Modeling and Simulation of Lower Limb Rehabilitation Exoskeletons: A Comparative Analysis for Dynamic Model Validation and Optimal Approach Selection

Abstract

Accurate modeling and simulation of lower limb rehabilitation exoskeleton (LLRE) enables effective control resulting in enhanced performance and ensuring efficient rehabilitation. There are two primary objectives of this study. First is to validate the existing models and second is to identify the optimal modeling approach for exoskeletons. For validation, firstly a lower limb rehabilitation exoskeleton is modeled using three different modeling approaches which include analytical modeling, bond graph modeling, and modeling through Simscape (SS). Thereafter, dynamic responses of analytical and graphical modeling are compared with SS model using key dynamic response parameters, including rise time, peak time, and others. The SS-based physical model can be employed for validation because SS, unlike mathematical modeling, uses unit-consistent physical domain data and, therefore, serves as an intermediate step between mathematical modeling and hardware validation. Secondly, to identify the most suitable modeling approach, a structured and comprehensive comparison of different modeling approaches based on aspects such as control domain, complexity, ease of use, and other relevant factors is carried out. The results highlight the qualitative strengths and limitations of the three approaches. Previous studies focus on individual methods and lack such comparison. This work contributes to the validation of models and identification of an efficient and effective modeling methodology for LLRE. The findings reveal that Simscape™ is the most suitable approach for modeling LLREs as it provides multidisciplinary system modeling and supports real-time simulation. The validated model can now be employed for advancements in model-based control design. Moreover, the identified optimal approach provides an insight to researchers and engineers for model selection in early-stage design and control development of complex mechatronic systems. Future work includes comparison of dynamic responses with actual hardware responses to experimentally validate the effectiveness of the model for real-world patient assistance and mobility restoration.

1. Introduction

The world population is increasing and along with it, the number of patients with musculoskeletal and neuro-motor dysfunction is rising at a rapid pace. Both dysfunctions cause lower limb-related disabilities that limit patients’ mobility. Immobility of patients makes them vulnerable to other diseases such as cardio-vascular and diabetes. Old age is another factor that causes immobility. Most people in old age experience common diseases such as sarcopenia, weak bones, and reduced joint performance, resulting in poor health conditions and restriction in their movements [1,2]. The number of people aged 65 or above in 2050 is estimated to reach around 1549 million, which is almost a 120% increase from the current figures [3]. During the COVID-19 pandemic, elderly and disabled people faced difficulties in seeking help due to strict lockdowns and social distancing policies. Elderly people were at greater risk while seeking assistance as they were more susceptible to the corona virus [4,5,6]. Due to the abovementioned facts, proper rehabilitation and care methods are necessary to improve patients’ mobility and reduce dependence on foreign assistance.

Traditional rehabilitation methods have a few drawbacks, including lack of versatility to address patients’ specific needs, and successful rehabilitation is bound by the experience of physicians. Patients typically opt to exercise at home due to the extended duration of therapy and hefty expenditures, but this type of training lacks scientific guidelines. Under these circumstances and without proper rehabilitation, many patients lose their ability to stand up on their feet again. The use of advanced rehabilitation equipment is an effective option for regaining lower limb function and control. Rehabilitation using state-of-the-art techniques can help alleviate pain, increase productivity, boost self-confidence, and improve overall quality of life.

As a result, there is a growing demand for more convenient rehabilitation robots to aid in the recovery process. Advancements in robotic technology offer an opportunity for lower extremity rehabilitation to potentially boost therapeutic dosage while relieving some of the physiotherapist’s responsibilities [7,8]. Wearable lower limb exoskeleton has become a well sought mechatronic system for research due to its promising benefits in terms of assistance and rehabilitation for disabled and elderly people. Robotic systems can facilitate joint mobility, provide sensory feedback during movement, and offer a secure environment for controlled parameter variations [9]. While patients have a positive attitude towards the use of robotic exoskeletons in rehabilitation [7], a therapist’s willingness to employ robotic technology in their practice may be influenced by the learning curve required to grow accustomed to using the device and the lack of scientific data supporting its therapeutic benefits [10,11].

Lower limb rehabilitation exoskeletons (LLREs) can serve a wide range of patients suffering from spinal cord injury, stroke, cerebral palsy, Parkinson, multiple sclerosis, and traumatic brain injury. Rehabilitation robots, such as LLREs, can assist therapists in reducing their workload, analyzing training data, and evaluating recovery in a controlled and reproducible manner [12]. The exoskeletons have been designed and studied since the 1960s, but technical limitations prevented them from achieving their intended objectives. Functional classification of exoskeletons based on their applications related to rehabilitation of the lower limb is shown in Figure 1. Within the medical domain, exoskeletons are designed for upper limb and lower limb rehabilitation. LLREs are further categorized based on their usage in two primary settings: stationary and mobile. Stationary exoskeletons refer to exoskeletons integrated into fixed platforms combined with treadmill and body weight support systems whereas mobile exoskeletons refer to rehabilitation systems which provide on-ground training either independently or with mobile platforms. This research focuses on the modeling and simulation of mobile LLREs. The category of mobile exoskeletons is highlighted in Figure 1 to emphasize the scope of the present work.

Researchers have developed various hardware models for lower limb rehabilitation exoskeletons (LLREs) with the goal of developing better exoskeletons for improved rehabilitation results. Some earlier models, such as those in [13,14,15], only had 2 degrees of freedom (DOF) for each leg (hip and knee), with the ankle fixed in place, making it easier to control but less conformant to the human body. More recent models, such as those in [16,17,18,19,20,21], have 3 DOF for each leg, including an ankle joint, which better conforms to human anatomy and enhances balance during assisted walking.

DC motors have been the preferred choice for actuation in most proposed models due to their compact size, ease of control, accuracy, and repeatability. However, a few models in [22,23] have used hydraulic and pneumatic actuators, which can deliver higher torque but are less accurate, require bulky equipment, and are difficult to control.

To develop the exoskeleton model, dynamic modeling of the physical system is carried out. In previous studies, it has been observed that researchers have performed dynamic modeling of various LLREs by using different modeling techniques. They have performed dynamic modeling and simulation of rehabilitation exoskeletons using Euler–Lagrange (EL)/Euler–Newton (EN) method and bond graph modeling technique [13,16,24,25,26,27,28,29,30].

Table 1. Overview of dynamic modeling approaches.

Sr. No.	Modeling Approaches	Exoskeletons	Bio-Mechatronic Systems
1	Conventional Modeling	[13,24,25,26,27,28,29,30]	[31,32,33,34,35]
2	Bond Graph Modeling	[36,37,38,39,40,41,42]	[43,44,45,46,47,48,49,50,51,52,53]
3	SimscapeTM	*	[54,55,56,57]

* This table reflects modeling approaches used in published LLRE studies up to the year 2021, the period during which this research was conducted. Subsequent work using Simscape™ has since emerged, which supports the relevance and timeliness of this study’s focus on structured modeling comparison [58,59].

summarizes the modeling techniques used in the past for obtaining the mathematical model of LLREs and other bio-mechatronic systems.

In the Euler–Lagrange method, the difference between the kinetic and potential energies is used to generate a system model while relating the associated forces to the generalized coordinates. Although a controller can be designed for the modeled system to obtain the desired output, this method involves complex computations to precisely model the errors and inaccuracies. In addition to that, the EL method is less flexible in terms of handling any changes or modifications in mechanical hardware. For any change in hardware model, the modeling process requires re-derivation of all equations, increasing the computational load. An alternative modeling approach for manipulators is the bond graph (BG) technique which provides a simple and flexible model of the system [36,37,38,39,40,41] as compared to the conventional Lagrangian method. Using BG method, a comprehensive modeling of any mechatronics system can be performed. In research works [40,41,42], modeling of exoskeletons for medical and industrial purposes is performed using the BG method.

Modeling of exoskeletons can be carried out using the Simscape (SS). It is a modeling tool that allows efficient modeling of complex physical systems within the Simulink environment. The modeling process is facilitated using physical connections, and the models built are based on physical components and their interactions through energy flow. The physical network approach provides an intuitive and comprehensible way of modeling complex systems involving multiple physical domains such as electrical, hydraulic, mechanical, thermal, and more. Furthermore, there are a variety of pre-built components that can be easily assembled into models. The pre-built components save time and effort when developing a model and ensure its accuracy and reliability.

Once a model has been created in SS, it can be simulated and analyzed using the same methods and tools as Simulink. SS offers a variety of analysis tools for investigating the behavior of a system, such as time-domain simulation, frequency-domain analysis, and optimization.

Several modeling approaches, i.e., analytical, graphical, and physical, have been reviewed above. Despite significant achievements in the modeling of exoskeleton systems, previous studies primarily focused on individual modeling and lacked a structured and comprehensive comparison of modeling approaches. Such a comparison is essential in the identification of the most suitable modeling approach for LLRE. Consequently, it remains an open research challenge to select the most suitable modeling approach for multidomain systems that ensure accuracy, ease of use, and computational efficiency. Firstly, this work addresses the highlighted gap by providing a systematic and structured comparison of modeling approaches based on technical parameters, thereby validating the existing models. Secondly, through the comparison of qualitative aspects, this work contributes the identification of the optimized approach for the modeling of the LLRE. Together, these contributions reflect the novelty of this research work. The research methodology followed in this work is presented in Figure 2.

Exoskeleton modeling in this study was carried out using three modeling approaches, and the responses and approaches are compared based on technical parameters and qualitative aspects, respectively, resulting in the validation of existing models and identification of optimized modeling approach for LLRE. The validated model and the identified optimized approach will support researchers in the field of LLRE modeling and control in selecting a suitable modeling approach, enabling them to design efficient control algorithms.

This paper is organized as follows: The introduction section provides an overview of the different exoskeletons and modeling and simulation approaches for LLREs. Section 2 details the modeling and simulation of LLRE using three different modeling approaches. Section 3 presents the results and comparative analysis of the three approaches based on technical parameters and qualitative aspects. It further discusses the findings, highlights the advantages and limitations of each approach and ultimately identifies the most suitable modeling approach. Section 4 outlines the conclusions and future research directions.

2. Modeling and Simulation of LLRE

In this section, the details of the hardware of LLRE have been presented in detail, followed by a comprehensive modeling of the LLRE using three different approaches.

2.1. LLRE Hardware Components

The proposed LLRE has three DOF and consists of three links, where link 1 is aligned to the femur, link 2 with the fibula and tibia and link 3 with the foot. The hip, knee, and ankle joints are designed for flexion and extension. The maximum range of motion for the joints has been set at 30°/212° for the hip, 60°/210° for the knee, and 13°/220° for the ankle to ensure physiological comfort and safety. These joint angles belong to a local coordinate system of exoskeleton with neutral alignment set at 180°. Upon transforming these angles into clinical joint space, the ranges correspond to anatomical limits as established in [60,61] for hip, knee, and ankle joints. The exoskeleton system has 1 DOF movement for each joint (hip, knee, and ankle) with linear actuators. The CAD model and hardware of LLRE are shown in Figure 3.

In the case of the motor mounted at the hip joint, the dynamic contributions of the connected structure—including two distal links (link 2 and link 3), two motors (motors for knee and ankle joint actuation), and the mass of a human leg—can be lumped and reflected towards the hip joint for modeling as a combined load, i.e., a spring–mass–damper acting on the hip motor. Such a model is an equivalent (reduced) dynamic representation and is referred to as a “composite model.” This abstraction captures the dynamic interaction between the actuator and the mechanical load for modeling and control purposes.

The composite model for hip joint control is considered. The schematic diagrams of the complete LLRE model and composite model are shown in Figure 4 and Figure 5, respectively. Figure 4 shows the schematic diagram of the LLRE. Three motors (mhj, mkj, and maj, i.e., motors for actuation of hip joint, knee joint, and ankle joint, respectively) along with rack and pinion mechanisms for achieving linear actuation are connected in parallel to the links which are connected using hinge joints. The damping and stiffness of the hinge joints are added in the model. Figure 5 shows the schematic diagram of the composite model presenting the linear actuator for hip joint actuation along with the rack and pinion and combined mass, damping, and stiffness from the distal links and motors as seen by the mhj. For the dynamic modeling of exoskeleton, two cases, i.e., exoskeletons with different values of system variables have been considered. The purpose of presenting two cases is to evaluate the dynamic behavior and robustness for different load conditions. The parameters for both cases are shown in

Table 2. Parameters for dynamic modeling of LLRE.

Variables	Discipline	Description	Case-I	Case-II
$k_t$	Electrical	Motor Torque Constant	$0.010\mathrm{N}\mathrm{m}/\mathrm{A}$	$0.010\mathrm{N}\mathrm{m}/\mathrm{A}$
$k_b$	Electrical	Motor Back EMF Constant	$0.010\mathrm{V}/(\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s})$	$0.010\mathrm{V}/(\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s})$
$L_a$	Electrical	Armature Inductance	$1\mathrm{\mu }\mathrm{H}$	$0.9\mathrm{m}\mathrm{H}$
$R_a$	Electrical	Armature Resistance	$0.1\mathrm{\Omega }$	$1\mathrm{m}\mathrm{\Omega }$
$J_a$	Mechanical	Armature Inertia	$0.5728\mathsf{\mu }\mathrm{k}\mathrm{g}{\mathrm{m}}^2$	$0.01\mathsf{\mu }\mathrm{k}\mathrm{g}{\mathrm{m}}^2$
$D_a$	Mechanical	Armature Damping	$2\mathrm{\mu }\mathrm{N}\mathrm{m}/(\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s})$	$0.01\mathsf{\mu }\mathrm{N}\mathrm{m}/(\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s})$
$M$	Mechanical	Combined Mass of Exoskeleton Link and Human Limb Segment	$1\mathrm{k}\mathrm{g}$	$100\mathrm{k}\mathrm{g}$
$R$	Mechanical	Radius of Pinion	$0.1\mathrm{m}$	$0.1\mathrm{m}$
$f_v$	Mechanical	Damping Coefficient	$4\mathrm{\mu }\mathrm{N}/(\mathrm{m}/\mathrm{s})$	$5.94\mathsf{\mu }\mathrm{N}/(\mathrm{m}/\mathrm{s})$
$K$	Mechanical	Stiffness of Leg	$0.1\mathsf{\mu }\mathrm{N}/\mathrm{m}$	$0.01\mathsf{\mu }\mathrm{N}/\mathrm{m}$

.

2.2. Conventional Modeling of LLRE

In this section, a conventional model of the LLRE is developed using the laws of physics to deduce the equations for the system. The mechatronic system comprises a motor with gears, rack and pinion, and mass–spring–damper load as shown in Figure 5.

By applying Newton’s second law:

$$\begin{gathered} \left(\sum Impedances at {\theta }_m\right){\theta }_m+\left(\sum Impedances at X\right)X\times r \\ =\sum Applied Torques at {\theta }_m \end{gathered}$$

$$(J_a{s}^2+D_{a}s){\theta }_m+F(s)\times r=T_m(s)$$

(1)

where ‘$J_a$’ is moment of inertia of armature shaft, ‘$D_a$’ is damping of armature shaft, ‘$T_m$’ is the motor torque. For the spring–mass–damper system,

$$(M{s}^2+f_{v}s+K)X(s)=F(s)$$

(2)

where ‘$M$’ is the mass of the load attached to rack, ‘$f_v$’ is viscous damping of the rack and pinion and ‘$K$’ is the stiffness of the rack. $X\left(s\right)$ can be written as

$$X(s)=r{\theta }_m(s)$$

(3)

In Equation (3), ‘$X\left(s\right)$’ is the linear displacement of the load, ‘$r$’ represents the radius of the pinion and ‘${\theta }_m\left(s\right)$’ represents the angular displacement of the pinion. Substituting values from Equations (2) and (3) in Equation (1):

$$(J_a{s}^2+D_{a}s){\theta }_m(s)+(M{s}^2+f_{v}s+K)r{\theta }_m\times r=T_m(s)$$

(4)

Using Kirchhoff’s Voltage Law (KVL),

$$E_a(s)=R_a{I}_a(s)+L_{a}s{I}_a(s)+V_b(s)$$

(5)

where ‘$E_a$’ is applied armature voltage, ‘$V_b$’ is back EMF of motor, ‘$R_a$’ and ‘$L_a$’ are armature resistance and inductance, respectively. The back EMF voltage can be determined as

$$V_b(s)=k_{b}s{\theta }_m$$

(6)

where ‘$k_b$’ is the back EMF constant of DC motor. The torque generated by the motor can be expressed as

$$T_m(s)=k_t{I}_a(s)$$

(7)

where ‘$T_m$’ is the motor’s torque, ‘$k_t$’ is the motor’s torque constant, and ‘$I_a$’ is the motor’s armature current. Equation (7) can be written as

$$I_a(s)={\displaystyle \frac{T_m(s)}{k_t}}$$

(8)

After substituting values in Equation (5) and re-writing,

$${\displaystyle \frac{(R_a+L_{a}s)T_m(s)}{k_t}}+k_{b}s{\theta }_m(s)=E_a(s)$$

(9)

Equation (9) becomes

$${\displaystyle \frac{(R_a+L_{a}s)\{J_a{s}^2+D_{a}s){\theta }_m(s)+(M{s}^2+f_{v}s+K)r^2{\theta }_m\}}{k_t}}+k_{b}s{\theta }_m(s)=E_a(s)$$

(10)

$${\displaystyle \frac{{\theta }_m(s)}{E_a(s)}}={\displaystyle \frac{k_t}{(R_a+L_{a}s)\{J_a{s}^2+D_{a}s){\theta }_m(s)+(M{s}^2+f_{v}s+K)r^2+k_b{k}_{t}s}}$$

(11)

The transfer function of the system in terms of the velocity of the load attached to the rack of the system and the voltage applied to the system is determined as Equation (12).

$${\displaystyle \frac{V_L(s)}{E_a(s)}}={\displaystyle \frac{k_{t}sr}{(L_a{J}_a+L_a{r}^{2}M)s^3+(R_a{J}_a+R_a{r}^{2}M+L_a{D}_a+f_v{r}^2{L}_a)s^2+(R_a{D}_a+f_v{R}_a{r}^2+k{L}_a{r}^2+k_t{k}_b)s+R_{a}k{r}^2}}$$

(12)

For the transfer function of exoskeleton case-I, the values of the variables from Table 2 are substituted into Equation (12). The open-loop and closed-loop transfer functions become

$${\displaystyle \frac{V(s)}{E_a(s)}}={\displaystyle \frac{9.998\times {10}^{4}s}{s^3+1\times {10}^5{s}^2+1\times {10}^{4}s+0.0099}}$$

(13)

$${\displaystyle \frac{V(s)}{E_a(s)}}={\displaystyle \frac{9.998\times {10}^4{s}^4+9.998\times {10}^9{s}^3+1\times {10}^9{s}^2+999.6s}{s^6+2\times {10}^5{s}^5+1\times {10}^4{s}^4+1.2\times {10}^{10}{s}^3+1.1\times {10}^9{s}^2+1200s+9.996\times {10}^{-5}}}$$

(14)

Similarly, for LLRE case-II, the transfer functions of the system become

$${\displaystyle \frac{V(s)}{E_a(s)}}={\displaystyle \frac{1.111s}{s^3+1.111s^2+0.1111s+1.111\times {10}^{-10}}}$$

(15)

$${\displaystyle \frac{V(s)}{E_a(s)}}={\displaystyle \frac{1.111s^4+1.235s^3+0.1235s^2+1.235\times {10}^{-10}s}{s^6+2.222s^5+2.568s^4+1.481s^3+0.1358s^2+1.481\times {10}^{-10}s+1.235\times {10}^{-20}}}$$

(16)

2.3. Bond Graph Modeling of LLRE

In the previous section, the dynamic model in the form of transfer functions has been obtained in Equations (13)–(16). The relationship between the voltage ‘$v$’ provided to the exoskeleton and the resulting limb displacement ‘$x$’ can be analyzed. In this section, LLRE is modeled using the BG technique. BG enables researchers to better understand and evaluate the dynamics of the system by offering a more comprehensible and visual description of the system. Bond graph modeling further enables the inclusion of elements from multiple physical domains, such as mechanical, electrical, and hydraulic, providing a more flexible environment for system modeling. Bond graph modeling technique has been used in [45,46,47,48,49,50,51,52,53] for the modeling of various robotic systems. This approach is especially helpful for LLRE as the effects of parameter changes on the system’s behavior can be analyzed efficiently as compared to conventional modeling. 20-Sim software has been used as a tool for the generation of bond graph and for obtaining system equations in the form of state space. The word bond graph of the proposed exoskeleton is presented in Figure 6, illustrating the flow of electrical power from the source to the motor, followed by the flow of mechanical power for the desired rotation of the links.

To obtain accurate modeling, real-time parametric values for both electrical and mechanical hardware have been employed. These include various parameters, such as the armature resistance of the motor, damping and inertia of the motor, and damping of links and joints. The BG of the complete LLRE and the composite model are shown in Figure 7 and Figure 8. The parametric values of exoskeleton hardware are listed in Table 2 with the two cases being considered. The transfer function of the bio-mechatronics system case-I is obtained after frequency response analysis in 20-Sim and is presented in Equation (17).

$${\displaystyle \frac{V(s)}{E_a(s)}}={\displaystyle \frac{9.998\times {10}^{4}s}{s^3+1\times {10}^5{s}^2+1\times {10}^{4}s+0.0099}}$$

(17)

$${\displaystyle \frac{V(s)}{E_a(s)}}={\displaystyle \frac{9.998\times {10}^4{s}^4+9.998\times {10}^9{s}^3+1\times {10}^9{s}^2+999.6s}{s^6+2\times {10}^5{s}^5+1\times {10}^4{s}^4+1.2\times {10}^{10}{s}^3+1.1\times {10}^9{s}^2+1200s+9.996\times {10}^{-5}}}$$

(18)

$${\displaystyle \frac{V(s)}{E_a(s)}}={\displaystyle \frac{1.111s}{s^3+1.111s^2+0.1111s+1.111\times {10}^{-10}}}$$

(19)

$${\displaystyle \frac{V(s)}{E_a(s)}}={\displaystyle \frac{1.111s^4+1.235s^3+0.1235s^2+1.235\times {10}^{-10}s}{s^6+2.222s^5+2.568s^4+1.481s^3+0.1358s^2+1.481\times {10}^{-10}s+1.235\times {10}^{-20}}}$$

(20)

The OL and CL transfer functions of case-I and case-II in Equation (13) through Equations (16) and (17) through Equation (20) are the same and will present similar system responses to the various given input signals.

2.4. Simscape^TM Model of LLRE

In this section, LLRE is modeled using Simscape. Multidomain components with physical signals are attached as per the schematic diagram and the system model is obtained. The controlled voltage source supplies the input voltage to the DC motor component which, according to the parametric values ($R_a,L_a,k_b,k_t,J_a$ and $D_a$), converts the electrical energy to mechanical energy. The SS rack and pinion block converts the rotational motion to translational motion. The block named, “Simulink-PS Converter” converts the unitless numerical Simulink signal to a unit-based physical signal. Hence, bridging the gap between mathematical simulation and real hardware. The SS model of the complete LLRE is shown in Figure 9. The parametric values are obtained from Table 2, which were used for conventional mathematical modeling and bond graph modeling earlier. The equivalent composite model of the exoskeleton developed using SS is presented in Figure 10.

3. Results, Discussion, and Comparative Analysis

In this section, the dynamic responses of both cases of the LLRE obtained from CM, BG, and SS modeling approaches are presented and analyzed. Thereafter, the error between the responses obtained from CM and BG is compared with the SS modeling approach. This comparison will identify the differences between the models and, hence, will validate the existing model. Figure 11 and Figure 12 show the OL and CL step, impulse, and sinusoidal responses of the LLRE case-I and case-II. For CL, the feedback configuration employed is unity. No additional scaling or controller design is introduced at this stage as the objective is to compare the responses from different modeling approaches for validation. The comparison of OL and CL step, impulse, and sinusoidal (time-varying signal) responses will highlight the tracking performance, response to sudden disturbances, and phase lag, respectively, in models with and without feedback. The reason for analyzing the system dynamics for these signals is because they represent typical scenarios in rehabilitation robotics and efficient control design.

3.1. Results and Discussion

3.1.1. Step Response—Open Loop and Closed Loop

The open-loop results provide significant insights into the inherent dynamic characteristics of the system without external control influence. As shown in Figure 11 and Figure 12, both models exhibit stable step responses without feedback control, reaching final values of 9.191 and 9.951, respectively. However, the OL response is slower in both cases, with a rise time of 21.2990 and 19.214 secs. The steady-state error and other values of the response parameters are shown in

Table 3. Details of response/performance parameters.

Parameters	Conventional Modeling			Bond Graph			Simscape™
Input Type	Step	Impulse	Sinusoidal	Step	Impulse	Sinusoidal	Step	Impulse	Sinusoidal
Open-Loop Responses for LLRE Case-I
Rise Time (sec)	21.2990	0.7918	0.3940	21.2990	0.7918	0.3940	21.2990	0.7918	0.3940
Settling Time (sec)	50	50	40	50	50	40	50	50	40
Peak Time (sec)	-	1	3	-	1	3	-	1	3
Steady-State Value	9.919	0.0077	1 (−0.9716)	9.919	0.0077	1 (−0.9716)	9.919	0.0077	1 (−0.9716)
Peak Value	-	0.9541	1.7270	-	0.9541	1.7270	-	0.9541	1.7270
Steady-State Error	−8.919	0.0459	0 (0.03)	−8.919	0.0459	0 (0.03)	−8.919	0.0459	0 (0.03)
Closed-Loop Responses for LLRE Case-I
Rise Time (sec)	1.9000	0.7648	0.4456	1.9000	0.7648	0.4456	1.9000	0.7648	0.4456
Settling Time (sec)	9.394	4.773	5.4370	9.394	4.773	5.4370	9.394	4.773	5.4370
Peak Time (sec)	-	1	2.2820	-	1	2.2820	-	1	2.2820
Steady-State Value	0.9090	0.0009	0.6713 (−0.6713)	0.9090	0.0009	0.6713 (−0.6713)	0.9090	0.0009	0.6713 (−0.6713)
Peak Value	-	0.6064	0.7091	-	0.6064	0.7091	-	0.6064	0.7091
Steady-State Error	0.0910	0.3936	0.3287 (−0.3287)	0.0910	0.3936	0.3287 (−0.3287)	0.0910	0.3936	0.3287 (−0.3287)
Open-Loop Responses for LLRE Case-II
Rise Time (sec)	19.214	0.7428	1.0400	19.214	0.7428	1.0400	19.214	0.7428	1.0400
Settling Time (sec)	50	50	47.89	50	50	47.89	50	50	47.89
Peak Time (sec)	-	3	3.735	-	3	3.735	-	3	3.735
Steady-State Value	9.951		0.7811 (−0.7691)	9.951		0.7811 (−0.7691)	9.951		0.7811 (−0.7691)
Peak Value	-		1.579	-		1.579	-		1.579
Steady-State Error		0.1596			0.1596			0.1596
Closed-Loop Responses for LLRE Case-II
Rise Time (sec)	1.4890	0.6711	1.0240	1.4890	0.6711	1.0240	1.4890	0.6711	1.0240
Settling Time (sec)		8.9990	18.6480		8.9990	18.6480		8.9990	18.6480
Peak Time (sec)	-	1.708	28.0520	-	1.708	28.0520	-	1.708	28.0520
Steady-State Value	1.055		0.9799 (−0.9817)	1.055		0.9799 (−0.9817)	1.055		0.9799 (−0.9817)
Peak Value	-		0.9805	-		0.9805	-		0.9805
Steady-State Error	−0.0550	0.4809		−0.0550	0.4809		−0.0550	0.4809

.

With unity feedback control, the CL response demonstrates faster rise time (9.394 s), reduced steady-state error (0.0910), and improved stability. The CL system outperforms the OL system, indicating better tracking performance and enhanced stability.

3.1.2. Impulse Response—Open Loop and Closed Loop

Both models peak at 1 s with a small difference in rise times (0.7918 and 0.7648 s). However, the OL system takes significantly longer (50 s) to settle at zero. The open-loop impulse response highlights the system’s natural damping characteristics and its ability to reject disturbances without feedback. The CL response shows faster settling (4.773 s).

3.1.3. Sinusoidal Response—Open Loop and Closed Loop

The OL system response to time-varying sinusoidal signal demonstrates poor trajectory tracking with noticeable amplitude deviations and phase lag.

The CL system exhibits better tracking accuracy with minimal phase lag and amplitude deviation, making it more suitable for trajectory tracking in rehabilitation applications. Closed-loop control consistently enhances system performance across step, impulse, and sinusoidal inputs by reducing steady-state error, improving settling time, and ensuring better tracking accuracy.

3.2. Comparative Analysis of Modeling Approaches

3.2.1. Dynamic Responses upon Technical Parameters

The comparative study of dynamic responses, i.e., of Euler–Lagrange vs. Simscape™ and bond graph vs. Simscape™ quantitatively confirms negligible error between the models. The values of all parameters are considered up to four decimal places, while error values fall in the range of exponent −7 and −13 as can be seen in Figure 13 and Figure 14, leading to their consideration as zero. The insignificance of minimal differences such as zero beyond seven decimal places fall within acceptable tolerances for LLRE. As for LLRE, reference trajectory tracking error is normally reduced to zero or less. While it has no specific value of precision, for factual accuracy, it can be correlated with a clinically acceptable range of errors in rehabilitation. Similarly, the clinically acceptable error is also not a universal number and is purpose specific. Therefore, an error of zero or less is acceptable in all circumstances. Furthermore, through robust, intelligent, and adaptive control strategies, an error of less than zero degree and precise up to three decimal places (i.e., $\times {10}^{-3}$) can be achieved as presented in [62]. For even higher precision, further research and analysis can be conducted to identify the potential sources of discrepancy and refine the models accordingly. Therefore, the results confirm that the responses obtained from different modeling approaches do not vary from each other, thus validating the existing models.

Each modeling approach has its strengths and limitations. The Euler–Lagrange method is widely used in classical mechanics, robotics, and control applications, offering a strong mathematical framework and requiring strong mathematical background. Bond graph modeling is greater in flexibility compared to EL and provides an intuitive visual representation of multidomain systems and requires familiarity with bond graph theory. Simscape™ is more flexible than graphical modeling and provides real-time simulation environment, making it a promising alternative for modeling of mechatronics systems such as LLRE.

3.2.2. Modeling Approaches upon Qualitative Aspects

For the comparative analysis of modeling approaches based on qualitative aspects, the qualitative aspects of the modeling approaches are first presented; thereafter, the aspects for each approach are scored using the Likert scale. Finally, the overall score for each approach is calculated. The qualitative aspects include domain, complexity, software support, flexibility, simulation speed, and ease of use.

Table 4. Qualitative aspects of modeling approaches.

Aspects	Newton–Euler or Euler–Lagrange Method	Bond Graph	Simscape™
Domain	Classical mechanics, robotics, and control theory	Graphical modeling of dynamic systems	Mechatronics, controls, and physical system modeling [59]
Complexity	Handles complex systems with multiple constraints	Suitable for complex systems but requires expertise [63]	Efficient for complex systems, generalization needed for large models
Software Support	MATLAB R2023a, Python 3.10, and other tools	Specialized software like 20-sim, EASY5	Integrated into MATLAB/Simulink
Flexibility	Flexible for various physical systems	Flexible across energy domains [63]	Pre-defined blocks enable ease of modeling [59]
Simulation Speed	Computationally expensive for large systems [58]	Efficiency depends on model complexity	Resource-intensive but efficient for medium-sized systems
Ease of Use	Requires strong mathematical background	Needs understanding of bond graph theory	Intuitive for users familiar with Simulink

presents the advantages and disadvantages of each modeling approach over the other for the different aspects.

Table 5. Likert scale score of qualitative aspects of modeling techniques with descriptions.

Aspects	Domain	Complexity	Software Support	Flexibility	Simulation Speed	Ease of Use
Euler–Lagrange Method	4 Lacks Modularity and real-time support	2 Becomes unmanageable with more degrees of freedom.	4 Less modular and harder to port for hardware-in-loop systems	3 Requires re-derivation and manual implementation for each change	4 Very efficient due to low overhead; ideal for linear models and fast controller testing	2 Requires knowledge of system dynamics, Laplace transforms, and manual coding
Bond Graph	3 Lacks Integration with hardware control	3 Needs strong domain knowledge and manual causality assignment.	2 Tools are not deeply integrated with control toolboxes	4 Energy-based modeling is domain-independent [63], but modeling structure becomes complex for larger systems	3 Performance drops with complex causality or stiff systems	2 Requires specialized knowledge of bond graph modeling and causality assignment
Simscape™	5 Support real-time simulation	5 Simscape models physical constraints (e.g., hard stops, joint limits) and multidomain systems very well [59]	5 Seamless integration with MATLAB toolboxes, real-time targets, Simulink coder, and Simscape add-ons [58,59]	5 Simscape provides plug-and-play blocks for multidomain physical systems with automatic domain connectivity [58,59]	3 Becomes slower as model complexity and domain interactions grow	5 Drag-and-drop interface; no advanced math needed; highly intuitive for Simulink users

shows the Likert scale score according to the rubrics defined in

Table 6. Likert scale rubrics for different aspects of modeling techniques.

Score	Real-Time Capability and Model Modularity	Complexity	Software Support	Flexibility	Simulation Speed	Ease of Use
1	No real-time support and fully monolithic design; Difficult to reuse or integrate	Very complex; not scalable; High manual effort	No integration with standard tools; limited support	Rigid; not adaptable to other domains	Very slow; unsuitable for repeated runs	Very hard to use; steep learning curve; poor docs
2	Limited real-time functionality and poor modularity; Customization needed for reuse or deployment	Complex and hard to generalize	Limited compatibility; niche or outdated tools needed	Limited flexibility; reuse requires major changes	Slow for medium/large systems; needs solver tuning	Difficult; poor GUI; scarce examples
3	Moderate support for real-time simulation; some modular structure but reuse requires effort	Moderate effort; manageable for simple systems	Moderate support; can be integrated with effort	Reusable in similar systems; some adaptation possible	Moderate simulation time; acceptable for small models	Usable with experience; manageable learning curve
4	Good real-time compatibility and fairly modular design; Subsystems reusable with minimal adaptation	Good abstraction; scalable with effort	Good support for most toolchains; some built-in integration	Adaptable across multiple domains with minor adjustments	Efficient simulations with optimization	Easy to learn; good GUI and documentation
5	Full real-time support and highly modular architecture; Ideal for scalable, Deployable applications	Highly modular; easily scalable for complex systems	Excellent integration with MATLAB/ Simulink, toolboxes, and real-time applications	Highly flexible and cross-domain; supports modular, reusable modeling	Very fast; suitable for rapid prototyping and iterative simulations	Very intuitive interface; rich examples; ideal for beginners

. The description of each score is also presented. The rubrics show that the scoring is based on a five-point scale corresponding to deficiency, limited, moderate, good, and excellent, respectively.

The Likert scale analysis employed in Table 5, Table 6 and

Table 7. Overall score of different metrics for modeling techniques.

Metric	TF/ODE	Bond Graph	Simscape™
Mean Score	3.00	2.73	4.53
Standard Deviation (σ)	1.18	0.91	1.23
Max Domain Score	5	4	5
Min Domain Score	2	1	1
Number of Criteria Rated	6	6	6

evaluates and determines the relative strengths and limitations of each modeling approach through scoring. This scoring is based on reference-backed assessments that consider both merits and trade-offs and not through data-driven statistical analysis. This reflects the scope of Likert scale analysis which is intended to provide a structured qualitative comparison.

All three modeling approaches were evaluated using six criteria and an overall score based on different metrices is shown in Table 7. A mean score of 4.53 along with a maximum domain score of 5 shows the superiority of the Simscape™-based physical modeling approach as compared to conventional and bond graph modeling. Simscape™ reduces the need for extensive and recursive complex calculations with flexibility in the modification of system components, resulting in faster and easier model development. It streamlines the modeling process by utilizing pre-built components within MATLAB/Simulink as compared to the Euler–Lagrange method, which relies on a rigorous mathematical formulation and bond graph modeling, which provides a graphical energy-based representation.

Ultimately, this novel comparison confirms that Simscape™ is the practical, efficient, and effective modeling approach, successfully achieving both objectives of this research, i.e., validation of existing models and identification of optimized methodology. Computational efficiency, ease of use, and compatibility with MATLAB^® makes this modeling approach the most effective choice for LLRE modeling and control design. This work is a step forward for comprehensive and accurate modeling of a system with physical signal resulting in effective hardware implementation to achieve better rehabilitation results.

4. Conclusions

The first primary objective of the research work, i.e., validation of the developed models by comparing with the physical unit-based signal Simscape™ model was achieved and the errors between the different comparisons ranged between exponent −7 and −13 (${10}^{-7}$ to ${10}^{-13}$) which are negligible differences in error values, i.e., effectively zero beyond five decimal places. This shows that there is no difference between the dynamic responses, and hence, validating the models. These models, being closer to the hardware implementation, can now be used for real-time controller design. Secondly, the qualitative comparison of different aspects based on Likert scale for each domain characteristics and evaluated using statistical parameters showed a mean score of 4.53 out of 5 for the Simscape™ model. The analysis proved that the Simscape™ model qualifies as the most promising approach for modeling LLREs. It further proves that the Euler–Lagrange and bond graph methods, despite being robust modeling techniques, were outperformed by Simscape™ in terms of computational efficiency, ease of implementation, flexibility, and real-time simulation capabilities. Hence, establishing Simscape™ as the optimal modeling approach along with its pivotal role in the validation of developed models for LLRE systems.