1. Introduction
Robots have been identified as a possible way to improve and automate patient’s access to rehabilitation therapy and also to provide new tools for therapists. Few decades before, human limbs rehabilitation was done using conventional techniques which take long time, consume more energy, and the results are not always successful. Nowadays, with the increase in the number of patients, in fact, recent statistics showed that more than 13.7 million new strokes are recorded each year. Classic rehabilitation is not sufficient to treat all patients effectively, and here, exoskeleton robots come into play, capable of assisting disabled people in recovering control of their limbs [Reference Jebri, Madani, Djouani and Benallegue1–Reference Onose, Cârdei, Crăciunoiu, Avramescu, Opriş, Lebedev and Constantinescu5]. Furthermore, exoskeletons or wearable robots have been used also in military domains [Reference Proud, Lai, Mudie, Carstairs, Billing, Garofolini and Begg6, Reference Yuan, Wang, Ma and Gong7].
PSO Algorithm.
Mechanical structure of ULEL.
While exoskeleton utilization for therapy purposes has not widely spread yet in industry, several researches have been conducted in order to control wearable robots for passive, assistive, and active rehabilitation modes using different control laws and multiple techniques such as Sliding Mode Control (SMC) [Reference Jellali, Madani, Khraief and Belghith2, Reference Riani, Madani, Benallegue and Djouani3, Reference Islam, Rahmani and Rahman8], neural network control [Reference Jebri, Madani, Djouani and Benallegue1, Reference Daachi, Madani, Daachi and Djouani4, Reference Wu, Wang, Chen and Wu9–Reference He, Ge, Li, Chew and Ng12], adaptive PID control [Reference Belkadi, Oulhadj, Touati, Khan and Daachi13–Reference Chen, Yang and Sun15], EMG-based, and impedance control [Reference Khoshdel, Akbarzadeh, Naghavi, Sharifnezhad and Souzanchi-Kashani16–Reference Tsai, Wang, Hsu, Fu and Lai20]. A review on different control strategies that has been applied for upper limb exoskeleton robots is established in ref. [Reference Proietti, Crocher, Roby-Brami and Jarrasse21].
Mechanical properties of ULEL.
Controller diagram.
Despite the existence of several robust control techniques, and when it comes to reliability, the Super Twisting SMC (ST-SMC) may be at the top of the list of controllers dedicated to non-linear systems due to its robustness and effectiveness that has been proven in many studies such [Reference Jian, Zhitao and Hongye22–Reference Imine, Fridman and Madani27]. To overcome the lack and problem of the classic ST-SMC which consist of setting properly the controller parameters, this paper presents an adaptive ST-SMC algorithm that improves the performances of the classical SMC algorithm via the optimal online tuning of its parameters, by using the Particle Swarm Optimization (PSO) algorithm. Similar control strategies have been applied on few non-linear systems such as BLDC motor [Reference Gülbaş, Hameş and Furat28] which is simulation-based; nevertheless, it still suffers from chattering in control effort and the gains are tuned offline. Another experiment offline-based has been performed on a 7 DOF lower limb exoskeleton robot by applying an impedance control [Reference Mokhtari, Taghizadeh and Mazare29]. Next, an application simulation-based has been applied on 3 DOF lower limb exoskeleton robot by applying an adaptive control [Reference Faraj, Maalej, Derbel, Naifar and Rossomando30] which optimizes the controller’s parameters offline. Also, valuable techniques are developed for upper limb rehabilitation using neuromuscular electrical stimulation (NMES). The work presented in ref. [Reference Oliveira, Costa, Catunda, Pino, Barbosa and de Souza31] addresses the application of the time-scaling-based SMC approach to control arm movements using NMES technique. The developed SM strategy helps mitigate chattering due to the use of an integrator in the input control signal. Furthermore, no differentiator is applied to construct the sliding surface. In ref. [Reference Paz, Oliveira, Pino and Fontana32], a stochastic extremum seeking (ES) approach is used to adapt the gains of a PID controller in order to apply an appropriate functional NMES. The approach allows PID controller parameters to be adjusted in real time using the ES method to minimize a cost function and provides the desired performances. The proposed approach in this paper addresses the problems related to exoskeleton rehabilitation, where it provides an online controlling which allows the studied non-linear system to be used at any dependent circumstances. Finally, the stability analysis in sense of Lyapunov and the finite time convergence of the proposed controller has been carried out in the same way as [Reference Moreno33–Reference Moreno and Osorio36]. In order to demonstrate the performance and efficiency of the proposed controller, a comparison with the classic ST-SMC has been performed and a real-time experiment has been carried out with two healthy subjects using an exoskeleton robot called ULEL.Footnote 1
Field of motion of ULEL [Reference Riani39].
Parameter values.
External disturbances efforts applied on joint 1 and 2.
This work presents a challenge in developing a function that allows the parameters to be optimized smoothly online, avoid heavy offline calculation problem, extend the life cycle of the actuators, and give the optimal unique paramters for each patient. Also, fixing the parameters for many users makes the performance depend on the subject wearing the exoskeleton. Therefore, optimizing the parameters online will certainly improve the performance of the exoskeleton robot.
Evolution of the objective function.
Controller parameters obtained with PSO in good
$K$
and bad
$K'$
initial guesses cases.
Position trajectory tracking of joint 1 of the proposed controller AST-SMC and classic ST-SMC, respectively: (a) Position trajectory tracking; (b) Position tracking error.
Position trajectory tracking of joint 2 of the proposed controller AST-SMC and classic ST-SMC, respectively: (a) Position trajectory tracking; (b) Position tracking error.
Velocity trajectory tracking of joint 1 of the proposed controller AST-SMC and classic ST-SMC, respectively: (a) Velocity trajectory tracking; (b) Velocity tracking error.
Velocity trajectory tracking of joint 2 of the proposed controller AST-SMC and classic ST-SMC, respectively: (a) Velocity trajectory tracking; (b) Velocity tracking error.
Input control torques for joints 1 and 2 with external disturbances: (a) Represent joint 1; (b) represent joint 2.
Position and velocity tracking error RMS.
This paper will proceed as follows, Section 2 is reserved for the considered 3 DOF exoskeleton robot. Section 3 is dedicated to adaptive controller. Section 4 presents the simulation and experiment results. In the last section, a conclusion is given with future works.
2. Considered system
The system considered in this study is formed by the wearer and the exoskeleton ULEL designed by RB3D company for the LISSIFootnote 2 Laboratory. ULEL has 3 active revolute joints allowing the flexion/extension movements of the shoulder, elbow, and wrist joints. In addition, an adjustable passive ball joint can be used to settle the position of the exoskeleton’s arm as depicted in Fig. 1. The characteristics of the three active joints of ULEL are summarized in Table II.
2.1. Dynamic model
The dynamic model of the examined system that consists of the exoskeleton and the wearer can be expressed by using Euler-Lagrange formalism as follows:
\begin{equation} M\!\left (q\right )\ddot{q}+H\!\left (q,\dot{q}\right )=\tau _{exo}+\tau _{hum} \end{equation}
with
\begin{align*} H\!\left (q,\dot{q}\right )=C\!\left (q,\dot{q}\right )\dot{q}+G\!\left (q\right )+F\!\left (\dot{q}\right ) \end{align*}
and
\begin{align*} u=\tau _{exo} \end{align*}
where
$q\in \mathbb{R}^{n}\text{, }\dot{q}\in \mathbb{R}^{n}\text{ and }\ddot{q}\in \mathbb{R}^{n}$
are the joints positions, velocities, and accelerations, respectively;
$M\!\left (q\right )\in \mathbb{R}^{n\times n}$
is the nonsingular inertia matrix;
$C\!\left (q,\dot{q}\right )\in \mathbb{R}^{n\times n}$
is the Coriolis and centrifugal matrix;
$G\!\left (q\right )\in \mathbb{R}^{n}$
represents the gravity forces vector;
$F\!\left (\dot{q}\right )\in \mathbb{R}^{n}$
is the dissipation term;
$\tau _{exo}\in \mathbb{R}^{n}\text{ and }\tau _{hum}\in \mathbb{R}^{n}$
are the exoskeleton and human torques, respectively;
$u\in \mathbb{R}^{n}$
is the control signal input.
The dynamic model of the system 1 can be written as
\begin{equation} \ddot{q}=f\!\left (q,\dot{q}\right )+g\!\left (q\right )u+d(t) \end{equation}
where
$f\!\left (q,\dot{q}\right )$
,
$g\!\left (q\right )$
and
$d(t)$
are given by
\begin{align*} f\!\left (q,\dot{q}\right )=-M\!\left (q\right )^{-1}H\!\left (q,\dot{q}\right ) \end{align*}
\begin{align*} g\!\left (q\right )=M\!\left (q\right )^{-1} \end{align*}
\begin{align*} d(t)=M\!\left (q\right )^{-1}\tau _{hum} \end{align*}
where
$d(t)\in \mathbb{R}^{n}$
is considered as the unknown uncertainties and external bounded disturbances vector, which will be compensated by the controller.PSO
The following assumptions are considered:
Assumption 1.
The positions
$q\!\left (t\right )$
and the velocities
$\dot{q}\!\left (t\right )$
are measured.
Assumption 2.
The disturbance
$d(t)$
is differentiable with respect to time and its derivative
$\dot{d}(t)$
is bounded.
3. Adaptive super twisting sliding mode controller
In this study, the Adaptive ST-SMC (AST-SMC) has been used in order to handle the disadvantage of the classical super twisting algorithm which are located in parameters adaptation almost for each user. The major reason of choosing a second-order controller like ST-SMC algorithm rather than, for example, classical SMC or PID is the superiority of its robustness and reliability on both of them, which comprises of multiple points such as:
-
• Ensure high-quality trajectory tracking in position and velocity
-
• Ability of reaching the equilibrium point in finite time for both, sliding surface and its time derivative
-
• Easily and smoothly rid of external disturbances in finite time
-
• Provide comfort and security for the wearer.
In this section, a controller design and the online adaptation PSO-based algorithm are presented. The proof of the stability analyses and the finite time convergence of the proposed controller is given in A.1.
3.1. Controller design
The control law has been designed in two steps: first, selecting the sliding surface, and then designing the controller.
Let the sliding surface of the super twisting controller be defined as:
\begin{equation} S=\dot{e}+\lambda e \end{equation}
where
$S^{T}=[S_{1},\ldots,S_{n}]\in \mathbb{R}^{n}$
is the sliding manifold,
$\lambda \in \mathbb{R}$
is a positive constant;
$e,\dot{e}\in \mathbb{R}^{n}$
are the joint position and velocity errors, respectively, such as
$e=q_{d}-q$
and
$\dot{e}=\dot{q}_{d}-\dot{q}$
with
$q_{d},\dot{q}_{d}\in \mathbb{R}^{n}$
are the desired position and velocity joints separately.
Remark 1.
$\lambda$
has been chosen as a positive constant in order to get exponential convergence of the error dynamics in the sliding mode
$(S=0)$
.
The proposed AST-SMC that has derived from [Reference Jian, Zhitao and Hongye22, Reference Imine, Fridman and Madani27] is given as the following:
\begin{equation} u=g\!\left (q\right )^{-1}\!\left (\ddot{q}_{d}-f\!\left (q,\dot{q}\right )+\lambda \,\dot{e}+K_{1}\sqrt{\!\left |S\right |}sign\!\left (S\right )+\intop K_{2}sign\!\left (S\right )dt\right ) \end{equation}
where
$K_{1}=diag\!\left (K_{1,1},\ldots,K_{1,n}\right )\in \mathbb{R}^{n\times n}$
and
$K_{2}=diag\!\left (K_{2,1},\ldots,K_{2,n}\right )\in \mathbb{R}^{n\times n}$
are the bounded controller’s parameters according to the boundaries in the proof section A.1, and they will be optimized by the PSO algorithm as explained in section 3.2 and depicted in Fig. 2, and these parameters are formed by positive diagonal matrices.
$\sqrt{\!\left |S\right |}$
is a matrix such
$\sqrt{\!\left |S\right |}=diag\!\left (\sqrt{\!\left |S_{1}\right |},\ldots,\sqrt{\!\left |S_{n}\right |}\right )\in \mathbb{R}^{n\times n}$
, whereas
$sign\!\left (S\right )$
is a vector where
$sign\!\left (S\right )=\!\left [sign\!\left (S_{1}\right ),\ldots,sign\!\left (S_{n}\right )\right ]^{T}\in \mathbb{R}^{n}$
and
$\ddot{q}_{d}\in \mathbb{R}^{n}$
are the desired acceleration.
Substituting the control law (4) into the Equation (2) gives:
\begin{align*} \ddot{q}\begin{array}[t]{l} =f\!\left (q,\dot{q}\right )+g\!\left (q\right )u+d(t)\\[5pt] =\ddot{q}_{d}+\lambda \,\dot{e}+K_{1}\sqrt{\!\left |S\right |}\,sign\!\left (S\right )+\intop K_{2}\,sign\!\left (S\right )dt+d(t) \end{array} \end{align*}
Then it comes:
\begin{equation} \ddot{q}_{d}-\ddot{q}+\lambda \,\dot{e}+K_{1}\sqrt{\!\left |S\right |}\,sign\!\left (S\right )+\intop K_{2}\,sign\!\left (S\right )dt+d(t)=0 \end{equation}
Since the derivative of the sliding surface
$S$
is given by
$\dot{S}=\!\left (\ddot{q}_{d}-\ddot{q}+\lambda \,\dot{e}\right )$
, then the Equation (5) can be written the following Super Twisting (ST) form:
Experimental setup.
\begin{equation} \dot{S}=-K_{1}\,\sqrt{\!\left |S\right |}\,sign\!\left (S\right )-\intop K_{2}\,sign\!\left (S\right )dt-d(t) \end{equation}
The Equation (6) can still be written in the following system:
\begin{equation} \!\left \{ \begin{array}{l} \dot{S}=-K_{1}\,\sqrt{\!\left |S\right |}\,sign\!\left (S\right )+Z\\[5pt] \dot{Z}=-K_{2}\,sign\!\left (S\right )-\dot{d}(t) \end{array}\right . \end{equation}
A recall of the Lyapunov stability analysis of the ST algorithm is given in the Appendix A.1. Therefore, the system (7) which represents the closed loop dynamics is stable and its finite time convergence is guaranteed.
3.2. PSO algorithm
Because the super twisting algorithm contains many parameters, choosing a meta-heuristic algorithm like PSO to handle their eigenvalues is adequate method to upgrade any controller and make it adaptive. However, evaluation criteria must be well studied and more suitable for the specified application.
The PSO algorithm has been introduced for the first time in ref. [Reference Kennedy and Eberhart37]. Its main idea is based on the co-working of agents called “particles,” which represent each individual in the swarm that tends to find a solution of a complex optimization problem, through minimizing an objective function. The basic working behavior of the PSO is, firstly, the random initialization of the swarm that contains a population number related to wanted solutions. Secondly, updating and sharing particle data is in an iterative manner for the goal of finding the global best solutions. In order to do that, setting a significant value of inertia coefficient
$w$
$\in \mathbb{R}$
is required for searching the global area “exploration,” then after the algorithm linearly reduces its eigenvalue using Equation (10), the PSO focuses the search on local area “exploitation.” This algorithm defines each
$i$
-th individual in the swarm by its velocity
$V_{i}\in \mathbb{R}^{m}$
and position
$X_{i}\in \mathbb{R}^{m}$
vectors as illustrated in Equations (8) and (9), respectively. For this work, the used PSO algorithm is based on ref. [Reference Dang and Hoshino38] and given by the following equations:
\begin{equation} V_{i}^{j+1}=w^{j}\,V_{i}^{j}+c_{1}\!\left (P_{i}^{j}-X_{i}^{j}\right )+c_{2}\!\left (G^{j}-X_{i}^{j}\right )+c_{3}\frac{r^{j}}{||V_{i}^{j}||^{2}} \end{equation}
Subjects used in experiments.
Position trajectory tracking of joint 1, trial without a subject and with subjects 1 and 2: (a) Position trajectory tracking; (b) Position tracking error.
Position trajectory tracking of joint 2, trial without a subject and with subjects 1 and 2: (a) Position trajectory tracking; (b) Position tracking error.
Velocity trajectory tracking of joint 1, trial without a subject and with subjects 1 and 2: (a) Position trajectory tracking; (b) Position tracking error.
Position trajectory tracking of joint 2, trial without a subject and with subjects 1 and 2: (a) Position trajectory tracking; (b) Position tracking error.
Input control torques for joints 1 and 2 without any external perturbations: (a) Represent joint 1; (b) Represent joint 2.
Input control torques for joints 1 and 2 with the presence of external disturbances: (a) Represent joint 1; (b) Represent joint 2.
Position trajectory tracking of joint 1 with the presence of disturbances, trial with subjects 1 and 2: (a) Position trajectory tracking; (b) Position tracking error.
\begin{equation} X_{i}^{j+1}=X_{i}^{j}+V_{i}^{j+1} \end{equation}
where
$j$
is the iteration and
-
•
$c_{1},\,c_{2},\,c_{3}\in \mathbb{R}$
are positive constants. -
•
$r^{j}\in \mathbb{R}^{m}$
is a random positive vector at the iteration
$j$
. -
•
$P_{i}^{j}\in \mathbb{R}^{m}$
is the personal best vector of the
$i$
-th particle at the iteration
$j$
. -
•
$G^{j}\in \mathbb{R}^{m}$
is the global best vector of all particles at the iteration
$j$
. -
•
$X_{i}^{j},\,X_{i}^{j+1}\in \mathbb{R}^{m}$
are the current and updated position vectors separately of the
$i$
-th particle. -
•
$V_{i}^{j},\,V_{i}^{j+1}\in \mathbb{R}^{m}$
are the current and updated velocity vectors, respectively of the
$i$
-th particle. -
•
$w^{j}\in \mathbb{R}$
is a positive weigh computed by
\begin{equation} w^{j}=w_{max}-\!\left (\frac{\!\left (w_{max}-w_{min}\right )}{j_{max}}\right )\times j \end{equation}
where
$w_{min}$
and
$w_{max}$
are the minimal and the maximal value and
$j_{max}$
is the maximal value of the iteration
$j$
.
The parameters are calculated using the following equation:
\begin{align*} \theta (t+dt)=\theta (t)+G^{j_{max}} \end{align*}
where
$dt$
is the sampling time,
$\theta (t)=[K_{1,1}(t),\ldots,K_{1,n}(t),K_{2,1}(t),\ldots,K_{2,n}(t)]^{T}\in \mathbb{R}^{2n}$
is the parameter vector at the moment
$t$
, and
$G^{j_{max}}\in \mathbb{R}^{m}$
is the global best vector of all particles at the final iteration
$j_{max}$
.
Before defining the objective function, let us introduce the following hypothesis.
Assumption 3.
Let
$T\in \mathbb{\mathbb{R}}^{+}$
be a short period of time such
$u\!\left (t\right )$
and
$\ddot{q}\!\left (t\right )$
remain constants for
$t\in \!\left [t,\,t+T\right ]$
.
The equation of motion will be defined by
\begin{equation} \dot{q}\!\left (t+T\right )=\ddot{q}\!\left (t\right )T+\dot{q}\!\left (t\right ) \end{equation}
and
\begin{equation} q\!\left (t+T\right )=\frac{1}{2}\ddot{q}\!\left (t\right )T^{2}+\dot{q}\!\left (t\right )T+q\!\left (t\right ) \end{equation}
The objective function that has been utilized in this paper uses the norms of the predictive sliding surface
$S\!\left (t+T\right )$
, the control signal
$u\!\left (t\right )$
, and the swarm position
$X(t)$
in order to avoid high gains and reduce power consumption. It can be defined as the following:
\begin{equation} J=\alpha \!\left \Vert S\!\left (t+T\right )\right \Vert +\beta \!\left \Vert u\!\left (t\right )\right \Vert +\gamma \!\left \Vert X(t)\right \Vert \end{equation}
with
\begin{equation} S\!\left (t+T\right )=\!\left (\dot{q}_{d}\!\left (t+T\right )-\dot{q}\!\left (t+T\right )\right )+\lambda \!\left (q_{d}\!\left (t+T\right )-q\!\left (t+T\right )\right ) \end{equation}
where
$t\in \mathbb{R}$
is the current time;
$\alpha,\,\beta,\,\gamma \in \mathbb{R}$
are positive constant scalars.
A summary of the PSO algorithm optimization is shown as an algorithm in Table 1. Also, the control diagram of the new proposed controller is presented in Fig. 2.
4. Results and analysis
To verify and validate the performance of the proposed online adaptive controller, both simulation and real-time experiment have been performed on 3 DOF right upper limb exoskeleton robot called ULEL, designed with the mechanical characteristics as shown in Tables II and III by RB3D company for arm rehabilitation. In this study, the tests have been applied on 2 DOF (Shoulder and elbow) only, allowing flexion/extension movements. The parameter values of the PSO that have been used are summarized in Table IV.
4.1. Simulation results
Initially, simulation has been done using the dynamic model of the exoskeleton A.2. In order to evaluate the robustness of the proposed approach, a sinusoidal resistive and assistive external disturbances forces have been applied in software as shown in Fig. 3 and described by Equation (15). These efforts represent the assistive and resistive rehabilitation therapy, respectively, where the patient has to be completely passive in the first stages of therapy. After a certain period, he will be asked to resist or assist the robot in order to accelerate the recovery of the brain control. The amplitude of the disturbances represents more than 20% of the maximum applied torques on both joints. A comparison between the proposed control law AST-SMC with the classical ST-SMC algorithm has been introduced in order to show the performance of the proposed controller.
\begin{equation} \begin{array}{c} d_{1}\!\left (t\right )=\!\left \{ \begin{array}{l@{\quad}l} -5\,\sin \!\left (0.5\,t-\frac{\pi }{2}\right ) & \text{ if }t\in \!\left [20,40\right ]\\[3pt] +5\,\sin \!\left (0.5\,t-\frac{\pi }{2}\right ) & \text{ if } t\in \!\left [60,80\right ]\\[3pt] 0 & \text{otherwise} \end{array}\right .\\[3pt] \\[2pt] d_{2}\!\left (t\right )=\!\left \{ \begin{array}{l@{\quad}l} -2\,\sin \!\left (0.7\,t+\frac{\pi }{2}\right ) & \text{ if }t\in \!\left [20,40\right ]\\[3pt] +2\,\sin \!\left (0.7\,t+\frac{\pi }{2}\right ) & \text{ if }t\in \!\left [60,80\right ]\\[3pt] 0 & \text{otherwise} \end{array}\right . \end{array} \end{equation}
The classical super twisting control law is the same as the adaptive one 4, except that it differs by its non-adaptive parameters (fixed)
$K_{1}$
and
$K_{2}$
such:
\begin{align*} K_{1}=\!\left [\begin{array}{c@{\quad}c} 6 & 0\\[5pt] 0 & 4 \end{array}\right ],K_{2}=\!\left [\begin{array}{c@{\quad}c} 6 & 0\\[5pt] 0 & 4 \end{array}\right ] \end{align*}
The desired trajectories that have been used are sinusoidal and expressed as the following:
\begin{equation} \!\left \{ \begin{array}{c} q_{d1}\!\left (t\right )=\frac{\pi }{10}\,\sin \!\left (0.5t-\frac{\pi }{2}\right )+\frac{\pi }{6}\\[5pt] q_{d2}\!\left (t\right )=\frac{\pi }{10}\,\sin \!\left (0.7t+\frac{\pi }{2}\right )+\frac{\pi }{6} \end{array}\right . \end{equation}
4.1.1. Discussion:
Fig. 4 shows the minimization of the objective function and its convergence almost to zero, which means the optimal parameters are found. In other words, the figure illustrates the capability of PSO algorithm that have been used in finding the proper eigenvalues needed for the controller.
Fig. 5 depicts the behavior of the controller parameters when they have been in search online. Even though the parameters
$K_{1}$
and
$K_{2}$
have been initialized with random numbers between
$\!\left [1\;5\right ]$
and
$\!\left [1\;3\right ]$
, respectively, the proposed technique is able to reach the optimum values as demonstrated in the figure. Nevertheless, starting with proper values will make reaching the optimal parameters faster. Although
$K_{1}'$
and
$K_{2}'$
parameters have started with bad initial guesses which gives a relatively poor tracking performance, Fig. 4 shows the convergence to the optimal values.
For joint position trajectory tracking, Figs. 6 and 7 cover and illustrate tracking position and its error, which shows the superiority and performance of the proposed controller compared with the classical ST-SMC in terms of trajectory tracking and robustness with respect to external disturbances. In fact, moments 20s and 80s show that the error of the classic controller is almost double the error of the proposed one in shoulder joint, which confirms its efficiency over the classic controller.
The joint velocity trajectory tracking is represented in Figs. 8 and 9, where they illustrate and prove the performance of the proposed approach. In fact, the onset and cessation times of external disturbances highlight the effectiveness of the proposed adaptive controller compared to other controller.
Fig. 10 shows torques applied to shoulder and elbow joints. Despite the resistive and assistive perturbations, the proposed PSO algorithm allows the proposed controller to produce an adequate and optimal torques.
The root mean square (RMS) of steady state of tracking errors in position and velocity is illustrated in Fig. 11. It shows the effectiveness of the proposed adaptive controller over the classical ST-SMC, knowing that the classic ST-SMC algorithm parameters were carefully chosen.
4.2. Experimental results
Experiment validation has been carried out in order to prove and verify the effectiveness of the proposed adaptive controller. The exoskeleton actuators use a current controlled DC motors of MAXON company. An encoder has been used to measure the real position of each joint. Limit switches, current limit, mechanical stop, and all safety requirements were considered and activated during the experiments. The first-order Euler’s solver has been used with sufficiently small sampling time of 0.004s. The experimental setup depicted in Fig. 12 is based on a computer equipped with a dSpace DS1103 PPC real-time controller card and an installed MATLAB/Simulink software with the dSpace control desk application.
Before the experiments on a real subject start, the following safety protocol has been considered and taken into account:
-
1. Run an experimentation testing without a subject.
-
2. Mechanical stops adjustments, in order to avoid unwanted positions.
-
3. Set the limits of the electrical currents of the actuator’s motors, in order to bound the applied torques.
-
4. Emergency stop buttons are always available near the handle.
-
• Experimental scenarios
Tests have been applied on the ULEL exoskeleton alone without any wearer and then with two subjects that have the details described in Table V.
Position trajectory tracking of joint 2 with the presence of disturbances, trial with subjects 1 and 2: (a) Position trajectory tracking; (b) Position tracking error.
Velocity trajectory tracking of joint 1 with the presence of disturbances, trial with subjects 1 and 2: (a) Position trajectory tracking; (b) Position tracking error.
Velocity trajectory tracking of joint 2 with the presence of disturbances, trial with subjects 1 and 2: (a) Position trajectory tracking; (b) Position tracking error.
During tests, the subjects were told to:
-
• First, remain completely passive without applying any resistive or assistive efforts :
$\tau _{hum}=0$
-
• Second, exert an assisting muscular effort :
$\tau _{hum}\neq 0$
applied in the same direction of
$\tau _{exo}$
and a resistive muscular effort :
$\tau _{hum}\neq 0$
applied in the opposite direction of
$\tau _{exo}$
in certain times duration.
4.2.1. Discussion:
Figs. 13 and 14 show a sustaining trajectory tracking in position for both joints 1&2. Although human subjects have been told to remain completely passive, it is not doable in reality. In fact, moments in intervals [40, 60] seconds and [100, 120] seconds show that subjects were not completely passive. Even though the impassiveness of the subjects, the error is negligible, which demonstrates the quality and efficiency of the proposed controller.
Figs. 15 and 16 describe the trajectory tracking in velocity for both joints 1 and 2. The quality and efficacy of pursuing the desired velocity despite the presence of the measurement noise can be seen from these figures.
The control inputs depicted in Fig. 17 explain efforts applied by the controller on the actuators and illustrate clearly the impact of the subjects on the control torques. Despite the varying sizes and weights of the wearers, the proposed controller generates smooth torques for each subject.
Since the controller has been applied to couple joints (shoulder and elbow), it is difficult for the wearers to perform the same external resistive and assistive efforts simultaneously on both articulations, due to that, external disturbances impact on applied torques as shown in Fig. 18 which might be unclear. However, it is clear in position and velocity trajectory tracking.
Figs. 19 and 20 show position trajectory tracking of the joint 1 and joint 2 with the presence of the external disturbances. Despite the strong perturbation forces especially in joint 2, the proposed controller remains stable and keeps the position trajectory tracking smoother.
Figs. 21 and 22 show velocity trajectory tracking of joint 1 and joint 2 with the external disturbances applied. It can be seen clearly that the proposed controller can easily deal with the external disturbances with different amplitudes, which emphasizes the performance of the proposed controller.
Fig. 18 shows the generated torques in joint 1 and joint 2 for each subject. From these results, it can be seen that the proposed controller responded in the same manner as the testing subjects. In addition, the interactions with the exoskeleton were different for both wearers and the controller held the smoothness of the torques, which explains the optimization of the torques in parallel with the controller parameters. All this proves the efficiency and performance of the proposed controller.
5. Conclusion
In this paper, a real-time adaptive controller has been proposed for an upper limb exoskeleton rehabilitation robot. This approach is based on the super twisting algorithm where its parameters are calculated using a new online PSO algorithm. The new proposed technique lies in defining the cost function of the optimization criterion. Furthermore, the adaptation is based on optimizing the variation of the parameters and not the parameters themselves. The parameters of the novel algorithm are adjusted according to the conditions of the closed loop stability analysis. This approach has been tested in simulation than in real-life experiment with two healthy subjects. The simulation obtained results show the superiority of the proposed controller over the classical one. Therefore, experiments were taken only using the proposed controller. Indeed, two cases have been studied, whether the subjects remain passive or exert external resistive/assistive forces. The satisfactory experimental results with the context of rehabilitation confirm the effectiveness of the proposed approach and show great robustness against external disturbances compared with the previous studies that use an offline/non-adaptive controllers. As a follow-up to this present work, applying impedance control to support the interaction forces between the wearer and the exoskeleton will be investigated.
Author contributions
All the authors have contributed evenly to make this research paper.
Financial support
This research received no specific grant from any funding agency, commercial or not-for-profit sectors.
Competing interests
The authors declare no competing interests exist.
Ethical approval
None.
A. Appendix
A.1. Lyapunov stability analysis of the super-twisting algorithm
Consider the standard super twisting algorithm (ST) with a perturbation term
\begin{equation} \!\left \{ \begin{array}{l} \dot{\sigma }_{1}=-k_{1}\sqrt{\!\left |\sigma _{1}\right |}sign\!\left (\sigma _{1}\right )+\sigma _{2}\\[5pt] \dot{\sigma }_{2}=-k_{2}\,sign\!\left (\sigma _{1}\right )+\xi (t,\sigma ) \end{array}\right . \end{equation}
where
$\sigma _{1},\sigma _{2}\in \mathbb{R}$
and the perturbation term
$\xi$
is uniformly bounded (
$\!\left |\xi \right |\lt \delta$
).
Let us prove the stability of the equilibrium point (
$\sigma _{1},\sigma _{2})=(0,0)$
using the work stated in ref. [Reference Moreno33, Reference Moreno and Osorio36].
Consider the following Lyapunov function:
\begin{equation} V=\zeta ^{T}P\zeta \end{equation}
where
$\zeta =[\sqrt{\!\left |\sigma _{1}\right |}sign\!\left (\sigma _{1}\right ),\sigma _{2}]^{T}$
and
$P$
is a positive definite matrix.
Notice that
$V(\zeta,t)$
is continuous and differentiable except when
$\sigma _{1}=0.$
In fact, when
$\sigma _{1}\neq 0$
,
$\dot{V}$
exists and is negative definite. However, before arriving at the equilibrium point
$(\sigma _{1},\sigma _{2})=(0,0)$
, the solution of system A1 crosses the line
$\sigma _{1}=0$
when
$\sigma _{2}\neq 0$
. This means that the derivative of the Lyapunov function exists almost everywhere while
$V(t)$
decreases until the system reaches the equilibrium. As presented in ref. [Reference Moreno and Osorio36],
$V(t)$
is a strong Lyapunov function for A1 in the form of Equation (A2).
Moreover, this Lyapunov function is positive definite but radially unbounded
\begin{equation} \gamma _{min}(P)\!\left \Vert \zeta \right \Vert ^{2}\leq V\leq \gamma _{max}(P)\!\left \Vert \zeta \right \Vert ^{2} \end{equation}
where
$\!\left \Vert \zeta \right \Vert _{2}^{2}=\!\left |\sigma _{1}\right |+\sigma _{2}^{2}$
represents the Euclidian norm of
$\zeta$
.
The construction of suitable positive definite matrices
$P=P^{T}$
, provided in ref. [Reference Moreno and Osorio36], is based on the following algebraic LMI equation:
\begin{equation} \!\left [\begin{array}{cc} A^{T}P+PA+\epsilon P+\delta ^{2}C^{T}C & PB\\[5pt] B^{T}P & -1 \end{array}\right ]\lt 0 \end{equation}
where
\begin{align*} A=\!\left [\begin{array}{c@{\quad}c} -\frac{1}{2}k_{1} & \frac{1}{2}\\[5pt] -k_{2} & 0 \end{array}\right ];\; B=\!\left [\begin{array}{c} 0\\[5pt] 1 \end{array}\right ];\; C=\!\left [\begin{array}{cc} 1 & 0\end{array}\right ] \end{align*}
with
$k_{1}$
and
$k_{2}$
are positive gains.
Using the vector
$\zeta =[\sqrt{\!\left |\sigma _{1}\right |}sign\!\left (\sigma _{1}\right ),\sigma _{2}]^{T}$
, the system A1 can be rewritten as
\begin{equation} \dot{\zeta }=\frac{1}{\!\left |\zeta _{1}\right |}(A\zeta +B\tilde{\xi }(t)) \end{equation}
where the transformed perturbation
$\tilde{\xi }(t,\zeta )=\!\left |\zeta _{1}\right |\xi (t,\sigma )$
satisfies
$\!\left |\tilde{\xi }(t,\zeta )\right |\leq \delta \!\left |\xi _{1}\right |$
. As a consequence,
$\omega (\tilde{\xi },\zeta )=-\tilde{\xi }^{2}(t,\zeta )+\delta ^{2}\zeta _{1}^{2}\geq 0$
.
As
$k_{1}$
and
$k_{2}$
are positive gains, the system (A,B,C) is observable and controllable, so we can use the bounded-real [Reference Boyd, Ghaoui, Feron and Balakrishnan40]. It is shown that the linear matrix Inequality A4 is feasible if and only if the linear system defined by
$H(s)=\delta C(sI-A)^{-1}B$
is nonexpansive, that is,
\begin{align*} \underset{\omega }{max}\!\left |H(j\omega )\right |\lt 1 \end{align*}
This implies the following condition:
\begin{align*} \underset{\omega }{max}\!\left |G(j\omega )\right |\lt \frac{1}{\delta } \end{align*}
where
\begin{align*} G(s)=C(sI-A)^{-1}B=\frac{\frac{1}{2}}{s^{2}+\frac{1}{2}k_{1}s+\frac{1}{2}k_{2}} \end{align*}
The previous equality yields two conditions for gains. By choosing one of them
\begin{align*} \underset{\omega }{max}\!\left |G(j\omega )\right |=\frac{1}{k_{2}}ifk_{1}^{2}\gt 4k_{2} \end{align*}
We can then deduce conditions on gains
$k_{1}$
and
$k_{2}$
as follows:
\begin{align*} \begin{array}{c} k_{2}\gt \delta \\[5pt] k_{1}^{2}\gt 4k_{2} \end{array} \end{align*}
Consider the Lyapunov function defined by Equation (A2). Its derivative writes
\begin{align*} \dot{V}(\zeta )\begin{array}[t]{l} =\frac{1}{\!\left |\zeta _{1}\right |}\!\left [\zeta \tilde{\xi }\right ]^{T}\!\left [\begin{array}{cc} A^{T}P+PA & PB\\[9pt] B^{T}P & 0 \end{array}\right ]\!\left [\zeta \tilde{\xi }\right ]\\[9pt] \leq \frac{1}{\!\left |\zeta _{1}\right |}\!\left \{ \!\left [\zeta \tilde{\xi }\right ]^{T}\!\left [\begin{array}{cc} A^{T}P+PA & PB\\[9pt] B^{T}P & 0 \end{array}\right ]\!\left [\zeta \tilde{\xi }\right ]+\omega (\tilde{\xi },\zeta )\right \} \\[9pt] \leq \frac{1}{\!\left |\zeta _{1}\right |}\!\left [\zeta \tilde{\xi }\right ]^{T}\!\left [\begin{array}{cc} A^{T}P+PA+\delta ^{2}C^{T}C & PB\\[9pt] B^{T}P & -1 \end{array}\right ]\!\left [\zeta \tilde{\xi }\right ]\\[9pt] \leq \frac{1}{\!\left |\zeta _{1}\right |}\!\left [\zeta \tilde{\xi }\right ]^{T}\!\left [\begin{array}{cc} A^{T}P+PA+\epsilon P-\epsilon P+\delta ^{2}C^{T}C & PB\\[9pt] B^{T}P & -1 \end{array}\right ]\!\left [\zeta \tilde{\xi }\right ]\\[9pt] \leq -\frac{\epsilon }{\!\left |\zeta _{1}\right |}\zeta ^{T}P\zeta \end{array} \end{align*}
Finally,
\begin{equation} \dot{V}\leq -\frac{\epsilon }{\!\left |\zeta _{1}\right |}\zeta ^{T}P\zeta =-\frac{\epsilon }{\!\left |\zeta _{1}\right |}V(\zeta ) \end{equation}
From Equation (A3), we deduce the following inequality:
\begin{align*} \!\left |\zeta _{1}\right |\leq \!\left \Vert \zeta \right \Vert _{2}\leq \frac{V^{{1}/{2}}(\zeta )}{\gamma _{min}^{{1}/{2}}\!\left \{ P\right \} } \end{align*}
We can then conclude that
$\dot{V}$
satisfies
\begin{align*} \dot{V}\leq -\alpha V^{{1}/{2}}(\zeta ) \end{align*}
where
\begin{equation} \alpha =\epsilon \gamma _{min}^{{1}/{2}}\!\left \{ P\right \} \end{equation}
The previous result guarantees the finite convergence of vector
$\sigma =\!\left [\sigma _{1},\sigma _{2}\right ]^{T}$
to zero. This time is bounded by
\begin{equation} T_{0}=\frac{2V^{{1}/{2}}(\zeta (0))}{\alpha } \end{equation}
where
$\zeta (0)$
is the initial value of
$\zeta$
and
$\alpha$
is given by Equation (A7).
A.2. The dynamic model of the ULEL exoskeleton:
The following dynamic model and characteristics represent the empty exoskeleton.
Model parameters:
Masses
$(kg)$
:
\begin{align*} \!\left \{ \begin{array}{c} m_{1}=5.5058\\[5pt] m_{2}=5.2909 \end{array}\right . \end{align*}
Length
$(m)$
:
\begin{align*} l_{1}=0.2999 \end{align*}
Gravity centers according to articulations references
$(m)$
:
\begin{align*} \!\left \{ \begin{array}{c} x_{m_{1}}=0.1050,y_{m_{1}}=0\\[5pt] x_{m_{2}}=0.0945,y_{m_{2}}=-0.0500 \end{array}\right . \end{align*}
The moment of inertia according to the rotational axes
$(kg.m^{2})$
:
\begin{align*} \!\left \{ \begin{array}{c} I_{1}=0.1142\\[5pt] I_{2}=0.1221 \end{array}\right . \end{align*}
Viscous friction
$(N.m.s/rad)$
:
\begin{align*} \!\left \{ \begin{array}{c} v_{f_{1}}=10.4000\\[5pt] v_{f_{2}}=5.2000 \end{array}\right . \end{align*}
Dry friction
$(N.m)$
:
\begin{align*} \!\left \{ \begin{array}{c} d_{f_{1}}=3.9000\\[5pt] d_{f_{2}}=1.6193 \end{array}\right . \end{align*}
Trigonometric transformation:
\begin{align*} \!\left \{ \begin{array}{c} S1=\sin (q_{1})\\[5pt] C1=\cos (q_{1})\\[5pt] S12=\sin (q_{1}+q_{2})\\[5pt] C12=\cos (q_{1}+q_{2}) \end{array}\right . \end{align*}
Inertia matrix
$M$
:
\begin{align*} \begin{array}{l} M_{11}=\begin{array}[t]{l} 2.l_{1}.m_{2}.(x_{m_{2}}.C1+y_{m_{2}}S1).C12-2.l_{1}.m_{2}.(y_{m_{2}}.C1-x_{m_{2}}.S1).S12+\\[5pt] (l_{1}^{2}+x_{m_{2}}^{2}+y_{m_{2}}^{2}).m_{2}+m_{1}.(x_{m_{1}}^{2}+y_{m_{1}}^{2})+I_{1}+I_{2} \end{array}\\[5pt] M_{12}=l_{1}.m_{2}.(x_{m_{2}}.C1+y_{m_{2}}S1).C12+l_{1}.m_{2}.(x_{m_{2}}.S1-y_{m_{2}}.C1).S12+m_{2}.(x_{m_{2}}^{2}+y_{m_{2}}^{2})+I_{2}\\[5pt] M_{21}=M_{12}\\[5pt] M_{22}=I_{2}+m_{2}.(x_{m_{2}}^{2}+y_{m_{2}}^{2}) \end{array} \end{align*}
Coriolis and centrifugal matrix
$C$
:
\begin{align*} \begin{array}[t]{l} C_{11}=l_{1}\,\dot{q_{2}}\,((x_{m_{2}}S1-y_{m_{2}}C1)\,C12-S12\,(x_{m_{2}}C1+y_{m_{2}}S1))\,m_{2}\\[5pt] C_{12}=\begin{array}[t]{l} (-l_{1}\,m_{2}\,(x_{m_{2}}C1+y_{m_{2}}S1)\,S12-l_{1}\,m_{2}\,(y_{m_{2}}\,C1+x_{m_{2}}S1)\,C12)\,\dot{q_{1}}+\\[5pt] (-l_{1}\,m_{2}\,(x_{m_{2}}C1+y_{m_{2}}S1)\,S12+l_{1}\,m_{2}\,(x_{m_{2}}\,S1-y_{m_{2}}C1)\,C12)\,\dot{q_{2}} \end{array}\\[5pt] C_{21}=\begin{array}[t]{l} (2\,l_{1}\,m_{2}\,(y_{m_{2}}C1-x_{m_{2}}S1)\,C12+l_{1}\,m_{2}\,(x_{m_{2}}\,S1-y_{m_{2}}C1)\,C12+\\[5pt] l_{1}\,m_{2}\,(x_{m_{2}}C1+y_{m_{2}}S1)\,S12\,\dot{q_{1}}) \end{array}\\[5pt] C_{22}=0 \end{array} \end{align*}
Gravity vector
$G$
:
\begin{align*} \begin{array}{l} G_{1}=m_{1}.g.(x_{m_{1}}.S1+y_{m_{1}}.C1)+m_{2}.g.(x_{m_{2}}.S12+y_{m_{2}}.C12+l_{1}.S1)\\[5pt] G_{2}=m_{2}.g.(x_{m_{2}}.S12+y_{m_{2}}.C12) \end{array} \end{align*}
where
$g=9.81\,m/s{{}^2}$
Dissipation vector
$F$
:
\begin{align*} \begin{array}{l} F_{1}=v_{f_{1}}.\dot{q_{1}}+d_{f_{1}}.sign(\dot{q_{1}})\\[5pt] F_{2}=v_{f_{2}}.\dot{q_{1}}+d_{f_{2}}.sign(\dot{q_{2}}) \end{array} \end{align*}
