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Robust LQR-Based Neural-Fuzzy Tracking Control for a Lower Limb Exoskeleton System with Parametric Uncertainties and External Disturbances

Robust LQR-Based Neural-Fuzzy Tracking Control for a Lower Limb Exoskeleton System with... Hindawi Applied Bionics and Biomechanics Volume 2021, Article ID 5573041, 20 pages https://doi.org/10.1155/2021/5573041 Research Article Robust LQR-Based Neural-Fuzzy Tracking Control for a Lower Limb Exoskeleton System with Parametric Uncertainties and External Disturbances Jyotindra Narayan and Santosha K. Dwivedy Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India Correspondence should be addressed to Jyotindra Narayan; n.jyotindra@gmail.com Received 23 February 2021; Revised 14 April 2021; Accepted 15 May 2021; Published 12 June 2021 Academic Editor: Fahd Abd Algalil Copyright © 2021 Jyotindra Narayan and Santosha K. Dwivedy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The design of an accurate control scheme for a lower limb exoskeleton system has few challenges due to the uncertain dynamics and the unintended subject’sreflexes during gait rehabilitation. In this work, a robust linear quadratic regulator- (LQR-) based neural-fuzzy (NF) control scheme is proposed to address the effect of payload uncertainties and external disturbances during passive-assist gait training. Initially, the Euler-Lagrange principle-based nonlinear dynamic relations are established for the coupled system. The input-output feedback linearization approach is used to transform the nonlinear relations into a linearized state-space form. The architecture of the adaptive neuro-fuzzy inference system (ANFIS) and used membership function are briefly explained. While varying mass parameters up to 20%, three robust neural-fuzzy datasets are formulated offline with the joint error vector and LQR control input. Thereafter, to deal with external interferences, an error dynamics with a disturbance estimator is presented using an online adaptation of the firing strength matrix. The Lyapunov theory is carried out to ensure the asymptotic stability of the coupled human-exoskeleton system in view of the proposed controller. The gait tracking results for the proposed control scheme (RLQR-NF) are presented and compared with the exponential reaching law-based sliding mode (ERL-SM) controller. Furthermore, to investigate the robustness of the proposed control over LQR control, a comparative performance analysis is presented for two cases of parametric uncertainties and external disturbances. The first case considers the 20% raise in mass values with a trigonometric form of disturbances, and the second case includes the effect of the 30% increment in mass values with a random form of disturbances. The simulation runs have shown the promising gait tracking aspects of the designed controller for passive-assist gait training. 1. Introduction systematic yet comprehensive review has been carried out on the state-of-the-art developments of such multijoint and Over the last two decades, an increasing number of neurolog- single-joint exoskeleton devices for gait rehabilitation, ical disorders such as stroke, spinal cord injury, and Parkin- mobility aid, and strength amplification. A well-known treadmill-oriented exoskeleton, LOPES son’s disease have been observed in different age groups. The World Health Organization (WHO) reported “stroke” [5], has been developed with a 2D translatable pelvis seg- as one of the principal reasons for nearly 5 million people’s ment, two active hip joints, and an active knee joint for lower fatality through 2000-2016 and the third pioneering source limb rehabilitation. The system was controlled to supervise of debility throughout the world [1]. To address the concerns or follow the subjects using “robot-in-charge” and “subject- in-charge” modes. Bortole et al. [6] designed a 6-DOF lower of motor functionality in the lower body caused by neurolog- ical disorders, researchers have developed many robot-based limb exoskeleton for overground training of stroke subjects lower limb exoskeleton devices to produce therapeutic effects with a body height of 1.50-1.95 m and a body mass of during walking [2, 3]. In a recent work by Kalita et al. [4], a 100 kg. Hsieh et al. [7] proposed a soft exoskeleton design 2 Applied Bionics and Biomechanics control, Long et al. [20] presented a hybrid strategy where for preswing gait training of subjects with weak muscles, where a single actuator with a pulley-slider arrangement is SMC is augmented with a cerebellar model articulation con- used to drive the lower limb joints. The prototype is devel- troller (CMAC) to predict the motion intent of the subject. The optimized sliding surface of the SMC is estimated using oped and clinically investigated with seven subjects. In a study on a parallel mechanism-based lower limb rehabilita- the genetic algorithm to improve the effectiveness of the tion, Rastegarpanah et al. [8] investigated the performance proposed control scheme. Liu et al. [21] introduced an of a 6-DOF robot by executing foot trajectories of 20 healthy event-triggered SMC for effective tracking of the reference subjects. Furthermore, the same prototype was tested for trajectory using a lower limb exoskeleton system with PUEDs. In another work to address the model uncertainties eight poststroke patients while carrying out three exercises, i.e., hip flexion/extension, ankle dorsiflexion/plantarflexion, and the unintended subject’s response, Wu et al. [22] pro- and marching [9]. Aggogeri et al. [10] presented a modular posed an adaptive control scheme for a 3-DOF lower extrem- and reconfigurable mechanism for rehabilitating ankle joints ity rehabilitation device. Working on the decoupled control of different subjects. Cestari et al. [11] introduced the ATLAS strategy, Sun et al. [23] designed a reduced-order adaptive fuzzy approach and implemented it on a two-link exoskele- exoskeleton to assist the children during flexion/extension of the hip, knee, and ankle joints. At the preliminary level, a ton system for lower limb rehabilitation. dummy with body features of a 10-year-old child was used Furthermore, in recent times, robust intelligent control to test the exoskeleton system. Patané et al. [12] proposed a schemes have gain popularity to address the adverse effects multijoint exoskeleton, WAKE-up, to rehabilitate the knee, of PUEDs with effective approximation features. A neural network (NN) along with a time-delay evaluation-based con- ankle, and foot of the pediatric subjects with neurological dis- orders. The device was tested with four healthy children and trol scheme is proposed by Zhang et al. [24] to realize the three children with cerebral palsy. To amplify human endur- desired gait trajectory for a simulated model of a 10-DOF ance while carrying heavy loads, BLEEX [13] (7-DOF/limb) exoskeleton. The performance of the designed control was developed with intelligent and adaptable robot-based scheme is investigated by comparing the classical PD control scheme. Narayan and Dwivedy [25] proposed a neuro-fuzzy strategies where linear hydraulic actuators were used for the actuation of 4-DOF. Recently, Ji et al. [14] introduced a wear- compensator for PID control to deal with the system’s known able exoskeleton, SIAT-WEXv2, to support the user’s waist and unknown uncertainties during passive gait rehabilitation of a human child. The controller is found to be more robust and bones while lifting heavy objects in construction and logistic industries by providing an assistive output of 28 N. towards external disturbances over payload uncertainties. Chen et al. [26] proposed a disturbance estimator-based To augment the rehabilitation devices’ performance, the appropriate control schemes are designed by the researchers subject-cooperative control for a weight-reinforced active- for executing repetitive gait movements. The control archi- assist rehabilitation device. They computed the interaction torques using a backpropagation neural network-aided tecture required for the exoskeleton systems poses extra com- plexity over the conventional robotic arm control due to the disturbance observer and proved the stability using the Lyapunov theory. In a recent work by Han et al. [27], time- sophisticated mechanical configuration, complex motion trajectory, and human involvement. The researchers, in the delay estimator-aided computed torque control is designed literature, have regarded the predefined gait tracking control to deal with PUEDs of a lower limb exoskeleton system. Moreover, an adaptive radial basis function neural network as the basis of every control scheme for exoskeleton systems, where the joint movements of the lower limb could be (RBFNN) is utilized to compensate for the time-delay error. On the other hand, few researchers have explored the estimated using gait analysis experiments [15, 16, 17]. Although the exoskeleton systems exploit the gait of healthy optimal control, especially the linear quadratic regulator humans to replicate the same using predefined trajectory (LQR), to realize the natural gait [28, 29, 30, 31]. The LQR scheme with full-state feedback yields control measures con- control schemes, however, in practice, they are unable to attain the proper gait trajectory because of the parametric cerning the whole body compared to PD control for every uncertainties and external disturbances (PUEDs). Therefore, independent joint [28]. In addition to that, the relative prom- various robust control strategies have been designed to deal inence of curtailing the tracking error and minimalizing the with the limitations of classical trajectory tracking control control torque can be regulated by computing optimal values of time-varying gain based on the design parameters of a sin- in lower limb exoskeleton systems [18, 19, 20, 21, 22, 23]. Ajayi et al. [18] proposed a bounded control scheme for the gle controller. Furthermore, LQR as a linear control scheme rehabilitation of the knee ankle joint of a user in a sitting might be exploited for nonlinear system dynamics by position. The stability of the control law and convergence approximating the linear time-varying form and significantly analysis of the gain observer is validated with the Lyapunov mitigating the computational complexity involved in several nonlinear controllers. Ajjanaromvat and Parnichkun [29] theory. The simulation results are presented without and with the effect of the human interaction torque. Yang et al. proposed an iterative online learning-based LQR control [19] presented a sliding mode control (SMC) scheme where scheme for a treadmill-appended exoskeleton to investigate a second-order command filter-aided backstepping is incor- the robustness analysis. Moreover, the proposed control porated to avert the “explosion of complexity.” Moreover, scheme is aided with an adaptive iterative learning control to address tracking errors. Gupta et al. [30] presented the the fuzzy logic is exploited to counter the chattering issues of the control scheme during the estimation of structured LQR control for lower limb exoskeleton systems by consider- and unstructured uncertainties. In another work on robust ing the 4-DOF human gait model in the Single Support Phase Applied Bionics and Biomechanics 3 (SSP). They exploited the nondominated sorting genetic tem with different heights of the subjects augments the algorithm to find out the optimal weighing matrix. However, feature of cost-effectiveness. Considering the subject’s physi- the formulation work has not considered the uncertain fac- ological safety, all possible degrees of freedom should be tors in system dynamics. Castro et al. [31] proposed an avoided at the initial phases of rehabilitation training. Invok- integral-aided LQR (LQRi) and unknown input disturbance ing the design features, authors have designed a low-cost observer (UIO) to address external interferences of the lower stand-alone module-aided lower limb exoskeleton system limb exoskeleton system. The results of the proposed control for pediatric rehabilitation in their previous work [32]. The are compared with proportional-derivative control and CAD model of the designed exoskeleton system is shown in found to be more effective. Figures 1(a) and 1(b). A 3-DOF multilink mechanism for Although the hybrid form of sliding mode control can be each leg was intended to carry out hip flexion/extension, knee considered a highly robust control strategy, chattering always flexion/extension, and ankle dorsiflexion/plantarflexion affects the performance of exoskeleton systems. On the other motions. The placements of the joint actuators were made hand, the LQR is the most optimal control scheme and lacks to avoid any physical interference with the subject’s body. to resolve uncertain exoskeleton dynamics. Therefore, in this To serve subjects of different heights, a telescopic link joint work, a new robust LQR-based neural-fuzzy control scheme arrangement was designed around the knee joint of the exo- is designed for the lower limb exoskeleton system with para- skeleton system. Moreover, a detailed structural analysis of metric uncertainties and external disturbances during passive the stand-alone module was carried for maximum loading gait rehabilitation training. The key highlights of the present conditions at the hip joint [32]. work are as follows: The mechanical configuration of the exoskeleton system is intended for children of 8-12 years of age, 25-40 kg weight, (i) The input-output feedback linearization approach is and 115-125 cm height. The possible range of motion represented to linearize the nonlinear dynamics of (ROM) for three joints of the exoskeleton system in the sagittal ° ° ° ° the lower limb exoskeleton system plane is as follows: 30 /-12 (hip-f/e), 60 /-10 (knee-f/e), and ° ° 13 /-20 (ankle-d/p). To avoid any undesirable actions beyond (ii) A robust offline LQR-based neural-fuzzy control the ROM, an emergency stop option is provisioned at the scheme is designed to deal with payload uncertainties software interface during simulation runs. In this work, an (iii) A disturbance estimator is proposed using an online eight-year-old male subject’s anthropometric and kinematic parameters (body mass: 30 kg and body height: 1.22 m) are adaptation of firing strength in offline designed LQR-NF architecture considered input parameters to the control architecture. The breakdown of input parameters for the lower limb exoskeleton (iv) The simulation results are carried out for the RLQR- and subject is shown in Table 1, where the length of the NF control scheme and compared with an exponential thigh and shank link is kept constant at 0.27 m and reaching law-based sliding mode control (ERL-SM) to 0.30 m, respectively. track the desired gait trajectory during passive thera- Furthermore, an affordable wireless Labview-aided peutic training Kinect setup was established to conduct the experimental gait analysis. With necessary approval, the child subject was (v) The robustness performance of the proposed control asked to follow an inclined path over the ground in front of scheme (RLQR-NF) is investigated by varying pay- the experimental setup for 1.6-2.0 seconds. The angle estima- load parameters and inducing different forms of tion algorithm comprehended the information about the external disturbances lower limb joint angles from the skeleton model in Labview. The rest of the paperwork is structured as follows. The The angle estimation algorithm exploits the relation between mechanical description of the lower limb exoskeleton system joint triples using vector algebra. and the estimation of control input parameters are presented The detailed procedure of performing gait analysis, as in Section 2. In Section 3, the nonlinear dynamic relation is shown in Figure 2(a), is based on the work by Narayan et al. formulated using the Euler-Lagrange principle, and thereaf- [33]. The skeleton form of the subject during the gait analysis ter, input-output linearization of the nonlinear form is is illustrated in Figure 2(b). The desired lower limb joint explained. Section 4 presents the concept of ANFIS architec- angles attained from the experiment are presented in ture with the selected membership function. Section 5 Figure 3(a) and the corresponding trajectory in Figure 3(b). describes a detailed design procedure of the proposed control The ROM for the hip, knee, and ankle joints are recorded ° ° ° ° ° ° strategy. In Section 6, the Lyapunov theory of stability is pre- as 22.16 to -8.98 , 58.26 to 1.21 , and 5.84 to -7.94 for an sented. The control results are simulated and discussed in eight-year-old child, respectively. Section 7. The complete paperwork is concluded in Section 8. 3. Dynamic Model of the Coupled Human- 2. Mechanical Configuration of the Lower Limb Exoskeleton System Exoskeleton System In this section, the Euler-Lagrange principle is used to The main criteria for the mechanical design of a lower limb formulate the nonlinear dynamics of the coupled human- exoskeleton system are to ensure its strength and stability exoskeleton system. Thereafter, the input-output feedback of the subject’s safety. Moreover, the adaptability of the sys- linearization approach is exploited to linearize the nonlinear 4 Applied Bionics and Biomechanics (a) (b) Figure 1: CAD model of (a) LLES (labels: (1) thigh link, (2) shank link, (3) foot link, (4) hybrid stepper motor, (5) lead screw actuator, (6) stepper motor, (7) timing belt, (8) support module, (9) wheels, and (10) telescopic link joint connector) and (b) LEES with a human dummy [32]. where Table 1: Specifications of the lower limb exoskeleton system and child dummy. Part Mass (kg) Length (m) COM (m) 1 1 T _ _ ð3Þ K = 〠 m s_ s_ + θ I θ , i i i i i i Lower limb exoskeleton system 2 2 i=1 e e e m =4:75 l =0:25‐0:30 l =0:12‐0:15 Thigh link 1 1 c1 e e e m =1:60 l =0:30‐0:35 l =0:14‐0:17 Shank link 2 2 c2 e e m =0:85 l =0:05 l =0:02 Foot link 3 3 c3 P = 〠 m gh : ð4Þ ðÞ Child (age 8 years, body weight 30 kg, and body height 122 cm) i ci i=1 h h h Thigh m =3:50 l =0:27 l =0:13 1 1 c1 h h h Shank m =2:25 l =0:30 l =0:15 2 2 c2 In the abovementioned relations, θ represents the gener- h h h Foot m =0:65 l =0:04 l =0:02 alized coordinate of the human-exoskeleton system. The 3 c3 kinetic and potential energy about the i-link is denoted by K and P, respectively. In Equation (3), θ , s_ , m , and I signify i i i i behavior of the dynamical system. The transformed linear the angular velocity, speed of the center of mass in transla- state-space relation is established for the dynamics of the tional direction, mass, and inertia corresponding to the i lower limb exoskeleton system. -link. The acceleration due to the gravitational effects is referred by g, and the distance between the i-link’s center point forming the gravitational vector and the origin is 3.1. Nonlinear Dynamic Formulation. Among various denoted by h as illustrated in Equation (4). ci methods for expressing applied joint torques and angular Referring to Equations (2)–(4) to solve Equation (1), the acceleration, the Euler-Lagrange principle is well appreciated nonlinear dynamics of the coupled dynamical system can be by the research communities [34]. Invoking the Euler- articulated as follows: Lagrange principle, which employs kinetic and potential energy, the nonlinear representation of the coupled human- exoskeleton dynamics is obtained. A multilink structure of € _ _ τ = MðÞ θ θ + C θ, θ θ + GðÞ θ , ð5Þ the coupled system with a collaboration effect is shown in Figures 4(a) and 4(b). A generalized formulation to estimate the joint torques using the Lagrangian L is as follows: where τ = τ + τ + τ , d ∂L ∂L > a eh he τ = − , ð1Þ > e h dt _ ∂θ ∂θ > i MðÞ θ = M ðÞ θ + M ðÞ θ , i < ð6Þ e h _ _ _ C θ, θ = C θ, θ + C θ, θ , e h GðÞ θ = G ðÞ θ + G ðÞ θ , L = K − P, ð2Þ Applied Bionics and Biomechanics 5 Display panel (Modify Loop run at 16fps Comparison joint triples) of vector 1-2 Joint 1 Joint 2 Kinect initialization Comparison Joint 3 of vector 2-3 Angle evaluation Kinect Kinect configuration block read Initialize 3D display Kinect Cluster processor While loop Kinect close (a) (b) Figure 2: Gait analysis experiment. (a) Schematic diagram of the detailed procedure. (b) A child subject with the skeleton model during the experiment. 60 –0.35 –0.4 –0.45 –0.5 –0.55 Starting point at t = 0 –20 –0.6 0 0.5 1 1.5 2 –0.2 0 0.2 0.4 Y (m) Time (sec) Hip Desired gait trajectory Knee Ankle (a) (b) Figure 3: Experimental gait data. (a) Desired joint angular trajectory. (b) Desired gait trajectory. 2 3 The matrix form of inertial, Coriolis-centrifugal, and M M M 11 12 13 gravity effects of the coupled dynamical system is signified 6 7 6 7 MðÞ θ = M M M , by MðθÞ, Cðθ, θÞ, and GðθÞ, respectively. In Equation (6), 21 22 23 4 5 e h M ðθÞ and M ðθÞ represent the inertial dynamics of the exo- M M M 31 32 33 skeleton and human leg in the matrix form, respectively. The 2 3 Coriolis-centrifugal matrix of the exoskeleton and human leg C C C 11 12 13 e h 6 7 _ _ is represented by C ðθ, θÞ, and C ðθ, θÞ, respectively. The 6 7 C θ, θ = C C C , ð7Þ 21 22 23 4 5 gravity matrix of the exoskeleton and human leg is referred e h C C C by G ðθÞ and G ðθÞ, respectively. τ implies the actuator tor- 31 32 33 a 2 3 que while driving the joint of a human’s lower limb. The col- laboration torque is indicated by τ and τ for collaboration eh he 6 7 6 7 of exoskeletons with humans and vice versa, respectively. G θ = , ðÞ G 4 5 During exoskeleton-human interaction, splints are exploited to keep the exoskeleton link and human leg Desired joint angles (deg) X (m) 6 Applied Bionics and Biomechanics c1 1 k Exoskeleton link c2 Human segment c3 (a) (b) Figure 4: Coupled human-exoskeleton configuration. (a) A simplified linkage model. (b) Interaction dynamics of the coupled human- exoskeleton system. attached, which induces the collaboration torques (τ and ing, the actuator dynamics can be formulated using Kirchh- eh τ ). However, in passive gait rehabilitation, as considered off’s law to obtain the control voltage (U ) as follows: he in the present work, these collaborations are withdrawn by assuming rigid connections and matching joint angles for ˇ τ R ˇ _ _ U = + i L + Ξ θ, ð11Þ exoskeletons and humans. As illustrated in Figure 4(b), the m m m e interaction dynamics is formulated as below: where R , S , i , L , and Ξ denote the armature resistance, T T _ _ m m m m e τ = −τ = J f = J kΔx + cΔx_ = k θ − θ + c θ − θ , ðÞ ðÞ eh he co h e h e torque sensitivity, current, armature inductance, and back EMF constant of the DC motor. These parameters are ð8Þ selected from the specification sheet provided by Bholanath Precision Engineering Private Limited [35]. where f represents the collaboration force between the exo- co To imitate the realistic cases, the actuator saturation skeleton and the human, k and c signify the mechanical stiff- should be considered in the design of the control law to avoid ness and damping factors of the used splints, Δx denotes the the hysteresis cycle and maintain the linearity of the actuator. Cartesian coordinate disparity between the human leg and Moreover, this ensures closed-loop stability by limiting the the exoskeleton link, and ðθ − θ Þ refers to the joint angular h e large control signals. Based on the saturation theory, the disparity between the human leg and the exoskeleton link. Furthermore, in the presence of parametric uncertainties control signal ðU Þ from Equation (11) can be further and external disturbances, Equation (5) can be rewritten as defined as below: ˇ ˇ ˇ ˇ € _ _ τ = M ðÞ θ θ + C θ, θ θ + G ðÞ θ , ð9Þ ℧   U >℧ , > m m m ˇ ˇ ˇ U = U   U ≤℧ , ð12Þ where m m m m 8 > τ = ητ + D, > −℧   U < −℧ , > m m m M ðÞ θ = MðÞ θ + η MðÞ θ , ð10Þ where ℧∈ℝ denotes a vector with positive elements. The _ _ _ C θ, θ = C θ, θ + η C θ, θ , upper and lower saturation bound is denoted by ℧ and > m : −℧ , respectively. G ðÞ θ = GðÞ θ + η GðÞ θ , 3.2. Input-Output Feedback Linearization. The main objec- where D denotes the external disturbances applied by the tive of the feedback linearization is to correctly linearize the subjects to the system; η is the uncertain scaling factor when nonlinear dynamics with suitable modifications in state- considering the same amount of variation in dynamic space coordinates using an inner loop control [34]. Thereaf- parameters. ter, an outer loop control with a new set of coordinates can be After considering joint torques (τ) equivalent to joint formed to establish a linear relationship between the output actuator torques (τ ) in case of passive rehabilitation train- vector (y) and the input vector (u) and validate the cost a Applied Bionics and Biomechanics 7 functions of the control design. Consider the nonlinear Now, a linear relationship between inputs and outputs is multiple-input and single-output (MISO) dynamic relation to be established by performing the differentiation of the with n as the order and p as the total number of inputs as well outputs ðy Þ till the input terms appear in the formulation. ðr Þ as outputs, defined in the affine state: Considering r is the smallest integer, ðy Þ can be evaluated j j 8 with a complete term of inputs as follows: x _ t = Ψ x t + 〠 ðÞ ðÞ ðÞ ΠðÞ xðÞ t U ðÞ t , i m ð13Þ i=1 r r ðÞ j j j−1 > ˇ y = L Λ x + 〠 L L Λ x U , i, j =1,2, ⋯, p, : ðÞ ðÞ j Ψ j Π Ψ j m i i y ðÞ t = ΛðÞ xðÞ t , i=1 i i ð19Þ where x = ½x , x , ⋯, x  ∈ ℝ denotes the state vector, 1 2 n i i ˇ ˇ ˇ ˇ where L Λ and L Λ signify the ith Lie derivatives of Λ ðxÞ U = ½U , U , ⋯, U  ∈ ℝ signifies the control input j j j Ψ Π m m m m i 1 2 p T in the direction of Ψ and Π, respectively. In Equation (19), r vector, and y = ½y , y , ⋯, y  ∈ ℝ indicates the output 1 2 p denotes the relative degree for the output y which provides vector. information about the number of derivatives required to n n carry out at least one of the inputs in the formulation [37, Theorem 1. Suppose Ψ : ℝ ⇒ ℝ signifies a smooth vector n n n 38]. The sum of every relative degree from Equation (19) field on ℝ and Λ : ℝ ⇒ ℝ denotes a scalar function. Then, constitutes the total relative degree ðrÞ which needs to be less the Lie derivative of Λ to Ψ, referred as L Λ, is expressed as than or equal to the system’s order. follows [36, 37]: ∂Λ ∂Λ r = 〠 r ≤ n: ð20Þ L Λ = ΨðÞ x = 〠 ΨðÞ x : ð14Þ j=1 ∂x ∂x i=1 Furthermore, rewriting Equation (19) and expressing the Similarly, the Lie derivative of L Λ with respect to Ψ is nonlinear control law U to form the linear relationship defined as m between the input and the output as follows, one can get L Λ = L L Λ : ð15Þ ðÞ Ψ Ψ Ψ hi ð21Þ y , ⋯, y = δ x + σ x · U , ðÞ ðÞ 1 m In general, hi T r γ γ−1 0 ˇ ˇ ˇ ˇ ð22Þ u = u , u , ⋯, u = y , ⋯, y , 1 2 p 1 L Λ = L L Λ  with L Λ = Λ,∀γ = 1, ⋯, p: ð16Þ Ψ Ψ Ψ where n n Theorem 2. The function Φ : ℝ ⇒ ℝ , specified in a region 2 3 Y ⊂ ℝ , is termed as diffeomorphism if the function Φ along > L Λ ðÞ x > Ψ 1 −1 6 7 with the inverse Φ (if it exists) is smooth, i.e., differentiable > 6 ⋮ 7 6 7 δðÞ x = 6 7, everywhere [36, 37]. > > 6 7 4 5 However, as the global diffeomorphism is rare, one > L Λ x > ðÞ Ψ p should check for local diffeomorphisms, i.e., transformations > 2   3 ðÞ r ðÞ r −1 ðÞ r −1 1−1 1 1 defined in a limited neighborhood of a specified point [37]. L L Λ ðÞ x L L Λ ðÞ x ⋯ L L Λ ðÞ x Π Ψ 1 Π Ψ 1 Π Ψ 1 1 2 p 6 7 With the concept of diffeomorphism, we transform a nonlin- 6 7 6 7 > ðÞ r −1 ðÞ r −1 ðÞ r −1 2 2 2 6 7 > L L Λ ðÞ x L L Λ ðÞ x ⋯ L L Λ ðÞ x ear system into another one by changing the variables in the > Π Ψ 2 Π Ψ 2 Π Ψ 2 1 2 p > 6 7 > σ x = : ðÞ 6 7 following form: > 6 7 > ⋮ ⋮ ⋮ > 6 7 > 6 7 > 4 r −1 r r −1 5 ðÞ ðÞ ðÞ > p p−1 p > L L Λ ðÞ x L L Λ ðÞ x ⋯ L L Λ ðÞ x T : Π p Π p Π p Ψ Ψ Ψ 1 2 p z = z , z , ⋯, z = Φ x , ð17Þ ½ ðÞ 1 2 n ð23Þ where ΦðxÞ characterizes n variables as Assuming σðxÞ is not singular, the input transferred 2 3 2 3 hi r −1 1 1 form, i.e., the nonlinear control law, can be possibly Λ  L Λ   ⋯ L Λ 1 Ψ 1 Ψ 1 6 7 6 7 defined as 6 Φ 7 6 7 6 7 6 7 ΦðÞ x = = , 6 7 6 7 −1 4 ⋮ 5 4 5 ˇ hi U = σ x −δ x +u ˇ , ð24Þ T ðÞðÞ ðÞ r −1 m Λ  L Λ   ⋯ L Λ p Ψ p Ψ p ˇ ˇ ˇ ˇ where u ˇ = ½u ˇ , u ˇ , ⋯, u ˇ  and U = ½U , U , ⋯, U  . ð18Þ 1 2 p m m m m 1 2 p In Equation (24), u ˇ denotes the new input vector, U with x = ½x , x , ⋯, x  . refers to the decoupling control law, σðxÞ signifies an 1 2 n 8 Applied Bionics and Biomechanics invertible matrix of order p × p, and δðxÞ represents a 4. Adaptive Neural-Fuzzy Inference System decoupling matrix of the system. The adaptive neural-fuzzy inference system (ANFIS), Furthermore, the linearizing law from Equation (24) is colloquially known as the neural-fuzzy or neuro-fuzzy (NF) applied for transforming the nonlinear dynamics of the system, was proposed by Jang and Sun [40] by augmenting coupled human-exoskeleton system (Equation (9)) into the the benefits of adaptive neural networks and fuzzy reasoning. linear state-space representation as follows: In the NF system, IF-THEN-based fuzzy logic inferences are constructed to form the learning rules with a defined input- ˇ ˇ output dataset and reproduce the output vector with zero z _ =Az +Bu ˇ , ð25Þ error tolerance. A neural-fuzzy system exploits the fuzzy y =Cz, input variables and input-dependent nonfuzzy output vari- ables given by Takagi and Sugeno [41]. For instance, If the acceleration of the robot send − effector is high,then ˇ ˇ where A =A + ΔA, B =B + ΔB, and u = ðΔ +1Þu + d. ˇ ˇ A and B are the state-weight factor matrices with the ft ðÞ = c × acceleration of the robot send − effector : ð27Þ effect of parametric uncertainties; A, B,and C denote the state-weight factor matrices evaluated with nominal system In Equation (27), high signifies a fuzzy label with the mem- parameters, i.e., in the absence of parametric uncertainties; Δ bership function (MF), representing the acceleration of the is the uncertain scaling factor in the linearized state-space form robot send − effector intheruleproposition. Therulesubse- related to state-weight factor matrices and can be expressed in quent with nonfuzzy behavior is formulated according to the terms of dynamic parameters of the given system [39]. The input variable of the premise step, acceleration of the robot s input vector u carries the effectof parametervariationsand end − effector. external disturbances. The input vector in the absence of PUEDs is denoted byu. The disturbance applied to the system 4.1. ANFIS Architecture. A neural-fuzzy system inherently after feedback linearization is denoted by d, being analogous to exploits the five layers. The primary network architecture D from Equation (9). The linearized dynamics in Equation (25) with two input vectors ðz : x, yÞ and one output vector stands valid with the following assumptions: ðf ðzÞÞ is considered to show the generalized process of ˇ ˇ A1:the ðA,BÞ is in the controllable form. ANFIS. As shown in Figure 5, two kinds of nodes are employed A2: the disturbance d is in the bounded form. in the architecture: first, a square node for adaptation of the In the present work, the state vector and output vector parameters, and second, a circular node that behaves as a fixed related to the hip, knee, and ankle joints of the coupled dynam- node with no parameter. The layer-by-layer development of _ _ _ ical system is considered z = ½θ  θ  θ  θ  θ  θ  the ANFIS structure is given below [42, 43]. H K A H K A and y = ½θ  θ  θ  , respectively. The input vector H K A Layer 1. This layer acts as a conversion function for the crisp with nominal system parameters is defined as u = T value of the input vector into an appropriate MF-based fuzzy ½u  u  u  . Moreover, the respective state-weight fac- H K A language, depicted as follows: 6×6 6×3 3×6 tor matrices (A ∈ ℝ , B ∈ ℝ , andC ∈ ℝ ) can be for- mulated as below. The entries of these matrices are extensively = Ω ðÞ x , j A presented in the appendix. j ð28Þ O = Ω ðÞ y , j B 0 1 > ½ 0 I > 3×3 3 > C > α α α 1 B 41 42 43 C > where O represents the jth node output for the first layer. B C > j B α ~ α ~ α ~ C 41 42 43 B C Ω ðxÞ and Ω ðyÞ denote the membership weightage of > B C A = α α α α α α α α α , > A B 51 51′ 51′′ 52 52′ 52′′ 53 53′ 53′′ j j > B C > − + − + + − > B ½ C > 3×3 α ~ α ~ α ~ α ~ α ~ α ~ α ~ α ~ α ~ > B ′ ′ ′ ′ ′ ′ C 51 51 51 ′ 52 52 52 ′ 53 53 53 ′ respective input variables, defined for A - and B -type MF. B C j j > @ α α α α α α α α α A > 61 61′ 61′′ 62 62′ 62′′ 63 63′ 63′′ − − − − − − > In this work, the generalized bell membership function is > α ~ α ~ α ~ α ~ α ~ α ~ α ~ α ~ α ~ ′ ′ ′ ′ ′ ′ > 61 61 61 ′ 62 62 62 ′ 63 63 63 ′ 6×6 chosen to replicate a proper probability distribution behavior 0 1 > ½ 0 > 3×3 and expressed as follows: > B C B β β β C 41 42 43 B C B ~ ~ ~ C > β β β > B 41 42 43 C > B C > B C bell z : a , b , c = Ω ðÞ x = Ω ðÞ y = : >B = β β β , j j j A B hi B 51 52 53 C j j b > 2 B C > ~ ~ ~ B C 1+ z − c /a > β β β j j > 51 52 53 B C B C @ β β β A 61 62 63 ð29Þ > ~ ~ ~ > β β β > 61 62 63 6×3 0 1 1 0 0 0 0 0 > In Equation (29), “a ” and “b ” signify the width and B C j j B C >C = : 0 1 0 0 0 0 @ A shape parameters of the membership function. The value of 0 0 1 0 0 0 3×6 “b ” is generally positive; however, it can be considered neg- ð26Þ ative in case of inverted shape. “c ” indicates the center j Applied Bionics and Biomechanics 9 Layer 5 Layer 2 Layer 1 Layer 3 Layer 4 x y A1 w w 1 2 Π N w f 1 1 A2 f (z) B1 w f N 2 2 y Π B2 2 x y Figure 5: ANFIS architecture. position of the membership function. Having an extra 5. Robust Design of the LQR-Based Neural- parameter compared to Gaussian MF, the generalized bell Fuzzy Control has the added advantage of tuning the steepness at cross- over positions. The design procedure of RLQR-NF control is organized into two parts: first, the offline training of a robust LQR-based Layer 2. This layer evaluates the firing strength for every rule ANFIS training dataset to deal with parametric uncertainties, and second, the online training of the LQR-based ANFIS using a product of incoming signals from each circular node. architecture using the adaptive law of weights to compensate It is designated by notation Π in the ANFIS architecture. The following expression is used to estimate the firing strength for the external disturbances. In both parts, the effects of parametric uncertainties and external disturbances are (w ) as follows: explicitly considered. Thereafter, the stability of the proposed control strategy in the presence of PUEDs is addressed by the O = w = Ω ðÞ x = Ω ðÞ y , j =1,2: ð30Þ j j A B j j Lyapunov theory in the next section. 5.1. Offline Training of the RLQR-NF Dataset for Parametric Layer 3. In this layer, the normalization of the node’s firing Uncertainties. As shown in Figure 6, this subsection is further strength is carried out by dividing the jth rule firing strength presented into two stages: first, the formulation of a robust to all rules’ total firing strength. This layer is designated by LQR-NF dataset by varying mass parameters, and second, notation N in the ANFIS architecture. The firing strength the stepwise layout of offline training of a dataset using (w  ) is normalized as follows: ANFIS parameters to design the robust control strategy. 3 5.1.1. Stage I: Formulation of the RLQR-NF Training Dataset. O = w  = , j =1,2: ð31Þ j j w + w The training dataset, having multiple-input and single- 1 2 output (MISO), is formed by employing the concepts of the LQR control strategy as shown in Equation (25). The LQR Layer 4. This layer, having the square nodes, is used to cost function is considered a minimization problem while estimate the rule’s involvement by defuzzification of input applying the optimality conditions and is expressed as fol- variables and produce the respective output as follows: lows [44]: O = w  ∠ = w  p x + q y + r , ð32Þ j j j j j j j ð ˇ ˇ J = z Qz +u Ru dt, ð34Þ where w  indicates the normalized firing strength and p , q , j j and r signify the subsequent limits. where Q and R denote the user-defined state-weight matrix and control cost matrix, respectively. An appropriate selec- Layer 5. This layer, having circular shape nodes with the tion of both matrices directly influences the performance designation ∑, processes the final output using the characteristics of the controller. summation of all incoming signals from the preceding layer. The generalized input (u ˇ ) to the control system is artic- Mathematically, it can be expressed as follows: ulated by regulating the error vector (ℯ) as follows: ∑ w ∠ j j j O =〠w ∠ = : ð33Þ j j j ∑ w j j ˇ j u =Kℯ =KzðÞ −z , ð35Þ des 10 Applied Bionics and Biomechanics Stage I Stage II Mathematical Defining input Selection of Offline formulation and output ANFIS training of of RLQR-NF training training RLQR-NF dataset vectors parameters controller Figure 6: Flowchart representation of the stage-wise design procedure. where z and z represent the desired and actual state vec- denote the controller output for the hip, knee, and ankle des tors, respectively. The optimal state gain matrix (K) can be joints of the exoskeleton device, respectively. In Equation expressed in terms of the control cost matrix (R) and the (41), ℯ , ℯ , and ℯ signify the hip, knee, and ankle joint θ θ θ H K A state-weight factor matrix (B) as follows: angular errors, respectively. ℯ , ℯ , and ℯ represent the _ _ _ θ θ θ H K A respective errors of the hip, knee, and ankle joint angular velocities. −1 K = R B P, ð36Þ The expanded structure of the robust dataset ðS Þ is rd finally depicted as below: where P is the answer for the algebraic form of the Riccati equation expressed in the form of state-weight factor matri- S = ℯ u ˇ , ð42Þ rd m ,m ,m t c hf ˇ ˇ ces (A and B), state-weight matrix (Q), and control cost matrix (R) as given below [44, 45]: where T T −1 ˇ ˇ ˇ ˇ PA +A P −PBR B P +Q = 0: ð37Þ S = S 1 S 2 S 3 : ð43Þ ðÞ ðÞ ðÞ rd rd rd rd It is truly evident from Equation (36) that the state gain ˇ ˇ matrix (K) is regulated by A, B, Q, and R matrices where 5.1.2. Stage II: Stepwise Layout of Offline Training of the ˇ ˇ RLQR-NF Dataset. The layout and execution of the proposed A and B are reliant on the mechanical arrangement and control strategy for the exoskeleton device are presented in dynamic parameters of the requisite system. the following steps. Exploiting the controller’s gain (K), a generalized dataset (S ) with the error vector (ℯ) and respective input Step 1. The state gain matrix (K ) is evaluated by solv- (u ˇ ) to the control system can be created in the following m ,m ,m t c hf ing Equations (36) and (37) for a different set of coupled form [44]: thigh ðm Þ, calf ðm Þ, and heel-foot ðm Þ masses, as shown t c hf in Table 2. The parametric variation is incorporated by S = ℯ u ˇ : ð38Þ increasing the nominal mass values up to 20%, with an increment of 0.3, 0.15, and 0.06 kg for the thigh, calf, and In this work, the dataset ðS Þ is expanded into a robust heel-foot. After performing several numerical experiments, form by evaluating the controller’s gain (K ) for a m ,m ,m t c hf the state-weight matrix and control cost matrix are selected bounded variation of the coupled thigh ðm Þ, calf ðm Þ, and t c as Q = diag ð400000, 4000, 8000, 800, 8000, 800Þ and R = heel-foot ðm Þ masses of the human-exoskeleton system. hf eyeð3, 3Þ. Thereafter, the controller input ðu Þ for the hip, knee, m ,m ,m t c hf and ankle joints is formulated as follows: Step 2. Apply Equation (39) to compute the controller input ðu Þ for the operating range of state variables in the m ,m ,m t c hf u ˇ =K ℯ, ð39Þ error vector as shown in Table 3. The structure of three robust m ,m ,m m ,m ,m t c hf t c hf datasets is formed by exploiting Equations (42) and (43). where Step 3. The training of robust datasets is carried out using the ANFIS approach. The first six columns of every dataset are hi ð40Þ u ˇ = u ˇ u ˇ u ˇ , inherently considered the input set. The last column of every m ,m ,m H K A t c hf m ,m ,m m ,m ,m m ,m ,m t c hf t c hf t c hf dataset is regarded as the output set. The input set comprises the error vector ðℯÞ, and the output set contains the controller hi ð41Þ input vector ðu ˇ Þ. The three ANFIS architectures are ℯ = ℯ ℯ ℯ ℯ ℯ ℯ : _ _ _ θ θ θ m ,m ,m θ θ θ t c hf H K A H K A formed, trained, and saved as anfis1.fis, anfis2.fis, and anfis3.- ˇ ˇ ˇ In Equation (40), u , u , and u fis for three controller inputs. Several simulation runs are H K A m ,m ,m m ,m ,m m ,m ,m t c t c t c hf hf hf Applied Bionics and Biomechanics 11 Table 2: Variation in lower limb mass parameters of the coupled Table 4: Training parameters of ANFIS architectures. human-exoskeleton system. Training parameters anfis1.fisanfis2.fisanfis3.fis Thigh, m (kg) Calf, m (kg) Heel-foot, m (kg) t c hf MF type Gaussian Gaussian Gaussian 8.25 3.85 1.50 MF number 5 12 3 8.58 4.00 1.56 Error tolerance 0.00001 0.0001 0.001 8.91 4.16 1.62 Epochs 10 15 5 9.24 4.31 1.68 Learning model Hybrid Hybrid Hybrid 9.57 4.47 1.74 9.90 4.62 1.80 Substituting Equation (44) into Equation (22), the linear- ized dynamic model can be written as Table 3: Operating range of the error in state variables. ˇ ˇ ˇ z _ = A −BK z +B d + d : ð45Þ Variables in the error Minimum Maximum Units vector value value Now, the error dynamics of the control design in the time -60 60 Degree domain can be expressed as below: -60 60 Degree -30 30 Degree ˇ ˇ ˇ ̂ ℯ_ðÞ t = A −BK ℯðÞ t +B dt ðÞ −ft ðÞ , ð46Þ _ -90 90 Degree/sec -90 90 Degree/sec where -60 60 Degree/sec ˇ ˇ ˇ ˇ Bft ðÞ =z ðÞ t −Bdt ðÞ − A −BK z ðÞ t : ð47Þ des des performed by varying the number of MF from 1 to 50 and Employing assumption A2, the f ðtÞ can be considered a epochs from 1 to 30. Thereafter, the training parameters function with an upper limit. Therefore, the effectiveness of are selected based on the zero error tolerance between the coupled human-exoskeleton system can be augmented the desired and predicted output vectors. In general, by estimating the f ðtÞ from dðtÞ. Using the ANFIS architec- ANFIS utilizes two optimization methods: backpropagation ture mentioned in Section 4, the f ðtÞ is approximated as and hybrid, to establish the learning between the input and follows [46, 47]: output vectors. A gradient descent model is employed to evaluate the node error in the backpropagation method. In contrast, a least square algorithm along with the gradi- f z = W Ω z + ϱ z , ð48Þ ðÞ ðÞ ðÞ ent descent model is exploited to regulate the errors in the hybrid method. In this work, the hybrid method is used where W denotes the ideal normalized firing strength matrix; with all three datasets for training the neural-fuzzy net- ΩðzÞ signifies the membership function vector; and ϱðzÞ works. The complete details of training parameters are represents an error of approximation with the condition given in Table 4. After generating training files (anfis1.fis, ϱðzÞ ≤ ζ, where ζ is a constant factor. anfis2.fis,and anfis3.fis) as the desired robust LQR-based Now utilizing Equations (46) and (48), the error dynam- neural-fuzzy controller, the respective signals are inputted to ics can be rewritten as the nonlinear dynamics of the coupled human-exoskeleton system. ˇ ˇ ˇ ℯ_ t = A −BK ℯ t +B dt − W Ω z − ϱ z : ð49Þ ðÞ ðÞ ðÞ ðÞ ðÞ 5.2. Online Training of the RLQR-NF Dataset for External Disturbances. Considering assumption A1, the control input From the above equation, the disturbance estimator dðtÞ with uncertain parameters ðu ̂ Þ can be further expressed in is defined as follows to design the control law [47]: terms of the state feedback ðKzÞ and disturbance observer ðdÞ as ̂ ̂ d = W Ω ðÞ z − ζ sign ℯ PB , i =1, 2, ⋯, p and j =1,2, ⋯, l, i ji j i u ̂ =ðÞ Δ +1 u = d −Kz, ð44Þ ð50Þ where d denotes the estimated disturbance by ANFIS where p and l denote the number of inputs and network architecture, and K signifies the state feedback matrix. nodes in the hidden layer. 12 Applied Bionics and Biomechanics In the above expression, the estimated strength matrix T T T _ ˇ V = −ℯ Q +K RK ℯ +2〠 ℯ PB ðW Þ is updated according to the following law: ji i=1 ̂ ~ × d − W Ω ðÞ z + W Ω ðÞ z − ϱðÞ z ð58Þ i ji j ji j i _ T W = −Γ Ω z ℯ PB , ð51Þ ðÞ ji i j i −1 _ +2〠 W Γ W : ji i ji where Γ denotes a positive definite matrix with the sym- i=1 metric property. Now substituting Equations (50) and (51) into Equation 6. Stability Analysis of the LQR-Based Neural- (58), one can obtain Fuzzy Control In this section, the Lyapunov function is presented to analyze T T V = −ℯ Q +K RK ℯ +2〠 ℯ PB the global stability of the proposed control scheme under the i=1 effect of parametric uncertainties and external disturbances. × d − W Ω ðÞ z − ϱðÞ z Consider the Lyapunov candidate function as follows: i ji j i ð59Þ T T p ≤−ℯ Q +K RK ℯ T −1 ~ ~ V = ℯ Pℯ + 〠 W Γ W , ð52Þ ji ji i 2 ≤ 0: i=1 with Invoking the above equation, it can be concluded that the error dynamics of the coupled human-exoskeleton system is = W − W , ð53Þ ji ji ji asymptotically stable. Therefore, the proposed controller carries out the asymptotic tracking with error ℯ⟶ 0 (as t⟶ ∞) following the disturbance estimator (Equation where W denotes the estimation error between the estimated ji (50)) and firing strength adaptation law (Equation (51)). Fur- strength matrix and the ideal constant strength matrix. thermore, the local stability of the proposed controller can be Differentiating Equation (52) and employing Equations analyzed using the pole placement theory, where the conver- (49) and (37), one can obtain gence rate can be investigated by keeping the poles on the left side of the s-plane [45]. T T T V = −ℯ Q +K RK ℯ +2ℯ PB 7. Results and Discussion T _ −1 ~ ~ × d − W Ω ðÞ z − ϱðÞ z +2〠 W Γ W : i j i ji ji ji i i=1 In this section, the simulation results and analyses are pre- sented to evaluate the effectiveness of the proposed control ð54Þ strategy (RLQR-NF) for an exoskeleton device during pas- sive gait rehabilitation measures. The block representation Now utilizing Equation (53), the error difference between for the proposed control strategy is schematically shown the actual value and the desired value by the designed neural- in Figure 7. At first, the performance of the proposed con- fuzzy network can be expressed as trol strategy is compared with a contrast control strategy without parametric uncertainties and external disturbances. ~ ̂ W Ω z = W Ω z − W Ω z : ð55Þ ðÞ ðÞ ðÞ ji j ji j ji j In this work, the exponential reaching law-based sliding mode (ERL-SM) control is used as a contrast control strat- Moreover, as the firing strength matrix ðW Þ is a con- ji egy [48]. Thereafter, two cases are contemplated to demon- stant matrix, Equation (53) holds the following relation after strate the controller’s robustness: the first case, increasing differentiation: the coupled segment masses by 20% with a trigonometric form of disturbances, and the second case, increasing the _ _ ~ ̂ W = W : ð56Þ coupled segment masses by 30% with a random form of ji ji disturbances. Reconstituting Equation (54) using Equations (55) and 7.1. Simulation Results without Parametric Uncertainties and (56), one can obtain External Disturbances. In this subsection, a comparative analysis between the RLQR-NF and ERL-SM control strate- T T T V = −ℯ Q +K RK ℯ +2ℯ PB gies is presented for desired gait tracking during passive gait rehabilitation measures. In the absence of parametric uncer- ̂ ~ × d − W Ω ðÞ z + W Ω ðÞ z − ϱðÞ z i ji j ji j i tainties and external disturbances, i.e., Δ =0 and d =0 in ð57Þ Equation (25), the nominal mass values of lower limb seg- −1 _ ~ ̂ +2〠 W Γ W , ments ðm =8:25, m =3:85, and m =1:5Þ are taken into ji i ji t c hf i=1 account for drawing the state of comparison. Using Equation Applied Bionics and Biomechanics 13 W = –Γ 𝛺 (z)e PB ji i j i Online disturbance estimator d = W 𝛺 (z)– sign (e PB ) i j i j i Offline trained 𝜃 – RLQR-NF model External (u , u , u ) H K A disturbances –0.35 Desired gait trajectory –0.4 –0.45 –0.5 𝜃 (t) –0.55 Starting point (at t = 0) –0.6 –0.2 –0.1 0 0.1 0.2 0.3 0.4 + X (m) – 𝜃 (t), 𝜃 (t) a a 𝜃 (t) 𝛥 u Kinematics 𝛥 t 𝜃 (t), 𝜃 (t) a a (H, K, A) (H, K, A) Desired joint angle (𝜃 ) Potentiometer/IMU (hip: H, knee: K and ankle: A) Figure 7: Schematic representation of the implemented proposed RLQR-NF controller. –0.4 –0.45 –0.5 –0.55 –0.6 –0.65 –0.2 –0.1 0 0.1 0.2 0.3 0.4 X (m) Desired ERL-SM RLQR-NF Figure 8: Desired trajectory tracking for ERL-SM and RLQR-NF control schemes with nominal mass values. (36), the optimal state gain matrix for the nominal mass eters into the proposed control (RLQR-NF) and ERL-SM parameters ðK Þ is calculated as below: control strategy. A time period of 2 seconds is considered 8:25,3:85,1:5 to complete one gait cycle. The starting Cartesian position (X, Y: 0.25 m, -0.60 m) of the desired gait trajectory is illus- 0 1 448:63 −2:72 −0:57 17:84 −0:31 −0:127 trated in black color. The actual trajectories in the Cartesian B C B C coordinate frame are presented by the dashed blue line K = 7:65 12:40 2:49 3:20 20:27 0:82 : 8:25,3:85,1:5 @ A (RLQR-NF) and the green line (ERL-SM). 8:72 −1:92 20:17 0:33 −2:56 6:40 Figures 9(a) and 9(b) depict the tracking error in both ð60Þ directions, i.e., X- and Y-directions (ℯ and ℯ ). The maxi- x y mum absolute deviation in the X-direction ðjℯ j Þ for the max ERL-SM and RLQR-NF control schemes is 0.013 m and In the ERL-SM control strategy [48], the control law 0.008 m, respectively. In the Y-direction, the respective devi- parameters are used as c = diag ð50, 50, 50Þ, ε =0:5× I , and ation ðjℯ j Þ is observed to be 0.009 m and 0.006 m for the k = I . As shown in Figure 8, the healthy gait trajectory is y max e 3 ERL-SM and RLQR-NF control strategies. tracked by incorporating the kinematic and dynamic param- Y (m) Robust LQR based neural-fuzzy controller e (t), e (t) Coupled dynamical system State feedback 14 Applied Bionics and Biomechanics 0.01 0.01 0.005 0.005 –0.005 –0.005 –0.01 –0.01 –0.015 –0.02 –0.015 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time (s) Time (s) ERL-SM RLQR-NF (a) (b) Figure 9: Position tracking error for ERL-SM and RLQR-NF control schemes with nominal mass values. (a) X-direction. (b) Y-direction. 30 60 10 20 0 𝜃 0 𝜃 –5 –10 –10 –20 –20 012 012 Time (s) Time (s) Time (s) Desired ERL-SM RLQR-NF (a) (b) (c) Figure 10: Joint angle tracking for ERL-SM and RLQR-NF control schemes with nominal mass values. (a) Hip joint. (b) Knee joint. (c) Ankle joint. The tracking of desired joint angles with applied control through repetitive gait rehabilitation exercises. With the strategies is illustrated in Figures 10(a)–10(c). The angular ERL-SM control scheme, the peak values of the hip, knee, deviations (ℯ , ℯ , and ℯ ) from desired joint trajectories and ankle signals are estimated as 32.98 V, 19.88 V, and θ θ θ H K A are shown in Figures 11(a)–11(c). Considering the hip joint, 1.8 V. On the other hand, with the RLQR-NF control it is observed that the maximum absolute deviation ðjℯ j Þ scheme, the respective values of control signals are found max ° ° for respective controllers is 0.78 (ERL-SM) and 0.51 to be 30.25 V, 18.1 V, and 1.25 V. It is evident from the (RLQR-NF). In the knee joint, the deviation ðjℯ j Þ is results that the proposed control strategy (RLQR-NF) out- max ° ° found to be 1.15 and 1.16 for the system with the ERL-SM performs the contrast control strategy (ERL-SM) to track and RLQR-NF control strategies, respectively. For the ankle the desired gait trajectory, however, with a marginal differ- joint, the respective deviations ðjℯ j Þ are estimated as ence. Therefore, to demonstrate the effectiveness of the pro- A max ° ° posed control when dealing with PUEDs, variations in mass 0.81 (ERL-SM) and 0.32 (RLQR-NF). parameters and the form of disturbances are considered Figures 12(a)–12(c) demonstrate the generated control further. signals (u , u , and u ) following the desired trajectory H K A (deg) e (m) (deg) e (m) (deg) A Applied Bionics and Biomechanics 15 1 1.5 1 0.5 0.5 0.5 0 0 𝜃 𝜃 𝜃 –0.5 –0.5 –0.5 –1 –1.5 –1 –1 0 1 2 0 1 2 0 1 2 Time (s) Time (s) Time (s) ERL-SM RLQR-NF (a) (b) (c) Figure 11: Joint tracking error for ERL-SM and RLQR-NF control schemes with nominal mass values. (a) Hip joint. (b) Knee joint. (c) Ankle joint. –50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) (a) –30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) (b) –3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) ERL-SM RLQR-NF (c) Figure 12: Control signals for ERL-SM and RLQR-NF control schemes with nominal mass values. (a) Hip joint. (b) Knee joint. (c) Ankle joint. 7.2. Simulation Results for Parametric Variations and proposed control scheme over the contrast control External Disturbances. In order to realize the robust- scheme. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ness of the proposed control strategy (RLQR-NF), the root mean square error (RMSE) is computed and RMSE = 〠 ℯ , kk compared with the ERL-SM control strategy for para- a=1 ð61Þ metric uncertainties and external disturbances. More- RMSE − RMSE ERL‐SM RLQR‐NF over, based on RMSE values, the performance index PI = × 100%, RMSE (PI) is calculated to analyze the improvement of the ERL‐SM e (deg) u (volt) u (volt) u (volt) A K H e (deg) e (deg) A 16 Applied Bionics and Biomechanics 1.2 1.5 1 0.5 0.5 0 0 0 𝜃 𝜃 –0.5 –0.5 –1 –1.2 –1.5 –1 012 012 012 Time (s) Time (s) Time (s) ERL-SM RLQR-NF (a) (b) (c) Figure 13: Joint tracking error for ERL-SM and RLQR-NF control schemes with the first case of PUEDs. (a) Hip joint. (b) Knee joint. (c) Ankle joint. 1.5 1.5 1 0.5 0.5 0 0 0 𝜃 𝜃 𝜃 –0.5 –0.5 –1 –1.5 –1.5 –1 012 012 0 12 Time (s) Time (s) Time (s) ERL-SM RLQR-NF (a) (b) (c) Figure 14: Joint tracking error for ERL-SM and RLQR-NF control schemes with the second case of PUEDs. (a) Hip joint. (b) Knee joint. (c) Ankle joint. where ℯ : ℯ represents the error between the desired RMSE values for the control strategies are recorded as a θ ° ° and actual joint angles, and N is the size of the error 0.321 and 0.224 . vector. RMSE and RMSE signify the root Considering the second case of PUEDs (Case II), where ERL‐SM RLQR‐NF the system masses are increased by 30% ðm =10:73, m = mean square errors related to ERL-SM and RLQR-NF t c control strategies. 5:00, and m =1:95Þ along with a random form of distur- hf Considering the first case of PUEDs (Case I), where the bances ðD = ð5 × randomð1ÞÞ, D = ð3 × randomð1ÞÞ, and 1 2 system masses are increased by 20% ðm =9:90, m =4:62 t c D = ð2 × randomð1ÞÞ, the joint angular errors (ℯ , ℯ , and 3 θ θ H K , and m =1:80Þ along with a trigonometric form of dis- hf ℯ ) for the proposed and contrast control strategies are turbances ðD = ð6 sin ð4πtÞÞ, D = ð5 sin ð3πtÞÞ, andD = 1 2 3 shown in Figures 14(a)–14(c). For the hip joint, the values ð3 sin ð2πtÞÞÞ, the joint angular errors (ℯ , ℯ , and ℯ ) θ θ θ H K A ° of RMSE and RMSE are found to be 0.613 ERL‐SM RLQR‐NF for the applied control strategies are shown in and 0.287 , respectively. The respective RMSE values related Figures 13(a)–13(c). For the hip joint, RMSE and ERL‐SM ° ° ° ° to the knee joint tracking are obtained as 0.742 and 0.434 . RMSE are estimated as 0.578 and 0.283 , respec- RLQR‐NF The following RMSE values for the ankle joint are found to tively. The respective RMSE values for the knee joint are ° ° ° ° found to be 0.672 and 0.42 . In ankle joint tracking, the be 0.334 and 0.228 . e (deg) e (deg) e (deg) e (deg) e (deg) e (deg) A A Applied Bionics and Biomechanics 17 Table 5: Comparative performance analysis of the proposed control over the contrast control. Joint name RMSE (ERL-SM) (deg) RMSE (RLQR-NF) (deg) PI (%) Case I: with a 20% increment in mass parameters and a trigonometric form of external disturbancesðÞ m =9:90, m =4:62, and m =1:80 t c hf ðÞ D =ðÞ 6 × sinðÞ 4πt , D =ðÞ 5 × sinðÞ 3πt , andD =ðÞ 3 × sinðÞ 2πt 1 2 3 Hip 0.578 0.283 51.04 Knee 0.672 0.420 37.50 Ankle 0.321 0.224 30.21 Case II: with a 30% increment in mass parameters and a random form of external disturbancesðÞ m =10:73, m =5:00, and m =1:95 t c hf ðÞ D =ðÞ 6 × randomðÞ 4 , D =ðÞ 5 × randomðÞ 3 , andD =3 ð × randomðÞ 2 1 2 3 Hip 0.613 0.287 53.19 Knee 0.742 0.434 41.51 Ankle 0.334 0.228 31.73 state gain matrix, robust ANFIS training datasets have been Table 6: Settling time of ERL-SM and RLQR-NF control schemes formed with a variation of system parameters. The operating for convergence analysis. range of the error vector and control responses have been Settling time (sec) regarded as the training input and output vectors. The Control Lower Nominal Case I of Case II of ANFIS architectures have been trained offline to deal with scheme limb joint system mass PUEDs PUEDs the effect of parametric uncertainties. Thereafter, the online Hip 1.971 1.986 1.993 adaptation law of firing strength in ANFIS architectures has ERL-SM Knee 1.962 1.974 1.988 been incorporated to deal with external disturbances. The asymptotic stability of the coupled dynamics while applying Ankle 1.927 1.943 1.969 the proposed control has been ensured using the Lyapunov Hip 1.951 1.967 1.991 theory. Finally, the effectiveness of the proposed controller RLQR-NF Knee 1.946 1.959 1.987 has been investigated by comparing it to the exponential Ankle 1.914 1.933 1.958 reaching law-based sliding mode control. The robustness analysis has been carried out by varying mass parameters and inducing different forms of external disturbances. The Table 5 presents the performance index (PI) of the pro- simulation results have shown the potential of the proposed posed control over the contrast control. The proposed con- robust tracking control for passive gait rehabilitation using trol is promising in desired gait tracking compared to the an exoskeleton system. In the future, the effect of human contrast control, subjected to PUEDs. Moreover, as observed involvement will be considered to design an “assist-as- from Table 5, the performance index (PI) is improved by needed” control strategy during active rehabilitation. 2.15%, 4.01%, and 1.52% in Case II as compared to Case I. During rehabilitation exercises, this performance investiga- Appendix tion allows the lower limb exoskeleton system to carry out repetitive movements with greater accuracy under the pres- The elements of the state-weight factor matrices ðA,BÞ can ence of PUEDs. be evaluated using the following derived formulations. The convergence of both control schemes is investigated For the A matrix, by evaluating the settling time, i.e., the time lapsed for the error to drop within 2% of the final value. The settling time α = −12l l l g 2m +4m m +3m m +2m m , 41 1 2 3 2 1 2 1 3 2 3 for the error in the hip, knee, and ankle joints for every set 2 2 2 2 of mass values is presented in Table 6. The low values of α ~ =2l l 12l m +12l m m +16l m m +12l m m 41 2 3 1 1 2 3 1 1 2 1 1 3 settling time indicate the faster convergence of the proposed − 48l l m m − 36l l m Þ, controller (RLQR-NF) over the contrast controller (ERL-SM) 2 3 2 3 2 3 3 before achieving the full stable state. 2 2 α =72l l l gm + m +2m m , 42 1 2 3 2 3 2 3 α ~ = α ~ , 42 41 8. Conclusions α = −18gl m m , 43 1 2 3 α ~ In this work, a robust LQR-based neural-fuzzy control has 41 α ~ = , been proposed to follow the natural gait trajectory using an 2l l 2 3 exoskeleton system during passive rehabilitation measures. α =12g 6l m +3l m +4l m +3l m ðÞ 51 1 2 1 3 2 2 2 3 Primarily, a linearized state-space form of the nonlinear l m l m l m 1 1 2 2 3 3 human-exoskeleton has been established via the input- l m + l m + l m + + + , 1 2 1 3 2 3 2 2 2 output feedback linearization method. Employing the LQR 18 Applied Bionics and Biomechanics β = − 144l l m +72l l m +96l l m +72l l m , α ~ = α ~ = α ~ = α ~ = α ~ = α ~ = α ~ = α ~ = α ~ ðÞ ′ ′ ′ ′ ′ 42 1 3 2 1 3 3 2 3 2 2 3 3 51 51 ′ 52 52 ′ 53 53 ′ 63 52 53 ~ ~ ~ α ~ β = β = β , 42 43 41 = α ~ = , 63 ′ 2l β =72l l m + 144l l m +72l l m , 43 1 2 2 1 3 2 1 3 3 α =6gl m +2l m + l m ðÞ β = −ðÞ 72l m +36l m +48l m +36l m , 51 2 2 2 3 3 3 1 2 1 3 2 2 2 3 2 2 2 2 2 4l m +12l m +3l m +4l m +3l m +12l l m 1 1 1 2 1 3 2 2 2 3 1 2 2 β = , 2l +6l l m − 12l l m Þ, 1 2 3 2 3 3 2 2 2 2 2 β =48l m + 144l m +36l m +48l m +36l m 1 1 1 2 1 3 2 2 2 3 l α ~ 2 41 α = , ′ + 144l l m +72l l m − 144l l m , 1 2 2 1 2 3 2 3 3 2l 2 2 2 2 α =6gm 6l l m +3l l m +9l l m +4l l m ′ 3 2 1 1 2 2 2 3 1 51 ′ 1 2 1 1 l β 2 41 β = , 2 2 2 2 +12l l m +3l l m − 12l l m − 18l l m 2l 1 3 2 1 3 3 3 2 3 2 3 3 2 2 2 2 β = − 72l l m +36l l m + 108l l m +48l l m +6l l l m +3l l l m Þ, 53 1 2 1 2 1 2 1 2 2 1 3 1 1 2 3 2 1 2 3 3 2 2 2 2 + 144l l m +36l l m − 144l l m − 216l l m 1 3 2 1 3 3 3 2 3 2 3 3 α =6glðÞ m +2l m + l m 52 2 2 2 3 3 3 +72l l l m +36l l l m Þ, 1 2 3 2 1 2 3 3 ðÞ 6l m +3l m +4l m +3l m , 1 2 1 3 2 2 2 3 l β α =6gl m +2l m + l m 2 41 ðÞ ~ 52 2 2 2 3 3 3 β = , 2 2 2 2 2 4l m +12l m +3l m +4l m +3l m 1 1 1 2 1 3 2 2 2 3 β =36l l m +2l m + l m , ðÞ 61 1 2 2 3 2 3 3 +12l l m +6l l m − 12l l m Þ, 1 2 2 1 2 3 2 3 3 ~ β = , α = α = α = α , 2 ′ ′ 52 ′ 51 ′ 53 63 β = β , 62 53 α =6gl mðÞ 6l m +3l m +4l m +3l m , ′ 3 3 1 2 1 3 2 2 2 3 ~ ~ 2 2 2 2 2 β = β , 62 53 α =6gl m 4l m +12l m +3l m +4l m +3l m ′ 3 3 1 2 3 2 3 53 ′ 1 1 1 2 2 2 2 2 2 2 2 2 2 2 β =36l l m + 144l l m m +48l l m m + 144l l m m 63 1 2 2 1 2 2 3 1 2 1 2 1 2 1 3 +12l l m +6l l m − 12l l m Þ, 1 2 2 1 2 3 2 3 3 2 2 2 2 + 216l l l m m + 144l l l m m + 144l l m m 2 2 2 1 2 3 2 3 1 2 3 1 3 1 3 2 3 α =6glðÞ m +2l m + l m 6l l m +3l l m +9l l m 61 2 2 2 3 3 3 1 2 1 2 1 2 1 2 2 2 2 2 2 2 3 3 2 +36l l m +48l l m m − 144l l m m − 432l l m 1 3 3 1 3 1 3 2 3 2 3 2 3 3 2 2 2 2 2 +4l l m +12l l m +3l l m − 12l l m − 18l l m 1 3 1 1 3 2 1 3 3 3 2 3 2 3 3 2 2 3 2 2 − 432l l m − 144l l m , 2 3 3 3 3 +6l l l m +3l l l m Þ, 1 2 3 2 1 2 3 3 l l m β 2 3 3 41 l α ~ 2 41 β = : ðA:2Þ α ~ = α ~ = α ~ = α ~ = α ~ = , 61 ′ 62 ′ ′ 61 62 63 2 2 2 2 2 2 2 2 2 α =6g 3l l m +12l l m m +4l l m m +12l l m m ′ 2 3 1 2 1 3 61 1 2 2 1 2 1 2 1 2 Data Availability 2 2 2 2 +18l l l m m +12l l l m m +12l l m m 1 2 3 2 3 1 2 3 1 3 1 3 2 3 2 2 2 2 3 3 2 2 The data used to support the findings of the study are +3l l m +4l l m m − 12l l m m − 36l l m 1 3 3 1 3 1 3 2 3 2 3 2 3 3 included within the article. 2 2 2 3 2 − 36l l m − 12l l m Þ, 2 3 3 3 3 Conflicts of Interest α =36gl l m +2l m + l m ðÞ 61 ′ 1 2 2 3 2 3 3 l m l m l m The authors declare that they have no conflicts of interest. 1 1 2 2 3 3 l m + l m + l m + + + , 1 2 1 3 2 3 2 2 2 Acknowledgments α ~ = α ~ = , ′ ′ 61 ′ 62 ′ The authors acknowledge the Department of Scientific and α = α , 62 61 Industrial Research, India, for establishing the PRISM (Pro- moting Innovations in Individuals, Start-ups, and MSMEs) α = α = α , ′ ′ ′ 62 61 63 scheme under which this project work is carried out. 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Robust LQR-Based Neural-Fuzzy Tracking Control for a Lower Limb Exoskeleton System with Parametric Uncertainties and External Disturbances

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Copyright © 2021 Jyotindra Narayan and Santosha K. Dwivedy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Applied Bionics and Biomechanics Volume 2021, Article ID 5573041, 20 pages https://doi.org/10.1155/2021/5573041 Research Article Robust LQR-Based Neural-Fuzzy Tracking Control for a Lower Limb Exoskeleton System with Parametric Uncertainties and External Disturbances Jyotindra Narayan and Santosha K. Dwivedy Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India Correspondence should be addressed to Jyotindra Narayan; n.jyotindra@gmail.com Received 23 February 2021; Revised 14 April 2021; Accepted 15 May 2021; Published 12 June 2021 Academic Editor: Fahd Abd Algalil Copyright © 2021 Jyotindra Narayan and Santosha K. Dwivedy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The design of an accurate control scheme for a lower limb exoskeleton system has few challenges due to the uncertain dynamics and the unintended subject’sreflexes during gait rehabilitation. In this work, a robust linear quadratic regulator- (LQR-) based neural-fuzzy (NF) control scheme is proposed to address the effect of payload uncertainties and external disturbances during passive-assist gait training. Initially, the Euler-Lagrange principle-based nonlinear dynamic relations are established for the coupled system. The input-output feedback linearization approach is used to transform the nonlinear relations into a linearized state-space form. The architecture of the adaptive neuro-fuzzy inference system (ANFIS) and used membership function are briefly explained. While varying mass parameters up to 20%, three robust neural-fuzzy datasets are formulated offline with the joint error vector and LQR control input. Thereafter, to deal with external interferences, an error dynamics with a disturbance estimator is presented using an online adaptation of the firing strength matrix. The Lyapunov theory is carried out to ensure the asymptotic stability of the coupled human-exoskeleton system in view of the proposed controller. The gait tracking results for the proposed control scheme (RLQR-NF) are presented and compared with the exponential reaching law-based sliding mode (ERL-SM) controller. Furthermore, to investigate the robustness of the proposed control over LQR control, a comparative performance analysis is presented for two cases of parametric uncertainties and external disturbances. The first case considers the 20% raise in mass values with a trigonometric form of disturbances, and the second case includes the effect of the 30% increment in mass values with a random form of disturbances. The simulation runs have shown the promising gait tracking aspects of the designed controller for passive-assist gait training. 1. Introduction systematic yet comprehensive review has been carried out on the state-of-the-art developments of such multijoint and Over the last two decades, an increasing number of neurolog- single-joint exoskeleton devices for gait rehabilitation, ical disorders such as stroke, spinal cord injury, and Parkin- mobility aid, and strength amplification. A well-known treadmill-oriented exoskeleton, LOPES son’s disease have been observed in different age groups. The World Health Organization (WHO) reported “stroke” [5], has been developed with a 2D translatable pelvis seg- as one of the principal reasons for nearly 5 million people’s ment, two active hip joints, and an active knee joint for lower fatality through 2000-2016 and the third pioneering source limb rehabilitation. The system was controlled to supervise of debility throughout the world [1]. To address the concerns or follow the subjects using “robot-in-charge” and “subject- in-charge” modes. Bortole et al. [6] designed a 6-DOF lower of motor functionality in the lower body caused by neurolog- ical disorders, researchers have developed many robot-based limb exoskeleton for overground training of stroke subjects lower limb exoskeleton devices to produce therapeutic effects with a body height of 1.50-1.95 m and a body mass of during walking [2, 3]. In a recent work by Kalita et al. [4], a 100 kg. Hsieh et al. [7] proposed a soft exoskeleton design 2 Applied Bionics and Biomechanics control, Long et al. [20] presented a hybrid strategy where for preswing gait training of subjects with weak muscles, where a single actuator with a pulley-slider arrangement is SMC is augmented with a cerebellar model articulation con- used to drive the lower limb joints. The prototype is devel- troller (CMAC) to predict the motion intent of the subject. The optimized sliding surface of the SMC is estimated using oped and clinically investigated with seven subjects. In a study on a parallel mechanism-based lower limb rehabilita- the genetic algorithm to improve the effectiveness of the tion, Rastegarpanah et al. [8] investigated the performance proposed control scheme. Liu et al. [21] introduced an of a 6-DOF robot by executing foot trajectories of 20 healthy event-triggered SMC for effective tracking of the reference subjects. Furthermore, the same prototype was tested for trajectory using a lower limb exoskeleton system with PUEDs. In another work to address the model uncertainties eight poststroke patients while carrying out three exercises, i.e., hip flexion/extension, ankle dorsiflexion/plantarflexion, and the unintended subject’s response, Wu et al. [22] pro- and marching [9]. Aggogeri et al. [10] presented a modular posed an adaptive control scheme for a 3-DOF lower extrem- and reconfigurable mechanism for rehabilitating ankle joints ity rehabilitation device. Working on the decoupled control of different subjects. Cestari et al. [11] introduced the ATLAS strategy, Sun et al. [23] designed a reduced-order adaptive fuzzy approach and implemented it on a two-link exoskele- exoskeleton to assist the children during flexion/extension of the hip, knee, and ankle joints. At the preliminary level, a ton system for lower limb rehabilitation. dummy with body features of a 10-year-old child was used Furthermore, in recent times, robust intelligent control to test the exoskeleton system. Patané et al. [12] proposed a schemes have gain popularity to address the adverse effects multijoint exoskeleton, WAKE-up, to rehabilitate the knee, of PUEDs with effective approximation features. A neural network (NN) along with a time-delay evaluation-based con- ankle, and foot of the pediatric subjects with neurological dis- orders. The device was tested with four healthy children and trol scheme is proposed by Zhang et al. [24] to realize the three children with cerebral palsy. To amplify human endur- desired gait trajectory for a simulated model of a 10-DOF ance while carrying heavy loads, BLEEX [13] (7-DOF/limb) exoskeleton. The performance of the designed control was developed with intelligent and adaptable robot-based scheme is investigated by comparing the classical PD control scheme. Narayan and Dwivedy [25] proposed a neuro-fuzzy strategies where linear hydraulic actuators were used for the actuation of 4-DOF. Recently, Ji et al. [14] introduced a wear- compensator for PID control to deal with the system’s known able exoskeleton, SIAT-WEXv2, to support the user’s waist and unknown uncertainties during passive gait rehabilitation of a human child. The controller is found to be more robust and bones while lifting heavy objects in construction and logistic industries by providing an assistive output of 28 N. towards external disturbances over payload uncertainties. Chen et al. [26] proposed a disturbance estimator-based To augment the rehabilitation devices’ performance, the appropriate control schemes are designed by the researchers subject-cooperative control for a weight-reinforced active- for executing repetitive gait movements. The control archi- assist rehabilitation device. They computed the interaction torques using a backpropagation neural network-aided tecture required for the exoskeleton systems poses extra com- plexity over the conventional robotic arm control due to the disturbance observer and proved the stability using the Lyapunov theory. In a recent work by Han et al. [27], time- sophisticated mechanical configuration, complex motion trajectory, and human involvement. The researchers, in the delay estimator-aided computed torque control is designed literature, have regarded the predefined gait tracking control to deal with PUEDs of a lower limb exoskeleton system. Moreover, an adaptive radial basis function neural network as the basis of every control scheme for exoskeleton systems, where the joint movements of the lower limb could be (RBFNN) is utilized to compensate for the time-delay error. On the other hand, few researchers have explored the estimated using gait analysis experiments [15, 16, 17]. Although the exoskeleton systems exploit the gait of healthy optimal control, especially the linear quadratic regulator humans to replicate the same using predefined trajectory (LQR), to realize the natural gait [28, 29, 30, 31]. The LQR scheme with full-state feedback yields control measures con- control schemes, however, in practice, they are unable to attain the proper gait trajectory because of the parametric cerning the whole body compared to PD control for every uncertainties and external disturbances (PUEDs). Therefore, independent joint [28]. In addition to that, the relative prom- various robust control strategies have been designed to deal inence of curtailing the tracking error and minimalizing the with the limitations of classical trajectory tracking control control torque can be regulated by computing optimal values of time-varying gain based on the design parameters of a sin- in lower limb exoskeleton systems [18, 19, 20, 21, 22, 23]. Ajayi et al. [18] proposed a bounded control scheme for the gle controller. Furthermore, LQR as a linear control scheme rehabilitation of the knee ankle joint of a user in a sitting might be exploited for nonlinear system dynamics by position. The stability of the control law and convergence approximating the linear time-varying form and significantly analysis of the gain observer is validated with the Lyapunov mitigating the computational complexity involved in several nonlinear controllers. Ajjanaromvat and Parnichkun [29] theory. The simulation results are presented without and with the effect of the human interaction torque. Yang et al. proposed an iterative online learning-based LQR control [19] presented a sliding mode control (SMC) scheme where scheme for a treadmill-appended exoskeleton to investigate a second-order command filter-aided backstepping is incor- the robustness analysis. Moreover, the proposed control porated to avert the “explosion of complexity.” Moreover, scheme is aided with an adaptive iterative learning control to address tracking errors. Gupta et al. [30] presented the the fuzzy logic is exploited to counter the chattering issues of the control scheme during the estimation of structured LQR control for lower limb exoskeleton systems by consider- and unstructured uncertainties. In another work on robust ing the 4-DOF human gait model in the Single Support Phase Applied Bionics and Biomechanics 3 (SSP). They exploited the nondominated sorting genetic tem with different heights of the subjects augments the algorithm to find out the optimal weighing matrix. However, feature of cost-effectiveness. Considering the subject’s physi- the formulation work has not considered the uncertain fac- ological safety, all possible degrees of freedom should be tors in system dynamics. Castro et al. [31] proposed an avoided at the initial phases of rehabilitation training. Invok- integral-aided LQR (LQRi) and unknown input disturbance ing the design features, authors have designed a low-cost observer (UIO) to address external interferences of the lower stand-alone module-aided lower limb exoskeleton system limb exoskeleton system. The results of the proposed control for pediatric rehabilitation in their previous work [32]. The are compared with proportional-derivative control and CAD model of the designed exoskeleton system is shown in found to be more effective. Figures 1(a) and 1(b). A 3-DOF multilink mechanism for Although the hybrid form of sliding mode control can be each leg was intended to carry out hip flexion/extension, knee considered a highly robust control strategy, chattering always flexion/extension, and ankle dorsiflexion/plantarflexion affects the performance of exoskeleton systems. On the other motions. The placements of the joint actuators were made hand, the LQR is the most optimal control scheme and lacks to avoid any physical interference with the subject’s body. to resolve uncertain exoskeleton dynamics. Therefore, in this To serve subjects of different heights, a telescopic link joint work, a new robust LQR-based neural-fuzzy control scheme arrangement was designed around the knee joint of the exo- is designed for the lower limb exoskeleton system with para- skeleton system. Moreover, a detailed structural analysis of metric uncertainties and external disturbances during passive the stand-alone module was carried for maximum loading gait rehabilitation training. The key highlights of the present conditions at the hip joint [32]. work are as follows: The mechanical configuration of the exoskeleton system is intended for children of 8-12 years of age, 25-40 kg weight, (i) The input-output feedback linearization approach is and 115-125 cm height. The possible range of motion represented to linearize the nonlinear dynamics of (ROM) for three joints of the exoskeleton system in the sagittal ° ° ° ° the lower limb exoskeleton system plane is as follows: 30 /-12 (hip-f/e), 60 /-10 (knee-f/e), and ° ° 13 /-20 (ankle-d/p). To avoid any undesirable actions beyond (ii) A robust offline LQR-based neural-fuzzy control the ROM, an emergency stop option is provisioned at the scheme is designed to deal with payload uncertainties software interface during simulation runs. In this work, an (iii) A disturbance estimator is proposed using an online eight-year-old male subject’s anthropometric and kinematic parameters (body mass: 30 kg and body height: 1.22 m) are adaptation of firing strength in offline designed LQR-NF architecture considered input parameters to the control architecture. The breakdown of input parameters for the lower limb exoskeleton (iv) The simulation results are carried out for the RLQR- and subject is shown in Table 1, where the length of the NF control scheme and compared with an exponential thigh and shank link is kept constant at 0.27 m and reaching law-based sliding mode control (ERL-SM) to 0.30 m, respectively. track the desired gait trajectory during passive thera- Furthermore, an affordable wireless Labview-aided peutic training Kinect setup was established to conduct the experimental gait analysis. With necessary approval, the child subject was (v) The robustness performance of the proposed control asked to follow an inclined path over the ground in front of scheme (RLQR-NF) is investigated by varying pay- the experimental setup for 1.6-2.0 seconds. The angle estima- load parameters and inducing different forms of tion algorithm comprehended the information about the external disturbances lower limb joint angles from the skeleton model in Labview. The rest of the paperwork is structured as follows. The The angle estimation algorithm exploits the relation between mechanical description of the lower limb exoskeleton system joint triples using vector algebra. and the estimation of control input parameters are presented The detailed procedure of performing gait analysis, as in Section 2. In Section 3, the nonlinear dynamic relation is shown in Figure 2(a), is based on the work by Narayan et al. formulated using the Euler-Lagrange principle, and thereaf- [33]. The skeleton form of the subject during the gait analysis ter, input-output linearization of the nonlinear form is is illustrated in Figure 2(b). The desired lower limb joint explained. Section 4 presents the concept of ANFIS architec- angles attained from the experiment are presented in ture with the selected membership function. Section 5 Figure 3(a) and the corresponding trajectory in Figure 3(b). describes a detailed design procedure of the proposed control The ROM for the hip, knee, and ankle joints are recorded ° ° ° ° ° ° strategy. In Section 6, the Lyapunov theory of stability is pre- as 22.16 to -8.98 , 58.26 to 1.21 , and 5.84 to -7.94 for an sented. The control results are simulated and discussed in eight-year-old child, respectively. Section 7. The complete paperwork is concluded in Section 8. 3. Dynamic Model of the Coupled Human- 2. Mechanical Configuration of the Lower Limb Exoskeleton System Exoskeleton System In this section, the Euler-Lagrange principle is used to The main criteria for the mechanical design of a lower limb formulate the nonlinear dynamics of the coupled human- exoskeleton system are to ensure its strength and stability exoskeleton system. Thereafter, the input-output feedback of the subject’s safety. Moreover, the adaptability of the sys- linearization approach is exploited to linearize the nonlinear 4 Applied Bionics and Biomechanics (a) (b) Figure 1: CAD model of (a) LLES (labels: (1) thigh link, (2) shank link, (3) foot link, (4) hybrid stepper motor, (5) lead screw actuator, (6) stepper motor, (7) timing belt, (8) support module, (9) wheels, and (10) telescopic link joint connector) and (b) LEES with a human dummy [32]. where Table 1: Specifications of the lower limb exoskeleton system and child dummy. Part Mass (kg) Length (m) COM (m) 1 1 T _ _ ð3Þ K = 〠 m s_ s_ + θ I θ , i i i i i i Lower limb exoskeleton system 2 2 i=1 e e e m =4:75 l =0:25‐0:30 l =0:12‐0:15 Thigh link 1 1 c1 e e e m =1:60 l =0:30‐0:35 l =0:14‐0:17 Shank link 2 2 c2 e e m =0:85 l =0:05 l =0:02 Foot link 3 3 c3 P = 〠 m gh : ð4Þ ðÞ Child (age 8 years, body weight 30 kg, and body height 122 cm) i ci i=1 h h h Thigh m =3:50 l =0:27 l =0:13 1 1 c1 h h h Shank m =2:25 l =0:30 l =0:15 2 2 c2 In the abovementioned relations, θ represents the gener- h h h Foot m =0:65 l =0:04 l =0:02 alized coordinate of the human-exoskeleton system. The 3 c3 kinetic and potential energy about the i-link is denoted by K and P, respectively. In Equation (3), θ , s_ , m , and I signify i i i i behavior of the dynamical system. The transformed linear the angular velocity, speed of the center of mass in transla- state-space relation is established for the dynamics of the tional direction, mass, and inertia corresponding to the i lower limb exoskeleton system. -link. The acceleration due to the gravitational effects is referred by g, and the distance between the i-link’s center point forming the gravitational vector and the origin is 3.1. Nonlinear Dynamic Formulation. Among various denoted by h as illustrated in Equation (4). ci methods for expressing applied joint torques and angular Referring to Equations (2)–(4) to solve Equation (1), the acceleration, the Euler-Lagrange principle is well appreciated nonlinear dynamics of the coupled dynamical system can be by the research communities [34]. Invoking the Euler- articulated as follows: Lagrange principle, which employs kinetic and potential energy, the nonlinear representation of the coupled human- exoskeleton dynamics is obtained. A multilink structure of € _ _ τ = MðÞ θ θ + C θ, θ θ + GðÞ θ , ð5Þ the coupled system with a collaboration effect is shown in Figures 4(a) and 4(b). A generalized formulation to estimate the joint torques using the Lagrangian L is as follows: where τ = τ + τ + τ , d ∂L ∂L > a eh he τ = − , ð1Þ > e h dt _ ∂θ ∂θ > i MðÞ θ = M ðÞ θ + M ðÞ θ , i < ð6Þ e h _ _ _ C θ, θ = C θ, θ + C θ, θ , e h GðÞ θ = G ðÞ θ + G ðÞ θ , L = K − P, ð2Þ Applied Bionics and Biomechanics 5 Display panel (Modify Loop run at 16fps Comparison joint triples) of vector 1-2 Joint 1 Joint 2 Kinect initialization Comparison Joint 3 of vector 2-3 Angle evaluation Kinect Kinect configuration block read Initialize 3D display Kinect Cluster processor While loop Kinect close (a) (b) Figure 2: Gait analysis experiment. (a) Schematic diagram of the detailed procedure. (b) A child subject with the skeleton model during the experiment. 60 –0.35 –0.4 –0.45 –0.5 –0.55 Starting point at t = 0 –20 –0.6 0 0.5 1 1.5 2 –0.2 0 0.2 0.4 Y (m) Time (sec) Hip Desired gait trajectory Knee Ankle (a) (b) Figure 3: Experimental gait data. (a) Desired joint angular trajectory. (b) Desired gait trajectory. 2 3 The matrix form of inertial, Coriolis-centrifugal, and M M M 11 12 13 gravity effects of the coupled dynamical system is signified 6 7 6 7 MðÞ θ = M M M , by MðθÞ, Cðθ, θÞ, and GðθÞ, respectively. In Equation (6), 21 22 23 4 5 e h M ðθÞ and M ðθÞ represent the inertial dynamics of the exo- M M M 31 32 33 skeleton and human leg in the matrix form, respectively. The 2 3 Coriolis-centrifugal matrix of the exoskeleton and human leg C C C 11 12 13 e h 6 7 _ _ is represented by C ðθ, θÞ, and C ðθ, θÞ, respectively. The 6 7 C θ, θ = C C C , ð7Þ 21 22 23 4 5 gravity matrix of the exoskeleton and human leg is referred e h C C C by G ðθÞ and G ðθÞ, respectively. τ implies the actuator tor- 31 32 33 a 2 3 que while driving the joint of a human’s lower limb. The col- laboration torque is indicated by τ and τ for collaboration eh he 6 7 6 7 of exoskeletons with humans and vice versa, respectively. G θ = , ðÞ G 4 5 During exoskeleton-human interaction, splints are exploited to keep the exoskeleton link and human leg Desired joint angles (deg) X (m) 6 Applied Bionics and Biomechanics c1 1 k Exoskeleton link c2 Human segment c3 (a) (b) Figure 4: Coupled human-exoskeleton configuration. (a) A simplified linkage model. (b) Interaction dynamics of the coupled human- exoskeleton system. attached, which induces the collaboration torques (τ and ing, the actuator dynamics can be formulated using Kirchh- eh τ ). However, in passive gait rehabilitation, as considered off’s law to obtain the control voltage (U ) as follows: he in the present work, these collaborations are withdrawn by assuming rigid connections and matching joint angles for ˇ τ R ˇ _ _ U = + i L + Ξ θ, ð11Þ exoskeletons and humans. As illustrated in Figure 4(b), the m m m e interaction dynamics is formulated as below: where R , S , i , L , and Ξ denote the armature resistance, T T _ _ m m m m e τ = −τ = J f = J kΔx + cΔx_ = k θ − θ + c θ − θ , ðÞ ðÞ eh he co h e h e torque sensitivity, current, armature inductance, and back EMF constant of the DC motor. These parameters are ð8Þ selected from the specification sheet provided by Bholanath Precision Engineering Private Limited [35]. where f represents the collaboration force between the exo- co To imitate the realistic cases, the actuator saturation skeleton and the human, k and c signify the mechanical stiff- should be considered in the design of the control law to avoid ness and damping factors of the used splints, Δx denotes the the hysteresis cycle and maintain the linearity of the actuator. Cartesian coordinate disparity between the human leg and Moreover, this ensures closed-loop stability by limiting the the exoskeleton link, and ðθ − θ Þ refers to the joint angular h e large control signals. Based on the saturation theory, the disparity between the human leg and the exoskeleton link. Furthermore, in the presence of parametric uncertainties control signal ðU Þ from Equation (11) can be further and external disturbances, Equation (5) can be rewritten as defined as below: ˇ ˇ ˇ ˇ € _ _ τ = M ðÞ θ θ + C θ, θ θ + G ðÞ θ , ð9Þ ℧   U >℧ , > m m m ˇ ˇ ˇ U = U   U ≤℧ , ð12Þ where m m m m 8 > τ = ητ + D, > −℧   U < −℧ , > m m m M ðÞ θ = MðÞ θ + η MðÞ θ , ð10Þ where ℧∈ℝ denotes a vector with positive elements. The _ _ _ C θ, θ = C θ, θ + η C θ, θ , upper and lower saturation bound is denoted by ℧ and > m : −℧ , respectively. G ðÞ θ = GðÞ θ + η GðÞ θ , 3.2. Input-Output Feedback Linearization. The main objec- where D denotes the external disturbances applied by the tive of the feedback linearization is to correctly linearize the subjects to the system; η is the uncertain scaling factor when nonlinear dynamics with suitable modifications in state- considering the same amount of variation in dynamic space coordinates using an inner loop control [34]. Thereaf- parameters. ter, an outer loop control with a new set of coordinates can be After considering joint torques (τ) equivalent to joint formed to establish a linear relationship between the output actuator torques (τ ) in case of passive rehabilitation train- vector (y) and the input vector (u) and validate the cost a Applied Bionics and Biomechanics 7 functions of the control design. Consider the nonlinear Now, a linear relationship between inputs and outputs is multiple-input and single-output (MISO) dynamic relation to be established by performing the differentiation of the with n as the order and p as the total number of inputs as well outputs ðy Þ till the input terms appear in the formulation. ðr Þ as outputs, defined in the affine state: Considering r is the smallest integer, ðy Þ can be evaluated j j 8 with a complete term of inputs as follows: x _ t = Ψ x t + 〠 ðÞ ðÞ ðÞ ΠðÞ xðÞ t U ðÞ t , i m ð13Þ i=1 r r ðÞ j j j−1 > ˇ y = L Λ x + 〠 L L Λ x U , i, j =1,2, ⋯, p, : ðÞ ðÞ j Ψ j Π Ψ j m i i y ðÞ t = ΛðÞ xðÞ t , i=1 i i ð19Þ where x = ½x , x , ⋯, x  ∈ ℝ denotes the state vector, 1 2 n i i ˇ ˇ ˇ ˇ where L Λ and L Λ signify the ith Lie derivatives of Λ ðxÞ U = ½U , U , ⋯, U  ∈ ℝ signifies the control input j j j Ψ Π m m m m i 1 2 p T in the direction of Ψ and Π, respectively. In Equation (19), r vector, and y = ½y , y , ⋯, y  ∈ ℝ indicates the output 1 2 p denotes the relative degree for the output y which provides vector. information about the number of derivatives required to n n carry out at least one of the inputs in the formulation [37, Theorem 1. Suppose Ψ : ℝ ⇒ ℝ signifies a smooth vector n n n 38]. The sum of every relative degree from Equation (19) field on ℝ and Λ : ℝ ⇒ ℝ denotes a scalar function. Then, constitutes the total relative degree ðrÞ which needs to be less the Lie derivative of Λ to Ψ, referred as L Λ, is expressed as than or equal to the system’s order. follows [36, 37]: ∂Λ ∂Λ r = 〠 r ≤ n: ð20Þ L Λ = ΨðÞ x = 〠 ΨðÞ x : ð14Þ j=1 ∂x ∂x i=1 Furthermore, rewriting Equation (19) and expressing the Similarly, the Lie derivative of L Λ with respect to Ψ is nonlinear control law U to form the linear relationship defined as m between the input and the output as follows, one can get L Λ = L L Λ : ð15Þ ðÞ Ψ Ψ Ψ hi ð21Þ y , ⋯, y = δ x + σ x · U , ðÞ ðÞ 1 m In general, hi T r γ γ−1 0 ˇ ˇ ˇ ˇ ð22Þ u = u , u , ⋯, u = y , ⋯, y , 1 2 p 1 L Λ = L L Λ  with L Λ = Λ,∀γ = 1, ⋯, p: ð16Þ Ψ Ψ Ψ where n n Theorem 2. The function Φ : ℝ ⇒ ℝ , specified in a region 2 3 Y ⊂ ℝ , is termed as diffeomorphism if the function Φ along > L Λ ðÞ x > Ψ 1 −1 6 7 with the inverse Φ (if it exists) is smooth, i.e., differentiable > 6 ⋮ 7 6 7 δðÞ x = 6 7, everywhere [36, 37]. > > 6 7 4 5 However, as the global diffeomorphism is rare, one > L Λ x > ðÞ Ψ p should check for local diffeomorphisms, i.e., transformations > 2   3 ðÞ r ðÞ r −1 ðÞ r −1 1−1 1 1 defined in a limited neighborhood of a specified point [37]. L L Λ ðÞ x L L Λ ðÞ x ⋯ L L Λ ðÞ x Π Ψ 1 Π Ψ 1 Π Ψ 1 1 2 p 6 7 With the concept of diffeomorphism, we transform a nonlin- 6 7 6 7 > ðÞ r −1 ðÞ r −1 ðÞ r −1 2 2 2 6 7 > L L Λ ðÞ x L L Λ ðÞ x ⋯ L L Λ ðÞ x ear system into another one by changing the variables in the > Π Ψ 2 Π Ψ 2 Π Ψ 2 1 2 p > 6 7 > σ x = : ðÞ 6 7 following form: > 6 7 > ⋮ ⋮ ⋮ > 6 7 > 6 7 > 4 r −1 r r −1 5 ðÞ ðÞ ðÞ > p p−1 p > L L Λ ðÞ x L L Λ ðÞ x ⋯ L L Λ ðÞ x T : Π p Π p Π p Ψ Ψ Ψ 1 2 p z = z , z , ⋯, z = Φ x , ð17Þ ½ ðÞ 1 2 n ð23Þ where ΦðxÞ characterizes n variables as Assuming σðxÞ is not singular, the input transferred 2 3 2 3 hi r −1 1 1 form, i.e., the nonlinear control law, can be possibly Λ  L Λ   ⋯ L Λ 1 Ψ 1 Ψ 1 6 7 6 7 defined as 6 Φ 7 6 7 6 7 6 7 ΦðÞ x = = , 6 7 6 7 −1 4 ⋮ 5 4 5 ˇ hi U = σ x −δ x +u ˇ , ð24Þ T ðÞðÞ ðÞ r −1 m Λ  L Λ   ⋯ L Λ p Ψ p Ψ p ˇ ˇ ˇ ˇ where u ˇ = ½u ˇ , u ˇ , ⋯, u ˇ  and U = ½U , U , ⋯, U  . ð18Þ 1 2 p m m m m 1 2 p In Equation (24), u ˇ denotes the new input vector, U with x = ½x , x , ⋯, x  . refers to the decoupling control law, σðxÞ signifies an 1 2 n 8 Applied Bionics and Biomechanics invertible matrix of order p × p, and δðxÞ represents a 4. Adaptive Neural-Fuzzy Inference System decoupling matrix of the system. The adaptive neural-fuzzy inference system (ANFIS), Furthermore, the linearizing law from Equation (24) is colloquially known as the neural-fuzzy or neuro-fuzzy (NF) applied for transforming the nonlinear dynamics of the system, was proposed by Jang and Sun [40] by augmenting coupled human-exoskeleton system (Equation (9)) into the the benefits of adaptive neural networks and fuzzy reasoning. linear state-space representation as follows: In the NF system, IF-THEN-based fuzzy logic inferences are constructed to form the learning rules with a defined input- ˇ ˇ output dataset and reproduce the output vector with zero z _ =Az +Bu ˇ , ð25Þ error tolerance. A neural-fuzzy system exploits the fuzzy y =Cz, input variables and input-dependent nonfuzzy output vari- ables given by Takagi and Sugeno [41]. For instance, If the acceleration of the robot send − effector is high,then ˇ ˇ where A =A + ΔA, B =B + ΔB, and u = ðΔ +1Þu + d. ˇ ˇ A and B are the state-weight factor matrices with the ft ðÞ = c × acceleration of the robot send − effector : ð27Þ effect of parametric uncertainties; A, B,and C denote the state-weight factor matrices evaluated with nominal system In Equation (27), high signifies a fuzzy label with the mem- parameters, i.e., in the absence of parametric uncertainties; Δ bership function (MF), representing the acceleration of the is the uncertain scaling factor in the linearized state-space form robot send − effector intheruleproposition. Therulesubse- related to state-weight factor matrices and can be expressed in quent with nonfuzzy behavior is formulated according to the terms of dynamic parameters of the given system [39]. The input variable of the premise step, acceleration of the robot s input vector u carries the effectof parametervariationsand end − effector. external disturbances. The input vector in the absence of PUEDs is denoted byu. The disturbance applied to the system 4.1. ANFIS Architecture. A neural-fuzzy system inherently after feedback linearization is denoted by d, being analogous to exploits the five layers. The primary network architecture D from Equation (9). The linearized dynamics in Equation (25) with two input vectors ðz : x, yÞ and one output vector stands valid with the following assumptions: ðf ðzÞÞ is considered to show the generalized process of ˇ ˇ A1:the ðA,BÞ is in the controllable form. ANFIS. As shown in Figure 5, two kinds of nodes are employed A2: the disturbance d is in the bounded form. in the architecture: first, a square node for adaptation of the In the present work, the state vector and output vector parameters, and second, a circular node that behaves as a fixed related to the hip, knee, and ankle joints of the coupled dynam- node with no parameter. The layer-by-layer development of _ _ _ ical system is considered z = ½θ  θ  θ  θ  θ  θ  the ANFIS structure is given below [42, 43]. H K A H K A and y = ½θ  θ  θ  , respectively. The input vector H K A Layer 1. This layer acts as a conversion function for the crisp with nominal system parameters is defined as u = T value of the input vector into an appropriate MF-based fuzzy ½u  u  u  . Moreover, the respective state-weight fac- H K A language, depicted as follows: 6×6 6×3 3×6 tor matrices (A ∈ ℝ , B ∈ ℝ , andC ∈ ℝ ) can be for- mulated as below. The entries of these matrices are extensively = Ω ðÞ x , j A presented in the appendix. j ð28Þ O = Ω ðÞ y , j B 0 1 > ½ 0 I > 3×3 3 > C > α α α 1 B 41 42 43 C > where O represents the jth node output for the first layer. B C > j B α ~ α ~ α ~ C 41 42 43 B C Ω ðxÞ and Ω ðyÞ denote the membership weightage of > B C A = α α α α α α α α α , > A B 51 51′ 51′′ 52 52′ 52′′ 53 53′ 53′′ j j > B C > − + − + + − > B ½ C > 3×3 α ~ α ~ α ~ α ~ α ~ α ~ α ~ α ~ α ~ > B ′ ′ ′ ′ ′ ′ C 51 51 51 ′ 52 52 52 ′ 53 53 53 ′ respective input variables, defined for A - and B -type MF. B C j j > @ α α α α α α α α α A > 61 61′ 61′′ 62 62′ 62′′ 63 63′ 63′′ − − − − − − > In this work, the generalized bell membership function is > α ~ α ~ α ~ α ~ α ~ α ~ α ~ α ~ α ~ ′ ′ ′ ′ ′ ′ > 61 61 61 ′ 62 62 62 ′ 63 63 63 ′ 6×6 chosen to replicate a proper probability distribution behavior 0 1 > ½ 0 > 3×3 and expressed as follows: > B C B β β β C 41 42 43 B C B ~ ~ ~ C > β β β > B 41 42 43 C > B C > B C bell z : a , b , c = Ω ðÞ x = Ω ðÞ y = : >B = β β β , j j j A B hi B 51 52 53 C j j b > 2 B C > ~ ~ ~ B C 1+ z − c /a > β β β j j > 51 52 53 B C B C @ β β β A 61 62 63 ð29Þ > ~ ~ ~ > β β β > 61 62 63 6×3 0 1 1 0 0 0 0 0 > In Equation (29), “a ” and “b ” signify the width and B C j j B C >C = : 0 1 0 0 0 0 @ A shape parameters of the membership function. The value of 0 0 1 0 0 0 3×6 “b ” is generally positive; however, it can be considered neg- ð26Þ ative in case of inverted shape. “c ” indicates the center j Applied Bionics and Biomechanics 9 Layer 5 Layer 2 Layer 1 Layer 3 Layer 4 x y A1 w w 1 2 Π N w f 1 1 A2 f (z) B1 w f N 2 2 y Π B2 2 x y Figure 5: ANFIS architecture. position of the membership function. Having an extra 5. Robust Design of the LQR-Based Neural- parameter compared to Gaussian MF, the generalized bell Fuzzy Control has the added advantage of tuning the steepness at cross- over positions. The design procedure of RLQR-NF control is organized into two parts: first, the offline training of a robust LQR-based Layer 2. This layer evaluates the firing strength for every rule ANFIS training dataset to deal with parametric uncertainties, and second, the online training of the LQR-based ANFIS using a product of incoming signals from each circular node. architecture using the adaptive law of weights to compensate It is designated by notation Π in the ANFIS architecture. The following expression is used to estimate the firing strength for the external disturbances. In both parts, the effects of parametric uncertainties and external disturbances are (w ) as follows: explicitly considered. Thereafter, the stability of the proposed control strategy in the presence of PUEDs is addressed by the O = w = Ω ðÞ x = Ω ðÞ y , j =1,2: ð30Þ j j A B j j Lyapunov theory in the next section. 5.1. Offline Training of the RLQR-NF Dataset for Parametric Layer 3. In this layer, the normalization of the node’s firing Uncertainties. As shown in Figure 6, this subsection is further strength is carried out by dividing the jth rule firing strength presented into two stages: first, the formulation of a robust to all rules’ total firing strength. This layer is designated by LQR-NF dataset by varying mass parameters, and second, notation N in the ANFIS architecture. The firing strength the stepwise layout of offline training of a dataset using (w  ) is normalized as follows: ANFIS parameters to design the robust control strategy. 3 5.1.1. Stage I: Formulation of the RLQR-NF Training Dataset. O = w  = , j =1,2: ð31Þ j j w + w The training dataset, having multiple-input and single- 1 2 output (MISO), is formed by employing the concepts of the LQR control strategy as shown in Equation (25). The LQR Layer 4. This layer, having the square nodes, is used to cost function is considered a minimization problem while estimate the rule’s involvement by defuzzification of input applying the optimality conditions and is expressed as fol- variables and produce the respective output as follows: lows [44]: O = w  ∠ = w  p x + q y + r , ð32Þ j j j j j j j ð ˇ ˇ J = z Qz +u Ru dt, ð34Þ where w  indicates the normalized firing strength and p , q , j j and r signify the subsequent limits. where Q and R denote the user-defined state-weight matrix and control cost matrix, respectively. An appropriate selec- Layer 5. This layer, having circular shape nodes with the tion of both matrices directly influences the performance designation ∑, processes the final output using the characteristics of the controller. summation of all incoming signals from the preceding layer. The generalized input (u ˇ ) to the control system is artic- Mathematically, it can be expressed as follows: ulated by regulating the error vector (ℯ) as follows: ∑ w ∠ j j j O =〠w ∠ = : ð33Þ j j j ∑ w j j ˇ j u =Kℯ =KzðÞ −z , ð35Þ des 10 Applied Bionics and Biomechanics Stage I Stage II Mathematical Defining input Selection of Offline formulation and output ANFIS training of of RLQR-NF training training RLQR-NF dataset vectors parameters controller Figure 6: Flowchart representation of the stage-wise design procedure. where z and z represent the desired and actual state vec- denote the controller output for the hip, knee, and ankle des tors, respectively. The optimal state gain matrix (K) can be joints of the exoskeleton device, respectively. In Equation expressed in terms of the control cost matrix (R) and the (41), ℯ , ℯ , and ℯ signify the hip, knee, and ankle joint θ θ θ H K A state-weight factor matrix (B) as follows: angular errors, respectively. ℯ , ℯ , and ℯ represent the _ _ _ θ θ θ H K A respective errors of the hip, knee, and ankle joint angular velocities. −1 K = R B P, ð36Þ The expanded structure of the robust dataset ðS Þ is rd finally depicted as below: where P is the answer for the algebraic form of the Riccati equation expressed in the form of state-weight factor matri- S = ℯ u ˇ , ð42Þ rd m ,m ,m t c hf ˇ ˇ ces (A and B), state-weight matrix (Q), and control cost matrix (R) as given below [44, 45]: where T T −1 ˇ ˇ ˇ ˇ PA +A P −PBR B P +Q = 0: ð37Þ S = S 1 S 2 S 3 : ð43Þ ðÞ ðÞ ðÞ rd rd rd rd It is truly evident from Equation (36) that the state gain ˇ ˇ matrix (K) is regulated by A, B, Q, and R matrices where 5.1.2. Stage II: Stepwise Layout of Offline Training of the ˇ ˇ RLQR-NF Dataset. The layout and execution of the proposed A and B are reliant on the mechanical arrangement and control strategy for the exoskeleton device are presented in dynamic parameters of the requisite system. the following steps. Exploiting the controller’s gain (K), a generalized dataset (S ) with the error vector (ℯ) and respective input Step 1. The state gain matrix (K ) is evaluated by solv- (u ˇ ) to the control system can be created in the following m ,m ,m t c hf ing Equations (36) and (37) for a different set of coupled form [44]: thigh ðm Þ, calf ðm Þ, and heel-foot ðm Þ masses, as shown t c hf in Table 2. The parametric variation is incorporated by S = ℯ u ˇ : ð38Þ increasing the nominal mass values up to 20%, with an increment of 0.3, 0.15, and 0.06 kg for the thigh, calf, and In this work, the dataset ðS Þ is expanded into a robust heel-foot. After performing several numerical experiments, form by evaluating the controller’s gain (K ) for a m ,m ,m t c hf the state-weight matrix and control cost matrix are selected bounded variation of the coupled thigh ðm Þ, calf ðm Þ, and t c as Q = diag ð400000, 4000, 8000, 800, 8000, 800Þ and R = heel-foot ðm Þ masses of the human-exoskeleton system. hf eyeð3, 3Þ. Thereafter, the controller input ðu Þ for the hip, knee, m ,m ,m t c hf and ankle joints is formulated as follows: Step 2. Apply Equation (39) to compute the controller input ðu Þ for the operating range of state variables in the m ,m ,m t c hf u ˇ =K ℯ, ð39Þ error vector as shown in Table 3. The structure of three robust m ,m ,m m ,m ,m t c hf t c hf datasets is formed by exploiting Equations (42) and (43). where Step 3. The training of robust datasets is carried out using the ANFIS approach. The first six columns of every dataset are hi ð40Þ u ˇ = u ˇ u ˇ u ˇ , inherently considered the input set. The last column of every m ,m ,m H K A t c hf m ,m ,m m ,m ,m m ,m ,m t c hf t c hf t c hf dataset is regarded as the output set. The input set comprises the error vector ðℯÞ, and the output set contains the controller hi ð41Þ input vector ðu ˇ Þ. The three ANFIS architectures are ℯ = ℯ ℯ ℯ ℯ ℯ ℯ : _ _ _ θ θ θ m ,m ,m θ θ θ t c hf H K A H K A formed, trained, and saved as anfis1.fis, anfis2.fis, and anfis3.- ˇ ˇ ˇ In Equation (40), u , u , and u fis for three controller inputs. Several simulation runs are H K A m ,m ,m m ,m ,m m ,m ,m t c t c t c hf hf hf Applied Bionics and Biomechanics 11 Table 2: Variation in lower limb mass parameters of the coupled Table 4: Training parameters of ANFIS architectures. human-exoskeleton system. Training parameters anfis1.fisanfis2.fisanfis3.fis Thigh, m (kg) Calf, m (kg) Heel-foot, m (kg) t c hf MF type Gaussian Gaussian Gaussian 8.25 3.85 1.50 MF number 5 12 3 8.58 4.00 1.56 Error tolerance 0.00001 0.0001 0.001 8.91 4.16 1.62 Epochs 10 15 5 9.24 4.31 1.68 Learning model Hybrid Hybrid Hybrid 9.57 4.47 1.74 9.90 4.62 1.80 Substituting Equation (44) into Equation (22), the linear- ized dynamic model can be written as Table 3: Operating range of the error in state variables. ˇ ˇ ˇ z _ = A −BK z +B d + d : ð45Þ Variables in the error Minimum Maximum Units vector value value Now, the error dynamics of the control design in the time -60 60 Degree domain can be expressed as below: -60 60 Degree -30 30 Degree ˇ ˇ ˇ ̂ ℯ_ðÞ t = A −BK ℯðÞ t +B dt ðÞ −ft ðÞ , ð46Þ _ -90 90 Degree/sec -90 90 Degree/sec where -60 60 Degree/sec ˇ ˇ ˇ ˇ Bft ðÞ =z ðÞ t −Bdt ðÞ − A −BK z ðÞ t : ð47Þ des des performed by varying the number of MF from 1 to 50 and Employing assumption A2, the f ðtÞ can be considered a epochs from 1 to 30. Thereafter, the training parameters function with an upper limit. Therefore, the effectiveness of are selected based on the zero error tolerance between the coupled human-exoskeleton system can be augmented the desired and predicted output vectors. In general, by estimating the f ðtÞ from dðtÞ. Using the ANFIS architec- ANFIS utilizes two optimization methods: backpropagation ture mentioned in Section 4, the f ðtÞ is approximated as and hybrid, to establish the learning between the input and follows [46, 47]: output vectors. A gradient descent model is employed to evaluate the node error in the backpropagation method. In contrast, a least square algorithm along with the gradi- f z = W Ω z + ϱ z , ð48Þ ðÞ ðÞ ðÞ ent descent model is exploited to regulate the errors in the hybrid method. In this work, the hybrid method is used where W denotes the ideal normalized firing strength matrix; with all three datasets for training the neural-fuzzy net- ΩðzÞ signifies the membership function vector; and ϱðzÞ works. The complete details of training parameters are represents an error of approximation with the condition given in Table 4. After generating training files (anfis1.fis, ϱðzÞ ≤ ζ, where ζ is a constant factor. anfis2.fis,and anfis3.fis) as the desired robust LQR-based Now utilizing Equations (46) and (48), the error dynam- neural-fuzzy controller, the respective signals are inputted to ics can be rewritten as the nonlinear dynamics of the coupled human-exoskeleton system. ˇ ˇ ˇ ℯ_ t = A −BK ℯ t +B dt − W Ω z − ϱ z : ð49Þ ðÞ ðÞ ðÞ ðÞ ðÞ 5.2. Online Training of the RLQR-NF Dataset for External Disturbances. Considering assumption A1, the control input From the above equation, the disturbance estimator dðtÞ with uncertain parameters ðu ̂ Þ can be further expressed in is defined as follows to design the control law [47]: terms of the state feedback ðKzÞ and disturbance observer ðdÞ as ̂ ̂ d = W Ω ðÞ z − ζ sign ℯ PB , i =1, 2, ⋯, p and j =1,2, ⋯, l, i ji j i u ̂ =ðÞ Δ +1 u = d −Kz, ð44Þ ð50Þ where d denotes the estimated disturbance by ANFIS where p and l denote the number of inputs and network architecture, and K signifies the state feedback matrix. nodes in the hidden layer. 12 Applied Bionics and Biomechanics In the above expression, the estimated strength matrix T T T _ ˇ V = −ℯ Q +K RK ℯ +2〠 ℯ PB ðW Þ is updated according to the following law: ji i=1 ̂ ~ × d − W Ω ðÞ z + W Ω ðÞ z − ϱðÞ z ð58Þ i ji j ji j i _ T W = −Γ Ω z ℯ PB , ð51Þ ðÞ ji i j i −1 _ +2〠 W Γ W : ji i ji where Γ denotes a positive definite matrix with the sym- i=1 metric property. Now substituting Equations (50) and (51) into Equation 6. Stability Analysis of the LQR-Based Neural- (58), one can obtain Fuzzy Control In this section, the Lyapunov function is presented to analyze T T V = −ℯ Q +K RK ℯ +2〠 ℯ PB the global stability of the proposed control scheme under the i=1 effect of parametric uncertainties and external disturbances. × d − W Ω ðÞ z − ϱðÞ z Consider the Lyapunov candidate function as follows: i ji j i ð59Þ T T p ≤−ℯ Q +K RK ℯ T −1 ~ ~ V = ℯ Pℯ + 〠 W Γ W , ð52Þ ji ji i 2 ≤ 0: i=1 with Invoking the above equation, it can be concluded that the error dynamics of the coupled human-exoskeleton system is = W − W , ð53Þ ji ji ji asymptotically stable. Therefore, the proposed controller carries out the asymptotic tracking with error ℯ⟶ 0 (as t⟶ ∞) following the disturbance estimator (Equation where W denotes the estimation error between the estimated ji (50)) and firing strength adaptation law (Equation (51)). Fur- strength matrix and the ideal constant strength matrix. thermore, the local stability of the proposed controller can be Differentiating Equation (52) and employing Equations analyzed using the pole placement theory, where the conver- (49) and (37), one can obtain gence rate can be investigated by keeping the poles on the left side of the s-plane [45]. T T T V = −ℯ Q +K RK ℯ +2ℯ PB 7. Results and Discussion T _ −1 ~ ~ × d − W Ω ðÞ z − ϱðÞ z +2〠 W Γ W : i j i ji ji ji i i=1 In this section, the simulation results and analyses are pre- sented to evaluate the effectiveness of the proposed control ð54Þ strategy (RLQR-NF) for an exoskeleton device during pas- sive gait rehabilitation measures. The block representation Now utilizing Equation (53), the error difference between for the proposed control strategy is schematically shown the actual value and the desired value by the designed neural- in Figure 7. At first, the performance of the proposed con- fuzzy network can be expressed as trol strategy is compared with a contrast control strategy without parametric uncertainties and external disturbances. ~ ̂ W Ω z = W Ω z − W Ω z : ð55Þ ðÞ ðÞ ðÞ ji j ji j ji j In this work, the exponential reaching law-based sliding mode (ERL-SM) control is used as a contrast control strat- Moreover, as the firing strength matrix ðW Þ is a con- ji egy [48]. Thereafter, two cases are contemplated to demon- stant matrix, Equation (53) holds the following relation after strate the controller’s robustness: the first case, increasing differentiation: the coupled segment masses by 20% with a trigonometric form of disturbances, and the second case, increasing the _ _ ~ ̂ W = W : ð56Þ coupled segment masses by 30% with a random form of ji ji disturbances. Reconstituting Equation (54) using Equations (55) and 7.1. Simulation Results without Parametric Uncertainties and (56), one can obtain External Disturbances. In this subsection, a comparative analysis between the RLQR-NF and ERL-SM control strate- T T T V = −ℯ Q +K RK ℯ +2ℯ PB gies is presented for desired gait tracking during passive gait rehabilitation measures. In the absence of parametric uncer- ̂ ~ × d − W Ω ðÞ z + W Ω ðÞ z − ϱðÞ z i ji j ji j i tainties and external disturbances, i.e., Δ =0 and d =0 in ð57Þ Equation (25), the nominal mass values of lower limb seg- −1 _ ~ ̂ +2〠 W Γ W , ments ðm =8:25, m =3:85, and m =1:5Þ are taken into ji i ji t c hf i=1 account for drawing the state of comparison. Using Equation Applied Bionics and Biomechanics 13 W = –Γ 𝛺 (z)e PB ji i j i Online disturbance estimator d = W 𝛺 (z)– sign (e PB ) i j i j i Offline trained 𝜃 – RLQR-NF model External (u , u , u ) H K A disturbances –0.35 Desired gait trajectory –0.4 –0.45 –0.5 𝜃 (t) –0.55 Starting point (at t = 0) –0.6 –0.2 –0.1 0 0.1 0.2 0.3 0.4 + X (m) – 𝜃 (t), 𝜃 (t) a a 𝜃 (t) 𝛥 u Kinematics 𝛥 t 𝜃 (t), 𝜃 (t) a a (H, K, A) (H, K, A) Desired joint angle (𝜃 ) Potentiometer/IMU (hip: H, knee: K and ankle: A) Figure 7: Schematic representation of the implemented proposed RLQR-NF controller. –0.4 –0.45 –0.5 –0.55 –0.6 –0.65 –0.2 –0.1 0 0.1 0.2 0.3 0.4 X (m) Desired ERL-SM RLQR-NF Figure 8: Desired trajectory tracking for ERL-SM and RLQR-NF control schemes with nominal mass values. (36), the optimal state gain matrix for the nominal mass eters into the proposed control (RLQR-NF) and ERL-SM parameters ðK Þ is calculated as below: control strategy. A time period of 2 seconds is considered 8:25,3:85,1:5 to complete one gait cycle. The starting Cartesian position (X, Y: 0.25 m, -0.60 m) of the desired gait trajectory is illus- 0 1 448:63 −2:72 −0:57 17:84 −0:31 −0:127 trated in black color. The actual trajectories in the Cartesian B C B C coordinate frame are presented by the dashed blue line K = 7:65 12:40 2:49 3:20 20:27 0:82 : 8:25,3:85,1:5 @ A (RLQR-NF) and the green line (ERL-SM). 8:72 −1:92 20:17 0:33 −2:56 6:40 Figures 9(a) and 9(b) depict the tracking error in both ð60Þ directions, i.e., X- and Y-directions (ℯ and ℯ ). The maxi- x y mum absolute deviation in the X-direction ðjℯ j Þ for the max ERL-SM and RLQR-NF control schemes is 0.013 m and In the ERL-SM control strategy [48], the control law 0.008 m, respectively. In the Y-direction, the respective devi- parameters are used as c = diag ð50, 50, 50Þ, ε =0:5× I , and ation ðjℯ j Þ is observed to be 0.009 m and 0.006 m for the k = I . As shown in Figure 8, the healthy gait trajectory is y max e 3 ERL-SM and RLQR-NF control strategies. tracked by incorporating the kinematic and dynamic param- Y (m) Robust LQR based neural-fuzzy controller e (t), e (t) Coupled dynamical system State feedback 14 Applied Bionics and Biomechanics 0.01 0.01 0.005 0.005 –0.005 –0.005 –0.01 –0.01 –0.015 –0.02 –0.015 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time (s) Time (s) ERL-SM RLQR-NF (a) (b) Figure 9: Position tracking error for ERL-SM and RLQR-NF control schemes with nominal mass values. (a) X-direction. (b) Y-direction. 30 60 10 20 0 𝜃 0 𝜃 –5 –10 –10 –20 –20 012 012 Time (s) Time (s) Time (s) Desired ERL-SM RLQR-NF (a) (b) (c) Figure 10: Joint angle tracking for ERL-SM and RLQR-NF control schemes with nominal mass values. (a) Hip joint. (b) Knee joint. (c) Ankle joint. The tracking of desired joint angles with applied control through repetitive gait rehabilitation exercises. With the strategies is illustrated in Figures 10(a)–10(c). The angular ERL-SM control scheme, the peak values of the hip, knee, deviations (ℯ , ℯ , and ℯ ) from desired joint trajectories and ankle signals are estimated as 32.98 V, 19.88 V, and θ θ θ H K A are shown in Figures 11(a)–11(c). Considering the hip joint, 1.8 V. On the other hand, with the RLQR-NF control it is observed that the maximum absolute deviation ðjℯ j Þ scheme, the respective values of control signals are found max ° ° for respective controllers is 0.78 (ERL-SM) and 0.51 to be 30.25 V, 18.1 V, and 1.25 V. It is evident from the (RLQR-NF). In the knee joint, the deviation ðjℯ j Þ is results that the proposed control strategy (RLQR-NF) out- max ° ° found to be 1.15 and 1.16 for the system with the ERL-SM performs the contrast control strategy (ERL-SM) to track and RLQR-NF control strategies, respectively. For the ankle the desired gait trajectory, however, with a marginal differ- joint, the respective deviations ðjℯ j Þ are estimated as ence. Therefore, to demonstrate the effectiveness of the pro- A max ° ° posed control when dealing with PUEDs, variations in mass 0.81 (ERL-SM) and 0.32 (RLQR-NF). parameters and the form of disturbances are considered Figures 12(a)–12(c) demonstrate the generated control further. signals (u , u , and u ) following the desired trajectory H K A (deg) e (m) (deg) e (m) (deg) A Applied Bionics and Biomechanics 15 1 1.5 1 0.5 0.5 0.5 0 0 𝜃 𝜃 𝜃 –0.5 –0.5 –0.5 –1 –1.5 –1 –1 0 1 2 0 1 2 0 1 2 Time (s) Time (s) Time (s) ERL-SM RLQR-NF (a) (b) (c) Figure 11: Joint tracking error for ERL-SM and RLQR-NF control schemes with nominal mass values. (a) Hip joint. (b) Knee joint. (c) Ankle joint. –50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) (a) –30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) (b) –3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) ERL-SM RLQR-NF (c) Figure 12: Control signals for ERL-SM and RLQR-NF control schemes with nominal mass values. (a) Hip joint. (b) Knee joint. (c) Ankle joint. 7.2. Simulation Results for Parametric Variations and proposed control scheme over the contrast control External Disturbances. In order to realize the robust- scheme. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ness of the proposed control strategy (RLQR-NF), the root mean square error (RMSE) is computed and RMSE = 〠 ℯ , kk compared with the ERL-SM control strategy for para- a=1 ð61Þ metric uncertainties and external disturbances. More- RMSE − RMSE ERL‐SM RLQR‐NF over, based on RMSE values, the performance index PI = × 100%, RMSE (PI) is calculated to analyze the improvement of the ERL‐SM e (deg) u (volt) u (volt) u (volt) A K H e (deg) e (deg) A 16 Applied Bionics and Biomechanics 1.2 1.5 1 0.5 0.5 0 0 0 𝜃 𝜃 –0.5 –0.5 –1 –1.2 –1.5 –1 012 012 012 Time (s) Time (s) Time (s) ERL-SM RLQR-NF (a) (b) (c) Figure 13: Joint tracking error for ERL-SM and RLQR-NF control schemes with the first case of PUEDs. (a) Hip joint. (b) Knee joint. (c) Ankle joint. 1.5 1.5 1 0.5 0.5 0 0 0 𝜃 𝜃 𝜃 –0.5 –0.5 –1 –1.5 –1.5 –1 012 012 0 12 Time (s) Time (s) Time (s) ERL-SM RLQR-NF (a) (b) (c) Figure 14: Joint tracking error for ERL-SM and RLQR-NF control schemes with the second case of PUEDs. (a) Hip joint. (b) Knee joint. (c) Ankle joint. where ℯ : ℯ represents the error between the desired RMSE values for the control strategies are recorded as a θ ° ° and actual joint angles, and N is the size of the error 0.321 and 0.224 . vector. RMSE and RMSE signify the root Considering the second case of PUEDs (Case II), where ERL‐SM RLQR‐NF the system masses are increased by 30% ðm =10:73, m = mean square errors related to ERL-SM and RLQR-NF t c control strategies. 5:00, and m =1:95Þ along with a random form of distur- hf Considering the first case of PUEDs (Case I), where the bances ðD = ð5 × randomð1ÞÞ, D = ð3 × randomð1ÞÞ, and 1 2 system masses are increased by 20% ðm =9:90, m =4:62 t c D = ð2 × randomð1ÞÞ, the joint angular errors (ℯ , ℯ , and 3 θ θ H K , and m =1:80Þ along with a trigonometric form of dis- hf ℯ ) for the proposed and contrast control strategies are turbances ðD = ð6 sin ð4πtÞÞ, D = ð5 sin ð3πtÞÞ, andD = 1 2 3 shown in Figures 14(a)–14(c). For the hip joint, the values ð3 sin ð2πtÞÞÞ, the joint angular errors (ℯ , ℯ , and ℯ ) θ θ θ H K A ° of RMSE and RMSE are found to be 0.613 ERL‐SM RLQR‐NF for the applied control strategies are shown in and 0.287 , respectively. The respective RMSE values related Figures 13(a)–13(c). For the hip joint, RMSE and ERL‐SM ° ° ° ° to the knee joint tracking are obtained as 0.742 and 0.434 . RMSE are estimated as 0.578 and 0.283 , respec- RLQR‐NF The following RMSE values for the ankle joint are found to tively. The respective RMSE values for the knee joint are ° ° ° ° found to be 0.672 and 0.42 . In ankle joint tracking, the be 0.334 and 0.228 . e (deg) e (deg) e (deg) e (deg) e (deg) e (deg) A A Applied Bionics and Biomechanics 17 Table 5: Comparative performance analysis of the proposed control over the contrast control. Joint name RMSE (ERL-SM) (deg) RMSE (RLQR-NF) (deg) PI (%) Case I: with a 20% increment in mass parameters and a trigonometric form of external disturbancesðÞ m =9:90, m =4:62, and m =1:80 t c hf ðÞ D =ðÞ 6 × sinðÞ 4πt , D =ðÞ 5 × sinðÞ 3πt , andD =ðÞ 3 × sinðÞ 2πt 1 2 3 Hip 0.578 0.283 51.04 Knee 0.672 0.420 37.50 Ankle 0.321 0.224 30.21 Case II: with a 30% increment in mass parameters and a random form of external disturbancesðÞ m =10:73, m =5:00, and m =1:95 t c hf ðÞ D =ðÞ 6 × randomðÞ 4 , D =ðÞ 5 × randomðÞ 3 , andD =3 ð × randomðÞ 2 1 2 3 Hip 0.613 0.287 53.19 Knee 0.742 0.434 41.51 Ankle 0.334 0.228 31.73 state gain matrix, robust ANFIS training datasets have been Table 6: Settling time of ERL-SM and RLQR-NF control schemes formed with a variation of system parameters. The operating for convergence analysis. range of the error vector and control responses have been Settling time (sec) regarded as the training input and output vectors. The Control Lower Nominal Case I of Case II of ANFIS architectures have been trained offline to deal with scheme limb joint system mass PUEDs PUEDs the effect of parametric uncertainties. Thereafter, the online Hip 1.971 1.986 1.993 adaptation law of firing strength in ANFIS architectures has ERL-SM Knee 1.962 1.974 1.988 been incorporated to deal with external disturbances. The asymptotic stability of the coupled dynamics while applying Ankle 1.927 1.943 1.969 the proposed control has been ensured using the Lyapunov Hip 1.951 1.967 1.991 theory. Finally, the effectiveness of the proposed controller RLQR-NF Knee 1.946 1.959 1.987 has been investigated by comparing it to the exponential Ankle 1.914 1.933 1.958 reaching law-based sliding mode control. The robustness analysis has been carried out by varying mass parameters and inducing different forms of external disturbances. The Table 5 presents the performance index (PI) of the pro- simulation results have shown the potential of the proposed posed control over the contrast control. The proposed con- robust tracking control for passive gait rehabilitation using trol is promising in desired gait tracking compared to the an exoskeleton system. In the future, the effect of human contrast control, subjected to PUEDs. Moreover, as observed involvement will be considered to design an “assist-as- from Table 5, the performance index (PI) is improved by needed” control strategy during active rehabilitation. 2.15%, 4.01%, and 1.52% in Case II as compared to Case I. During rehabilitation exercises, this performance investiga- Appendix tion allows the lower limb exoskeleton system to carry out repetitive movements with greater accuracy under the pres- The elements of the state-weight factor matrices ðA,BÞ can ence of PUEDs. be evaluated using the following derived formulations. The convergence of both control schemes is investigated For the A matrix, by evaluating the settling time, i.e., the time lapsed for the error to drop within 2% of the final value. The settling time α = −12l l l g 2m +4m m +3m m +2m m , 41 1 2 3 2 1 2 1 3 2 3 for the error in the hip, knee, and ankle joints for every set 2 2 2 2 of mass values is presented in Table 6. The low values of α ~ =2l l 12l m +12l m m +16l m m +12l m m 41 2 3 1 1 2 3 1 1 2 1 1 3 settling time indicate the faster convergence of the proposed − 48l l m m − 36l l m Þ, controller (RLQR-NF) over the contrast controller (ERL-SM) 2 3 2 3 2 3 3 before achieving the full stable state. 2 2 α =72l l l gm + m +2m m , 42 1 2 3 2 3 2 3 α ~ = α ~ , 42 41 8. Conclusions α = −18gl m m , 43 1 2 3 α ~ In this work, a robust LQR-based neural-fuzzy control has 41 α ~ = , been proposed to follow the natural gait trajectory using an 2l l 2 3 exoskeleton system during passive rehabilitation measures. α =12g 6l m +3l m +4l m +3l m ðÞ 51 1 2 1 3 2 2 2 3 Primarily, a linearized state-space form of the nonlinear l m l m l m 1 1 2 2 3 3 human-exoskeleton has been established via the input- l m + l m + l m + + + , 1 2 1 3 2 3 2 2 2 output feedback linearization method. Employing the LQR 18 Applied Bionics and Biomechanics β = − 144l l m +72l l m +96l l m +72l l m , α ~ = α ~ = α ~ = α ~ = α ~ = α ~ = α ~ = α ~ = α ~ ðÞ ′ ′ ′ ′ ′ 42 1 3 2 1 3 3 2 3 2 2 3 3 51 51 ′ 52 52 ′ 53 53 ′ 63 52 53 ~ ~ ~ α ~ β = β = β , 42 43 41 = α ~ = , 63 ′ 2l β =72l l m + 144l l m +72l l m , 43 1 2 2 1 3 2 1 3 3 α =6gl m +2l m + l m ðÞ β = −ðÞ 72l m +36l m +48l m +36l m , 51 2 2 2 3 3 3 1 2 1 3 2 2 2 3 2 2 2 2 2 4l m +12l m +3l m +4l m +3l m +12l l m 1 1 1 2 1 3 2 2 2 3 1 2 2 β = , 2l +6l l m − 12l l m Þ, 1 2 3 2 3 3 2 2 2 2 2 β =48l m + 144l m +36l m +48l m +36l m 1 1 1 2 1 3 2 2 2 3 l α ~ 2 41 α = , ′ + 144l l m +72l l m − 144l l m , 1 2 2 1 2 3 2 3 3 2l 2 2 2 2 α =6gm 6l l m +3l l m +9l l m +4l l m ′ 3 2 1 1 2 2 2 3 1 51 ′ 1 2 1 1 l β 2 41 β = , 2 2 2 2 +12l l m +3l l m − 12l l m − 18l l m 2l 1 3 2 1 3 3 3 2 3 2 3 3 2 2 2 2 β = − 72l l m +36l l m + 108l l m +48l l m +6l l l m +3l l l m Þ, 53 1 2 1 2 1 2 1 2 2 1 3 1 1 2 3 2 1 2 3 3 2 2 2 2 + 144l l m +36l l m − 144l l m − 216l l m 1 3 2 1 3 3 3 2 3 2 3 3 α =6glðÞ m +2l m + l m 52 2 2 2 3 3 3 +72l l l m +36l l l m Þ, 1 2 3 2 1 2 3 3 ðÞ 6l m +3l m +4l m +3l m , 1 2 1 3 2 2 2 3 l β α =6gl m +2l m + l m 2 41 ðÞ ~ 52 2 2 2 3 3 3 β = , 2 2 2 2 2 4l m +12l m +3l m +4l m +3l m 1 1 1 2 1 3 2 2 2 3 β =36l l m +2l m + l m , ðÞ 61 1 2 2 3 2 3 3 +12l l m +6l l m − 12l l m Þ, 1 2 2 1 2 3 2 3 3 ~ β = , α = α = α = α , 2 ′ ′ 52 ′ 51 ′ 53 63 β = β , 62 53 α =6gl mðÞ 6l m +3l m +4l m +3l m , ′ 3 3 1 2 1 3 2 2 2 3 ~ ~ 2 2 2 2 2 β = β , 62 53 α =6gl m 4l m +12l m +3l m +4l m +3l m ′ 3 3 1 2 3 2 3 53 ′ 1 1 1 2 2 2 2 2 2 2 2 2 2 2 β =36l l m + 144l l m m +48l l m m + 144l l m m 63 1 2 2 1 2 2 3 1 2 1 2 1 2 1 3 +12l l m +6l l m − 12l l m Þ, 1 2 2 1 2 3 2 3 3 2 2 2 2 + 216l l l m m + 144l l l m m + 144l l m m 2 2 2 1 2 3 2 3 1 2 3 1 3 1 3 2 3 α =6glðÞ m +2l m + l m 6l l m +3l l m +9l l m 61 2 2 2 3 3 3 1 2 1 2 1 2 1 2 2 2 2 2 2 2 3 3 2 +36l l m +48l l m m − 144l l m m − 432l l m 1 3 3 1 3 1 3 2 3 2 3 2 3 3 2 2 2 2 2 +4l l m +12l l m +3l l m − 12l l m − 18l l m 1 3 1 1 3 2 1 3 3 3 2 3 2 3 3 2 2 3 2 2 − 432l l m − 144l l m , 2 3 3 3 3 +6l l l m +3l l l m Þ, 1 2 3 2 1 2 3 3 l l m β 2 3 3 41 l α ~ 2 41 β = : ðA:2Þ α ~ = α ~ = α ~ = α ~ = α ~ = , 61 ′ 62 ′ ′ 61 62 63 2 2 2 2 2 2 2 2 2 α =6g 3l l m +12l l m m +4l l m m +12l l m m ′ 2 3 1 2 1 3 61 1 2 2 1 2 1 2 1 2 Data Availability 2 2 2 2 +18l l l m m +12l l l m m +12l l m m 1 2 3 2 3 1 2 3 1 3 1 3 2 3 2 2 2 2 3 3 2 2 The data used to support the findings of the study are +3l l m +4l l m m − 12l l m m − 36l l m 1 3 3 1 3 1 3 2 3 2 3 2 3 3 included within the article. 2 2 2 3 2 − 36l l m − 12l l m Þ, 2 3 3 3 3 Conflicts of Interest α =36gl l m +2l m + l m ðÞ 61 ′ 1 2 2 3 2 3 3 l m l m l m The authors declare that they have no conflicts of interest. 1 1 2 2 3 3 l m + l m + l m + + + , 1 2 1 3 2 3 2 2 2 Acknowledgments α ~ = α ~ = , ′ ′ 61 ′ 62 ′ The authors acknowledge the Department of Scientific and α = α , 62 61 Industrial Research, India, for establishing the PRISM (Pro- moting Innovations in Individuals, Start-ups, and MSMEs) α = α = α , ′ ′ ′ 62 61 63 scheme under which this project work is carried out. 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