Design of Optimized Desired Optimal Controller (DOC)-Based Multivariable Model Reference Adaptive Control (MRAC) Algorithm for Robotic Human Knee Flexion During Gait

Recently, there is a surge in robotics research focused on observing how humans move, abnormal walking patterns, and developing models and limb structures that mimic human capabilities [1]. As growing in the number of people suffering from limb loss due to accidents, illnesses, injuries, and disasters, there is a growing urgency to find techniques to retrieve their physical abilities. This is important for ensuring their well-being and ability to lead independent lives [2]. The robotic knee prosthesis is an important part to help people walk. It operates by controlling the position of the lower leg, generating the necessary knee force, and reducing movement during the leg swing. This helps to stabilize the lower limb and allows for walking [2].

Globally, about 1.1 billion people suffer from disability and technology that helps people with disabilities can improve their lives significantly. It helps them to be more independent and involved in society [3]. Exoskeletons are designed to help walking implement diverse control strategies on lower limbs. Powered lower-limb orthotic devices, commonly referred to as powered exoskeletons, are widely recognized as rehabilitation and gait assistance tools [4].

To enable the functionality of a robotic knee prosthesis, a prosthesis controller module receives sensor readings from joint encoders, force sensors, and an inertial measurement unit. It utilizes this information to transmit commands to the prosthesis device, such as the desired motor position or torque [5]. The integration of physical hardware, software control algorithms, and user interaction poses several challenges in effectively assisting individuals with impaired locomotion using robotic devices. Overcoming these challenges involves enabling users to employ various gait patterns and developing robust evaluation methods to assess the effectiveness and performance of assistive devices [6].

Following femoral amputation, numerous active prostheses have been developed to rehabilitate gait function, capable of producing and controlling knee torque. Various strategies have been proposed to control active femoral prostheses. For instance, Rifaï et al. [7] presented an adaptive control approach for knee joint orthosis considering a user's effort, Chevalier et al. [8] discussed a theoretical model for shank movement around the knee joint and employed two design techniques, model-based design, and auto-tuning, to design a PID controller for controlling the shank angle.

Furthermore, Rifaï et al. [7] developed an adaptive sliding mode controller based on super-twisting for knee joint orthosis. Their approach incorporated a human-driven model, Bkekri et al. [9] proposed a sliding mode controller based on the theory of Lyapunov stability.

Considering a mathematical model of a nonlinear system combining human shank dynamics and knee joint orthosis, Kohli et al. [10] designed a feedback linearization controller. Khalaf et al. [11] proposed a design and control method by experimentally evaluating the regenerated energy across a femoral prosthesis during the swinging phase of a gait cycle.

In recent years, researchers have shown great interest in nature-inspired algorithms. Swarm intelligence (SI) algorithms have gained prominence. These algorithms emulate the operation of natural systems, such as animal or plant behavior, to develop effective optimization techniques plants [12, 13]. Particle swarm optimization (PSO) [14], artificial bee colony algorithm (ABC) [15], ant colony optimization (ACO) [16], grey wolf optimization (GWO) [17], krill herd algorithm (KHA) [18], brainstorm optimization (BSO) [19], and whale optimization algorithm (WOA) [20] are all examples of popular SI algorithms.

The Whale Optimization Algorithm (WOA) [20] is a new kind of optimization algorithm. The natural hunting techniques of humpback whales served as inspiration for this algorithm. WOA uses three operators—"encircle prey," "research prey," and "attack prey"—to track down its meals. WOA has undergone significant change. These studies can be broken down into two broad categories. Using the WOA to solve real-world problems is one of them. The other is to implement measures that enhance WOA's functionality [21].

Another swarm intelligence search algorithm inspired by the coordinated foraging behavior of krill is known as the "Krill Herd" (KH) algorithm [18]. Individual krill navigate a three-dimensional region in search of nearby food and a dense herd, as part of a population-based strategy involving a very large number of krill. With KH as an optimization algorithm, the distance of a krill from its food source is treated as a design variable. There are three types of algorithms that can be compared to the KH algorithm: (1) Evolutionary algorithms (2), a bacterial foraging algorithm, and (3), swarm intelligence [22, 23].

In order to underscore the pragmatic ramifications of WOA, a case study [24] concerning the real-time PID controller optimization of a micro robotics system demonstrates WOA's exceptional performance relative to Grey Wolf Optimization. The study provides evidence that WOA exhibits superior performance compared to alternative methods, as determined by its ISTES cost function, which decreases error by 47.5%. WOA is suggested by these empirical results for adjusting PID parameters in micro robotic systems.

Previous research does not address the nonlinearity of robotic knee flexion during walking, with a particular emphasis on enhancing mobility and rehabilitation for patients. Therefore, in this context, this paper aims to address the design of an optimized Desired Optimal Controller (DOC)-based multivariable Model Reference Adaptive Control (MRAC) algorithm for robotic human knee flexion during gait. This algorithm assists the robotic knee joints to achieve better performance to match how people walk and their body's needs.  Gives help to people who have moving problems to feel natural and move in an efficient manner. The aim is to design a controller that exhibits fast rise time with no overshoot and the steady-state error approaches zero. By incorporating concepts from DOC and MRAC, it will give these desired characteristics smoothness, energy efficiency, stability, and compatible with the patterns of the user's gait. Hence, the objective of the proposed investigation is to:

1. Maximizing the Performance of Robotic Knee Joints:

·Design of a control system which ensures the response of the robotic knee joints to be appropriate for different inputs and conditions.

·Enhance the capacity to adapt to various locomotion patterns and the physiological circumstances.

2. Attaining Precise Control Attributes:

·Make sure that the flexion of the knee is seamless and with no overshoot.

·Achieve a rapid onset of motion and eradicate steady-state error while walking.

3. Desired Optimal Control and MRAC Integration:

·Blend the tenets of Model Reference Adaptive Control and Desired Optimal Control.

4. Improvements to Controller Attributes:

·Ensure the energy efficiency, fluidity, and stability of the control system.

·Customized assistance can be achieved by aligning the controller with the user's unique locomotion patterns.

5. Development and Validation of Algorithms:

·Construct a comprehensive MRAC algorithm utilizing DOC.

·Conduct experiments and simulations to evaluate the efficacy of the algorithm.

6. Investigation of Methods for Combined Optimization:

·Explore the potential for integration between WOA and KHO.

·Evaluate the advantages of integrating WOA-KHO in contrast to conventional optimization algorithms.

This paper will further explore the development, simulation, and experimental validation of the algorithm in the following sections. By doing so, it will make a valuable contribution to the field of robotic knee prosthesis technology, which aims to enhance gait assistance and rehabilitation.

This paper provides a comprehensive overview of WOA and KHA, shedding light on their significance and potential applications in solving optimization problems. The subsequent sections will delve into the details of these algorithms, elucidating their mechanisms and showcasing their contributions to the advancement of optimization techniques.

3.1 The whale optimization algorithm (WOA)

The key algorithm is inspired by the habits of whales. WOA's basic performance consists of three stages: prey encirclement, bubble-net assault, and prey discovery. In what follows, it will discuss the mathematical models of three distinct strategies [12, 20, 21]:

Cascading its victim The humpback whales encircle their prey once they identify its location. In the absence of prior knowledge regarding the location of the optimal design in the search space, the WOA algorithm operates under the assumption that the current best candidate solution is either the target prey or is near the optimum, thereby indicating its suitability as the search agent. As a result, the remaining agents will endeavor to revise their stances regarding this agent. The other search agents update their locations by moving onward to the best-fitted agent which is followed by the below equations:

$\vec{Ð}=\left|\vec{ Ç } \cdot \overrightarrow{X^*}(t)-\vec{X}(t)\right|$       (5)

$\vec{X}(t+1)=\overrightarrow{X^*}(t)-\overrightarrow{\text { Á }} \cdot \vec{Ð}$       (6)

where, $X^*, \vec{X}$, and $t$ represent the location coordinate of the optimal solution, the current each iteration, and the position vector format correspondingly. Coefficient vectors, denoted as $\vec{ Á }$ and $\text { Ç }$ are known as coefficient vectors and calculated by using the below equations:

$\vec{Á }=2 \vec{á} \cdot \vec{r}-\vec{á}$         (7)

$\vec{Ç}=2 \vec{r}$        (8)

where, $\vec{r}$ represents the random vector within the range $[0,1]$, and with each iteration, the value of $\vec{á}$ decreases linearly from 2 to 0.

(b) Exploitation stage (i.e., the bubble-net attack); the two operations comprising the exploitation stage.

(1) The implementation of a diminishing encircling mechanism involves the reduction of a value as specified in Eq. (4). Note that $\vec{á}$  is a random value between $[-\vec{á}, \vec{á}]$.

(2) The spiral update positional method is utilized to compute the distance separating the cetacean and its prey. The following spiral equation is used to simulate the helix-shaped motion:

$\vec{X}(t+1)=\overrightarrow{\mathrm{Ð}}^l e^{b l} \cdot \cos (2 \pi l)+\overrightarrow{X^*}(t)$         (9)

In the given interval [−1, 1], denoted as "l," a random number, and "b" a constant. A probability of 50% is postulated regarding the selection between the spiral model and the shrinking encircling mechanism. The mathematical model is therefore expressed as follows:

$\vec{X}(t+1)=\left\{\begin{array}{c}\overrightarrow{X^*}(t)-\vec{Á} \cdot \vec{\ominus}, \quad P<0.5 \\ \vec{Ð}^l e^{b l} \cdot \cos (2 \pi l)+\overrightarrow{X^*}(t), \quad \mathrm{P} \geq 0.5\end{array}\right.$        (10)

The symbol $P$ denotes a number chosen at random from a uniform distribution.

(c) Stage of exploration (i.e., pursuit of the prey). The exploration for target method utilizes a variation of $\vec{Á}$ that is either greater than 1 or less than −1. As of now, during the exploitation phase, update the location of a search agent. The mathematical model shown below is employed to conduct a global search for $|\vec{Á}|>1$.

$\overrightarrow{\mathrm{Ð}}=\left|\vec{Ç} \cdot \vec{X}_{\text {rand }}-\vec{X}\right|$        (11)

$\vec{X}(t+1)=\vec{X}_{\text {rand }}-\overrightarrow{\mathrm{Á}} \cdot \stackrel{\rightharpoonup}{\mathrm{Ð}}$       (12)

where, $\vec{X}_{\text {rand }}$  is a random location vector which is chosen from existing population.

Algorithm 1 shows the pseudo-code of the original WOA algorithm.

WOA

Initialize a population of n random whales or search agents X_i (i=1, 2, 3, …, n).

Evaluate each search agent B the best search agent.

While (t< max - epoch)

for each search agent in the population

Update WOA parameters $(\vec{á}, \vec{Á}, \vec{Ç}, l, P)$

if $(\mathrm{P}<0.5)$

if $(|\vec{Á}|<\boldsymbol{l})$

Update the current search agent by $\vec{X}(t+1)=B-\vec{Á} \cdot \vec{Ð}$

else if $(|\bar{Á}| \geq l)$

select a random search agent $\left(\vec{X}_{\text {rand }}\right)$

update the current search agent by $\vec{X}(t+1)=\left(\vec{X}_{\text {rand }}\right)-\vec{Á} \cdot \vec{Đ}$

end if

else if $(P \geq 0.5)$

Update the current search agent by $\vec{X}(t+1)=\vec{Ð}^l \boldsymbol{e}^{\boldsymbol{b} \boldsymbol{l}} \cdot \cos (2 \pi \boldsymbol{l})+B$

End if

End for

Evaluate the search agent

Update B if there is the better solution in the population

T=t+1

End while

Return B

3.2 Krill herd algorithm (KHA)

The KHA was originally suggested by Gandomi et al. [18] to model the movement of krill in the ocean. Local and global search are balanced adequately, which is one of the primary benefits of KHA that improves its discovery capability. Upon calculating the objective functional for every krill, this algorithm selects the finest krill [25]. The input layer of the krill herd algorithm is the krill position, and the objective function is sustenance. As shown in Eq. (13), the position of each krill $\left(\mathcal{V}_{\mathrm{ɍ}, i}^{i t r}\right)$ in its life cycle is determined by the factor of induction, the factor of foraging, and the factor of diffusion [22, 23, 25-29] by the following equations:

$\mathcal{V}_{\mathrm{ɍ},i}^{i t r}=\mathcal{V}_{\text {induction}, i}^{i t r}+\mathcal{V}_{\text {foraging}, i}^{i t r}+\mathcal{V}_{\text {diffusion}, i}^{i t r}$          (13)

The factor of induction $\left(\mathcal{V}_{\text {induction}, i}^{\text {itr }}\right)$: The motion of each krill as follows:

$\mathcal{V}_{\text {induction}, i}^{i t r}=\alpha_{\text {induction}, i} \mathcal{V}_{\text {induction}, i}^{\max }+\omega_i \mathcal{V}_{\text {induction}, i}^{i t r-1}$        (14)

where, $\mathcal{V}_{\text {induction}, i}^{\max }$ is the utmost forced speed produced in the absence of other krill, $\mathcal{V}_{\text {induction}, i}^{i t r-1}$ is the last forced activity, ω_i is the moment of inertia of the forced motion [0-1], and α_induction,i is the start of the forced move, according to Eq. (15):

$\alpha_{\text {induction,i }}=$$\alpha^{\text {local }}{ }_{\text {induction,i }}+$$\alpha^{\text {target }}{ }_{\text {induction,i }}$     (15)

The local influence given by i neighboring krill is denoted by $\alpha_{\text {local }}^{\text {induction,} \mathrm{i}}$, and the target direction effect formed by $\alpha^{\text {target }}{ }_{\text {induction, i}}$ the ideal krill individual position is denoted by i et.

The factor of foraging $\left(\mathcal{V}_{\text {foraging}, i}^{i t r}\right)$: Each krill adjusts its position based on where it has found food recently and in the past as:

$\mathcal{V}_{\text {foraging}, i}^{\text {itr }}=\mathcal{V}_f \beta_{\text {foraging}, i}+\omega_f \mathcal{V}_{\text {foraging}, i}^{\text {old }}$      (16)

$\mathcal{V}_f$ is the foraging velocity, $\beta_{\mathrm{foraging}, i}$ is the foraging motion their source, $\omega_f$ is the foraging inertial weight between $[0,1]$, and $\mathcal{V}_{\text {foraging,} i}^{\text {old }}$ is the last foraging activity.

$\beta_{\mathrm{foraging}, i}=\beta_{\text {foraging}, i}^{\text {food}}+\beta_{\mathrm{foraging}, i}^{\text {best}}$          (17)

$\beta_{\mathrm{foraging}, i}^{\text {food }}$ stands for i's interest in food, and $\beta_{\mathrm{forag}}^{\text {best}}$ stands for i's interest in being in peak physical condition.

The factor of diffusion ($\mathcal{V}_{\text {diffusion}, i}^{\text {itr}}$): The population's variation is guaranteed by a standard random operator, as in the case of random diffusion:

$\mathcal{V}_{\text {diffusion}, i}^{i t r}=\mathcal{V}^{\max } \delta$         (18)

where, $\mathcal{V}^{\max }$ is the maximum rate of diffusion, δ is a uniformly distributed random vector, and the matrices it generates are numbers in the interval [1, 1]. Combining these three patterns, we can determine an individual krill's location vector from t-to-t Dt by solving Eq. (19) [28]:

$x_i^{i t r+1}=X_i^{i t r}+$$Ḉ_{ɍ, i}^{i t r}$$\times \sum_{j=1}^S\left(U_j-L_j\right)$         (19)

The krill's position at time $\mathrm{t}$ is denoted by $X_i^{i t r}$, and at next time by $\mathcal{X}_i^{i t r+1}$. The maximum and bottom bounds of the variables $\mathcal{U}_j$ and $L_j$, the overall collection of parameters $\mathrm{S}$, and the characteristic of the interval [0,2] are given by $\mathcal{U}_j$ and $L_j$, respectively.

KHA: Idealizing the mobility properties of the krill individuals allows for the creation of a wide variety of algorithms. The following is a high-level overview of the KH algorithm.

Data Structures—Simple Bounds Defined, Algorithm Parameters Determined, etc.

Beginning Steps: Generating the Search Space's Starting Population at Random.

Position-based fitness assessment: rating each krill in the population.

Determining the Movement:

Induced movement.

Foraging

Diffusion at Random

Include the use of genetic operators.

A krill's position in the search space is being updated.

Step III must be repeated until the termination conditions are met.

End.

Figure 5 depicts the Simulink model of the human-robotic knee plant as described by Eq. (2) and using the parameter values described in the preceding section. Figure 6 explains the unit step response of the open loop system for the nonlinear model as specified in Figure 5 that employs Eq. (1) which includes the nonlinear parameters of the Knee joint plant dynamic model as listed in Table 1 and for the simplified model as denoted in Eq. (4) that represent the second order knee joint plant model. Based on the step response characteristics, the two responses reveal a steady-state error (e_s.s) of 0.19 and 0.2 for the nonlinear and simplified models, respectively, indicate that the system is slightly underdamped. For the nonlinear model, the unit step response of the closed-loop robotic human knee system is shown in Figure 7. Observations from this figure highlight certain characteristics of the closed-loop response. Notably, it shows a relatively high e_s.s of 0.493, accompanied by a settling time t_s of 0.545 Sec. with an error tolerance of 0.02%., and a rise time t_r of 0.106 Sec. These findings imply a system response with slow behavior with overshoot.

In this study, Figure 8 illustrates the Simulink model of the designed system that uses the desired DOC-based MRAC algorithm for controlling the robotic human knee. The adaptive gains k_e, k_y, k_u are allocated the values 0.6, 0.001, and 0.001 to optimize the controller. The PID controller's parameters k_p, k_i, and k_d are fine-tuned to accomplish enhanced system response, with specific values of 0.80, 1.29, and 0.90, respectively. By taking ζ=1 and ω_n=10 and substituting these values in Eq. 20, the transfer function of the reference model is explained in Eq. (26).

$\mathrm{TF}(\mathrm{s})=\frac{100}{s^2+20 s+100}$       (26)

Figure 9 explains the response of the proposed controller. This figure demonstrates that the controller's response stabilizes at torque =50.97 (Newton. Sec.). This value is well within the accepted range of control signals, making it the optimal controller for the investigation's robotic human leg plant.

5.png

Figure 5. Simulink model of the robotic human knee plant

6.png

Figure 6. The unit step response of the open loop system robotic human knee for nonlinear and simplified models

7.png

Figure 7. The unit step response of the closed loop for the robotic human knee system

Figure 10 depicts the unit-step response of the robotic human knee system using the proposed DOC-based MRAC algorithm. Implementing the proposed controller enhances system response and eliminates e_s.s, resulting in a e_s.s of zero. However, the system's t_s=2.52 Sec. at an error tolerance of 0.02%, indicating a relatively slow response due to the nonlinearity of the dynamic knee joint model as specified in Eq. (1), and a rise time t_r=1.602 Sec.

8.png

Figure 8. Simulink model of the system using the desired DOC-based MRAC algorithm for the robotic human knee

9.png

Figure 9. The response of the desired DOC-based MRAC controller

10.png

Figure 10. The unit step response of the system using the desired DOC-based MRAC algorithm for the robotic human knee

To enhance the system's performance and achieve optimal response, an optimization technique is introduced to reduce the system's settling time t_s and accelerate its overall response. Figure 11 depicts the unit step response of the robotic human knee system using the proposed DOC-based MRAC algorithm optimized with GA. The simulation results demonstrate a not significant improvement in the system's performance when the GA is employed to tune the parameters of the proposed controller. The optimization technique implemented through GA effectively reduces the system's settling time t_s to 0.56 Sec. with an error tolerance of 0.02%. Simultaneously, it maintains a rise time t_r of 0.45 Sec. and a small acceleration in the overall response. Yet, a steady-state error e_s.s of 0.1 persists.

11.png

Figure 11. The unit step response of the system using GA for tuning parameters of  the desired DOC-based MRAC

Figure 12 presents the unit step response of the robotic human knee system using the proposed DOC-based MRAC algorithm optimized with PSO, demonstrating a slight improvement in system performance. The optimization technique reduces settling time t_s to 0.26 Sec. with an error tolerance of 0.02%. Simultaneously, it maintains a rise time t_r of 0.15 Sec. and accelerates the overall response without minimizing steady-state error e_s.s.

12.png

Figure 12. The unit step response of the system using PSO for tuning parameters of  the desired DOC-based MRAC

13.png

Figure 13. The unit step response of the system using WOA for tuning parameters of the desired DOC-based MRAC

Figure 13, the unit step response of the robotic human knee system using the proposed DOC-based MRAC algorithm optimized with WOA. The simulation results demonstrate significant improvements in the system's performance when the WOA is employed to tune the parameters of the proposed controller. The optimization technique reduces settling time effectively reduces the system's settling time t_s to 0.082 Sec. with an error tolerance of 0.02%. Simultaneously, it maintains a rise time t_r of 0.07  Sec. and accelerates the overall response while minimizing steady-state error e_s.s to zero. These findings indicate the successful optimization of the controller's parameters through the WOA-based approach, leading to enhanced efficiency and precision in controlling robotic knee flexion during gait.

Figure 14 introduces the unit step response of the robotic human knee system utilizing the proposed DOC-based MRAC algorithm optimized with the Combined WOA and KHO, exhibiting further improvement. The joint optimization approach outperforms the use of WOA alone, achieving reduced settling time t_s to 0.043  Sec. at an error tolerance of 0.02%, Simultaneously, it maintains a rise time t_r of 0.029 Sec and accelerates the overall response with and zero steady-state error e_s.s. The synergistic effect of these nature-inspired algorithms enabled a comprehensive exploration of the parameter space, leading to the identification of an optimal parameter set for the controller.

14.png

Figure 14. The unit step response of the system using WOA-KHO for tuning parameters of the desired DOC-based MRAC

In general, the simulation outcomes unequivocally illustrate the efficacy of both the WOA-based and Combined WOA-KHO methodologies in enhancing the performance of the system. By integrating optimization techniques, the controller's parameters were refined, resulting in improved gait control and a walking experience that is more comfortable and natural for people with mobility impairments. Furthermore, further enhancements in the system's response characteristics were observed with the integration of WOA and the Combined WOA-KHO optimization approach. These improvements encompassed decreases in steady-state error, settling time, rise time, and overshoot.

The integration of KHO and WOA yields distinctive aspects that set it apart from the GA, PSO, and WOA algorithms individually. WOA draws inspiration from the foraging behavior of whales to optimize exploration in space. On the other hand, KHO uses swarm intelligence that simulates collaborative behaviors noticed in nature to enhance its capabilities of exploration.

The incorporation of the WOA-KHO approach utilizes the characteristics of both WOA and KHO to make it easier to thoroughly study and adjust the settings of the controller effectively. Consequently, the developed optimization algorithm makes it more thorough and effective, gives better control accuracy and system behavior. This optimization algorithm combines multiple algorithms to improve robot performance. It reduces errors over time, speeds up how quickly the robot becomes stable, and optimizes the time it takes to reach its peak performance. As a result, this approach makes robots more effective and responsive, making it a valuable tool for advanced robot control.

In this paper, the controller is designed to use a model of the MRAC algorithm-based DOC. For optimizing this algorithm, advanced optimization algorithms are used. The results of the simulation explain that the combination of these algorithms enhances the performance of the controller and efficiency. This gives an exception of the  controller to be validated using experimental data, which will give further evidence of its accuracy.