A system of dual quaternion matrix equations with its applications

We employ the M-P inverses and ranks of quaternion matrices to establish the necessary and sufficient conditions for solving a system of the dual quaternion matrix equations $(AX, XC) = (B, D)$, along with providing an expression for its general solution. Serving as an application, we investigate the solutions to the dual quaternion matrix equations $AX = B$ and $XC=D$, including $\eta$-Hermitian solutions. Lastly, we design a numerical example to validate the main research findings of this paper

Extrinsic Infrastructure Calibration Using the Hand-Eye Robot-World Formulation

We propose a certifiably globally optimal approach for solving the hand-eye robot-world problem supporting multiple sensors and targets at once. Further, we leverage this formulation for estimating a geo-referenced calibration of infrastructure sensors. Since vehicle motion recorded by infrastructure sensors is mostly planar, obtaining a unique solution for the respective hand-eye robot-world problem is unfeasible without incorporating additional knowledge. Hence, we extend our proposed method to include a-priori knowledge, i.e., the translation norm of calibration targets, to yield a unique solution. Our approach achieves state-of-the-art results on simulated and real-world data. Especially on real-world intersection data, our approach utilizing the translation norm is the only method providing accurate results.Comment: Accepted at 2023 IEEE Intelligent Vehicles Symposiu

Yet a better closed-form formula for the 3D nearest rotation matrix problem

This technical report complements the results recently presented in [1] showing that they can be extended to define an efficient and robust method to determine the rotation matrix nearest to an arbitrary 3 × 3 matrix. This problem arises in different areas of robotics that range from the simple case in which we have to restore the orthogonality of a noisy rotation matrix to point-cloud registration or hand-eye calibration. We show that the new method compares favorably with the classical approaches to address this problem and also with more recent methodsPreprin

A regularization-patching dual quaternion optimization method for solving the hand-eye calibration problem

The hand-eye calibration problem is an important application problem in robot research. Based on the 2-norm of dual quaternion vectors, we propose a new dual quaternion optimization method for the hand-eye calibration problem. The dual quaternion optimization problem is decomposed to two quaternion optimization subproblems. The first quaternion optimization subproblem governs the rotation of the robot hand. It can be solved efficiently by the eigenvalue decomposition or singular value decomposition. If the optimal value of the first quaternion optimization subproblem is zero, then the system is rotationwise noiseless, i.e., there exists a ``perfect'' robot hand motion which meets all the testing poses rotationwise exactly. In this case, we apply the regularization technique for solving the second subproblem to minimize the distance of the translation. Otherwise we apply the patching technique to solve the second quaternion optimization subproblem. Then solving the second quaternion optimization subproblem turns out to be solving a quadratically constrained quadratic program. In this way, we give a complete description for the solution set of hand-eye calibration problems. This is new in the hand-eye calibration literature. The numerical results are also presented to show the efficiency of the proposed method

Motion, Unit Dual Quaternion and Motion Optimization

We introduce motions as real six-dimensional vectors. A motion means a rotation and a translation. We define a motion operator which maps unit dual quaternions to motions, and a UDQ operator which maps motions to unit dual quaternions. By these operators, we present the formulation of motion optimization, which is actually a real unconstrained optimization formulation. Then we formulate two classical problems in robot research, i.e., the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem as motion optimization problems. This opens a new way to solve these problems via real unconstrained optimization

Dual quaternion-based inverse kinematics of dexterous finger

The inverse kinematics solution of a dexterous robotic finger has a significant impact on the real-time control of the robotic hand. Therefore a rapid method for solving is needed. The classical homogeneous matrix transformation is the most popular method used in robot kinematics. However, for the multi degree-of-freedom (DOF) robotic finger, the matrix parameters cost much storage and the inverse matrix calculation requires a large amount of computational cost. So it is not conducive to the real-time control of the robotic hand. Therefore, a method based on dual quaternions is presented for analysing the kinematics of a multi-DOF (4-DOF) robotic thumb. Firstly, the kinematics equation is expressed by dual quaternions. Then the multivariate kinematic equations are converted to binary quadratic equations with methods of separating variables and variable substitution, which is relatively easy to obtain the closed-form solution of the inverse kinematics. Finally, it proves that the dual quaternions method has advantages over the homogeneous matrix transformation in storage and computational cost by the specific numbers for the robotic thumb, which is conducive to the real-time control of robotic hand