# Variable Admittance Control in Sliding Mode for Robust Physical Human–Robot Interaction

^{*}

## Abstract

**:**

## 1. Introduction

- Limitations of conventional SMC techniques applied to VAC were thoroughly investigated from perspectives of tracking performance, adaptation to time-varying admittance parameters, reachability condition, and chattering removal.
- A new indirect tracking approach was proposed with a new universal sliding surface, corresponding control law, and theoretical proof, yielding improved VAC tracking performance.
- Reachability condition of the proposed SMC approach for VAC was proved, where the challenge of deriving an analytical solution for Coriolis and centrifugal matrix used in Laypnov stability proof for a higher DoF system was addressed.
- Acceleration feedback was included in equivalent control, and its capability to improve tracking performance was verified in numerical simulation and experiments.

## 2. Human-Intention-Based VAC and PER

#### 2.1. Variable Admittance Control

#### 2.2. Power Envelope Regulation

## 3. Reachability in VAC

#### 3.1. Conventional Sliding Surface

#### 3.2. Sliding Surface in Admittance Control

**w**’ represents a fixed Cartesian coordinate at the base of the manipulator in world space, and the mapping from joint to Cartesian space of the end-effector is simply the Jacobian-based operation:

**,**while leaving $\mathit{H}$, $\mathit{C}$, and $\mathit{G}$ as functions of the original system states, is incorrect and creates confusion. A new sliding surface is proposed with theoretical proof to address the above limitations.

#### 3.3. Reachability in VAC

**,**that is

**,**i.e., ${\mathit{C}}_{\mathit{w}}^{\mathit{*}}$, guarantee overall reachability for all solutions of ${\mathit{C}}_{\mathit{w}}$? To address this issue, the unique ${\mathit{S}}_{\mathit{w}}\left(\mathit{x},\dot{\mathit{x}}\right)$ is adopted, and the term ${\mathit{C}}_{\mathit{w}}^{\mathit{*}}$ is modeled as a source of uncertainty (refer to the following proof). Furthermore, time-varying admittance parameters are taken into account.

**Control law for SMC in VAC.**The system is stable in the sense of Lyapunov with the control law considering uncertainties and disturbances:

**Proof.**

**,**${\widehat{\mathit{S}}}_{\mathit{w}}\left(\mathit{x}\right)$, and ${\widehat{\mathit{G}}}_{\mathit{w}}\left(\mathit{x}\right)$ include kinematic parameters and inertial terms. As long as the modeling uncertainties are bounded, the above assumption is valid. Substituting Equations (37) and (38) into Equation (36) arrives

#### 3.4. Chattering Removal

#### 3.5. Acceleration Feedback in SMC

## 4. Experimental Validation

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of Universal Characteristic of the New Sliding Surface in Human-Following Case:**

## Appendix B

**Derivation of Variable Sliding Mode Gain with Acceleration Feedback:**

## Appendix C

**Assumption**

**A1.**

**Assumption**

**A2.**

**,**${\mathit{S}}_{\mathit{w}}=-{{\mathit{J}}^{-1}}^{\mathit{T}}\mathit{H}\left(\mathit{q}\right){\mathit{J}}^{-1}\dot{\mathit{J}}{\mathit{J}}^{-1}+{{\mathit{J}}^{-1}}^{\mathit{T}}\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right){\mathit{J}}^{-1}$, which contributes to deriving the sliding mode gain. In Equation (39),${\mathit{C}}_{\mathit{w}}^{\mathit{*}}$is bounded as long as the entire system is stable. Moreover,$\mathit{\sigma}$is also bounded and approaches zero. Therefore, it is fair to assume that${\mathit{C}}_{\mathit{w}}^{\mathit{*}}\mathit{\sigma}$is bounded, such that it can be modeled as bounded uncertainty.

**Assumption**

**A3.**

## References

- Krüger, J.; Lien, T.K.; Verl, A. Cooperation of human and machines in assembly lines. CIRP Ann.
**2009**, 58, 628–646. [Google Scholar] [CrossRef] - Mörtl, A.; Lawitzky, M.; Kucukyilmaz, A.; Sezgin, M.; Basdogan, C.; Hirche, S. The role of roles: Physical cooperation between humans and robots. Int. J. Robot. Res.
**2012**, 31, 1656–1674. [Google Scholar] [CrossRef] - Solanes, J.E.; Gracia, L.; Munoz-Benavent, P.; Miro, J.V.; Carmichael, M.G.; Tornero, J. Human–robot collaboration for safe object transportation using force feedback. Robot. Auton. Syst.
**2018**, 107, 196–208. [Google Scholar] [CrossRef] - Xing, H.; Torabi, A.; Ding, L.; Gao, H.; Li, W.; Mushahwar, V.K.; Tavakoli, M. Human-robot collaboration for heavy object manipulation: Kinesthetic teaching of the role of wheeled mobile manipulator. In Proceedings of the 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Prague, Czech Republic, 27 September–1 October 2021; pp. 2962–2969. [Google Scholar]
- Xing, H.; Torabi, A.; Ding, L.; Gao, H.; Deng, Z.; Mushahwar, V.K.; Tavakoli, M. An admittance-controlled wheeled mobile manipulator for mobility assistance: Human–robot interaction estimation and redundancy resolution for enhanced force exertion ability. Mechatronics
**2021**, 74, 102497. [Google Scholar] [CrossRef] - Xing, H.; Ding, L.; Gao, H.; Li, W.; Tavakoli, M. Dual-user haptic teleoperation of complementary motions of a redundant wheeled mobile manipulator considering task priority. IEEE Trans. Syst. Man Cybern. Syst.
**2022**, 52, 6283–6295. [Google Scholar] [CrossRef] - Liu, Z.; Hao, J. Intention recognition in physical human-robot interaction based on radial basis function neural network. J. Robot.
**2019**, 2019, 4141269. [Google Scholar] [CrossRef] - Park, J.S.; Park, C.; Manocha, D. I-planner: Intention-aware motion planning using learning-based human motion prediction. Int. J. Robot. Res.
**2019**, 38, 23–39. [Google Scholar] [CrossRef] - Tortora, S.; Michieletto, S.; Stival, F.; Menegatti, E. Fast human motion prediction for human-robot collaboration with wearable interface. In Proceedings of the 2019 IEEE International Conference on Cybernetics and Intelligent Systems (CIS) and IEEE Conference on Robotics, Automation and Mechatronics (RAM), Bangkok, Thailand, 18–20 November 2019; pp. 457–462. [Google Scholar]
- Li, S.; Zhang, L.; Diao, X. Deep-learning-based human intention prediction using RGB images and optical flow. J. Intell. Robot. Syst.
**2020**, 97, 95–107. [Google Scholar] [CrossRef] - Krishnan, R.H.; Pugazhenthi, S. Mobility assistive devices and self-transfer robotic systems for elderly, a review. Intell. Serv. Robot.
**2014**, 7, 37–49. [Google Scholar] [CrossRef] - Yan, T.; Cempini, M.; Oddo, C.M.; Vitiello, N. Review of assistive strategies in powered lower-limb orthoses and exoskeletons. Robot. Auton. Syst.
**2015**, 64, 120–136. [Google Scholar] [CrossRef] - Windrich, M.; Grimmer, M.; Christ, O.; Rinderknecht, S.; Beckerle, P. Active lower limb prosthetics: A systematic review of design issues and solutions. Biomed. Eng. Online
**2016**, 15, 5–19. [Google Scholar] [CrossRef] - Han, J.H.; Lee, S.J.; Kim, J.H. Behavior hierarchy-based affordance map for recognition of human intention and its application to human–robot interaction. IEEE Trans. Hum.-Mach. Syst.
**2016**, 46, 708–722. [Google Scholar] [CrossRef] - Chen, J.; Ro, P.I. A Conceptual Approach of Passive Human-Intention-Orientated Variable Admittance Control using Power Envelope. In Proceedings of the 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Prague, Czech Republic, 27 September–1 October 2021; pp. 7300–7306. [Google Scholar]
- Chen, J.; Ro, P.I. Human intention-oriented variable admittance control with power envelope regulation in physical human-robot interaction. Mechatronics
**2022**, 84, 102802. [Google Scholar] [CrossRef] - Müller, F.; Janetzky, J.; Behrnd, U.; Jäkel, J.; Thomas, U. User force-dependent variable impedance control in human-robot interaction. In Proceedings of the 2018 IEEE 14th International Conference on Automation Science and Engineering (CASE), Munich, Germany, 20–24 August 2018; pp. 1328–1335. [Google Scholar]
- Cacace, J.; Finzi, A.; Lippiello, V. Enhancing shared control via contact force classification in human-robot cooperative task execution. In Human Friendly Robotics; Springer: Cham, Switzerland, 2019; pp. 167–179. [Google Scholar]
- Hogan, N. Impedance control: An approach to manipulation. In Proceedings of the 1984 American Control Conference, San Diego, CA, USA, 6–8 June 1984; pp. 304–313. [Google Scholar]
- Ott, C.; Mukherjee, R.; Nakamura, Y. Unified impedance and admittance control. In Proceedings of the 2010 IEEE International Conference on Robotics and Automation, Anchorage, AK, USA, 3–7 May 2010; pp. 554–561. [Google Scholar]
- Keemink, A.Q.; van der Kooij, H.; Stienen, A.H. Admittance control for physical human–robot interaction. Int. J. Robot. Res.
**2018**, 37, 1421–1444. [Google Scholar] [CrossRef] - Duchaine, V.; Gosselin, C. Safe, stable and intuitive control for physical human-robot interaction. In Proceedings of the 2009 IEEE International Conference on Robotics and Automation, Kobe, Japan, 12–17 May 2009; pp. 3383–3388. [Google Scholar]
- Li, Z.; Huang, B.; Ye, Z.; Deng, M.; Yang, C. Physical human–robot interaction of a robotic exoskeleton by admittance control. IEEE Trans. Ind. Electron.
**2018**, 65, 9614–9624. [Google Scholar] [CrossRef] - Li, K.; Chen, R.; Nuchkrua, T.; Boonto, S. Dual loop compliant control based on human prediction for physical human-robot interaction. In Proceedings of the 2019 58th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE), Hiroshima, Japan, 10–13 September 2019; pp. 459–464. [Google Scholar]
- Lecours, A.; Mayer-St-Onge, B.; Gosselin, C. Variable admittance control of a four-degree-of-freedom intelligent assist device. In Proceedings of the 2012 IEEE International Conference on Robotics and Automation, Saint Paul, MN, USA, 14–18 May 2012; pp. 3903–3908. [Google Scholar]
- Sharkawy, A.N.; Koustoumpardis, P.N.; Aspragathos, N. A neural network-based approach for variable admittance control in human–robot cooperation: Online adjustment of the virtual inertia. Intell. Serv. Robot.
**2020**, 13, 495–519. [Google Scholar] [CrossRef] - Li, Y.; Ge, S.S. Human-Robot Collaboration Based on Motion Intention Estimation. IEEE/ASME Trans. Mechatron.
**2013**, 19, 1007–1014. [Google Scholar] [CrossRef] - Itadera, S.; Kobayashi, T.; Nakanishi, J.; Aoyama, T.; Hasegawa, Y. Impedance control based assistive mobility aid through online classification of user’s state. In Proceedings of the 2019 IEEE/SICE International Symposium on System Integration (SII), Paris, France, 14–16 January 2019; pp. 243–248. [Google Scholar]
- Dimeas, F.; Aspragathos, N. Online stability in human-robot cooperation with admittance control. IEEE Trans. Haptics
**2016**, 9, 267–278. [Google Scholar] [CrossRef] - Kronander, K.; Billard, A. Stability considerations for variable impedance control. IEEE Trans. Robot.
**2016**, 32, 1298–1305. [Google Scholar] [CrossRef] - Ferraguti, F.; Talignani Landi, C.; Sabattini, L.; Bonfè, M.; Fantuzzi, C.; Secchi, C. A variable admittance control strategy for stable physical human–robot interaction. Int. J. Robot. Res.
**2019**, 38, 747–765. [Google Scholar] [CrossRef] - Park, J.; Choi, Y. Input-to-state stability of variable impedance control for robotic manipulator. Appl. Sci.
**2020**, 10, 1271. [Google Scholar] [CrossRef] - Slotine, J.J.E.; Li, W. Applied Nonlinear Control; Prentice Hall: Englewood Cliffs, NJ, USA, 1991; Volume 199, p. 705. [Google Scholar]
- Bucak, İ.Ö. An In-Depth Analysis of Sliding Mode Control and Its Application to Robotics. In Automation and Control; IntechOpen: London, UK, 2020. [Google Scholar]
- Lu, Z.; Goldenberg, A.A. Robust impedance control and force regulation: Theory and experiments. Int. J. Robot. Res.
**1995**, 14, 225–254. [Google Scholar] - Tu, Y.; Zhu, A.; Song, J.; Shen, H.; Shen, Z.; Zhang, X.; Cao, G. An adaptive sliding mode variable admittance control method for lower limb rehabilitation exoskeleton robot. Appl. Sci.
**2020**, 10, 2536. [Google Scholar] [CrossRef] - Torabi, M.; Sharifi, M.; Vossoughi, G. Robust adaptive sliding mode admittance control of exoskeleton rehabilitation robots. Sci. Iran.
**2018**, 25, 2628–2642. [Google Scholar] [CrossRef] - Liu, J.; Wang, X. Advanced Sliding Mode Control for Mechanical Systems; Springer: Berlin, Germany, 2012; pp. 31–35. [Google Scholar]
- Fazli, E.; Rakhtala, S.M.; Mirrashid, N.; Karimi, H.R. Real-time implementation of a super twisting control algorithm for an upper limb wearable robot. Mechatronics
**2022**, 84, 102808. [Google Scholar] [CrossRef] - Han, J.D.; Wang, Y.C.; Tan, D.L.; Xu, W.L. Acceleration feedback control for direct-drive motor system. In Proceedings of the 2000 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2000) (Cat. No. 00CH37113), Takamatsu, Japan, 31 October–5 November 2000; Volume 2, pp. 1068–1074. [Google Scholar]
- Sedghi, B.; Bauvir, B.; Dimmler, M. Acceleration feedback control on an AT. In Ground-Based and Airborne Telescopes II; SPIE: Bellingham, WA, USA, 2008; Volume 7012, pp. 704–715. [Google Scholar]
- Ma, J.; Yao, Y.; Liu, K. Sliding mode control with angular acceleration feedback for a flight motion simulator. In Proceedings of the 2009 41st Southeastern Symposium on System Theory, Tullahoma, TN, USA, 15–17 March 2009; pp. 190–194. [Google Scholar]
- Aung, M.T.S.; Shi, Z.; Kikuuwe, R. A new parabolic sliding mode filter augmented by a linear low-pass filter and its application to position control. J. Dyn. Syst. Meas. Control.
**2018**, 140, 041005. [Google Scholar] [CrossRef] - Studenny, J.; Belanger, P.R. Robot manipulator control by accelaration feedback: Stability, design and performance issues. In Proceedings of the 1986 25th IEEE Conference on Decision and Control, Athens, Greece, 10–12 December 1986; pp. 80–85. [Google Scholar]
- Buss, S.R. Introduction to inverse kinematics with jacobian transpose, pseudoinverse and damped least squares methods. IEEE J. Robot. Autom.
**2004**, 17, 16. [Google Scholar] - Xing, H.; Torabi, A.; Ding, L.; Gao, H.; Li, W.; Tavakoli, M. Enhancing kinematic accuracy of redundant wheeled mobile manipulators via adaptive motion planning. Mechatronics
**2021**, 79, 102639. [Google Scholar] [CrossRef]

**Figure 1.**Conventional SMC is implemented on a 2-DoF planar manipulator that appeared in (

**d**); (

**a**,

**b**) display direct tracking in joint and Cartesian space, respectively. The reaching phase is observed in (

**a**) when the initial position differs from the reference; (

**c**) displays the interaction force. (

**e**,

**f**) present tracking under admittance control for human-leading and human-following, respectively. In (

**e**), tracking is between actual states and desired response of admittance. In contrast, two tracking goals are required in (

**f**): (i), responses of admittance system track referenced trajectory; (ii), actual trajectory tracks responses of admittance system. Subscription ‘1′ represents the x-axis dimension.

**Figure 3.**SMC with the new sliding surface being applied: (

**a**) displays the same force profile in Figure 1. Admittance response, ${\mathit{x}}_{c}$, does not explicitly appear in the controller; nonetheless, it is tracked by actual states accurately. In comparing (

**b**) to Figure 1e, an improvement is observed that suggests the efficiency of the proposed method. Moreover, the same sliding surface works well for both human-leading and human-following cases as appeared in (

**c**).

**Figure 4.**(

**a**) Simcape model of the manipulator under a planar 3-DoF configuration and (

**b**) Physical human–robot interaction scene with an experimental setup that has a similar configuration.

**Figure 5.**Implementation of SMC in VAC for Kinova Gen3. (

**a**) displays the interaction force. The nominal force in Equation (2) is defined as 5 N, and the maximum safe operating speed is selected as 0.1 m/s in Equation (5). Effort-saving behavior is reflected by the observation that higher velocity is maintained even if the interaction force restores the nominal level from −7 N between 4–5 s. Safe interaction is reflected by the constrained velocity within 0.1 m/s under higher force. In (

**b**), a sliding mode gain is used without considering time-varying admittance parameters, resulting in oscillation around 4 s. In contrast, the proposed control law, i.e., Equations (30)–(32), leads to an improvement in performance in (

**c**). The modeling error is purposely designed to be large to demonstrate the robustness of the controller, where the mass of each link in modeling is designed as ${\widehat{m}}_{1}=0.93\text{}\mathrm{k}\mathrm{g}$, ${\widehat{m}}_{2}=0.27\text{}\mathrm{k}\mathrm{g}$, and ${\widehat{m}}_{3}=60\text{}\mathrm{k}\mathrm{g}$, compared to the exact value, ${m}_{1}=9.3\text{}\mathrm{k}\mathrm{g}$, ${m}_{2}=2.7\text{}\mathrm{k}\mathrm{g}$, and ${m}_{3}=6.0\text{}\mathrm{k}\mathrm{g}$, respectively.

**Figure 6.**Chattering phenomenon is observed when boundary layer thickness is decreased from 0.1 in Figure 5c to 0.03.

**Figure 7.**The variable boundary layer approach is applied in implementing SMC in VAC, in which the same VAC models and force profile are employed as in Figure 5c. Tracking is relatively accurate, and the boundary layer thickness does not need to be defined by trial and error. It is also observed that the new sliding mode gain is simply a low-pass filtered signal of Equation (31). The modeling errors are designed the same as in Figure 5.

**Figure 8.**Implementation of SMC in VAC for Kinova Gen3, where acceleration feedback is included. (

**a**) displays the interaction force. (

**b**) illustrates the control in human leading case. A smooth and perfect tracking is observed compared to Figure 5b. In (

**c**), stiffness is updated proportional to the human interaction force, then is regulated by PER. As a result, system passivity is preserved, i.e., $\xbd\text{}{\dot{K}}_{{d}_{\text{}}}^{i}\left(t\right){\stackrel{~}{x}}_{i}^{2}\le \mu {D}_{{d}_{\text{}}}^{i}{\dot{\stackrel{~}{x}}}_{i}^{2}$. Moreover, tracking of ${\mathit{x}}_{\mathit{r}}$ by ${\mathit{x}}_{\mathit{c}}$ and ${\mathit{x}}_{\mathit{c}}$ by $\mathit{x}$ are both realized. The modeling errors are designed the same as in Figure 5.

**Figure 9.**Tracking performance of applying indirect tracking approach for human-leading VAC with a constant sliding mode gain, ${K}_{smc}^{i}=20$.

**Figure 10.**Tracking performance of applying indirect tracking approach for human-leading VAC with a constant sliding mode gain, ${K}_{smc}^{i}=70$.

**Figure 11.**Tracking performance of applying indirect tracking approach for human-leading VAC with a constant sliding mode gain, ${K}_{smc}^{i}=20$. The mass of links 1 and 3 are modified to be ${m}_{1}=2.3\text{}\mathrm{k}\mathrm{g}$ and ${m}_{3}=0.65\text{}\mathrm{k}\mathrm{g}$ in the controller.

**Figure 12.**Tracking performance of applying indirect tracking approach without AFC for human-leading VAC. The proposed variable sliding mode gain and variable boundary layer thickness methods are employed. The mass of links 1 and 3 are modified to be ${m}_{1}=2.3\text{}\mathrm{k}\mathrm{g}$ and ${m}_{3}=0.65\text{}\mathrm{k}\mathrm{g}$ in the controller.

**Figure 13.**Tracking performance of indirect tracking approach with AFC for human-leading VAC. The proposed variable sliding mode gain and variable boundary layer thickness methods are employed. Acceleration error gain is selected as ${Q}_{i}=3.0$. The mass of links 1 and 3 are modified to be ${m}_{1}=2.3\text{}\mathrm{k}\mathrm{g}$ and ${m}_{3}=0.65\text{}\mathrm{k}\mathrm{g}$ in the controller.

**Figure 14.**Tracking performance of indirect tracking approach with AFC for human-leading VAC. The proposed variable sliding mode gain and variable boundary layer thickness methods are employed. The structure composed of a second-order low-pass filter and M2-PSMF filter is applied to acceleration signals. Acceleration error gain is selected as ${Q}_{i}=3.0$. The mass of links 1 and 3 are modified to be ${m}_{1}=2.3\text{}\mathrm{k}\mathrm{g}$ and ${m}_{3}=0.65\text{}\mathrm{k}\mathrm{g}$.

**Figure 15.**Filtered measured acceleration and equivalent acceleration signals using the M2-PSMF filter.

**Figure 16.**Tracking performance of indirect tracking approach with AFC for human-leading VAC. The proposed variable sliding mode gain and variable boundary layer thickness methods are employed. The M2-PSMF filter is applied to acceleration signals. Acceleration error gain is selected as ${Q}_{i}=3.0$. The mass of links 1 and 3 are modified to be ${m}_{1}=2.3\text{}\mathrm{k}\mathrm{g}$ and ${m}_{3}=2.75\text{}\mathrm{k}\mathrm{g}$ in the controller.

**Figure 17.**Statistical overview of repeatable experimental results of using indirect tracking approach with constant ${K}_{smc}$, variable ${K}_{smc}$ without AFC, and variable ${K}_{smc}\text{}$with AFC methods. Each method corresponds to the method being employed to obtain results in Figure 9, Figure 12 and Figure 14, respectively.

**Figure 18.**Tracking performance of indirect tracking approach with AFC for human-following VAC. The M2-PSMF filter is applied to acceleration signals. Acceleration error gain is selected as ${Q}_{i}=3.0$. The mass of links 1 and 3 are modified to be ${m}_{1}=2.3\text{}\mathrm{k}\mathrm{g}$ and ${m}_{3}=0.65\text{}\mathrm{k}\mathrm{g}$.

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**MDPI and ACS Style**

Chen, J.; Ro, P.I.
Variable Admittance Control in Sliding Mode for Robust Physical Human–Robot Interaction. *Appl. Sci.* **2023**, *13*, 11219.
https://doi.org/10.3390/app132011219

**AMA Style**

Chen J, Ro PI.
Variable Admittance Control in Sliding Mode for Robust Physical Human–Robot Interaction. *Applied Sciences*. 2023; 13(20):11219.
https://doi.org/10.3390/app132011219

**Chicago/Turabian Style**

Chen, Jingdong, and Paul I. Ro.
2023. "Variable Admittance Control in Sliding Mode for Robust Physical Human–Robot Interaction" *Applied Sciences* 13, no. 20: 11219.
https://doi.org/10.3390/app132011219