Deviated Pursuit Guidance

• Deviated pursuit guidance is a geometric control strategy that intentionally offsets the line-of-sight to optimize intercept trajectories and engagement geometry.
• It encompasses formulations such as cyclic pursuit, delayed-information schemes, and consensus-based methods for synchronizing time-to-go across multiple agents.
• Recent studies demonstrate that this approach enhances interception performance and robustness in dynamic, uncertain, and cooperative network environments.

Searching arXiv for papers on deviated pursuit guidance and closely related formulations. Deviated pursuit guidance is a family of guidance constructions in which a pursuer, interceptor team, or agent network does not simply chase a target along the instantaneous line of sight, but instead regulates a deliberate geometric, temporal, or informational deviation to shape the engagement. In the recent literature, that deviation appears as a rotated pursuit vector in cyclic pursuit, a deviation angle relative to the line of sight, a prescribed terminal angular separation among multiple pursuers, a delayed-information surrogate state, or an asymmetric visibility sector for vision-based interception. Across these formulations, the common feature is that the guidance objective is achieved by controlling the geometry of pursuit rather than by direct pursuit alone (Segall et al., 2020).

1. Canonical geometric formulations

A canonical formulation appears in deviated linear cyclic pursuit. For agents $p_i(t)\in\mathbb{R}^2$, ordered cyclically, the autonomous law is

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$

with

$$R(\theta) = \left [ \begin{matrix} \cos(\theta) & \sin(\theta)\ -\sin(\theta) & \cos(\theta) \end{matrix} \right ].$$

Here the deviation angle $\theta$ rotates the local pursuit vector away from direct pursuit. The associated critical angle is

$\theta_c=\frac{\pi}{n},$

and the collective regime depends on whether $|\theta|$ is below, equal to, or above $\theta_c$ (Segall et al., 2020).

In interceptor guidance, the same idea is expressed through a line-of-sight-relative deviation angle. For the $i$-th interceptor in a planar engagement,

$\delta_i = \gamma_{M_i} - \theta_i,$

with relative kinematics

$$V_{r_i} = \dot{r}_i = V_T \cos(\gamma_T - \theta_i) - V_{M_i} \cos\delta_i,$$

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$0

Against a constant-velocity, non-maneuvering target, deviated pursuit yields an exact time-to-go expression,

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$1

and this exactness is the basis for several cooperative guidance laws (Gopikannan et al., 18 Sep 2025).

A third geometric usage arises in cooperative interception with prescribed relative intercept angles. In that setting, the deviation is not a constant line-of-sight offset during the engagement, but a terminal angular separation,

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$2

imposed among adjacent pursuers at intercept. This makes deviated pursuit a tool for terminal geometry design as well as for line-of-sight shaping (Wang et al., 2024).

2. Pursuit–evasion engagements and single-interceptor guidance

One recent pursuit–evasion formulation considers an escape flight vehicle (EFV) guided by deep reinforcement learning against a pursuit flight vehicle (PFV) guided by proportional navigation. The PFV uses

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$3

with overload commands converted to aerodynamic angles through

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$4

The EFV, by contrast, learns guidance commands to maximize residual velocity subject to an evasion-distance constraint, with capture declared when the distance drops below $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$5 m. The resulting optimization is described as an irregular dynamic max-min problem because the optimal stopping time is unknown, the residual velocity depends on the full command sequence, the recursion is complex and nonlinear, and the aerodynamic forces are not available in simple closed form. The state used by the policy has 8 variables—relative position, relative velocity, and the previous EFV guidance commands—and the action is the change in guidance commands rather than the absolute commands. In simulation, PPO achieved residual velocity $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$6 m/s with evasion distance $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$7 m, while ES-enhanced PPO achieved $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$8 m/s with evasion distance $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$9 m, improving PPO by about $$R(\theta) = \left [ \begin{matrix} \cos(\theta) & \sin(\theta)\ -\sin(\theta) & \cos(\theta) \end{matrix} \right ].$$0 (Hu et al., 2024).

This EFV–PFV setting is asymmetric: only the escaping vehicle learns, while the pursuer is a fixed-rule adversary. That asymmetry preserves the classical pursuit structure on the pursuer side and uses deviation on the evader side to distort the pursuit geometry. A plausible implication is that deviated pursuit guidance need not belong exclusively to the pursuer; it can also be used as an escape-guidance principle when the objective is to force an unfavorable line-of-sight evolution for the pursuer (Hu et al., 2024).

A control-theoretic variant appears in input-output feedback-linearized interception. There the outputs are the line-of-sight angular rates,

$$R(\theta) = \left [ \begin{matrix} \cos(\theta) & \sin(\theta)\ -\sin(\theta) & \cos(\theta) \end{matrix} \right ].$$1

and the baseline objective is

$$R(\theta) = \left [ \begin{matrix} \cos(\theta) & \sin(\theta)\ -\sin(\theta) & \cos(\theta) \end{matrix} \right ].$$2

The key result is that output regulation alone does not guarantee interception because

$$R(\theta) = \left [ \begin{matrix} \cos(\theta) & \sin(\theta)\ -\sin(\theta) & \cos(\theta) \end{matrix} \right ].$$3

To remove the non-intercepting zero-dynamics branch, the proposed Closing Alignment Toggle Scheme switches the sign of the linearizing control according to the sign of the pursuer velocity projection on the LOS, thereby enforcing the closing branch and guaranteeing interception over a broad class of initial geometries for which the decoupling matrix is invertible (Dorsey et al., 4 May 2026).

3. Cooperative interception, consensus, and prescribed timing

In cooperative salvo interception, deviated pursuit is used because it provides an exact time-to-go variable that can be synchronized across interceptors. For a moving, non-manoeuvring target, the cooperative law in one formulation is built on

$$R(\theta) = \left [ \begin{matrix} \cos(\theta) & \sin(\theta)\ -\sin(\theta) & \cos(\theta) \end{matrix} \right ].$$4

with consensus dynamics defined on the time-to-go errors. The communication topology is a pseudo-undirected graph, and the consensus value depends on the left null vector of the weighted Laplacian: $$R(\theta) = \left [ \begin{matrix} \cos(\theta) & \sin(\theta)\ -\sin(\theta) & \cos(\theta) \end{matrix} \right ].$$5 With positive weights, the consensus impact time lies in the convex hull of the initial time-to-go values; with one negative edge weight chosen within Nyquist-based gain-margin bounds, the common impact time can be pushed outside that convex hull. In the reported five-interceptor simulations, a cycle graph with positive weights yielded a consensus impact time of $$R(\theta) = \left [ \begin{matrix} \cos(\theta) & \sin(\theta)\ -\sin(\theta) & \cos(\theta) \end{matrix} \right ].$$6 s, and a negative-weight perturbation increased it to $$R(\theta) = \left [ \begin{matrix} \cos(\theta) & \sin(\theta)\ -\sin(\theta) & \cos(\theta) \end{matrix} \right ].$$7 s; a star graph case gave $$R(\theta) = \left [ \begin{matrix} \cos(\theta) & \sin(\theta)\ -\sin(\theta) & \cos(\theta) \end{matrix} \right ].$$8 s with positive weights and $$R(\theta) = \left [ \begin{matrix} \cos(\theta) & \sin(\theta)\ -\sin(\theta) & \cos(\theta) \end{matrix} \right ].$$9 s with one negative weight (Sinha et al., 2024).

A related framework addresses seeker-limited interceptor teams. Only a subset of interceptors directly observes the target, so seeker-less agents use a fixed-time distributed observer for

$\theta$0

with convergence in fixed time $\theta$1. The exact deviated-pursuit time-to-go, computed from estimated quantities, has relative degree one with respect to lateral acceleration: $\theta$2 That structure supports a higher-order sliding mode consensus law, and the work states that the interceptors establish consensus in time-to-go within finite time $\theta$3 (Gopikannan et al., 18 Sep 2025).

Deviated pursuit also appears in three-agent cooperative defense. An evader and defender cooperate against a pursuer. The evader drives

$\theta$4

to zero, effectively luring the pursuer toward a collision course, while the defender uses deviated pursuit toward either the pursuer or the evader, depending on whether the stance is aggressive or defensive. The timing objective is encoded in sliding surfaces such as

$\theta$5

or

$\theta$6

and fixed-time convergence is obtained with sliding-mode terms of the form

$\theta$7

The defender thereby maintains a constant deviation-angle pursuit path while controlling engagement duration (Sinha et al., 2021).

A separate cooperative formulation imposes relative intercept angles through nonlinear optimal control. With

$\theta$8

and terminal constraints

$\theta$9

Pontryagin’s maximum principle gives the explicit optimal command

$\theta_c=\frac{\pi}{n},$0

For the reported $\theta_c=\frac{\pi}{n},$1 case, about $\theta_c=\frac{\pi}{n},$2 trajectories were generated, a feedforward neural network with three hidden layers of 20 neurons was trained, and inference took about $\theta_c=\frac{\pi}{n},$3 ms on an MYC-Y6ULY2 CPU at 528 MHz. The implementation switches to proportional navigation below $\theta_c=\frac{\pi}{n},$4 m because the terminal mapping is not one-to-one near intercept (Wang et al., 2024).

4. Swarm-level deviated pursuit and broadcast guidance

In multi-agent systems, deviated pursuit guidance has a collective rather than target-intercept meaning. In deviated linear cyclic pursuit with broadcast guidance, agent $\theta_c=\frac{\pi}{n},$5 senses only the relative position of agent $\theta_c=\frac{\pi}{n},$6, all agents share a common orientation frame, and an external velocity signal $\theta_c=\frac{\pi}{n},$7 is detected only by a random subset of agents called ad-hoc leaders. The closed-loop dynamics are

$\theta_c=\frac{\pi}{n},$8

or in stacked form

$\theta_c=\frac{\pi}{n},$9

with $|\theta|$0 and $|\theta|$1 (Segall et al., 2020).

The deviation angle governs the stability class of the collective dynamics.

Condition on $|\theta|$2	Autonomous behavior	Guided behavior
$|\theta|$3	Convergence to centroid	Translation with velocity $|\theta|$4
$|\theta|$5	Circular motion	Moving circular orbit
$|\theta|$6	Unstable spreading	Unstable behavior

For $|\theta|$7, the group asymptotically moves with velocity

$|\theta|$8

If all agents detect the signal, the formation collapses to a single point moving with $|\theta|$9: $\theta_c$0 If only a subset detects the signal, the asymptotic positions become

$\theta_c$1

so the agents align along a line rotated by $\theta_c$2 relative to $\theta_c$3. For $\theta_c$4, the same detection asymmetry shifts the centers of the common circular orbits rather than eliminating them. The spectral characterization is explicit: the nonzero eigenvalues of $\theta_c$5 lie in the open left-half plane for $\theta_c$6, two lie on the imaginary axis for $\theta_c$7, and at least two lie in the open right-half plane for $\theta_c$8 (Segall et al., 2020).

This networked formulation broadens the meaning of deviated pursuit guidance. The deviation no longer indicates a single interceptor’s offset from line of sight; it determines the asymptotic collective regime, the effect of partial broadcast reception, and the emergent translation-orbit structure of the swarm. This suggests that the term covers both interception laws and formation-level geometric control.

5. Estimation, uncertainty, and decision-aware deviation

Under uncertainty, deviated pursuit guidance can be driven by the posterior distribution of the engagement state rather than by a point estimate. A Bayesian decision-theoretic framework modifies the perfect-information DGL1 law by treating the game-space region as a multi-hypothesis decision problem. The hypotheses are whether the state is above the singular region, inside it with target mode $\theta_c$9, inside it with target mode $i$0, or below it. The posterior is represented by an interacting multiple model particle filter with 1000 particles, 500 per target mode, and the resulting stochastic guidance law has five modes. When the Bayesian decision is nonunique, the controller deliberately shapes the trajectory toward the singular-region boundary to improve future estimation quality; this is called information-enhancement trajectory shaping. In Monte Carlo simulation, the information-enhancing version reduced the required warhead lethality radius from $i$1 m to $i$2 m for a late maneuver at $i$3 s, about a $i$4 improvement over the estimation-aware variant without trajectory shaping, and achieved about a $i$5 improvement over regular DGL1 at $i$6 kill probability (Mudrik et al., 11 Feb 2026).

A closely related delayed-information formulation replaces instantaneous pursuit of the current target state with pursuit of a delayed, smoothed surrogate. The information state is

$i$7

with two time-varying delays satisfying

$i$8

The delayed-information center of the ZEM uncertainty set, $i$9, becomes the decisive pursuit variable. A particle-based fixed-lag smoother supplies the delayed states, and semi-Markov modeling of target maneuver modes estimates the delays online. In Monte Carlo comparison, the resulting TV-DGLCC law required a lethality radius of $\delta_i = \gamma_{M_i} - \theta_i,$0 m at $\delta_i = \gamma_{M_i} - \theta_i,$1, compared with $\delta_i = \gamma_{M_i} - \theta_i,$2 m for DGLC and $\delta_i = \gamma_{M_i} - \theta_i,$3 m for DGL1, corresponding to reported improvements of $\delta_i = \gamma_{M_i} - \theta_i,$4 over DGL1 and $\delta_i = \gamma_{M_i} - \theta_i,$5 over DGLC (Mudrik et al., 5 Mar 2026).

Another robust formulation uses disturbance attenuation and measurement feedback. The guidance law is

$\delta_i = \gamma_{M_i} - \theta_i,$6

with feasibility requiring

$\delta_i = \gamma_{M_i} - \theta_i,$7

The analysis shows that if $\delta_i = \gamma_{M_i} - \theta_i,$8 is held fixed, higher measurement noise can increase the feedback gain, whereas tuning $\delta_i = \gamma_{M_i} - \theta_i,$9 near the critical disturbance-attenuation ratio $$V_{r_i} = \dot{r}_i = V_T \cos(\gamma_T - \theta_i) - V_{M_i} \cos\delta_i,$$0 reverses that trend. It also shows that trajectory shaping through $$V_{r_i} = \dot{r}_i = V_T \cos(\gamma_T - \theta_i) - V_{M_i} \cos\delta_i,$$1 alters both the control and estimation Riccati equations and can create an additional local minimum in $$V_{r_i} = \dot{r}_i = V_T \cos(\gamma_T - \theta_i) - V_{M_i} \cos\delta_i,$$2, so $$V_{r_i} = \dot{r}_i = V_T \cos(\gamma_T - \theta_i) - V_{M_i} \cos\delta_i,$$3 must be reselected jointly with trajectory shaping (Or et al., 2020). This suggests a second, non-geometric sense of deviation: the pursuit path can be shaped by minimax robustness requirements and estimator structure, not only by explicit line-of-sight offsets.

6. Pursuit-inspired path following and visibility-constrained interception

Several works transfer deviated pursuit ideas from target interception to path following. One robust look-ahead pursuit law for fixed-wing UAVs treats the path as a sequence of moving virtual target points

$$V_{r_i} = \dot{r}_i = V_T \cos(\gamma_T - \theta_i) - V_{M_i} \cos\delta_i,$$4

and defines look-ahead angle errors

$$V_{r_i} = \dot{r}_i = V_T \cos(\gamma_T - \theta_i) - V_{M_i} \cos\delta_i,$$5

The nominal commands are

$$V_{r_i} = \dot{r}_i = V_T \cos(\gamma_T - \theta_i) - V_{M_i} \cos\delta_i,$$6

and disturbance compensation is added through $$V_{r_i} = \dot{r}_i = V_T \cos(\gamma_T - \theta_i) - V_{M_i} \cos\delta_i,$$7 and $$V_{r_i} = \dot{r}_i = V_T \cos(\gamma_T - \theta_i) - V_{M_i} \cos\delta_i,$$8. Under the stated compensation inequality, the settling time bound is

$$V_{r_i} = \dot{r}_i = V_T \cos(\gamma_T - \theta_i) - V_{M_i} \cos\delta_i,$$9

For a large disturbance level $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$00, the reported distance error between UAV and virtual target converged almost linearly and stabilized to zero in less than about 10 seconds on average (Sheng et al., 22 May 2025).

A three-dimensional bounded-input path-following law adopts a similar pseudo-target viewpoint but works directly with 3D relative kinematics, lead angles $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$01, range $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$02, and saturated speed and angular-rate dynamics. The design objective is

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$03

in fixed time, and the fixed-time bounds are of the form

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$04

with analogous expressions for the pitch and yaw lead-angle subsystems. The guidance strategy is explicitly described as drawing inspiration from pursuit guidance while removing dependence on the detailed geometry of the path (Kumar et al., 2024).

A more literal visibility-constrained variant is Planar-Sector LOS guidance for interception of agile aerial targets with a lifting-wing quadcopter. Instead of a symmetric conic field-of-view constraint, the LOS is constrained to the asymmetric sector

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$05

The sector tightly constrains lateral image error while relaxing longitudinal image error. Under the lifting-wing quadcopter model, the reported effect is nearly $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$06 more available thrust near the LOS direction than conventional conic LOS constraints. Theorem 1 states that if the initial LOS satisfies $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$07, then the trajectory remains in $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$08 and $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$09. Outdoor experiments reported successful interceptions at ranges up to $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$10 m, with $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$11 success for static targets and $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$12 for dynamic targets, while maintaining continuous visual tracking throughout the engagement (Liu et al., 9 Jun 2026).

These path-following and LOS-sector formulations retain the essential structure of deviated pursuit guidance: the vehicle does not simply align with the geometric centerline of the target or path, but pursues a deliberately shaped surrogate—virtual target, lead-angle manifold, or asymmetric LOS sector—that better matches actuation limits, disturbance rejection, or visibility constraints.

7. Related analytical and computational frameworks

Several adjacent frameworks do not define a deviated pursuit law directly but provide analytical machinery for such laws. Time-optimal collaborative guidance via the generalized Hopf formula casts multi-pursuer pursuit–evasion as a bounded-control Hamilton–Jacobi–Isaacs problem on a joint state space. The terminal set is a union of individual capture sets,

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$13

with terminal cost

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$14

and the generalized Hopf formula computes the value function pointwise without a grid. The reported simulations exhibit coordinated paths in which pursuers separate to surround and contain the evader rather than maintaining simple straight-line chase. This suggests that optimal differential-game formulations naturally produce curved or deviated pursuit trajectories even when the law is not parameterized by an explicit deviation angle (Kirchner et al., 2017).

A hypersonic pursuit formulation uses a similar logic in local form. A nominal open-loop saddle-point trajectory is computed offline, the nonlinear hypersonic dynamics are linearized about that reference, and an auxiliary finite-horizon zero-sum LQDG yields feedback corrections

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$15

The actual applied input is the nominal control plus the deviation-correcting feedback, clipped to bounds. Against three target strategies, the reported miss distances were $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$16 m, $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$17 m, and $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$18 m, respectively (Lee et al., 2021).

Circular pursuit dynamics furnish a further baseline. In a planar pure circular pursuit model with target moving on a circle and pursuer always pointing at the target, the reduced system is

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$19

The analysis shows equilibria only for $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$20, a stable-focus to stable-node transition near $\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$21, and a force-limited extension

$\dot{p}_i(t) =R(\theta) (p_{i+1}(t) - p_i(t)),$22

Although this framework is explicitly not a deviated-pursuit law, it provides a low-dimensional relative-motion template and bifurcation methodology that can be adapted when deviation angles or biased heading laws are introduced (Shekhawat et al., 10 Apr 2026).

Taken together, these related results indicate that deviated pursuit guidance is not a single closed-form rule. It is a broader design principle spanning explicit angular deviation, exact time-to-go coordination, distributed and broadcast pursuit, estimator-aware trajectory shaping, and pursuit-inspired path following. The literature therefore treats deviation not as an error term to be eliminated, but as a controlled degree of freedom through which capture, escape, simultaneity, observability, and maneuverability are organized.