Design of a Sliding Mode Controller for a Prosthetic Human Hand’s Finger

Handling editor: Muhsin J. Jweeg


H I G H L I G H T S A B S T R A C T
• The prosthetic finger is move by tendon instead of motor. • The system is non-linear dynamic model. • Sliding mode control can drive the model to desired position. • Classical Sliding mode control suffers from chattering.
In this research paper, the modeling and control of a tendon driven, instead of joint motors, prosthetic finger that mimics the actual human index finger were deliberated. Firstly, the dynamic model of the prosthetic finger is developed based on a 3-degree of freedom (DOF) articulated robot structure and utilizing the Lagrange equation. Then, the classical sliding mode control (CSMC) strategy was implemented to control the finger motion. To overcome the cons of CSMC, such as the chattering problem, an adaptive sliding mode controller (ASMC) was developed. MATLAB Simuphalange was used to perform the simulation after the necessary equations were derived. The obtained results showed that the ASMC superior to the CSMC in depressing the chattering and fast response.

Introduction
In our society, robots play an important role. They are currently commonly used for labor-intensive operations in various industrial applications that involve a high degree of precision and repetition. In the entertainment industry, robots can be used in the form of toys and animations. The role of robots in society is continually evolving. This research aims to bring them further into domestic aid, medicine, military, search, and rescue. The robot must perform only one particular task in many of these applications and be an engineered for a single operation. However, as robots' possible use increases, they will need to communicate with objects in their environments. The creation of end effectors that accumulate and use various objects as resources is an important challenge in robots' production. The human hand is considered to be the most dexterous end effector, with a total of 26 DOF [1]. The construction of a highly advanced prosthesis hand needs to deal with two significant problems: the development of a mechanical design that would permit enough movement freedom and how to build controllers be robust and capable of handling the associated mechanical design, which is commonly so complicated. The human body movement is based on the muscles that apply forces to the skeleton by the tendons. In the literature, with the robot and prosthetic hand, the human hand's motion system is commonly applied to imitate the natural movement [2].
The human hand tendon arrangements were tested to ensure that the robot's hands have the maximum total muscle strength requirements. From this analysis, it can be inferred that the robotic hand has force values highly related to that of the human hand. Weghe et al. [3] Created a human hand tested that was anatomically correct to determine its mechanism, function, and power. In recent years, many papers have investigated the serial robots assembled based on the tendon-driven mechanisms, which are composed of pulley-belt configurations. Kawanishi, et al. [4] have developed a system considering the fuzzy logic approach for fingertip position control of a 4 DOF tendon-driven robotic finger. The tendon's elasticity was also taken into consideration. Hristu et al. [5] made a comparison between the efficiency of fuzzy logic and traditional PID controllers for a multi-finger robotic hand. The aim was to control both the fingertip's position and the forces exerted on the fingertip. It was established that the fuzzy logic control approach superior to the PID controller. Muscles will exercise pulling effort only, and 258 muscle forces are sent through the tendons to the finger bones. The muscles of the forearm stretch to the hand more than fifteen tendons. When the finger is extended, one set of muscles and tendons exercise the finger's expanding motion, and a separate set of muscles and tendons create the bending motion. Tendon structure in hands is problematic and, however, leads to the high functionality of human hand movements through this sophisticated tendon arrangement. Hand tendons that lie at the back of the human hand straighten the fingers and flexor tendons and turn the fingers on the palm side of the hand. In this research, a 3-degree of freedom chain robot mechanism closely mimics the human index finger's size is modeled. On the palm and the back of the hand, the phalanges' tendon fixing points are reduced to three.
However, the considered problem in this work is the control of an under actuated finger, which is a very challenging task than the fully actuated one, and how to guarantee the overall stability of such a system. The work suggests first designing a classical sliding mode controller (CSMC), which is theoretically is able to reject the bounded matched perturbation altogether. The following points summarize the main steps that are conducted in this research to obtain this work's aim: To analyze of dynamic model and state space representation of finger contact grasped object for the human prosthetic hand. To the development of conventional sliding mode control (SMC) algorithm of the prosthetic finger. To analyze the prosthetic finger's stability controlled by the conventional SMC scheme based on the sliding surface defined in the state space representation, the control action is responsible for maintaining and constructing the sliding motion. Figure 1 represents the finger of a human prosthetic hand, on which the controller methods are applied. A prosthesis model of the two dimensions (2D) finger was considered a movie chain consisting of three cylindrical phalanges that mimic the inertial characteristics of the finger index's proximal, central and distal phalanges, as demonstrated in Figure 2. The articulations of the model to reflect the joint masses of the finger were supplemented with spherical components.
Using the Lagrange-Euler formulation that is shown in Equation 25 below, the system equations of motion can be obtained for each system coordinate[ 1 2 2 ], which give the applied torque on each joint, as follows: Where is the torque that each phalange experiences, and ̈ is the angular acceleration? However, the system generalized equations of motion in the matrix form is given by the following nonlinear form: Where ( ) is an 3 × 3 mass matrix, ( ,̇ ) is an 3 × 1 vector contains the Coriolis and centrifugal terms, ( ) is an 3 × 1 gravity vector, and is an 3 × 1 generalized input torque vector; The terms VG 1 , VG 2 and VG 3 constitute the lumped expressions of Carioles/centrifugal forces and gravity consequences. We assume that phalange 1 is attached to phalanges 2 and 3 through the pulley-belt machine approach.

Classical sliding mode controller (SMC)
One of the successful methods to monitor nonlinear systems involving matched disturbances is sliding mode control (SMC), which has provided efficiency in design has been widely used in industrial [7]. In this control scheme, states of the system are handled toward a sliding surface and forced to stay on (or near) it. However, two parts are there within the layout of this class of controllers. Firstly, within the system's state space model, a sliding surface is introduced; secondly, to construct and maintain the sliding motion, the control law, which is responsible for that, is found [8]. Thus, the sliding surface ( ) can be presented as: (31) Where  is a constant and it is > 0. However, let assume that 1 = , 2 =̇ where is the error and ̇ is the dervitive of the error and = 1, hence the sliding surface can be rewritten as: The complete control law can be defined as follows: represent the nominal control part, and the discontinuous control part [9], which is defined as: ) K ( ) is a discontinuous gain, and n ( ) is known as a signup function, which can mathematically be expressed as in Equation 35 below.
Therefore, the control action equation can be written as [10]: The structure of the SMC is shown in Figure 4. This structure has one input: the motor's torque and the output is the finger's desired position. To reduce the chattering problem, the saturation function is used instead of ( ) function in Equation 36; thus, it can be rewritten as: Where the saturation function is: Where φ expresses the boundary layer's width, as shown in Figure 5 (saturation function).

Adaptive Sliding Mode Control (ASMC) Design
The biggest challenges to applying SMC scheme are two interrelated phenomena: chattering and high control action activity. The amplitude of the chattering is proportional to the magnitude of the discontinuous total control. These two issues can be solved at the same time if the magnitude is decreased to a minimum permissible level represented by the conditions for the existing sliding mode [9]. In this study, ASMC is proposed to deal with the two issues mentioned above. ASMC has repressed the switching gain to a low level. When the gains are decreased, the effects of control actions are minimized, which causes attenuating the chattering. Reduce the chattering leads to good tracking performance, which is decreasing the thermal and mechanical losses in the systems [11].The ASMC is more compact and relaxed in the architecture than the classical SMC. As well as, the system reliability is achieving with a limited control effort while using the ASMC [12].
Where, is the control action to be configured, and k (t) is the adaptive controller gain that is described as in the following:    ], > 0 and > 0, and k(t) is chosen depending on the following rules [11].
is the lowest acceptable value of k(t), the highest acceptable value of k(t) can the system handle it, and μ (0) is the initial start point of the gain (t). To provide more information and illustrations on the rule of the adaptive control and how to apply the conditions of Equation 42, the following flowchart is presented. Finally, the controller law will be updated with a saturation function as opposed to the signup function, as described: The main purpose of the ASMC technique is to design a robust controller that can be able to Orient the sliding variable toward the manifold surface and keep the system trajectory in this required sliding surface [14].

Simulation Results
In this work, two techniques were used to control a prosthetic finger. In the current simulation results, the adaptive sliding mode control shows more effectiveness than the classical sliding mode control by solving the chattering problem concerned with high control effort and high control gain. The system has been simplified into six states, which have the initial states as follow: (rad), 2 (0) = 4 (rad), 3 Table 2. Figures 6 and 7 illustrate the state of trajectory from the initial to the end point for the three phalanges. Using the reaching condition of SMC. These trajectories reach approximately zero, and it will make the system asymptotic stable. Figures 8 and 9 show the tracking performance between the actual and desired position, which does not exceed 2.5 sec for both CSMC and ASMC. Figures 10 -11 show the sliding variables for the three-phalange; it is clear that the CSMC suffers from some chattering issue, while the ASMC has a smooth sliding surface. Figure 12 illustrates the control gain for both CSMC and ASMC; the ASMC can reduce the gain to an acceptable value compared with CSMC, where the high gain value is the reason for high charting in the torque action. Figure 13 shows the torque action for the CSMC and ASMC; however, it is evident that utilizing the CSMC on the system, the model will suffer from chattering while applying the ASMC minimizes the chattering phenomenon.

Conclusions
The modeling and control of a tendon controlled 3-degree of freedom prosthetic finger was presented in this research. After obtaining the dynamic modeling, the classic (CSMC) and adaptive (ASMC) sliding mode control techniques were then applied to control the finger's motion. It was noticed that the chattering phenomenon is a severe problem of CSMC. However, the comparison between the CSMC and ASMC which is clarified in Table 3, demonstrated that the ASMC could reduce the controllers' gain value to be reasonable and minimal, thus reducing the control and chattering magnitude. It was also concluded that the ASMC reduces the gain to an acceptable value compared with CSMC, which produces high gain leading to high chattering in the torque action. This was evident from the obtained results where the model system suffered from chattering while applying the ASMC minimizes the chattering phenomenon.