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144 Robot Dynamics

Then we have

(3.4.8)

and the kinetic energy in terms of p(t) is
(3.4.9)

It is worth noting that
(3.4.10)

Defining the manipulator Hamiltonian by
(3. 4.11)

Hamilton’s equations of motion are

(3.4.12)

(3.4.13)

Note that

(3.4.14)

Evaluating (3.4.13) yields

which may be expressed (see the Problems) as

(3.4.15)

where G(q) is the gravity vector and ⊗ is the Kronecker product (see Section
3.3).
Defining the state vector as x R as
2n
(3.4.16)

we see that the arm dynamics may be expressed as

(3.4.17)

Copyright © 2004 by Marcel Dekker, Inc.