Eliminating Chattering in Prosthetic Fingers Using Classic Synergetic Control

A prosthetic finger is a custom-made device to replace a missing or dysfunctional finger, restoring function and appearance [1]. It can range from simple cosmetic covers to advanced robotic prostheses with sophisticated functionalities [2]. Advancements in materials science, biomechanics, and prosthetic technology have led to sophisticated prosthetic fingers using materials like silicone, carbon fiber, and lightweight metals, though challenges such as size, weight, and stiffness complicate controller design and accurate mathematical modeling; advanced prosthetics often incorporate electronics, sensors, and microprocessors to enable natural movement and precise control [3]. Several studies have shown that prosthetic fingers significantly improve hand function and psychosocial well-being, highlighting their importance in enhancing the quality of life for individuals with finger amputations [4-7]. Scholars have proposed various feasible control strategies for prosthetic finger control, and past researchers have utilized different approaches in MATLAB simulations.

Despite significant advancements in control strategies for prosthetic fingers, effectively controlling of prosthetic fingers and mitigating the chattering effect inherent in robust methods like Sliding Mode Control (SMC) and Adaptive Sliding Mode Control (ASMC) remains challenging. Existing approaches, such as Proportional Integral Derivative (PID), Fuzzy Logic Control (FLC), and computed torque control, have benefits and limitations but do not fully address chattering or the complexities of nonlinear systems and input uncertainties, necessitating further research for robust, stable, and precise control strategies.

Arslan et al. [2, 4] compared the path-tracking efficiency of various control strategies, including FL, SMC, and PID controllers. They evaluated these different strategies using a biomimetic robot hand-finger model, analyzing simulation results quantitatively. They found that caution is needed when interpreting these findings as they are based on simulations rather than real-world applications. Therefore, the study emphasized the analysis of experimental results, including tendon forces and prosthetic finger motion.

Xu et al. [8] presented findings from a project using two synergistic inputs to achieve precise movements with a prosthetic hand. They established postural synergy on a phantom hand, kinematically identical to a real hand, to better replicate intended poses for manipulation tasks. They developed and assembled the prosthetic hand, qualitatively verifying its motion ranges. This technique allows imaginary hand positions to be accurately replicated by the prosthetic hand using contralateral postural synergies and synergistic inputs, enabling continuous transition between positions.

Lysenko et al. [9] proposed a mathematical model for a biomechanical finger prosthesis, focusing on its dynamics during a broader range of movements. They concluded that the control system's mathematical model was accurately constructed, ensuring regulator output values stayed within permissible limits and control time closely matched the engine's constant time.

Hamidi et al. [10] introduced classical and adaptive control designs based on synergetic theory for tracking control of a single-link robot arm actuated by artificial muscles. The ASMC addresses uncertainties in muscle parameters and ensures system stability. Simulated results indicate that the optimized control design provides superior tracking performance in terms of transient and steady-state characteristics compared to non-optimal designs, despite higher power consumption. In the presence of parameter uncertainties, the adaptive control system outperforms non-adaptive systems.

Al-Hussein et al. [11] studied the suppression of chaotic oscillations in a three-conductor power system. They utilized an ASMC algorithm along with a Static Synchronous Compensator (STATCOM) and energy storage devices in their control unit design. The system's dynamics displayed critical chaotic oscillations influenced by specific parameters, which can jeopardize power system bus voltage stability. Chaos in power systems may lead to system collapse, significantly impacting the quality of commercial power services.

The Shadow Robot Company [12] created the Shadow Dexterous Hand, a humanoid robotic hand system with 24 movements that closely resembles the dexterity and kinematics of a human hand. PID controllers are configured during setup, and control strategies are implemented via EtherCAT. Depending on the launch file selected, software testing on both simulated and real hardware can be done seamlessly using the same interface thanks to the drivers provided by the shadow robot EtherCAT stack and a simulated hand model available in the Gazebo simulator.

Fan et al. [13] suggested an adaptive grasp technique that uses error estimation compensation and collaboration control to mitigate object pose uncertainty. The error compensation method estimates the object's pose error based on finger tactile information and uses multiple techniques to compensate for it based on different error ranges. The outcomes demonstrate that this adaptive grasp strategy greatly raises the success rate of grasping objects in uncertain poses.

Li et al. [14] proposed a new object-level impedance control framework aimed at improving grasp force and quality. By dynamically sliding from the initial to the final grasp configuration, optimal grasp quality is achieved. The proposed controller allows the object to be maneuvered in hand while responding to external forces. Proper design parameter selection, as verified by a Lyapunov function, ensured the closed-loop robotic system's effective control performance.

Shahriari et al. [15] developed a single-arm control policy that applies to all robots in a multi-manual system, including an adaptive force-impedance controller based on a modeled control objective. Two experiments were carried out: the first involved a bi-manual system grasping and moving an object along a desired path, and the second tested a teleoperation setup that allowed remote control of the object. The findings demonstrated the approach's robustness and practicability.

Zhang et al. [16] proposed adaptive sliding mode friction indemnification adaptive impedance controllers for a stereophonic artificial hand. To satisfy the demands of a force-tracking impedance controller, they created a five-fingered hand with multisensory abilities and DSP-based control hardware and software. According to experimental results, in uncertain environments with unknown stiffness and position, the adaptive controller with friction indemnification was able to achieve accurate force-tracking and stable torque/force response.

Jalani et al. [17] offered a novel approach to active compliance control using Integral Sliding Mode Control (ISMC), which combines a virtual mass-spring damper for compliant control with a model reference approach. The task controller demonstrated excellent tracking performance in spite of high friction and stiction, according to the results. Furthermore, by using the posture controller for the index and thumb fingers in conjunction with spherical and cylindrical coordinates, the fingers were able to move around the object without colliding, highlighting the grasping task.

Herrmann et al. [18] proposed a new approach to using ISMC for active compliance control. The use of spherical and cylindrical coordinates, combined with the posture controller for the thumb and index finger, ensured that both fingers moved around the object without colliding, allowing for practical and powerful grasping. Furthermore, BERUL fingers with tactile pressure sensors reduced the force applied to objects. An automated tuning procedure demonstrated the effectiveness of this compliant control in grasping similar objects at various required force levels.

Labbadi and Cherkaoui [19] examined a robust, adaptive controller for tracking and stabilizing the flight path of quadrotor Unmanned Aerial Vehicles (UAVs). They used the Newton-Euler method to determine the dynamics of the quadrotor and created two sturdy controllers to control parametric uncertainties. This design combines SMC methods with the Adaptive Backstepping approach.

Mohd Zaihidee et al. [20] investigated cutting-edge SMC applications for permanent magnet synchronous motor (PMSM) speed control. The study identified several areas that require additional research, such as the application of sliding mode observers in conjunction with other controllers, SMC for PMSM Direct Torque Control (DTC), and sensorless speed control applications utilizing sliding mode controllers.

A neuro-sliding mode control approach was tested using a reference compensation technique in a non-model-based control framework [21]. A Radial Basis Function (RBF) network, similar to a Multi-Layer Perceptron (MLP), was used to account for uncertainties in a three-link robot manipulator. A stability analysis of the neuro-sliding mode control scheme was carried out, and simulation results showed that it outperformed traditional sliding mode control in tracking performance.

Majeed et al. [22] focused on modeling and controlling a tendon-driven prosthetic finger resembling a human index finger. They employed CSMC and ASMC techniques to manage finger movements, aiming to overcome CSMC's chatter issue. Their findings demonstrated that ASMC outperformed CSMC, offering quicker response times and reduced chatter during prosthetic finger movements.

Analysis of existing literature identified a gap in controlling of prosthetic fingers, which poses greater challenges than fully actuated ones. Many existing control strategies, including PID, FLC, and computed torque control, have limitations that prevent them from fully addressing the complexities and issues such as the chattering effect and system uncertainties, necessitating further research for more robust solutions.

Previous studies have not investigated the use of CSC in prosthetic finger motion, despite its potential to improve stability and reduce chattering, which could otherwise damage the prosthetic finger. Research on CSC specifically for prosthetic finger control is lacking. Where the CSC offers several advantages: it provides robust control against disturbances and parameter variations, enables precise regulation of system states, manages nonlinear dynamics effectively, maintains system stability in dynamic environments, and can be adapted to various system configurations.

Therefore, this paper proposes a novel anti-interference control strategy for trajectory tracking in the joint space of a grasp-driven prosthetic finger. It integrates CSC to enhance stability and eliminate chattering during finger motion, aiming to prevent potential damage to the prosthetic finger. Wherefore, this work presents a CSC to control the variables of a constrained prosthetic finger, aiming to maintain system robustness amid parameter variations and disturbances. Also, it involves analyzing the dynamic model and state space representation of the robotic manipulator, developing CSC algorithms, and conducting stability analysis based on the control law to ensure consistent and stable motion of the manipulator.

The primary advantages of the CSC approach include:

• CSC offers robust control against parameter variations, ensuring stable performance without chattering in dynamic and uncertain environments.
• CSC algorithms enable precise regulation of system states, which is essential for controlling complex and underactuated robotic systems.
• CSC efficiently manages nonlinear dynamics, making it ideal for systems with complex interactions and constraints.
• CSC contributes to system stability by analyzing the dynamic model and state-space representations.
• Designing rules for CSC is based on straightforward mathematical principles, making it simple to implement even in complex systems.
• CSC can be customized to meet specific system configurations and requirements, increasing its applicability across a wide range of robotic systems and manipulations.

This paper will include the following sections, in sequential order:

• The dynamics and control model outlines the mathematical model of a prosthetic finger actuated by CSC.
• Results and Discussion Show simulation results and discuss the control system and model response.
• Finally, the paper is concluded with a section titled Conclusion. The methodology of this paper is described in the block diagram shown in Figure 1, which depicts the sequence of presenting the contents of this research.

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Figure 1. Block diagram of the methodology

Current control theory faces difficulties in addressing the complexity of intricate macro-systems, which involve multi-dimensional, nonlinear dynamics and interconnected subsystems exchanging information, matter, and power. This complexity makes it challenging to identify universal control laws for system synthesis [32]. Comprehensive CSC is an approach designed to manage nonlinear systems by integrating multiple control mechanisms to ensure stability and resilience despite uncertainties and nonlinearities. This offers a solution for inherently unstable systems [33]. A control strategy has been developed to track the angular position in a prosthetic finger system using the CSC approach to create the CSC algorithm for a prosthetic finger system, following specific steps and assuming no external disturbances [34].

The first step in the design is to define the error $e_1$ is the difference between the desired angle position $\left(x_{1 d}=\theta_{1 d}\right)$ and the actual angle position $\left(x_1=\theta_1\right)[35]$.

$e_1=x_1-x_{1 d}$     (21)

$e_2=x_3-x_{3 d}$     (22)

$e_3=x_5-x_{5 d}$     (23)

$e_2$ is the difference between the desired angle position $\left(x_{3 d}=\theta_{2 d}\right)$ and the actual angle position $\left(x_3=\theta_2\right)$ and $e_3$ is the difference between the desired angle position $\left(x_{5 d}=\theta_{3 d}\right)$ and the actual angle position $\left(x_5=\theta_3\right)$.

By calculating the first, second and third derivatives, one can obtain the desired outcome.

$\dot{e}_1=\dot{x}_1-x_{1 d}=x_2-x_{1 d}$     (24)

$\dot{e}_2=\dot{x}_3-x_{3 d}=x_4-x_{3 d}$     (25)

$\dot{e}_3=\dot{x}_5-x_{5 d}=x_6-x_{5 d}$     (26)

$\begin{aligned} \ddot{e}_1=\dot{x}_2-x_{1 d}^{\ddot{ }}= & \frac{1}{y}\left[\left(M_{22} M_{33}-M_{32} M_{23}\right) u_1+\left(M_{32} M_{13}-M_{12} M_{33}\right) u_2+\left(M_{12} M_{23}-M_{22} M_{13}\right) u_3+\left(M_{32} M_{23}-M_{22} M_{33}\right) C_1\right. \\ & +\left(M_{32} M_{23}-M_{22} M_{33}\right) G_1+\left(M_{12} M_{33}-M_{13} M_{33}\right) C_2+\left(M_{12} M_{33}-M_{13} M_{32}\right) G_2+\left(M_{22} M_{13}-M_{12} M_{23}\right) C_3 \\ & \left.+\left(M_{22} M_{13}-M_{12} M_{23}\right) G_3\right]x_{1 d}^{\ddot{ }}\end{aligned}$     (27)

$\begin{aligned} \ddot{e_2}=\dot{x_4}-x_{3 d}^{\ddot{ }}= & \frac{1}{y}\left[\left(M_{31} M_{23}-M_{21} M_{33}\right) u_1+\left(M_{11} M_{33}-M_{13} M_{31}\right) u_2+\left(M_{13} M_{23}-M_{11} M_{23}\right) u_3+\left(M_{21} M_{33}-M_{31} M_{23}\right) C_1\right. \\ & +\left(M_{21} M_{33}-M_{31} M_{23}\right) G_1+\left(M_{13} M_{31}-M_{11} M_{33}\right) C_2+\left(M_{13} M_{31}-M_{11} M_{33}\right) G_2+\left(M_{11} M_{23}-M_{21} M_{13}\right) C_3 \\ & \left.+\left(M_{11} M_{23}-M_{21} M_{13}\right) G_3\right]-x_{3 d}^{\ddot{ }}\end{aligned}$     (28)

$\begin{aligned} \ddot{e_3}=\dot{x}_6-x_{5 d}^{\ddot{ }}= & \frac{1}{y}\left[\left(M_{32} M_{21}-M_{31} M_{22}\right) u_1+\left(M_{31} M_{12}-M_{32} M_{11}\right) u_2+\left(M_{11} M_{22}-M_{21} M_{12}\right) u_3+\left(M_{31} M_{22}-M_{32} M_{21}\right) C_1\right. \\ & +\left(M_{31} M_{22}-M_{32} M_{21}\right) G_1+\left(M_{32} M_{11}-M_{31} M_{12}\right) C_2+\left(M_{32} M_{11}-M_{31} M_{12}\right) G_2+\left(M_{21} M_{12}-M_{11} M_{22}\right) C_3 \\ & \left.+\left(M_{21} M_{12}-M_{11} M_{22}\right) G_3\right]-x_{5 d}^{\ddot{ }}\end{aligned}$     (29)

where,

$y=M_{31} M_{23} M_{12}-M_{21} M_{33} M_{12}+M_{21} M_{13} M_{32}+M_{11} M_{22} M_{33}-M_{11} M_{32} M_{23}+M_{13} M_{31} M_{22}$     (30)

The dynamic equation of the Marco variable $\sigma$ is described as

$\sigma_1=c e_1+\dot{e}_1$     (31)

$\sigma_2=c e_2+\dot{e}_2$     (32)

$\sigma_3=c e_3+\dot{e_3}$     (33)

In this context, the scalar design for synergetic control is denoted as $c$, where $c$ is a positive value.

$\dot{\sigma}_1=c \dot{e}_1+\ddot{e}_1$     (34)

$\dot{\sigma}_2=c \dot{e}_2+\ddot{e}_2$     (35)

$\dot{\sigma}_3=c \dot{e}_3+\ddot{e}_3$     (36)

The symbol $\dot{\sigma}_1, \dot{\sigma}_2$ and $\dot{\sigma}_3$ denote the manifold equation variable as defined within the context.

$T \dot{\sigma}_1+\sigma_1=0$     (37)

$T \dot{\sigma}_2+\sigma_2=0$     (38)

$T \dot{\sigma}_3+\sigma_3=0$     (39)

where, $T$ is greater than 0 , isthe converging ratio of $\sigma$ to manifold with $\dot{\sigma}$.

By substituting Eqs. (34)-(36) into Eqs. (37)-(39), to obtain:

$T\left(c \dot{e}_1+\ddot{e}_1\right)+\sigma_1=0$     (40)

$T\left(c \dot{e}_2+\ddot{e}_2\right)+\sigma_2=0$     (41)

$T\left(c \dot{e}_3+\ddot{e}_3\right)+\sigma_3=0$     (42)

Additionally, by utilizing Eqs. (40)-(42), one can derive:

$\begin{gathered}T\left(c \dot{e}_1+\left(\frac{1}{y}\left[\left(M_{22} M_{33}-M_{32} M_{23}\right) u_1+\left(M_{32} M_{13}-M_{12} M_{33}\right) u_2+\left(M_{12} M_{23}-M_{22} M_{13}\right) u_3+\left(M_{32} M_{23}-M_{22} M_{33}\right) C_1+\right.\right.\right. \\ \left(M_{32} M_{23}-M_{22} M_{33}\right) G_1+\left(M_{12} M_{33}-M_{13} M_{33}\right) C_2+\left(M_{12} M_{33}-M_{13} M_{32}\right) G_2+\left(M_{22} M_{13}-M_{12} M_{23}\right) C_3+\left(M_{22} M_{13}-\right. \\ \left.\left.\left.\left.M_{12} M_{23}\right) G_3\right]-x_{1 d}^{\ddot{ }}\right)\right)+\sigma_1=0\end{gathered}$     (43)

$\begin{gathered}T\left(c \dot{e}_2+\left(\frac{1}{y}\left[\left(M_{31} M_{23}-M_{21} M_{33}\right) u_1+\left(M_{11} M_{33}-M_{13} M_{31}\right) u_2+\left(M_{13} M_{23}-M_{11} M_{23}\right) u_3+\left(M_{21} M_{33}-M_{31} M_{23}\right) C_1+\right.\right.\right. \\ \left(M_{21} M_{33}-M_{31} M_{23}\right) G_1+\left(M_{13} M_{31}-M_{11} M_{33}\right) C_2+\left(M_{13} M_{31}-M_{11} M_{33}\right) G_2+\left(M_{11} M_{23}-M_{21} M_{13}\right) C_3+\left(M_{11} M_{23}-\right. \\ \left.\left.\left.\left.M_{21} M_{13}\right) G_3\right]-x_{3 d}^{\ddot{ }}\right)\right)+\sigma_2=0\end{gathered}$     (44)

$\begin{gathered}T\left(c \dot{e}_3+\left(\frac{1}{y}\left[\left(M_{32} M_{21}-M_{31} M_{22}\right) u_1+\left(M_{31} M_{12}-M_{32} M_{11}\right) u_2+\left(M_{11} M_{22}-M_{21} M_{12}\right) u_3+\left(M_{31} M_{22}-M_{32} M_{21}\right) C_1+\right.\right.\right. \\ \left(M_{31} M_{22}-M_{32} M_{21}\right) G_1+\left(M_{32} M_{11}-M_{31} M_{12}\right) C_2+\left(M_{32} M_{11}-M_{31} M_{12}\right) G_2+\left(M_{21} M_{12}-M_{11} M_{22}\right) C_3+\left(M_{21} M_{12}-\right. \\ \left.\left.\left.\left.M_{11} M_{22}\right) G_3\right]-x_{5 d}^{\ddot{ }}\right)\right)+\sigma_3=0\end{gathered}$     (45)

The synergetic control law for a prosthetic finger system can be derived from Eqs. (43)-(45), as demonstrated below:

$u_1=M_{11}\left(-c \dot{e_1}+x_{1 d}^{\ddot{ }}-\frac{\sigma_1}{T}\right)+M_{12} \dot{x_4}+M_{13} \dot{x_6}+C_1 x_2+G_1$     (46)

$u_2=M_{21} \dot{x_2}+M_{22}\left(-c \dot{e_2}+x_{3 d}^{\ddot{ }}-\frac{\sigma_2}{T}\right)+M_{23} \dot{x_6}+C_2 x_4+G_2$    (47)

$u_3=M_{31} \dot{x_2}+M_{32} \dot{x_4}+M_{33}\left(-c \dot{e_3}+x_{5 d}^{\ddot{ }}-\frac{\sigma_3}{T}\right)+C_3 x_6+G_3$     (48)

Figure 3 illustrates the schematic representation of the CSC design for a robot arm.

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Figure 3. Schematic diagram of CSC

To validate the results of the methodology used in this study, a comparison will be made with the reference [22], as this study is closer to the system used in this work in terms of dynamic analysis, but differs in the approach used, employing CSMC on the prosthetic finger. Therefore, the comparison will be conducted to ensure the accuracy of the algorithms used in this study.

Figure 4 illustrates the angular position results of the prosthetic finger compared with the reference [22]. Through Figure 4, it is evident that classical synergetic control has proven its efficiency and superior strength over CSMC, where observed a significant improvement in tracking angular position smoothly, without chatter, and in slightly less time than the CSMC algorithm.

Figure 5 demonstrates the arrangement of phalanx for the model controlled in CSC. It was observed that the tracking response, achieving the desired position, occurred within a maximum of 9 sec for CSC, the time taken to connect to the desired position is favorable compared to the previous study.it is intelligible that the CSC controller was able to successfully track the desired position for each phalange and stabilize, and distinguished by their efficiency, flexibility, and smoothness in terms of performance.

Figure 6 illustrates the control action of a CSC, where respectively the first, second and third phalanx control action reaches peaks were 1.559 N/m, 0.542 N/m and 0.014 N/m, and they stabilize simultaneously at 1.5 sec and demonstrated monotonous and smooth control signal behaviors for CSC controller. This behavior has enhanced the use of this type of control in various applications. It was observed that the applied torque is zero in the steady state for each controller.

Figure 7 shows the angular velocity for CSC. The first, second and third angular velocities peak at 2.354 and stabilize at 5.1 sec.

The simulation results can be seen in Figure 8 for the tracking error the CSC prosthetic finger system.

The velocity error behavior for CSC is shown in Figure 9 the system can be describing was global asymptotic stability to the by the convergence the of angular position to each phalange to the desired position. This controller CSC can make the prosthetic finger asymptotically stable by forcing the error and the derivative of the error to reach zero at a good time in the final trajectory, resulting in precise tracking of the movement of the prosthetic finger.

4a.png

(a) CSMC

4b.png

(b) CSC

Figure 4. Tracking performance between the desired and existing position

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Figure 5. Tracking performance between the desired and existing position of CSC algorithm

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Figure 6. Control action response using CSC algorithm to the prosthetic finger

7.png

Figure 7. Angular velocity response for using CSC

8.png

Figure 8. Position error response for CSC

9.png

Figure 9. Velocity error response for CSC