1. Basilio Bona DAUIN – Politecnico di Torino
07/08/2018
ROBOTICS 01PEEQW
Basilio Bona
DAUIN – Politecnico di Torino
di 23

2. Planar 2 DOF manipulator Kinematics

3. Planar two-arm manipulator – 1
This simple manipulator
cannot provide 3 DOF,
since it has only two joints
This is the TCP orientation assumed as the third Euler angle
We will compute below
the kinematics functions of the reduced DOFs
Basilio Bona
ROBOTICS 01PEEQW /2016

4. Planar two-arm manipulator – 2
Direct position KF, assuming
Direct velocity KF
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ROBOTICS 01PEEQW /2016

5. Planar two-arm manipulator – 3
Analytical Jacobian
Geometric Jacobian, assuming
All joints are revolute, hence:
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ROBOTICS 01PEEQW /2016

6. Planar two-arm manipulator – 4
Geometric Jacobian
Remember: all vectors are represented in
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ROBOTICS 01PEEQW /2016

7. Planar two-arm manipulator – 5
Geometric Jacobian
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ROBOTICS 01PEEQW /2016

8. Planar two-arm manipulator – 6
Geometric Jacobian
These rows are canceled
since they add no information
In conclusion
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ROBOTICS 01PEEQW /2016

9. Inverse position – 1
Inverse position; the solution can be obtained in different ways; this is one of them:
hence:
To avoid inverting cos(.), we can find the solution computing
Basilio Bona
ROBOTICS 01PEEQW /2016

10. Inverse position – 2 hence
The square root produces two solutions: elbow up and elbow down
elbow up
elbow down
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ROBOTICS 01PEEQW /2016

11. Inverse position – 3 Angle is obtained computing
Hence, solving the system
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ROBOTICS 01PEEQW /2016

12. Inverse position – 4 An alternative solution Basilio Bona
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13. Inverse velocity – 1
First we reduce the Jacobian to a square matrix, deleting the angle-related row
The determinant is
The inverse is
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ROBOTICS 01PEEQW /2016

14. Singularity – 1 Singularity arises when Basilio Bona
ROBOTICS 01PEEQW /2016

15. Singularity – 2 SINGULARITY
These two vectors span a subspace of dimension 1
NON SINGULARITY
These two vectors span a subspace of dimension 2
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ROBOTICS 01PEEQW /2016

16. Singularity – 3 Now we compute the various subspaces when Basilio Bona
ROBOTICS 01PEEQW /2016

17. Singularity – 4 Now we compute the various subspaces when Basilio Bona
ROBOTICS 01PEEQW /2016

18. Singularity – 5 Now we compute the various subspaces when Basilio Bona
ROBOTICS 01PEEQW /2016

19. Singularity – 6 Numerical example 3 1 2 4 1 2 Basilio Bona
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20. Geometry of singularity – 14 1
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21. Geometry of singularity – 23
Basilio Bona
ROBOTICS 01PEEQW /2016