Smooth Interpolation of Orientation by Rolling and Wrapping for Robot Motion Planning Yueshi Shen ∗ , Knut H¨ uper † , and F´ atima Silva Leite ‡ ∗ Department of Information Engineering, Research School of Information Sciences and Engineering Australian National University, Canberra ACT 0200 Australia, Email: yueshi.shen@anu.edu.au † National ICT Australia Limited, Systems Engineering And Complex Systems Program Locked Bag 8001, Canberra ACT 2601 Australia, Email: knut.hueper@nicta.com.au ‡ Department of Mathematics and Institute of Systems and Robotics University of Coimbra, Coimbra 3001-454 Portugal, Email: fleite@mat.uc.pt Abstract— This paper investigates a novel procedure to cal- culate smooth interpolation curves of the rotation group SO3, which is commonly considered as the standard representation of rigid-body’s orientations. The algorithm is a combination of rolling and wrapping with the pull back/push forward technique. One remarkable advantage of this approach is that interpolation curves will be given in closed form, which brings convenience for implementations on real-time control systems. A numerical example along with some visualization results is presented as well. I. I NTRODUCTION Smooth interpolation in Euclidean spaces has many applica- tions in robot motion planning (for example, interpolation of end-effector’s linear position, or of joint traversing points), and has been well studied ever since the outset of robotics research. The usual approach for solving this category of problems is to apply cubic splines or even higher order polynomials, whose coefficients can be calculated through a linear system [1][2]. On the other hand, smooth interpolation in non-Euclidean spaces has comparatively attracted less roboticists’ attentions, although it is in fact an interesting theoretical problem with applications to path planning for mechanical systems whose configuration spaces contain components which are Lie groups or symmetric spaces. For example, smooth interpolation of orientation (namely, the rotation or Special Orthogonal group SO 3 ) is particularly useful in robotics (e.g., motion planning of rigid body); computer graphics (e.g., animation of 3D objects); satellite attitude control; and so on. As far as we notice, most of the existing literature on SO 3 curve design can be classified into two major methodologies: i) extension of the De Casteljau algorithm; and ii) coordinate parametrization. The first class of work focuses on gener- alization of B´ ezier curves for Lie groups. Park and Ravani † National ICT Australia is funded by the Australian Department of Communications, Information Technology and the Arts and the Australian Research Council through Backing Australia’s Ability and the ICT Centre of Excellence Programs. ‡ This work was initiated while the third author visited NICTA in early 2005. Financial support from Calouste Gulbenkian Foundation, ISR-Coimbra, and NICTA is acknowledged. show how the De Casteljau algorithm can be extended to Riemannian manifolds, and the mathematical elegance of Lie groups is utilized for constructing B´ ezier curves efficiently [3]. Later Crouch et al. suggest a modified De Casteljau algorithm to generate cubic splines on connected and compact Lie groups. Details for SO 3 appear in [4]. However, De Casteljau-like algorithms are in general com- putationally expensive and quite cumbersome for interpolation applications [5]. More recent research has adopted a different approach for SO 3 ’s interpolation, which first re-parameterizes rotation matrices (by rotation axes and angles, for instance), then performs cubic spline interpolation based on such repre- sentations. Kang and Park’s paper collects a few popular SO 3 interpolation algorithms of this class, also the trajectory dis- tortion caused by various coordinate parametrization schemes has been comparatively studied and discussed [5]. To eliminate the distortion of interpolation curves intro- duced by local diffeomorphisms (i.e., by the process of simple pull back/push forward [6]), H¨ uper and Silva Leite recently proposed a novel interpolation method combining pull back/push forward with rolling and wrapping on smooth manifolds, in particular on S 2 [7]. Although many of the ideas in [7] can be directly applied here, contrary to the two-sphere situation, the geometric intuition of SO 3 is less obvious. Building on [8], the contribution of this paper is to specialize the previous work to SO 3 . The rest of the paper is organized as follows: Section 2 gives the mathematical formulation of our SO 3 interpolation problem; Section 3 introduces a few relevant differential ge- ometry concepts and then presents the major algorithm which calculates an interpolation curve of SO 3 , given in an explicit form; Section 4 shows a numerical example of interpolating three orientations, and some visualization results are attached at the end of this paper. II. PROBLEM DESCRIPTION As we know, SO 3 can be considered as a smooth 3-dim sub-manifold of R 3×3 . Therefore, for all R ∈ SO 3 , the affine tangent space T aff R SO 3 can be considered as an affine subspace of R 3×3 (see Fig.1). Proceedings of the 2006 IEEE International Conference on Robotics and Automation Orlando, Florida - May 2006 0-7803-9505-0/06/$20.00 ©2006 IEEE 113