail selecting a grasp, planning a collision-free trajectory, and tracking the resulting trajectory via a feedback controller. Pose-estimation error during execution or scene motion can invalidate this open-loop commitment and trigger replanning. We thus present Grasp Distance Fields (GDFs), smooth softmin distances to finite sets of arm-hand grasp configurations. Using their negative gradients as feedback, we jointly select and execute grasps without planning a trajectory. A CBF-CLF quadratic program (QP) enforces self-collision, workspace, object, and obstacle-clearance constraints, while its CLF slack quantifies obstruction of task progress. We bound the softmin approximation error by $\log N/ρ$ and prove forward invariance of the filtered safe set. To handle changes in contact topology, we combine a hysteretic contact-mode transition with a wrench-quality CBF that limits degradation of the realized force-closure margin relative to hold onset. Using our method, a fixed-base manipulator and a Unitree G1 equipped with the same underactuated hand grasp and lift 46 of 50 test objects amid clutter and moving obstacles. The realized grasps also retain a median 94% of their synthesized quality margin, and each QP solve requires 0.09 ms within a 20 ms control interval. Project page: www.clintonenwerem.com/gdf.