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Active isotropic compliance in redundant manipulators

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Abstract

The isotropic compliance property is examined in the Special Euclidean Group SE(3) in the case of redundant manipulators. The redundancy problem is solved by means of the QR decomposition of the transposed Jacobian matrix, and the compliance property is achieved by means of active stiffness regulation. Thanks to the defined control matrices, the control system realizes the isotropy condition. The local optimization of the joint torques is discussed. In particular, the joint control torques work is minimized obtaining an analytic solution through a Lyapunov equation. The proposed approach is applied to a 7R and to a 9R serial manipulator, and verified by means of multibody dynamics simulations.

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Authors and Affiliations

  1. Department of Mechanical, Energy, Management and Transportation Engineering, University of Genoa, 16145, Genoa, Italy

    Matteo Verotti

  2. Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, 20156, Milano, Italy

    Pierangelo Masarati & Marco Morandini

  3. Department of Engineering, University of Roma Tre, 00146, Rome, Italy

    Nicola P. Belfiore

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  1. Matteo Verotti
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  2. Pierangelo Masarati
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  3. Marco Morandini
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  4. Nicola P. Belfiore
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Correspondence to Matteo Verotti.

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Appendix

Appendix

Fig. 12
figure 12

Seven-DoF manipulator in the posture q a : end-effector displacements and rotations, with and without control action, for the simulation time from 3 s to 6 s (force applied in the \(y\)-axis direction)

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Fig. 13
figure 13

Seven-DoF manipulator in the posture q a : end-effector displacements and rotations, with and without control action, for the simulation time from 6 s to 9 s (force applied in the \(z\)-axis direction)

Full size image
Fig. 14
figure 14

Seven-DoF manipulator in the posture q a : end-effector displacements and rotations, with and without control action, for the simulation time from 9 s to 12 s (moment applied in the \(x\)-axis direction)

Full size image

The first posture ( q a ) end-effector response, in the case of a force applied along the \(y\) direction with and without control action, is reported in Fig. 12. Analogously, the cases regarding the force on the \(z\) direction, the moment about the \(x\) direction, and the moment about the \(z\) direction are reported in Fig. 13, Fig. 14, and Fig. 15, respectively. The correspondent end-effector responses for the second case ( q b ) are reported in Fig. 16, Fig. 17, Fig. 18 and Fig. 19.

Fig. 15
figure 15

Seven-DoF manipulator in the posture q a : end-effector displacements and rotations, with and without control action, for the simulation time from 15 s to 18 s (moment applied in the \(z\)-axis direction)

Full size image
Fig. 16
figure 16

Seven-DoF manipulator in the posture q b : end-effector displacements and rotations, with and without control action, for the simulation time from 3 s to 6 s (force applied in the \(y\)-axis direction)

Full size image
Fig. 17
figure 17

Seven-DoF manipulator in the posture q b : end-effector displacements and rotations, with and without control action, for the simulation time from 6 s to 9 s (force applied in the \(z\)-axis direction)

Full size image
Fig. 18
figure 18

Seven-DoF manipulator in the posture q b : end-effector displacements and rotations, with and without control action, for the simulation time from 9 s to 12 s (moment applied in the \(x\)-axis direction)

Full size image
Fig. 19
figure 19

Seven-DoF manipulator in the posture q b : end-effector displacements and rotations, with and without control action, for the simulation time from 15 s to 18 s (moment applied in the \(z\)-axis direction)

Full size image

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Verotti, M., Masarati, P., Morandini, M. et al. Active isotropic compliance in redundant manipulators. Multibody Syst Dyn 49, 421–445 (2020). https://doi.org/10.1007/s11044-020-09724-2

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