Tham khảo tài liệu 'advanced robotics - control of interactive robotic interfaces volume 29 part 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Lie Groups and Rigid Motions 209 It is well known that the configuration of a rigid body in the space can be described using homogeneous coordinates by the homogeneous matrix h h ỵooo ụ -A 2i where R is a 3 X 3 orthogonal matrix with determinant equal to 1 which represents the orientation of the rigid body while p is a vector that represent its position. The set of matrices of this form is called special Euclidean group in R3 and is denoted by SE 3 . It can be easily seen that the set SE 3 has the structure of manifold. The structure of group can be checked taking as operation the matrix multiplication. Thus SE A is both a manifold and a group and therefore it is a Lie group. Consider two rigid bodies i and j the relative configuration of i and j is represented by an homogeneous matrix hị. The Lie algebra of SE 3 is denoted by se 3 and it is the set of matrices of the form ỵooo oj A 22 1 where w is 3 X 3 skew-symmetric matrix and V G R3. The Lie brackets of the Lie algebra are given by A B AB - BA A B e se 3 An element of se 3 is called a twist. The vector space se 3 has dimension 6 and is isomorphic to R6. The relative motion of two rigid bodies can be described by a curve hị í on SE 3 . The generalized relative velocity is hị G ThjyySEtA . The most general instantaneous motion of a rigid body in the space is a screw motion Chasles theorem 290 272 . a rototranslation around an instantaneous axis in the space. Thanks to the tangent maps of the left or right translation map it is possible to describe the generalized velocity through a twist. In particular tị G se 3 represents the instantaneous velocity of the motion the skew symmetric matrix w represent the rotation around the instantaneous axis and V the velocity along the instantaneous axis. Dually the most general instantaneous system of forces that can be applied to a rigid body is given by a pure momentum and a pure translation around an instantaneous spatial axis Poinsot s theorem 290
