Yuanqing Wu

Inversion Symmetry of the Euclidean Group: Theory and Application to Robot Kinematics

Just as the 3-D Euclidean space can be inverted through any of its points, the special Euclidean group SE(3) admits an inversion symmetry through any of its elements and is known to be a symmetric space. In this paper, we show that the symmetric submanifolds of SE(3) can be systematically exploited to study the kinematics of a variety of kinesiological and mechanical systems and, therefore, have many potential applications in robot kinematics. Unlike Lie subgroups of SE(3), symmetric submanifolds inherit distinct geometric properties from inversion symmetry. They can be generated by kinematic chains with symmetric joint twists. The main contribution of this paper is: 1) to give a complete classification of symmetric submanifolds of SE(3); 2) to investigate their geometric properties for robotics applications; and 3) to develop a generic method for synthesizing their kinematic chains.

The 2D Orientation Interpolation Problem: A Symmetric Space Approach

In this paper, we propose a novel construction of Bezier curves of two-dimensional ((2)D) orientations using the geometry of real projective plane (mathrm{mathbb {R}P^{2}}). Unlike the commonly adopted unit 2-sphere model (S^{2}), (mathrm{mathbb {R}P^{2}}) is naturally embedded in the (3)D special orthogonal group (mathrm{SO(3)}). It is also a symmetric space that is equipped with a particular class of isometries called geodesic symmetry, which allows us to generate any geodesics using the exponential map of (mathrm{SO(3)}). We implement the generated geodesics to construct Bezier curves for direction interpolation.

Comparative study of robot kinematic calibration algorithms using a unified geometric framework

In this paper, we conduct a comparative study of three well known robot kinematic calibration algorithms, namely the Denavit-Hartenberg (DH) parameter algorithm, the product of exponentials (POE) algorithm, and the local POE (LPOE) algorithm. To cope with distinct formulations associated to different algorithms, we propose a unified geometric framework which is based on POE kinematics and a novel Adjoint error model. The Adjoint error model offers us an extremely efficient way to benchmark the aforesaid calibration algorithms, and also compare them to a novel calibration algorithm based on the Adjoint error model.

POE-Based Robot Kinematic Calibration Using Axis Configuration Space and the Adjoint Error Model

The product of exponential model based robot calibration approach eliminates parameter discontinuity and simplifies coordinate frame setup, but demands extra effort to normalize twist coordinates and differentiate parameter-varying exponential maps. In this paper, we show that such an endeavor can be exempted by respecting the nonlinear geometry of the joint axis configuration space (ACS), the set of all possible axis locations. We analyze the geometry of the ACS models for prismatic and revolute joints, and treat the errors as Adjoint transformations on joint twists. We propose a novel robot kinematic calibration algorithm based on the ACS and Adjoint error model. It is geometrically intuitive, computationally efficient, and can easily handle additional assumptions on joint axes relations. We present a comparative study with simulations and experiments to show that our algorithm outperforms the existing ones in various aspects.

Motion Interpolation in Lie Subgroups and Symmetric Subspaces

We show that a map defined by Pfurner, Schrocker and Husty, mapping points in 7-dimensional projective space to the Study quadric, is equivalent to the composition of an extended inverse Cayley map with the direct Cayley map, where the Cayley map in question is associated to the adjoint representation of the group SE(3). We also verify that subgroups and symmetric subspaces of SE(3) lie on linear spaces in dual quaternion representation of the group. These two ideas are combined with the observation that the Pfurner-Schrocker-Husty map preserves these linear subspaces. This means that the interpolation method proposed by Pfurner et al. can be restricted to subgroups and symmetric subspaces of SE(3).

Design of a Novel 3-DoF Serial-Parallel Robotic Wrist: A Symmetric Space Approach

For the past forty years, design of robotic wrists in the robot industry has been dominated by a serial kinematics architecture, which parameterizes the end-effector orientation space by Euler angles. Such a design suffers from stationary (or dead-centre) configurations, as well as a weak third axis due to gear train backlash. It was once believed that the study of parallel kinematics mechanisms could result in viable alternatives overcoming the shortcomings of serial wrists. However, this did not happen, probably due to the limited workspace, complex kinematics, and inherent singularities characterizing parallel architectures. In this paper, we propose a novel class of serial-parallel 3-DoF robotic wrists, based on a particular geometry usually found in constant-velocity (CV) shaft couplings. The theory of CV couplings originated with Myard’s study and culminated with Hunt’s work. We have gone one step further, by fully decrypting and completing Hunt’s development using symmetric space theory. The latter allows us to provide an easy-to-follow procedure for synthesizing a unique type of parallel wrists with interconnections. Such novel wrists entail analytic direct and inverse kinematic analyses, and their singularities can be easily identified using the so-called half-angle property, which holds for all symmetric subspaces of the special Euclidean group. By conveniently choosing geometric parameters, the proposed wrists can achieve a singularity-free pointing cone of (180^circ ), in addition to an unlimited rolling.

Parallel Dynamics Computation Using Prefix Sum Operations

A new parallel framework for fast computation of inverse and forward dynamics of articulated robots based on prefix sums (scans) is proposed. We first re-investigate the well-known recursive Newton–Euler formulation for robot dynamics and show that the forward–backward propagation process for robot inverse dynamics is equivalent to two scan operations on certain semigroups. Then, we showed that state-of-the-art forward dynamic algorithms can also be cast into a sequence of scan operations almost completely, with unscannable parts clearly identified. This suggests a serial–parallel hybrid approach for systems with a moderate number of links. We implement our scan-based algorithms on Nvidia CUDA platform with performance compared with multithreading CPU-based recursive algorithms; a computational acceleration is demonstrated.

Symmetric subspaces of SE(3)

Abstract Being a Lie group, the group SE(3) of orientation preserving motions of the real Euclidean 3-space becomes a symmetric space (in the sense of O. Loos) when endowed with the multiplication µ(g, h) = gh−1g. In this note we classify all connected symmetric subspaces of SE(3) up to conjugation. Moreover, we indicate some of its important applications in robot kinematics.

Identifiability and Improvement of Adjoint Error Approach for Serial Robot Calibration

In this paper, we first analyze the identifiability of POE based Adjoint error approach. By carefully examining the linear dependence between calibration Jacobian columns, it is proved that joint offsets and Adjoint errors cannot be identified simultaneously, and the maximum dimension of identifiable parameters is 4r + 2t + 6. Some more scenarios are considered to augment the Adjoint error approach. To satisfy the constraints on joint relations, constrained method and projection method are proposed. Moreover, we present the identifiability of reduction ratios and joint pitches. Simulations of a 6 Degree-of-Freedom robot and a SCARA robot are given to illustrate and compare our methods. It shows that the constrained method can handle such situations effectively and yields better results.

Unified GPU-Parallelizable Robot Forward Dynamics Computation Using Band Sparsity

This letter proposes a unified GPU-parallelizable approach for robot forward dynamics (FD) computation based on the key fact that parallelism of prevailing FD algorithms benefits from the essential band sparsity of the joint space inertia (JSI) matrix or its inverse. The existing FD algorithms are categorized into three classes: direct JSI algorithms, propagation algorithms, and constraint force algorithms. Their associated systems of linear equations are transformed into a set of block bidiagonal (the first and second classes) and tridiagonal (the third class) linear systems, which can be conveniently and efficiently parallelized over the existing CPU–GPU programming platforms using various state-of-the-art parallel algorithms, such as parallel all-prefix sum (scan) and odd–even elimination. This high-level perspective allows unified and efficient implementation of all three classes of algorithms and also other potentially efficient algorithms, with the bonus that different algorithms can be swiftly compared to recommend problem-specific solutions.