Motor Algebra for 3D Kinematics: The Case of the Hand-Eye Calibration

In this paper we apply the Clifford geometric algebra for solving problems of visually guided robotics. In particular, using the algebra of motors we model the 3D rigid motion transformation of points, lines and planes useful for computer vision and robotics. The effectiveness of the Clifford algebra representation is illustrated by the example of the hand-eye calibration. It is shown that the problem of the hand-eye calibration is equivalent to the estimation of motion of lines. The authors developed a new linear algorithm which estimates simultaneously translation and rotation as components of rigid motion.

[1]  Yiu Cheung Shiu,et al.  Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX=XB , 1989, IEEE Trans. Robotics Autom..

[2]  P. Lounesto.  Clifford Algebras and Spinors , 1997 .

[3]  Jack C. K. Chou,et al.  Finding the Position and Orientation of a Sensor on a Robot Manipulator Using Quaternions , 1991, Int. J. Robotics Res..

[4]  Nikos A. Aspragathos,et al.  A comparative study of three methods for robot kinematics , 1998, IEEE Trans. Syst. Man Cybern. Part B.

[5]  Mengxiang Li,et al.  Head-eye calibration , 1995, Proceedings of IEEE International Conference on Computer Vision.

[6]  William H. Press,et al.  Numerical recipes in C , 2002 .

[7]  A. T. Yang,et al.  Application of Dual-Number Matrices to the Inverse Kinematics Problem of Robot Manipulators , 1985 .

[8]  W. Hoffman.  The Lie algebra of visual perception , 1966 .

[9]  David Hestenes,et al.  Space-time algebra , 1966 .

[10]  Eduardo Bayro-Corrochano.  The geometry and algebra of kinematics , 2001 .

[11]  J Rooney.  A Comparison of Representations of General Spatial Screw Displacement , 1978 .

[12]  P. Dodwell.  The Lie transformation group model of visual perception , 1983, Perception & psychophysics.

[13]  Fadi Dornaika,et al.  Hand-Eye Calibration , 1995, Int. J. Robotics Res..

[14]  J. Y. S. Luh,et al.  Dual-number transformation and its applications to robotics , 1987, IEEE Journal on Robotics and Automation.

[15]  H. Grassmann.  Der Ort der Hamilton'schen Quaternionen in der Ausdehnungslehre , 1877 .

[16]  D. Chevallier,et al.  Lie algebras, modules, dual quaternions and algebraic methods in kinematics , 1991 .

[17]  Gerald Sommer,et al.  The global algebraic frame of the perception-action cycle , 1999 .

[18]  J. D. Everett.  A Treatise on the Theory of Screws , 1901, Nature.

[19]  Clifford,et al.  Applications of Grassmann's Extensive Algebra , 1878 .

[20]  Eduardo Bayro-Corrochano,et al.  The dual quaternion approach to hand-eye calibration , 1996, Proceedings of 13th International Conference on Pattern Recognition.

[21]  H.H. Chen,et al.  A screw motion approach to uniqueness analysis of head-eye geometry , 1991, Proceedings. 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[22]  Yiu Cheung Shiu,et al.  Comments on 'Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX=XB' [with reply] , 1991, IEEE Trans. Robotics Autom..

[23]  D. Hestenes,et al.  Projective geometry with Clifford algebra , 1991 .

[24]  Michael W. Walker.  Manipulator kinematics and the epsilon algebra , 1988, IEEE J. Robotics Autom..

[25]  Jitendra Malik,et al.  Tracking people with twists and exponential maps , 1998, Proceedings. 1998 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No.98CB36231).

[26]  J. M. McCarthy,et al.  Dual Orthogonal Matrices in Manipulator Kinematics , 1986 .

[27]  J. Rooney,et al.  On the Three Types of Complex Number and Planar Transformations , 1978 .

[28]  David Hestenes.  New Foundations for Classical Mechanics , 1986 .

[29]  C. Barus.  A treatise on the theory of screws , 1998 .

[30]  Roger Y. Tsai,et al.  A new technique for fully autonomous and efficient 3D robotics hand/eye calibration , 1988, IEEE Trans. Robotics Autom..

[31]  I. M. Yaglom,et al.  Complex Numbers in Geometry , 1969, The Mathematical Gazette.

[32]  A. S. Solodovnikov,et al.  Hypercomplex Numbers: An Elementary Introduction to Algebras , 1989 .

[33]  Eduardo Bayro-Corrochano,et al.  A new methodology for computing invariants in computer vision , 1996, Proceedings of 13th International Conference on Pattern Recognition.

[34]  Vijay R. Kumar,et al.  Kinematics of robot manipulators via line transformations , 1990, J. Field Robotics.

[35]  Eduardo Bayro-Corrochano,et al.  Geometric algebra: a framework for computing point and line correspondences and projective structure using n uncalibrated cameras , 1996, Proceedings of 13th International Conference on Pattern Recognition.

[36]  K. Zindler.  Geometrie der Dynamen , 1903 .

[37]  D. Hestenes,et al.  Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics , 1984 .

[38]  Richard P. Paul,et al.  A computational analysis of screw transformations in robotics , 1990, IEEE Trans. Robotics Autom..

[39]  Jake K. Aggarwal,et al.  Estimation of motion from a pair of range images: A review , 1991, CVGIP Image Underst..

[40]  J. Michael McCarthy,et al.  Introduction to theoretical kinematics , 1990 .

[41]  Eduardo Bayro-Corrochano,et al.  Selforganizing Clifford neural network , 1996, Proceedings of International Conference on Neural Networks (ICNN'96).

[42]  Ching-Cheng Wang,et al.  Extrinsic calibration of a vision sensor mounted on a robot , 1992, IEEE Trans. Robotics Autom..

[43]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .