# Model-Free Tracking Control with Prescribed Performance for a Shape Memory Alloy-Based Robotic Hand
## 1 Introduction
Currently, two control methods are considered to improve the control precision of the SMA-based robotic hand system, namely, applying a neural network (NN) to establish the dynamic model of the robotic hand and designing a controller to reduce nonlinearities of the system. In [25], the neural network is used to model the SMA-based robotic hand system in real time, and then predictive control is used to predict the system so as to control the movement of the SMA hand. However, it takes a lot of time to collect the input and output data, and the model trained by the neural network also has errors. In [26], based on the step response signal, the transfer function of the SMA based robotic hand model is established by the system identification method. Then, the sliding mode control method is used to control the system. However, the SMA-based robotic hand is a nonlinear system, and there is a large error between the system identification model and the actual system. In [27], a fuzzy logic control method is designed to realize the motion control of the SMA-based robotic hand. The visual camera is used to track the position of the finger joint instead of the joint angle sensor, which reduces the complexity of the system. In [28], the motion of the SMA-based robotic hand is controlled by the fuzzy PID control method, in which fuzzy logic is used to adjust the parameters of PID. The experimental results show that the PID fuzzy controller has good performance in reducing the maximum overshoot and ensuring control stability. In addition, the control method based on SMA resistance feedback is also used in the control of the SMA-based robotic hand [29; 30; 31]. The key point of adopting NN methods is that the procedure is laborious for gaining the input/output (I/O) data for training the NN off-line. Moreover, it is uncertain that the training condition of building models is in accordance with the condition of the controlled plants in practise. The designing controller method is not taking the performance analysis or accuracy analysis into account. Therefore, it is imperative to design a controller with a simple structure and reliable performance for the SMA-based robotic hand, to obtain high tracking precision and ensure an advanced tracking performance. For a practical system, a control method that is robust, with a simple structure and high precision is preferable. Hence, combined with a new dynamic linearization technique (DLT) derived from Deng [32], we propose a novel control method without the information of the controlled plant. The essence of DLT is that the discrete time nonlinear system can be substituted by an equivalent dynamical linearization data model, which is established at each dynamic execution time point of the closed-loop system. Furthermore, the dynamic linearization technique is a data-driven method, which is merely dependent on I/O data. In order to solve the disadvantages of conventional DLT, such as excessive parameters, the complications of parameter selection, and unclear physical significance for parameters, the new DLT is proposed by applying Taylor series expansion and the differential mean value theorem. It is generally known that it is inescapable to have to deal with the unmodeled dynamics and robustness problems. However, the new DLT can eliminate the above problems efficiently and effectively. In recent years, prescribed performance control (PPC) has been a promising method for constraining the tracking error into a preassigned zone [33]. More specifically, the constrained tracking error can be converted into another form of unconstrained variable by a transformed error function. Then, PPC is designed to stabilize the unconstrained variable to further restrict the tracking error in a preassigned zone. For application, it is highly essential to investigate PPC for controlled plants, which are subjected to constraints generally, such as the steady state error and the convergence rate. Hence, PPC has been a research focus, attracting the interest of researchers [34; 35; 36].