Date: August, 2017. Location: Quito, Pichincha, Ecuador.
|Lunes 14, 8:31 am
|We’re gonna be talking about rigid bodies. The first thing you wanna know about rigid bodies is besides the fact that they’re rigid, you wanna think about modelling them. And in order to model them, we attach reference frames to them. So here I show reference frame A, which is essentially an origin O and a set of axes, x, y, and z. We would also need a set of unit vectors that are parallel to these axes, a1, a2, and a3. So these vectors are mutually orthogonal.|
And in fact, it’s these vectors that are more important than the axes. In fact, I’m gonna get rid of those axes and just keep the unit vectors, a1, a2, and a3. You have to remember that these vectors are attached to the rigid body as is the origin O.
Likewise for frame B, we have another set of unit vectors, b1, b2,and b3. And an origin P that’s attached to the rigid body.
So, we have two sets of basis vectors. Each set of basis vectors consists of mutually orthogonal unit vectors. The as are attached to the frame A, the bs are attached to the frame B.
Now I have a different set of components, q1 prime, q2 prime and q3 prime. Clearly, they are different from q1, q2, and q3 because b1, b2, and b3 are different from a1, a2, and a3. And yet, we wanna find a relationship between q1, q2, and q3 on one side and q1 prime, q2 prime, and q3 prime on the other side. This relationship is called a rigid body transformation because you’re talking about the same point. And looking at it from the vantage point of two different frames, each frame attached to a different rigid body.
So how do we relate q1, q2, and q3 to q1 prime, q2 prime, and q3 prime? Well, you can write down the vectors pictorially, and you see that immediately that picture suggests that you can use the triangle law of vector addition. The vector from O to Q is simply the sum of the vector from O to P, and the vector from P to Q. I can write this down in terms of components, as we’ve discussed before.
So now, I have a vector equation.I could even try to write this equation in terms of 3 by 1 vectors.
But you cannot simply add these 3 by 1 vectors.
Instead, you should take the vector of components q1 prime, q2 prime, and q3 prime, pre-multiplied by suitable transformation matrix, so that the resulting set of components is along a1, a2, and a3. Well, this transformation is essentially due to a rotation. And the matrix in front is a 3 by 3 rotation matrix. It’s denoted by the boldface symbol R with subscripts A and B. Suggesting that you are transforming components from frame B into frame A.
How do you write the components of a rotation matrix? Well, it’s a 3 by 3 matrix, and if you look carefully at the vector equation and the equation with 3 by 1 vectors, you can see that the rotation matrix is simply a collection of dot products or scalar products. You’re taking all possible combinations of the basis vectors, b1, b2, and b3, with the basis vectors a1, a2, and a3. In fact, if you look at the first row, it is simply the components of the basis vector b1 written in frame A.
Now we’re gonna collapse everything into a single matrix, the homogenous transformation matrix. Again, the same equation that essentially describes the triangle law of vector addition in terms of components, components written in terms of a1, a2, and a3. Let’s use homogeneous coordinates where we append the regular xyz coordinates by the number 1 as the fourth element. So this set of four numbers essentially give you a vector which is a representation of the position vector, but in projector coordinates. To relate these two sets of 4 by 1 vectors, all you need is a homogenous transformation matrix that includes elements of the rotation matrix that we’ve just described and the translation from O to P given by the components p1, p2, and p3. The last row, which consists of 0s and 1s, is simply inserted to make sure that the matrix multiplication reflects the triangle law of vector addition.
Well, this 4 by 4 matrix, we’re gonna denote by the boldface symbol T with subscripts A and B. And again, the subscripts denote the fact that you’re transforming position vectors from the second letter, B, to the first letter, A. This is our 4 by 4 homogeneous transformation matrix.
For an example see: Transformation example in Matlab
Written by: Larry Francis Obando – Technical Specialist
Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.
Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.
Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.
Contact: Ecuador (Quito, Guayaquil, Cuenca)
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