# 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

## References

- Ekkelenkamp, R.; Veneman, J.; van der Kooij, H. LOPES: Selective Control of Gait Functions During the Gait Rehabilitation of CVA Patients. In Procedings of the 9th International Conference on Rehabilitation Robotics, Chicago, IL, USA, 28 June–1 July 2005; pp. 361–364.
- Herr, H.M.; Grabowski, A.M. Bionic ankle-foot prosthesis normalizes walking gait for persons with leg amputation. Proc. Biolog. Sci.
**2012**, 279, 457–464. [Google Scholar] [CrossRef] [PubMed] - Lens, T.; Kunz, J.; Stryk, O.V.; Trommer, C.; Karguth, A. BioRob-Arm: A Quickly Deployable and Intrinsically Safe , Light-Weight Robot Arm for Service Robotics Applications. In Proceedings of the 41st International Symposium on Robotics, Munich, Germany, 7–9 June 2010; pp. 905–910.
- Duschau-Wicke, A.; von Zitzewitz, J.; Caprez, A.; Lunenburger, L.; Riener, R. Path control: A method for patient-cooperative robot-aided gait rehabilitation. IEEE Trans. Neural Syst. Rehabilit. Eng.
**2010**, 18, 38–48. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pratt, J.; Chew, C.M.; Torres, A.; Dilworth, P.; Pratt, G. Virtual model control: An intuitive approach for bipedal locomotion. Int. J. Rob. Res.
**2001**, 20, 129–143. [Google Scholar] [CrossRef] - Vallery, H.; van Asseldonk, E.H.F.; Buss, M.; van der Kooij, H. Reference trajectory generation for rehabilitation robots: complementary limb motion estimation. IEEE Trans. Neural Syst. Rehabilit. Eng.
**2009**, 17, 23–30. [Google Scholar] [CrossRef] [PubMed] - Hogan, N. Impedance Control: An Approach to Manipulation: Part I,II,III. Journal of dynamic systems, measurement and control
**1985**, 107, 1–24. [Google Scholar] [CrossRef] - Kong, K.; Member, S.; Bae, J. Control of Rotary Series Elastic Actuator for Ideal Force-Mode Actuation in Human-Robot Interaction Applications. IEEE/ASME Int. Conf. Mechatr.
**2009**, 14, 105–118. [Google Scholar] [CrossRef] - Calanca, A.; Piazza, S.; Fiorini, P. Force Control System for Pneumatic Actuators of an Active Gait Orthosis. In Proceedings of the 2010 3rd IEEE RAS and EMBS International Conference on Biomedical Robotics and Biomechatronics (BioRob), Tokyo, Japan, 26–29 September 2010; pp. 849–854.
- Calanca, A.; Piazza, S.; Fiorini, P. A motor learning oriented, compliant and mobile Gait Orthosis. Appl. Bionic. Biomech.
**2012**, 9, 15–27. [Google Scholar] [CrossRef] - Vallery, H. Stable and User-Controlled Assistance of Human Motor Function. Ph.D. Thesis, University of Munchen, Munich, Germany, 2009. [Google Scholar]
- Buerger, S.; Hogan, N. Relaxing Passivity for Human-Robot Interaction. In Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, Beijing, China, 9–15 October 2006; pp. 4570–4575.
- Buerger, S.; Hogan, N. Complementary stability and loop shaping for improved human-robot interaction. IEEE Trans. Robot.
**2007**, 23, 232–244. [Google Scholar] [CrossRef] - Colgate, E. The Control of Dynamically Interacting Systems. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 1988. [Google Scholar]
- Hogan, N. Controlling impedance at the man/machine interface. In Proceedings of the International Conference on Robotics and Automation, Scottsdale, AZ, USA, 14–19 May 1989; pp. 1626–1631.
- Pratt, G.; Williamson, M.; Dillworth, P. Stiffness isn’t everything. In Proceedings of the Fourth International Symposium on Experimental Robotics, Stanford, CA, USA, 30 June–2 July 1995.
- Tagliamonte, N.L.; Accoto, D. Passivity Constraints for the Impedance Control of Series Elastic Actuators. J. Syst. Contr. Eng.
**2013**, 228, 138–153. [Google Scholar] [CrossRef] - Calanca, A.; Fiorini, P. Human-Adaptive Control of Series Elastic Actuators. Robotica
**2014**, in press. [Google Scholar] - Vallery, H.; Veneman, J.; van Asseldonk, E.H.F.; Ekkelenkamp, R.; Buss, M.; van Der Kooij, H. Compliant actuation of rehabilitation robots. IEEE Robot. Automa. Mag.
**2008**, 15, 60–69. [Google Scholar] [CrossRef] - Pratt, G.A.; Willisson, P.; Bolton, C. Late motor processing in low-impedance robots: Impedance control of series-elastic actuators. In Proceedings of the American Control Conference, Boston, MA, USA, 30 June–2 July 2004; pp. 3245–3251.
- Markus, G.; Roman, M.; Konigorski, U. Model Based Control of Series Elastic Actuators. In Proceedings of the IEEE RAS/EMBS International Conference on Biomedical Robotics and Biomechatronics, Rome, Italy, 24–27 June 2012; pp. 538–543.
- Bae, J.; Kong, K.; Tomizuka, M. Gait Phase-Based Smoothed Sliding Mode Control for a Rotary Series Elastic Actuator Installed on the Knee Joint. In Proceedings of the American Control Conference, Baltimore, MD, USA, 30 June–2 July 2010; pp. 6030–6035.
- Utkin, V.I. Sliding Modes in Control and Optimization; Communications and Control Engineering; Springer-Verlag: Berlin, Germany, 1992; p. 286. [Google Scholar]
- Slotine, J.J.E.; Li, W. Applied Nonlinear Control; Prentice Hall: Englewood Cliffs, NJ, USA; Vol. 62, 1991; pp. 174–174. [Google Scholar]
- Damme, M.V.; Vanderborght, B. Proxy-based sliding mode control of a planar pneumatic manipulator. Int. J. Rob. Res.
**2009**, 28, 266–284. [Google Scholar] [CrossRef] - Beyl, P.; Van Damme, M.; Van Ham, R.; Vanderborght, B.; Lefeber, D. Design and control of a lower limb exoskeleton for robot-assisted gait training. Appl. Bionic. Biomech.
**2009**, 6, 229–243. [Google Scholar] [CrossRef] - Frisoli, A.; Sotgiu, E.; Procopio, C.; Bergamasco, M.; Rossi, B.; Chisari, C. Design and Implementation of a Training Strategy in Chronic Stroke with an Arm Robotic Exoskeleton. In Proceedings of the International Conference on Rehabilitation Robotics, Zurich, 29 June–1 July 2011; pp. 1127–1134.
- Kong, K. Proxy-based impedance control of a cable-driven assistive system. Mechatronics
**2013**, 23, 147–153. [Google Scholar] [CrossRef] - Kikuuwe, R.; Yasukouchi, S.; Fujimoto, H.; Yamamoto, M. Proxy-Based Sliding Mode Control: A Safer Extension of PID Position Control. IEEE Trans. Rob.
**2010**, 26, 670–683. [Google Scholar] [CrossRef] - Edwards, C.; Spurgeon, S. Sliding Mode Control: Theory And Applications; Taylor & Francis: London, UK, 1998. [Google Scholar]
- Bartolini, G.; Ferrara, A.; Usai, E. Chattering avoidance by second-order sliding mode control. IEEE Trans. Autom. Contr.
**1998**, 43, 241–246. [Google Scholar] [CrossRef] - Levant, A. Higher-order sliding modes, differentiation and output-feedback control. Int. J. Contr.
**2003**, 76, 924–941. [Google Scholar] [CrossRef] - Francis, B.; Wonham, W. The internal model principle of control theory. Automatica
**1976**, 12, 457–465. [Google Scholar] [CrossRef] - Calanca, A.; Capisani, L.; Ferrara, A.; Magnani, L. MIMO closed loop identification of an industrial robot. IEEE Trans. Contr. Syst. Technol.
**2011**, 19, 1214–1224. [Google Scholar] [CrossRef] - Calanca, A.; Capisani, L.; Fiorini, P.; Ferrara, A. Improving Continuous Approximation of Sliding Mode Control. In Proceedings of the 16th International Conference on Advanced Robotics, Montevideo, Uruguay, 25–29 November 2013.
- Pratt, G.A.; Williamson, M. Series Elastic Actuators. In Proceedings of the International Conference on Intelligent Robots and Systems, Pittsburgh, PA, USA, 5–9 August 1995; pp. 399–406.

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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