A practical algorithm for smooth interpolation between different angular positions

Abstract This paper proposes a new methodology for the interpolation of a given set of 3D rotation poses that have to be reached in successive times by preserving continuity in orientation, angular velocity and angular acceleration. The discussed algorithm ensures the generation of smooth angular trajectories without singularities. The distinctive features of the proposed approach are the straightforward formulation, the reduced computational burden and the lack of iterative procedures. The presented methodology has applications in the generation of spatial motion of mechanical systems (e.g. robotics, flying devices) or in 3D computer graphics. After a theoretical introduction, the proposed algorithm is compared with other methods available in literature and some possible applications are presented.

[1]  G. Legnani,et al.  A homogeneous matrix approach to 3D kinematics and dynamics — I. Theory , 1996 .

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

[3]  Hammad Mazhar,et al.  Chrono: An Open Source Multi-physics Dynamics Engine , 2015, HPCSE.

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

[5]  Ravi Ramamoorthi,et al.  Fast construction of accurate quaternion splines , 1997, SIGGRAPH.

[6]  Paolo Righettini,et al.  A homogeneous matrix approach to 3D kinematics and dynamics—II. Applications to chains of rigid bodies and serial manipulators , 1996 .

[7]  Huang Xiang,et al.  Trajectory Planning Algorithm Based on Quaternion for 6-DOF Aircraft Wing Automatic Position and Pose Adjustment Method , 2010 .

[8]  Quaternion calculus and fast animation , 1987 .

[9]  Debasish Roy,et al.  Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam , 2008 .

[10]  Yaoyao Shi,et al.  C2-Continuous Orientation Planning for Robot End-Effector with B-Spline Curve Based on Logarithmic Quaternion , 2020, Mathematical Problems in Engineering.

[11]  Peilin Hong,et al.  Smooth orientation interpolation using parametric quintic-polynomial-based quaternion spline curve , 2018, J. Comput. Appl. Math..

[12]  Research on tool path planning for five-axis machining , 2009, 2009 IEEE International Conference on Industrial Engineering and Engineering Management.

[13]  Olivier A. Bauchau,et al.  Interpolation of rotation and motion , 2014 .

[15]  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.

[16]  Andreas Müller,et al.  Smooth orientation path planning with quaternions using B-splines , 2015, 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

[17]  Daniel Pletinckx,et al.  Quaternion calculus as a basic tool in computer graphics , 2005, The Visual Computer.

[18]  V. Bolotnikov.  Polynomial interpolation over quaternions , 2014, 1405.3707.

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

[20]  Ignacio Romero,et al.  The interpolation of rotations and its application to finite element models of geometrically exact rods , 2004 .

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

[22]  Giovanni Legnani,et al.  Kinematics Analysis of a Class of Spherical PKMs by Projective Angles , 2018, Robotics.

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

[24]  Erik B. Dam,et al.  Quaternions, Interpolation and Animation , 2000 .

[25]  W. Hamilton.  II. On quaternions; or on a new system of imaginaries in algebra , 1844 .

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