1 Composite Rotations

Two frames A and B are initially coincident. Frame B then undergoes the following sequence of transformations:

1. a rotation of /4 about the y axis (fixed);
2. a rotation of /2 about the x axis (fixed);
3. a rotation of /6 about the z axis (moving);
4. a rotation of /3 about the x axis (fixed);
5. a rotation of /3 about the y axis (moving).

Write the final rotation matrix ^A_BR describing the orientation of B with respect to A.

Note: you do not need to compute the final matrix by performing all intermediate multiplications. All that matters here is the order, so you can leave matrices in their symbolic form (as long as it is correct).

2 Transformation Matrices

Two frames A and B are initially coincident. Frame B then undergoes the following transformations:

1. a rotation of /2 about the x axis;
2. a translation of 3 units about the y;
3. a rotation of /2 about the z axis (fixed frame).

Write the transformation matrices ^A_BT and ^B_AT.

3 Quaternions to Rotations

Let q = a + bi + cj + dk be a unit quaternion. In the lecture notes it is stated that its associated rotation matrix is

R .

Show that R is a rotation matrix.

4 Change of Coordinates

Three robots are operating in a shared space. Let A, B, and C the three frames attached to the robots, and let W be a world frame. Assume that the following transformation matrices are known: ^BT, ^C_WT, ^B_CT, ^W_B T Assume robot A perceives a point of interest whose coordinates are ^Ap. Can _A

you determine any of the following: ^Bp, ^Cp. ^Wp? For each of the required points, if the answer is positive, show how it can be computed, and if the answer is negative explain why it cannot be computed.

5 Quaternions

Quaternions can be multiplied following rules similar to those we follow for complex numbers. A fundamental thing to remember is that quaternions product is not commutative. When multiplying two quaternions, keep in mind the following definitions about products between their imaginary coefficients i,j,k:

• i² = j² = k² = ijk = 1
• ij = k, ji = k
• jk = i, kj = i
• ki = j, ik = j

Consider the following two quaternions:

• = 1 + 2i 3k
• = 5 + 4j + 2

Compute:

1. the product pq.
2. the norm of the product pq.

Note: show the intermediate steps; if you just write the result, you will not get any point.