The pursuit-evasion game is a kind of differential game demonstrated by Isaacs (1967). It is an important class of two-player zero-sum differential games with perfect information and is useful in military applications, especially in missile guidance applications. In the game, the pursuer and the evader try to minimize and maximize the intercept time or the miss-distance as a payoff function. Among previous studies, Guelman et al. (1988) studied a simple case of pursuit-evasion game that can be solved without addressing a two-point boundary value problem. Breitner et al. (1993) proved that a multiple shooting method is able to precisely solve practical pursuit-evasion games subject to state constraints. In addition to these studies, a variety of parameter optimization methods such as those described in (Hargraves and Paris, 1987) can be extended to solve realistic pursuit-evasion games. In this study, we propose a direct method for solving pursuit-evasion games (Tahk et al., 1998a, b).

We first summarize the algorithm proposed in (Tahk et al., 1998a, b) for the reader's convenience. The algorithm is a direct method based on discretization of control inputs for solving pursuit-evasion games for which the intercept time is the payoff function of the game. Every iteration of the algorithm has two features: update and correction. The update step improves the evader's control to maximize the intercept time, and modifies the pursuer's control to satisfy the terminal condition. After applying the update step several times, the correction step is used to minimize the pursuer's flight time.

Note that the solution of the pursuit-evasion game can be used for guidance of the pursuer and the evader if the solution is obtained in real time. For this purpose, the iteration number of the solver is limited to reduce the computation time, at the cost of solution accuracy (Kim et al., 2006). In this study, the work of Kim et al. (2006) was refined for better numerical efficiency. Also, the capture set of the proposed guidance method is compared with that of proportional navigation to confirm that the proposed differential game guidance laws provide a smaller no-escape envelope.

The problem is a two-dimensional pursuit-evasion game with the final time as a payoff function. The dynamics of the pursuer and the evader are expressed as follows:

The admissible control inputs u_p (t) and u_e (t) are assumed to be piecewise-continuous functions subject to the following constraints:

Our pursuit-evasion problem is written as

subject to the constraints in Eq. (2) and the final constraint,

To develop a numerical algorithm, we used a direct approach based on parameterization of the control inputs. The control inputs of both players are separated into the following form:

where u_p,k and u_e,k , which are assumed to be constant in the k-th interval, should satisfy the constraints in Eq. (2) as follows:

The solution technique proposed in (Tahk et al., 1998b) is composed of two procedures: update and correction. The update procedure determines δ u_e , a small perturbation of u_e , in the direction of maximizing t_f ^* . It also includes an algorithm for the computation of a new u_p ^* . Instead of finding the minimum time solution, this algorithm computes δ u_p, the smallest variation of u_p ^*, to satisfy the final constraint (Eq. 4). If u_p * is far from the minimum time solution, the correction procedure executes and provides an adequate tuning of u_p ^* to satisfy the optimality within the error bounds defined by the user. The aforementioned procedures, illustrated in Fig. 1, are implemented repeatedly until there is no improvement in t_f ^* . In (Kim et al., 2006), the update and correction procedures are called step 1 and step 2 to avoid confusion of terminologies.

We assume that a capture within a finite time is guaranteed. Any perturbations of both players' control input, δ u_e and δ u_p , are said to be admissible if they satisfy following conditions:

The capture condition is expressed as

where P and E are defined as

and v_p and v_e are the sensitivities of r_p (t f ) and r_e (t f ), respectively, to the perturbation of t _f. We define v_r as follows:

Then, Eq. (8) can be written as

Let n₁ and n₂ denote two orthonormal unit vectors in the two-dimensional space. We define n₁ as

Now, P , E , and (특수문자)r_f are expressed as

where p₁ , p₂ , e₁ , and e₂ are N'1 column vectors and d₁ and d₂ are scalars. For a given δ u_e , let δ u_p ^o , which is orthogonal to the null space of p^T ₂ , be the minimum norm solution of the combination of Eqs. (11) and (13), which is expressed as1

If δ u_p ^o is admissible, thenδ u_p ^*=δ u_p ^o. For u_p=u_p^* and t_f=t_f^*, δ u_p satisfying the capture condition can be written as

where δ u_p ⁿ is the vector in the null space of p^T ₂ , which can be chosen freely without affecting the capture condition. After some manipulations of Eqs. (14) and (15), we obtain

where

The evader's control input is then updated as follows:

where j denotes the iteration number, and subscript k is kth element of the separated evader's control variables.

When the deviation of u_p and t _f from the minimum norm solution grows too much, an optimal δ u_p ⁿ should be found to correct u_p and t _f . δ u_p ⁿ , an undefined term of Eq. (16), is calculated as follows. We write δ u_p ⁿ as

where δ u_p ^v is the amount of constraint violation of u_p+δ u_p ^o+δ u_p' ⁿ, and δ u_p ^r is used to make δ u_p ⁿ which belongs to the null space of p^T ₂ . Given δ u_p ^v , δ u_p ^r can be expressed as

where δ u_p^ro belongs to the null space of , and δ u_p ^rn is to be determined in an optimal way.

For minimizing (수식삽입), we find the δ u_p^ro that minimizes the norm of -δ u_p ^v +δ u _p ^ro. The norm of δ u_p ⁿ=-δ u_p ^v +δ u _p ^rois minimized by

After calculating δ u_p ⁿ , each component of is examined to check whether or not any violations of control input constraints occur. As soon as δ δ u_p ⁿ is determined, δ u_p is updated using the following relationship:

Once u_e and u_p are u_pdated, t _f is u_pdated using

In Eqs. 22) and (23), i denotes the iteration number.

In step 1, the pursuer's control input is determined to satisfy the capture condition, and u_p +δ u_p is assumed to close to the optimal control variable for the case of small perturbations of both players' control variables. If the pursuer's control input u_p is far from the optimum to minimize a payoff function, we optimize the pursuer's control variable during the correction procedure.

Assume that the pursuer's control input is subject to variation, but the evader's control input is fixed; that is, δ u_e = 0 . The first variation necessary condition for optimality is given by for all admissible δ u_p . To inspect the optimality condition at a later time, we define I_S as the set of saturated parameters. It is represented as

where K_S and s(i) denote the number of elements in I_S, and its i th element. We construct a N × K_s matrix S as

The pursu_er's control variable is restricted to constraints, and a saturated control variable can be perturbed only in one direction, as shown by

Then, let q₁ denotes the component of p₁ normal to p₂ , which is given by

Hence, the Kuhn-Tucker condition is written as

where m_k > 0 for all . We also derive the following necessary conditions for optimality:

For the correction procedure, we set dδu_e = 0 as previously mentioned. Then, we substitute all elements of q₁ that do not satisfy the necessary conditions for optimality by zero to obtain (특수문자), and we compute (특수문자), the component of (특수문자) normal to (특수문자):

We choose du_p^rn as

to reduce t_f because is the null space of p₂^T . We compute δu_p^rn and the total correction of du_p given by the following relations, starting from δu_p^o=δu_p^v=δu_p^ro, as follows:

If there is any violation of the control input constraint, we use the u_pdate procedure of u_p as explained in step 1.

In this section, we explain the method we used to construct the guidance law proposed in (Kim et al., 2006). The algorithm introduced in the preceding section consists of two steps. This algorithm initializes both players' control inputs, and then checks the optimality conditions. If the pursuer's control input and final time do not satisfy the optimality conditions, step 2 is performed; otherwise, step 1 is executed. The evader tries to maximize the final time or to avoid capture by the pursuer, and the pursuer tries to intercept the evader. After performing step 1, we examine whether a difference in the positions of the players at the final time is within the error bound. If it is within the error bound, a variation of the evader's position at the final time is examined. If the variation is within the error bound, the algorithm is terminated and gives optimal trajectories. Otherwise, step 1 is repeated. If the difference in the positions of the players at the final time is not within the error bound, the optimality conditions of the pursuer's control input and the final time are checked again.

The initializations of both players' control inputs are the same as the algorithm proposed in (Tahk et al., 1998a, b). In this case, the algorithm is started with step 2. At the proposed algorithm, step 2 is executed when a violation of the optimality condition has occurred and step 1 is repeated when a variation of the evader's position at the final time is not within the error bound. To construct a guidance law, this procedure is changed to execute step 2 once and step 1 ten times. The dynamics of both players are integrated with specific integration time interval δt using the initial value of the game solution after performing step 2. Then, state variables changed by integration are considered as initial conditions. The termination condition is the same as the proposed algorithm. The algorithm as a guidance law is described in Fig. 2.

With integration, this modified algorithm changes control inputs. First, control input elements of both players are got rid of the control input sets after an integration and the others are replaced previous elements of each control inputs. The reason is that the first control input elements of the pursuer and the evader are used during integration, and both players' positions are updated by integration. By reason of integration with δt , both players have new initial conditions. After executing integration and replacement (movement) of control inputs, the control input elements of both players are improved during step 1 as previously mentioned. In this respect, the proposed guidance law is similar to the approach of model predictive control (MPC). A schematic diagram of the movement of control input elements is shown in Fig. 3. In addition, we reduced the integration time interval δt when t _f / N was smaller than the specific integration time interval δt . In this case, time interval δt was reduced to t _f / N . If δt is used consistently, pursuit-evasion game solutions are calculated with the small time interval compared to δt . In this case, integration with δt can result in a large miss distance because the dynamics of both players integrate with a much longer time interval than with a discrete time interval of their control inputs near the final time. Thus, we should reduce the integration time interval, which guarantees that integration time interval δt is close to t _f / N . If the integration time interval continu_es to decrease, δt becomes infinitesimal and the iteration executes infinitely. To prevent this, the integration time interval is fixed the specific value near the final time and the iteration number of the solver is limited. Applying this technique, we can improve the interception precision at the final stage. This guidance law is called the differential game guidance law.

The differential game guidance law proposed in (Kim et al., 2006) wasapplied to the planar pursuit-evasion game problem illustrated in Fig. 4. The equations of motion of the evader are described as

In these equations, x_e and y_e denote the position in Cartesian coordinates, γ_e is the flight path angle, v_e is the speed, u_e is the non-dimensional control input variable, R_e is the minimum turn radius, and c is the drag coefficient. If the drag coefficient is zero, the evader does not produce a drag force and its speed does not decrease.

The equations of motion of the pursuer, which are analogous to those of the evader, are as follows:

where R_p is the minimum turn radius, andx x_pr, y_pr, γ_pr, v_pr, and u_p are analogous to the evader model. The c-3ptnstant a, which is composed of the zero-lift-drag coefficient C_D0 and maximum lift coefficient C_Lmax, is defined as

The constant b consists of the maximum lift coefficient and the induced drag factor K:

The engagement scenario for the numerical example is shown in Fig. 5. The evader is initially 6,724.97 m apart from the pursuer along the x-axis, and moves with v_e(0) = 300m / s and (0) = 131.86°. The pursuer initially moves with v_p(0) = 920.83m / s and (0) = 85.36°. The parameters used in this engagement model are taken from (Guelman et al., 1988) as R_e = 600m, R_p = 1515.15m, a = 0.0875, b = 0.4. For a realistic evader, the drag coefficient was chosen as 0.4.

The differential game guidance law parameters were chosen as N = 50 and ε₁ = 0.002m. The algorithm proposed in (Tahk et al., 1998a, b) includes the parameter ε₁, but the differential game guidance law does not define it. The reason is that the proposed guidance law; i.e., the differential game guidance law, does not require it because of the difference in the execution control between step 1 and step 2, as mentioned in the previous section.

The trajectories of the pursuer and the evader are shown in Fig. 6. The pursuer intercepts the evader at 18.565769 seconds. The time histories of the control input of both players are illustrated in Fig. 7. The figure shows that an initial control input of the evader is a zero. The figure also shows that the control inputs of both players are constrained by Eq. (2).

To compare the performance of the differential game guidance law with other guidance laws, proportional navigation guidance (PNG) was adopted as a guidance law of the pursuer. In spite of using PNG, we assumed that the evader knows the maneuvers of the pursuer which are optimal. In this case, the pursuer's control inputs are calculated from

where N' denotes the effective navigation ratio, and V_c and are the closing velocity and line-of-sight rate, respectively. From a_cmd, the control input of the pursu_er is determined as follows:

It is also constrained by the first expression in Eq. (2). Using PNG as a guidance law, the termination condition is replaced by the miss distance. If the miss distance is smaller than 1 m, we believe that the pursu_er intercepts the evader.

Trajectories of the pursuer using PNG as a guidance law, and of the evader, are given in Fig. 8. The pursuer captured the evader at 23.339497 seconds. Time histories of both control inputs are shown in Fig. 9, and are similar to Fig. 7. The pursuer's initial control input is a few different between the differential game guidance and PNG. This is because a control input of the pursuer using the proposed guidance law is initialized by PNG, and then it is improved by step 2.

Table 1 is a summary of the simulation results. The intercept time of the differential game guidance law is shorter than the time using PNG. In the case of using PNG, a computation time is approximately three times longer than that using pursuit-evasion game solution. The number of iterations needed to capture the evader is nearly two times greater than that of the proposed guidance law.

To compare the interception performance of both methods, which are both players using the differential game guidance law and the pursuer using PNG, the capture set was calculated. For simplicity, we considered only the initial condition for the previous example except that the initial flight path angles of the pursuer and the evader varied from 0° to 20° and from 0° to 350°, respectively. The reason why we selected the initial flight path angle ranges of the pursuer is that seeker's gimbals used typical guided missiles adopt lock-on type. The conventional field of view of a seeker is limited from 2° to 5°, but the field of view of seeker-adopted lock-on type gimbals is larger than 15°. The field of view of gimbaled seekers, however, is not much larger than 15°. Thus, the initial flight path angle ranges of the pursuer were chosen as 0° to 20°.

Figures 10 and 11 show the capture sets of the proposed guidance law, and the case using PNG as a guidance law of the pursuer. In these figures, the initial conditions marked with (특수문자) are those for which the pursuer can capture the evader within a finite time. The other symbols indicate the termination of the program without a capture. In Fig. 10, the symbol ★ indicates a circumstance for which the pursuer cannot capture the evader under the termination condition of the differential game guidance law, but is able to intercept the evader under the termination condition when using PNG as a guidance law of the pursuer. The symbol × implies that the pursuer is not able to intercept the evader under both termination conditions. In Fig. 11, the symbol × indicates that the miss distance is larger than 1 meter, so interception by the pursuer does not occur. From Figs. 10 and 11, we know that the capture set of the proposed guidance law is larger than one using PNG as a guidance law of the pursuer.

We carried out a performance analysis of the proposed guidance law based on pursuit-evasion game solutions that were sought by the pursuit-evasion game solver. The derivation of the proposed pursuit-evasion game solver was described, and construction of the proposed guidance law was explained in detail. The guidance law proposed in (Kim et al., 2006) consists of two procedures, update and correction, called step 1 and step 2. Control inputs of the pursuer and the evader were

initialized in the differential game guidance law, and then improved by step 1 and 2. Step 2 executes first that this procedure provides a way to optimize the pursuer trajectory when it deviates excessively from the optimal trajectory. Then, the dynamics of both players were integrated with a specific time interval. Step 1 performs that control inputs of the evader were updated to maximize the increment in the capture time, while the pursuer tried to minimize it. This procedure iterated until satisfying the termination condition. The differential game guidance law was applied to solve a numerical example. For a comparison of the interception performance, PNG was adopted to a guidance law of the pursuer. Simulation results of the engagement scenario were tabulated for each case--the differential game guidance law, and using PNG as the guidance law of the pursuer. Capture sets of both cases were calculated for performance analysis. Based on our numerical example simulation results, we know that the differential game guidance law provided a smaller no-escape envelop than PNG.