![]() This we refer to as a homogeneous transformation. It's got A and B and an additional element which is set to 1. The homogeneous vector has got three elements in it. We have the familiar representation of a vector in a 2-dimensional plane, two numbers A and B. The elements of the matrix include the rotation matrix, which is the function of the orientation Theta, and the translation between the origin of coordinate frame A and coordinate frame B. The point is described with respect to coordinate frame A by a homogeneous vector which we can obtain from the homogeneous vector of the point with respect to coordinate frame B multiplied by this matrix. It has got three elements not two, and we'll talk about that in just a moment. The notation, the tilde, above the little p indicates that it is a homogeneous factor. We can write this even more succinctly in this form. And our vectors which is to have 2 elements now have three elements. Now we see that that's embedded into the top left corner of a 3x3 matrix. What we've down now is added some zeros and some ones to the bottom row of the matrix. Let's make things a little bit more symmetric and write it like this. Now this expression has got a two-element vector and a three-element vector over here. If you're unsure about this depth, expand out the matrix and convince yourself that they are exactly the same. Now I can pull a little bit of the trick and now write it like this. Let's expand the vectors and matrix in this expression and write them in terms of their elements. And again because these two coordinate frames are parallel I can write this. I can substitute one equation into the other and write this. ![]() We can do that because these two vectors are described with respect to coordinate frames whose axis are parallel to one another. Now we're allowed to add vectors so we can add this vector ATV to the vector VP to obtain the vector that describes the point P with respect to coordinate frame A. Now let's describe the origin of coordinate frame V with respect to coordinate frame A and we do that with a vector. And as a sanity check that we've written this expression correctly, we noticed that these two Bs are next to each other. #Webots compound motion rotate translation how to#We've already talked about how to rotate a vector from one coordinate frame to another using a rotation matrix. ![]() Now we can rotate that vector into a new frame, coordinate frame V, and coordinate frame V has axis which are parallel to coordinate frame A. We can represent the point P by a vector with respect to frame B and that vector can be described in terms of its components along the XB and YB axis. Coordinate frame A, Coordinate frame B which is translated and rotated with respect to coordinate frame A and a point P. Now let's bring translation into the picture. ![]()
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