Launching Point Estimation Using Inverse First-Order Pitch Programming

Abstract

This paper presents an estimation of the launching points of unmanned aerial vehicles (UAVs) equipped with boosters. When UAVs are detected and tracked by sensors such as radars, the fast identification of vehicle launching points is needed for air traffic control and defense systems. When UAVs in a boosting phase are controlled by first-order pitch programming, this paper leverages inverse first-order pitch programming to analytically solve launching points. Furthermore, a two-point robust measurement selection (T-RMS) scheme is developed to reduce errors such as random noise and bias by utilizing multiple moving average filters. The proposed work is verified by various simulation results.

1. Introduction

Target tracking has been researched for aircraft and unmanned aerial vehicles (UAVs) over several decades. Estimating multiple vehicle trajectories from sensor measurements is critical to air traffic control [1]. In addition, launching point estimation is necessary for defense systems [2]. Since reducing the computational time of estimation is much more preferred for decision-making, highly accurate estimation in the shortest time is as yet an unresolved problem in the area of target tracking.

This paper proposes the estimation schemes of UAV launching points with first-order pitch programming, which is used for a UAV guidance law. We analytically solve vehicle launching points by utilizing the inverse of first-order pitch programming. This scheme enables the shortest computational time of estimation. In addition, we develop a two-point robust measurement selection (T-RMS) scheme. Since an analytic solution of a launching point needs two-point measurements, we choose the best two-point measurements to decrease estimation errors.

1.1. Related Work

Target tracking algorithms are mainly developed with two classes: model-based and model-free. A model-based approach in [3] uses target dynamics and measurement process represented by typical state space models. By converting each state variable into independent dynamic representation, target positions are estimated with a regularized least-square optimization algorithm of an auto-regressive moving average model. Researchers in [4] develop a maximum likelihood estimator for trajectory estimation and impact point prediction of a thrusting or ballistic object with unbiased measurements. The interacting multiple model techniques in [5] employ boost and Keplerian prediction models in order to handle staging and burnout events. A smoothing integrated probabilistic data association algorithm estimates a launching point from cluttered measurements of a projectile with a constant axial force model [6]. An iterative least-square algorithm estimates a launching point of a constant-speed projectile by linearizing a nonlinear regression model, including parameters of a kinematic model [7]. For other applications such as electric vehicles [8] and a robot manipulator [9], linear matrix inequality, which is a type of optimization algorithm based on models, is employed to design a quantizer and distributed filters. However, the use of such models requires high computational complexity, such as iterative computations.

Meanwhile, model-free approaches play an active role in research. In [10], a target motion is modeled by a time-polynomial function whose coefficients are estimated by a weighted least square algorithm. With the line-of-sight (LOS) angle measured vertically and horizontally, a n-th order polynomial function is fitted by a least-square method [11]. From a trajectory database, a k-NN search algorithm is developed to compare a transformed sensed trajectory and trajectories in a sub-trajectory database [12]. Single and multiple long–short term memory (LSTM) networks are utilized to estimate launching points with pre-filtered data generated by an unscented Kalman filter and a LSTM network, respectively [13,14]. However, most model-free approaches deal with unbiased measurements.

1.2. Major Contribution

We develop schemes that estimate the launching points of UAVs in a boosting phase. We leverage the framework of first-order pitch programming to analytically derive launching points, because first-order pitch programming controls vehicle trajectories. We address the decreases in estimation errors from the analytic solutions of the launching points. Selecting two-point measurements filtered by multiple moving averages enables us to reduce estimation errors such as random noise and bias.

Pitch programming is one type of guidance law that controls vehicle attitude in a vertical plane [15]. Since UAVs in a boosting phase employ first-order pitch programming to achieve the required vehicle speed after burn-out while maintaining low aerodynamic load, vehicle position and speed are generated and controlled by first-order pitch programming. Hence, we estimate the parameters of first-order pitch programming from vehicle position and speed. We call this framework inverse first-order pitch programming. In the framework of inverse first-order pitch programming, we combine position and angle measurements for preventing the degeneracy of a measurement matrix. This fused information allows for the estimation of first-order pitch-programming parameters, including a launching point with two-point measurements. Furthermore, we address random noise and bias by leveraging multiple moving average filters. A moving average filter inherently alleviates the effect of random noise by averaging multiple measurements. We devise a two-point robust measurement selection (T-RMS) scheme that chooses the best measurement among measurements filtered by multiple moving average filters at two points in order to reduce bias.

1.3. Organization

The remainder of this paper has been organized into the following sections: in Section 2, we describe first-order pitch programming. In Section 3, we present a launching point estimation of UAVs. In Section 4, we describe a two-point robust measurement selection (T-RMS) scheme. In Section 5, we demonstrate mathematical simulation. In Section 6, a conclusion and the details of future work are provided.

2. Problem Formulation

Here, we describe first-order pitch programming of unmanned aerial vehicles (UAVs) in 2-D space. Although first-order pitch programming is widely represented by a function of time-polynomial, we define first-order pitch programming as a function of the down-range-polynomial, because UAV trajectories generated by position-based first-order pitch programming are insensitive to the thrust variation of UAVs [16].

Let x and y be the down-range and altitude of an UAV, respectively. Let i and j be the individual indexes of two different measurements. Let V and $\theta$ be vehicle speed and pitch angle, respectively. Let $x_L$ be a launching point. Let $O-XY$ be a coordinate system of a guidance frame, called O frame. Let $O^{\prime }-X{Y}^{\prime }$ be a coordinate system of an estimation frame, called $O^{\prime }$ frame. A trajectory generated by the first-order pitch programming is depicted in Figure 1.

The black dash line in Figure 1 represents a vehicle trajectory that sensors cannot detect and track, whereas the red solid line shows the available positional and velocity data from sensor measurements. Because we have no knowledge about the origin point of the guidance frame, we select the origin point of the estimation frame as the down-range of the first measurement. Let $a_0$ and $a_1$ be parameters of first-order pitch programming. We simply drop off the subscript indexes of x and $\theta$ because we do not lose generality in terms of one measurement and one pitch angle. Then, we propose first-order pitch programming as follows:

$$\begin{array}{c}\hfill \theta =ta{n}^{-1}\left(2a_0(x-x_L)+a_1\right).\end{array}$$

(1)

Assumption 1. 

Fight path angles of UAVs are approximately equal to vehicle-pitch angles.

Assumption 2. 

Vehicle-pitch angles are approximately equal to the commands of vehicle-pitch angles.

Assumption 3. 

$a_0$ is negative.

Remark 1. 

Assumption 1 shows that the angle of attack is small when compared with flight path angle. Because speed maximization and attitude stabilization are the main focuses of UAVs in a boosting phase, UAVs do not highly maneuver during thrusting; thus, the angle of attack is small. Assumption 2 indicates that the UAV autopilot can control a pitch angle that follows the pitch angle command; hence, pitch angle commands are approximately equal to pitch angles. Assumption 3 means that the rates of pitch-angle commands generated by first-order pitch programming keep decreasing when down-range positions keep increasing.

From Assumptions 1 and 2, we derive vehicle velocities with a pitch angle produced by first-order pitch programming as follows:

$$\begin{array}{c}\hfill \dot{y}=tan\theta \dot{x}.\end{array}$$

(2)

Then, we obtain vehicle trajectories by combining and integrating Equations (1) and (2) as follows:

$$\begin{array}{c}\hfill y=a_0{\left(x-x_L\right)}^2+a_1\left(x-x_L\right).\end{array}$$

(3)

Remark 2. 

Equation (3) represents a vehicle trajectory with a down-range polynomial, incorporating a launching point. This strategy is different from previous work [10,11] that represents positional data composed of a time-polynomial function. Moreover, the vehicle trajectory is insensitive to timing errors such as time delays due to the down-range polynomial, regardless of time.

3. Analytic Solution of Launching Point

Let $\mathbf{z}\in R^3$ be a measurement vector. Let $\mathbf{x}\in R^3$ be a vector including unknown parameters of first-order pitch programming and a launching point. Let $\mathit{ϵ}$ be a vector of measurement error. Then, the measurement equation is as follows:

$$\begin{array}{c}\hfill \mathbf{z}=\mathbf{H}\mathbf{x}+\mathit{ϵ},\end{array}$$

(4)

where

$$\mathbf{z}=\left(\begin{array}{c}tan{\theta }_i\\ tan{\theta }_j\\ y_i\end{array}\right),\mathbf{H}=\left(\begin{array}{ccc}2x_i& 1& 0\\ 2x_j& 1& 0\\ x_i^2& x_i& 1\end{array}\right),\mathbf{x}=\left(\begin{array}{c}{a}_0\\ a_1-2a_0{x}_L\\ a_0{x}_L^2-a_1{x}_L\end{array}\right).$$

(5)

We analytically solve Equation (4) by the following theorem.

Theorem 1. 

Suppose that $x_i\ne 0$, $x_i\ne x_j$, $x_i<x_j$, ${\theta }_i>{\theta }_j$, and $x_L<0$. Let cost function J be ${ϵ}^2$. Let ${\widehat{x}}_L$, ${\widehat{a}}_0$ and ${\widehat{a}}_1$ be estimates of unknown parameters and a launching point. Then, $\widehat{\mathit{x}}$, which is composed of ${\widehat{x}}_L$, ${\widehat{a}}_0$, and ${\widehat{a}}_1$, minimizes cost function J. We obtain ${\widehat{x}}_L$, ${\widehat{a}}_0$, and ${\widehat{a}}_1$ as follows:

$$\begin{array}{cc}\hfill {\widehat{x}}_L& ={\displaystyle \frac{x_{j}tan{\theta }_i-x_{i}tan{\theta }_j}{tan{\theta }_i-tan{\theta }_j}}\hfill \\ \hfill & -{\displaystyle \frac{\sqrt{{tan}^2{\theta }_i{(x_i-x_j)}^2-2y_i(x_i-x_j)(tan{\theta }_i-tan{\theta }_j)}}{tan{\theta }_i-tan{\theta }_j}}.\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\widehat{a}}_0& ={\displaystyle \frac{1}{2(x_i-x_j)}}(tan{\theta }_i-tan{\theta }_j).\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\widehat{a}}_1& ={\displaystyle \frac{1}{2(x_i-x_j)}}(-x_{j}tan{\theta }_i+x_{i}tan{\theta }_j)+2{\widehat{a}}_0{\widehat{x}}_L.\hfill \end{array}$$

(8)

Proof. 

We have $\widehat{\mathbf{x}}={\left({\mathbf{H}}^{\top }\mathbf{H}\right)}^{-1}{\mathbf{H}}^T\mathbf{z}$ in the sense of the least square for more equations than unknowns. However, in this case, H is a square matrix. The determinant of H defined in [17] is $2(x_i-x_j)$, which is nonzero due to $x_i\ne x_j$. In addition, both $x_i\ne x_j$ and $x_i\ne 0$ make the adjoint of H have full rank by the procedures of row echelon. Hence, the solution of $\widehat{\mathbf{x}}$ is unique, by computing $\widehat{\mathbf{x}}={\mathbf{H}}^{-1}\mathbf{z}$. We have a quadratic equation in terms of ${\widehat{x}}_L$ through combining the second and third rows of $\widehat{\mathbf{x}}$. From Assumption 3 and $x_i<x_j$, we have $tan{\theta }_i-tan{\theta }_j>0$. Thus, the discriminant of the quadratic equation is positive. In addition, $x_{j}tan{\theta }_i-x_{i}tan{\theta }_j>0$ because $x_i<x_j$ and ${\theta }_i>{\theta }_j$. Since negative $x_L$ always exist from Equation (3) and the formation of the coordinate system ‘O − XY’, we acquire the solution of quadratic Equation (3). Thus, the solutions of $\widehat{\mathbf{x}}={\mathbf{H}}^{-1}\mathbf{z}$ are represented by Equations (6)–(8).  □

Equation (5) shows that measurement matrix H has down-range information. Although measurement error $\mathit{ϵ}$ is minimized by $\widehat{\mathbf{x}}$ in the sense of the least square, down-range measurement errors still negatively affect the estimation of launching points. The next section will describe a scheme that decreases the errors of the measurement matrix H through the use of moving averaging filters.

4. Two-Point Robust Measurement Selection (T-RMS)

Moving average filters are used for random noise reduction. Here, we employ multiple moving average filters to decrease both random noise and bias error. Let N be a window size that is a variable. Let k be a time index, where k is an integer greater than one. Let m be a measurement and $M_F$ represents an output of a moving average filter. Let $\overline{m}$ be a true value of a measurement. Let $\Delta m$ be a bias of a measurement. Let v be a random noise of a measurement. Then, a trailing moving average filter according to N greater than and equal to one is represented by

$$\begin{array}{c}\hfill M_{F_k}\left(N\right)={\displaystyle \frac{1}{N}}\sum _{p=0}^{N-1}{m}_{k-p},\end{array}$$

(9)

where $m_{k-p}={\overline{m}}_{k-p}+\Delta m+v_{m_{k-p}}$ and $N\le k$. When we consider the mean value of a random noise is zero, we have

$$\begin{array}{c}\hfill M_{F_k}\left(N\right)={\displaystyle \frac{1}{N}}\left(\sum _{p=0}^{N-1}{\overline{m}}_{k-p}\right)+\Delta m.\end{array}$$

(10)

In addition, estimating one step backward is derived as follows:

$$\begin{array}{c}\hfill M_{F_{k-1}}\left(N\right)={\displaystyle \frac{1}{N}}\sum _{p=1}^N{m}_{k-p},\end{array}$$

(11)

where $N\le k-1$. For the zero mean value of a random noise, we have

$$\begin{array}{c}\hfill M_{F_{k-1}}\left(N\right)={\displaystyle \frac{1}{N}}\left(\sum _{p=1}^N{\overline{m}}_{k-p}\right)+\Delta m.\end{array}$$

(12)

Note that we establish a trailing average filter of individual information by changing a variable name of Equations (9) and (11). For example, the moving average filters of a pitch angle are ${\Theta }_{F_k}\left(N\right)={\displaystyle \frac{1}{N}}{\sum }_{p=0}^{N-1}{\theta }_{k-p}$ and ${\Theta }_{F_{k-1}}\left(N\right)={\displaystyle \frac{1}{N}}{\sum }_{p=1}^N{\theta }_{k-p}$, where ${\theta }_{k-p}={\overline{\theta }}_{k-p}+\Delta \theta +v_{{\theta }_{k-p}}$.

The following figure shows a diagram of two-point robust measurement selection (T-RMS) based on Equations (9) and (11). The yellow box represents multiple trailing moving average filters at time index $k-1$ and the green box represents trailing multiple moving average filters at time index k. The multiple trailing moving average filters are indicated according to a window size ranged from ${\xi }_1$ to ${\xi }_{\tau }$. Our goal is to find a trailing moving average filter that decreases measurement errors among multiple trailing moving average filters. The red box in Figure 2 represents the best trailing moving average filters for current estimation and one step backward estimation. We show a condition to find the best trailing moving average filters with the following theorem:

Theorem 2. 

Let $X_{F_k}$, $Y_{F_k}$, and ${\Theta }_{F_k}$ be the outputs of the trailing moving average filters for the down-range, altitude, and pitch angle at time index k, respectively. Let ${\overline{x}}_k$, ${\overline{y}}_k$, ${\overline{\theta }}_k$, and ${\overline{x}}_L$ be the true values of the down-range, altitude, pitch angle, and launching point, respectively. Let $\Delta x$, $\Delta y$, and $\Delta \theta$ be measurement biases for down-range, altitude, and pitch angle, respectively. Suppose ${\overline{x}}_k{\overline{\theta }}_{k-N}-{\overline{x}}_{k-N}{\overline{\theta }}_k\approx 0$, ${\Theta }_{F_k}\left(N\right)\approx {\overline{\theta }}_{k-N}$, $Y_{F_k}\left(N\right)\approx {\overline{y}}_{k-N}$, and ${\overline{\theta }}_k$ is small. If we find N to make $X_{F_k}\left(N\right){\Theta }_{F_{k-1}}\left(N\right)-X_{F_{k-1}}\left(N\right){\Theta }_{F_k}\left(N\right)\approx 0$, then ${\widehat{x}}_L\approx {\overline{x}}_L$.

Proof. 

We choose two-point true information of the down-range, altitude, and pitch angle at time indexes $k-N$ and k and insert them into Equation (6). Since ${\overline{\theta }}_k$ is small, we have

$$\begin{array}{cc}\hfill {\overline{x}}_L& ={\displaystyle \frac{{\overline{x}}_k{\overline{\theta }}_{k-N}-{\overline{x}}_{k-N}{\overline{\theta }}_k}{{\overline{\theta }}_{k-N}-{\overline{\theta }}_k}}\hfill \\ \hfill & -{\displaystyle \frac{\sqrt{{\overline{\theta }}_{k-N}^2{({\overline{x}}_{k-N}-{\overline{x}}_k)}^2-2{\overline{y}}_{k-N}({\overline{x}}_{k-N}-{\overline{x}}_k)({\overline{\theta }}_{k-N}-{\overline{\theta }}_k)}}{{\overline{\theta }}_{k-N}-{\overline{\theta }}_k}}.\hfill \end{array}$$

(13)

In addition, we select two consecutive data filtered by moving average filters represented by Equations (9) and (11) and put them into Equation (6) to estimate a launching point. The launching point estimate is

$$\begin{array}{cc}\hfill & {\widehat{x}}_L={\displaystyle \frac{X_{F_k}\left(N\right){\Theta }_{F_{k-1}}\left(N\right)-X_{F_{k-1}}\left(N\right){\Theta }_{F_k}\left(N\right)}{{\Theta }_{F_{k-1}}\left(N\right)-{\Theta }_{F_k}\left(N\right)}}\hfill \\ \hfill & -{\displaystyle \frac{\sqrt{{\Theta }_{F_{k-1}}^2\left(N\right){(X_{F_{k-1}}\left(N\right)-X_{F_k}\left(N\right))}^2-2Y_{F_{k-1}}\left(N\right)(X_{F_{k-1}}\left(N\right)-X_{F_k}\left(N\right))({\Theta }_{F_{k-1}}\left(N\right)-{\Theta }_{F_k}\left(N\right))}}{{\Theta }_{F_{k-1}}\left(N\right)-{\Theta }_{F_k}\left(N\right)}}.\hfill \end{array}$$

(14)

By using Equations (10), (12) and (14), we derive the following equation:

$$\begin{array}{cc}\hfill {\widehat{x}}_L& ={\displaystyle \frac{X_{F_k}\left(N\right){\Theta }_{F_{k-1}}\left(N\right)-X_{F_{k-1}}\left(N\right){\Theta }_{F_k}\left(N\right)}{{\overline{\theta }}_{k-N}-{\overline{\theta }}_k}}\hfill \\ \hfill & -{\displaystyle \frac{\sqrt{{\Theta }_{F_{k-1}}^2\left(N\right){({\overline{x}}_{k-N}-{\overline{x}}_k)}^2-2Y_{F_{k-1}}\left(N\right)({\overline{x}}_{k-N}-{\overline{x}}_k)({\overline{\theta }}_{k-N}-{\overline{\theta }}_k)}}{{\overline{\theta }}_{k-N}-{\overline{\theta }}_k}}.\hfill \end{array}$$

(15)

Through assumptions, we have ${\overline{x}}_k{\overline{\theta }}_{k-N}-{\overline{x}}_{k-N}{\overline{\theta }}_k\approx 0$, ${\Theta }_{F_k}\left(N\right)\approx {\overline{\theta }}_{k-N}$, and $Y_{F_k}\left(N\right)\approx {\overline{y}}_{k-N}$. When we compare Equations (13) and (15), ${\widehat{x}}_L\approx {\overline{x}}_L$ if we find N to make $X_{F_k}\left(N\right){\Theta }_{F_{k-1}}\left(N\right)-X_{F_{k-1}}\left(N\right){\Theta }_{F_k}\left(N\right)\approx 0$.  □

Remark 3. 

We refer to $X_{F_k}\left(N\right){\Theta }_{F_{k-1}}\left(N\right)-X_{F_{k-1}}\left(N\right){\Theta }_{F_k}\left(N\right)$ as one condition for two-point robust measurement selection (T-RMS). Theorem 2 implies that measurement errors are reduced by choosing N that minimize the T-RMS condition. Since a single trailing moving average filter embedding a fixed window size cannot always satisfy the T-RMS condition, we employ multiple trailing moving average filters and choose the best estimate of the filters. In other words, we select a trailing moving average filter with a different window size at every time.

5. Simulation Results

We present various simulations to verify the proposed filtering schemes, which are composed of an analytic solution of a launching point and multiple trailing moving average filters, including two-point robust measurement selection (T-RMS). The initial positions in the X-axis and Y-axis are 0 m and 200 m, respectively. The initial pitch angle and vehicle speed is 32.5 deg and 10 m/s, respectively. The constant vehicle acceleration is 10 m/s². The true position of a launching point, which is $x_L$, is −300 m. For T-RMS, the minimum and maximum window sizes of multiple trailing moving average filters are 3 and 10, respectively. The fixed window size of a single trailing average filter is 10 for comparing the performance of the proposed schemes. Let ${\sigma }_X$ and ${\sigma }_Y$ be standard deviations of X and Y positions, respectively. Let ${\sigma }_{\theta }$ be a standard deviation of pitch angle. Let ${\mu }_X$ and ${\mu }_Y$ are means of X and Y positions, respectively. Let ${\mu }_{\theta }$ be a mean value of the pitch angle. For measurement errors, we generate random noise with a normal distribution of ${\mu }_X=0$ and ${\sigma }_X=30$ m for X-axis position. The random noise of Y-axis position follows the same normal distribution of the random noise of X-axis position. We consider two scenarios according to standard deviation of pitch-angle random noise: the first scenario is a normal distribution of ${\mu }_{\theta }=0$ and ${\sigma }_{\theta }=0.2$ m rad, and the second scenario is ${\mu }_{\theta }=0$ and ${\sigma }_{\theta }=0.5$ m rad. A bias error of 10 m is individually applied to both the X and Y positions. Figure 3, Figure 4, Figure 5 and Figure 6 represent the simulation results of the first scenario.

Figure 3 and Figure 4 show the position data of a vehicle trajectory. True positional information is represented by integrating velocities along the X-axis and Y-axis after vehicle speed is produced by integrating constant accelerations. Positional measurements are represented by combining true information and measurement errors of the first scenario. In the figures, a single trailing moving average filter with fixed window size, called MAF, and the multiple trailing moving average filters with T-RMS, called T-RMS, decrease measurement errors when we compare true positional information and positional measurements. In addition, the measurement errors of pitch angles are reduced by MAF and T-RMS in Figure 5.

Figure 6 describes the estimates of a launching point from MAF and T-RMS. Because the T-RMS condition represented by Theorem 2 is met by utilizing the multiple trailing moving average filters with different window size, T-RMS has the better estimate of the launching point than MAF. Figure 7, Figure 8, Figure 9 and Figure 10 represent the simulation results of the second scenario.

Since the standard deviation of pitch angle error of the second scenario is twice as large as that of the first scenario, the measurement errors of Figure 9 are larger than those of Figure 5. Figure 10 shows that T-RMS is a much better estimate of the launching point than MAF. The reason why T-RMS has better estimates is that the best window size at every time is selected to satisfy the T-RMS condition. To quantitatively analyze MAF and T-RMS, we compute root-mean-square errors according to scenarios as follows.

Table 1. Root-Mean-Square Error.

Standard Deviation of Angle Noise	No Filter	MAF	T-RMS
0.2 m rad	59.4 m (19.8%)	23.9 m (8.0%)	21.9 m (7.3%)
0.5 m rad	62.0 m (20.7%)	28.8 m (9.6%)	18.7 m (6.2%)

shows root-mean-square error of no filter, MAF, and T-RMS. When the standard deviation of angle noise increases, the estimation error of the launching point increases for no filter and MAF. However, T-RMS decreases the estimation error of the launching point when the standard deviation of angle noise increases. Furthermore, T-RMS is superior to no filter and MAF in terms of root-mean-square error regardless of the standard deviation of angle noise. This implies that the use of T-RMS is advantageous for a better estimate of a launching point.

6. Conclusions

This paper presents an estimation of the launching points of unmanned aerial vehicles (UAVs) in a boosting phase. By analytically solving inverse first-order pitch programming of UAVs, a vehicle launching point is estimated when positional information and pitch angles derived by vehicle velocities are given. This paper leverages multiple trailing moving average filters to address measurement errors such as random noise and bias. The development of two-point robust measurement selection (T-RMS) enables us to employ two consecutive data processed by multiple trailing moving average filters for decreasing measurement errors such as random noise and bias. Future work will deal with the non-boosting phases of UAVs.