Page 93 - The Mechatronics Handbook

0066_Frame_C07 Page 3 Wednesday, January 9, 2002 3:39 PM

FIGURE 7.1 Sketch of a rigid body with position vector, velocity, and angular velocity vectors.

FIGURE 7.2 Multiple link robot manipulator arm.

to use these generalized motions {q k : k = 1,…,K} to describe the dynamics. It is sometimes useful to
define a vector or matrix, J(q k ), called a Jacobian, that relates velocities of physical points in the machine
{}
to the generalized velocities q ˙ k . If the position vector to some point in the machine is r P (q k ) and is
determined by geometric constraints indicated by the functional dependence on the {q k (t)}, then the
velocity of that point is given by

v P ∑ --------q r = Jq ˙ ⋅ (7.2)
=
∂ r P
˙
∂q r
where the sum is on the number of generalized degrees of freedom K. The three-by-K matrix J is called
q ˙
a Jacobian and is a K × 1 vector of generalized coordinates. This expression can be used to calculate
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