Compare the forward kinematics of a nonholonomic robot to those of a holonomic robot.

1. Compare the forward kinematics of a nonholonomic robot to those of a holonomic robot.

2. Note that our textbook Craig will provide a very useful and general way to develop the forward kinematics of a Holonomic robot.

3. EP AP

4. Recall, that a holonomic robot will (kinematically) “repeat”.

5. Recall, that a holonomic robot will (kinematically) “repeat”. • In general, by returning the internal, servo-controlled angles to earlier, taught values, you will bring about a return of the borne load to its corresponding location in space.

6. Recall, that a holonomic robot will (kinematically) “repeat”.

7. However there is a subtle distinction between holonomic robots and nonholonomic robots that can be appreciated with this animation.

8. The nonholonomic robot will not repeat by merely returning the internal angles of rotation to their previous values.

9. What is the nature of the forward kinematics of this nonholonomic robot?

10. Let’s consider a “bird’s eye view” of the axis that connects the wheelchair’s two wheels.

11. Bird’s eye view of axle.

12. Bird’s eye view of axle.

14. Development of nonholonomic equations for numerical integration.

15. Imperfect tracking won’t affect the holonomic robot’s terminal pose.

16. Imperfect tracking won’t affect the holonomic robot’s terminal pose.

17. Imperfect tracking won’t affect the holonomic robot’s terminal pose.

18. Imperfect tracking won’t affect the holonomic robot’s terminal pose.

19. But imperfect tracking has a cumulative effect on the nonholonomic robot.

20. But imperfect tracking has a cumulative effect on the nonholonomic robot.

21. But imperfect tracking has a cumulative effect on the nonholonomic robot.

22. But imperfect tracking has a cumulative effect on the nonholonomic robot.

23. For all their disadvantages, nonholonomic, wheeled robots offer some distinct advantages.

24. For all their disadvantages, nonholonomic, wheeled robots offer some distinct advantages.

25. The wheelchair can get away with a three-degree-of-freedom movement using just two servomechanisms, one for each wheel, precisely because it is a nonholonomic system.

26. How will we control such a robot if teach/repeat, the standard for holonomic robots, cannot be applied?

27. One possibility is to track a line in the floor.

28. One possibility is to track a line in the floor. But this can get messy!

29. Also tracking of a physical line doesn’t permit deviation from the path in the event of an obstacle.

30. And line tracking doesn’t permit pivoting in tight spaces.

31. You could track a wall instead. • Many early developments of this kind. • Shakey • The Kent floor-cleaning robot

32. Alternatively, create sensor-based autonomy and reasoning, such as “simultaneous localization and mapping”, SLAM.

33. So far SLAM methods have not produced much fruit. Hard to achieve absent humans’ ability of object recognition. The “correspondence problem”.

34. Random motion.

35. Intelligent random motion.

36. Compare Jacobians for our two two-degree-of-freedom robots.

46. EP AP

47. Z-Y-X Euler Angles

48. Z-Y-X Euler Angles - Just three numbers are needed to specify the orientation of one set of axes relative to another.

49. Z-Y-X Euler Angles Just three numbers are needed to specify the orientation of one set of axes relative to another. One possible set of these numbers is the Z-Y-X Euler angles

50. Consider the {A} and {B} frames shown below.