What is the solution to part A Specifically how do you deter

What is the solution to part A? Specifically how do you determine the equivalent motor torque?

A) Show that an equation of the form X=A*u-B can be used to represent the dynamics of the first link. Where u is the control signal.

Figure 4 shows the first link of a small Cartesian manipulator. The other li of the manipulator are modelled as a 10kg mass. This mass is driven vertically through a rack and pinion gear by a DO motor. Parameter values are: Motor constant K. motor tergue giver by ten Knuj H08 k torment inertia 0.2 m rev Rack & pinion gear rati 9.R Gravitational con a) show that an equation of the form Au-H can be used to represent the dymamics of the first link, where u is the control signal b) Determine the closed loop natural froquency and damping ratio if the feedforward proportional plus derivative (PD) controller of Figure s is used. c) Using the same controller, what is the steady stale pasition error due to the weight of the 10kg mass?

Solution

solution:

1)here given manipulator using a motor to raise another arm of wiegth 10 kg,hence torque require to doing so is

Trequired=m*g*radius of pinion=10*9.81*r=98.1 r N m

2)where our torque production by is control by PD controller,here PD controller worksss on error signal e

here error=e=X-Xd

but for t=0, Xd=0

hence error=e=X

where displacement X=velocity*time=V*t

hence for PD controller output contro signal is given by

u=Kpep+Kd(dep/dt)+Uo

as initial signal is zero

Uo=0

hence contro signal turns

u=Kpep+Kd(dep/dt)

on putting X=Vt ,Kp=900,Kd=50

we get

u=900X+50d(X)/dt

on putting value we get

u=900x+50V

hence output position at any instant is given by

X=(1/900)u-(50/900)V

if velocity is constant second block is constant ,then we write

X=Au-B

hence for first link to control output motion signal provided is governed by above equation

2)motor equivalent torque is given by

Tm=Km*u=10(900x+50V)=9000X+500V

3)here for system to run properly equation must satisfy following equation

Tm-I*w\'-Cw=Trequired=98.1 r

Tm-I*w\'-Cw=98.1 r

where C=damping coefficient

I=moment of inertia

w=angular velocity

r=radius of pinion