3D Kinematics Consists of two parts 3D rotation 3D translation  The same as 2D 3D rotation is more complicated than 2D rotation (restricted to z- axis) Next, we will focus on the spatial (3D) rotation 2D (rigid body) kinematics C.M. translation and rotation Fall 20121

3D Rotation Representations Euler angles Axis-angle 3X3 rotation matrix Unit quaternion Learning Objectives Representation (uniqueness) Perform rotation Composition Interpolation Conversion among representations … Fall 20122

Euler Angles Specify orientation in rotation along 3 axes Variation: which axes? global or local? Fall 20123

   Write a program … From Mason’s book Fall 20124

Euler Angles Roll, pitch, yaw Ref: Gimbal lock: reduced DOF due to overlapping axes Why gimbal lock a problem? Fall 20125

Axis-Angle Representation Rot(n,  ) n: rotation axis (global)  : rotation angle (rad. or deg.) follow right-handed rule Rot(n,  )=Rot (-n,-  ) Problem with null rotation: rot(n,0), any n Perform rotation Rodrigues formula Interpolation/Composition: poor Rot(n 2,  2 )Rot(n 1,  1 ) =?= Rot(n 3,  3 ) Fall 20126

Rodrigues Formula v ’ =R v  r v v’v’ References: ml Fall 20127

Rotation Matrix Meaning of three columns Perform rotation: linear algebra Composition: trivial orthogonalization might be required due to FP errors Interpolation: ? Fall 20128

Gram-Schmidt Orthogonalization If 3x3 rotation matrix no longer orthonormal, metric properties might change! Verify! Fall 20129

Quaternion A mathematical entity invented by Hamilton Definition i j k Fall

Quaternion (cont) Operators Addition Multiplication Conjugate Length Fall

Unit Quaternion Define unit quaternion as follows to represent rotation Example Rot(z,90°)  q and – q represent the same rotation Why “ unit ” ? DOF point of view! Fall

quaternion → axis-angle Fall Use both values of sine and cosine to determine the angle!!

Ex: q and –q are the same! Fall

Example x y z x y z Rot (90, 0,0,1) OR Rot (-90,0,0,-1) Fall

Unit Quaternion (cont) Perform Rotation Composition Interpolation Linear Spherical linear Fall

Example x y,x ’ z,z ’ y’y’ Rot(z,90°) p(2,1,1) Fall

Example (cont) Fall

Example x y,x ’ z,z ’ y’y’ x,x ’ y z,y ’ z’z’ Fall

Matrix Conversion Fall

Matrix Conversion (cont) Find largest qi 2 ; solve the rest Fall

Slerp (Spherical Linear Interpolation) The computed rotation quaternion rotates about a fixed axis at constant speed References: q r unit sphere in R 4 Fall

Spatial Displacement Any displacement can be decomposed into a rotation followed by a translation Matrix Quaternion Fall

   Write a program … From Mason’s book Fall

   y z x x’’’ y’’’ z’’’ Fall

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From previous page Fall

From Lee and Koh (1995) Euler angles in ASF In v’=Mv convention Fall