Abstract
In this paper, an ultralocal modelbased prescribed performance assistasneeded control scheme (UPPAC) is presented for series elastic actuatorbased upper limb patientexoskeleton system (SULPES) subject to complex state constraints. The complex state constraints are introduced to adjust constraint on assistive torque change rate according to the assistive torque error, so as to avoid the rapid change of the assistive torque caused by overshoot and ensure the safety of the patient. In the outer impedance subcontrol loop, the desired assistive torque is calculated by an impedance controller. Then, to reduce dependence on precise model parameters, the inner torque subcontrol loop is proposed based on the ultralocal model and neural approximation. Combine integral barrier Lyapunov function with backstepping strategy, the prescribed constraints for tracking error and complex state constraints are satisfied. Finally, both theoretical proof and cosimulation illustrate the effectiveness of the proposed UPPAC.
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1 Introduction
For decades, with the deepening of population aging, stroke has been one of the most common diseases among elderly, which can lead to impaired movement ability. With the development of robotics technology, assisted rehabilitation training based on exoskeletons has become increasingly popular. Series elastic actuators (SEA) based exoskeletons have become the mainstream of rehabilitation exoskeletons because they incorporate elastic mechanisms, which can achieve torque control and effectively reduce the impact of mechanical shocks on patients [1]. There is a main obstacle for the control of series elastic actuatorbased upper limb patientexoskeleton system (SULPES), how to realise highaccuracy assistasneeded control in the presence of complex state constraints and challenging parameter identification.
In the early research of exoskeleton control algorithms, most of them used trajectory tracking control [2]. However, this method ignores the patient’s own movement ability. In order to cooperate with the patient’s own muscle strength and make the patient more actively participate in the rehabilitation task, assistasneeded controller is proposed, which can adjust the assistive torque according to the patient’s own movement ability [3]. To achieve assistasneeded control, it is necessary to enable the exoskeleton to follow the patient’s movements while providing assistance to the patient. Impedance control can adjust the relationship between interaction torque and position of exoskeletons, which is a very effective means to realise assistasneeded control [4]. In [5], the assistive torque is adjusted through the positiontracking error and interaction torque, and the exoskeleton can track the patient and provide the appropriate assistive torque by the impedance control. In [6], a task path is defined in the velocity domain, and assistive torque is calculated through the velocity field, then, an impedance controller for velocity tracking is proposed to achieve assistasneeded control. However, due to various mechanical limitations, output or state constraints are inevitable, the above methods [4,5,6] do not consider the existence of constraints.
While ensuring security at the hardware level through SEA, it is also necessary to ensure patient safety through control algorithms. In practical applications, when there is a large tracking error caused by factors such as overshoot, the controller will generate a large input signal, which will lead to an increase in the change rate of the assistive torque and cause damage to the patient. Therefore, it is necessary to constrain the change rate of the assistive torque, which can be regarded as state constraints. Barrier Lyapunov function (BLF) is an effective approach to address the constraint problems [7]. The most common constraint type is output constraint, logtype BLF is used to handle output constraints for exoskeleton [8]. In addition, tantype BLF [9] and cottype BLF [10] can also handle output constraints. For the fullstate constraints, BLF is used to make all signals in the closedloop system bounded in a finite time, which ensures that the deviation of the exoskeleton trajectory is within a bounded range [11]. Then, the integral BLF (iBLF) is also a good method to deal with fullstate constraints [12]. In comparison to the traditional BLF, iBLF reduces the steps of converting the output or state constraints into error constraints [13]. BLF can also be used to constrain errors [14]. In [15], tantype BLF is used to constrain the tracking errors of exoskeleton, so as to ensure that the joint angle is limited to a safe range. In addition, the concept of complex state constraints has been proposed [16]. Different from the traditional fullstate constraints, the complex state constraints are designed in a cascade manner, where the boundary of the state depends only on other states [17]. Although the above BLFs can effectively solve the constraint problem, these methods are designed based on dynamic model. However, in practical applications, parameter identification is often inaccurate, which will affect the accuracy of the controller.
To address the aforementioned challenges, the ultralocal model (ULM) is proposed through the concept of modelfree control [18]. The ULM is a simple linear model that approximates the original system and only requires input and output (I/O) data, thus avoiding the influence of inaccurate parameter identification. In ULM, the dynamic terms of the original system are transformed into an lumped term, which is typically estimated by timedelay estimator (TDE) [19]. Additionally, estimation methods such as disturbance observer [20] and neural networks [21] can be employed to estimate the lumped term. ULM can be combined with fractional order control [22], prescribedtime control [23], sliding mode control [24], and iterative learning control [25] to achieve more accurate control effect thanks to its simple structure. In addition, the establishment of ULM only requires I/O data so that it can be applied to a wide range of systems, including quadrotors [26], manipulators [27], exoskeletons [28], aerobic digestion process [29] and so on.
To address the problems mentioned above, an ULMbased prescribed performance assistasneeded controller (UPPAC) is proposed for SULPES under complex state constraints. In comparison to the previously mentioned works, this article incorporates the following improvements:

(1)
In order to increase compliance, make patients more comfortable, the state constraint is introduced to to limit the assistive torque change rate. The assistive torque is generated through SEA, so the state constraint on assistive torque change rate is achieved by constraining the motor angular velocity.

(2)
Motivated by the idea of complex state constraints [17], it is different from output constraints [8] or state constraints [11]. The complex constraint on motor angular velocity is adjusted based on the motor angle tracking error. When the tracking error is large, the constraint on motor angular velocity is reduced to prevent rapid changes in the assistive torque due to large control input signals caused by tracking errors.

(3)
Different from the BLFbased controllers mentioned above, the ULM is established to approximate the original system. Based on ULM, the iBLFbased prescribed performance controller is designed, which eliminates the dependence on precise model parameters. In addition, iBLF and error constraints are introduced, so that complex state constraints are satisfied and the control accuracy is further improved.
The remaining sections of this paper is organized as follows. The dynamic model of SULPES and the control objectives are presented in Sect. 2. Section 3 illustrates the design procedure of the UPPAC. Sections 4 and 5 are dedicated to simulation results and conclusions.
2 Dynamic modeling of SULPES and preliminaries
2.1 Dynamic modeling of SULPES
To realize assistasneeded control for SULPES under complex state constraints, dynamic modeling of the SULPES is necessary. The structure of the considered 2DOF SULPES is constructed in SolidWorks, as shown in Fig. 1.
From the energy analysis and using Hamilton’s principle [30], the dynamic equations of 2DOF SULPES can be modeled as follows:
where \({\mathbf{{\theta }}_m} = {[{\theta _{m,1}};{\theta _{m,2}}]}\) is the angle of the motor, \({\mathbf{{\theta }}_l} = {[{\theta _{l,1}};{\theta _{l,2}}]}\) is the angle of the patient’s joint, \({{{\tau }}_p} \in {\mathbb {R}}^2\) denotes the torque generated by patient, \(\mathbf{{\tau }}_s \in {\mathbb {R}}^2\) is the torque generated by SEA, \(\mathbf{{K}}_s\) and \(\mathbf{{B}}_s\) demonstrate the stiffness and damping of the SEA, \(\mathbf{{\tau }}_m\in {\mathbb {R}}^2\) denotes the torque of the motor, the matrix elements are given in our previous work [31].
2.2 Complex state constraints
In practical applications, when a large tracking error occurs due to factors such as overshoot, the controller will generate a large control signal, leading to rapid changes in the assistive torque. However, rapid changes in the assistive torque can cause harm to the patient. Therefore, to protect the patient’s safety, it is necessary to constrain the assistive torque change rate. The assistive torque is generated by SEA, so the constraint on motor angular velocity can be used to constrain the assistive torque change rate. The constraint on motor angular velocity can be adjusted in realtime based on angle tracking error. When the tracking error increases, the constraint on angular velocity decreases to limit its speed and prevent rapid changes in the assistive torque. As the tracking error decreases, the constraint on angular velocity will relax, which will not affect the normal assistance training.
The constrained region for the motor angular velocity of each joint with a complex boundary is described as
where \({\mu _{xl,i}}({z _{1,i}})\) and \({\mu _{xu,i}}({z _{l1,i}})\) denote the upper and lower bounds of \({{\dot{\theta }} _{m,i}}\) and are smooth functions with respect to \({\theta _{m,i}}\), \({z _{1,i}}={\theta _{m,i}}{\theta _{r,i}}\) is the tracking error of the motor angle, \({\theta _{r,i}}\) denotes the reference tracking trajectory of the motor, which will be provided later, \(i=1,2\).
Remark 1
For the selection of complex constraint boundaries, the first thing to consider is the assistive torque change rate required to complete the rehabilitation task normally. The maximum value of the boundary should not be too small to prevent affecting the patient’s normal rehabilitation task. In addition, the setting of the maximum value of the boundary should also consider the limitation in the actuator’s speed or torque capabilities to avoid excessive mechanical damage.
Remark 2
If only a simple saturation function is used to limit the maximum assistive torque, then when significant errors occur due to overshoot and other reasons, the assistive torque will instantly increase, causing impact and damage to the patient’s upper limbs. By using complex state constraints to limit the assistive torque change rate, the assistive torque can slowly increase when the above situation occurs. This not only ensures the safety of the patient but also enhances compliance, thereby improving the patient’s comfort during rehabilitation training.
Concerning the SULPES under complex state constraints, an UPPAC algorithm is developed via iBLF and the backstepping strategy to achieve the following two control objectives: (1) calculate the desired assistive torque for the patient through the angle and angular velocity of the patient’s upper limb, (2) control the assistive torque generated by the exoskeleton enabling the trajectory of patient’s upper limb \({\textbf {P}}_p=(\mathbf{{J}}^T(\theta _{l}))^{1}{\theta }_l\) to follow the desired rehabilitation trajectory \({\textbf {P}}_d\).
3 UPPAC design and stability analysis
For SULPES under complex state constraints, an ultralocal modelbased prescribed performance assistasneeded controller (UPPAC) is proposed to accomplish the control objectives outlined in the previous section. The proposed UPPAC has a dualloop structure as shown in Fig. 2. The outer impedance subcontrol loop will be designed in Sect. 3.1, then, the inner torque subcontrol loop will be proposed based on the admittance control and ultralocal model in Sect. 3.2, and the stability analysis will be given in Sect. 3.3.
3.1 Outer impedance subcontrol loop design
The torque of the upper limb of the patient is generally given as follows [32]
where \({\theta }_d=[{\theta }_{d,1};{\theta }_{d,2}]=\mathbf{{J}}^T(\theta _{l}){\textbf {P}}_d\) is desired trajectory of patient, \(\mathbf{{J}}^T(\theta _{l}) \in {\mathbb {R}}^{2 \times 2}\) is the Jacobian matrix, \(\mathbf{{K}}_p\) and \(\mathbf{{B}}_p\) are the stiffness and damping parameters of patient, \(k_p \in (0,1]\) represents the movement ability of patient, the smaller the \(k_p\) is, the muscle of patient is weak, and when \(k_p = 1\), the patient is healthy and can exercise freely.
Due to the insufficient muscle movement capability of the patient, the exoskeleton is required to provide the assistive torque for the patient to complete rehabilitation tasks. Inspired by [33], the desired assistive torque \({\tau }_r\) can be obtained by impedance control:
where \({{B}}_a\) and \({{K}}_a\) are the damping and stiffness parameters.
In this way, based on the patient’s desired trajectory \({\theta }_{d}\) and actual trajectory \({\theta }_{l}\), the desired assistive torque is obtained through impedance control algorithm, and how to track the desired assistive torque \({\tau _{r}}\) will be given in the next subsection.
3.2 Inner torque subcontrol loop design
To provide the desired assistive torque to the patient while ensuring safety, first, convert the desired assistive torque into the motor’s reference trajectory through admittance control. Then, develop a modelbased performance control (ULMPPC) based on ULM using iBLF and backstepping strategies to achieve tracking of the motor’s reference trajectory, enabling SEA to generate the desired assistive torque.
3.2.1 Admittance controller design
First of all, it is necessary to obtain the reference trajectory of the motor through admittance control algorithm.
The exoskeleton exerts assistive torque on the patient through SEA, and the torque generated by SEA is shown in equation (3).
Using the concept of admittance control, a reference trajectory \({\theta }_r\) for the motor is derived as follows:
with \({\Delta \theta }=[{\Delta \theta _{1}};{\Delta \theta _{2}}]\) and \({\tau _{r}}=[{\tau _{r,1}};{\tau _{r,2}}]\).
3.2.2 Ultralocal model design
Controllers designed by backstepping methods require accurate model parameters. However, parameter identification is often inaccurate, which can affect the precision of the controller. To address this issue, ULM is developed to approximate the system (3) [18, 34]:
where \(\mathbf{{u}}=\tau _m\), \(\mathbf{{ y}}=\theta _{m}\), \(\mathbf{{F}}\) indicates an unknown lumped term which encompasses all the dynamics characteristics, \(\alpha \) expresses a constant input gain.
Since \({\textbf {F}}\) is an unknown term in ULM, the timedelay estimator (TDE) is applied to estimate it [35]:
where \(\hat{\textbf{F}}\) is the estimation of \(\mathbf{{F}}\), \(\Delta \) D represents a small time delay, usually taking one sampling period.
There is an assumption to be made for the ULM (9).
Assumption 1
Due to the existence of mechanical constraints, the angle, velocity, and acceleration of the exoskeleton and the torque of the motor are all bounded. The unknown term \(\mathbf{{\textit{F}}}\) contains all the properties of the dynamics model, so \(\mathbf{{\textit{F}}}\) is also bounded.
Remark 3
The purpose of designing the ULM (9) is not to replace the dynamic model (1)(3) entirely, but rather to get rid of dependence on precise model parameters.
3.2.3 ULMbased prescribed performance controller design
The tracking error of the exoskeleton \(\mathbf{{z}}_1 =[z_{1,1},z_{1,2}]^T\) can be rewritten as follows:
where \({\mathbf{{y}}_d}={\theta }_{r}\).
To achieve better control accuracy, the prescribed constraints for tracking errors are given as follows:
where \(\mu _{zl0,i}>\mu _{zlf,i}>0\), \(\mu _{zu0,i}>\mu _{zuf,i}>0\), \(k_{zl}>0\) and \(k_{zu}>0\) are constants, \(i=1,2\).
To track the reference trajectory \({\theta }_r\) and satisfy the complex state constraints (4) and prescribed constraints for tracking error (12), an ULMPPC will be designed via the iBLF and backstepping strategy. The controller design process will be divided into the following three steps. In Step 1, the iBLF will be introduced to keep the tracking error in the prescribed constraints (12). Then, in Step 2, another iBLF will be utilized to satisfy the complex constraints (4) and the ULMPPC will be given. Finally, in step 3, the adaptive laws of neural approximations will be designed, which are used to estimate nonlinear terms and the unknown term \(\textbf{F}\). Analyze the ithjoint for simplicity, \(i = 1, 2\).
Step 1: In order to satisfy the inequality (12), consider an iBLF \({V_{b1,i}}\) on the region \(\Omega _1= \{ z_{1,i}: \mu _{zl,i}(t)< z_{1,i} < \mu _{zu,i}(t) \}\):
Differentiating the iBLF (15), one has:
Considering the first term of (16), it is obtained that:
where \(z_{2,i}=\dot{y}_{i}a_i\), \(a_i\) is a virtual control law, which will be provided later.
Considering the second term of equation (16), one can get that:
with \({\psi _{zl,i}} = \frac{1}{{{z_{1,i}}}}\left( (2{\mu _{zl,i}} + {\mu _{zu,i}})\ln \left( {\frac{{{\mu _{zl,i}}}}{{{\mu _{zl,i}} + {z_{1,i}}}}} \right) + {\mu _{zu,i}}\ln \left( {\frac{{{\mu _{zu,i}}}}{{{\mu _{zu,i}}  {z_{1,i}}}}} \right) \right) \), and \(\mathop {\lim }\limits _{{z_{1,i}} \rightarrow 0 } {\psi _{zl,i}}= ({\mu _{zl,i}}+{\mu _{zu,i}})/{\mu _{zl,i}}\).
The third term of (16) can be deduced as follows:
with \({\psi _{zu,i}} = \frac{1}{{{z_{1,i}}}}\left( {\mu _{zl,i}}\ln \left( {\frac{{{\mu _{zl,i}}}}{{{\mu _{zl,i}} + {z_{1,i}}}}} \right) + ({\mu _{zl,i}} + 2{\mu _{zu,i}})\ln \left( {\frac{{{\mu _{zu,i}}}}{{{\mu _{zu,i}}  {z_{1,i}}}}} \right) \right) \), and \(\mathop {\lim }\limits _{{z_{1,i}} \rightarrow 0 } {\psi _{zu,i}}= ({\mu _{zl,i}}+{\mu _{zu,i}})/{\mu _{zu,i}}\).
Substituting equation (17), (18), and (19) into (16), one has:
Define a nonlinear term as:
with \({X_{1,i}} = {[{y_{i}},{y_d},{\mu _{zl,i}},{{\dot{\mu }} _{zl,i}},{\mu _{zu,i}},{{\dot{\mu }} _{zu,i}}]^T}\).
Based on the neural approximation, \({W_{1,i}}({X_{1,i}}) \) can be approximated by:
where \(\zeta _{1,i} \in {\mathbb {R}}^n\) is the weight vector, \({Q}({X_{1,i}}) = {[{q_{1}}({X_{1,i}}),{q_{2}}({X_{1,i}}),\ldots ,{q_{n}}({X_{1,i}})]^T}\) is the radial basis vector, \({q_{j}}({X_{1,i}})\) is the Gaussian function, \(j=1,2,\ldots ,n\), \(n>1\) denotes the node number, \({\epsilon _{1,i}}({X_{1,i}})\) expresses the approximation error satisfying \(\left {\epsilon _{1,i}}({X_{1,i}})\right \le {{\bar{\epsilon }} _{1,i}}\), \({{\bar{\epsilon }} _{1,i}}\) is a positive constant.
Then, taking equations (21) and (22) into account, the equation (20) yields:
The virtual control law \(a_i\) can be obtained as follows:
where \({c_{1,i}} > 0\) is a constant, \({b_{1,i}} = \frac{{{\mu _{zu,i}}  {z_{1,i}}}}{{{\mu _{zl,i}} + {\mu _{zu,i}}}}{{\dot{\mu }} _{zl,i}}  \frac{{{\mu _{zl,i}} + {z_{1,i}}}}{{{\mu _{zl,i}} + {\mu _{zu,i}}}}{{\dot{\mu }} _{zu,i}}\), and \({\hat{\zeta }} _{1,i}\) is the estimation of \(\zeta _{1,i}\).
Using the Young’s inequality, one can obtain that
Putting the virtual control law (24) and equation (23) into (23), one gets:
where \({\tilde{\zeta }} _{1,i}= {\hat{\zeta }} _{1,i}{\zeta }_{1,i}\) denotes the estimation error of \({\zeta }_{1,i}\).
Step 2: After satisfying the prescribed constraints (12) and designing the virtual control law (23), to satisfy the complex state constraints in (4), consider an iBLF \(V_{b2,i}\) on the region \(\Omega _2=\{\mu _{x,i}<\dot{y}_i<\mu _{x,i}\}\), one has:
Then, the above derivation of equation (27) can be obtained as follow:
Consider the ULM (9), the first term of equation (28) can be obtained:
Then, taking the second term of equation (28) into consideration, one has:
with \({\varphi _i} = \frac{{({\mu _{xl,i}} + {\mu _{xu,i}})}}{{{z_{2,i}}}}\ln \frac{{({\mu _{xl,i}} + {a_i})({\mu _{xu,i}}  {a_i}  {z_{2,i}})}}{{({\mu _{xu,i}}  {a_i})({\mu _{xl,i}} + {a_i} + {z_{2,i}})}}\), and \(\mathop {\lim }\limits _{{z_{1,i}} \rightarrow 0 } {\varphi _{i}}= ({\mu _{xl,i}}+{\mu _{xu,i}})^2/(({\mu _{xl,i}}+a_i)*({\mu _{xu,i}}+a_i))\).
Considering the third term of (28), it can be deduced that
with \({\psi _{xl,i}} = \frac{1}{{{z_{2,i}}}}\left( (2{\mu _{xl,i}} + {\mu _{xu,i}} + {a_i})\ln \left( {\frac{{{\mu _{xl,i}} + {a_i}}}{{{\mu _{xl,i}} + {a_i} + {z_{2,i}}}}} \right) + ({\mu _{xu,i}}  {a_i})\ln \left( {\frac{{{\mu _{xu,i}}  {a_i}}}{{{\mu _{xu,i}}  {a_i}  {z_{2,i}}}}} \right) \right) \), and \(\mathop {\lim }\limits _{{z_{1,i}} \rightarrow 0 } {\psi _{xl,i}}= ({\mu _{xl,i}}+{\mu _{xu,i}})/({\mu _{xl,i}}+a_i)\).
The fourth term of equation (28) can be deduced as follows:
with \({\psi _{xu,i}} = \frac{1}{{{z_{1,i}}}}\left( ({\mu _{xl,i}} + {a_i})\ln \left( {\frac{{{\mu _{xl,i}} + {a_i}}}{{{\mu _{xl,i}} + {a_i} + {z_{2,i}}}}} \right) + ({\mu _{xl,i}} + 2{\mu _{xu,i}}  {a_i})\ln \left( {\frac{{{\mu _{xu,i}}  {a_i}}}{{{\mu _{xu,i}}  {a_i}  {z_{2,i}}}}} \right) \right) \), and \(\mathop {\lim }\limits _{{z_{1,i}} \rightarrow 0 } {\psi _{xu,i}}= ({\mu _{xl,i}}+{\mu _{xu,i}})/({\mu _{xu,i}}a_i)\).
Then, substituting the obtained above equations (29), (30), (31), and (32) into equation (28), one obtains:
where \({\tilde{F}}_i=F_i{\hat{F}}_i\) is the estimation error of \(F_i\).
Define a nonlinear term as:
with \({X_{2,i}} = [{y_{i}},{\dot{y}_{i}},{y_d},{\dot{y}_d},{\ddot{y}_d},{\mu _{zl,i}},{{\dot{\mu }} _{zl,i}},{\ddot{\mu }_{zl,i}}, {\mu _{zu,i}},{{\dot{\mu }} _{zu,i}},{\ddot{\mu }_{zu,i}},{\mu _{xl,i}},{{\dot{\mu }} _{xl,i}},{\mu _{xu,i}},{{\dot{\mu }} _{xu,i}},\) \({\hat{\zeta }} _{1,i}^{},u(t  \Delta )]^T\).
Based on the neural approximation, \({W_{2,i}}({X_{2,i}})\) can be approximated by:
where \(\zeta _{2,i} \in {\mathbb {R}}^n\) is the weight vector, \({\epsilon _{2,i}}({X_{2,i}})\) is the approximation error satisfying \(\left {\epsilon _{2,i}}({X_{2,i}})\right \le {{\bar{\epsilon }} _{2,i}}\), \({{\bar{\epsilon }} _{2,i}}\) is a positive constant.
Using the Young’s inequality, one can deduce that
Then, equation (33) can be rewritten as follows:
The ULMPPC can be designed as follows:
where \({c_{2,i}} > 0\), \({b_{2,i}} = \frac{{{\mu _{xu,i}}  {\dot{y}_{i}}}}{{{\mu _{xl,i}} + {\mu _{xu,i}}}}{{\dot{\mu }} _{xl,i}}  \frac{{{\mu _{xl,i}} + {\dot{y}_{i}}}}{{{\mu _{xl,i}} + {\mu _{xu,i}}}}{{\dot{\mu }} _{xu,i}}\), \(\hat{\zeta }_{2,i}\) is the estimation of \(\zeta _{2,i}\), which will be designed later.
Then, substituting the control law (38) into (37), it yields:
with \({\tilde{\zeta }} _{2,i}= {\hat{\zeta }} _{2,i} \zeta _{2,i}\).
Step 3: After satisfying complex state constraints (4) and designing ULMPPC (38), to obtain the adaptive laws of \(\hat{\zeta }_{1,i}\) and \(\hat{\zeta }_{2,i}\), the following Lyapunov function is considered:
where \({\lambda _1}\) and \({\lambda _2}\) are positive constants.
Then, the derivation of the Lyapunov function (40) yields:
The adaptive laws of \({\hat{\zeta }}_{1,i}\) and \({\hat{\zeta }}_{2,i}\) can be established as follows:
where \({\varsigma _1}\) and \({\varsigma _2}\) are positive constants.
Based on the adaptive laws (42) and (43), the equation (41) yields:
Using the Young’s inequation again, it holds that:
Inequation (44) can be rewritten as follows:
In summary, the proposed UPPAC can be composed based on an impedance controller (6), admittance controller (7)(8), ULMPPC (38), intermediate virtual control law (24), and neural approximation adaptive laws (42)(43).
Remark 4
For the inner torque subcontrol loop, controller design is conducted through backstepping. First of all, torque control is transformed into position control via admittance controller. Afterwards, considering that the control effect of the controller designed by the backstepping method will be affected by the accuracy of the model parameters, ULM is established to approximate the original system. Additionally, to achieve better tracking results, prescribed constraints for tracking errors have been introduced. Due to the consideration of both tracking error constraints and complex state constraints in this control method, iBLF and backstepping methods are used for controller design, where iBLF \(V_{b1,i}\) and \(V_{b2,i}\) can make the system satisfy error constraints and complex state constraints, respectively. Finally, neural approximations are employed to estimate the nonlinear terms in the controller design process and the unknown lumped terms in the ULM.
As for ULM, the controller design process is based on it. Because it can decouple the original system into two simple singleinput singleoutput subsystems, its role not only helps the controller reduce dependence on precise model parameters but also facilitates controller design by replacing the original system, thus reducing the complexity of the controller design process.
3.3 Stability analysis
In this part, the stability of the closedloop SULPES (1)(3) under complex state constraints with the proposed UPPAC will be analyzed. Firstly, several lemmas, which can be used in the stability analysis, will be given. Secondly, a theorem of the proposed UPPAC is introduced. Finally, the proof of the theorem is given, and the stability of the closedloop SULPES with UPPAC is analyzed.
The lemma given below is indispensable in stability analysis.
Lemma 1
[36]. For the following iBLFs
if \(\mu _{al}(t)<a<\mu _{au}(t)\) and \(\mu _{dl}< d<\mu _{du}\), \(d=b+c\), \(\forall t \ge 0\), \(V_{1}\) and \(V_{2}\) satisfy the inequalities
Lemma 2
[37]. For a positive and continuous function V(a) satisfying \(b_1(\left a\right )\le V(a) \le b_2(\left a\right )\), where \(b_1\) and \(b_2\) are nonlinear functions, if exist constants \(c_1>0\) and \(c_2>0\), one can obtain that
Then, the solution a(t) is uniformly bounded.
The stability theorem of the closedloop SULPES (1)(3) under complex state constraints with the proposed UPPAC is given as follows:
Theorem 1
Considering the SULPES (1)(3) under complex state constraints (4),applying the proposed UPPAC, for the initial value satisfying \(\mu _{zl,i}(0)<z_{l,i}(0)< \mu _{zu,i}(0)\) and \( \mu _{xl,i}<\dot{\theta }_{m,i}(0) < \mu _{xu,i}\), the parameters satisfying \({c_{1,i}}\), \({c_{2,i}}\), \(\lambda _1\), \(\lambda _2\), \(\varsigma _1\) and \(\varsigma _2\) are positive constants, \({b_{1,i}} = \frac{{{\mu _{zu,i}}  {z_{1,i}}}}{{{\mu _{zl,i}} + {\mu _{zu,i}}}}{{\dot{\mu }} _{zl,i}}  \frac{{{\mu _{zl,i}} + {z_{1,i}}}}{{{\mu _{zl,i}} + {\mu _{zu,i}}}}{{\dot{\mu }} _{zu,i}}\), \({b_{2,i}} = \frac{{{\mu _{xu,i}}  {\dot{y}_{i}}}}{{{\mu _{xl,i}} + {\mu _{xu,i}}}}{{\dot{\mu }} _{xl,i}}  \frac{{{\mu _{xl,i}} + {\dot{y}_{i}}}}{{{\mu _{xl,i}} + {\mu _{xu,i}}}}{{\dot{\mu }} _{xu,i}}\), then the following properties can be guaranteed that:
(1) The tracking error \({z_{1,i}}\) and the estimation errors \({\tilde{\zeta }_{1,i}}\), \({\tilde{\zeta }_{2,i}}\) are semiglobally uniformly bounded;
(2) The motor angular velocity \(\dot{\theta }_{m,i}\) satisfy the comlex state constraint (4), and the tracking error \({z_{1,i}}\) satisfies the prescribe constraints (12);
(3) The patient’s upper limb can track the desired rehabilitation trajectory \({\textbf {P}}_d\).
Proof
The proof process is mainly divided into the following three parts. The first part will prove the convergence of tracking error \(z_{1,i}\) and estimation errors \({\tilde{\zeta }_{1,i}}\), \({\tilde{\zeta }_{2,i}}\). Then, the second part will prove that (4) and (12) hold. Finally, the third part will prove the convergence of trajectory tracking error of the patient’s upper limb.
(1) Consider Lemma 1, the Lyapunov function (40) can be rewritten as follows:
Bearing in mind the inequation (45), the above inequality then can be derived that:
with \({s_0} = \min \{ {c_1},{c_2},{\varsigma _1},{\varsigma _2}\}\), and \({S_i} = \frac{{{\varsigma _1}{{\left\ {{\zeta _{1,i}}} \right\ }^2}}}{2} + \frac{{{\varsigma _2}{{\left\ {{\zeta _{2,i}}} \right\ }^2}}}{2} + \frac{1}{2}{{\bar{\varepsilon }}} _{1,i}^2 + \frac{1}{2}{{\bar{\varepsilon }}} _{2,i}^2\).
Multiplying \(e^{s_0 t}\) on both sides of (47), and integrating it over [0, t], one can obtain that:
Considering equation (48), Lemma 2, and the Lyapunov stability theorem in [37], one can obtain that \({z_{1,i}}\), \({\tilde{\zeta }_{1,i}}\), and \({\tilde{\zeta }_{2,i}}\) are semiglobally uniformly bounded.
(2) After the convergence proof of the tracking error and estimation errors, it will be proved that the proposed UPPAC can satisfy the complex state constraints (4) and prescribed constraints illustrated in (12).
The process of tracking error \(z_{1,i}\) satisfying the preset constraints (12) will be proven by using the method of contradiction.
Integrating the iBLF \(V_{b1,i}\), one can obtain that:
Assume that there exist a time \(t=T\) such that \( z_{1,i}(T) =\mu _{zl,i}(T)\) or \( z_{1,i}(T) =\mu _{zu,i}(T)\). Then, one can get that \({V_{b1,i}}(T)=\infty \). However, the boundedness of \(V_{0,i}\) has been proved. So, \(\left z_{1,i}(T)\right \ne \mu _{zl,i}(T)\) and \(\left z_{1,i}(T)\right \ne \mu _{zu,i}(T)\). And consider the definitions of \(\mu _{zl,i}\) and \(\mu _{zu,i}\), if the initial value satisfies \(\mu _{zl,i}(0)<z_{l,i}(0)< \mu _{zu,i}(0)\), then \(z_{1,i}\) satisfies the prescribed constraints (12).
Because the proof of \(\mu _{xl,i}<{\dot{\theta }}_{m,i}< \mu _{xu,i}\) is similar to the proof of \(\mu _{zl,i}<z_{1,i}< \mu _{zu,i}\), the proof process for motor angular velocity \({\dot{\theta }}_{m,i}\) satisfying complex state constraints (4) is omitted.
(3) In the second part of the proof, it can be concluded that the motor angle tracking error \(z_{1,i}\) satisfies the prescribed constraints (12), therefore, the torque generated by SEA \(\tau _s\) can accurately track the desired assistive torque \(\tau _{r}\). The impedance controller (6) can be regarded as a PD controller, and the convergence proof of the PD controller is detailed in [38]. Thus, the patient’s upper limb angle \(\theta _{l}\) can track the desired trajectory \(\theta _d\). Ultimately, the patient’s upper limbs can track the desired rehabilitation trajectory \({\textbf {P}}_d\).
In conclusion, the proposed UPPAC ensures the convergence of the tracking error \(z_{1,i}\) and satisfies the complex state constraints (4), and finally enables the achievement of assistasneeded control of SULPES under complex state constraints. \(\square \)
4 UPPAC application and cosimulation results
The SULPES in SolidWork is introduced into MATLAB/SimMechanics, and the joint simulation experiment of the proposed UPPAC is carried out. To illustrate the effectiveness of UPPAC, two cases involving the patient’s motor ability are considered in the cosimulation experiment, namely, the patient’s movement ability is weak and the patient’s movement ability is variable. The desired trajectory \({\textbf {P}}_d\) of the rehabilitation task in this cosimulation is set as a circular motion training task at the end of the human upper limb in Cartesian space, with a center point at [0.35, 0] and a radius of 0.15m. The parameters of patient’s torque are set as \({{{\textbf {K}}}} _p=\text {diag}\{85,85\}\) and \({{\textbf {{B}}}} _p=\text {diag}\{15.5\}\), the parameters of the SEA are set as \({{{\textbf {K}}}} _s=\text {diag}\{30,30\}\) and \({{{\textbf {B}}}} _s=\text {diag}\{2,2\}\).
Next, the following two simulation cases will be considered. In case 1, the patient’s motor ability is poor, and the effectiveness of UPPAC is proved by comparison with neural networkbased sliding mode control (NNSMC) and BLFbased TDE controller (BLFTC) methods. In case 2, the patient’s movement ability is variable to further validate that the proposed UPPAC can change the assistive force based on the patient’s movement ability.
4.1 Case 1: the patient’s movement ability is weak
In this case, the patient’s movement ability is weak, so set the parameter \(k_p=0.1\).
Then, to ensure the safety of patients, the complex constraints on the angular velocity of the exoskeleton are considered as follows:
with \({\mu _{a,1}}=1.5\), \({\mu _{a,2}}=1\), \({\mu _{b,1}}=6 \times 10^{3}\) and \({\mu _{b,2}}=9 \times 10^{3}\).
NNSMC [39] and BLFTC [33] are applied to illustrate the performance of UPPAC.
The NNSMC is defined as follow [39]:
And the BLFTC is defined as follow [33]:
The prescribed constraints for tracking error \({\textbf {z}}_{1}\) are considered as follows:
Other parameters of UPPAC are selected as \({\textbf {B}}_a=\text {diag} \{300,300\}\), \({\textbf {K}}_a=\text {diag} \{3000,3000\}\), \(\alpha =1\), \(c_{1,1}=100\), \(c_{1,2}=100\), \(c_{2,1}=500\), \(c_{2,2}=500\), \(\lambda _1=\lambda _2=10\), \(\varsigma _1=10\), \(\varsigma _2=100\), \(\Delta \) takes one sampling period. The parameters of NNSMC are chosen as \(r_1 = 50\), \(r_2 = 500\), \(r_3 = 10\). Then the parameters of BLFTC are chosen as \(k_e=0.002\), \(k_1=250\), \(k_2=1000\).
Remark 5
For the selection of controller parameters, firstly, regarding the complex state constraints, as the derivatives of their constraint boundaries are used in the controller, the boundaries are represented using the \(\cos \) function to ensure their continuity and smoothness. As for the selection of parameters \(\mu _{a,1}\) and \(\mu _{a,2}\), their values should not be too small to avoid impacting normal rehabilitation training. The selection of parameters \(\mu _{b,1}\) and \(\mu _{b,2}\) should not exceed the maximum value of the error constraint boundary. Ultimately, the complex state constraints are depicted as the black dashed lines in Fig. 4, where as the error approaches 0, the state constraint range widens, and as it approaches the error boundary \(\mu _{b,1}\) and \(\mu _{b,2}\), the state constraint range tightens.
For the prescribed constraints, it is necessary to consider the actual situation. Due to the existence of initial errors, the boundaries need to be set larger to prevent initial errors from exceeding the boundaries. Then, in order to obtain smaller steadystate errors, it is necessary to set the values of the boundaries to be small. For the selection of the ultralocal model parameter \(\alpha \), the value of \(\alpha \) will affect the convergence speed. The smaller the value of \(\alpha \), the faster the convergence speed but will cause a relatively large overshoot, so set \(\alpha =1\).For the values of parameters \(c_1\) and \(c_2\), generally speaking, the selection of these two parameters can refer to the parameters of the PD controller. For the selection of neural network parameters, parameters \(\lambda _1\) and \(\lambda _2\) do not need to be selected too large, we choose \(\lambda _1=\lambda _2=20\). And for parameters \(\varsigma _1\) and \(\varsigma _2\), we choose \(\varsigma _1=10\), \(\varsigma _2=100\) by trial and error during the simulation studies.
Under the aforementioned conditions, the corresponding simulation results are depicted in Figs. 3, 4, 5, 6 and 7 and Tables 1 and 2. In Fig. 3, the tracking performance of the reference trajectory \(\theta _{r}\) of each motor by UPPAC is shown as the dashed red lines, and the prescribed constraints of the tracking errors are illustrated by the dotted black lines. Motor 1 and 2 denote the motor in shoulder and elbow joint, respectively. Obviously, whether it is the motor of the shoulder joint or elbow joint, the tracking error using UPPAC can be kept within the prescribed constraints, and their steadystate errors can converge to the intervals of \((1 \times 10^{3},1.2\times 10^{3})\) and \((1 \times 10^{3},1.3\times 10^{3})\), respectively. Moreover, it can be seen from Tables 1 and 2 that the average absolute tracking errors of the reference trajectory \(\theta _{r}\) by the proposed UPPAC are only \(5.15 \times 10^{5}\)rad and \(2.52 \times 10^{4}\)rad, and the standard deviation values are \(6.16 \times 10^{5}\) and \(2.60 \times 10^{4}\). Therefore, the proposed UPPAC can achieve highprecision and smooth tracking for SULPES.
For the comparison of NNSMC, although it can also achieve good tracking results, due to the fact that this method does not consider the prescribed constraints of the tracking errors, it may result in the tracking errors of the reference trajectory exceeding the prescribed constraints, and their steadystate errors are all larger than the proposed UPPAC. And for the comparison of BLFTC, as this method requires a large value to keep the initial error inside the constraints, and the constraint on tracking errors is timeinvariant. Therefore, the error constraints cannot decrease over time, which may result in larger steadystate errors. Then, from Tables 1 and 2, it can be easily seen that the average absolute value tracking error of UPPAC is only 14.1% of NNSMC, and 27.6% of BLFTC, and the standard deviation of the tracking error by UPPAC is only 13.8% of NNSMC, and 35.3% of BLFTC, indicating that the proposed UPPAC has more precise and stable tracking performance.
Then, in Fig. 4, the complex constraints are demonstrated, the phase portrait of \({\textbf {z}}_{1}  {\dot{\theta }}_{m}\) by UPPAC are denoted as the dotted red lines, and the complex state constraints are illustrated as the dotted black lines. It can be concluded that although the angular velocity of UPPAC may be relatively high and close to the boundary due to overshoot at the beginning of operation, the proposed UPPAC can maintain the angular velocity within the boundary and satisfy the complex state constraints, enhancing the safety and flexibility of the system.
Due to the fact that the other two comparison methods do not consider the complex state constraints, the angular velocity of the compared NNSMC and BLFTC methods exceeds the complex state limit boundary at the beginning of rehabilitation task. This will cause an instantaneous increase in assistive torque at the beginning of the rehabilitation task, which will potentially cause injury to patients due to the impact. Moreover, the rapid change rate of assistive torque can lead to a decrease in compliance, thereby reducing patient comfort.
The assistive torque and its tracking error are shown in Fig. 5. It is obvious that the proposed UPPAC can achieve convergence in a short time, and has small steadystate errors and overshoots. For the compared NNSMC, due to the absence of reference trajectory error constraints, the overshoots are larger than those of UPPAC, with values of 6 and 15 respectively, and the steadystate error fluctuates within the interval of \((0.16, 0.12)\). For the comparative BLFTC, although this method considers the reference trajectory error constraints, the constraints are time invariant. Therefore, its overshoots are smaller than NNSMC, but the steadystate errors are larger than the proposed UPPAC. Then, from Table 1 and 2, the average absolute value of assistive torque tracking error by UPPAC is only 7.9% of NNSMC and 13.2% of BLFTC, and the standard deviation of assistive torque tracking error by UPPAC is less than 33.5% of NNSMC and 40.5% of BLFTC. Therefore, the proposed UPPAC algorithm has a small impact on patients and operates more stably, which can significantly improve enhance patients’ compliance, making patients more comfortable during the rehabilitation process.
The simulation results of the trajectory on the patient’s upper limb endpoint are illustrated in Fig. 6 and Tables 1 and 2. It is clear that all these three methods can achieve assistasneeded control of the SULPES, but the steadystate error of UPPAC are smaller than those of NNSMC and BLFTC, and the absolute average error of the xaxis and yaxis of UPPAC is only \(1.79 \times 10^{4}\) and \(5.19 \times 10^{4}\). From Table 1 and 2, it can be deduced that the average absolute value of position tracking error by UPPAC is only 18.4% of NNSMC and 28.3% of BLFTC, and the standard deviation of position tracking error by UPPAC is only 23.0% of NNSMC and 60.6% of BLFTC. In summary, the proposed UPPAC can enable patients to track the desired rehabilitation trajectory more accurately and can perform rehabilitation training more effectively. In addition, smaller tracking errors can ensure that the patient’s upper limbs remain around the desired trajectory, avoiding excessive deviation that may cause damage to the patient.
Figure 7 shows the update curves of the adaptive parameters \({\hat{\zeta }}_{1}\) and \({\hat{\zeta }}_{2}\). From equation (21), it can be seen that the value of \(W_1\) is mainly related to the tracking error \(z_1\) and derivative of the prescribed constraints boundary. At the beginning of operation, the values of tracking error and derivative of the boundary are both large, so the value of \(\zeta _{1}\) is also large. After the tracking error converges, both the tracking error and the derivative of the boundary approach 0, leading to a small value of \({\hat{\zeta }}_{1}\). For \({\hat{\zeta }}_{2}\), as derived from equation (34), the value of \(W_2\) is related to the TDE estimation error and derivative of the complex constraints. Since the TDE parameter \(\Delta \) takes one sampling period, the value of TDE estimation error is very small. In addition, the derivative value of the complex constraints is also small, this results in a low value for \({\hat{\zeta }}_{2}\). The value of \({\hat{\zeta }}_{2,1}\) fluctuates within the interval \((0.0025,0)\), and \({\hat{\zeta }}_{2,2}\) fluctuates within the interval \((0.002,0)\).
The control input signal results of the three control algorithms are shown in Fig. 8 and Table 2. As can be seen from Table 2, the standard deviations of the control input signals for the shoulder and elbow joints using the proposed UPPAC are 2.84 and 0.53, respectively, which are only 71.2% and 36.9% of those of the BLFTC method, and less than 74% and 59% of those of the NNSMC. Additionally, for the standard deviation of the derivative of the control input, the proposed UPPAC achieves values for the shoulder joint that are less than 11% of those under BLFTC and 25% of those under NNSMC, while for the elbow joint, the values are less than 6.3% of those under BLFTC and 17% of those under NNSMC.Therefore, the proposed UPPAC demonstrates stronger stability and smoother performance, which can lead to a more natural and comfortable experience, reducing patient discomfort.
4.2 Case 2: the patient’s movement ability is variable
In this case, the patient’s movement ability is variable to further validate that the proposed UPPAC can change the assistive force based on the patient’s movement ability. At the beginning, the patient’s movement ability is weak. Through rehabilitation training, the patient ’s movement ability is restored. After a period of time, the patient feels fatigue and the movement ability is weakened. So set the parameter \(k_p\) as follows
In addition, in real world rehabilitation scenarios, uncertainties in the system are inevitable, therefore the shaking caused by muscle fatigue and exoskeleton measurement noise are considered in this case. The shaking caused by muscle fatigue is represented as \([0.15\sin (10t);0.05\cos (10t)]\). The measurement noise is represented by a set of normally distributed random numbers with amplitudes of \([1 \times 10^{4},1 \times 10^{4}]\).
In this case, the parameters of UPPAC are selected as \({\textbf {B}}_a=\text {diag} \{300,300\}\), \({\textbf {K}}_a=\text {diag} \{3000,3000\}\), \(\alpha =1\), \(c_{1,1}=100\), \(c_{1,2}=100\), \(c_{2,1}=1000\), \(c_{2,2}=1000\), \(\lambda _1=\lambda _2=10\), \(\varsigma _1=10\), \(\varsigma _2=100\).
Then, the complex state constraints and prescribed constraints for tracking error \({\textbf {z}}_{1}\) of UPPAC are same as in case 1.
Under the above simulation conditions, the corresponding simulation results are demonstrated in Figs. 9, 10, 11, 12 and 13. In Fig. 9, the tracking results of each motor by UPPAC are depicted as the dotted red lines, and the dashed black lines denote the prescribed constraints of the tracking errors. Obviously, when the patient’s movement ability changes, the proposed UPPAC can still keep the tracking error within the prescribed constraints. However, due to the presence of measurement noise, the tracking error curve will shake within the interval \((0.001, 0.001)\).
Then, Fig. 10 demonstrates the complex state constraints, the dashed red lines denote the phase portrait of \({\textbf {z}}_1  {\dot{\theta }}_m\), and the dotted black lines denote the complex state constraints. It can be observed that under the proposed UPPAC, when the patient’s movement ability changes and there are system uncertainties, the angular velocity may fluctuate. However, the angular velocity can still be maintained within complex state constraints, thereby ensuring flexibility and safety.
Then, the tracking performance of assistive torque is given in Fig. 11, from the 5th to 15th seconds, as the patient’s movement ability recovers, the assistive torque gradually decreases. However, after the 15th seconds, as the patient’s movement ability declines, the assistive torque required to complete the rehabilitation task gradually increases. From the tracking performance of the assistive torque, it can be observed that the proposed UPPAC can adjust the assistive torque according to the patient’s movement ability. Between the 10th and 20th seconds, the peak value of the assistive torque is only half of that from the 0th to 5th seconds. Additionally, due to human body tremors and measurement noise, the assistive torque is affected, especially the assistive torque at the elbow joint, where the tracking error fluctuates within the range of \((0.3, 0.3)\).
From Fig. 12, it can be concluded that although the terminal tracking error is relatively large at the beginning of the rehabilitation task due to initial errors and overshoot, the steadystate error can be maintained within 0.001. Therefore, the proposed UPPAC can make patient’s upper limbs to move according to the desired rehabilitation training trajectory. Despite the uncertainties caused by human body shaking and measurement noise, which may lead to fluctuations in tracking error, their impact on the overall tracking performance is minimal.
Figure 13 shows the values of the adaptive laws \(\hat{\zeta }_1\) and \(\hat{\zeta }_2\). The value of \(\hat{\zeta }_1\) is mainly influenced by the tracking error \(z_1\) and the derivative of the prescribed constraints, so although it may exhibit small fluctuations due to measurement noise, it generally tends to be stable. The value of \(\hat{\zeta }_2\) is mainly related to the TDE estimation error and complex state constraints. Since the second derivative of the motor angle is used in the TDE, and this second derivative value fluctuates significantly due to measurement noise, the estimated value of \(\hat{\zeta }_2\) is also affected, resulting in significant fluctuations at certain moments.
Overall, with the application of the proposed UPPAC, SULPES can adaptively adjust the assistive torque according to changes in the patient’s movement abilities. However, the proposed method may produce minor fluctuations when uncertainties such as measurement noise are present in the system, particularly affecting the assistive torque at the elbow joint. Despite this, the patient’s upper limb can still follow the desired rehabilitation training trajectory. In future research, methods such as H\(\infty \) control [40] and observers [41] will be integrated to reduce the impact of measurement noise on control performance.
5 Conclusions
In this paper, an ultralocal modelbased prescribed performance assistasneeded controller is proposed for the SULPES under complex state constraints. The proposed UPPAC achieves assistasneeded control through the impedance control strategy, and its inner torque subcontrol loop is proposed based on the ultralocal model, which can get rid of the dependence on accurate model parameters. In addition, the introduction of iBLF ensures that UPPAC satisfies complex state constraints and prescribed constraints for tracking errors. Finally, by comparing with NNSMC and BLTC, the cosimulation results show that UPPAC can satisfy the complex state constraints, has smaller overshoot, shorter convergence time, and the average tracking error of UPPAC is only 14.1% of NNSMC and 27.6% of BLFTC. In addition, the proposed method can adjust the required assistive torque based on the patient’s movement ability.
From [42] and [43], it can be found that admittance parameters should be adaptively adjusted based on the severity of patients’ injuries. Additionally, by adjusting admittance parameters, patients can be encouraged to participate more actively in rehabilitation training, leading to improved rehabilitation outcomes. Therefore, in the subsequent work, patients’ movement abilities will be evaluated based on tracking errors of the upper limb trajectory and the magnitude of assistive torque. Admittance parameters will then be adaptively adjusted according to the evaluation results to achieve better rehabilitation outcomes. Furthermore, the speed of rehabilitation training is also a crucial indicator affecting its effectiveness [33]. Therefore, in the subsequent work, the training speed will be adjusted based on patients’ motor abilities. These avenues will be the primary focus of our further research.
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The research was funded in part by the National Natural Science Foundation of China under grant numbers (62173182).
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Wei, Y., Wang, H.P. & Tian, Y. Ultralocal modelbased prescribed performance assistasneeded control for series elastic actuatorbased upper limb patientexoskeleton system under complex state constraints. Nonlinear Dyn 112, 17183–17204 (2024). https://doi.org/10.1007/s11071024099287
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DOI: https://doi.org/10.1007/s11071024099287