1. Basilio Bona DAUIN – Politecnico di Torino
22/02/2019
ROBOTICS 01PEEQW
Basilio Bona
DAUIN – Politecnico di Torino
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2. Example 01
Jacobians

3. Example 1 – DH parameters
Denavit – Hartenberg parameters
Basilio Bona
ROBOTICS 01PEEQW /2016

4. Example 1
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ROBOTICS 01PEEQW /2016

5. Example 1 Eqn. 1 Euler angles eqn. (2.79) page 52 Basilio Bona
ROBOTICS 01PEEQW /2016

6. Example 1 Knowing the Euler angles, everything will be easy.
Assume we do not know them.
Squaring and adding
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ROBOTICS 01PEEQW /2016

7. Example 1
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ROBOTICS 01PEEQW /2016

8. Example 1 Linear velocities Angular velocities: analytical approach
We call these “eulerian velocities”
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ROBOTICS 01PEEQW /2016

9. Example 1 Analytic Jacobian (by differentiation) Eqn. 2a Eqn. 2b
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ROBOTICS 01PEEQW /2016

10. Example 1 Transformation matrix (see textbook) Basilio Bona
ROBOTICS 01PEEQW /2016

11. Example 1
From the previous slide we have the cartesian velocity in base RF we can now compute the Jacobian matrix
Eq. 3
This is the geometric angular Jacobian
Now we compute it in a different way
Basilio Bona
ROBOTICS 01PEEQW /2016

12. Example 1 First we compute the angular Jacobian
Both joints are rotoidal, therefore, considering results at page.92
First we compute the angular Jacobian
These two columns are equal to those in Eqn. 3
Basilio Bona
ROBOTICS 01PEEQW /2016

13. Example 1 This relation is obtained from direct KF – Eqn. 1
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ROBOTICS 01PEEQW /2016

14. Example 1 Then we compute
Instead of transforming it in RF 0 and after making the vector product,
we make the vector product in RF 1 and then we transform the result
to express it in RF 0
Basilio Bona
ROBOTICS 01PEEQW /2016

15. Example 1 Now we transform from RF 1 to RF 0
In conclusions, the two linear Jacobians are
It coincides with the results of Eqn. 2a
Basilio Bona
ROBOTICS 01PEEQW /2016

16. Example 1
Linear Jacobians are independent from the methods used to compute them
(since we use always Cartesian representation)
Instead, Angular Jacobians, depends on the conventions used to express
the TCP orientation
In conclusions:
Analytical Jacobian
Geometrical Jacobian
Basilio Bona
ROBOTICS 01PEEQW /2016