Читать книгу Robot Modeling and Control - Mark W. Spong - Страница 56

The projection technique described above scales nicely to the three-dimensional case. In three dimensions, each axis of the frame o₁x₁y₁z₁ is projected onto coordinate frame o₀x₀y₀z₀. The resulting rotation matrix R ∈ SO(3) is given by

As was the case for rotation matrices in two dimensions, matrices in this form are orthogonal, with determinant equal to 1 and therefore elements of SO(3).

Example 2.1.

Suppose the frame o₁x₁y₁z₁ is rotated through an angle θ about the z₀-axis, and we wish to find the resulting transformation matrix . By convention, the right hand rule (see Appendix B) defines the positive sense for the angle θ to be such that rotation by θ about the z-axis would advance a right-hand threaded screw along the positive z-axis.

From Figure 2.3 we see that

and

while all other dot products are zero. Thus, the rotation matrix has a particularly simple form in this case, namely

(2.3)

Figure 2.3 Rotation about z₀ by an angle θ.

The rotation matrix given in Equation (2.3) is called a basic rotation matrix (about the z-axis). In this case we find it useful to use the more descriptive notation instead of to denote the matrix. It is easy to verify that the basic rotation matrix has the properties

(2.4)

(2.5)

which together imply

(2.6)

Similarly, the basic rotation matrices representing rotations about the x and y-axes are given as (Problem 2–8)

(2.7)

(2.8)

which also satisfy properties analogous to Equations (2.4)–(2.6).

Example 2.2.

Consider the frames o₀x₀y₀z₀ and o₁x₁y₁z₁ shown in Figure 2.4.

Projecting the unit vectors x₁, y₁, z₁ onto x₀, y₀, z₀ gives the coordinates of x₁, y₁, z₁ in the o₀x₀y₀z₀ frame as

The rotation matrix specifying the orientation of o₁x₁y₁z₁ relative to o₀x₀y₀z₀ has these as its column vectors, that is,

Figure 2.4 Defining the relative orientation of two frames.