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Section 4.2: Generic Properties of Kinetostatic Transmission Elements
the rest of the operational model.
The transmission behaviour, or the \services", of a kinetostatic transmission element can
be best described by considering a simple element mapping a set of n scalar (kinematical)
input variables q 2 IRn , to a set of m scalar (kinematical) output variables q 0 2 IRm , as
displayed in Fig. 4.2.
q
q_
q
Q
input output
position
q 2 IRn q0 2 IRm
- q0
velocity
q_ 2 IRn q_ 0 2 IRm
- \map"
- q_ 0
acceleration q 2 IRn q0 2 IRm
- q0
Q0
force
Q 2 IRn Q0 2 IRm
b) input and output coordinates
a) transmission behaviour
Figure 4.2: Model of a kinetostatic transmission element
The transmission of motion involves the mapping of the input variables q to a corresponding set of output variables q0 . As well, the time derivatives q_ and q of the
input coordinates, i. e. the velocity and acceleration, are mapped to the corresponding
time derivatives q_ 0 and q0 of the output coordinates. Apart from the kinematic transmission functions, the kinetostatic transmission element also induces a transmission of
forces. This transmission is directed in opposite direction to the transmission of motion,
mapping the forces Q0 at the (kinematic) output of the kinetostatic transmission element
to corresponding forces Q at the (kinematic) input. The reasoning behind this kind of
modeling is explained in the following theoretical background information.
Theoretical background: Structure of the kinetostatic transmission functions
The transmission of motion is governed by the following formulas:
position:
q0 = ' ( q )
velocity:
q_ 0 = J q_
acceleration: q 0 = J q + J_ q_
where J represents the Jacobian of the transmission element
mn :
J = @
@q 2 IR
(4.1)
For the transmission of forces, one rst assumes that the transmission is ideal , i. e., that it neither
generates nor consumes power. Then, equality of virtual work at the input and output holds, and it
follows
qT Q = q0 T Q0 :
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