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Optimal variable stiffness control: formulation and application to explosive movement tasks

Abstract

It is widely recognised that compliant actuation is advantageous to robot control once dynamic tasks are considered. However, the benefit of intrinsic compliance comes with high control complexity. Specifically, coordinating the motion of a system through a compliant actuator and finding a task-specific impedance profile that leads to better performance is known to be non-trivial. Here, we propose an optimal control formulation to compute the motor position commands, and the associated time-varying torque and stiffness profiles. To demonstrate the utility of the approach, we consider an “explosive” ball-throwing task where exploitation of the intrinsic dynamics of the compliantly actuated system leads to improved task performance (i.e., distance thrown). In this example we show that: (i) the proposed control methodology is able to tailor impedance strategies to specific task objectives and system dynamics, (ii) the ability to vary stiffness can be exploited to achieve better performance, (iii) in systems with variable physical compliance, the present formulation enables exploitation of the energy storage capabilities of the actuators to improve task performance. We illustrate these in numerical simulations, and in hardware experiments on a two-link variable stiffness robot.

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Notes

  1. Regarding the ball throwing task, the reader may also refer to recent works that consider underactuated robots, and use optimal motion control, or employ unstable zero dynamics to achieve fast and accurate throw; see Mettin et al. (2010), Shoji et al. (2010).

  2. This is because on compliant actuators the joint torques do not correspond to the motor torques directly.

  3. Where .

  4. While in the present paper, the actuator torque is assumed to be position dependent (as is the case in the majority of VS actuators), the formulation remains valid for cases where the torque is velocity dependent (e.g., due to viscoelastic forces).

  5. This is indeed the case in many athletic disciplines (e.g., shot put, discus throw, hammer throw, javelin throw, high jump) executed through explosive actions.

  6. This two-point boundary value problem defines the stationary point of the optimal control problem (6) and (8). In this way, it provides a necessary condition to find a locally optimal solution.

  7. The lower indices in (9) and (10), denote partial derivatives with respect to the corresponding variables.

  8. If the control inputs are saturated (i.e., restricted with “hard constraints” (7)), the corresponding feedback gains may be set to zero, as suggested in Li and Todorov (2007). Alternatively, one could utilise penalty terms to embed the inequality constraints in the objective function (8), see Stengel (1994).

  9. This control law is only optimal in the neighbourhood of the optimal open-loop motion generated by: u=u (t).

  10. P q ∈ℝm×n, D q ∈ℝm×n, P θ ∈ℝm×m and D θ ∈ℝm×m.

  11. Note that the feedback correction on u is a PD-control performed with optimal position and velocity gains.

  12. Mechanically Adjustable Compliance and Controllable Equilibrium Position Actuators.

  13. In the present case the stiffness matrix has diagonal elements only. Off-diagonal elements in the stiffness matrix appear if the configuration change on one joint induces torque change on an another joint (e.g., see a human arm model of Mussa-Ivaldi et al. (1985) that incorporates bi-articular muscles).

  14. As an example, human peak performance as characterised by the rotation speed of the shoulder during a baseball pitch of a professional pitcher, is between 6900–9800/s (Herman 2007). This kind of high-performance task execution is not in the scope of present robotic systems.

  15. There is a transient at the start of the movement as the pre-tensioning motors move to the optimal (but fixed) commanded positions.

  16. The potential energy stored by the linear springs in the present actuators is computed as: where F is the spring force while K s =diag(κ 1,κ 2) is the matrix of the spring stiffness constants.

  17. During antagonistic co-activation in humans, muscles do no mechanical work but consume metabolic energy.

  18. Indeed, an explosive movement can be executed by feed-forward joint torque modulation, e.g., realised by optimal fixed stiffness control, as shown in Sects. 4.1 and 5.

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Acknowledgements

This work was funded by the EU Seventh Framework Programme (FP7) as part of the STIFF project. The authors gratefully acknowledge this support. We would like to thank Alexander Enoch for his work on the hardware design and Andrius Sutas for his contribution to the control interface. In addition, we thank Dr. Jun Nakanishi and Dr. Takeshi Mori for fruitful discussions regarding this work.

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Correspondence to David Braun.

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Braun, D., Howard, M. & Vijayakumar, S. Optimal variable stiffness control: formulation and application to explosive movement tasks. Auton Robot 33, 237–253 (2012). https://doi.org/10.1007/s10514-012-9302-3

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  • DOI: https://doi.org/10.1007/s10514-012-9302-3

Keywords

  • Variable impedance control
  • Optimal stiffness control
  • Dynamic task
  • Power amplification