RM4D: A Combined Reachability and Inverse Reachability Map for Common 6-/7-axis Robot Arms by Dimensionality Reduction to 4D

TL;DR

This work exploits commonalities of existing six and seven axis robot arms to reduce the dimension of the discretization from 6D to 4D, resulting in a much more compact map that can be constructed by an order of magnitude faster than existing maps, with no inversion overheads and no loss in accuracy.

Abstract

Knowledge of a manipulator's workspace is fundamental for a variety of tasks including robot design, grasp planning and robot base placement. Consequently, workspace representations are well studied in robotics. Two important representations are reachability maps and inverse reachability maps. The former predicts whether a given end-effector pose is reachable from where the robot currently is, and the latter suggests suitable base positions for a desired end-effector pose. Typically, the reachability map is built by discretizing the 6D space containing the robot's workspace and determining, for each cell, whether it is reachable or not. The reachability map is subsequently inverted to build the inverse map. This is a cumbersome process which restricts the applications of such maps. In this work, we exploit commonalities of existing six and seven axis robot arms to reduce the dimension of the discretization from 6D to 4D. We propose Reachability Map 4D (RM4D), a map that only requires a single 4D data structure for both forward and inverse queries. This gives a much more compact map that can be constructed by an order of magnitude faster than existing maps, with no inversion overheads and no loss in accuracy. Our experiments showcase the usefulness of RM4D for grasp planning with a mobile manipulator.

Figures (7)

• Figure 1: RM4D has a single data structure that can be queried as (a) reachability map and (b) inverse reachability map. The coordinate system indicates the desired TCP pose, with the blue axis being the approach vector.
• Figure 2: Illustration of our two assumptions: (a) Rotating the last wrist joint only affects the in-plane rotation. The TCP position and the approach vector (blue) remain constant. (b) Rotating the base gives an arc/circle.
• Figure 3: The steps in our canonical transformation, allowing the dimensionality reduction: (\ref{['fig:tf_step0']}) TCP frames for different angles of the base joint, (\ref{['fig:tf_step1']}) TCP frames are centered at origin, (\ref{['fig:tf_step2']}) TCP frames are rotated such that the approach vector $\Vec{r}_z$ (blue) is contained in the $x^{(+)}z$ half-plane. As a result, we get a canonical base position $(x^*, y^*)$ that is identical for all configurations in (\ref{['fig:tf_step0']}). With this, we can map the 6D pose to the 4D vector $(p_z, \theta, x^*, y^*)$, where $p_z$ is the z-coordinate of the TCP and $\theta$ the angle between $\Vec{r}_z$ and the world z-axis (up).
• Figure 4: Number of novel map elements visited per 1M samples throughout the map construction. The map is sufficiently filled as this value approaches zero. RM4D can be constructed with an order of magnitude fewer samples.
• Figure 5: Accuracy, True Positive Rate (TPR), and False Positive Rate (FPR) throughout the construction of the maps for the UR5e (top) and Franka (bottom). RM4D reaches a higher accuracy with fewer samples.