A planar two-aram manipulator is loaded with a weight W at its end effector. Sup

Question

A planar two-aram manipulator is loaded with a weight W at its end effector. Suppose that the joint motors are to provide enough holding torque to maintain static equilibrium. One way measuring the required torque is by measuring the individual joint torques via the current requirements. Suppose that the motor torques are proportional to the armature currents with T=K_ri where kT is the torque constant. If the current requirements are know to be i_1 and i_2 for the position shown, determine the weight being held. What relationship, if any, exists between i_1 and i_2? Neglect the weight kof the manipulator arms and motors. Determine the forward kinematics relationship between C and A to determine the two dimensional position of C using the angular positions. Show the Denavit-Hartenberg Matrix.

Explanation / Answer

Torque of the motor is governed by..

T = k i

   T1 = k i1……………………………(1)

   T2 = k i2……………………………(2)

Torque about point B is, T2 = W LcosO2…………………….(3)

Torque about point A is , T1 = W (Lcos O2+ LcosO1)……(4)

Also written as T1 = WLcosO1 +T2……………………(4)

From (1)&(4)

So ..it can be written as

    K i1 = WL(cosO1 +cosO2)

W = K i1/(cosO1+cosO2) this is the weight being held

from (4)

K i1 = WLcosO1 + Ki2

K i1 = {Ki1/(cosO1 +cosO2)}LcosO1 + Ki2

i1   = L i1 cosO1/(cosO2 + cosO1) + i2

Now,

position of joint b, T10 = (L, O1)

x= LcosO1, y =LsinO1

position of joint c ,T21 =(L, O2)

x=LcosO2, y =LsinO2

Position of C wrt to A

  X = LcosO1+L cosO2

Y = LsinO1+L sinO2

In Denavit Matrix form

T20 =T10 T21

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