# Exploring the Just Noticeable Interaction Stiffness Differences of an Impedance-Controlled Series Elastic Actuator

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## Abstract

**:**

## 1. Introduction

**Hypothesis 1 (H1).**

**Hypothesis 2 (H2).**

## 2. Materials and Methods

#### 2.1. Variable Torsional Stiffness Actuator

#### 2.2. Impedance Control

**Setting A:**${k}_{s}=50\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$, ${k}_{d}=150\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$,**Setting B:**${k}_{s}=150\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$, ${k}_{d}=50\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$.

#### 2.3. User Study

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

JND | Just Noticeable Difference |

DC | Direct Current |

VTS | Variable Torsional Stiffness |

IV | Independent Variable |

FFT | Fast Fourier Transform |

DFG | Deutsche Forschungsgemeinschaft |

${k}_{d}$ | Virtual Stiffness (of the impedance control) |

${k}_{s}$ | Real Stiffness (of the elastic element) |

${k}_{i}$ | Interaction Stiffness (Combination of virtual and real stiffness) |

${k}_{init}$ | Initial Value of Stiffness during an experiment |

${k}_{step}$ | Increase in the Value of Stiffness in between movements of a trial |

## References

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**Figure 1.**Picture of the VTS actuator with (1) DC motor with a gearbox (Actuation Unit 1), connected to the torsional rod via a coupling; (2) DC motor with gearbox (Actuation Unit 2), connected to torsional rod mechanism via a coupling; (3) Linear potentiometer measuring ${l}_{s}$; (4) Torsional rod with a mechanism to change its length ${l}_{s}$; (5) Torque sensor measuring ${\tau}_{s}$; (6) Lever for user interaction.

**Figure 2.**Mechanism to change effective length of the torsional rod. Actuation Unit 2 moves a counter bearing along the torsional rod that transmits the torque from the torsional rod to the lever, which is used for user interaction. Changing the position of the bearing thus changes the effective length ${l}_{s}$ of the torsional rod.

**Figure 3.**Structure of an impedance-controlled SEA with real stiffness ${k}_{s}$, load inertia ${J}_{l}$, and virtual impedance parameters: stiffness ${k}_{d}$, damping ${d}_{d}$, and inertia ${J}_{a,d}$. The system behaves as a two-mass torsional oscillator subject to the interaction torque ${\tau}_{int}$.

**Figure 4.**Interaction step responses. Deflection of $0.35\mathrm{rad}$ from a fixed lever position. Setting A: ${k}_{s}=50\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$, ${k}_{d}=150\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$; Setting B: ${k}_{s}=150\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$, ${k}_{d}=50\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$. Tests show that the expected interaction behavior, i.e., ${\tau}_{int}={k}_{i}{\tilde{\phi}}_{l}$, is obtained when implementing inertia shaping. Without it, unmodeled static friction components are noticeable and deform the interaction curve ${\tau}_{int}$ vs. ${\tilde{\phi}}_{l}$.

**Figure 5.**Periodic interaction examples with setting A: ${k}_{s}=50\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$, ${k}_{d}=150\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$. The periodic interaction signals are obtained by deflecting the position of the lever following a visual and auditory metronome. The plots show 15 cycles of measured torque and deflection measurements.

**Figure 6.**Measured and theoretical Bode magnitude plot (

**top**) and phase plot (

**bottom**) of stiffness transfer function $K\left(s\right)=\frac{{\tau}_{int}\left(s\right)}{{\tilde{\phi}}_{l}\left(s\right)}$. Setting A: ${k}_{s}=50\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$, ${k}_{d}=150\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$, Setting B: ${k}_{s}=150\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$, ${k}_{d}=50\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$. Natural frequencies of measured response match.

**Figure 7.**VTS experimental setup. The actuator is fully covered by black plexiglass and a noise cancelling headset is used during experiments.

**Figure 8.**Example run of one experiment. The varied stiffness is ${k}_{d}$, the initial value ${k}_{init}$ is 50 $\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$. The value of ${k}_{s}$ is 220 $\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$ for all trials. Reversals are shown as filled red circles. The resulting JND is calculated using the average value of ${k}_{step}$ of the last 10 trials.

**Figure 9.**Boxplot of the obtained absolute JND (

**left**) and normalized JND (

**right**). Condition 1: Varied stiffness ${k}_{s}$ (real), initial stiffness $100\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$. Condition 2: Varied stiffness ${k}_{s}$ (real), initial stiffness $50\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$. Condition 3: Varied stiffness ${k}_{d}$ (virtual), initial stiffness $100\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$. Condition 4: Varied stiffness ${k}_{d}$ (virtual), initial stiffness $50\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$.

Description | Symbol | Value | Unit |
---|---|---|---|

Mass of the lever | ${m}_{l}$ | 1.3 | $\mathrm{k}\mathrm{g}$ |

Distance to center of mass of the lever | ${l}_{l}$ | 0.4 | $\mathrm{m}$ |

Moment of inertia of the lever | ${J}_{l}$ | 0.29 | $\mathrm{k}\mathrm{g}{\mathrm{m}}^{2}$ |

Moment of inertia of Actuation Unit 1 | ${J}_{a}$ | 2 | $\mathrm{k}\mathrm{g}{\mathrm{m}}^{2}$ |

Viscous friction coefficient of Actuator unit 1 | ${B}_{a,v}$ | 4 | $\mathrm{N}\mathrm{m}\mathrm{s}{\mathrm{rad}}^{-1}$ |

Coulomb friction coefficient of Actuator unit 1 | ${B}_{a,c}$ | 5.5 | $\mathrm{N}\mathrm{m}$ |

Shaping factor for friction model | S | 5 | 1 |

Rod length range | ${l}_{s}$ | 0.0275–0.19 | $\mathrm{m}$ |

Rod stiffness (real) range | ${k}_{s}$ | 43–220 | $\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$ |

Control stiffness (virtual) range | ${k}_{d}$ | 43–220 | $\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$ |

Control damping (virtual) | ${d}_{d}$ | 0 | $\mathrm{N}\mathrm{m}\mathrm{s}{\mathrm{rad}}^{-1}$ |

Control inertia (shaped) | ${J}_{a,d}$ | 0.2 | $\mathrm{k}\mathrm{g}{\mathrm{m}}^{2}$ |

Cond. No. | Varied Stiffness | Initial Stiffness ${\mathit{k}}_{\mathbf{init}}$ in N m rad ^{−1} | ${\mathit{k}}_{\mathbf{step}}$ of the First Trial in N m rad ^{−1} | Stiffness Increase ${\mathbf{\Delta}}^{+}$ in N m rad ^{−1} | Stiffness Decrease ${\mathbf{\Delta}}^{-}$ in N m rad ^{−1} |
---|---|---|---|---|---|

1 | ${k}_{s}$ | 100 | 100 | 20 | 10 |

2 | ${k}_{s}$ | 50 | 50 | 10 | 5 |

3 | ${k}_{d}$ | 100 | 100 | 20 | 10 |

4 | ${k}_{d}$ | 50 | 50 | 10 | 5 |

Cond. No. | Varied Stiffness | Initial Stiffness ${\mathit{k}}_{\mathbf{init}}$ in N m rad${}^{-1}$ | Mean No. of trials | Absolute JND in N m rad${}^{-1}$ | Normalized JND in % of ${\mathit{k}}_{\mathbf{init}}$ |
---|---|---|---|---|---|

1 | ${k}_{s}$ | 100 | $27\pm 4$ | $16.40\pm 4.69$ | $16.40\pm 4.69$ |

2 | ${k}_{s}$ | 50 | $29\pm 4$ | $8.50\pm 2.44$ | $17.00\pm 4.88$ |

3 | ${k}_{d}$ | 100 | $27\pm 2$ | $16.00\pm 3.68$ | $16.00\pm 3.68$ |

4 | ${k}_{d}$ | 50 | $28\pm 4$ | $8.17\pm 2.18$ | $16.33\pm 4.35$ |

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## Share and Cite

**MDPI and ACS Style**

Velasco-Guillen, R.J.; Schofer, F.; Bliek, A.; Beckerle, P.
Exploring the Just Noticeable Interaction Stiffness Differences of an Impedance-Controlled Series Elastic Actuator. *Actuators* **2023**, *12*, 378.
https://doi.org/10.3390/act12100378

**AMA Style**

Velasco-Guillen RJ, Schofer F, Bliek A, Beckerle P.
Exploring the Just Noticeable Interaction Stiffness Differences of an Impedance-Controlled Series Elastic Actuator. *Actuators*. 2023; 12(10):378.
https://doi.org/10.3390/act12100378

**Chicago/Turabian Style**

Velasco-Guillen, Rodrigo J., Felix Schofer, Adna Bliek, and Philipp Beckerle.
2023. "Exploring the Just Noticeable Interaction Stiffness Differences of an Impedance-Controlled Series Elastic Actuator" *Actuators* 12, no. 10: 378.
https://doi.org/10.3390/act12100378