Explicit Representation of the Orientation: Exponential Coordinates

Explicit Representation of the Orientation: Exponential Coordinates

In the previous lesson, we became familiar with rotation matrices, and we saw that the nine-dimensional space of rotation matrices subject to six constraints (three unit norm constraints and three orthogonality constraints) could be used to implicitly represent the three-dimensional space of orientations. There are also methods to express the orientation with a minimum number of parameters (three in three-dimensional space). Exponential coordinates that define an axis of rotation and the angle rotated about that axis, the three-parameter Euler angles, the three-parameter roll-pitch-yaw angles, the Cayley-Rodrigues parameters, and the unit quaternions (use four variables subject to one constraint) are some of…
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Implicit Representation of the Orientation: a Rotation Matrix

Implicit Representation of the Orientation: a Rotation Matrix

In the previous lesson, we became familiar with the concept of the configuration for the robots, and we saw that the configuration of a robot could be expressed by the pair (R,p) in which R is the rotation matrix that implicitly represents the orientation of the body frame with respect to the reference frame and p is the position of the origin of the body frame relative to the space frame. In this lesson, we will focus on the orientation, and we will see that we can implicitly represent the orientation using powerful tools named rotation matrices, and we will…
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