Configuration Space Distance Fields for Manipulation Planning

TL;DR

This work introduces Configuration Space Distance Field (CDF), a distance-field representation defined in robot configuration space that preserves unit-gradient properties and enables direct gradient-based planning. By formulating $f_c(p,q)=\min_{q'}\|q-q'\|$ with the constraint $f_s(p,q')=0$, CDF enables one-step gradient projection for inverse kinematics and natural geodesics around obstacles, bridging task-space SDF techniques to configuration-space planning. The paper provides (i) a computation/fusion pipeline for CDF, (ii) a neural CDF variant using an MLP with a loss combining distance, gradient, eikonal, and tension terms, and (iii) extensive demonstrations on planar and 7-DoF Franka robots for whole-body IK and manipulation planning, showing substantial gains in speed, solution throughput, and success rates over SDF-based methods. The results establish CDF as a unified, differentiable, geometry-aware representation that can integrate with learning and optimization to improve manipulation planning in high-dimensional spaces. This framework offers a practical pathway to leveraging SDF-inspired techniques directly in configuration space, enabling efficient IK, planning, and learning-driven control for complex robotic systems.

Abstract

The signed distance field is a popular implicit shape representation in robotics, providing geometric information about objects and obstacles in a form that can easily be combined with control, optimization and learning techniques. Most often, SDFs are used to represent distances in task space, which corresponds to the familiar notion of distances that we perceive in our 3D world. However, SDFs can mathematically be used in other spaces, including robot configuration spaces. For a robot manipulator, this configuration space typically corresponds to the joint angles for each articulation of the robot. While it is customary in robot planning to express which portions of the configuration space are free from collision with obstacles, it is less common to think of this information as a distance field in the configuration space. In this paper, we demonstrate the potential of considering SDFs in the robot configuration space for optimization, which we call the configuration space distance field. Similarly to the use of SDF in task space, CDF provides an efficient joint angle distance query and direct access to the derivatives. Most approaches split the overall computation with one part in task space followed by one part in configuration space. Instead, CDF allows the implicit structure to be leveraged by control, optimization, and learning problems in a unified manner. In particular, we propose an efficient algorithm to compute and fuse CDFs that can be generalized to arbitrary scenes. A corresponding neural CDF representation using multilayer perceptrons is also presented to obtain a compact and continuous representation while improving computation efficiency. We demonstrate the effectiveness of CDF with planar obstacle avoidance examples and with a 7-axis Franka robot in inverse kinematics and manipulation planning tasks.

Figures (10)

• Figure 1: Differences between SDF and CDF. The colored level sets depict the distances to the object, where the black contours represent joint angles leading to robot-object surface contact. The zero-level-set of SDF and CDF is the same, but the other level sets of CDF are characterized by evenly expanded distances and unit norm gradient. This property leads to gradient projection that can directly be computed, which solves the inverse kinematics problem for the contact task in one step (see trajectories in cyan and pink), whereas SDF requires multiple iterations while encountering singularities, which can even fail due to gradient vanishing (see trajectory in yellow). Geodesics on the CDF naturally wrap around the shape of the object in configuration space (see trajectories in blue, green and purple).
• Figure 2: Illustration of the computation of CDF. During the offline phase, we initialize the workspace of the robot as a volumetric grid and compute zero-level-set joint configurations for each grid point. For online computation, given an object $\mathcal{O}$, we identify the closest configuration in the set $\bm Q$ associated with occupied grids to calculate the $\ell^2$ distance. We further encode the CDF with neural networks to obtain a compact and grid-free representation.
• Figure 3: Comparison between CDF and SDF in solving whole-body inverse kinematics problem. (a) The initial sampled joint configurations. (b) Gradient projection by CDF. (c) Task space visualization of feasible solutions in (b). (d) Results for distance query-based method with L-BFGS optimizer. We can see that with the baseline SDF approach, the system can get stuck when the gradient of the SDF vanishes or reaches the singularity.
• Figure 4: Gradient projection for whole-body inverse kinematics using neural CDF. The centers of the red spheres are target points, where the radius of spheres is set to $0.05m$.
• Figure 5: Goalkeeper task in simulation.