### Other Explicit Representation for the Orientation in Robotics: Euler Angles

In the lesson about the degrees of freedom of a robot, we learned that there are at least three independent parameters needed to express the orientation of a rigid body. In the previous lesson, we learned about the exponential coordinate representation for the orientation that is a three-parameter representation for a rotation matrix R, and parameterizes the rotation matrix using a unit axis of rotation and the angle of rotation about this axis. There are also other explicit representations that are useful in different applications when dealing with orientations. In this lesson, we will talk about Euler Angles and will…

### 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…