Encircling Singularities of a Serial Robot

Description

The folder Animations of Anthropomorphic Arm with Spherical Wrist contains animation of this wrist-separable robot encountering coincident overhead and wrist singularities. Counter to intuition, a displacement path where the arm motion is minimal at singularity encounter switches solution branches associated with the arm singularity, whereas an encounter where the arm swings through 180-deg remains on the same solution branch. The same holds for wrist motion and whether the wrist solution switches branches. In the animation files in two formats named RobotZeroSelfMotions, the wrist and arm appear nearly stationary during the singularity encounter, but that both the arm and the wrist have switched branches is apparent from both the arm and wrist rotation being reset at the repeat of the animation cycle. In RobotTwoSelfMotions, both the wrist and arm undergo large motions at singularity encounter, but that both wrist and arm remain on the same solution branch is demonstrated by the lack of a reset in either wrist or arm placement when an animation cycle repeats. This minimal-motion singularity encounter shown in RobotZeroSelfMotions computes the animation path automatically using the singularity encirclement procedure. Other animations show combinations of remaining on or switching between either or both of the wrist and arm solution branches. These singularity encounters use the encirclement procedure to reach the self-motion path of the wrist. Following the wrist self-motion path reaches the self-motion path of the arm. Where to switch between or exit either of these self-motion paths was determined manually from plots of finite-displacement departures from these paths. Automatically generating singularity encounters that enter and exit self-motion paths is considered for future work. Because the encirclement path results in small angle displacements of the robot, the portions of the animations on the encirclement path appear to casual viewing as hesitations in the robot motion. Careful scrutiny reveals these small motions, distinguishing the half-turn to transit the singularity encounter without following self-motion paths from the quarter-turn to reach the wrist self-motion. The folder Serial robot models contains solid models representing four serial robot architectures. The open-source Open SCAD software available at openscad.org allows viewing these robot models from different viewpoints. The file numerics.zip contains the QR matrix decomposition Java software used to conduct the encirclement procedure. This modification of the QR solver from the public-domain JAMA linear algebra package for Java https://math.nist.gov/javanumerics/jama/ has these enhancements. Adds option for numerically stable column pivoting for rank-revealing Reduces number of square-root evaluations in scaling operation Allows wide-rectangular matrix where rows m < columns n Calculates "full" in place of "thin" Q matrix when m > n

Steps to reproduce

The .scad files are for 3D CAD models of robots viewable with the OpenSCAD software package available from https://openscad.org/

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