1. Basilio Bona DAUIN – Politecnico di Torino
24/02/2019
ROBOTICS 01PEEQW
Basilio Bona
DAUIN – Politecnico di Torino
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2. 24/02/2019
Trajectory Planning 2
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3. Planning of the task space variables
24/02/2019
Planning of the task space variables
Not all motions are planned in the joint space
Very often the planning takes place in the task (cartesian) space, since it is more convenient for the operator
This happens when a particular geometric profile must be followed (as in the case of continuous welding or glue deposition, etc. ) or when obstacles in the task space must be avoided
Once the cartesian targets are determined, the inverse kinematics function must be invoked, in order to provide joint reference to the control algorithm
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016
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4. Inverse Kinematics Convex Combination Inverse Kinematics
Profile Generator
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ROBOTICS 01PEEQW /2016

5. Task space planning
In the following slides we will consider only the cartesian position planning
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

6. Task space planning: the position variables
The task space position planning starts with the formal definition of the required trajectory
The trajectory can be associated to a time law: e.g., it may be necessary to do a certain path with a prescribed velocity (in such tasks as continuous welding, glue deposition, etc.)
When the task requires an interaction with the environment, as in manipulation, deburring, or other tasks where the TCP is subject to external forces, it is necessary to plan the position and the force at the same time. This case will not be treated here
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

7. Task space planning: the position variables
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

8. Task space planning: the position variables
Case 1
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ROBOTICS 01PEEQW /2016

9. Task space planning: the position variables
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

10. Task space planning: the position variables
Case 2
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ROBOTICS 01PEEQW /2016

11. Example: arc in the plane
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ROBOTICS 01PEEQW /2016

12. Example: arc in the plane
Discrete time
where
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ROBOTICS 01PEEQW /2016

13. Example: simple arc in the plane
angle
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ROBOTICS 01PEEQW /2016

14. Example: another arc in the plane
Clockwise angle
Anti-clockwise angle
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

15. Orientation planning – 1
Orientation is defined by a rotation matrix; during the planning phase it must always remain an orthonormal matrix with positive unit determinant
Hence it is wrong to plan the orientation as
since
Other methods must be used; the most common are three
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

16. Orientation planning – 1
Axis –angle representation: one parameter is planned
Planar sliding: two parameters are planned
Euler angles: three parameters are planned
Initial data common to the three approaches:
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

17. Method 1: axis-angle Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

18. Method 1: axis-angle Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

19. Method 1: axis-angle axis u = -0.5774 theta = 120.0000
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

20. Method 2: planar sliding
Do not confuse k with k
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

21. Method 2: planar sliding
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

22. Method 2: planar sliding
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

23. Method 2: planar sliding
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

24. Method 2: planar sliding
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

25. Method 2: planar sliding
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

26. Method 2: planar sliding
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

27. Method 3: Euler angles Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

28. Method 3: Euler angles Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

29. Method 3: Euler angles axis u= -0.5774 0.5774 Delta_eul = 180 90
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

30. Axis-angle vs Euler comparison
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ROBOTICS 01PEEQW /2016

31. Axis-angle vs Euler comparison
This is the third planned reference frame
Axis-angle
Euler
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ROBOTICS 01PEEQW /2016

32. Characteristics of the methods
Axis –angle: is simple to implement and gives a moderately good geometrical insight in the movement performed by the robot
Planar sliding: is the most complex, but provides the best geometrical insight
Euler angles: is the most simple, but suffers of poor geometrical insight
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

33. Inverse kinematics Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

34. Inverse kinematics Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016

35. Actuators constraints
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ROBOTICS 01PEEQW /2016

36. Actuators constraints
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ROBOTICS 01PEEQW /2016

37. Micro-macro interpolation
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ROBOTICS 01PEEQW /2016

38. Micro-macro interpolation
Example
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ROBOTICS 01PEEQW /2016

39. Micro-macro interpolation
Zoom
Errors
True values
Approximate values
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW /2016