Visualizing High-Dimensional Configuration Spaces: A Comprehensive Analytical Approach

TL;DR

This work tackles the challenge of visualizing high-dimensional configuration spaces $\mathcal{C}$ for manipulators to qualitatively and quantitatively assess approximations $\hat{\mathcal{C}}$ without dimensionality reduction. It introduces a 2D visualization built on the manipulator's kinematic chain, combining 2D joint projections with parallel-coordinate labeling and color-based indexing, including $S^1$-based arc representations, to preserve inter-joint dependencies. The approach enables visual identification of $\mathcal{C}_{free}$ regions, qualitative comparison of different $\hat{\mathcal{C}}$s, and numerical metrics that reflect accuracy and coverage, validated on a 7-DOF Baxter arm with GAN-based sampling guided by workspace constraints. Results show the visualization aligns with collision-checker outcomes and highlights regions where high-dimensional representations diverge, offering actionable insights for SBMP design and evaluation.

Abstract

The representation of a Configuration Space C plays a vital role in accelerating the finding of a collision-free path for sampling-based motion planners where the majority of computation time is spent in collision checking of states. Traditionally, planners evaluate C's representations through limited evaluations of collision-free paths using the collision checker or by reducing the dimensionality of C for visualization. However, a collision checker may indicate high accuracy even when only a subset of the original C is represented; limiting the motion planner's ability to find paths comparable to those in the original C. Additionally, dealing with high-dimensional Cs is challenging, as qualitative evaluations become increasingly difficult in dimensions higher than three, where reduced-dimensional C evaluation may decrease accuracy in cluttered environments. In this paper, we present a novel approach for visualizing representations of high-dimensional Cs of manipulator robots in a 2D format. We provide a new tool for qualitative evaluation of high-dimensional Cs approximations without reducing the original dimension. This enhances our ability to compare the accuracy and coverage of two different high-dimensional Cs. Leveraging the kinematic chain of manipulator robots and human color perception, we show the efficacy of our method using a 7-degree-of-freedom CS of a manipulator robot. This visualization offers qualitative insights into the joint boundaries of the robot and the coverage of collision state combinations without reducing the dimensionality of the original data. To support our claim, we conduct a numerical evaluation of the proposed visualization.

Figures (11)

• Figure 1: 2D visualization of a $\mathcal{C}_{free}$ for a 7-degree-of-freedom (DOF) robot manipulator.
• Figure 2: Classical representations of $\mathcal{C}$ from the first 3 joints of a 7-DOF Baxter robot. Fig. \ref{['fig:3DCS']} shows 5000 $\mathcal{C}_{free}$-states sampled; with joints $\theta_i \in [-\pi,\pi],i \in \mathbb{N}$. Using parallel coordinates in Fig. \ref{['fig:parallel3D']} can effectively capture the constraints within individual joints by representing each independent state as a unique colored line but fails to illustrate the complex interactions between them given the produced clutter. The projection of $\mathcal{C}$ in WS from Fig. \ref{['fig:projectionws']} is useful for one sample of $\mathcal{C}$; the evaluation of the whole $\mathcal{C}$ is difficult given that only some boundaries can be visualized when trying to show the whole distribution, the rest is occluded by the 2D projection in the 3D WS. The 2D projections in Fig. \ref{['fig:projection2d']}, $\theta_0 \times \theta_1$ and $\theta_1 \times \theta_2$ of $\mathcal{C}$ capture the interactions between adjacent joints, such as the presence of a 3D hole by projecting it in two dimensions, but the simultaneous interactions across individual samples in all dimensions can be challenging to discern in specific states/regions.
• Figure 3: A 3 DOF robot. We can easily observe the constraints of each joint of the robot by looking at the coordinate $\theta_i$ from each $\mathcal{C}_{free}$-state $\bm{q}$, with $j_i,k_i \in \mathbb{R}$ and $i \in \mathbb{N}$. Each unconstrained joint can be seen as a circle that is isomorphic to a line segment between $-\pi$ and $\pi$. It is not possible to observe the dependencies between the joints.
• Figure 4: Discretization effect when the continuous values from $\mathcal{C}$ are displayed on a discrete output device.
• Figure 5: Each parent joint $\theta_{i}$ can be ordered given the interval $[-\pi,\pi]$. Then we assign a color to each tree $\theta_{i+1}|\theta_{i}^j$ by distributing uniformly the value of the joint $\theta_i^j$ given the number of samples in the color space. All the colored children from $\theta_{i}|\theta_{i+1}^j$ are assigned a radius $r_{i+1}^j$ by an increasing function $g(\theta_{i+1}|\theta_i^j)$ and are plotted in their respective 2D positions $g(\theta_{i+1}|\theta_i^j)cos(\theta_{i+1}^j),g(\theta_{i+1}|\theta_i^j)sin(\theta_{i+1}^j)$. The length of each circumference is given by the maximum and minimum $\theta_i$ from each $\theta_{i+1}$