Analysis of Motion Planning by Sampling in Subspaces of Progressively Increasing Dimension

Abstract

Despite the performance advantages of modern sampling-based motion planners, solving high dimensional planning problems in near real-time remains a challenge. Applications include hyper-redundant manipulators, snake-like and humanoid robots. Based on the intuition that many of these problem instances do not require the robots to exercise every degree of freedom independently, we introduce an enhancement to popular sampling-based planning algorithms aimed at circumventing the exponential dependence on dimensionality. We propose beginning the search in a lower dimensional subspace of the configuration space in the hopes that a simple solution will be found quickly. After a certain number of samples are generated, if no solution is found, we increase the dimension of the search subspace by one and continue sampling in the higher dimensional subspace. In the worst case, the search subspace expands to include the full configuration space -making the completeness properties identical to the underlying sampling-based planer. Our experiments comparing the enhanced and traditional version of RRT, RRT-Connect, and BidirectionalT-RRT on both a planar hyper-redundant manipulator and the Baxter humanoid robot indicate that a solution is typically found much faster using this approach and the run time appears to be less sensitive to the dimension of the full configuration space. We explore important implementation issues in the sampling process and discuss its limitations.

Multilevel Motion Planning: A Fiber Bundle Formulation

“…which reflects an exponential increase of samples in higher dimensions. This idea is similar to the selection of bundle spaces using a geometric progression (Xanthidis et al 2018). Note also the close connection to multilevel monte carlo (Giles 2015) and sparse grid methods (Bungartz and Griebel 2004).…”

“…There are two main approaches. First, we can select sequences of subspaces of the state space (Xanthidis et al 2018), then sample them by selecting the subspaces based on the density of samples. Second, we can use workspace sampling (Van den Zucker et al 2008;Rickert et al 2014;Luna et al 2020), where state space samples are taken from the restriction of collision-free sets in workspace.…”

Multilevel motion planning: A fiber bundle formulation

“…We can compute an exponential importance as 1/|V k | 1/d + 1 which reflects an exponential increase of samples in higher dimensions. This idea is similar to the selection of bundle spaces using a geometric progression (Xanthidis et al, 2018). This is also related to multilevel monte carlo (Giles, 2015) and sparse grid methods (Bungartz and Griebel, 2004).…”

“…There are two main approaches. First, we can select sequences of subspaces of the state space (Xanthidis et al, 2018), then sample them by selecting the subspaces based on the density of samples. Second, we can use workspace sampling Zucker et al, 2008;Rickert et al, 2014;Luna et al, 2020), where state space samples are taken from the restriction of collision-free sets in workspace.…”

“…We can do workspace sampling by focusing on narrow passages , or by selecting promising points on the robot and guiding them through the workspace (Luna et al, 2020). Our approach is similar in that we use lower-dimensional sampling on the base space and we select base spaces based on a density criterion (Xanthidis et al, 2018). However, we differ by smoothly changing between path and graph restriction sampling, and by using a recursive path section method to efficiently find solution paths.…”