Visualizing High-Dimensional Configuration Spaces: A Comprehensive Analytical Approach

Abstract

The representation of a Configuration Space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> 's representations through limited evaluations of collision-free paths using the collision checker or by reducing the dimensionality of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> for visualization. However, a collision checker may indicate high accuracy even when only a subset of the original <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> is represented; limiting the motion planner's ability to find paths comparable to those in the original <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> . Additionally, dealing with high-dimensional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> s is challenging, as qualitative evaluations become increasingly difficult in dimensions higher than three, where reduced-dimensional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> evaluation may decrease accuracy in cluttered environments. In this paper, we present a novel approach for visualizing representations of high-dimensional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> s of manipulator robots in a 2D format. We provide a new tool for qualitative evaluation of high-dimensional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> s approximations without reducing the original dimension. This enhances our ability to compare the accuracy and coverage of two different high-dimensional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> s. Leveraging the kinematic chain of manipulator robots and human color perception, we show the efficacy of our method using a 7-degree-of-freedom <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> 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.