FRACTIONAL MULTI-LOOP ACTIVE DISTURBANCE REJECTION CONTROL FOR A LOWER KNEE EXOSKELETON SYSTEM

. Rehabilitation Exoskeleton is becoming more and more important in physiotherapists’ routine work. To improve the treatment performance, such as reducing the recovery period and/or monitoring and reacting to unpredictable situations, the rehabilitation manipulators need to help the patients in various physical trainings. A special case of the active disturbance rejection control (ADRC) is applied to govern a proper realisation of basic limb rehabilitation trainings. The experimental study is performed on a model of a flexible joint manipulator, whose behaviour resembles a real exoskeleton rehabilitation device (a one-degree-of-freedom, rigid-link, flexible-joint manipulator). The fractional (FADRC) is an unconventional model-independent approach, acknowledged as an effective controller in the existence of total plant uncertainties, and these uncertainties are inclusive of the total disturbances and unknown dynamics of the plant. In this work, three FADRC schemes are used, the first one using a fractional state observer (FSO), or FADRC1, second one using a fractional proportional-derivative controller (FPD), or FADRC2, and the third one a Multi-loop fractional in PD-loop controller and the observer-loop (Feedforward and Feedback), or FADRC3. The simulated Exoskeleton system is subjected to a noise disturbance and the FADRC3 shows the effectiveness to compensate all these effects and satisfies the desired position when compared with FADRC1 and FADRC2. The design and simulation were carried out in MATLAB/Simulink.


out in MATLA
/Simulink.

Introduction

An exoskeleton is an electromechanical framework that reflects the form and function of the body of an operator wearing it.In rehabilitation, robotics is a new topic that is believed to be a promising way to automate training.Robotic rehabilitation can take the place of a therapist's physical training activities, allowing for more rigorous repeated motions and lower therapy costs [1].Fixed trajectory controllers move the user's limb along a predefined movement path.They provide the smallest amount of freedom because user engagement is not taken into account [2].There has been a substantial amount of research in several sectors required for developing and increasing the performance of these devices, and there are numerous obstacles in this area, one of the most important of them being the system control.The exoskeleton control scheme can be divided into position, torque/force, and force interaction controllers based on physical factors.To ensure that the exoskeleton joints revolve at the proper angle, the position control technique is typically used.Some exoskeleton axes have fixed joint positions due to rehabilitation goals.The PD position controller is used for these axes, and the axes are locked at a specified angle pos tion [3][4][5].

Exoskeletons with many degrees of freedom have been implemented with proportional-integralderivative (PID) controllers to control the leg's motion, despite its ease of implementation, the usage of PID control is limited by the convergence analysis and coefficient adjustments [6].A backstepping controller based on the shank-orthosis system was proposed, where backstepping can commonly address stability, monitoring, and robustness control problems under less restrictive settings than other methods [7,8].

In recent years, a fuzzy controller with a bang-bang controller with a high accuracy and fast response has been proposed for the output torque control.An adaptive self-organising fuzzy controller is developed for the rehabilitation Exoskeleton in this study, its fuzzy sliding surface can help to reduce the number of fuzzy rule [9].The self-organising learning mechanism is employed to modify fuzzy rules in real time.A collection of fuzzy IF-THEN rules is used to create an intelligent lower extremity rehabilitation training system controlled by fuzzy or adaptive fuzzy controllers [10,11].As a result, another intelligent control system based on a neuro-fuzzy controller for upper limbs was suggested in [12], to construct a power assist exoskeleton.In order to achieve an accurate control performance, ANN-based model predictive control (MPC) methods are used, despite the ability to approximate nonlinear properties, real-time performance is constrained, and all of these control applications are lim ted [13].

Any system uncertainties, such as exogenous disruptions, unmodelled dynamics, and parameter perturbations, have a significant impact on the performance of a control system.The development of the any controller that attempts to fulfil these objectives while also assuring disturbance rejection and strong tracking performance in the face of huge uncertainty is complicated [14].Because of this, anti-disturbance approaches are used for both external and internal loop controllers and estimators have been widely employ d [15].

Regarding the biomechanics of the exoskeleton, a sliding mode control (SMC) may be an appropriate solution due to its robustness to both internal and external system uncertainties.One of these solutions proposed a control scheme with a model-free decentralised output feedback adaptive high-order sliding mode control to solve the trajectory tracking problem in each degree of freedom of the exoskeleton.In this work, a second-order adaptive sliding mode controller based on the super-twisting algorithm drives the exoskeleton articulations to track the proposed reference trajectories [16].The main problem of the SMC is the chattering.Many studies partly solved this thanks to the ability of the Exoskeleton to follow the optimised trajectories [17].To achieve an optimal performance, the SMC parameters should be chosen carefully.Genetic Algorithm GA [18], Particle Swarm Optimization PSO [19], Grey-wolf optimization [20], and Ant colony optimization [21] are examples of common optimisation methods that are used in exoskeleton devices.GA is used to determine the optimal sliding surface and the sliding control law.It is easy to use and capable of finding global optima, which can be used to improve the structure of optimisation systems [22].
Many works also deal with the reduction of chattering by employing a synergetic control (SC) technique.This control methodology has been used to govern highly coupled and complicated nonlinear systems.The SC is based on state-space theory.An independent manifold is developed to fulfil the necessary control criteria in the situation of parametric and nominal uncertainty, and a controller constructed based on SC might drive the system's state variables in such a manner as to track it.It eliminates chattering while maintaining the same control systems as SMC [23].

Th ADRC is a modern robustness regulation conceptual model based on the standard PID control algorithm, first introduced in [24], and further developed in [25].The ADRC has recently been used in the field of rehabilitation systems [26,27].The ADRC was created by merging ESO with a variety of control approaches.

Frac ional order controllers, like CRONE control and expanded fractional control theories based on adaptive control and sliding mode control, have produced improved results in both theory and practice [28,29].The fractional order control has been investigated in robotic and engineering systems and used to obtain reliable performances in industrial systems, despite its inherent complexity [30][31][32][33].Later, utilising various tuning techniques and numerical optimisation, several researchers focused on the design and synthesis of FOPID controllers [34].

The classical extended states observer (ESO) is generalised to a fractional order extended states observer (FESO) in FADRC [35], using fractional calculus.Numerous studies show that the fraction-order active disturbance rejection controller (FADRC) outperforms the integer-order ADRC in terms of robustness, disturbance rejection, and parameter variation uncertainty capacity [36][37][38].

In ter s of robustness against noisy environments and perturbation, some researchers have suggested a FADRC control system that consists of a proportional controller and a fraction-order ESO [39].In contrast, other researchers have suggested an ADRC and fraction-order PID (FOPID) direct torque control technique for the hydro-turbine speed governor system that is load-disturbance tolerant, since the integer order ESO (IESO) has weak high-frequency disturbance prediction capabilities [40].In order to improve the performance of the system's control, a FADRC typically offered a control technique based on a fraction-order ESO (FESO) [35].The paper's contribution can be highlighted by the following points: I.In order to increase the lower limb exoskeleton system's ability to reject disturbances and achieve limited convergence, FADRC has been introduced in this research.


II.

There

s ev
dence for the controlled system's robustness and better convergence properties to satisfy the stability.


III.

A co

ariso
study have been conducted between the proposed FADRC3 and two configurations of FADRC, focusing on error performance indices and the control torque required with minimal chattering.

The paper i organised into several sections.Section 2 introduces the structure, mathematical model calculations of the proposed exoskeleton, and ADRC methodology.Stability analysis is presented in Section 3. Section 4 shows the simulation results and discussion about the system.Conclusions and future work are discussed in Section 5.


Materials and

ethods

The fundamenta
ideas underlying the knee-joint mathematical model, the ADRC control components, and the proposed controllers are established in this part.


Modelling of

wer Limb Robotic Rehabilitation Exoskeleton

The active exo
keleton electromechanical device described in this study is worn by a human operator and is intended to improve the wearer's physical performance.Direct motor driving through reducers provides the action of the hip and knee.The exoskeleton's swing leg is the primary focus of the effort.

Generally, the d namic model of the lower knee joint motion is [41][42][43]:
J θ = −τ g cosθ − osition θ is the knee joint angle between the actual position of the shank and the full extension position, θ and θ are the knee joint angular velocity and acceleration, respectively.J, A, B, τ g , τ c , τ h are the Inertia, solid friction coefficient, viscous friction coefficient, gravity torque, controller torque and human torque, respectively.Based on the mathematical equation above, designers can see that the system is inherently nonlinear due to nonlinear terms like cos and sign, which makes it difficult to evaluate; nevertheless, the ADRC Handle System has the advantage of being able to control system dynamics as a linear controller.

The single control variable in this work is the position (θ), which must be regulated according to a specified trajectory.Subjects rested upright on a treatment table in a fixed state, with the shank hanging off the table at roughly -45 degrees, which is the rest position.

The formal testing consisted of the following trials that were completed in a specific sequence.moving to an angle of -45 degrees of knee flexion to reach -90 degrees, and then returning to -45 degrees, after that, moving to an angle of 0 degrees of knee flexion and returning to -45 degrees (Initial condition).This training was repeated every 4 seconds.The flat series of Maxon's EC90 brushless DC motors was used for the exoskeleton.It can provide a constant torque of up to 560 mN•m.This motor was chosen because it is among the cheapest and smallest models on the market, making it suitable for use in the exoskeleton.

Figure 1 shows the structure of the exoskeleton.The parameters of the exoskeleton system used in this study has been listed in Table 1 [41].


ADRC Methodolog

The primary conc
pt of ADRC is disturbance rejection, which is based on the lack of a precise mathematical model of the system [43,44].The ESO is utilised in the ADRC framework to estimate the disturbance so that it can later be terminated in the control rule.

The ADRC is a for of nonlinear control approach that comes in two types.


Linear ADRC (LAD

)

The ADRC is mainl
composed of tracking differential (TD), extended state observer (ESO), and state error feedback (SEF).A general nth-order SISO dynamical Table 1.Parameters of the exoskeleton [41].

system, which may be used to describe a wide range of physical systems, can be formulated as a fundamental input-output representation for simplicity and without aba . , y + W + b 0 u . (2)
According to Equ (t, θ, θ, W ) + b 0 u , (3)
where u represents the control input (to be designed), (y) represents the only observable system output signal, W denotes the unidentified external disturbance, f (.) demonstrates probably the most frequently unknown internal (state-dependent and feasibly nonlinear) dynamics of the process, and (b 0 ) signifies the unknown input correction factor.The purpose of the control is to create a system input signal (u) that allows the system output (y) to track the target value (y d ), an irrespective influence of unmolded/unknown system components (seen as an acting perturbation).A new term (δ) can be introduced to tackle both the internal modelling mismatch f (.) and the unknown external disturbance (W ), resulting in a required generalised controllable canonical 0 u , δ = f (.) + W ,(4)
where (δ) is assumed to be the total disturbance.The modelling uncertainty related to (b) can also be treated as part of the total disturbance, so that Equatio 3 no.3/2023

Fractional Multi-loop Active Disturbance Rejection Control
δ )
The plant model can be described using the statespace representation as:
ẋ = Ax + Bu + Eδ , y = Cx . (6)
The plant matrices ar elling the total disturbances, then
x = [x 1 , x 2 , . . . , x n , x n+1 ] ∼ = [y, ẏ, . . . , y n , δ] . (7)
The form of the most widely use = C x . (8)
The aim is (x → x), so that
x = [x 1 , x2 , . . . . . . , xn , xn+1 ] ∼ = ŷ, ŷ, . . . . . . , δ ,(9)
where, L = [l 1 , l 2 , . . ., l n , l n+1 ] represents the observer gains and depends on the observer's bandwidth (w 0 ).If one chooses w 0 = 4w c , where w c is the controller bandwidth, then it is easy to calculate the elements of the observer matrix gains (l 1 = 3w o , l 2 = 3w 2 o , l 3 = w 3 o ) according to [25].The second part of th ne by linear proportional-derivative controller (PD) (10) where K p is the proportional gain and K d is the derivative gain.These values are calculated from the controller bandwidth (w c ) and damping ratio (ς) gn specifications [45],
u 0 = K p ϵ + K d ε = K p (r − x1 ) + K d ( ṙ − x2 ) ,K p = w 2 c , K d = 2ςw c . (11)
In the ADRC technique, the control rule is composed of a controller (u 0 ), which is in charge of ed task (i.e.y → y d ), and a disturbance rejection estimation term ( δ), as shown below:
u = u 0 − δ b 0 , (12)
where δ = x3 is the estimated total disturbance (in the system under test) and (u 0 ) is the proposed control signal for the currently disturbance-free system, which is designed to match some predetermined closed-loop requirements.The suggested control rule can be introduced in the extend stimating loop is well tuned (i.e.δ = δ), and this gives, theoretically:
y n = δ + b 0 (u 0 − δ) b 0 = u 0 . (13)
This reduces the system to a basic cas

de of integrators, makin
it easier to govern due to the system's fundamental adaptability to any perturbation.


Fractional ADRC (FADRC)

In this work, we use three approaches of FADRC, the first approach, FADRC1, depends on an ESO modified to a fractional order extended states observer FESO, the second approach, FARC2, depends on a Frac

onal PD controll
r, and finally, the third one is the proposed controller, FADRC3, combining FADRC1 and FADRC2.


FADRC1 approach

All the Equations ( 3)-( 12) for LADRC, can be used only by FESO, instead of linear LESO, so that we deal with modified ADRC (FADRC1).The definition of fractional calculus proposed by Caputo is widely used in fractional order control [46,47 tor:
D α =                      d α d α t f orα ≻ 0 1 f orα = 0 , t a (dt) −α f orα ≺ 0 (14)
where a and t denote the limits of the operation and (α) denotes the fractional order, which lies between (0 to 1).The important key in FADRC is the ESO, so t           D α1 x1 = x2 + l 1 (y − x1 ) D α2 x2 = x3 + Bu + l 2 (y − x1 ) , D α3 x3 = l 3 (y − x1 )(15)
where the observers gains L = [l 1 , l 2 , l 3 ] are determined by rules in [45], or using any optimisation methods.Figure 2. shows the general structure of FADRC1 for the Exoskeleton control system.The nonlinear structure (fractional term) is used in ESO to comprehensively estimate the un-modelled errors and external disturbances.In order for the FESO to estimate the state more quickly and accurately, and ultimately for the FADRC to achieve a superior control performance, the fractional-order integration has an additional weight function and the initial stage has a bigger integral response value.The FESO assesses both the overall disturbance and the variable structure dynamic states, resulting in a smaller observer bandwidth.Pursuing the analysis from [45], the bandwid d to settling time (τ s ) of the closed-loop system, according to the following formula:
w c = 10 τ s . (16)
In this application, the specification of the settling time of the controlled system is chosen to be τ s = 0.408 s.The observer and PD controller gains can be calculated according to the above equation with (w c = 24.5 rad•sec −1 ).The best fractional observer terms (α 1 , α 2 , α 3 ) values can be obtained by any optimisation algorithm [48][49][50].In this work, PSO parameters are chosen according to the trial and rror method as follows: Iterations = 30; inertia = 1.5; c1 = 2; c2 =2; swarm_size = 30; no_of_param = 3;

The PSO optimisation result gives the bes

value of Fractio
al observer integrators terms (α 1 , α 2 , α 3 ) as (0.5243, 1, 0.4513), respectively.


FADRC2 approach

In this section, we use y a PD controller modified by using fractional term of controller derivative:
u 0 = K p ϵK d D αϵ ϵ . (17)
The controller parameters (K p , K d ) are calculated according to Equation (11).One parameter to optimise for FADRC2 is the fractional term of derivative (α c ).

The PSO optimisation result gives the best value of Fractional controller te

(α c ) as (0.14
9). Figure 3 shows the general structure of FADRC2 for the Exoskeleton control system.


FADRC3 approach

In this section, we combine the two approaches, FADRC1 for minimisation of the chattering and FADRC2 for reducing the error, to construct a multifractional approach FADRC3 for improving the trajectory performance and increase th response accuracy.The same design parameters for FADRC1 and FADRC2 are used in this proposed approach.

Integral of the absolute error (IAE), Integral square error (ISE), Integral square of the control signal (ISU), Integral absolute of the control signal (IAU), and root mean square error (R.M.S.E) are the performance indices chosen d − y)|dt ISE = t 0 (y d − y) 2 dt R.M.S.E = 1 n n 1 (y d − y) 2 , ISU = t 0 u 2 dt IAU = t 0 |u|dt(18)
where, (y d ) is the reference input signal, (y) is the output of the system, (y d − y) denotes the error of the system, and (u) is the control output.The values were determined based on the minimum value of the index, which recommends the best performance [51,52].The IAU performance index provides a measure of chatter reduct

n in the control si
nal [53], whereas ISU refers to the controlling effort needed for a controller.


Stability analysis

One of the main issues in the control system design is how to guarantee the stability of the controlled sy tem [54].In this part, the stability analysis will be conducted based on the pole-placement theory.

Lemma 1: Considering the system of Equation ( 3) with the parametric uncertainties f , the control law developed based on FADRC can lead to an asymptotic convergence of tracking error to zero for a given desired trajectory.This will give the condition of bounde -input bounded-output (BIBO) stability of the closed-loop system in terms of roots proposed control algorithm is initiated by letting:
ϵ α i = x α i − x ∧α , i = 1, 2, 3, . .

n is selected sufficien
ly large, the FESO is BIBO stable and the whole closed-loop system is BIBO.


Results and Discussion

To verify the controller's robustness, FADRC will be simulated for two different cases in this part.The simulation results will be compared using many disturbances and noise criteria.The desired trajectory for training is a sinusoidal input with the starting angle of the exoskeleton of (-45 deg, or -0.785 rad), and the maximum angle of (-

deg, or -1.57rad),
t knee flexion, while to a maximum of (0 deg, or 0 rad), at knee extension.


No Disturbance case

To assess the controller's performance, it must be ran without any payload (no human torque impact τ h = 0) and without any disruptions or noises.FADRC1, FADRC2 and FADRC3 performances are shown in Figure 4. (Desired vs Real output).Figure 5 depicts the difference in the knee position between the desired and actual positions for all controllers.The FADRC3 control approach achieves the least tracking error, proving its effectiveness and superiority to FRADC1 and FADRC2, as shown in Table 2, with the smallest R.M.S.E (0.0039).When comparing the performance index (R.M.S.E) for the best case of FADRC2 and the proposed FADRC3, a reduction of 48 % is observed.It can be stated that the FADRC3 controller outperforms the FADRC1 and FADRC2 controllers due to the additional degree of freedom.

Figure 6 depicts the control efforts required to study the control torque (τ c ) or u(t) for all control systems.When compared to FADRC1 and FADRC2, the experimental results reveal that the FADRC3 control approach produces the smallest control effort required for a controller (ISU = 108) and the largest measure of chattering reduction in the control signal index (IAU = 39.63)due to the presence of two fractional terms, one at the

edback path (FESO) and
the other at the feedforward path (FPD), as can be seen in Table 3.


With Disturbance case

In this case, the performance of the proposed controller under disturbance application is assessed.The joint position is measured practically by an absolute encoder mounted at the rotational shaft of knee joint, which may cause a noise in the measurement.A noise can also be produced by the exoskeleton user during the training.It considers a condition of disturbance by a payload of 0.5 kg, which is introduced in between the flexion/extension at 2 s cycle as shown in Figure 7.

Figure 8 shows the position trajectory performance and how the FADRC3 compensates this effect and returns the trajectory to the desired path in a period of less than 0.2 s. Figure 9 shows the trajectory error, where the FADRC3 approach, at a steady-state condition, has the smallest error (R.M.S.E = 0.0091) when compared to FADRC1 and FADRC2, as shown in Table 4. Figure 10 depicts the control efforts required to study the control torque (τ c ) or u(t) for all control systems.When compared to FADRC1 and FADRC2, the experimental results reveal that the FADRC3 control approach produces the smallest control effort required for a controller (ISU = 115.74)and the largest measure of chatter reduction in the control signal index (IAU = 46.82).Table 5 shows the comparative experiment results.Since FADRC3 is the better approach for a more accurate tracking when compared with other approaches, we can focus on the effectives of ESO (which is the key aspect of ADRC), Figure 11 shows how the estimate x 3 (t) follows its total disturbance target (δ ∧ ).It is clearly shown that estimate x 3 (t) tracks total disturbances very closely, especially at a steady-state condition.As seen from Table 4 and Table 5, all indices are increased with disturbance, which is the desired behaviour due to the disturbance effect.


Conclusions

An FADRC controller has been devised in this research to reduce the lower exoskeleton tracking error for a knee joint motion.To compensate for the effec

of unstruct
red disturbances, the ADRC controller includes a fractional term in both paths, feedback and feedforward (FADRC3).There This study can be continued by implementing the suggested control method in a real-time context with FPGA [56].Utilising modern optimisation techniques to adjust the design parameters for the suggested controller's optimum performance is another continuation of this work [57,58].Additionally, the performance of the suggested controller can be compared to alternative control methods [59][60][61][62][63][64].

Figure 1 .
1
Figure 1.Lower Limb Robotic Rehabilitation Exoskeleton [41].


Figure 2 .
2
Figure 2. The general idea of FADRC1 for Exoskeleton control system.


Figure 3 .
3
Figure 3.The general idea of FADRC2 for Exoskeleton control system.


Figure 4 .
4
Figure 4. Knee position trajectory for comparison of all controllers.


Figure 5 .
5
Figure 5. Knee position error for comparison of all controllers.


Figure 6 .
6
Figure 6.Control torque required comparison of all controllers.


Figure 7 .
7
Figure 7. Constant disturbance payload.


Figure 8 .
8
Figure 8. Knee position trajectory for comparison of all controllers with disturbance.


Figure 9 .
9
Figure 9. Knee position error for comparison of all controllers with disturbance.


Figure 10 .
10
Figure 10.Control torque required comparison of all controllers with disturbance.


Figure 11 .
11
Figure 11.Total disturbances and its estimation of FADRC3 with disturbance.




is a comparison of the tracking responses of the proposed FADRC3 and FADRC controllers, one with the fractional term in the feedforward path (FADRC1) and the second one with the fractional term the in feedback path (FADRC2).The suggested control methods are implemented in the Exoskeleton Intelligently Communicating and Sensitive to Intention (EICoSI) model of a knee exoskeleton.As a result, many findings were reached.This technique combines the advantages of the fractional order robustness with the efficiency of the ADRC controller.A single-leg flexible exoskeleton was used to test this technology.The dynamic model was reorganised to become an extended state space.A fractional order ADRC was utilised to actively estimate this disturbance via ESO, with a FPD feedback controller subsequently compensating it.The suggested controller outperforms the typical ADRC controller due to the extra degree of freedom in the Fractional (PD and ESO).The simulation results demonstrate that the tracking errors obtained with the suggested FADRC3 are lower than those obtained with the typical FADRC1 and FADRC2 controllers.The usefulness of the suggested con