# Command Filter Backstepping Sliding Model Control for Lower-Limb Exoskeleton

Authors: Peng Yang; Gaowei Zhang; Jie Wang; Xiaozhou Wang; Lili Zhang; Lingling Chen
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

A command filter adaptive fuzzy backstepping control strategy is proposed for lower-limb assisting exoskeleton. Firstly, the human-robot model is established by taking the human body as a passive part, and a coupling torque is introduced to describe the interaction between the exoskeleton and human leg. Then, Vicon motion capture system is employed to obtain the reference trajectory. For the purpose of obviating the “explosion of complexity” in conventional backstepping, a second-order command filter is introduced into the sliding mode control strategy. The fuzzy logic systems (FLSs) are also applied to handle with the chattering problem by estimating the uncertainties and disturbances. Furthermore, the stability of the closed-loop system is proved based on the Lyapunov theory. Finally, simulation results are presented to illustrate the effectiveness of the control strategy.


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## Body

## 1. Introduction

Recently, the exoskeleton is increasingly used for power-assisting in industrial [1, 2], medical [3–5], and military [6, 7] areas. The human operator provides locomotion intention and a small muscle strength while the robot can help human complete the desired action with a suitable torque through replicating the operator’s movements [2]. The robot in such applications has been actively researched since 1990s, and some of them have been applied for various environments [8]. However, there are challenging problems in exoskeleton research such as the establishment of coupling model of human-robot system and designing of control strategy.From the modeling point of view, it has been proved that some methods used now are effective such as Newton-Euler equations and Lagrange dynamics [9, 10]. Based on aforementioned work, series of improvements have been proposed to increase the accuracy of human-robot system model. The inverse dynamic model is established and the singular points are avoided using damped least squares in [2]. In order to approximate the actual situation of human legs, a variety of musculoskeletal models are developed [11, 12]. The comparison between Hill-type and proportional model for human muscle is illustrated in [12], and Hill-type models are proved to be more appropriate.Nowadays, there have been several published papers on control strategies of human-robot cooperative control [13–18]. For instance, a sensitivity amplification control is proposed in [19] which could track the desired trajectory by minimizing the interaction torque. In [20], a robust sliding mode controller is proposed to guarantee the stability in disturbance situation, and the boundary layer is introduced to reduce the chattering problem of sliding model control. Additionally, an adaptive sliding model control based on state observer is proposed in [21], which could update the controller parameters online to improve the safety of the system.However, most of the control methods mentioned have limitations. In modeling, taking the human effects as disturbance [22], widely used in exoskeleton researches, is unreasonable. Human body is a part of the system obviously. Additional, the system has high-order features when taking the human body as a passive part [19]. Hence, the control strategies above are not available because of the “explosion of complexity” [23].In this paper, a command filtered backstepping sliding model control equipped with FLSs is proposed based on a human-robot model. Compared with the analogous literature, the main contributions are summarized as follows:(i)
The proposed human-robot model is established by taking the human leg as a passive part and the coupling torque between human and robot is introduced. Compared with the model built in [19], this paper converts the transfer to the state space form and introduces uncertainties and disturbances which are more actual and complicated.(ii)
For the purpose of testing the controller performance, an experiment is implanted to obtain the actual trajectory of the hip-joint. Compared with the sine curve, the upper bound increases rapidly after a few derivative operations, which may cause the system uncontrollable.(iii)
The command filter backstepping sliding model control is proposed. By using the command filter, the analytical derivate is unnecessary and the “explosion of complexity” in the controller design process is avoided [24–26]. Besides, the problem mentioned in (ii) is solved. Furthermore, the FLSs are used to approximate the uncertainties and disturbances and provide real-time compensations for the system.The paper is organized as follows: in Section2, the high-order human-robot model is introduced based on the linearized models of human leg and exoskeleton. Then the reference trajectory of hip-joint is obtained through experiments. In Section 3, considering the demands of controller design, a model in state space form with uncertainties and disturbances is obtained. Based on that work, the human-robot controller is designed with a command filter adaptive backstepping sliding mode and, by utilizing the Lyapunov theory, closed-loop system stability is analyzed. Simulation results are discussed in Section 4 and the conclusions are provided in Section 5.


## 2. System Modeling and Trajectory Generation

### 2.1. Dynamics of the Human-Robot System

As shown in Figure1, the human-robot system can be expressed as a person wearing a lower-limb exoskeleton which provides a back support to assist hip-joint motion. By passing the leg’s gravity to the waist, the muscle effort needed for human walking could be reduced to a low level.Figure 1
1-DOF lower-limb exoskeleton.In order to describe the system, an elementary model is used in this paper which consists of linearized one-degree-of-freedom (1-DOF) models for the human leg and the exoskeleton. In the process of modeling the physical interaction between the human and exoskeleton, a coupling torque, expressed as combination of a linear spring and a damper, is introduced.Then, the ideal dynamics of the human-robot system are given as follows [19]:(1)Ihθ¨ht+bhθ˙ht+khθht=τht-τctIeθ¨et+beθ˙et+keθet=τet+τct,where Ih,bh,kh are, respectively, the moment of inertia, joint damping coefficient, and joint stiffness coefficient of the human leg; θh(t) is the hip-joint angle; τh(t) is the net muscle torque acting on the joint. Ie,be,ke are, respectively, the moment of inertia, joint damping coefficient, and joint stiffness coefficient of the exoskeleton; θe(t) is the exoskeleton joint angle; τe(t) is the actuator torque.And the coupling torque is defined as follows:(2)bcθ˙ht-θ˙et+kcθht-θet=τct,where bc,kc are, respectively, the equivalent damping coefficient and stiffness coefficient of the interaction torque; τc is the coupling torque.


### 2.2. Trajectory Generation

The main way that the exoskeleton helps human complete the locomotion is tracking the human gait cycle. So a reasonable desired position trajectory is an essential factor for testing the model and controller. An experimenter (girl aged 25 years with mass of 52 kg and stature of 165 cm) volunteered to participate in the gait experiment with Vicon motion capture system, device provided by National Research Center for Rehabilitation Technical Aids (Figure2).Figure 2
Vicon motion capture system.Special trackers are fixed at the particular marks on the experimenter which can be captured by cameras distributed in reasonable location in experiment space. The information from different cameras are combined, and the actual hip-joint angle signals are obtained through data processing. From the considerable data obtained, the most reliable, representative, and authentic data is selected and reorganized. To ensure the smoothness of the trajectory, the Fourier series is introduced to describe the actual gait cycle. Note that the human locomotion satisfies the smooth characteristics.Mathematical expression of the desired trajectory can be expressed as follows [14]:(3)θr=θ0+∑k=1Naksin⁡kwnt+bkcos⁡kwnt,where θ0 is the initial value of θ2; ak and bk are the sine and cosine amplitudes of Fourier series; wn is the fundamental frequency; k is the harmonic order.The parameters of the trajectory are obtained by the curve fit toolbox of MATLAB. In order to approximate the actual curve and simplify the calculation process, fundamental frequency to third harmonics, that is,k=3 in (3), of the Fourier series is used in this paper. The parameters are illustrated in Table 1.Table 1
Parameter identification of the hip joint trajectory.
θ
0
9.50 (9.46, 9.55)
a
1
-
18.74 (-18.8, -18.7)
a
2
1.957 (1.9, 2.0)
a
3
0.7747 (0.72, 0.83)
w
n
3.775 (3.772, 3.778)
b
1
-
8.97 (-9.0, -8.9)
b
2
-
5.026 (-5.09, -4.96)
b
3
0.9843 (0.93, 1.04)According to (3) and the parameters given in Table 1, a reliable hip-joint trajectory (expressed as x1d) can be obtained and shown in Figure 3.Figure 3
The actual hip-joint trajectory obtained by experiment.The control objectives for the human-robot system are illustrated as follows:(i)
An adaptive controller for high-order human-robot system is designed, such that the position of human legx1 and exoskeleton x3 can track the actual trajectory x1d obtained from experiment.(ii)
The prescribed output tracking errorx~1=x1-x1c is always bounded. Besides when uncertainties and disturbances exist in the system, the tracking error can converge to a neighborhood of the origin in a short time.


## 2.1. Dynamics of the Human-Robot System

As shown in Figure1, the human-robot system can be expressed as a person wearing a lower-limb exoskeleton which provides a back support to assist hip-joint motion. By passing the leg’s gravity to the waist, the muscle effort needed for human walking could be reduced to a low level.Figure 1
1-DOF lower-limb exoskeleton.In order to describe the system, an elementary model is used in this paper which consists of linearized one-degree-of-freedom (1-DOF) models for the human leg and the exoskeleton. In the process of modeling the physical interaction between the human and exoskeleton, a coupling torque, expressed as combination of a linear spring and a damper, is introduced.Then, the ideal dynamics of the human-robot system are given as follows [19]:(1)Ihθ¨ht+bhθ˙ht+khθht=τht-τctIeθ¨et+beθ˙et+keθet=τet+τct,where Ih,bh,kh are, respectively, the moment of inertia, joint damping coefficient, and joint stiffness coefficient of the human leg; θh(t) is the hip-joint angle; τh(t) is the net muscle torque acting on the joint. Ie,be,ke are, respectively, the moment of inertia, joint damping coefficient, and joint stiffness coefficient of the exoskeleton; θe(t) is the exoskeleton joint angle; τe(t) is the actuator torque.And the coupling torque is defined as follows:(2)bcθ˙ht-θ˙et+kcθht-θet=τct,where bc,kc are, respectively, the equivalent damping coefficient and stiffness coefficient of the interaction torque; τc is the coupling torque.


## 2.2. Trajectory Generation

The main way that the exoskeleton helps human complete the locomotion is tracking the human gait cycle. So a reasonable desired position trajectory is an essential factor for testing the model and controller. An experimenter (girl aged 25 years with mass of 52 kg and stature of 165 cm) volunteered to participate in the gait experiment with Vicon motion capture system, device provided by National Research Center for Rehabilitation Technical Aids (Figure2).Figure 2
Vicon motion capture system.Special trackers are fixed at the particular marks on the experimenter which can be captured by cameras distributed in reasonable location in experiment space. The information from different cameras are combined, and the actual hip-joint angle signals are obtained through data processing. From the considerable data obtained, the most reliable, representative, and authentic data is selected and reorganized. To ensure the smoothness of the trajectory, the Fourier series is introduced to describe the actual gait cycle. Note that the human locomotion satisfies the smooth characteristics.Mathematical expression of the desired trajectory can be expressed as follows [14]:(3)θr=θ0+∑k=1Naksin⁡kwnt+bkcos⁡kwnt,where θ0 is the initial value of θ2; ak and bk are the sine and cosine amplitudes of Fourier series; wn is the fundamental frequency; k is the harmonic order.The parameters of the trajectory are obtained by the curve fit toolbox of MATLAB. In order to approximate the actual curve and simplify the calculation process, fundamental frequency to third harmonics, that is,k=3 in (3), of the Fourier series is used in this paper. The parameters are illustrated in Table 1.Table 1
Parameter identification of the hip joint trajectory.
θ
0
9.50 (9.46, 9.55)
a
1
-
18.74 (-18.8, -18.7)
a
2
1.957 (1.9, 2.0)
a
3
0.7747 (0.72, 0.83)
w
n
3.775 (3.772, 3.778)
b
1
-
8.97 (-9.0, -8.9)
b
2
-
5.026 (-5.09, -4.96)
b
3
0.9843 (0.93, 1.04)According to (3) and the parameters given in Table 1, a reliable hip-joint trajectory (expressed as x1d) can be obtained and shown in Figure 3.Figure 3
The actual hip-joint trajectory obtained by experiment.The control objectives for the human-robot system are illustrated as follows:(i)
An adaptive controller for high-order human-robot system is designed, such that the position of human legx1 and exoskeleton x3 can track the actual trajectory x1d obtained from experiment.(ii)
The prescribed output tracking errorx~1=x1-x1c is always bounded. Besides when uncertainties and disturbances exist in the system, the tracking error can converge to a neighborhood of the origin in a short time.


## 3. Controller Design

### 3.1. System Description and Control Strategy

For the exoskeleton, the human leg is a passive part and fulfills the locomotion with the interaction torque between the human and robot. For the dynamics shown in (1), when replacing τc with (2), the system is given by(4)Ihθ¨h+bh+bcθ˙h+kh+kcθh-bcθ˙e-kcθe=τhIeθ¨e+be+bcθ˙e+ke+kcθe-bcθ˙h-kcθh=τe.Note that all the states and the torques are time-varying variables and time flags are omitted for convenience.Taking the lump uncertainties, parametric/unmodeled uncertainties as well as the external disturbances, into account, (4) can be transformed into state space as(5)x˙1=x2x˙2=Ih-1τh-kc+khx1-bc+bhx2+kcx3+bcx4+Δ1x˙3=x4x˙4=Ie-1τe+kcx1+bcx2-kc+kex3-bc+bex4+Δ2,where x1=θh, x2=θ˙h, x3=θe, x4=θ˙e are the state vector of the system; the lump uncertainties in human-robot system are(6)Δ1=Ih-1-Δkc+khx1-Δbc+bhx2+Δkcx3+Δbcx4+d1Δ2=Ie-1Δkcx1+Δbcx2-Δkc+kex3-Δbc+bex4+d2.Note that the specific parameters of human leg are hard to be measured and the actuator of the exoskeleton includes mechanical errors which cannot be described precisely. Hence, the uncertainties of the system existed and are inevitable.Considering that the system has high-order features, a backstepping sliding model method is introduced to solve the complex problem with a recursive form [27]. To avoid the “explosion of complexity” in the controller design process, a second-order nonlinear command filter is employed in this paper. The command filter ensures that the desired command and its derivative satisfy the same magnitude and rate constrains [24].In order to handle the system chattering problem caused by the uncertainties and disturbances, fuzzy logic systems (FLSs) are equipped to estimate the upper bounds of the lump uncertainties. FLSs provide real-time compensations for the human-robot system to reduce the switching items of the sliding model control.Being equipped with second-order command filter and FLSs, a backstepping sliding mode control strategy for the human-robot system with uncertainties and disturbances is proposed in this paper.


### 3.2. Basic Assumptions and FLS

Some reasonable and useful assumptions are given at first which ensure the stability of the system.Assumption 1.
There exist constantsc1,c2>0, such that the inequality Δ1<c1, Δ2<c2 holds.Assumption 2 (Lipschitz).
For∀x1,x2∈Rn,∃L∈R+, the inequality holds as follows:(7)Fx1-Fx2≤Lx1-x2,where x1-x2 represents the 2-norm of (x1-x2). And all the desired commands xid, x˙id(i=1,2,3,4) satisfy the Lipschitz continuity.The design of the FLS consists of two steps. First, the fuzzy rule base should be made up as follows:Rl:
ifx1 is A1l and x2 is A2l and…and xn is Anlthenf^(x) is Bl,whereAil and Bl are fuzzy sets in R, l=1,2,…,N; i=1,2,…,n.The second step is defuzzification. Center average defuzzification operator is applied in this paper which can be expressed as(8)f^x=∑l=1Nf-l∏i=1nuAilxi∑l=1N∏i=1nuAilxi,where f-l=maxf∈RuBl(f), and uAil(xi), uBl are the membership functions.Define the fuzzy basis vector as(9)φx=∏i=1nuAilxi∑l=1N∏i=1nuAilxi.DenoteθT=[f-1,f-2,…,f-N] and φ(x)=[φ1(x),φ2(x),…,φn(x)]T; then the FLS can be expressed as(10)f^x=θTφx.The optimal parameterθ∗T can be defined by(11)θ∗T=argminθ∈Rnsupx∈Rnfx-θ∗Tφx.Lemma 3 (Wang [28]).
For∀f(x), which is continuous function and defined over a compact Ω, for any a constant ε∈R+, there exist an FLS and a parameter θ∗ such that (12)supx∈Ωfx-θ∗Tφx≤ε.


### 3.3. Controller Design

Consider the characteristics of the human-robot system, a backstepping sliding model control with second-order command filter is proposed in this paper. The controller design process is shown in this section.The output tracking errors and compensated tracking errors of the subsystems are defined, respectively, as(13)x~i=xi-xic,i=1,2,3,4.zi=x~i-ξi,i=1,2,3,4, where xi and xic represent the system states and filtered commands of the ith subsystem, respectively.Theξi(i=1,2,3,4) signals can be obtained by(14)ξ˙1=-k1ξ1+x2c-x2d+ξ2ξ˙2=-k2ξ2+Ih-1kcx3c-x3d+Ih-1kcξ3ξ˙3=-k3ξ3+x4c-x4d+ξ4ξ˙4=-k4ξ4+Ie-1τec-τedwith ξi(0)=0.Remark 4.
Equation (14) is used to achieve the filtering value ξi which are designed to compensate the errors caused by the command filters. Note that they can be computed with integrating processes to avoid the differential operations.For the purpose of eliminating the “explosion of complexity,” a second-order nonlinear command filter is designed to calculatexic,x˙ic. So the command filter is shown as follows:(15)r˙i,1r˙i,2=ri,2-2ζωnri,2-ωn2ri,1-xidxicx˙ic=ri,1ri,2,where ωn is the natural frequency of the filter and typically satisfies ωn>ki, i=(1,2,3,4) to ensure the tracking accuracy. ζ∈(0,1] is the damping ratio of the filter system. xid, (i=2,3,4) are the virtual control signals and x1d is the desired trajectory.The filter initial conditions arexic(0)=0, x˙ic(0)=0. Every command filter is designed to compute the filtered commands without differential operation. Furthermore, xic will track xid by choosing the suitable parameters.In order to find the optimal parameters of the FLSs, the adaptive laws are chosen forθ^1 and θ^2 as follows:(16)θ^˙1=γ1z2φ1θ^˙2=γ2z4φ2,where γ1,γ2 are the positive adaptive coefficients.Considering the compensating errors and the closed-loop dynamics, the virtual law can be defined as follows:(17)x2d=-k1x~1+x˙1c(18)x3d=kc-1Ih-k2x~2-z1+x˙2c-θ^1Tφ1-η1satsz2-τh+kc+khx1+bc+bhx2-bcx4(19)x4d=-k3x~3-kcIh-1z2+x˙3c(20)τed=Ie-k4x~4+x˙4-z3-θ^2Tφ2-η2satsz4-kcx1-bcx2+kc+kex3+bc+bex4,where θ^1, θ^2 are the estimated values of optimal parameters and φ1, φ2 are the basis vectors of FLSs. ki>0, (i=1,2,3,4) are control gains specified by the designer. η1, η2 are constants that ensure that the inequality η1>ε1, η2>ε2 holds.Thesats(x) is the switching function that satisfies(21)satsx=1,x>xΔkx,x≤xΔ-1,x<-xΔ.


### 3.4. Stability Analysis

Theorem 5.
For the system illustrated in (5), there exist a range of values for the gains k1,k2,k3,k4, and the adaptive coefficients γ1, γ2, such that the tracking error x~1 and compensation errors zi(i=1,2,3,4)can converge to zero with the compensations provided by FLSs.Proof.
The tracking errorx~i and the compensated tracking error zi are given first. (22)x~˙1=x˙1-x˙1c=x2-x2c+x2c-x2d+x2d-x˙1c=x~2+x2c-x2d+x2d-x˙1c=-k1x~1+x~2+x2c-x2dx~˙2=x˙2-x˙2c=Ih-1τh-kc+khx1-bc+bhx2+kcx3-x3c+x3c-x3d+x3dbcx4+Δ1-x˙2c=Ih-1τh-kc+khx1-bc+bhx2+kcx3d+kcx~3+bcx4+Δ1+Ih-1kcx3c-x3d-x˙2c=-k2x~2-z1+Ih-1kcx~3+Ih-1kcx3c-x3d+θ~1Tφ1+ε1-η1satsz2x~˙3=x˙3-x˙3c=x4-x4c+x4c-x4d+x4d-x˙3c=x~4+x4c-x4d+x4d-x˙3c=-k3x~3+x~4+x4c-x4d-kcIh-1z2x~˙4=x˙4-x˙3c=Ie-1τe-τec+τec-τed+τed+kcx1+bcx2-kc+kex3-bc+bex4+Δ2-x˙3c=Ie-1τ~e-τec-τed+τed+kcx1+bcx2-kc+kex3-bc+bex4+Δ2-x˙3c=-k4x~4-z3+Ie-1τ~e-τec-τed+θ~2Tφ2+ε2-η2satsz4.Due to the fact thatzi=x~i-ξi for i=1,2,3,4, the dynamics of compensated tracking error can be expressed as follows:(23)z˙1=x~˙1-ξ˙1=-k1x~1+x~2+x2c-x2d--k1ξ1+x2c-x2d+ξ2=-k1z1+z2z˙2=x~˙2-ξ˙2=-k2x~2-z1+Ih-1kcx~3+Ih-1kcx3c-x3d+θ~1Tφ+ε1-η1satsz2--k2ξ2+Ih-1kcx3c-x3d+Ih-1kcξ3=-k2z2-z1+Ih-1kcz3+θ~1Tφ+ε1-η1satsz2z˙3=x~˙3-ξ˙3=-k3x~3+x~4+x4c-x4d-kcIh-1z2--k3ξ3+x4c-x4d+ξ4=-k3z3+z4-kcIh-1z2z˙4=x~˙4-ξ˙4=-k4x~4-z3+Ie-1τ~e-τec-τed+θ~2Tφ2+ε2-η2satsz4--k4ξ4+Ie-1τec-τed=-k4z4-z3+θ~2Tφ2+ε2-η2satsz4+Ie-1τ~e,where θ~1=θ1∗-θ^1 and θ~2=θ2∗-θ^2.Remark 6.
Note the fact that the commandτec will directly output to control plant, so the conclusion is easily obtained that τe=τec.Then, define the control Lyapunov function for the closed-loop system as(24)Vt=12∑i=14zi2+γ1-1θ~1Tθ~1+γ2-1θ~2Tθ~2.Then the time derivative of the Lyapunov function (24) becomes(25)V˙t=∑i=14ziz˙i-γ1-1θ~1Tθ^˙1-γ2-1θ~2Tθ^˙2.Substitutez˙i with (23):(26)V˙t=z1-k1z1+z2+z2-k2z2-z1+Ih-1kcz3+θ~1Tφ+ε1-η1satsz2+z3-k3z3+z4-kcIh-1z2+z4-k4z4-z3+θ~2Tφ2+ε2-η2satsz4-γ1-1θ~1Tθ^˙1-γ2-1θ~2Tθ^˙2.The simplification can be written as(27)V˙t=-k1z12-k2z22-k3z32-k4z42+z2θ~1Tφ+ε1-η1satsz2+z4θ~2Tφ2+ε2-η2satsz4-γ1-1θ~1Tθ^˙1-γ2-1θ~2Tθ^˙2.Replaceθ^˙1, θ^˙2 with the adaptive law (16)(28)V˙t=-k1z12-k2z22-k3z32-k4z42+z2ε1-η1satsz2+z4ε2-η2satsz4≤-k1z12-k2z22-k3z32-k4z42+z2ε1-η1+z4ε2-η2.Letη1>ε1, η2>ε2; define the tracking error vector as z=[z1,z2,z3,z4]; then(29)V˙t≤-k1z12-k2z22-k3z32-k4z42≤-kminz2,where kmin=mink1,k2,k3,k4. According to (29), the error vector z is uniformly bounded, and limt→∞V(t)=V(∞). When integrating both sides of inequality (29), then(30)V∞-V0≤-kmin∫0∞z2dt.That is(31)kmin∫0∞z2dt≤V0-V∞.According to Barbalat Lemma proposed in [29], when the time variable t tends to infinity, the error vector z tends to zero.The block diagram of controller design is shown in Figure4.Figure 4
Block diagram of the controller design. (CF represents the command filter defined by (15); VCL represents the virtual control law of the four subsystems defined by (17)–(20); FLS represents the fuzzy logic system.)


## 3.1. System Description and Control Strategy

For the exoskeleton, the human leg is a passive part and fulfills the locomotion with the interaction torque between the human and robot. For the dynamics shown in (1), when replacing τc with (2), the system is given by(4)Ihθ¨h+bh+bcθ˙h+kh+kcθh-bcθ˙e-kcθe=τhIeθ¨e+be+bcθ˙e+ke+kcθe-bcθ˙h-kcθh=τe.Note that all the states and the torques are time-varying variables and time flags are omitted for convenience.Taking the lump uncertainties, parametric/unmodeled uncertainties as well as the external disturbances, into account, (4) can be transformed into state space as(5)x˙1=x2x˙2=Ih-1τh-kc+khx1-bc+bhx2+kcx3+bcx4+Δ1x˙3=x4x˙4=Ie-1τe+kcx1+bcx2-kc+kex3-bc+bex4+Δ2,where x1=θh, x2=θ˙h, x3=θe, x4=θ˙e are the state vector of the system; the lump uncertainties in human-robot system are(6)Δ1=Ih-1-Δkc+khx1-Δbc+bhx2+Δkcx3+Δbcx4+d1Δ2=Ie-1Δkcx1+Δbcx2-Δkc+kex3-Δbc+bex4+d2.Note that the specific parameters of human leg are hard to be measured and the actuator of the exoskeleton includes mechanical errors which cannot be described precisely. Hence, the uncertainties of the system existed and are inevitable.Considering that the system has high-order features, a backstepping sliding model method is introduced to solve the complex problem with a recursive form [27]. To avoid the “explosion of complexity” in the controller design process, a second-order nonlinear command filter is employed in this paper. The command filter ensures that the desired command and its derivative satisfy the same magnitude and rate constrains [24].In order to handle the system chattering problem caused by the uncertainties and disturbances, fuzzy logic systems (FLSs) are equipped to estimate the upper bounds of the lump uncertainties. FLSs provide real-time compensations for the human-robot system to reduce the switching items of the sliding model control.Being equipped with second-order command filter and FLSs, a backstepping sliding mode control strategy for the human-robot system with uncertainties and disturbances is proposed in this paper.


## 3.2. Basic Assumptions and FLS

Some reasonable and useful assumptions are given at first which ensure the stability of the system.Assumption 1.
There exist constantsc1,c2>0, such that the inequality Δ1<c1, Δ2<c2 holds.Assumption 2 (Lipschitz).
For∀x1,x2∈Rn,∃L∈R+, the inequality holds as follows:(7)Fx1-Fx2≤Lx1-x2,where x1-x2 represents the 2-norm of (x1-x2). And all the desired commands xid, x˙id(i=1,2,3,4) satisfy the Lipschitz continuity.The design of the FLS consists of two steps. First, the fuzzy rule base should be made up as follows:Rl:
ifx1 is A1l and x2 is A2l and…and xn is Anlthenf^(x) is Bl,whereAil and Bl are fuzzy sets in R, l=1,2,…,N; i=1,2,…,n.The second step is defuzzification. Center average defuzzification operator is applied in this paper which can be expressed as(8)f^x=∑l=1Nf-l∏i=1nuAilxi∑l=1N∏i=1nuAilxi,where f-l=maxf∈RuBl(f), and uAil(xi), uBl are the membership functions.Define the fuzzy basis vector as(9)φx=∏i=1nuAilxi∑l=1N∏i=1nuAilxi.DenoteθT=[f-1,f-2,…,f-N] and φ(x)=[φ1(x),φ2(x),…,φn(x)]T; then the FLS can be expressed as(10)f^x=θTφx.The optimal parameterθ∗T can be defined by(11)θ∗T=argminθ∈Rnsupx∈Rnfx-θ∗Tφx.Lemma 3 (Wang [28]).
For∀f(x), which is continuous function and defined over a compact Ω, for any a constant ε∈R+, there exist an FLS and a parameter θ∗ such that (12)supx∈Ωfx-θ∗Tφx≤ε.


## 3.3. Controller Design

Consider the characteristics of the human-robot system, a backstepping sliding model control with second-order command filter is proposed in this paper. The controller design process is shown in this section.The output tracking errors and compensated tracking errors of the subsystems are defined, respectively, as(13)x~i=xi-xic,i=1,2,3,4.zi=x~i-ξi,i=1,2,3,4, where xi and xic represent the system states and filtered commands of the ith subsystem, respectively.Theξi(i=1,2,3,4) signals can be obtained by(14)ξ˙1=-k1ξ1+x2c-x2d+ξ2ξ˙2=-k2ξ2+Ih-1kcx3c-x3d+Ih-1kcξ3ξ˙3=-k3ξ3+x4c-x4d+ξ4ξ˙4=-k4ξ4+Ie-1τec-τedwith ξi(0)=0.Remark 4.
Equation (14) is used to achieve the filtering value ξi which are designed to compensate the errors caused by the command filters. Note that they can be computed with integrating processes to avoid the differential operations.For the purpose of eliminating the “explosion of complexity,” a second-order nonlinear command filter is designed to calculatexic,x˙ic. So the command filter is shown as follows:(15)r˙i,1r˙i,2=ri,2-2ζωnri,2-ωn2ri,1-xidxicx˙ic=ri,1ri,2,where ωn is the natural frequency of the filter and typically satisfies ωn>ki, i=(1,2,3,4) to ensure the tracking accuracy. ζ∈(0,1] is the damping ratio of the filter system. xid, (i=2,3,4) are the virtual control signals and x1d is the desired trajectory.The filter initial conditions arexic(0)=0, x˙ic(0)=0. Every command filter is designed to compute the filtered commands without differential operation. Furthermore, xic will track xid by choosing the suitable parameters.In order to find the optimal parameters of the FLSs, the adaptive laws are chosen forθ^1 and θ^2 as follows:(16)θ^˙1=γ1z2φ1θ^˙2=γ2z4φ2,where γ1,γ2 are the positive adaptive coefficients.Considering the compensating errors and the closed-loop dynamics, the virtual law can be defined as follows:(17)x2d=-k1x~1+x˙1c(18)x3d=kc-1Ih-k2x~2-z1+x˙2c-θ^1Tφ1-η1satsz2-τh+kc+khx1+bc+bhx2-bcx4(19)x4d=-k3x~3-kcIh-1z2+x˙3c(20)τed=Ie-k4x~4+x˙4-z3-θ^2Tφ2-η2satsz4-kcx1-bcx2+kc+kex3+bc+bex4,where θ^1, θ^2 are the estimated values of optimal parameters and φ1, φ2 are the basis vectors of FLSs. ki>0, (i=1,2,3,4) are control gains specified by the designer. η1, η2 are constants that ensure that the inequality η1>ε1, η2>ε2 holds.Thesats(x) is the switching function that satisfies(21)satsx=1,x>xΔkx,x≤xΔ-1,x<-xΔ.


## 3.4. Stability Analysis

Theorem 5.
For the system illustrated in (5), there exist a range of values for the gains k1,k2,k3,k4, and the adaptive coefficients γ1, γ2, such that the tracking error x~1 and compensation errors zi(i=1,2,3,4)can converge to zero with the compensations provided by FLSs.Proof.
The tracking errorx~i and the compensated tracking error zi are given first. (22)x~˙1=x˙1-x˙1c=x2-x2c+x2c-x2d+x2d-x˙1c=x~2+x2c-x2d+x2d-x˙1c=-k1x~1+x~2+x2c-x2dx~˙2=x˙2-x˙2c=Ih-1τh-kc+khx1-bc+bhx2+kcx3-x3c+x3c-x3d+x3dbcx4+Δ1-x˙2c=Ih-1τh-kc+khx1-bc+bhx2+kcx3d+kcx~3+bcx4+Δ1+Ih-1kcx3c-x3d-x˙2c=-k2x~2-z1+Ih-1kcx~3+Ih-1kcx3c-x3d+θ~1Tφ1+ε1-η1satsz2x~˙3=x˙3-x˙3c=x4-x4c+x4c-x4d+x4d-x˙3c=x~4+x4c-x4d+x4d-x˙3c=-k3x~3+x~4+x4c-x4d-kcIh-1z2x~˙4=x˙4-x˙3c=Ie-1τe-τec+τec-τed+τed+kcx1+bcx2-kc+kex3-bc+bex4+Δ2-x˙3c=Ie-1τ~e-τec-τed+τed+kcx1+bcx2-kc+kex3-bc+bex4+Δ2-x˙3c=-k4x~4-z3+Ie-1τ~e-τec-τed+θ~2Tφ2+ε2-η2satsz4.Due to the fact thatzi=x~i-ξi for i=1,2,3,4, the dynamics of compensated tracking error can be expressed as follows:(23)z˙1=x~˙1-ξ˙1=-k1x~1+x~2+x2c-x2d--k1ξ1+x2c-x2d+ξ2=-k1z1+z2z˙2=x~˙2-ξ˙2=-k2x~2-z1+Ih-1kcx~3+Ih-1kcx3c-x3d+θ~1Tφ+ε1-η1satsz2--k2ξ2+Ih-1kcx3c-x3d+Ih-1kcξ3=-k2z2-z1+Ih-1kcz3+θ~1Tφ+ε1-η1satsz2z˙3=x~˙3-ξ˙3=-k3x~3+x~4+x4c-x4d-kcIh-1z2--k3ξ3+x4c-x4d+ξ4=-k3z3+z4-kcIh-1z2z˙4=x~˙4-ξ˙4=-k4x~4-z3+Ie-1τ~e-τec-τed+θ~2Tφ2+ε2-η2satsz4--k4ξ4+Ie-1τec-τed=-k4z4-z3+θ~2Tφ2+ε2-η2satsz4+Ie-1τ~e,where θ~1=θ1∗-θ^1 and θ~2=θ2∗-θ^2.Remark 6.
Note the fact that the commandτec will directly output to control plant, so the conclusion is easily obtained that τe=τec.Then, define the control Lyapunov function for the closed-loop system as(24)Vt=12∑i=14zi2+γ1-1θ~1Tθ~1+γ2-1θ~2Tθ~2.Then the time derivative of the Lyapunov function (24) becomes(25)V˙t=∑i=14ziz˙i-γ1-1θ~1Tθ^˙1-γ2-1θ~2Tθ^˙2.Substitutez˙i with (23):(26)V˙t=z1-k1z1+z2+z2-k2z2-z1+Ih-1kcz3+θ~1Tφ+ε1-η1satsz2+z3-k3z3+z4-kcIh-1z2+z4-k4z4-z3+θ~2Tφ2+ε2-η2satsz4-γ1-1θ~1Tθ^˙1-γ2-1θ~2Tθ^˙2.The simplification can be written as(27)V˙t=-k1z12-k2z22-k3z32-k4z42+z2θ~1Tφ+ε1-η1satsz2+z4θ~2Tφ2+ε2-η2satsz4-γ1-1θ~1Tθ^˙1-γ2-1θ~2Tθ^˙2.Replaceθ^˙1, θ^˙2 with the adaptive law (16)(28)V˙t=-k1z12-k2z22-k3z32-k4z42+z2ε1-η1satsz2+z4ε2-η2satsz4≤-k1z12-k2z22-k3z32-k4z42+z2ε1-η1+z4ε2-η2.Letη1>ε1, η2>ε2; define the tracking error vector as z=[z1,z2,z3,z4]; then(29)V˙t≤-k1z12-k2z22-k3z32-k4z42≤-kminz2,where kmin=mink1,k2,k3,k4. According to (29), the error vector z is uniformly bounded, and limt→∞V(t)=V(∞). When integrating both sides of inequality (29), then(30)V∞-V0≤-kmin∫0∞z2dt.That is(31)kmin∫0∞z2dt≤V0-V∞.According to Barbalat Lemma proposed in [29], when the time variable t tends to infinity, the error vector z tends to zero.The block diagram of controller design is shown in Figure4.Figure 4
Block diagram of the controller design. (CF represents the command filter defined by (15); VCL represents the virtual control law of the four subsystems defined by (17)–(20); FLS represents the fuzzy logic system.)


## 4. Simulation Results

In this section, a simulation of an 1-DOF lower-limb exoskeleton is established. All the parameters of the 1-DOF exoskeleton are illustrated in Table2.Table 2
Parameters of the human-robot system.
PARM
Value
I
e
0.01178 kg m2
b
e
0.345 N m s/rad
k
e
0.339 N m/rad
I
h
3.38 kg m2
b
h
3.5 N m s/rad
k
h
54.7 N m/rad
b
c
9.47 N m s/rad
k
c
1905 N m/radThe PARM is short for parameter, and the notation is omitted for convenience through the paper as long as special notation is not required.Remark 7.
All the parameters are cited in [19]. The parameters, just for calculating, are obtained from real experiment and useful for controller simulation.The lump uncertainties are chosen as follows:(32)Δkckc=Δbcbc=Δkhkh=Δbhbh=0.1,d1=d2=0.2sin⁡t.To ensure the stability of the system, the specific parameters of the controller are tuned in a trial-and-error procedure and shown in Table3.Table 3
The parameters of the controller.
PARM
Value
k
1
50
k
2
50
k
3
50
k
4
50
η
1
0.05
η
2
0.05
γ
1
150
γ
2
30
w
n
500
ς
0.9The simulation results are given in Figures5–10. It can be seen that the controller designed can guarantee the uncertainties and external disturbances.Figure 5
Position tracking of human-robot system.Figure 6
Tracking errors of human-robot system.Figure 7
Control inputs of every subsystem.Figure 8
Control input of the human-robot system.Figure 9
Compensated tracking error.Figure 10
The estimate of the uncertainties and disturbances with the FLSs.Figures5 and 6 illustrate the position tracking and the tracking errors of the human-robot system. In Figure 5, the desired trajectory x1d is represented as solid line, the human leg position x1 is shown as dash-dotted line, and the position of exoskeleton x3 is described as dashed line. Just as it is shown, the controlled plant can steadily track the hip curve in a satisfactory way. Figure 6 shows the errors of the trajectory tracking. It is easy to get that the proposed command filtered fuzzy adaptive backstepping controller could make the errors finally kept in the neighborhood of the origin from the figure. However there are still differences between the two errors because of the filtered errors.Figures7 and 8 show the virtual control signals xid as solid line and the filtered commands xic, which are produced by passing the virtual control signals through the second-order filter, as the dashed line. As expected, the control signals of all the subsystems are smooth and bounded. Moreover, the virtual control signals and the corresponding filtered commands satisfy the same magnitude, rate, and bandwidth constrains, which is different from the traditional first-order linear filter. The closed-loop tracking errors could converge to a tiny range around the origin in a very short time (about 0.1 s) just as shown in Figure 9.Figure10 illustrates the system’s lump uncertainties and estimated values obtained by FLSs. The lump uncertainties Δ1,Δ2 are expressed as solid lines and the estimates Δe1,Δe2 are represented as dashed lines. It can be seen that the estimated value (dash) reaching the real value (solid) in less than 0.2 s. Therefrom, upper bounds of the switch terms in sliding model control could be much smaller because of the compensations provided by FLSs.All the signals remain bounded in a reasonable range during the process. Obviously, the proposed control strategy with command filters and FLSs can be a suitable method for lower-limb exoskeleton.


## 5. Conclusion

A human-robot cooperative control strategy based on a convincing high-order model is proposed for a lower-limb assisting exoskeleton. A second-order command filter backstepping method is employed to determine the time derivatives of virtual control signals without differential operations. The FLSs are used to approximate the uncertainties and disturbances and compensate the system timely. In addition, the stability of the system is proved based on the Lyapunov theory. Finally, simulation results are presented to verify the effectiveness of the proposed command filter adaptive fuzzy control strategy.Future work will focus on the performance of the control strategy in actual experiment and the filtering errors should be proved to converge rigorously. Besides, further research on the state constrain control for high-order nonlinear systems [30] is also needed.



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