Basilio Bona DAUIN – Politecnico di Torino 22/02/2019 ROBOTICS 01PEEQW Basilio Bona DAUIN – Politecnico di Torino di 23

Example 01 Jacobians

Example 1 – DH parameters Denavit – Hartenberg parameters Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Example 1 Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Example 1 Eqn. 1 Euler angles eqn. (2.79) page 52 Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Example 1 Knowing the Euler angles, everything will be easy. Assume we do not know them. Squaring and adding Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Example 1 Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Example 1 Linear velocities Angular velocities: analytical approach We call these “eulerian velocities” Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Example 1 Analytic Jacobian (by differentiation) Eqn. 2a Eqn. 2b Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Example 1 Transformation matrix (see textbook) Basilio Bona ROBOTICS 01PEEQW - 2015/2016

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 - 2015/2016

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 - 2015/2016

Example 1 This relation is obtained from direct KF – Eqn. 1 Basilio Bona ROBOTICS 01PEEQW - 2015/2016

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 - 2015/2016

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 - 2015/2016

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 - 2015/2016