Mark W. Spong

Robot Modeling and Control


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      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

      (2.5)numbered Display Equation

      which together imply

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

       Example 2.2.

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

numbered Display Equation numbered Display Equation

The 3D rotation matrices illustrate an example of rotations in three dimensions.

      2.3 Rotational Transformations

The 3D rotation matrices illustrate how a coordinate frame is attached to a rigid body.

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      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

numbered Display Equation

      Combining these two equations we obtain

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      Thus, the rotation matrix can be used not only to represent the orientation of coordinate frame o1x1y1z1 with respect to frame o0x0y0z0, but also to transform the coordinates of a point from one frame to another. If a given point is expressed relative to o1x1y1z1 by coordinates , then represents the same point expressed relative to the frame o0x0y0z0.