Feedback Strategies for Hypersonic Pursuit of a Ground Evader

Trajectory optimization for hypersonic re-entry vehicles is in general a challenging problem that does not admit analytic solutions due to the vehicle’s highly nonlinear dynamics. For this reason, most research efforts in this area have focused on developing or enhancing numerical optimal control techniques, including direct [1] and indirect [2] methods, for efficient trajectory optimization. In the presence of parametric uncertainty or external disturbances, however, trajectories pre-computed offline may no longer be optimal during flight, which raises the need for a hypersonic vehicle to have an inner-loop feedback guidance system for stabilization or tracking. Traditionally, neighboring optimal (or extremal) control [3] has played an important role in aerospace applications and control of nonlinear systems which are subject to perturbations such as within initial states or terminal constraints. For the applications of neighboring optimal control in hypersonic trajectory design, one can refer to, for instance, [4, 5].

Robust control and guidance problems for hypersonic vehicles have been addressed by several different methods in the literature including LQR-based feedback guidance [6], desensitized optimal control [7], and deep learning method [8] to name but a few. Most of the recent work in hypersonic guidance are, however, focused on steering a hypersonic vehicle to a target that is either static or moves in a known/deterministic fashion but is not capable of maneuvering or evading. If the target can evade, the hypersonic trajectory optimizaiton problem evolves into a differential game [9], in which we are interested in finding a feedback strategy for the hypersonic vehicle to reshape its trajectory on the fly in response to the evading actions of the target.

Differential game theory, first proposed by Isaacs in his pioneering work [9], has been widely used in aerospace, defense, and robotics applications as a framework to model adversarial interactions between two or more players. Pursuit-evasion games, a special class of differential games in which a group of players attempt to capture another group of players, provide a powerful tool to obtain the worst-case strategies for intercepting a maneuvering target [10, 11]. Applying differential game theory to the problem of intercepting an evader by a hypersonic pursuer, however, to our best knowledge, has not yet been proposed in the relevant literature. This is due to analytical and computational complexities of the problem which can be attributed to the nonlinear dynamics of the hypersonic vehicle and the dependence of the atmospheric density upon the altitude. Although numerical methods for computing an open-loop representation of the feedback saddle point solution in two-player differential games have been well studied [12], such open-loop solutions can be ineffective against the target’s unexpected evasion.

The main contribution of this work is a novel formulation of the problem of intercepting a maneuvering ground target by a hypersonic pursuit vehicle as a tractable pursuit-evasion game. After solving for the target’s optimal evasion strategy based on a simplified version of the original nonlinear differential game, we reduce the latter game into an optimal control problem which can readily be solved with existing numerical techniques (either direct or indirect). Thereafter, we linearize the nonlinear hypersonic dynamics around the reference game trajectory and in turn formulate an auxiliary differential game whose feedback saddle point solution can be obtained analytically and computed quickly (real-time) and efficiently compared to the original noninear pursuit-evasion game.

The rest of this paper is structured as follows. In Section 2, the dynamical system models of an unpowered hypersonic pursuit vehicle and a maneuvering ground target are introduced to formulate a pursuit-evasion game. In Section 3, an approximation of the saddle point strategy of the target is computed and subsequently, the pursuit-evasion game is reduced to an optimal control problem whose solution will serve as a reference pursuit trajectory. In Section 4, the game dynamics are linearized with respect to the reference trajectory and an auxiliary two-player zero-sum LQDG is constructed. In Section 5, numerical simulations are presented. Lastly, in Section 6, some concluding remarks are provided.

In this section, the state space models that describe the motion of a hypersonic pursuit vehicle ($P$) and a maneuvering ground target ($T$) are introduced. Subsequently, the corresponding interception problem is formulated as a pursuit-evasion game.

In this section, we first derive the optimal strategy for $T$ in Problem 1 by imposing a few additional assumptions on the dynamical behaviors of $P$ and $T$ in order to obtain an approximated value function. With this strategy we simplify the game in Problem 1 into $P$’s one-sided optimal control problem and compute his optimal trajectory.

The OLPS, albeit optimal in an open-loop sense, is in fact not effective against $T$ when the latter makes sub-optimal and / or decisions which are inconsistent with the OLES. This is because the OLPS cannot take into account the deviation in $T$’s state. In other words, as opposed to our discussion on (12), $P$ cannot take advantage of $T$’s sub-optimal play with the OLPS. To cope with the possible unexpected maneuvers of $T$, some auxiliary feedback input is needed such that the pursuit trajectory can be continuously reshaped in response to the state deviation of $T$.

Fortunately, in light of the fact that $P$ and $T$ have a large speed difference, we may further assume that any resulting pursuit-evasion trajectory caused by $T$’s evading maneuvers (whether optimal or not) will not significantly differ from the OLSPT. This particular assumption will allow us to linearize the dynamics of the players around the OLPTS and construct an auxiliary differential game in the neighborhood of this trajectory. Thereafter, we can derive an auxiliary feedback strategy for $P$ and combine it with the OLPS.

In this section, we present numerical simulation results of the proposed game-theoretic terminal guidance law. The vehicle parameters of $P$ are specified in Table 1, whereas the initial conditions of $P$ and $T$ are listed in Table 2. Other parameters that are necessary for the simulations are selected as: surface air density $\rho_{0}=1.2~{}kg/m^{3}$, scale height $H=7,500~{}m$, and maximum target speed $v_{T}=20~{}m/s$. The angle of attack, $\alpha$, takes values in the compact set $\mathcal{U}_{P}:=[-\pi/\mathrm{18},\pi/\mathrm{18}]$. The capture radius $\epsilon$ is assumed to be zero, which means that the game terminates if the positions of $P$ and $T$ coincide.

We first solve Problem 2 by utilizing a numerical optimal control technique (for instance, direct collocation [14]) to obtain the OLPS. Note that one can use any numerical method here (either direct or indirect), and in fact the OLPS and the corresponding reference trajectory do not have to be optimal (i.e., they may be sub-optimal) in order to apply the aggregate guidance laws proposed in Section 4; however, the closer this reference trajectory is to the optimal one, the smaller $t_{f}^{*}$ one can obtain. From Table 2 we see that $T$ is initially located to the right of $P$ (i.e., $x_{P}^{0}<x_{T}^{0}$), which implies that the OLES, according to (3.1), is given by $u_{T}^{*}=1$. In the following simulation results, we refer to $P$’s optimal trajectory, ${\bm{x}}_{P}^{*}$, as the Time Optimal (TO) trajectory, which is illustrated as a dash line in Figure 2. In addition, the nominal time of capture is obtained from the numerical solver as $t_{f}^{*}=16.1625(s)$.

Given the TO trajectory and the nominal time of capture, we can compute the time-varying matrices ${\mathbf{A}}(t)$ and ${\mathbf{B}}(t)$ as well as $\bm{{\mathcal{A}}}(t)$ and $\bm{{\mathcal{B}}}(t)$ prior to the commencement of game, all of which are defined over the finite time horizon $[0,t_{f}^{*}]$. Given these matrices, the MRDE (35) can be solved numerically (backward as said) to compute the feedback gains in (33) and (34), also prior to the commencement of game. Proper selection of the weighted coefficients, $w_{1}$, $w_{2}$, and $w_{3}$, is the most critical process of this guidance algorithm, which must be done via trial and error. The tuning procedures we have adopted are as follows; first, we find a sufficiently smaller value of $w_{1}$ from its current value such that the auxiliary control input $\nu_{P}$ does not override the OLPS $u_{P}^{*}$, in which case $P$ can easily stall. Once this is done, the final position of $P$ may be located at the desired $x$ position but not at the proper $h$ position (non-zero altitude). For this reason, we next increase the value of $w_{2}$ such that $\delta h_{P}\rightarrow 0$. Our final choice for the weighted coefficients are as follows: $w_{1}=3\cdot 10^{-5}$, $w_{2}=10^{3}$, and $w_{3}=10^{3}$.

In order to investigate the performance of our guidance law, namely (38), against unpredictable evasion of $T$, we apply it against $T$ who employs one of the following three evasion strategies: The first strategy, named $E1$, is the optimal evasion, i.e., $u_{T}(t)=u_{T}^{*}$ for all $t\in[0,t_{f}^{*}]$. In this case, we expect that $P$ will simply follow (or track) its reference trajectory since the deviation of $T$’s horizontal position ($\delta x_{T}$) will always be zero. The second strategy, $E2$, is evasion in the opposite direction of $E1$, i.e., $u_{T}(t)=-u_{T}^{*}$ for all $t\in[0,t_{f}^{*}]$. From the game-theoretic point of view, this strategy would be considered the worst strategy for $T$ since it would yield the smallest time of capture, only if $P$ knew its feedback saddle point strategy. Since $P$ employs, however, the proposed guidance law which is not necessarily optimal in Problem 1, $E2$ can be used to test the robustness of the latter guidance law. Lastly, the third strategy, named $E3$, is random evasion which is meant to confuse $P$ by changing the evasion direction randomly; in particular, $T$ periodically chooses her control input to be a random sample from the (discrete) uniform distribution over the set $\{-1,0,1\}$.

In Figure 2, the trajectories of $T$ corresponding to $E1$, $E2$, and $E3$ are shown as red curves, whereas the trajectories of $P$ in response to these strategies are shown as black curves with different shapes of markers. The miss distances are listed in Table 3, all of which turn out to be reasonably small. Since our terminal guidance law is prescribed with a fixed final time, in all three cases $P$ reaches his final position at $t_{f}^{*}$. It is interesting to observe that $P$’s trajectory against $E1$ is, as opposed to our earlier expectation, is notably different from the TO trajectory. This is presumably because $P$ employs the OLPS, originally a smooth function obtained from the numerical solver, as a piece-wise continuous function in the numerical simulations. Despite the numerical error that comes from such discretization, however, it is shown that $P$ is able to intercept $T$ eventually with a reasonably small miss distance. Finally, the input and state trajectories of $P$ are shown in Figures 3 and 4, respectively.

In this paper, we have proposed a terminal feedback guidance law that is based on a linearized approximation of the nonlinear hypersonic dynamics and the saddle point solution to a two-player zero-sum linear quadratic differential game. In the proposed guidance algorithm, given that the target’s optimal evasion is confined in one axis and under a few simplifying assumptions, an open-loop reference pursuit trajectory have been computed by solving an optimal hypersonic trajectory optimization problem. The latter trajectory has then been used to linearize the nonlinear game dynamics and construct an auxiliary linear quadratic differential game whose saddle point solution is combined with the open-loop input. We have also presented numerical simulation results to verify that, as long as the weighted coefficients are properly selected via trial and error, our proposed guidance law performs well and guarantees the hypersonic pursuit vehicle to intercept the maneuvering (evading) target in a finite time with reasonably small miss distances. One of the limitations of the proposed scheme is that it is only applicable to a scenario in which the target’s motion is constrained along a line. It is also required that the target’s position must be known to the hypersonic pursuit vehicle at every time instant in order to compute the feedback control input derived from the auxiliary differential game along a reference pursuit trajectory. Possible future directions of this work include consideration of scenarios in which the target may evade on a plane and the information given to the hypersonic pursuit vehicle on the target’s position is stochastic rather than deterministic.