.

and , respectively. It can now be seen directly that the transformation given in Equation (2.58) is equivalent to the (homogeneous) matrix equation

      (2.70)

      A set of basic homogeneous transformations generating SE(3) is given by

      (2.71)

      (2.72)

      (2.73)

for translation and rotation about the x, y, z-axes, respectively.

      The most general homogeneous transformation that we will consider may be written now as

      (2.74)

      In the above equation n = (nx, ny, nz) is a vector representing the direction of x₁ in the o₀x₀y₀z₀ frame, s = (sx, sy, sz) represents the direction of y₁, and a = (ax, ay, az) represents the direction of z₁. The vector d = (dx, dy, dz) represents the vector from the origin o₀ to the origin o₁ expressed in the frame o₀x₀y₀z₀. The rationale behind the choice of letters n, s, and a is explained in Chapter 3.

      The same interpretation regarding composition and ordering of transformations holds for 4 × 4 homogeneous transformations as for 3 × 3 rotations. Given a homogeneous transformation H⁰₁ relating two frames, if a second rigid motion, represented by H ∈ SE(3) is performed relative to the current frame, then

      whereas if the second rigid motion is performed relative to the fixed frame, then

       Example 2.10.

      The homogeneous transformation matrix that represents a rotation by angle α about the current x-axis followed by a translation of b units along the current x-axis, followed by a translation of d units along the current z-axis, followed by a rotation by angle θ about the current z-axis, is given by

      2.6.2 Exponential Coordinates for General Rigid Motions

Just as we represented rotation matrices as exponentials of skew-symmetric matrices, we can also represent homogeneous transformations as exponentials using so-called twists.

       Definition 2.3.

      Let v and k be vectors in with k a unit vector. A twist ξ defined by k and v is the 4 × 4 matrix

      (2.75)

      We define se(3) as

      (2.76)

      se(3) is the vector space of twists, and a similar argument as before in Section 2.5.4 can be used to show that, given any twist ξ ∈ se(3) and angle , the matrix exponential of ξθ is an element of SE(3) and, conversely, every homogeneous transformation (rigid motion) in SE(3) can be expressed as the exponential of a twist. We omit the details here.

      2.7 Chapter Summary

      In this chapter, we have seen how matrices in SE(n) can be used to represent the relative position and orientation of two coordinate frames for n = 2, 3. We have adopted a notional convention in which a superscript is used to indicate a reference frame. Thus, the notation represents the coordinates of the point p relative to frame 0.

      The relative orientation of two coordinate frames can be specified by a rotation matrix, R ∈ SO(n), with n = 2, 3. In two dimensions, the orientation of frame 1 with respect to frame 0 is given by

      in which θ is the angle between the two coordinate frames. In the three-dimensional case, the rotation matrix is given by

In each case, the columns of the rotation matrix are obtained by projecting an axis of the target frame (in this case, frame 1) onto the coordinate axes of the reference frame (in this case, frame 0).

      The set of n × n rotation matrices is known as the special orthogonal group of order n, and is denoted by SO(n). An important property of these matrices is that R^{− 1} = RT for any R ∈ SO(n).

      Rotation matrices can be used to perform coordinate transformations between frames that differ only in orientation. We derived rules for the composition of rotational transformations as

for the case where the second transformation, R, is performed relative to the current frame and

      for the case where the second transformation, R, is performed relative to the fixed frame.

      In the three-dimensional case, a rotation matrix can be parameterized using three angles. A common convention is to use the Euler angles (ϕ, θ, ψ), which correspond to successive rotations about the z, y, and z-axes. The corresponding rotation matrix is given by

      Roll, pitch, and yaw angles are similar, except that the successive rotations are performed with respect to the fixed, world frame instead of being performed with respect to the current frame.

      Homogeneous