Total Least Square Identification of Parallel Robots

1. Total Least Square Identification of Parallel Robots Sébastien Briot and Maxime Gautier IRCCyN – Nantes Nantes

2. Introduction • Applications of parallel robots • Force control, haptic devices, Flight/train/driving simulators • Require a correct reconstruction of ouput efforts • Force/torque sensors can sometimes be used => costly • Off line identification of the robot dynamic parameters c • IDIM requires: actual joints position, velocity and acceleration • actual joint torques t • available data: • joint position (encoders) q, sampling + band-pass filtering  • Current motor reference vt of the current amplifier • Joint j: , = the total drive gain , • Nj gear ratio (Nj > 50), Ktj torque constant, Gj gain of the current amplifier

3. Introduction • Usual identification of gtj • N, Gi and Kt identified separately by heavy tests on amplifier, motor • Needs to open the drive chain • Small errors on Gi and Kt involves large errors (N>50) on the drive gain gti • New method • Global identification of dynamic parameters + the total drive gains gtj • Use the accurate mass of a payload, weighed with a balance • Coupled identification, all joints together • Experimental results: improvement of the robot parameter dynamic identification

4. Dynamic Modeling • Parallel robots can be seen in dynamics as • A tree structure + platform • Loops are closed using constraints equations (Jacobian matrices)

5. Inverse Dynamic Identification Model (IDIM) • The IDIM of the tree structure can be written under the form • tidmtree is the vector of the virtual input torques, Fsttree is the jacobian matrix • qtreethe vector of joint coordinates • csttree is the vector of the standard dynamic parameters • Platform reactions • xthe platform position, v platform twist • cpl is the vector of the standard dynamic parameters

6. Inverse Dynamic Identification Model (IDIM) • Use of the constraint equations • Obtention of kinematic relationships • IDIM of the parallel robot • Minimal IDIM • Obtained by elimination of columns of Fthat are linearly related.

7. Inverse Dynamic Identification Model (IDIM) • Use of a straightforward way for the derivation of Jtree • express the kinematic relation between the platform twist v and the velocities vtk of all leg extremities Cmk,k. • express the kinematic relation between the velocities vtk of all leg extremities Cmk,k and the velocities of all joints of the tree structure • combine these two relations with • to obtain

8. Drive Gain identification • Requires the scaling using known parameters => the payload • IDIM with the payload • Fu,kL the jacobian matrices corresponding to the payload parameters • cuL is the vector of the unknown payload dynamic parameters • ckL is the vector of the unknown payload dynamic parameters • Because of perturbations • e is the vector of errors • Sampled and filtered IDIM • Y is the vector regrouping all torques samples, W is the observation matrix, r is the vector of errors

9. Drive Gain Identification • Payload identification: carrying out two trajectories • First line of W => trajectory without a payload • Second line of W => trajectory with a payload • Can be rewritten as • Can be solved using Total Least Square Techniques • Without perturbations, r = 0 and Wtot is rank deficient • In reality, not the case => find the matrix such that • This matrix minimizes the Frobenius norm

10. Total Least Square Solution • It can be proven that a solution can be obtained as • Vend is the last column of matrix V obtained via the SVD of • The robot dynamic parameters can be obtained as • In this work,ckL : payload mass ML only(very accurate value)

11. Case Study • Orthoglide robot • Acceleration: 2g • Workspace: • Cube edge of 25 cm • Isotropy properties • Manufacturer’s drive gains: 2000 • Payload • ML = 1.983 Kg± 0.005 Kg

12. DHm Parameters

13. Identification Results • Results cross-validation • Identification of a new mass of 1.136 Kg • With manufacturer’s gains: id. Mass = 1.09 • With new gains: id. Mass = 1.14

14. Conclusions • New approach for drive gain identification • Global identification of all joint drive gains and robot dynamic parameters • Method is very simpler to implement than previous ones • Obtained results show the importance of the drive gains identification • Future works • Force control, use of ground efforts for identification, etc.