Mark W. Spong - Robot Modeling and Control

Figure 2.3 Rotation about z 0by 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 o 0 x 0 y 0 z 0and o 1 x 1 y 1 z 1shown in Figure 2.4.

Projecting the unit vectors x 1, y 1, z 1onto x 0, y 0, z 0gives the coordinates of x 1, y 1, z 1in the o 0 x 0 y 0 z 0frame as

The rotation matrix specifying the orientation of o 1 x 1 y 1 z 1relative to o 0 x 0 y 0 z 0has these as its column vectors, that is,

Figure 2.4 Defining the relative orientation of two frames.

Figure 2.5shows a rigid object S to which a coordinate frame o 1 x 1 y 1 z 1is attached. Given the coordinates of the point p (in other words, given the coordinates of p with respect to the frame o 1 x 1 y 1 z 1), we wish to determine the coordinates of p relative to a fixed reference frame o 0 x 0 y 0 z 0. 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 o 0 x 0 y 0 z 0, 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 coordinate frame o 1 x 1 y 1 z 1with respect to frame o 0 x 0 y 0 z 0, but also to transform the coordinates of a point from one frame to another. If a given point is expressed relative to o 1 x 1 y 1 z 1by coordinates , then represents the same pointexpressed relative to the frame o 0 x 0 y 0 z 0.

We can also use rotation matrices to represent rigid motions that correspond to pure rotation. For example, in Figure 2.6(a) one corner of the block is located at the point pa in space. Figure 2.6(b) shows the same block after it has been rotated about z 0by the angle π . The same corner of the block is now located at point pb in space. It is possible to derive the coordinates for pb given only the coordinates for pa and the rotation matrix that corresponds to the rotation about z 0. To see how this can be accomplished, imagine that a coordinate frame is rigidly attached to the block in Figure 2.6(a), such that it is coincident with the frame o 0 x 0 y 0 z 0. After the rotation by π , the block’s coordinate frame, which is rigidly attached to the block, is also rotated by π . If we denote this rotated frame by o 1 x 1 y 1 z 1, we obtain

In the local coordinate frame o 1 x 1 y 1 z 1, the point pb has the coordinate representation . To obtain its coordinates with respect to frame o 0 x 0 y 0 z 0, we merely apply the coordinate transformation Equation ( 2.9), giving