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.
