# Model-Free Adaptive Sliding Mode Robust Control with Neural Network Estimator for the Multi-Degree-of-Freedom Robotic Exoskeleton

Authors: Xiangjian Chen; Di Li; Pingxin Wang; Xibei Yang; Hongmei Li
Journal: Complexity
 (2020)

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## Abstract

A new model-free adaptive robust control method has been proposed for the robotic exoskeleton, and the proposed control scheme depends only on the input and output data, which is different from model-based control algorithms that require exact dynamic model knowledge of the robotic exoskeleton. The dependence of the control algorithm on the prior knowledge of the robotic exoskeleton dynamics model is reduced, and the influence of the system uncertainties are compensated by using the model-free adaptive sliding mode controller based on data-driven methodology and neural network estimator, which improves the robustness of the system. Finally, real-time experimental results show that the control scheme proposed in this paper achieves better control performances with good robustness with respect to system uncertainties and external wind disturbances compared with the model-free adaptive control scheme and model-free sliding mode adaptive control scheme.


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## Body

## 1. Introduction

Exoskeleton robots can significantly improve the motor ability and quality of life of people with reduced limb function. With the acceleration of the aging process, the number of patients with stroke, Parkinson’s disease, spinal injury, and lower extremity paraplegia will continue to increase. Even after professional rehabilitation or treatment, the body functions of this group of people can be restored to a certain degree, but it is difficult to return to their original state. Most people’s physical control ability becomes lower, and walking will become a very difficult task. Activities and quality seriously affect their daily lives. From the point of view of population aging trend and the growth of the potential beneficiary population, the study of full-body-assisted exoskeleton robot for the elderly and disabled can not only improve the motor ability, but also promote the social harmonious development. However, existing exoskeleton robots are generally larger in size and heavier in weight. Adaptation to the ground and flexibility of movement still need to be further improved. The working mode of most exoskeleton systems is that the wearer is carried by the robot, and the gait of different users is not fully coordinated. Although there are several products that can meet the requirements of exoskeleton robot control, the coupling performance of human exoskeleton robot used to help the elderly and disabled is still very insufficient.One therapist Exoskeleton robots in this paper are designed to be worn to provide rehabilitation therapy for the stroke patients [1–4]. The effective control strategies are so important for the exoskeleton to operate coordinately with the human upper limb. General control methods [5, 6] are usually used for the control of manipulators.In order to help the stroke patient, one therapist Exoskeleton robots were designed in this paper [1–4]; this robot is so complicated to be operated control to work with human upper limb, so the general control methods [5, 6] are designed, and the other methods used the gravity and friction compensation controller also gained effective control performances [7–9]. Recently, researchers began to use more advanced control methods. But most of the control methods are designed based on the model of robot. Since the interacting model of this robot is time-varying with uncertainty, which makes the controller based on model schemes not enough effective. With the development of the technology, the data-driven method now is an alternative effective way, which does not need the model of interacting model of exoskeleton, and the information can be obtained directly form the recorded process [10].Data-driven Control (DDC) method does not contain the mathematical model information of the controlled process in designing process explicitly or implicitly, which only uses online or offline process data to design the controller. DDC has characteristics of convergence, stability, and robust under certain assumed conditions. In literature [11], they stated the necessity of the data-driven control theory in detail from three aspects, which are the control theory, the application of control theory, and the development of control theory. Meanwhile, many artificial intelligent methods [12–15] are used as supplementary methods for Data-driven control technology.As a kind of data-driven control technology, Model-free Adaptive Control (MFAC) has been popular among a lot of scholars due to its complete theoretical analysis. The general idea of this algorithm can be summarized as follows. A new pseudopartial derivative concept is introduced under a certain assumed condition. At each working point of the discrete system, an equivalent virtual dynamic linearization model is established, and then the dynamic linearization model is used to design the controller and the adaptive law, to analyze stability, etc. The existence of pseudopartial derivatives is guaranteed by the Cauchy mean value theorem and the solution of some numerical equations, which is insensitive to those deterministic factors, such as the time-varying parameters, the time-varying structures, and the change of the system rank. Therefore, model-free adaptive control avoids the influence of unmodelled dynamics, and its uncertainty is mainly caused by disturbances and incomplete data. The model-free adaptive control was applied to the linear motor control, which was compared with PID control in terms of control accuracy in literature [16]. In literature [17], model-free adaptive control was combined with neural network control, and the parameters of the controller are adjusted online real-time through neural network, and then this algorithm was applied in the experiment of three capacities water tank system. In literature [18], the author proposed the data-based model-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeleton, but the control concise and control speed still can be influenced by the errors, so this paper introduced the neural network to deal with this problem.The innovations of this paper are shown as follows: (1) the controller is designed only using the input and output data and without using the dynamic model of the system, which avoids the influence of model dynamics and system parameter uncertainty; (2) transverse cyclic pitch, longitudinal cyclic pitch, propeller collective pitch are used as control inputs, which avoids the use of unmeasured measurement of rotor flapping angle. It is necessary to achieve flight control; (3) the new controller can effectively enhance the anti-interference capability of the system; (4) the algorithm proposed has the characteristics of reliability and is verified in the comparison with other two algorithms.


## 2. Dynamic Model and Dynamic Transformation

In dynamic modeling, the dynamics of such a robotic system can be approximated by the following form:(1)τk=Mkq¨k+Ckq˙k+Gk+dk,where q∈Rn denotes the vector of joint variables, τ∈Rn denotes the inputs, Mk∈Rn×n represents the inertia, Ck,k˙∈Rn×n denotes the centrifugal and Coriolis state, and Gk∈Rn denotes the gravitational torques. The generalized force d∈Rn represents the disturbances, if the input can be represented as uk=τk, and the output then can be rewritten as yk=q˙k, and the kth moment discrete-time form of this model is(2)yk+1=I−TCkMkyk+TMkuk−TGk+dkMk.For the dynamic system equation (1) with Δyk,ΔukT≠0 for each fixed k, existing Φk can form a dynamic linearization data model:(3)Δyk+1=ΦkΔyk,ΔukT,where Φk=ϕ1k,ϕ2k,ϕ1k,ϕ2k∈Rn×n and Φk<b, b is a positive constant. For each k, considering the following equation:(4)ψk=HkΔyk,ΔukT,Hk∈Rn×n is one numerical matrix. If condition Δyk,ΔukT≠0 to be satisfied, then the above equation must have at least one solution Hk; in fact, it must have infinite number of solutions for each k.(5)Φk=H∗k+∂yk+1∂yk,∂yk+1∂uk.Now the dynamic system can be rewritten as(6)yk+1=ϕ1kΔyk+ϕ2kΔuk+yk,where ϕ1k,ϕ2k are dynamic changed.


## 3. Control Design

The controller design is divided into three parts: first, to apply the model-free adaptive control theory to designing the controller; second, to enhance the robustness of the system and compensate the influence of unknown external disturbance, this paper combines the mode-free adaptive control with both the sliding mode control and neural network estimator to compost the final controller; and finally, to prove that the convergence of the control error and the stability of the closed loop system with theoretical ways.


### 3.1. Designing Model-Free Adaptive Controller

There is the general nonlinear discrete-time MIMO system as follows:(7)yk+1=fyk,…,yk−ny,uk,…,uk−nuwhere uk,yk∈Rn are, respectively, the input and output vectors of the system, f…∈Rn is smooth nonlinear function, and ny,nu are the order of system.According to Theorem 1, the system equation (5) can be rewritten into(8)Δyk+1=ΦkΔuk,where(9)Φk=φ1,φ2,…,φn.Note 1.
In equation (9), Φk represents a time-varying dynamic linearization parameter. It can be computed by estimation algorithm such as least square algorithm, projection algorithm. In this paper, the estimation algorithm in paper [17] is used to calculate the value of the parameters.
Considering the system described by equation (9) and combining the definition Theorem 1, the controlled system can be rewritten as(10)y1k+1y2k+1…ynk+1=y1ky2k…ynk+ϕ11k0ϕ22k…0ϕnnku1ku2k…unk+fNL1′kfNL2′k…fNLn′k,(11)fNLi′=∑j=1nϕijuj−ϕiiui,i=1,2,…,n.
The coupling effect between systems is considered as a disturbance, and it can be compensated estimating by neural network.
First, the pseudomatrixϕk of the Jacobi matrix is calculated by the method given in [17].(12)Φ^Tk=Φ^Tk−1+ηΔuk−1u+Δuk−12×ΔyTk−Δuk−1Φ^Tk−1.
Takingδϕiik=ϕiikϕ^iik,i=1,2,…,n, it represents the error of parameter estimation.
Combining equations (11) and (12), we can get(13)y1k+1y2k+1…ynk+1=y1ky2k…ynk+ϕ^11k0ϕ^22k…0ϕ^nnkΔu1kΔu2k…Δunk+fNL1′kfNL2′k…fNLn′k.
From this, we can know thatfNLik includes unmodeled dynamics, and the coupling effect between systems includes time-varying parameter estimation error. It can be regarded as generalized disturbance signal and estimated by the neural network.


### 3.2. Design of Sliding Mode Controller

Define theith loop tracking error:(14)eik+1=yrik+1−yik+1,i=1,2,…,n.Define theith planar function of loop sliding mode controller:(15)sik=ciTEik.where ciT=1,cθi,cθi s the constant greater than 0.E
i
k
T
=
e
i
k
,
e
i
k
−
1
,
y
r
i
k is the bounded reference input.The discrete convergence law is(16)sik+1=1−qisik−εisgnsik,εi>0,0<qi<1.Equivalent control can be obtained by(17)sik+1=1−qisik−εisgnsik=ciTEik+1.From equations (16)–(19), we can get(18)ciTEik+1=eik+1+c0ieik=1−qisik−εisgnsik,eik+1=1−qisik−εisgnsik−c0ieik,which is(19)yrik+1−yik+1=1−qisik−εisgnsik−c0ieik,Φ⌢ikuik=yrik+1−yik−1−qisik+εisgnsik+c0ieik−fNLik,u⌢ik=1Φ⌢ik+σyrik+1−yik−c¯iTEik−fNLik+εisgnciTEik,where c¯iT=1−qi−c01,1−qic01.Note 2.
When is very small, the control input equation (12) may becomes very large or even unbounded. In order to avoid this phenomenon, we use a very small positive number.Note 3.
The control law equation (12) contains the unknown term fNLik. In this paper, the radial basis neural network is used to estimate this part.


### 3.3. Neural Network Estimator

In this paper, we used the radial basis function (RBF) neural network, which uses neural network approximation to predict disturbance and other uncertainties, and then combined with adaptive sliding mode control to realize the control of disturbed discrete nonlinear systems. The introduction of neural network estimator is designed to improve the dynamic performance of the system. The structure diagram of the control system is shown in Figure1 (take the case of two inputs/two outputs as an example).Figure 1
Structure of the control strategy.Δ
u
,
Δ
u
,
Δ
y
r are system control signals, output signals, and expected signals. fNNi is the output of the ith neural network predictor, that is, the estimate of the fNLi. The neural network structure and algorithm used in this paper are the same as reference [17], so it is omitted.


### 3.4. Stability Analysis

Suppose 1: The system has a globally asymptotically stable zero dynamics. Based on the approximation property of neural network knowledge, after properly selecting network structure and its network parameters,fNNi can approximate the generalized disturbance signal fNLi with arbitrary precision, so there is an arbitrarily small positive number ςi and the prediction error is always less than ζ, which is the output yNNi of neural network. Meet the following conditions:(20)fNNik−fNLik≤ςi,i=1,2,…,n.Definition of positive definite functionVik is(21)Vik=ciTEik2+1∑j−1kςi2j,i=1,2,…,n.According to the definition ofVik, we can know Vik>0. And(22)V˜ik+1=Vik+1−Vik=ciTEik+12+1∑j=1k+1ςi2j−ciTEik2−1∑j=1kςi2j=1−qiciTEik−εisgnciTEik2−ciTEik2+∑j=1kςi2j−∑j=1k+1ςi2j∑j=1k+1ςi2j∑j=1kςi2j=−2qiciTEik2+qi2ciTEik2−2εi1−qiciTEiksgnciTEik+εi2−ςi2k+1∑j=1k+1ςi2j∑j=1kςi2j=−qi2−qiciTEik2−2εi1−qiciTEiksgnciTEik−ςi2k+1∑j=1k+1ςi2j∑j=1kςi2j+εi2.As shown in equation (22), the first 3 items on the right side are less than 0. So if selecting εi to be small enough, make sure that ΔVik+1<0 is set up. It is known that Vi0 is bounded by the previous definition and the system initial value, which means that the system is stable under the control law, and the tracking error of the system converges to the neighborhood of zero.Definition of tracking error:(23)eik=yrik−yik.It is known that the expected tracking trajectoryyrik is bounded. It is also known from the previous proof that eik is bounded. So yrik is bounded.It is also known from the previous proof thatyik,i=1,2,…,n is bounded. Then by equation (16), we know that the system has a global asymptotically stable zero dynamics. Therefore, the existence of constants M1,M2, and k0 satisfies the following condition:(24)uik−1≤M1max0≤τ≤k1≤i≤nyiτ+M2,i=1,2,…,n.Obviously,uik,i=1,2,…,n, which is also a bounded sequence.


## 3.1. Designing Model-Free Adaptive Controller

There is the general nonlinear discrete-time MIMO system as follows:(7)yk+1=fyk,…,yk−ny,uk,…,uk−nuwhere uk,yk∈Rn are, respectively, the input and output vectors of the system, f…∈Rn is smooth nonlinear function, and ny,nu are the order of system.According to Theorem 1, the system equation (5) can be rewritten into(8)Δyk+1=ΦkΔuk,where(9)Φk=φ1,φ2,…,φn.Note 1.
In equation (9), Φk represents a time-varying dynamic linearization parameter. It can be computed by estimation algorithm such as least square algorithm, projection algorithm. In this paper, the estimation algorithm in paper [17] is used to calculate the value of the parameters.
Considering the system described by equation (9) and combining the definition Theorem 1, the controlled system can be rewritten as(10)y1k+1y2k+1…ynk+1=y1ky2k…ynk+ϕ11k0ϕ22k…0ϕnnku1ku2k…unk+fNL1′kfNL2′k…fNLn′k,(11)fNLi′=∑j=1nϕijuj−ϕiiui,i=1,2,…,n.
The coupling effect between systems is considered as a disturbance, and it can be compensated estimating by neural network.
First, the pseudomatrixϕk of the Jacobi matrix is calculated by the method given in [17].(12)Φ^Tk=Φ^Tk−1+ηΔuk−1u+Δuk−12×ΔyTk−Δuk−1Φ^Tk−1.
Takingδϕiik=ϕiikϕ^iik,i=1,2,…,n, it represents the error of parameter estimation.
Combining equations (11) and (12), we can get(13)y1k+1y2k+1…ynk+1=y1ky2k…ynk+ϕ^11k0ϕ^22k…0ϕ^nnkΔu1kΔu2k…Δunk+fNL1′kfNL2′k…fNLn′k.
From this, we can know thatfNLik includes unmodeled dynamics, and the coupling effect between systems includes time-varying parameter estimation error. It can be regarded as generalized disturbance signal and estimated by the neural network.


## 3.2. Design of Sliding Mode Controller

Define theith loop tracking error:(14)eik+1=yrik+1−yik+1,i=1,2,…,n.Define theith planar function of loop sliding mode controller:(15)sik=ciTEik.where ciT=1,cθi,cθi s the constant greater than 0.E
i
k
T
=
e
i
k
,
e
i
k
−
1
,
y
r
i
k is the bounded reference input.The discrete convergence law is(16)sik+1=1−qisik−εisgnsik,εi>0,0<qi<1.Equivalent control can be obtained by(17)sik+1=1−qisik−εisgnsik=ciTEik+1.From equations (16)–(19), we can get(18)ciTEik+1=eik+1+c0ieik=1−qisik−εisgnsik,eik+1=1−qisik−εisgnsik−c0ieik,which is(19)yrik+1−yik+1=1−qisik−εisgnsik−c0ieik,Φ⌢ikuik=yrik+1−yik−1−qisik+εisgnsik+c0ieik−fNLik,u⌢ik=1Φ⌢ik+σyrik+1−yik−c¯iTEik−fNLik+εisgnciTEik,where c¯iT=1−qi−c01,1−qic01.Note 2.
When is very small, the control input equation (12) may becomes very large or even unbounded. In order to avoid this phenomenon, we use a very small positive number.Note 3.
The control law equation (12) contains the unknown term fNLik. In this paper, the radial basis neural network is used to estimate this part.


## 3.3. Neural Network Estimator

In this paper, we used the radial basis function (RBF) neural network, which uses neural network approximation to predict disturbance and other uncertainties, and then combined with adaptive sliding mode control to realize the control of disturbed discrete nonlinear systems. The introduction of neural network estimator is designed to improve the dynamic performance of the system. The structure diagram of the control system is shown in Figure1 (take the case of two inputs/two outputs as an example).Figure 1
Structure of the control strategy.Δ
u
,
Δ
u
,
Δ
y
r are system control signals, output signals, and expected signals. fNNi is the output of the ith neural network predictor, that is, the estimate of the fNLi. The neural network structure and algorithm used in this paper are the same as reference [17], so it is omitted.


## 3.4. Stability Analysis

Suppose 1: The system has a globally asymptotically stable zero dynamics. Based on the approximation property of neural network knowledge, after properly selecting network structure and its network parameters,fNNi can approximate the generalized disturbance signal fNLi with arbitrary precision, so there is an arbitrarily small positive number ςi and the prediction error is always less than ζ, which is the output yNNi of neural network. Meet the following conditions:(20)fNNik−fNLik≤ςi,i=1,2,…,n.Definition of positive definite functionVik is(21)Vik=ciTEik2+1∑j−1kςi2j,i=1,2,…,n.According to the definition ofVik, we can know Vik>0. And(22)V˜ik+1=Vik+1−Vik=ciTEik+12+1∑j=1k+1ςi2j−ciTEik2−1∑j=1kςi2j=1−qiciTEik−εisgnciTEik2−ciTEik2+∑j=1kςi2j−∑j=1k+1ςi2j∑j=1k+1ςi2j∑j=1kςi2j=−2qiciTEik2+qi2ciTEik2−2εi1−qiciTEiksgnciTEik+εi2−ςi2k+1∑j=1k+1ςi2j∑j=1kςi2j=−qi2−qiciTEik2−2εi1−qiciTEiksgnciTEik−ςi2k+1∑j=1k+1ςi2j∑j=1kςi2j+εi2.As shown in equation (22), the first 3 items on the right side are less than 0. So if selecting εi to be small enough, make sure that ΔVik+1<0 is set up. It is known that Vi0 is bounded by the previous definition and the system initial value, which means that the system is stable under the control law, and the tracking error of the system converges to the neighborhood of zero.Definition of tracking error:(23)eik=yrik−yik.It is known that the expected tracking trajectoryyrik is bounded. It is also known from the previous proof that eik is bounded. So yrik is bounded.It is also known from the previous proof thatyik,i=1,2,…,n is bounded. Then by equation (16), we know that the system has a global asymptotically stable zero dynamics. Therefore, the existence of constants M1,M2, and k0 satisfies the following condition:(24)uik−1≤M1max0≤τ≤k1≤i≤nyiτ+M2,i=1,2,…,n.Obviously,uik,i=1,2,…,n, which is also a bounded sequence.


## 4. Simulation and Results

The mechanical structure of this robotic exoskeleton has been shown in Figure2.Figure 2
Model of the robot.In this simulation experiment, the wrist two joints keep unchanged, and the other three joints will be controlled by the different controllers, which include model-free adaptive sliding mode control combined with neural network (MFASMCNN), model-free adaptive sliding mode control (MFASMC), and model-free adaptive control (MFAC). The simulation results will demonstrate the superiority of the proposed algorithm. In the last simulation part, the five joints of the robot were all under control of MFASMCNN and MFASMC approaches to verify the good performance of MFASMCNN when the simulation time is 10 s and the discrete-time sample periodT = 0.001 s.


### 4.1. Controling Three Joints of Robot

The desired angle velocities [1, 2] of the robot have been set as follows:(25)yd1=−0.4,yd1<−0.4,2π10cos2π10t,−0.4≤yd1≤0.4,0.4,yd1>0.4,yd2=−0.6,yd2<−0.6,2π7cos2π7t,−0.6≤yd2≤0.6,0.6,yd2>0.6,yd3=−0.8,yd3<−0.8,2π5cos2π5t,−0.8≤yd3≤0.8,0.8,yd3>0.8.


### 4.2. Performance of Model-Free Adaptive Control

Figure3 demonstrates the simulation results used by model-free adaptive control scheme, where Figures 3(a)–3(c) show the performances of outputs y1,y2,y3, Figure 3(d) shows the control inputs of u1,u2,u3 respectively. From Figures 3(a)–3(c), we can see that control inputs have some fluctuation.Figure 3
The model-free adaptive control simulation results of three joints under control. (a) Tracking performance ofy1(k). (b) Tracking performance of y2(k). (c) Tracking performance of y3(k). (d) Control inputs of u1(k), u2(k), and u3(k).
(a)
(b)
(c)
(d)


### 4.3. Performance of Model-Free Adaptive Sliding Mode Control

Figure4 shows the control performance using the model-free adaptive sliding mode control method, where Figures 4(a)–4(c) show control performances of outputs of y1,y2,y3, form which we can see that the convergence of the method is slow. Figure 4(d) shows control inputs of u1,u2,u3, respectively, from which we can see that the control performance of the MFASMC on the robotic exoskeleton is tracking well, but have a big fluctuation at first for control inputs shown in Figure 4(d).Figure 4
The model-free adaptive sliding mode control simulation results of three joints under control. (a) Tracking performance ofy1(k). (b) Tracking performance of y2(k). (c) Tracking performance of y3(k). (d) Control inputs of u1(k), u2(k) and u3(k).
(a)
(b)
(c)
(d)


### 4.4. Performance of Model-Free Adaptive Sliding Mode Combined with Neural Network Control

This part shows the control performance of model-free sliding mode combined with neural network control method, the initial values shown as(26)yk−1=0,0,0Tuk−1=0,0,0T,yk−2=0,0,0Tuk−2=0,0,0T,φ1=diag1,1,1,φ2=diag1,1,1,The controller parameters areη=ρ=λ=μ=1, q=800, ε=10, Γ=diag4,4,1.Figure5 shows the control performance of model-free adaptive sliding mode combined with neural network method, where Figures 5(a)–5(c) show the performances of outputs y1,y2,y3, from which we can see that the tracking results perform quite satisfactory. Figure 5(d) presents the control inputs of u1,u2,u3, from which we can also see that control inputs are quite stable.Figure 5
Performance of model-free adaptive sliding model control combined with neural network control of three joints. (a) Tracking performance ofy1(k). (b) Tracking performance of y2(k). (c) Tracking performance of y3(k). (d) Control inputs of u1(k), u2(k) and u3(k).
(a)
(b)
(c)
(d)


### 4.5. Comparison Analysis

In this part, comparison has been done to demonstrate the control performance of the proposed method; for this, the root mean square (RMS) values of errors are created by the three kinds of controller, and the results are shown in Tables1–3, respectively, from comparison results of the RMS values, we can conclude that the convergence performance and speed of the proposed method is better than two other controllers although the proposed controller has a little overshoot at first time.Table 1
RMS values of errors by MFAC.
(0,2]
(2,4]
(4,6]
(6,8]
(8,10]
e1
0.0227
0.0090
0.0160
0.0155
0.0154
e2
0.0232
0.0204
0.0167
0.0179
0.0072
e3
0.0097
0.0103
0.0075
0.0058
0.0032Table 2
RMS values of errors by MFASMC.
RMS of error
(0,2]
(2,4]
(4,6]
(6,8]
(8,10]
e1
0.0292
0.0028
0.0019
0.0009
0.0021
e2
0.0411
0.0045
0.0031
0.0023
0.0015
e3
0.0230
0.0018
0.0016
0.0019
0.0019Table 3
RMS values of errors by MFASMCNN.
RMS of error
(0,2]
(2,4]
(4,6]
(6,8]
(8,10]
e1
0.0186
0.0014
0.0013
0.0007
0.0014
e2
0.0261
0.0035
0.0021
0.0016
0.0009
e3
0.0138
0.0012
0.0009
0.0011
0.0011


### 4.6. Five Joints under Control

The desired angle velocities of robot are set as follows:(27)yd1=−0.4,yd1<−0.4,2π10cos2π10t,−0.4≤yd1≤0.4,0.4,yd1>0.4,yd2=−0.5,yd2<−0.5,2π8cos2π8t,−0.5≤yd2≤0.5,0.5,yd2>0.5,yd3=−0.6,yd3<−0.6,2π7cos2π7t,−0.6≤yd3≤0.6,0.6,yd3>0.6,yd4=−0.7,yd4<−0.7,2π6cos2π6t,−0.7≤yd4≤0.7,0.7yd4>0.7,yd5=−0.8,yd5<−0.8,2π5cos2π5t,−0.8≤yd5≤0.8,0.8,yd5>0.8.The initial values of the proposed method are(28)yk−1=0,0,0,0,0Tuk−1=0,0,0,0,0T,yk−2=0,0,0,0,0Tuk−2=0,0,0,0,0T,φ1=diag1,1,1,1,1,φ2=diag1,1,1,1,1.The controller parameters areη=ρ=λ=μ=1,ε=10,q=800, Γ=diag4,4,4,0.2,0.2.Figure6 shows the control performance based on MAFSMC and MFASMCNN, where Figures 6(a)–6(e) show the performances of outputs y1,y2,y3,y4,y5, in which we can see that the convergence of the proposed method is quite well, and Figure 6(f) shows the control inputs of u1,u2,u3,u4,u5 respectively, from which we can see that the control is stable.Figure 6
Comparison results with five joints under control (“11”represents MFSMC,“22” represents MFASMCNN). (a) Tracking performance ofy1(k). (b) Tracking performance of y2(k). (c) Tracking performance of y3(k). (d) Tracking performance of y4(k). (e) Tracking performance of y5(k). (f) Control inputs of u1(k), u2(k), u3(k), u4(k) and u5(k) with MFSMC. (g) Control inputs of u1(k), u2(k), u3(k), u4(k) and u5(k) with MFASMCNN.
(a)
(b)
(c)
(d)
(e)
(f)
(g)From the above simulation results in Figures and3 and 4, we can see that the MFAC, MFASMC, and MFASMCNN methods are satisfactory under the parameter time-invariable system and also the control inputs of MFASMCNN method are more stable under the big fluctuation.


## 4.1. Controling Three Joints of Robot

The desired angle velocities [1, 2] of the robot have been set as follows:(25)yd1=−0.4,yd1<−0.4,2π10cos2π10t,−0.4≤yd1≤0.4,0.4,yd1>0.4,yd2=−0.6,yd2<−0.6,2π7cos2π7t,−0.6≤yd2≤0.6,0.6,yd2>0.6,yd3=−0.8,yd3<−0.8,2π5cos2π5t,−0.8≤yd3≤0.8,0.8,yd3>0.8.


## 4.2. Performance of Model-Free Adaptive Control

Figure3 demonstrates the simulation results used by model-free adaptive control scheme, where Figures 3(a)–3(c) show the performances of outputs y1,y2,y3, Figure 3(d) shows the control inputs of u1,u2,u3 respectively. From Figures 3(a)–3(c), we can see that control inputs have some fluctuation.Figure 3
The model-free adaptive control simulation results of three joints under control. (a) Tracking performance ofy1(k). (b) Tracking performance of y2(k). (c) Tracking performance of y3(k). (d) Control inputs of u1(k), u2(k), and u3(k).
(a)
(b)
(c)
(d)


## 4.3. Performance of Model-Free Adaptive Sliding Mode Control

Figure4 shows the control performance using the model-free adaptive sliding mode control method, where Figures 4(a)–4(c) show control performances of outputs of y1,y2,y3, form which we can see that the convergence of the method is slow. Figure 4(d) shows control inputs of u1,u2,u3, respectively, from which we can see that the control performance of the MFASMC on the robotic exoskeleton is tracking well, but have a big fluctuation at first for control inputs shown in Figure 4(d).Figure 4
The model-free adaptive sliding mode control simulation results of three joints under control. (a) Tracking performance ofy1(k). (b) Tracking performance of y2(k). (c) Tracking performance of y3(k). (d) Control inputs of u1(k), u2(k) and u3(k).
(a)
(b)
(c)
(d)


## 4.4. Performance of Model-Free Adaptive Sliding Mode Combined with Neural Network Control

This part shows the control performance of model-free sliding mode combined with neural network control method, the initial values shown as(26)yk−1=0,0,0Tuk−1=0,0,0T,yk−2=0,0,0Tuk−2=0,0,0T,φ1=diag1,1,1,φ2=diag1,1,1,The controller parameters areη=ρ=λ=μ=1, q=800, ε=10, Γ=diag4,4,1.Figure5 shows the control performance of model-free adaptive sliding mode combined with neural network method, where Figures 5(a)–5(c) show the performances of outputs y1,y2,y3, from which we can see that the tracking results perform quite satisfactory. Figure 5(d) presents the control inputs of u1,u2,u3, from which we can also see that control inputs are quite stable.Figure 5
Performance of model-free adaptive sliding model control combined with neural network control of three joints. (a) Tracking performance ofy1(k). (b) Tracking performance of y2(k). (c) Tracking performance of y3(k). (d) Control inputs of u1(k), u2(k) and u3(k).
(a)
(b)
(c)
(d)


## 4.5. Comparison Analysis

In this part, comparison has been done to demonstrate the control performance of the proposed method; for this, the root mean square (RMS) values of errors are created by the three kinds of controller, and the results are shown in Tables1–3, respectively, from comparison results of the RMS values, we can conclude that the convergence performance and speed of the proposed method is better than two other controllers although the proposed controller has a little overshoot at first time.Table 1
RMS values of errors by MFAC.
(0,2]
(2,4]
(4,6]
(6,8]
(8,10]
e1
0.0227
0.0090
0.0160
0.0155
0.0154
e2
0.0232
0.0204
0.0167
0.0179
0.0072
e3
0.0097
0.0103
0.0075
0.0058
0.0032Table 2
RMS values of errors by MFASMC.
RMS of error
(0,2]
(2,4]
(4,6]
(6,8]
(8,10]
e1
0.0292
0.0028
0.0019
0.0009
0.0021
e2
0.0411
0.0045
0.0031
0.0023
0.0015
e3
0.0230
0.0018
0.0016
0.0019
0.0019Table 3
RMS values of errors by MFASMCNN.
RMS of error
(0,2]
(2,4]
(4,6]
(6,8]
(8,10]
e1
0.0186
0.0014
0.0013
0.0007
0.0014
e2
0.0261
0.0035
0.0021
0.0016
0.0009
e3
0.0138
0.0012
0.0009
0.0011
0.0011


## 4.6. Five Joints under Control

The desired angle velocities of robot are set as follows:(27)yd1=−0.4,yd1<−0.4,2π10cos2π10t,−0.4≤yd1≤0.4,0.4,yd1>0.4,yd2=−0.5,yd2<−0.5,2π8cos2π8t,−0.5≤yd2≤0.5,0.5,yd2>0.5,yd3=−0.6,yd3<−0.6,2π7cos2π7t,−0.6≤yd3≤0.6,0.6,yd3>0.6,yd4=−0.7,yd4<−0.7,2π6cos2π6t,−0.7≤yd4≤0.7,0.7yd4>0.7,yd5=−0.8,yd5<−0.8,2π5cos2π5t,−0.8≤yd5≤0.8,0.8,yd5>0.8.The initial values of the proposed method are(28)yk−1=0,0,0,0,0Tuk−1=0,0,0,0,0T,yk−2=0,0,0,0,0Tuk−2=0,0,0,0,0T,φ1=diag1,1,1,1,1,φ2=diag1,1,1,1,1.The controller parameters areη=ρ=λ=μ=1,ε=10,q=800, Γ=diag4,4,4,0.2,0.2.Figure6 shows the control performance based on MAFSMC and MFASMCNN, where Figures 6(a)–6(e) show the performances of outputs y1,y2,y3,y4,y5, in which we can see that the convergence of the proposed method is quite well, and Figure 6(f) shows the control inputs of u1,u2,u3,u4,u5 respectively, from which we can see that the control is stable.Figure 6
Comparison results with five joints under control (“11”represents MFSMC,“22” represents MFASMCNN). (a) Tracking performance ofy1(k). (b) Tracking performance of y2(k). (c) Tracking performance of y3(k). (d) Tracking performance of y4(k). (e) Tracking performance of y5(k). (f) Control inputs of u1(k), u2(k), u3(k), u4(k) and u5(k) with MFSMC. (g) Control inputs of u1(k), u2(k), u3(k), u4(k) and u5(k) with MFASMCNN.
(a)
(b)
(c)
(d)
(e)
(f)
(g)From the above simulation results in Figures and3 and 4, we can see that the MFAC, MFASMC, and MFASMCNN methods are satisfactory under the parameter time-invariable system and also the control inputs of MFASMCNN method are more stable under the big fluctuation.


## 5. Conclusions

In this paper, we analyzed the transformation and linearization of the robotic exoskeleton dynamics, and a model-free adaptive sliding mode controller with neural network estimator based on data-driven methodology was designed. Then the convergence of control error and the stability of the closed loop system are proved. The experimental results show that the control design proposed in this paper performs well and has good robustness in system uncertainties and external disturbances compared with MFAC and MFASMC.



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*Source: 8327456-2020-03-17.xml*