The question for final

Question

The following network diagram is a position-position teleoperator:

em - effort in master side

es - effort in slave side

fm - flow in master side

fs - flow in slave side

eam - Actuator effort of the master side

eas - Actuator effort of the slave side

In our device effort is represented by force and flow is represented by velocity.

a) Find the corresponding control law of above position-position teleoperator.

b) Suppose the inputs to the teleoperator are em , fs, and the outputs are es , fm. Try to express the transfer matrix H from [em ; fs] to [es ; fm] by G, Zm, Zs.

c) In what condition this position-position teleoperator can approach to the ideal teleoperator? What is the physical meaning of this configuration?




Solution
a)
control law: eam = eas = G(fm + fs)


b)
From the control law in part (a) and diagram above, we can get the following equations:


em - zm * fm - eam = 0 (1)
es - zs * fs - eas = 0 (2)


Substitute eas and eam with the control law:


em - zm * fm - G(fm + fs) = 0 (3)
es - zs * fs - G(fm + fs) = 0 (4)


For equation (3):


fm = -(G*fs)/(zm + G)  + em/(zm + G)  (5)


For equation (4):


es = zs * fs + G(fm + fs)


Substitute Fs with equation (5),

es = (zs + G)fs + G(-G/(zm + G) * fs + 1/(zm + G)*em)

es = (zs + G (1 - G/(zm + G)))*fs + G/(zm + G)* em (6)

From (5) and (6), we can get matrix H as follow:


H = [G/(zm + G) , (zs + G (1 - G/(zm + G)));  1/(zm + G) , -G/(zm + G)]

c)
When zm and zs approach zreo, and the gain G goes to infinity. The master and slave are connected through rigid rod.