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The first transformation (Equation 1.14) can be used with Equation 1.13 to show that,

      (1.16)

      This is the same result as that shown in Equation 1.9.

Similarly, the second transformation (Equation 1.15) can be used with Equation 1.9 to show that,

      (1.17)

      Expanding this yields,

      (1.18)

      This is the same result as that shown in Equation 1.13.

      The transformations described in this example are typical of those used in dynamic analysis. Dynamicists are prone to using whatever coordinate system is appropriate at the time and, sometimes, there are many intermediate coordinate systems used in deriving the final system. Nevertheless, each coordinate system in the sequence must be right handed and must be generated by a simple plane rotation from the preceding system.

1.4 Two Dimensional Motion with Variable Length

Figure 1.4 shows a rigid body rotating in a plane about a fixed point . The body has a slot cut in it and a small object slides in this slot. Expressions for the velocity and acceleration of are desired.

Figure 1.4 A slider in a slot.

Since the distance from to changes with time, we start by defining a variable distance from to . A set of rotating unit vectors (, , ), fixed in the rotating body, as shown, is appropriate for this analysis since the position vector is aligned with , thereby making it easy to write.

      The angular velocity of the body is not specified in magnitude but the fact that the body rotates in a plane fixes the direction of the angular velocity to be . We assume that the angular velocity is,

      where is not constant so that (i.e. the rate of change of magnitude of the angular velocity vector) exists.

      The position of with respect to is then,

      and, differentiating this, we find the velocity of with respect to to be,

(1.19)

      The acceleration of the slider relative to point is defined to be,

or

(1.20)

      Since both the velocity and the acceleration are relative to the fixed or inertial point , they are in fact the absolute velocity and acceleration of point . We commonly write absolute velocities and accelerations without subscripts yielding,

      (1.21)

      and

      (1.22)

1.5 Three Dimensional Kinematics

Figure 1.5 shows a three degree of freedom robot. The horizontal arm is of fixed length and is free to rotate about a vertical axis through point with angular speed . Arm has a variable length and is free to rotate about an axis

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