The third chapter three-dimensional rigid motion description method: using a rotation matrix, the transformation matrix, quaternions, Euler angles and Eigen library. Because time is tight this week, look sketchy, if error also educated us, grateful.
The following records are scattered and fragmentary knowledge-based, finishing the line in the future.
1. The outer product of two vectors a X b can be seen as a multiplication of matrix and vector. Wherein a becomes a symmetric matrix.
2. Great Euler
(1) a vector of coordinates in the robot coordinate system to the world coordinate system coordinate conversion may have a translation plus a rotation composition. (Rigid body movement)
(2) rotation matrix. (Formula complement later) which each of the components of the two coordinate systems within the product groups.
SO (N) orthogonal group specific. n-dimensional. This set of n-dimensional space rotation matrix configuration.
(3) and a homogeneous coordinate transform matrix.
Add 1 end of the three-dimensional vector into a four-dimensional vector, that is, homogeneous coordinates. Then according to a '= Ra + t, derived transformation matrix.
SE (N) special Euclidean group. Rotating the upper left corner to the right translation vector, the lower right corner of 1 ;.
3. Euler angles and the rotation vector
(1) Problem: SO (3) rotation matrix nine amount, but only a rotation of 3 degrees of freedom, so that the redundant expression. 16 represents the amount of transformation matrices six degrees of freedom.
Rotation matrix must be an orthogonal matrix, determinant 1 with its own constraints, optimization estimation difficult.
Solution: therefore proposed with a rotational shaft and a rotation angle of the characterization transform.
A three-dimensional vector coincides with the direction of the rotary shaft, the length of the rotation angle, the rotation vector is the vector. And then a three-dimensional vector represented translation corresponds exactly six-dimensional transformation matrix.
Rodriguez formula:
Described rotation vector and rotation matrix R represents a relationship between the method 0n (where 0 represents the angle of game play it convenient .ubuntu below)
(2) Euler angles
yaw- pitch- roll;
Yaw, pitch, roll angle
z-axis x-axis y-axis
Gimbal lock problem (to be learning, teaching behind attached video link), that the singularity problem.
2D applies to the case with only one of the positioning.
4. quaternion
Multiplication rule using the properties of the complex domain. Slightly basic nature
p '= QPQ -1 where p is a point in space, represented by a dotted quaternion.
Remove the imaginary part p 'can be obtained coordinates of the point of rotation.