## 1.5 PRACTICE PROBLEMS

1.

a) A stepping motor is to be used to actuate one joint of a robot arm in a light duty pick and place application. The step angle of the motor is 10 degrees. For each pulse received from the pulse train source the motor rotates through a distance of one step angle.

i) What is the resolution of the stepper motor?

ii) Relate this value to the definitions of control resolution, spatial resolution, and accuracy, as discussed in class.

b) Solve part a) 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.

2. A stepping motor is to be used to drive each of the three linear axes of a cartesian coordinate robot. The motor output shaft will be connected to a screw thread with a screw pitch of 0.125”. It is desired that the control resolution of each of the axes be 0.025”

a) to achieve this control resolution how many step angles are required on the stepper motor?

b) What is the corresponding step angle?

c) Determine the pulse rate that will be required to drive a given joint at a velocity of 3.0”/sec.

3. For the stepper motor of question 6, a pulse train is to be generated by the robot controller.

a) How many pulses are required to rotate the motor through three complete revolutions?

b) If it is desired to rotate the motor at a speed of 25 rev/min, what pulse rate must be generated by the robot controller?

4. A stepping motor is to be used to actuate one joint of a robot arm in a light duty pick and place application. The step angle of the motor is 10 degrees. For each pulse received from the pulse train source the motor rotates through a distance of one step angle.

a) What is the resolution of the stepper motor?

b) Relate this value to the definitions of control resolution, spatial resolution, and accuracy, as discussed in class.

5. Find the forward kinematics for the robots below using geometry methods.

6. Consider the forward kinematic transformation of the two link manipulator below.

a) Given the position of the joints, and the lengths of the links, determine the location of the tool centre point using basic geometry.

b) Determine the inverse kinematics for the robot. (i.e., given the position of the tool, determine the joint angles of the robot.) Keep in mind that in this case the solution will have two different cases.

c) Determine two different sets of joint angles required to position the TCP at x=5”, y=6”.

d) What mathematical conditions would indicate the robot position is unreachable? Are there any other cases that are indeterminate?

7. 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.

8. 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.

9. Consider the following motion planning problem.

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.

10. We are designing motion algorithms for a 2 degree of freedom robot. To do this we are developing sample calculations to explore the basic process.

a) We want to move the tool in a straight line through space from (3”, 5”) to (8”, 7”). Develop equations that will give a motion that starts and stops smoothly. The motion should be complete in 1 second.

b) Find the velocity of the tool at t=0.5 seconds.

c) Plot out the tool position, joint positions and velocities as functions of time.

11. Why do robots not follow exact mathematical paths?

12. We are designing motion algorithms for a 2 degree of freedom robot. To do this we are developing sample calculations to explore the basic process. We want to move the tool in a straight line through space from (8”, 7”) to (3”, 5”). Develop equations that will give a motion that starts and stops smoothly. The motion should be complete in 2 seconds. Show all derivations.

13.