IEEE Proof IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS 1 Robust Ground Reaction Force Estimation and Control of Lower-Limb Prostheses: Theory and Simulation Vahid Azimi , Thang Tien Nguyen , Mojtaba Sharifi, Seyed Abolfazl Fakoorian, and Dan Simon Abstract—Ground reaction force (GRF) characteristics of 1 amputee walking are important for the analysis of clinical gait 2 data, and also to update model reference adaptive impedance 3 (MRAI) controllers. GRF estimation is a better alternative than 4 direct GRF measurement because of the disadvantages of load 5 cells, such as high cost, integration difficulties due to weight and 6 physical dimensions, the possibility of overload, and measurement 7 noise. This paper presents four robust MRAI observer/controller 8 combinations for GRF estimation-based control of a prosthe- 9 sis and a legged robot model in the presence of parametric 10 uncertainties and unmodeled dynamics, in which the robot 11 model is employed to mimic able-bodied walking. Since unknown 12 GRFs can reduce the performance of prosthesis control systems, 13 the proposed MRAI controllers are designed to compensate 14 their effects. All four proposed observer/controller combinations 15 enable the system to imitate the biomechanics of able-bodied 16 walking as defined by mechanical impedance, and provide flexi- 17 ble and smooth gait. Lyapunov analysis of the observer/controller 18 combinations is used to prove stability. The four MRAI con- 19 trollers are compared with each other from the perspective 20 of GRF estimation accuracy, prosthetic knee angle tracking 21 accuracy, parameter estimation accuracy, control effort, and com- 22 putational effort. Simulation results show that all four MRAI 23 controllers achieve good tracking performance, stable limit cycles, 24 and accurate GRF estimation. 25 Index Terms—Adaptive control, estimation, prosthetics, robust 26 control, sliding mode control. 27 I. I NTRODUCTION 28 T HE NUMBER of transfemoral amputees in the United 29 States is estimated at around 222 000 [1]. Amputees can 30 use a powered prosthetic leg to assist them with ambulation 31 Manuscript received March 8, 2018; accepted April 30, 2018. This work was supported by NSF under Grant 1344954. This paper was recommended by Associate Editor Z. Li. (Corresponding author: Thang Tien Nguyen.) V. Azimi is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA (e-mail: vahid.azimi@gatech.edu). T. T. Nguyen is with the Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam (e-mail: nguyentienthang@tdt.edu.vn). M. Sharifi is with the Department of Mechanical Engineering, Sharif University of Technology, Tehran 11365-11155, Iran (e-mail: mojtaba_sharifi@mech.sharif.edu). S. A. Fakoorian and D. Simon are with the Department of Electrical Engineering and Computer Science, Cleveland State University, Cleveland, OH 44115 USA (e-mail: s.fakoorian@csuohio.edu; d.j.simon@csuohio.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMC.2018.2836913 and participation in daily activities. In recent years, much pros- 32 thesis research has been conducted. In [2], a novel control 33 strategy for a powered prosthetic ankle based on a biomimetic 34 virtual constraint was suggested. In [3], a nonlinear real-time 35 optimization-based controller was used for a self-contained 36 powered transfemoral prosthesis. In that approach, control 37 Lyapunov function-based quadratic programs, coupled with 38 variable impedance control, were designed to improve both 39 tracking performance and energy efficiency. In [4], a controller 40 for a powered prosthesis provided variable upslope walking. 41 That control methodology enabled an above-knee amputee 42 to walk up slopes using a powered knee/ankle prosthesis. 43 In [5], a powered transtibial prosthesis with adaptable-stiffness 44 ankle and toe joints was designed using modified series-elastic 45 actuators. 46 Since the interaction between a system and unknown exter- 47 nal disturbances can reduce the performance of the closed-loop 48 controller, a robust controller can be designed to cancel dis- 49 turbance effects. Sliding mode control is one of the most 50 popular approaches to deal with disturbances and model 51 uncertainties in nonlinear systems. In recent years, sliding 52 mode control has been successfully applied to many robotic 53 systems [6]–[8]. 54 Adaptive sliding mode control [9]–[11] has been devel- 55 oped to control nonlinear systems while compensating for 56 both exogenous inputs and parametric uncertainties under the 57 assumption of known disturbance bounds. In general, the 58 bounds of unknown disturbances are not available, which moti- 59 vates the need to estimate the disturbances. In [12], sliding 60 mode control, in conjunction with the function approxima- 61 tion technique (FAT), was developed for nonlinear systems. 62 The time-varying uncertainties were approximated by finite- 63 term Fourier series, and the Fourier coefficients were updated 64 using a Lyapunov-based adaptation law. In [13], an adaptive 65 sliding mode controller was designed based on the FAT and 66 backstepping in the presence of unknown parameter uncertain- 67 ties and disturbances. In those two approaches, the number 68 of Fourier series terms needs to be large for accurate esti- 69 mate of the uncertainties and unknown inputs, which increases 70 computational complexity. 71 In [14], a second-order sliding mode (SM) observer 72 based on a modification of the super-twisting algorithm was 73 proposed to observe the states of mechanical systems with 74 unknown inputs. In that work, parameter identification, per- 75 turbation observation, and velocity estimation were achieved 76 2168-2216 c  2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.