Two-link lower limb exoskeleton model control enhancement using computed torque

ABSTRACT


INTRODUCTION
One of the leading causes of death and acquired long-term disability globally, the likelihood of stroke increases with aging [1]- [5].Mobility issues brought on by stroke or other related illnesses are becoming more common [6].When one cannot walk normally, they depend on wheelchairs or other mobility devices like orthotics or ankle braces.For many recovering from neurological illnesses, regaining the ability to walk is one of the key objectives.Robotic devices have been utilized in various treatments to help people retrain their motor functions and walk again [7], [8].Robotic gait/walking trainers including the G-EO, Gait Trainer GT II, Lexo, Lokomat ® , and Walkbot are available through market vendors, but the cost is still prohibitive [9].Innovation that can "do more with less" is important to provide significantly more commercial and social gain while using fewer finite resources like electricity, money, and time [10].
The robotic exoskeleton has been briefly described in several recent studies [11]- [15].Gait trainers, also known as lower limb rehabilitation exoskeleton robots, are an important class of rehabilitation robots that can control every joint's motion during training by wearing a wearable connection to the user's body.Exoskeleton robots for healing lower limbs were the focus of research in the 1960s [16], [17].Some researchers use proportional, integral, and derivative (PID) as controllers of these devices [18]- [20] because it is the most popular controller due to its high dependability and straightforward installation due to its great durability and simplicity.Despite their adaptability, current PID controllers have disadvantages that limit their performance, especially regarding robustness, disturbance rejection and tracking.PID controller alternatives that offer higher performance at a reasonable cost include fuzzy control, generalized linear control, and observer and state feedback.These alternatives frequently necessitate a thorough understanding of control theory, in addition to being more challenging to configure than a straightforward PID controller [21].Munadi et al. [22] researched that exoskeleton using PID as a control.The PID controller can achieve steady-state conditions at the set point, but there are still overshoots.
Many controllers for position control of exoskeleton robots are linear, which may be better for exoskeleton robots with complex structures and high levels of uncertainty [23].Because of factors like changes in load, friction, and outside interference, for an exoskeleton robot, a PID-only controller is insufficient [24], [25].Even if separate PID controls are sufficient for many set-point regulation issues, many tasks call for advanced trajectory-tracking tools.There are several suggestions to overcome these PID controller weaknesses.The first is to handle the non-linearity using feedback linearization [26].Second, improving trajectory tracking performance by employing a computed-torque control strategy that takes into account the manipulator's dynamic model [27].
There has been prior study on planar two-link exoskeletons and robotic manipulators that use PID, model predictive control (MPC), and linear quadratic (LQ) optimal control.PID control for a two-link arm manipulator was proposed by Robles [28] proposed, while the techniques proposed by Guechi et al. [29] (MPC control and LQ control) outperform the PID in terms of system performance.The contribution of this paper is a two-link lower limb exoskeleton model that can overcome the problem of uncertainties such as load and stiffness variations of the patient's lower limb.

METHOD
Figure 1 shows the general layout of a typical gait rehabilitation control system [30].The rehabilitation system hardware and multi-level control architecture can be divided into two main groups: physiotherapy algorithms (high level) and feedback control algorithms (low level).This research will focus on the low-level hardware and controller, major in the lower limb exoskeleton's overall performance.

Kinematics and dynamics
The kinematics and dynamics of the system must be taken into consideration while constructing a control system.Figure 2(a) shows the Lokomat ® , a robotic gait trainer product from Hocoma [31].While Figure 2(b) shows the low-cost version of the gait trainer built by the authors.Figure 2(c) depicts the kinematics of the two-link manipulator.

Kinematics
The exoskeleton for the lower limbs is identical to a planar RR arm with two links.Refer to Figure 2 to determine it is dynamics, where the masses of the link are assumed to be concentrated at it is ends [32].Parameter of joint is (1).(1) Together with the total vector of force being with the actuators supplying τ1 and τ2 torque kinetic and potential energy for link 1 is for link 2 we have for the velocity squared to be (9).
This means that link 2's kinetic energy equals.
In terms of potential energy stored in link 2: As a whole, the arm's Lagrangian equation is as (12).
The terms needed for (14) are Finally, the lower limb dynamics are provided by coupled nonlinear differential equations according to Lagrange's ( 13) and ( 14).6207

Dynamics
When the lower limb dynamics are written out in vector form, the result is This manipulator dynamics is expressed in its typical manner.

Control strategy of gait trainer 2.2.1. Selection of control algorithms
There are three types of control algorithms in the history of control theory: classical, modern, and intelligent.Traditional techniques such basic propositional derivative (PD) and PID linear control, while necessary, are ineffective for nonlinear systems.Motion control is an increasingly used type of modern control because it has a simple structure and superior control performance.The best performance can be achieved with intelligent control, but it is challenging to develop and requires much knowledge.Numerous innovative and sophisticated control strategies have lately been developed for nonlinear mechanical systems [33].For nonlinear mechanical systems, several cutting-edge and contemporary control strategies have been developed, including adaptive control, computed torque control, fuzzy control, neural network control, and sliding mode control [34].

Model learning
Techniques for building models from data have become increasingly valuable robotics tools in recent years.This occurrence is due to a multitude of factors.Standard models, such as rigid body dynamics, are only approximate because of the sophistication of today's robotic systems.This estimate does not accurately model unknown non-linear drive sources (hydraulic and motor saturation), passive elements such as cables, hydraulic hoses, cableways friction and friction sources [35].

Feedback linearization
In a control system, feedback linearization transforms a nonlinear plant-the process or system under control-into a linear plant.In a nonlinear system, this is primarily done to improve the effectiveness of a linear control strategy.However, compensating for nonlinearities and dynamic coupling must happen more quickly than the control speed.By including a linearizing and decoupling controller in the direct control loops, the feedback linearization technique (FLT) achieves this goal.
In the past, analog control circuit design hindered the creation of nonlinear functions from overcoming nonlinear generators due to escalating hardware complexity, which raises prices and reduces reliability [36].A nonlinear and coupled mechanical dynamic system's feedback linearization is described [37], such as a robotic manipulator.The term "model-based" refers to controllers built on dynamic models.Passivity-based and calculated torque controllers are two broad categories for model-based controllers [38].

Computed torque control
Gradually with time, a wide range of different control systems of a robot is being presented as options.Research from the 1970s gave rise to the computed torque controller, first proposed by Paul [39].The subsequent research has led to problems with its actual implementation (such as computation complexity and incorrect models) [40].Computed torque control is a method for generalizing feedback linearization to nonlinear systems.Known also as the inverse dynamics control, the computed torque control depends on the robot dynamics' inversion [41].Introducing the term servo, which takes the form of a PD controller, computed torque controls return a linearization to account for non-linearity in manipulator dynamics [41].The block diagram of the computed torque control technique is shown in Figure 3 feedback linearization can be time-and resource-intensive, so it might be difficult to apply it to change a nonlinear system into a linear one globally.Computed torque controllers provide robust performance characteristics, according to the experiment [42]- [46].Numerous studies using the computed torque control algorithm have been completed successfully [47]- [49].The disadvantage of computed torque controls, however, is the requirement for tracking-related real-time system dynamics computation [50].Consider the control input as (17).
This system, called torque calculation control, constituted of: i) an inner loop of correction for nonlinearity and ii) a second loop that receives an external control signal.This indicates that if these control principles are included in the robotic arm's dynamic model.

𝑞̈= 𝑣
This control input transforms the complex challenge of designing a nonlinear controller into a simple issue of designing a linear system with n subsystems.PD feedback is one method for controlling the outer loop v.  = ̈ +     ̇+     (18) This makes the total controlling input into  = () + (  ̈+     ̇+     ) + (, ̇)̇+   () which leads to linear error dynamics that are (20).
The theory of linear system predicts that the error in tracking would eventually reach zero.Computed torque control (CTC) for robot control systems is a robust motion control method that can provide global asymptotic stability [51].However, a conventional CTC can only be used efficiently if a precise and thorough dynamic model of a manipulator is provided [52].Shown in Table 2 are parameters of a two-link model.

Microcontroller and motor driving circuit
As shown in Figure 4, two DC motors move the two links in the mechanism, while encoders serve as position sensors.The Arduino Uno microcontroller and the Monster Moto Shield VNH30SP DC motor driver circuit are used to control the motors.Based on the ATmega 328 microcontroller, Arduino is the primary controller.The Monster Moto Shield VNH30SP motor driver module will enhance the current to the motor.

RESULTS AND DISCUSSION
Figures 5 and 6 show the output position using the PID controller, while Figures 7 and 8 show the output position using the computed torque controller.The obtained results of both controllers are compared.It can be seen from all the graphs that both PID and CTC can suppress the overshoot.Both controllers have good performances for downward movements but need better performances for upward movements because there is additional gravity as the disturbance.Figure 5 shows the plot of the position of motor 1 and motor 2 while moving downward using the PID controller.
The parameters for motor 1 are: Kp=0.5;Ki=0.01;Kd=1.1, while for motor 2: Kp=0.5;Ki=0.01;Kd=1.1.The PID controller worked well in that situation.At motor 1 the rise time is 25 ms, with a steady-state error of 0.0167%.Meanwhile, at motor 2 the rise time is 19 ms, with a steady-state error of 0.75%.
Figure 6 shows the position of motor 1 and motor 2 while moving upward using the PID controller.The parameters for motor 1 are: Kp=0.5;Ki=0.01;Kd=1.1, while for motor 2: Kp=0.5;Ki=0.01;Kd=1.1.In this situation, the PID controller did not work well at motor 2, and there is a steady-state error of more than 5%.There is a gravity disturbance that the PID controller cannot handle.At motor 1 the rise time is 21 ms, with a steady-state error of 0.1%.Meanwhile, at motor 2 the rise time is 17 ms, with a steady state error of 5.6%.
Figure 7 shows the plot of the position of motor 1 and motor 2 while moving downward using the CTC controller.The parameters for motor 1 are: Kp =4; Ki=0.001;Kd=0.0005, while for motor 2: Kp=0..001;Kd=0.0005.In this situation, the CTC controller worked well, even though at motor 2, there was still a little steady-state error.At motor 1 the rise time is 21 ms, with a steady state error of 1.04%.Meanwhile, at motor 2 the rise time is 30 ms, with a steady-state error of 2.175%.The plot of motor 1 and motor 2 positions during upward motion using CTC is shown in Figure 8.The parameters for motor 1 are: Kp=4; Ki=0.001;Kd=0.0005, while for motor 2: Kp=0.35;Ki=0.001;Kd=0.0005.CTC performed very well under these conditions.As mentioned, the computed torque controller method adjusts for gravity effects.There is a steady state inaccuracy of 1.74% with a rise time of 44 ms at motor 1.At the same time, motor 2 had a rise time of 25 ms and a steady-state error of 2.125%.The performance (rise time, steady state error, and overshoot) of CTC and PID controllers for downward and upward motion are summarized in Tables 3 and 4.Although CTC is better than PID controller in dealing with disturbance (gravity), PID controller has a faster rise time.The system performance index can be calculated or assessed to assess system performance.The root mean square error (RMSE) and the mean square error (MSE) might both be used in this case.The acronym MSE means the square of RMSE.A statistic known as the RMSE contrasts actual values with predictions made by a hypothetical model [53].MSE is represented as follows:

𝑉
V is quantity of data.The expression for RMSE is as follows:

𝑉
According to the RMSE and MSE, Tables 5 and 6 compare the PID and CTC responses numerically.It can be seen from the MSE and RMSE analysis that for upward movements, CTC always has better responses than PID.However, for the downward direction, the response of motor 2 (M2) using CTC is not as good as PID.

CONCLUSION
This study examines the usage of a computed torque controller on a straightforward 2-link exoskeleton model for a robotic gait trainer, then compares its performance with a PID controller.While the lower limb's movement is non-linear, the PID controller is linear.So, we need another control that can work on a non-linear system.It is the computed torque controller in this instance.
In robotics and mechatronics, a method called computed torque control is employed to control the motion of a robotic system.Using the system's equations of motion, the needed torques at the joints of the robotic system are computed to create the desired motion.In the case of a two-link lower limb exoskeleton of a robotic gait trainer, the computed torque control method would involve calculating the appropriate torques at the joints using the equations of motion for the two links and the joints linking them to accomplish the desired motion of the lower limb.
The Lagrangian technique has been used to examine the two-link mechanism's dynamics and describe its kinematics.A model of 2 links mechanism model has also been created.This model of 2 links lower limb exoskeleton model will be used to improve a robotic gait trainer model.The PID controller's performance was compared to the CTC's in both upward and downward directions.
From the result, the suggested computed torque controller for this two-link model has advantages in dealing with disturbance.However, the PID controller is still faster than CTC for the rise time.The experiments show that both the PID controller and CTC can suppress overshoot.The PID controller cannot compensate for gravity as the disturbance, but CTC can do that.As a result, in the downward direction, the steady-state error of the PID controller can be as high as 5.6%, but in the CTC, the controller can be reduced to 2.125%.The MSE and RMSE analysis show that CTC always responds better than PID.However, for downward movement, the response of motor 2 (M2) using CTC is better than PID.

Figure 1 .
Figure 1.The architecture of the gait rehabilitation control system

Int
lower limb exoskeleton model control enhancement using computed torque (Elang Parikesit)

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ISSN: 2088-8708 Int J Elec & Comp Eng, Vol. 13, No. 6, December 2023: 6204-6215 6210 With a sampling time of 10 ms, the encoder will provide data on the link's current position.The microcontroller will use the computed torque control technique to determine the pulse-width modulation (PWM) value to send to the driving circuit.

Figure 5 .Figure 6 .Figure 7 .
Figure 5.The plot of setpoint vs. PID output in the downward direction

Figure 8 .
Figure 8.The plot of setpoint vs. CTC output in the upward direction

Table 1 .
Table 1 lists the various model types (forward, inverse, mixed, and operator) and model learning architectures.An overview of the model types and how they relate to the various learning architectures and sample applications [27]

Table 2 .
Two-link parameters

Table 3 .
Performance of PID controller in the downward and upward direction

Table 4 .
Performance of CTC controller in the downward and upward direction

Table 5 .
Comparison of PID and CTC responses in an upward direction by numerical analysis Two-link lower limb exoskeleton model control enhancement using computed torque (Elang Parikesit) 6213

Table 6 .
Comparison of PID and CTC responses in a downward direction by numerical analysis