Robot Modeling and Control. Mark W. Spong

      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 transformations combine rotation and translation. 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 ∈

Скачать книгу