Novel polynomial shaping method for impact time and angle guidance law designs: Bézier curve approach

Abstract

One of the topics that intensive research is conducted on in aviation and space technologies is the development of algorithms that will direct an aircraft from one point to another. While it is possible to plan the route of the aircraft and develop control algorithms that will follow that route for vehicles with low speeds such as Unmanned Air Vehicles, guidance laws are used for vehicles with high speed and maneuverability such as missiles. If the requirements from engagement are to increase the damage to be given to the target or to make a salvo attack with multiple missiles,advanced guidance laws should be designed. The engagement requirements in the literature have been examined and it has been seen that impact time and impact angle of the missile are the two most important constraints. When the solution methods proposed for these problems are examined from literature, it is seen that analytical results are obtained using polynomial shaping methods in addition to control methods such as nonlinear control and optimal control. Within these studies, the studies created with polynomial shaping methods have been examined in depth as they form the basis of this study. As a result, the goal is to shape the Line of Sight as a polynomial and to obtain a guidance law that can control the impact time and angle together with the help of the obtained polynomial. B\'ezier curves, a polynomial shaping method, were used to shape the Line of Sight angle polynomial. In this method, polynomials are calculated as linear combinations of control points and basis functions. After analyzing engagement dynamics and making various adjustments, the resulting guidance law requires the second derivative of the LOS angle polynomial. Therefore, the curve to be created must be a curve that can be differentiated twice. By definition, this curve yields results for knot values between zero and one. Therefore, the curve to be created is arranged to provide the total impact time. The final value of the curve should be equal to the desired Line of Sight angle. The generated Line of Sight curve is in a nonlinear relation with the range and range rate values in the proposed guidance law. Therefore, one control point of the B\'ezier the curve is left free and calculated using an offline algorithm. The algorithm finds the parameter that minimizes the impact time error time using the upper and lower limits given as initial values with iterative simulations. With good initial estimations, the algorithm converges to the result in a few steps. The B\'ezier curve is completed after the determination of the final control point.Thus, the proposed method is completed with the completed curve. The proposed guidance law has been tested in several cases. In the first two cases impact time and angle capabilities are tested, it was observed that the impact time could be achieved for certain intervals for each engagement.The reason for this fact was stated to be the convergence of the look angle values to 90 degrees. The performance of the designed method was demonstrated in a salvo attack scenario where missiles with different initial flight path angles and at different initial ranges simultaneously approached the target at the same final line of sight angle. It was also demonstrated through simulation that the method developed for moving targets works with decent performance if the target acceleration value is known. In a cost analysis comparing the proposed method with a method in the literature, the proposed method was found to be more efficient. The proposed guidance law however, has limitations. The offline nature of the algorithm means that disturbances and uncertainties may affect its performance. Additionally, the initial guesses for the algorithm and the tuning parameter are areas for further development. In the future works of this areas are possible in addition to extensions to the 3D implementations. A guidance law that can adjust the impact time is also designed by shaping the range using B\'ezier curves. This guidance law does not require an offline algorithm .Two cases were conducted to test the impact time performance of the given method. The first one evaluates the impact time intervals to be achieved while the second one for testing the salvo attack performances. The simulations shows the effectiveness of the proposed law with high impact time intervals. Finally, a new method is proposed based on one of the guidance laws from the recent literature. In this method, Bézier curves are used to add the ability of controlling the impact time of the Impact Vector Guidance law which has only impact angle capabilities. In the proposed method, the missile follows a virtual target instead of the real target and the virtual target coincides with the real target at the desired impact time. The virtual target is designed to move on a Bézier curve, created using the boundary conditions of the engagement. To formulate the position of the virtual target along the curve, an S-shaped (sigmoid) function is proposed. This is necessary to remove the sudden jumps in the movement of the virtual target and ensures that it coincides with the real target at the end of the engagement. The alpha variable, which is specific to each case, is used in the design phase of the S-shaped function and is determined by an offline algorithm before the engagement. The lower value for alpha is fed to algorithm, then the algorithm increases it until the impact time is within the given tolerance. The alpha determines the saddle point of the S-shaped function. This method, which is still in the earlier phases, shows satisfactory results. In future studies, obtaining the online counterpart of the algorithm is possible. As an another future study, the offline algorithm can be used as an initial guesses for nonlinear receding horizon control method to deal with moving targets while compensating the disturbances and uncertainties.