Abstract
The lower extremity exoskeleton can enhance the ability of human limbs, which has been used in many fields. It is difficult to develop a precise force tracking control approach for the exoskeleton because of the dynamics model uncertainty, external disturbances, and unknown human–robot interactive force lied in the system. In this paper, a control method based on a novel recurrent neural network, namely zeroing neural network (ZNN), is proposed to obtain the accurate force tracking. In the framework of ZNN, an adaptive RBF neural network (ARBFNN) is employed to deal with the system uncertainty, and a fixed-time convergence disturbance observer is designed to estimate the external disturbance of the exoskeleton electrohydraulic system. The Lyapunov stability method is utilized to prove the convergence of all the closed-loop signals and the force tracking is guaranteed. The proposed control scheme’s (ARBFNN-FDO-ZNN) force tracking performances are presented and contrasted with the exponential reaching law-based sliding mode controller (ERL-SMC). The proposed scheme is superior to ERL-SMC with fast convergence speed and lower tracking error peak. Finally, experimental tests are conducted to verify the efficacy of the proposed controller for solving accurate force tracking control issues.
1. Introduction
When natural disasters such as earthquakes occur in remote mountainous areas, traffic is interrupted, which forces rescuers to enter the disaster area on foot [1]. The rescuers were unable to carry more relief supplies due to physical limitations. By combining the endurance of mechanical device and the intelligence of control system, the lower extremity exoskeleton makes up for human insufficiency in endurance, providing an effective way to accomplish the rescue task [2]. Therefore, the research on lower extremity exoskeleton has scientific significance and great social application requirement in the rescue [3] as well as rehabilitation [4, 5], aid to the disabled [6].
Under load-bearing conditions, each joint of the human lower limbs needs timely active assistance. A lower extremity exoskeleton is presented in Figure 1, where the exoskeleton has seven degree of freedoms (DOFs) for each leg. The exoskeleton directly connects the actuator in parallel to the corresponding joints of the lower limbs. Then, the lower extremity exoskeleton exerts force on the lower limbs through the connection point to achieve active assistance. Timely assistance to active joints is essential to reduce energy consumption of the pilot. As the control of one joint can be easily extended to that of the whole exoskeleton system, for simplicity, only the control problem of a knee joint is considered in this paper.
In rescue scenarios, the payloads carried by the operators are usually quite heavy that high-power supplies are expected. Hydraulic fluid power systems are well-known for their high-power density, many exoskeletons in this direction have been developed, for instance we can refer to the study by Chen et al. [7]. Among them, hydraulic actuators are usually used due to their large values of power/mass ratio [8–10]. However, the dynamics of hydraulic actuators is characteristic with high nonlinearity and time variation, which poses challenges to its corresponding controller design.
Force control is widely used in power augmentation exoskeleton robot due to the fact that its implicitly guarantees a safe and smooth operation for human–robot interaction [11]. Zhang et al. [12] proposed a hierarchical Lyapunov-based cascade adaptive control scheme for lower limb exoskeleton to follow the hydraulic force reference. In view of both cognitive and physical human–robot interaction forces, Lang et al. [13] exhibited a unified framework for scale force control of human-bearing augmentation exoskeleton. Cheng et al. [14] proposed a novel robust adaptive sliding mode control strategy of an electrohydraulic force loading system with consideration of external disturbances and parameter uncertainties. Chen et al. [15] presented an adaptive robust force control algorithm for a lower limb hydraulic exoskeleton to handle the problem of passive joints and holonomic constraints. Song et al. [16] implemented force tracking control of electrohydraulic servo system based on sliding mode control.
However, the force tracking control method mentioned above cannot fully cope with the time-varying challenges of the system. Zhang et al. [17, 18] have formally proposed a novel framework for solving time-varying nonlinear problems, namely zeroing neural network (ZNN). Much related works has been reported in the recent years. Li et al. [19] designed a ZNN model with predefined-time convergence to solve inequality constrained time-invariant quadratic problems. To solve the time-varying super determination problem, Zhang et al. [20] proposed a new variable–parameter convergence differential neural network, which can obtain the least squares solution. An error redefined neural network is proposed in [21] to control the mobile redundant manipulator to perform the tracking task with the redefined error monitoring function taken into account [20]. Zheng et al. [22] and Dai et al. [23] proposed a new controller design based on adaptive multilayer neurodynamics in the ZNN framework, implemented the time-varying trajectory tracking task with external interference and model uncertainty. Besides the predefined-time convergence ZNN [19], and intelligent fuzzy robustness ZNN [24], are also studied to solve the time-varying questions.
In addition, the presence of model uncertainties and the unknown external interference has an impact on the stability of the control system. Neural network methods are wildly utilized to deal with uncertainties in recent years [25]. RBFNN method was proposed to handle the problems of the unknown dynamic model of coordinated dual arms robot and the saturation nonlinearity of the motor [26]. The uncertainties of the robot was approximated by RBFNN [27]. As for the external disturbance, the disturbance observer technology is usually employed in the system for its clear physical significance and simplicity in the engineering implementation [28]. Song et al. [29] proposed a new adaptive neural network control method based on disturbance observer for hydraulic knee exoskeleton. A disturbance observer is designed and integrated into the controller to compensate for external perturbations and equivalent interaction forces acting on the piston rod of the hydraulic actuator.
Based on the analysis above, this paper proposes a new ZNN framework for the exoskeleton with an adaptive RBFNN (ARBFNN) and the fixed-time disturbance observer (FDO) employed to enhance the robustness and force tracking performance of the system. ARBFNN is used to compensate the uncertainty in the model, and the FDO is utilized to estimate the perturbation as well the ARBFNN estimation error.
The rest of this paper is organized into five sections. Section 2 presents the mathematical model of the servo system and the implicit zeroing dynamic of the system. Section 3 designs the control law with disturbance considered, in the meanwhile, the stability and other properties are also analyzed. Section 4 illustrates the simulation results of the proposed method. Section 5 shows the corresponding experimental results. Section 6 concludes the entire paper.
The main contributions of this paper are concluded as followed:(1)ARBFNN-FDO-ZNN control scheme is proposed for solving exoskeleton force tracking control issues, which builds the bridge among ZNN, observer, and RBFNN.(2)The problems of model uncertainties, external disturbances, and human–robot interactive force are all considered to enhance the robustness of the system.(3)The simulation results are carried out for the control scheme and compared with an exponential reaching law-based sliding mode control (ERL-SMC) to track the desired load force to perform its superiority.
2. Description of the Model
2.1. Mathematical Model
The electrohydraulic system shown in Figure 2 is composed of an oil pimp, a relief valve, a spool valve, a cylinder, an accumulator, etc. The oil pump provides the required high-pressure oil. The accumulator is designed to restore energy during high pressure and supplements the necessary oil to the hydraulic loop during low pressure. The relief valve is introduced to keep the oil pressure stable. A brief introduction to the principle of the hydraulic system is in the following. The piston is pushed by the high-pressure oil from the oil pump. The spool valve position controlled by voltage decides the piston in the cylinder heading left or right. The cylinder displacement reacts accurately as the spool valve displacement varies. In this way, the controller is designed to control the electric current u to gain our ends.
The control objective of the hydraulic exoskeleton is to regulate the voltage of the valve u so that the angle of the exoskeleton joint can track the pilot’s trajectory as closely as possible. In the servo system, the voltage u is the control input and the opening degree of the valve is based on it. And the load force is the output. The control voltage is adjusted according to the error between the expected value and the actual value, so that the load force reaches the desired value. Since it is not clear the change in oil temperature and the specific situation of pollution, some simplification is essential. It is assumed that the lower extremity exoskeleton can reach the corresponding angle when the force reaches the desired value. Besides, valves and fluids are considered ideal.
In the following, the model of the hydraulic system is introduced as follows [30]:where
The nomenclature appearing in Formula (1) is listed in Nomenclature Section.
2.2. Force Tracking Dynamic
In this paper, we explore force tracking control. Therefore, the load force is the subsystem. Let x be , and a subsystem Formula (2) is defined as follows:
Model (2) can be transformed into the following forms:where A is selected such that A is Hurwitz stable and the pair is observable.
An error function is given as . The evolution of the error function is defined as , where is the design parameter. Thus, a conventional ZNN model is completed.
3. Controller Design
However, the force tracking model is ideal. Actually, disturbance and errors often lie in the model implementation, such as differentiation error. Thus, the robustness of this subsystem should be taken into account.
In the noise-free case, the conventional ZNN converges to the desired result globally and exponentially (Theorem 1). Therefore, the polluted design formula of ZNN is explored in this paper. Besides, Figure 2 indicates the piston displacement is crucial to the whole hydraulic system and easily affected by the external factors. In this section, the control input u is designed with polluted ZNN design formula and the differentiation error of considered. Besides, the system uncertainty and external disturbances are inevitable in practical. The structure of force tracking system is shown in Figure 3.
3.1. Polluted ZNN Controller Design
u is implicited in Formula (4). For convenience, we perform the explicit conversion. The mathematical model introduced can be regarded as a one-dimension matrix. The selection of different variables has a direct effect on the derived control law. We define , , and . Then, the system Formula (1) is transformed into the following form with model uncertainties, external disturbances, and human–robot interaction force considered [5] as follows:where denotes the corresponding differentiation error of , , and denote the system uncertainty; and denote the unknown human–robot interactive force and the external disturbance, respectively.
Remark 1. During the movement, human–robot interaction force is inevitable in the system. Mathematically, the interactive force can be depicted as follows:where parameters and are constant values that amplify the differences between machine and human, is the desired force.
Assumption 1. For system Formula (5), there are bounded positive constants , , , and that satisfy the inequality , , , and .
First step: according to the design process of ZNN, the first Zhang function (ZF) is constructed and applied as follows:where denotes the desired force .where the design parameter scales the convergence rate and represents the bounded noise with each entry . Different values of puts various influences on the tracking performance as well .
Within the hydraulic system that can afford, the higher is set, the better the tracking performance is exhibited. is an activation function. In theory, an arbitrary odd monotonically increasing activation functions can be used for the ZF construction. Zhang et al. have discussed the different activation functions. Considering the simplicity of the controller, the linear function is selected. Then we can obtain
The design Formula (8) is applied, and we get
Since , we have
Second step: in general, and does not always equal zero when t = 0. Besides, no explicit expression of u is contained in Formula (11), the further consideration of the controller design is necessary. Then the second ZF is constructed as follows:
ZNN design Formula (8) is applied again. Then, we get
Then,
Substituting it into the equation above, the control law for the hydraulic system Formula (5) can be depicted belowwhere can be regarded as the internal disturbance in the system, is considered as the lumped external disturbance.
Theorem 1. (Global and exponential convergence without noise) Given the necessary parameters, if linear activation and design formula are used, the solution error converges to zero globally and exponentially at any randomly generated initial state .
Proof. ZNN design formula is as . Then, a Lyapunov function candidate for the system is defined as follows:
It is obvious that the Lyapunov function is positive-definite, because for any and . Besides, exists only for . Next, its time derivative can be obtained below. Since the linear activation function is utilized, we have,which guarantees the negative-definite of . The equation sign makes sense if and only if the condition that is satisfied. Based on Lyapunov theory, globally converges to zero. In view of this, the proof of global convergence is thus completed.
Next, we verify the exponential convergence. In consideration of ZNN error function, its analytic solution can be obtained as follows:
Evidently, with , the tracking error converges to zero exponentially with convergence rate , and the proof is done. It is worth pointing out that, it follows from Formula (18) that the larger is used, the faster convergence rate is obtained.
Theorem 2. (The error bound with noise) The polluted ZNN design Formula (7) is given. Whether the constant or time-varying noise, the absolute steady-state error of the system satisfies the following equality:
Proof. A Lyapunov function candidate is considered for the system Formula (5). Taking the derivation of it, we can obtain
where is the symbolic function. Besides, the following inequality yields
According to the results above, Formula (20) falls into the following two situations.(1)If , then , indicating the value of decrease over time. Such that with . Therefore, will return to its upper bound .(2)If , then the sign of is uncertain. If , the tracking error decreases or remains unchanged. Even though , the tracking error will increase with time. The tracking error does not exceed its upper bound . Once the tracking error exceeds the upper bound, it falls into the first situation. Then we can readily obtain .
The proof is thus completed.
3.2. Adaptive RBFNN and Fixed-Time Convergence Disturbance Observer Design
Clearly, the control law cannot achieve for the disturbance in the system.
First, the internal disturbance is approximated. It has been noted that a three-layer RBFNN with only one single-hidden layer is sufficient to approximate any degree of nonlinear system [31]. Then, RBFNN is introduced for estimating the unknown part of u. is bounded as well the system uncertainty and it can be substituted by the corresponding upper bound, which is conservative. Therefore, RBFNN is utilized to approximate g to enhance the performance of the controller.
Lemma 1. RBFNN can be utilized to approximate any continuous function arbitrarily [31].where is the weight matrix, is the input vector, is the radius basis function, is the approximation error, j is the number of neuron nodes. The ideal weight matrix is defined as follows:where is the effective estimation domain, M is the design parameter, and is the feasible region for states vector, respectively. is defined as follows:where c is the coordinate vector of the basis function center point, and b is the width of the basis function, .
According to Lemma 1, RBFNN is utilized to estimate the unknown function g as follows:where , are the perfect output weights vector and the input of RBFNN, respectively, is the neural network functional approximation error that is bounded, , is a small positive constant and in principle can be arbitrarily small by adjusting RBFNN, is RBFNN activation function. The output can be approximated bywhere is the actual weight matrix.
Next, we design the disturbance observer and the adaptive law for RBFNN weight matrix. The structure of ARBFNN is given in Figure 4.
The approximation error of RBFNN and the external disturbance consist of the compound disturbance. We let . Considering Assumption 1 and Lemma 1, the compound disturbance is bounded, that is to say, there is a positive constant which satisfies the inequality . Because D is unknown, the disturbance observer is designed to approximate this term.
First, define a variable where z is the auxiliary variable. And the derivate of z is shown as follows:where is a positive constant, L is the gain parameter. Based on Formula (27) and Formula (5), we have,where . The specific form of disturbance observer is given as follows:
The control law for u is given as follows:
Further, the observer error is defined as follows:
Lemma 2. If a continuous radial bounded function satisfies [32]:(1)(2)For any satisfying the inequalitythen the system is globally stable and converges in fixed time, where , , , and . The convergence time T satisfies the inequality as follows:where is a constant with
Theorem 3. (Fixed-time stable disturbance observer) If the disturbance observer Formula (29) is adopted, the compound disturbance D could be estimated in fixed time T, and the disturbance error is 0.
Proof. From Formula (31), it is obtained that if converges to zero, its corresponding derivative converges to 0 as well .
A Lyapunov candidate is chosen as follows:
The derivate of is as follows:
The adaptive law for RBFNN is given as follows:where is the design parameter. Such that,
For the analysis, r is taken as 1/3 in the paper. Notice, , and . Then, Formula (37) is rewritten as follows:
Clearly, . Besides, for the term , we have the following analysis.
We construct a continuous function .
Based on Lagrange mean value theorem, there exists a positive constant satisfying as follows:where .
Then, can be expressed in the following form:
Formula (38) is reformulated as follows:where . From Lemma 2, we can obtain the following inequality:where , respectively.
The proof is completed.
Theorem 4. (Robustness analysis) Considering the system Formula (5) with uncertainties and external disturbances, if RBFNN Formula (26), its adaptive law Formula (36), disturbance observer Formula (29) are employed, the system tracking error , are fixed time stable.
Proof. A Lyapunov function candidate is selected as follows:
The derivative of is described by
We construct a continuous function
It is common to find that is a monotonically decreasing function. Since , we have,where .
Then, Formula (45) is converted to
Therefore, the tracking error , are fixed time stable. From Lemma 2, the convergence time T1 satisfies as follows:where , respectively.
4. Simulation Studies
The simulation is presented in this section to demonstrate the effectiveness and superiority of the proposed control scheme. It has been discussed in Section 2 that the desired force is determined by the angle joint and the actuator displacement. There is a functional relationship between joint angle and load force . Considering the simplicity of the system, the desired load force is chosen as the reference quantity.
To avoid adverse effects caused by simulation parameters, the main parameters are selected as shown in Table 1. The considered disturbances in this paper are shown in Table 2, including four different common disturbances [33]. In this work, the ERL-SMC controller is used as a contrast control strategy. The corresponding controller without considering disturbances is given as follows:
where s is the sliding function, , , and k are positive constants, respectively. Similar to , k is used to adjust the convergence rate.
The simulation is repeated to observe the effects of the controller with various load forces and frequencies. We set the desired force as and , respectively. The external disturbance and interactive force related terms are given as and . The control law (30), ARBFNN (26), its adaptive law (36), and FDO (29) are employed. ZNN design parameter is set to be 100, 200, and 300, respectively. The structure of ARBFNN is 4-7-1. ARBFNN parameters are given as , , and . The disturbance observer parameters L, r are set to be 1 and 1/3, respectively. The initial condition is set to be . As for ERL-SMC controller, the corresponding parameters are selected as , , and . The sampling period is selected as 0.001 s. All simulation experiments are conducted on MATLAB/Simulink R2020b. The ode 3 solver is selected.
The blue dot dash line in Figure 5(a) shows that the proposed controller can track the desired force quickly. As presented in Figure 5(c) blue solid line, the maximum force tracking error for the proposed is 9.24 N. After 0.113 s, the tracking error reaches stable state. From Theorem 1, the maximum tracking error can be reduced by adjusting the design parameter . Then, is set to be 200 in Figure 6. Compared to Figure 5(c), the maximum tracking error of the proposed scheme reduces to 4.62 N in Figure 6(c), reduced by 50% and the convergence time drops from 0.113 to 0.47 s. Evidently, the increased leads to the diminishing peak value and convergence time.
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Furthermore, frequency can affect performance of the proposed controller. The blue dot dash line in Figures 6(c) and 7(c) indicates the increased frequency leads to a significant increase in maximum tracking error. Accordingly, the design parameter can be adjusted to enhance the control quality.
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To demonstrate the superiority of the ARBFNN-FDO-ZNN framework, we compare it with ERL-SMC in terms of maximum tracking error and convergence time. The key parameters and k are kept same for fair comparison. The comparative results are listed in Tables 3 and 4. Besides, partial enlargements (red solid line) in Figures 5(c)–8(c) illustrate the stable tracking error of ERL-SMC still oscillates with time. On the contrary, the tracking error of ARBFNN-FDO-ZNN is relatively smooth. There is no doubt the proposed control scheme performs its excellence. The evolution of the adaptation weights by ARBFNN are shown in Figure 9.
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5. Experimental Results
The effectiveness of the controller is further validated in experiments on the robot servo system. The block diagram is depicted in Figure 10. The proper hydraulic cylinder and servo valve are selected to meet the specific requirements. The computer disposition is Intel Core I7/2.8 G Hz, 8G memory, 250 G solid state disk with a 32-bit A/D converter, a 32-bit D/A converter, and an amplifier. The desired force signal is generated by the computer. The controller is adopted to control the servo system. The experimental results are demonstrated in Figures 11–14.
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Comparing Figure 11(c) with Figure 8(c), the maximum tracking error of experimental results is 1.38 N higher than that of simulation results, a increase of 14.9%. Considering the given reference is , the maximum tracking error of the algorithm is within the allowable range of engineering applications. To observe the influence of the design parameter, we increase the design parameters from 100 in Figure 11 to 200 in Figure 12. Correspondingly, it is found that the peak value is reduced from 10.62 in to 5.24 N.
Besides, we have investigated the effect of frequency variation on the control effect. All else being equal, the higher frequency has the negative impact on the experimental tracking performance (Figures 12 and 13), which is consistent with the simulation results. To resist the negative effects of frequency variations and enhance the control performance shown in Figure 13, we have increased the design parameter to 300. The result is presented in Figure 14. The maximum tracking error decreases from 25.54 to 16.82 N, a decrease of 34.14%.
Figures 11–14 demonstrate that the tracking error does not strictly converge to 0. First, the theoretical analysis indicates the tracking error converge to 0 with infinite time. Besides, the friction model established in this paper based on the existing friction theory is still different from the practical. Furthermore, due to the uncertainty and disturbance of the system parameters, although the nominal values have been given or estimated in the paper, the changes in the parameters are still unavoidable. Finally, some minor factors such as signal noise affect the performance of the system. Therefore, the actual tracking error and the theoretical differences, a normal phenomenon actually.
Generally, the experimental results indicate the effectiveness of the proposed framework.
6. Conclusions
In this paper, the ARBFNN-FDO-ZNN control scheme has been proposed for the accurate force tracking control of the exoskeleton. Primarily, the ARBFNN is introduced to solve the model uncertainty in the ZNN framework. Then, the FDO is adopted to handle the compound disturbance. The asymptotic stability of the coupled dynamics while applying the proposed control has been ensured using the Lyapunov theory. Compared with ERL-SMC, the simulation results present the superiority of the proposed scheme. Finally, the experimental results have been conducted to present the effectiveness of the scheme.
We incorporated ZNN with observer and RBFNN for the first time. From simulation and experimental results, it has been presented that the proposed approach is effective in fulfilling the force control objective in presence of model uncertainties, external disturbances, and human–robot interaction force. Future work will focus on the extension to the approach to switched control of the lower extremity exoskeleton.
Nomenclature
| : | Joint angle |
| : | Piston displacement |
| J: | System inertia |
| : | Mechanical time constant of the spool |
| M: | Moment arm |
| : | DC gain of the valve |
| : | Load force |
| u: | Control variable |
| : | Friction force |
| : | Areas of piston corresponding to A and B |
| m: | Mass |
| : | Pressure of chamber A and B |
| : | Mass center |
| : | Hydraulic internal and external leakage of the system |
| : | Spool displacement |
| : | Initial volume of cylinder |
| : | Viscous friction coefficient. |
Data Availability
The data used to support the findings of this study have not been made available because no new data were created or analyzed in this study.
Conflicts of Interest
The authors declare that they have no conflict of interest.
Acknowledgments
This work has been supported by the Basic Frontier Science and Technology Innovation Project with Army Engineering University KYGYJQZL2210.