Robot Modeling and Control. Mark W. Spong
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(2.12)
where
Example 2.4.
Henceforth, whenever convenient we use the shorthand notation cθ = cos θ, sθ = sin θ for trigonometric functions. Suppose frames o0x0y0z0 and o1x1y1z1 are related by the rotation
If A = Rz, θ relative to the frame o0x0y0z0, then, relative to frame o1x1y1z1 we have
In other words, B is a rotation about the z0-axis but expressed relative to the frame o1x1y1z1. This notion will be useful below and in later sections.
2.4 Composition of Rotations
In this section we discuss the composition of rotations. It is important for subsequent chapters that the reader understand the material in this section thoroughly before moving on.
2.4.1 Rotation with Respect to the Current Frame
Recall that the matrix
(2.13)
(2.14)
(2.15)
where each
(2.16)
Note that
(2.17)
Equation (2.17) is the composition law for rotational transformations. It states that, in order to transform the coordinates of a point p from its representation
We may also interpret Equation (2.17) as follows. Suppose that initially all three of the coordinate frames coincide. We first rotate the frame o1x1y1z1 relative to o0x0y0z0 according to the transformation