ABSTRACT
Department of Mathematics and Statistics, Universite´ Laval, Que´bec, Canada
Karim Oualkacha
Department of Mathematics, Universite´ du Que´bec A` Montre´al, Montre´al, Canada
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2 The One Axis Model in SE(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2.1 Rotation Matrices and Cardan Angles in SO(3) . . . . . . . . 113 6.2.2 A Geometric Construction of the One Axis Model . . . . . 113
6.3 Modeling Data from SE(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.4 Estimation of the Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4.1 The Rotation Only Estimator of the Rotation Axis A3 and B3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4.2 The Translation Only Estimator of the Parameters . . . . 117 6.4.3 The Rotation-Translation Estimator of the Parameters 118
6.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.5.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.5.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
This chapter considers the modeling of rotation translation pairs, (R, t), where R is an 3×3 rotation matrix and t ∈ <3 is a 3-dimensional translation vector. In rigid body kinematics, such a pair is used to characterize the position of a rigid object; t and R, respectively, give its location and its orientation. The set of all these pairs is denoted SE(3), see [19]. There are special cases of interest, the pure rotation (R, 0) and the pure translation (I3, t), where I3 is the 3×3 identity matrix. The set of 3×3 rotation matrices is denoted SO(3).
