## Description

1 Composite Rotations

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

transformations:

1. a rotation of π/4 about the y axis (fixed);

2. a rotation of π/2 about the x axis (fixed);

3. a rotation of π/6 about the z axis (moving);

4. a rotation of π/3 about the x axis (fixed);

5. a rotation of π/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 π/2 about the x axis;

2. a translation of 3 units about the y;

3. a rotation of π/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 =

2(a

2 + b

2

) − 1 2(bc − ad) 2(bd + ac)

2(bc + ad) 2(a

2 + c

2

) − 1 2(cd − ab)

2(bd − ac) 2(cd + ab) 2(a

2 + d

2

) − 1

.

Show that R is a rotation matrix4 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:

B

AT,

C

W T,

B

CT, W

B T Assume robot A perceives a point of interest whose coordinates are Ap. Can

you determine any of the following: Bp,

Cp. W p? 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:

• i

2 = j

2 = k

2 = ijk = −1

• ij = k, ji = −k

• jk = i, kj = −i

• ki = j, ik = −j

Consider the following two quaternions:

p = 1 + 2i − 3k

q = 5 + 4j + 2k.

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.