# Robust Force Control of Series Elastic Actuators

^{*}

## Abstract

**:**

## 1. Introduction

**Figure 1.**Typical rehabilitation robotics architecture based on force-controlled actuators. Symbols $C,\phantom{\rule{0.222222em}{0ex}}E$ and M represent the force controller, the electronics and the robot mechanics, respectively. The interaction force/torque is represented by τ and the interaction velocity by $\dot{\theta}$.

## 2. Sliding-Mode Control

#### 2.1. Sliding-Mode Concepts

#### 2.2. Continuous Approximation of SMC

#### 2.3. Higher Order SMC

## 3. Compliant Interaction Model

**Figure 2.**Representation of a human-robot interaction using a series elastic actuator (SEA). The human dynamics is considered unknown, and it is represented as the dashed area.

## 4. System Control

#### 4.1. Disturbance Analysis

#### 4.2. Control Design

#### 4.3. Improving the Accuracy within the Boundary

## 5. Experimental Results

**Figure 3.**The SEA prototype used as the testbed. The motor, M, is connected to the spring, S, and the angular quantities, ${\theta}_{m}$ and ${\theta}_{h}$, are measured by encoders ${E}_{1}$ and ${E}_{2}$, respectively. L is the arm support that is a pure inertial load when the system is not interacting with the human.

Parameter | Symbol | Value |
---|---|---|

Spring Stiffness | k | 1.040 Nm/rad |

Torque constant | ${k}_{t}$ | 0.42 Nm/A |

Motor inertia | ${J}_{m}$ | 0.00041 kg/m${}^{2}$ |

Wood load inertia | ${J}_{W}$ | 0.00025 kg/m${}^{2}$ |

Metal load inertia | ${J}_{M}$ | 0.00390 kg/m${}^{2}$ |

Control Law | Acron. | Sliding Surface | ${\widehat{\tau}}_{\mathrm{eq}}$ | k |
---|---|---|---|---|

$u={\widehat{\tau}}_{eq}-k\mathrm{sign}(w)$ | SM | standard (3) | (14) | $0.5$ |

$u={\widehat{\tau}}_{eq}-k\mathrm{sign}(w)$ | ISM | integral (18) | (19) | $0.5$ |

$u={\widehat{\tau}}_{eq}-k{\mathrm{h}}_{linear}(w)$ | ILA | integral (18) | (19) | $0.5$ |

$u={\widehat{\tau}}_{eq}-k\overline{\mathrm{h}}(w)$ | ILAR | integral (18) | (19) | $0.5$ |

Super Twisting | STW | integral (18) | (19) | $0.2$ * |

#### 5.1. Qualitative Analysis

**Figure 4.**Torque tracking of a sinusoidal reference with the SM algorithm. The load is held by hand and then released at about $t=1.1$ s.

**Figure 5.**Torque tracking of a sinusoidal reference with integral sliding-mode (ISM algorithm). The load is held by hand and then released at about $t=1.1$ s.

**Figure 6.**Torque tracking of a sinusoidal reference using a linear approximation (ILA algorithm). The load is held by hand and then released at about $t=1.4$ s.

**Figure 7.**Torque tracking of a sinusoidal reference using a linear approximation and a periodic task model (ILAR algorithm). The load is held by hand and then released at about $t=1.4$ s.

**Figure 8.**Torque tracking of a sinusoidal reference using the ILAR algorithm in a longer experiment to better see the resonator dynamics (sometimes, it is necessary for the resonator to acquire the oscillating energy). The load is held by hand and then released at about $t=3.2$ s.

**Figure 9.**Torque tracking of a sinusoidal reference using the STW algorithm. The load is held by hand and then released at about $t=1.4$ s.

**Figure 10.**Step response comparison among the SM (

**top**), ISM (

**middle**) and ILA (

**bottom**) algorithms. The left and right figures show the responses to a step of an amplitude of 0.5 and 1.0 Nm, respectively. Note that the reaching time of the SM algorithm is about 0.1 s on the left plot and 0.2 s on the right plot.

**Figure 11.**Torque tracking of a sinusoidal reference using a passive PID. The load is firstly held by hand and then released at about $t=3.5$ s. Note that the error scale is zoomed out with respect to the previous figures.

**Figure 12.**Torque tracking of a sinusoidal reference using a passive PD controller tuned on a free load condition. The load is firstly held by hand and then released at about $t=3$ s.

#### 5.2. Quantitative Analysis and Comparison

Adaptive | Sliding-Mode (ILA/ILAR) | |
---|---|---|

Stability | In the sense of boundedness | In the sense of boundedness (sliding boundary) |

Assumption for stability | Bounded human forces, second order human model | Bounded human accelerations |

Human model | Second order linear system | Any |

Accuracy | High accuracy is observed | A frequency model of the task is needed for high accuracy |

Sensors | Environment position not required (force sensor) | Environment position (or velocity) required |

Tuning | No tuning is necessary | Disturbance bound, boundary layer and task frequency |

High Impedance | Low Impedance | ||||||
---|---|---|---|---|---|---|---|

ILA | ILAR | AD | ILA | ILAR | AD | ||

$RMS$ | 13.9 | 5.3 | 6.0 | 21.0 | 9.5 | 6.2 | |

${\sigma}_{RMS}$ | 0.3 | 0.4 | 0.19 | 0.3 | 0.8 | 0.27 | |

$Max$ | 22.4 | 10.8 | 13.6 | 32.0 | 18.6 | 13.5 | |

${\sigma}_{Max}$ | 1.2 | 1.0 | 1.0 | 1.0 | 2.4 | 2.1 |

**Figure 13.**Graphical representation of the quantitative comparison with $3\sigma $ intervals to compare the performances in the high (green) and low (yellow) impedance conditions. The labels, PID and PD, are related to passive controllers, while AD indicates the adaptive controller.

**Figure 14.**The response of the adaptive control (

**top**) and ILAR algorithm (

**bottom**) in an intermittent contact test.

## 6. Conclusions

## A. Control Safety

## B. Adaptive Force Control

## Author Contributions

## Conflicts of Interest

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Calanca, A.; Capisani, L.; Fiorini, P.
Robust Force Control of Series Elastic Actuators. *Actuators* **2014**, *3*, 182-204.
https://doi.org/10.3390/act3030182

**AMA Style**

Calanca A, Capisani L, Fiorini P.
Robust Force Control of Series Elastic Actuators. *Actuators*. 2014; 3(3):182-204.
https://doi.org/10.3390/act3030182

**Chicago/Turabian Style**

Calanca, Andrea, Luca Capisani, and Paolo Fiorini.
2014. "Robust Force Control of Series Elastic Actuators" *Actuators* 3, no. 3: 182-204.
https://doi.org/10.3390/act3030182