# Stiffness Control of Variable Serial Elastic Actuators: Energy Efficiency through Exploitation of Natural Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. Actuator Modeling

#### 2.1. Dynamics Equations

#### 2.2. Natural Dynamics

## 3. Energy Consumption Analysis

#### 3.1. Energy Calculation

#### 3.2. Electrical Model Extensions

#### 3.3. Results

## 4. Stiffness Control

## 5. Experimental Evaluation

#### 5.1. Experimental Setup

#### 5.2. Sinusoidal Trajectory Experiments

## 6. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Consumed Energy per Oscillation in $\mathrm{J}$ of the ideal system (

**left**) and considering friction (

**right**).

**Figure 3.**Consumed Energy per Oscillation in $\mathrm{J}$ considering electrical losses (

**left**) and all losses (

**right**).

**Figure 6.**Simulation of adjustment to antiresonance while tracking a sinusoidal trajectory with 1.6 $\mathrm{Hz}$; link (blue) and actuator (red) positions in upper plot, counter bearing positions middle plot, mechanical power in lower plot.

**Figure 7.**Adjustment of stiffness while tracking different dual sinusoidal trajectories; link (blue) and actuator (red) positions; base trajectory: sine with 10${}^{\circ}$ and 1.6 $\mathrm{Hz}$; (

**top-left**) superimposed by sine with 5${}^{\circ}$, 0.8 $\mathrm{Hz}$; (

**top-right**) superimposed by sine with 5${}^{\circ}$, 2.4 $\mathrm{Hz}$; (

**bottom-left**) superimposed by sine with 10${}^{\circ}$, 0.8 $\mathrm{Hz}$; (

**bottom-right**) superimposed by sine with 10${}^{\circ}$, 2.4 $\mathrm{Hz}$.

**Figure 9.**Adjustment to antiresonance while tracking a sinusoidal trajectory with 1.6 $\mathrm{Hz}$; link (blue) and actuator (red) positions in upper plot, desired (red) and current (blue) counter bearing positions middle plot, mechanical power in lower plot.

**Figure 10.**Adjustment to antiresonance while tracking a dual sinusoidal trajectory with 0.8 $\mathrm{Hz}$ and 1.6 $\mathrm{Hz}$; link (blue) and actuator (red) positions in upper plot, desired (red) and current (blue) counter bearing positions middle plot, mechanical power in lower plot.

**Figure 11.**Energy-frequency behavior (blue) for sinusoidal trajectories at 75 $\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$ (

**left**) and 150 $\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$ (

**right**); antiresonance (black) at 1.63 $\mathrm{Hz}$ (

**left**) and 2.17 $\mathrm{Hz}$, respectively.

Mechanical Properties | |||
---|---|---|---|

Inertia link ${I}_{l}$ | 0.94 $\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$ | Inertia actuator ${I}_{a}$ | 1.15 $\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$ |

Mass link ${m}_{l}$ | 6.81 $\mathrm{k}\mathrm{g}$ | Length link ${l}_{l}$ | 0.362 $\mathrm{m}$ |

Coulomb fric. coeff. link ${\widehat{\tau}}_{f,l}$ | 3.3 × 10^{−2} $\mathrm{N}\mathrm{m}$ | Coulomb fric. coeff. actuator ${\widehat{\tau}}_{coul}$ | 2.4 $\mathrm{N}\mathrm{m}$ |

Viscous fric. coeff. $\sigma $ | −0.8 $\mathrm{N}\mathrm{m}\mathrm{s}$ | Stribeck fric. amplitude ${\widehat{\tau}}_{stri}$ | 376.1 $\mathrm{N}\mathrm{m}$ |

Stribeck form factor $\delta $ | −0.13 | Stribeck friction velocity ${V}_{S}$ | 3.6 × 10^{4} |

Gear ratio ${i}_{G}$ | 80 | ||

Electrical Properties | |||

Terminal resistance R | 0.4 Ω | Torque constant ${k}_{t}$ | 55 $\mathrm{m}\mathrm{N}\mathrm{m}\mathrm{A}$^{−1} |

Terminal inductance L | 0.8 $\mathrm{m}\mathrm{H}$ | Speed constant ${k}_{b}$ | 173.6 $\mathrm{rpm}\mathrm{V}$^{−1} |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Beckerle, P.; Stuhlenmiller, F.; Rinderknecht, S. Stiffness Control of Variable Serial Elastic Actuators: Energy Efficiency through Exploitation of Natural Dynamics. *Actuators* **2017**, *6*, 28.
https://doi.org/10.3390/act6040028

**AMA Style**

Beckerle P, Stuhlenmiller F, Rinderknecht S. Stiffness Control of Variable Serial Elastic Actuators: Energy Efficiency through Exploitation of Natural Dynamics. *Actuators*. 2017; 6(4):28.
https://doi.org/10.3390/act6040028

**Chicago/Turabian Style**

Beckerle, Philipp, Florian Stuhlenmiller, and Stephan Rinderknecht. 2017. "Stiffness Control of Variable Serial Elastic Actuators: Energy Efficiency through Exploitation of Natural Dynamics" *Actuators* 6, no. 4: 28.
https://doi.org/10.3390/act6040028