The application of the guided missile seeker is to provide stability to the sensor's line of sight toward a target by isolating it from the missile motion and vibration. The main objective of this paper is not only to present the physical modeling of two axes gimbal system but also to improve its performance through using fuzzy logic controlling approach. The paper is started by deriving the mathematical model for gimbals motion using Newton's second law, followed by designing the mechanical parts of model using SOLIDWORKS and converted to xml file to connect dc motors and sensors using MATLAB/SimMechanics. Then, a Mamdani-type fuzzy and a Proportional-Integral-Derivative (PID) controllers were designed using MATLAB software. The performance of both controllers was evaluated and tested for different types of input shapes. The simulation results showed that self-tuning fuzzy controller provides better performance, since no overshoot, small steady-state error and small settling time compared to PID controller.

Weapon history has evolved in tandem with human history. Recent weapon systems are evolving to cause the least amount of human harm and to neutralize military facilities. The primary goal of a missile is too accurately to shoot the moving and fixed targets. As a result, it has been steadily producing missiles that use different seeker technologies. The seeker system's function is to actively track up to the target by detecting and locking on to the object. In this system, there are many sources of noise and disturbances, which come from vibrations of the seeker and maneuvering of the missile while flighting, also there is a decoupling in the line of sight (LOS) between the seeker and the object. So, it is realized that the seeker consists of a two-axis gimbal platform to track stably and make a stabilization loop [

In the two-axis gimbal system, dc motors are connected to each axis as actuating parts, for the sensing part, two sensors are used a gyro is used to measure the speed and position sensor for angle measuring. To avoid disturbance and measure stable values, sensors are located on the inner gimbal. Most disturbances result from missile motions, gimbal system geometry, and gimbal system imperfections like mass unbalance [

On the other hand, many controllers were designed and studied for years to stabilize the gimbal system. Nonlinear controllers such as, in [

This paper is organized as follows, it started by derivation the mathematical model of the system by using Lagrange equations followed by 3D physical modeling of two-axis gimbal using MATLAB Simscape, after that fuzzy logic controller based Mamdani type designed for inner and outer loops. Finally, the simulation results of both fuzzy and PID controllers are presented.

Let the figure of under consideration system as shown as in

It contains a body with outer and inner gimbals such that the tracking sensor is mounted on the inner gimbal. In this regard, three references frames can be defined as body frame

Let

For the inertial angular velocities of frame

From the above equation, the relations between angular velocities of the outer gimbal, body, and inner gimbal are as follow:

From

The gimbal dynamics model can be derived from the torque relationships about the inner and outer gimbals. The equation of motion for the inner gimbal can be expressed as follows:

The inner gimbal motion equation about the pitch axis can be expressed as:

The relation

Then,

If the body is non-rotating, i.e., _{B}

The equation of motion for outer gimbal can be expressed as:

In which, all the parameters and coefficients are defined as a similar manner in the Section (2.1) and

The external undesired disturbance torques

where,

These terms of disturbance torque can be interpreted as follows. Suppose the rotation angle of

By using the relation

If the mechanical design is such that the condition

The last manipulation which should be done on the derived equation of the outer gimbal is to rewrite the equation according to the

By using relation

For a non-rotating body _{c}

Physical modeling is a way of modeling and simulating systems that consist of real physical components. It employs a physical network approach, where Simscape blocks correspond to physical elements, such as pumps, motors, and op-amps. By joining these blocks by lines corresponding to the physical connections that transmit power. This approach can describe the physical structure of a system, instead of linear and nonlinear equations. These virtual devices can drastically reduce the cost of testing control systems, software, and hardware. It can also improve the quality of the final product by enabling more complete testing of the entire system. SOLIDWORKS was used to design the gimbal, which consists of 3 parts as in

Fuzzy logic and fuzzy sets have been around now for more than 20 years. In 1965, Zadeh first proposed fuzzy sets, which are considered as an approach to processing data, and they became popular in the different fields of science. In (1974) Mamdani presented a fuzzy controller method and it gained high popularity in the engineering field [

Thus, it is easier to understand since its working principle depends on the linguistic statements. The fuzzy controller comprises four main phases, which are the fuzzification phase where the input values are converted to a fuzzy variable (linguistic variables). In this paper we used two input variables to control the dc motors, which are error (e) of the dc motor motion, and its derivative error (e^{.}) with three fuzzy subsets are both, which are [H M L] and [N Z P] respectively, by using three Gaussian membership functions as in

The output has five subsets represented using z-membership functions as shown in

The two axes gimbal system is validated using MATLAB/Simulink environment is depicted with SimMechanics model in

To confirm this efficiency, many comparison tests indicated using different inputs shapes.

In the following figures, a sin waves input is applied with different amplitudes for elevation and azimuth axes. The elevation axis response gave an error of 5.24 × 10^{−3} with PID control and 0.15 × 10^{−3} with fuzzy control while the azimuth response gave an error of 0.014 with PID control and 0.003 with fuzzy control. The error and control signal presented in

With the help of the control parameters of the electromechanical simulation model, the proposed fuzzy controller can track the command angle rapidly and accurately, by which high stabilization performance can be attained.

Due to fact, the elevation axis of the system is compatible with the azimuth axis to produce the final motion of the system. Therefore, it is necessary to test the control performance under movable base conditions by simulating the axes at various angles. In

Previous results clearly reflect the efficiency of the self-tuning fuzzy controller compared to the conventional PID. The test assures the superiority of the proposed fuzzy compared with the classical PID control. It has good tracking accuracy whether for azimuth gimbal or elevation gimbal.

In this paper, a two axes gimbal system was proposed and formulated utilizing Newton's second law. The equations for the gimbals’ motion were derived and introduced in two formulations according to the dynamic mass unbalance. The gimbal system was simulated using MATLAB/SimMechanics. A comparison between the system responses with different inputs shapes was made and the comparison results verified the proposed model. The responses have been analyzed, then the performance of the Self-tuning fuzzy controller has been tested using transient response analysis and a quantitative study of error analysis. Based on the obtained results, some observations can be cleared.

The proposed self-tuning fuzzy control provides good adaptivity to the gimbal system which offers high performance so that it can be utilized more efficiently in the dynamical environment that usually imposes large variable base rates. It is clear that the proposed fuzzy controller can reduce the response settling time as compared with the conventional PID controller.

Finally, the proposed fuzzy controller improves the closeness of System response and supports the system relative stability by reducing the response overshoot considerably without increasing the response rise time dramatically.

Taif University Researchers Supporting Project number (TURSP-2020/260), Taif University, Taif, Saudi Arabia.

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