Trajectory prediction-based guidance law

This paper proposes a guidance law based on trajectory prediction, aiming to address the difficulty of traditional guidance laws in meeting high-speed and highly maneuverable vehicles. The unscented Kalman filtering (UKF) technique is employed to estimate the target’s motion and predict the virtual impact point using the Singer model and measuring model. The midcourse guidance law is applied to the virtual target, taking into account the constraint of the intersection angle, while the terminal guidance utilizes modified proportional guidance. To mitigate the overload chattering in the transition sections of both midcourse and terminal guidance, the distance is used to modify the transition section of the terminal guidance. Simulation results demonstrate that the proposed guidance law effectively reduces both the encounter angle and the required maneuvering. Furthermore, to minimize midcourse guidance errors, the prediction results of the virtual target are continually updated during the trace process. This method can also be applied to trail other highly maneuverable targets.


Introduction
Due to the lower speed of aircraft compared to hypersonic targets, it is difficult to guarantee the accuracy of intersection by using traditional guidance law [1-3] .Dwivedi et al. [4] addressed this issue by designing an optimal mid-guidance law with front angle constraints to enable the seeker on the aircraft to detect and monitor the target effectively.Chen et al. [5] designed a medium guidance law based on the T-S fuzzy method and sliding mode control.Evcimen and Leblebicioglu [6] studied an improved PN guidance law for intercepting fast maneuvering targets based on the modified PID control method.Yamasaki et al. [7] proposed a unified design method of sliding mode surface considering and not the angle of attack constraint.Liang et al. [8] integrated integral sliding mode control (ISMC) and state-dependent Riccati equation (SDRE) to design a robust steering law.In [9], a three-dimensional nonlinear guidance law is designed based on the local sliding mode.Hexner and Pila [10] designed a stochastic optimal guidance law for aircraft overload constraints by using linear quadratic stochastic Gaussian optimal control theory and stochastic input description function for stochastic maneuvering targets.Hu et al. [11] developed a novel optimal guidance law for near-space hypersonic vehicles based on the multi-model theory and optimal control theory of structural random jump systems.
In order to effectively reduce the required overload and achieve successful intersection of vehicles, this paper investigates a new guidance law.The strategy is based on trajectory prediction, where the predicted trajectory of the target at the moment of impact is utilized as a virtual target.The intersection process involves mid-guidance using the virtual target and corrected proportional terminal guidance.

State modeling
The target kinematic equation can be expressed as: where ( ) ( ) ( ) ( ) û is the position, velocity, and acceleration of the target along the X, Y, and Z axes in the inertial system.A denotes the model description matrix, and ( ) stands for the process noise.And [ ] The measurement equation can be expressed as: ( ) where ( ) k h x stands for the position, elevation angle, and azimuth angle that the ground radar can measure to the target, and k v denotes the measurement noise.This paper uses the Singer model to describe the target's motion.It is assumed that the maneuvering acceleration ( ) t a is a first-order time-dependent process.Its time-dependent function is an exponential form, which is described as follows: The acceleration ( ) can be represented by a first-order time-dependent model with white noise input, and then we have: where ( ) t w is Gaussian white noise with zero mean and variance of ( ) The target Singer kinematics model in one direction can be expressed as: where x and x w are the position and acceleration noise of the target respectively.

Measurement modeling
where R  ,   , and   are measurement noise respectively.

Motion parameters estimation
The state parameters are estimated by the Unscented Kalman Filtering (UKF).The motion equation of the target is a continuous system, but the radar measurement value is discrete.Hence, this paper uses the discrete Kalman filtering to estimate the state parameters.
3.1.1.Discretization.By assuming that the sampling period of the radar is s T , the state transition matrix and process noise can be expressed as: ) 3.1.2.Initialization.The initialization value of the state parameter 0 x and its covariance matrix 0 P are:

Time Update Equation (k−1 to k).
The construction of Sigma points by Unscented Transform (UT) is as follows: [ ] is used to control the distribution of sigma points.
Then state propagation for each sigma point can be expressed as:

2(
) The covariance matrix of the prior estimate is calculated according to i m W : ) Then measurement propagation for each sigma point can be expressed as: ) The observed variables and their associated covariances can be solved by inverse UT transform: The measurement update of state and covariance by the standard Kalman filter equation is as follows:

Trajectory Prediction
An accurate initial value for the prediction comes from the estimation of the motion parameters.The target motion trajectory can be obtained iteratively by analytical method or numerical method.The approximation is good when using polynomials to approximate the trajectory of the target.This paper takes 2 n = mathematical models to describe the target ballistic inclination and ballistic deflection angle.
The coefficients of ( 1) and ( 20) are obtained by the least square method.The position information of the target can be updated by the equation of motion of the target: cos cos sin cos sin

Guidance Law
Based on the results of motion parameters estimation and trajectory prediction, a compensation algorithm for target maneuvering and trajectory prediction in the guidance law is designed to satisfy the multi-constraint conditions.By estimating the remaining flight time and predicting the target trajectory at the time of impact, the target at the time of predicted impact is used as a virtual target.The midcourse guidance is based on the virtual target, and considers the constraint intersection angle at the same time: where go t is the estimated remaining flight time; c v  is the relative velocity between the aircraft and the virtual target; 1 2 ˆ, q q and 1 2 ˆ, q q   are the predicted line-of-sight angle and line-of-sight rate between the aircraft and the virtual target; ˆt ( ) where c v is the relative velocity between the aircraft and the real target; 1 2 , q q   are the line-of-sight angles between the aircraft and the real target; yt k and zt k are the parameters of the terminal guidance law.
To avoid the overload chattering of the transition in the midcourse and terminal guidance section, the guidance law of the transition section is set as (1 ) (1 ) where When the distance is bigger than the distance of the transition period, the value is 1.It changes to 0 as the distance approaches the distance of the terminal guidance period.

Simulation
The engagement scenarios for simulation are detailed in

Target Trajectory Prediction Simulation
When the aircraft enters the section of the midcourse guidance, the aircraft starts to predict the target trajectory which can be applied to the guidance law.In this paper, the least square method is used to predict the target.The comparison between the prediction result and the real motion information of the target is shown in Figure 3 and Figure 4.The error of the predicted trajectory increases with time, so the initial prediction will cause a large error in guidance.To reduce the error of the midcourse guidance, the prediction result of the virtual target is updated in time during the intersection process.
With the decrease of the remaining time, the prediction accuracy of the midcourse guidance is gradually improved to enhance the terminal guidance.

Conclusion
In this paper, a new guidance strategy based on virtual targets is studied for the guidance law of hypersonic glide vehicles.UKF is used to estimate the motion information of the target.Then, the least square method is combined with the estimated motion information to fit the motion model coefficients and predict the flight-path angle and the heading angle of the predicted intercept point which is considered a virtual target point.Based on the virtual target, the midcourse guidance considers the constraint intersection angle at the same time.Compared to the traditional method, the proposed intersection strategy in this paper effectively limits the encounter angle and reduces the design complexity of the aircraft.

 and 0 ,
³ are undetermined parameters which determine the maneuvering characteristics of the target in the interval ( ) of the maneuvering acceleration.λ denotes the reciprocal of the maneuvering time constant, that is, the maneuvering frequency.Empirically,
n is the dimension of the state.
13)According to i k χ and the weight vector i m W , the prior estimate of the state at k is obtained:

1 m k and 2 mk
and ˆt  are the flight-path angle and the heading angle of the virtual target; are midcourse guidance law parameters.The modified proportional guidance (MPG) in the terminal guidance section is:

1 .
Performance simulation of target tracking filter The radar measurement errors of o R , o  , o  are 100 m, 0.1°, and 0.1° (1σ).EKF and UKF are used to track and estimate the hypersonic glide target respectively.According to Figure 2, compared with EKF, UKF can acquire closer target information to the real value and has higher filtering accuracy.

Figure 2 .
Figure 2. Errors of Position and Velocity Estimation.

Figure 3 .
Figure 3. Errors of Position and Velocity Prediction.

Figure 4 .
Figure 4. Prediction Trajectory.5.3.Guidance algorithm simulationUsing the guidance law designed in this paper (Trajectory Prediction-Based Intersection Guidance, TPIG) and the modified proportional guidance law, the intersection results and trajectories are shown in Figure5.It can be seen that the TPIG can meet the intersection requirements and achieve a smooth ballistic transition.

Figure 5 .
Figure 5. Relative Motion of Aircraft and Target.
.1.4.Measurement Update Process.Sigma points are generated based on prior estimates of state and covariance at time k: The velocity characteristics of the target are described by a third-order mathematical model: 5