Robot Modeling and Control. Mark W. Spong

      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.

      2.3 Rotational Transformations

Figure 2.5 shows a rigid object S to which a coordinate frame o₁x₁y₁z₁ is attached. Given the coordinates of the point p (in other words, given the coordinates of p with respect to the frame o₁x₁y₁z₁), we wish to determine the coordinates of p relative to a fixed reference frame o₀x₀y₀z₀. The coordinates satisfy the equation

Figure 2.5 Coordinate frame attached to a rigid body.

      In a similar way, we can obtain an expression for the coordinates by projecting the point p onto the coordinate axes of the frame o₀x₀y₀z₀, giving

      Combining these two equations we obtain

But the matrix in this final equation is merely the rotation matrix , which leads to

(2.9)

      Thus, the rotation matrix can be used not only to represent the orientation of

Скачать книгу