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1.5 PRACTICE PROBLEMS

1. For the Stanford arm below,

a) list the D-H parameters (Hint: extra “dummy” joints may be required)

b) Find the forward kinematics using homogenous matrices.

c) Find the Jacobian matrix for the arm.

d) If the arm is at θ1 = 45 degrees, θ2 = 45 degrees, r = 0.5m, find the speed of the TCP if the joint velocities are θ’1 = 1 degree/sec, θ’2 = 10 degrees/sec, and r’ = 0.01 m/sec.

2. Find a smooth path for a robot joint that will turn from θ= 75° to θ = -35° in 10 seconds. Do this by developing an equation then calculating points every 1.0 seconds along the path for a total motion time of 10 seconds.

3. A jointed arm robot has three rotary joints, and is required to move all three axes so that the first joint is rotated through 50 degrees; the second joint is rotated through 90 degrees, and the third joint is rotated through 25 degrees. Maximum speed of any of these rotational joints is 10 degrees/sec. Ignore effects of acceleration and deceleration and,

a) determine the time required to move each joint if slew motion (joint motion is independent of all other joints) is used.

b) determine the time required to move the arm to a desired position and the rotational velocity of each joint, if joint interpolated motion (all joints start and stop simultaneously) is used.

c) Solve question 4 under the condition that the three joints move at different rotational velocities. The first joint moves at 10 degrees/sec., the second joint moves at 25 degrees/sec, and the third joint moves at 30°/sec.

4. Find a smooth path for a robot joint that will turn from θ= 75° to θ = -35° in 10 seconds. Do this by developing an equation then calculating points every 1.0 seconds along the path for a total motion time of 10 seconds.

5.

a) A jointed arm robot has three rotary joints, and is required to move all three axes so that the first joint is rotated through 50 degrees; the second joint is rotated through 90 degrees, and the third joint is rotated through 25 degrees. Maximum speed of any of these rotational joints is 10 degrees/sec. Ignore effects of acceleration and deceleration and,

b) determine the time required to move each joint if slew motion (joint motion is independent of all other joints) is used.

c) determine the time required to move the arm to a desired position and the rotational velocity of each joint, if joint interpolated motion (all joints start and stop simultaneously) is used.