Rational motion design - a survey

Abstract In the last years many efforts have been made to transfer geometric CAD methods to motion design: The generation of a (one-parametric) motion with given properties. An object (arm of a robot…) is moved with respect to a fixed system. Methods from freeform curve design provide powerful tools to handle such problems. Some of these methods and results shall be presented in this survey. Here one can find a (geometric) approach including solutions of interpolation problems. First attempts are studied. Then we state two principal demands on the result of such a design process: The motions should be rational and invariant with respect to changes of coordinates in fixed and moving frame, respectively. We use quaternions to describe displacements and motions and give an overview on rational spline motions. Some known results on interpolating motions are reviewed. Then a third demand is presented: The design process should be ‘repeatable’ in the following sense: If a solution is computed the algorithm should give the same solution (including parametrisation), if we start with new input data generated by our first solution. The paper includes suggestions for further research.

[1]  Oswald Giering.  Vorlesungen über höhere Geometrie , 1982 .

[2]  Rae A. Earnshaw,et al.  Computer Graphics: Developments in Virtual Environments , 1995, Computer Graphics.

[3]  Jacques M. Hervé,et al.  The mathematical group structure of the set of displacements , 1994 .

[4]  Josef Hoschek,et al.  Fundamentals of computer aided geometric design , 1996 .

[5]  M. Husty.  An algorithm for solving the direct kinematics of general Stewart-Gough platforms , 1996 .

[6]  Wolfgang Rath.  Matrix groups and kinematics in projective spaces , 1993 .

[7]  Otto Röschel,et al.  Interpolation of helical patches by kinematic rational Bézier patches , 1990, Comput. Graph..

[8]  Ken Shoemake,et al.  Animating rotation with quaternion curves , 1985, SIGGRAPH.

[9]  Bert Jüttler.  Spatial rational motions and their application in computer aided geometric design , 1995 .

[10]  John F. Hughes,et al.  Smooth interpolation of orientations with angular velocity constraints using quaternions , 1992, SIGGRAPH.

[11]  Leo Joskowicz,et al.  Computational Kinematics , 1991, Artif. Intell..

[12]  K. Wohlhart,et al.  Motor Tensor Calculus , 1995 .

[13]  Bahram Ravani,et al.  Mappings of Spatial Kinematics , 1984 .

[14]  Michael G. Wagner.  Planar rational B-spline motions , 1995, Comput. Aided Des..

[15]  Bert Jüttler,et al.  An algebraic approach to curves and surfaces on the sphere and on other quadrics , 1993, Comput. Aided Geom. Des..

[16]  Bert Jüttler,et al.  Visualization of moving objects using dual quaternion curves , 1994, Comput. Graph..

[17]  Vincenzo Parenti-Castelli,et al.  Recent advances in robot kinematics , 1996 .

[18]  J. M. Hervé.  Intrinsic formulation of problems of geometry and kinematics of mechanisms , 1982 .

[19]  B. Joe,et al.  Orientation interpolation in quaternion space using spherical biarcs , 1993 .

[20]  Vijay Kumar,et al.  Planning of smooth motions on SE(3) , 1996, Proceedings of IEEE International Conference on Robotics and Automation.

[21]  G. Nielson.  Smooth Interpolation of Orientations , 1993 .

[22]  John K. Johnstone,et al.  A rational model of the surface swept by a curve * , 1995, Comput. Graph. Forum.

[23]  Bahram Ravani,et al.  Computer aided geometric design of motion interpolants , 1994 .

[24]  Sung Yong Shin,et al.  A general construction scheme for unit quaternion curves with simple high order derivatives , 1995, SIGGRAPH.

[25]  J. Schlag.  VIII.4 – USING GEOMETRIC CONSTRUCTIONS TO INTERPOLATE ORIENTATION WITH QUATERNIONS , 1991 .

[26]  Daniel Thalmann,et al.  Models and Techniques in Computer Animation , 2014, Computer Animation Series.

[27]  F. Park,et al.  Bézier Curves on Riemannian Manifolds and Lie Groups with Kinematics Applications , 1995 .

[28]  Sung Yong Shin,et al.  A C/sup 2/-continuous B-spline quaternion curve interpolating a given sequence of solid orientations , 1995, Proceedings Computer Animation'95.

[29]  Bahram Ravani,et al.  Geometric Construction of Bézier Motions , 1994 .

[30]  Takeo Kanade,et al.  Modelling and Planning for Sensor Based Intelligent Robot Systems [Dagstuhl Workshop, October 24-28, 1994] , 1995, Modelling and Planning for Sensor Based Intelligent Robot Systems.

[31]  Adolf Karger,et al.  Space kinematics and Lie groups , 1985 .

[32]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[33]  Jadran Lenarčič,et al.  Advances in Robot Kinematics and Computational Geometry , 1994 .

[34]  Myung Soo Kim,et al.  Hermite Interpolation of Solid Orientations Based on a Smooth Blending of Two Great Circular Arcs on SO(3) , 1995, Computer Graphics.

[35]  Takis Sakkalis,et al.  Pythagorean-hodograph space curves , 1994, Adv. Comput. Math..

[36]  Q. Jeffrey Ge.  An Inverse Design Algorithm for a G 2 Interpolating Spline Motion , 1994 .

[37]  Wilhelm Blaschke,et al.  Kinematik und Quaternionen , 1960 .

[38]  Bahram Ravani,et al.  Computational Geometry and Motion Approximation , 1993 .

[39]  Helmut Pottmann,et al.  Geometric Motion Design , 1994, Modelling and Planning for Sensor Based Intelligent Robot Systems.

[40]  Myung-Soo Kim,et al.  Interpolating solid orientations with circular blending quaternion curves , 1995, Comput. Aided Des..

[41]  James Arvo,et al.  Graphics Gems II , 1994 .

[42]  Bert Jüttler.  Zur Konstruktion Rationaler Kurven und Flächen auf Quadriken , 1993 .

[43]  Bahram Ravani,et al.  COMPUTER AIDED DESIGN OF ROBOT TRAJECTORIES USING RATIONAL MOTIONS , 1996 .

[44]  Sung-yong Shin,et al.  A Compact Differential Formula for the First Derivative of a Unit Quaternion Curve , 1996 .

[45]  M. G. Wagner,et al.  Computer-Aided Design With Spatial Rational B-Spline Motions , 1996 .

[46]  Myung-Soo Kim.  Pseudo Dynamic Keyframe Animation with Motion Blending on the Connguration Space of a Moving Mechanism , 1995 .

[47]  Dieter Lasser,et al.  Grundlagen der geometrischen Datenverarbeitung , 1989 .