Robot Modeling and Control. Mark W. Spong

      and

(2.46)

      These equations can be obtained by direct manipulation of the entries of the matrix given in Equation (2.44). The axis-angle representation is not unique since a rotation of − θ about − k is the same as a rotation of θ about k, that is,

      (2.47)

      If θ = 0 then is the identity matrix and the axis of rotation is undefined.

       Example 2.9.

      Suppose is generated by a rotation of 90° about z₀ followed by a rotation of 30° about y₀ followed by a rotation of 60° about x₀. Then

(2.48)

We see that and hence the equivalent angle is given by Equation (2.46) as

(2.49)

      The equivalent axis is given from Equation (2.46) as

(2.50)

      The above axis-angle representation characterizes a given rotation by four quantities, namely the three components of the equivalent axis k and the equivalent angle θ. However, since the equivalent axis k is given as a unit vector only two of its components are independent. The third is constrained by the condition that k is of unit length. Therefore, only three independent quantities are required in this representation of a rotation . We can represent the equivalent axis-angle by a single vector r as

(2.51)

      Note, since k is a unit vector, that the length of the vector r is the equivalent angle θ and the direction of r is the equivalent axis k.

      One should be careful to note that the representation in Equation (2.51) does not mean that two axis-angle representations may be combined using standard rules of vector algebra, as doing so would imply that rotations commute which, as we have seen, is not true in general.

      2.5.4 Exponential Coordinates

      In this section we introduce the so-called exponential coordinates and give an alternate description of the axis-angle transformation (2.44). We showed above in Section 2.5.3 that any rotation matrix R ∈ SO(3) can be expressed as an axis-angle matrix R_{k, θ} using Equation (2.44). The components of the vector are called exponential coordinates of R.

      To see why this terminology is used, we first recall from Appendix B the definition of so(3) as the set of 3 × 3 skew-symmetric matrices S satisfying

      (2.52)

      For let S(k) be the skew-symmetric matrix

      (2.53)

and let e^S(k)θ be the matrix exponential as defined in Appendix B

(2.54)

      Then we have the following proposition, which gives an important relationship between SO(3) and so(3).

       Proposition 2.1

      The matrix e^S(k)θ is an element of SO(3) for any S(k) ∈ so(3) and, conversely, every element of SO(3) can be expressed as the exponential of an element of so(3).

      Proof: To show that the matrix e^S(k)θ is in SO(3) we need to show that e^S(k)θ is an orthogonal matrix with determinant equal to + 1. To show this we rely on the following properties that hold for any n × n matrices A and B

      The first two properties above can be shown by direct calculation using the series expansion (2.54)

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