Consider the RP planar manipulator shown below. Angle OAE is a right angle. The

Question

Consider the RP planar manipulator shown below. Angle OAE is a right angle. The revolute joint rotates about origin O. The length of the first linkage is a constant parameter OA = a_1. The length of the second linkage is a variable AE = d_2 a. Obtain the forward kinematic equations relating the end-effecter position x_E, y_e and orientation phi_E to the joint displacements theta_1 and d_2. b. Given the end effector position (x_E, y_E), derive the inverse kinematics. Your answer should define (theta_1, d_2) as functions of the (x_E, y_E) position.

Explanation / Answer

given

OAE is a right angle

OA = a1 is a constant

AE = d2 is a variable

a. so, from trigonometry

xe = a1*cos(theta1) - d2*sin(theta1)

ye = a1*sin(theta1) + d2*cos(theta1)

orientation phie is the angle AE makes with horizontal

phie = 90 + theta1

b. inverting the relation

d2 = (a1*cos(theta1) - xe)/sin(theta1)

so ye = a1*sin(theta1) + (a1*cos(theta1) - xe)cos(theta1)/sin(theta1)

ye*sin(theta1) = a1*sin^2(theta1) + a1*cos^2(theta1) - xe*cos(theta1)

ye*sin(theta1) = a1 - xe*cos(theta1)

ye^2*sin^2(theta1) = a1^2 + xe^2cos^2(theta) - 2a1*xe*cos(theta1) = ye^2(1 - cos^2(theta))

cos^2(theta1)[xe^2 + ye^2] - 2*a1*xe*cos(theta1) + (a1^2 - ye^2) = 0

solvin for cos(theta1)

cos(theta1) = [2*a1*xe + sqroot(4*a1^2*xe^2 - 4(xe^2 + ye^2)(a1^2 - ye^2))]/2(xe^2 + ye^2)

cos(theta1) = [2*a1*xe + sqroot(4*a1^2*xe^2 - 4xe^2*a1^2 + 4xe^2*ye^2 - 4ye^2*a1^2 + 4ye^4)]/2(xe^2 + ye^2)

cos(theta1) = [a1*xe + ye*sqroot(xe^2- a1^2 + ye^2)]/(xe^2 + ye^2)

cos^2(theta) = [a1^2*xe^2 + ye^2[xe^2 - a1^2 + ye^2] + 2a1*xe*ye*sqroot(xe^2 - a1^2 + ye^2)]/(xe^2 + ye^2)^2

1 - sin^2(theta) = [a1^2*xe^2 + ye^2xe^2 - ye^2a1^2 + ye^4 + 2a1*xe*ye*sqroot(xe^2 - a1^2 + ye^2)]/(xe^2 + ye^2)^2

sin^2(theta) = 1 - [a1^2*xe^2 + ye^2xe^2 - ye^2a1^2 + ye^4 + 2a1*xe*ye*sqroot(xe^2 - a1^2 + ye^2)]/(xe^2 + ye^2)^2

sin^2(theta) = (xe^4 + ye^4 + 2xe^2*ye^2 - a1^2*xe^2 - ye^2xe^2 + ye^2a1^2 - ye^4 - 2a1*xe*ye*sqroot(xe^2 - a1^2 + ye^2)]/(xe^2 + ye^2)^2

sin^2(theta) = (xe^4 - a1^2*xe^2 + ye^2xe^2 + ye^2a1^2 - 2a1*xe*ye*sqroot(xe^2 - a1^2 + ye^2)]/(xe^2 + ye^2)^2

sin^2(theta) = (ye^2a1^2 + xe^2(xe^2 - a1^2 + ye^2) - 2a1*xe*ye*sqroot(xe^2 - a1^2 + ye^2)]/(xe^2 + ye^2)^2

sin(theta) = [a1*ye - xe*sqroot(xe^2 - a1^2 + ye^2)]/(xe^2 + ye^2)

hence

d2 = a1*[a1*xe + ye*sqroot(xe^2- a1^2 + ye^2)]/[a1*ye - xe*sqroot(xe^2 - a1^2 + ye^2)] - xe(xe^2 + ye^2)/[a1*ye - xe*sqroot(xe^2 - a1^2 + ye^2)]

d2 = [a1*[a1*xe + ye*sqroot(xe^2- a1^2 + ye^2)] - xe(xe^2 + ye^2)]/[a1*ye - xe*sqroot(xe^2 - a1^2 + ye^2)]

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