In the three-dimensional case, a homogeneous transformation has the form
The set of all such matrices comprises the set SE(3), and these matrices can be used to perform coordinate transformations, analogous to rotational transformations using rotation matrices.
Homogeneous transformation matrices can be used to perform coordinate transformations between frames that differ in orientation and translation. We derived rules for the composition of rotational transformations as
for the case where the second transformation, H, is performed relative to the current frame and
for the case where the second transformation, H, is performed relative to the fixed frame.
We also defined the vector spaces
and showed that elements of SO(3) and SE(3) can be expressed as matrix exponentials of elements of so(3) and se(3). Formally, SO(3) and SE(3) are Lie groups and so(3) and se(3) are their associated Lie algebras.
Problems
1 Using the fact that v1 · v2 = vT1v2, show that the dot product of two free vectors does not depend on the choice of frames in which their coordinates are defined.
2 Show that the length of a free vector is not changed by rotation, that is, that ‖v‖ = ‖Rv‖.
3 Show that the distance between points is not changed by rotation, that is, ‖p1 − p2‖ = ‖Rp1 − Rp2‖.
4 If a matrix R satisfies RTR = I, show that the column vectors of R are of unit length and mutually perpendicular.
5 If a matrix R satisfies RTR = I, thena) Show that b) Show that if we restrict ourselves to right-handed coordinate frames.
6 Verify Equations (2.4)–(2.6).
7 A group is a set X together with an operation * defined on that set such thatx1*x2 ∈ X for all x1, x2 ∈ X(x1*x2)*x3 = x1*(x2*x3)There exists an element I ∈ X such that I*x = x*I = x for all x ∈ XFor every x ∈ X, there exists some element y ∈ X such that x*y = y*x = IShow that SO(n) with the operation of matrix multiplication is a group.
8 Derive Equations (2.7) and (2.8).
9 Suppose A is a 2 × 2 rotation matrix. In other words ATA = I and . Show that there exists a unique θ such that A is of the form
10 Consider the following sequence of rotations:Rotate by ϕ about the world x-axis.Rotate by θ about the current z-axis.Rotate by ψ about the world y-axis.Write the matrix product that will give the resulting rotation matrix (do not perform the matrix multiplication).
11 Consider the following sequence of rotations:Rotate by ϕ about the world x-axis.Rotate by θ about the world z-axis.Rotate by ψ about the current x-axis.Write the matrix product that will give the resulting rotation matrix (do not perform the matrix multiplication).
12 Consider the following sequence of rotations:Rotate by ϕ about the world x-axis.Rotate by θ about the current z-axis.Rotate by ψ about the current x-axis.Rotate by α about the world z-axis.Write the matrix product that will give the resulting rotation matrix (do not perform the matrix multiplication).
13 Consider the following sequence of rotations:Rotate by ϕ about the world x-axis.Rotate by θ about the world z-axis.Rotate by ψ about the current x-axis.Rotate by α about the world z-axis.Write the matrix product that will give the resulting rotation matrix (do not perform the matrix multiplication).
14 If the coordinate frame o1x1y1z1 is obtained from the coordinate frame o0x0y0z0 by a rotation of about the x-axis followed by a rotation of about the fixed y-axis, find the rotation matrix R representing the composite transformation. Sketch the initial and final frames.
15 Suppose that three coordinate frames o1x1y1z1, o2x2y2z2, and o3x3y3z3 are given, and suppose Find the matrix .
16 Derive equations for the roll, pitch, and yaw angles corresponding to the rotation matrix R = (rij).
17 Verify Equation (2.44).
18 Verify Equation (2.46).
19 If is a rotation matrix show that + 1 is an eigenvalue of . Let k be a unit eigenvector corresponding to the eigenvalue + 1. Give a physical interpretation of k.
20 Let , θ = 90°. Find .
21 Show by direct calculation that given by Equation (2.44) is equal to given by Equation (2.48) if θ and k are given by Equations (2.49) and (2.50), respectively.
22 Compute the rotation matrix given by the product
23 Suppose represents a rotation of 90° about y0 followed by a rotation of 45° about z1. Find the equivalent axis-angle to represent . Sketch the initial and final frames and the equivalent axis vector k.
24 Find the rotation matrix corresponding to the Euler angles θ = 0, and . What is the direction of the x1 axis relative to the base frame?
25 Unit magnitude complex numbers a + ib with a2 + b2 = 1 can be used to represent orientation in the plane. In particular, for the complex number a + ib, we can define the angle θ = Atan2(a, b). Show that multiplication of two complex numbers corresponds to addition of the corresponding angles.
26 Show that complex numbers together with the operation of complex multiplication define a group. What is the identity for the group? What is the inverse for a + ib?
27 Complex numbers can be generalized by defining three independent square roots for − 1 that obey the multiplication rules Using these, we define a quaternion by Q = q0 + iq1 + jq2 + kq3, which is typically represented by the 4-tuple (q0, q1, q2, q3). A rotation by θ about the unit vector n = (nx, ny, nz) can be represented by the unit quaternion . Show that such a quaternion has unit norm, that is, q20 + q21 + q22 + q23 = 1.
28 Using , and the results from Section 2.5.3, determine the rotation matrix R that corresponds to the rotation represented by the quaternion (q0, q1, q2, q3).
29 Determine the quaternion Q that represents the same rotation as given by the rotation matrix R.
30 The quaternion Q = (q0, q1, q2, q3) can be thought of as having a scalar component q0 and a vector component q = (q1, q2, q3). Show that the product of two quaternions, Z = XY is given by Hint: Perform the multiplication (x0 + ix1 + jx2 + kx3)(y0 + iy1 + jy2 + ky3) and simplify the result.
31 Show that QI = (1, 0, 0, 0) is the identity element for unit quaternion multiplication, that is, QQI = QIQ = Q for any unit quaternion Q.
32 The conjugate Q* of the quaternion Q is defined as Show that Q* is the inverse of Q, that is, Q*Q = QQ* = (1, 0, 0, 0).
33 Let v be a vector whose coordinates are given by (vx, vy, vz).