Question

Assume the coordinate frame assignments shown in the figure for the Rhino robot used in the lab and the transformation matrix from the base frame to the end effector frame Ts is as given below Suppose we want to use the Rhino for a painting application where the end-effector should point at 4j degrees from the horizontal. In this case, we would need to align the end-effector frame Z axis (Z4) at 135 degrees to Z axis Given the desired end effector position coordinates, x, y, and z, write four equations relating joint angles, 0,, θ2.93 and θ4 to end effector coordinates and robot link parameters. (20 points) Solve the four equations for the four joint variables θι, θ-,83 and θ4 in terms of end effector coordinates x, y, and z and link parameters. (20 points) a. Z X X) b. a3 23

Answer 1

Solution:

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The equations are:

Take the last column of the matrix. It represents position vector of the end effector.

If Z-axis of End Effector makes 135⁰ with Base X-axis

Angle with Base Z axis is 360 - 135 - 90 = 135⁰

Square and add x, y, z:

Substituting $heta_3$ in

You get $heta_2$

Substituting $heta_2$ and $heta_3$    in

You get $heta_1$

You get $heta_4$ by using the equation: