Constant Bearing Pursuit Strategy

Updated 6 December 2025

Constant Bearing Pursuit Strategy is a feedback control law where follower agents maintain a constant bearing angle to the target, ensuring predictable paths and effective coordination.
It rigorously defines the reachable set of follower agents and quantifies geometric constraints under speed limitations, which is fundamental for safety-critical motion planning.
Extensions to cyclic, beacon-referenced, and three-dimensional frameworks enable scalable, decentralized solutions for UAVs, marine robotics, and sensor swarms.

The constant bearing pursuit (CB) strategy is a feedback control law for multi-agent systems in which each follower agent maintains a constant angle between its velocity vector and the line-of-sight (LOS) to a target (leader or neighbor). This guidance law is foundational in pursuit-evasion games, collision avoidance, collective motion, and coverage control frameworks, and is studied extensively in both planar and three-dimensional kinematic models. Recent works formalize its reachability properties, extend it to beacon-referenced and multi-agent cyclic graphs, and elucidate the optimization and geometric constraints of the follower’s accessible workspace.

1. Formal Definition and Kinematics

Consider a planar leader–follower pair with state variables

$\mathbf{x}_I(t) = \begin{bmatrix} x_I(t) \ y_I(t)\end{bmatrix},\quad \mathbf{u}_I(t) = v_I\begin{bmatrix}\cos\psi_I(t)\\sin\psi_I(t)\end{bmatrix}$

for the independent (leader) agent, and

$\mathbf{x}_D(t) = \begin{bmatrix} x_D(t) \ y_D(t)\end{bmatrix},\quad \mathbf{u}_D(t) = v_D\begin{bmatrix}\cos\psi_D(t)\\sin\psi_D(t)\end{bmatrix}$

for the dependent (follower) agent, with constant speeds $v_D > v_I > 0$ . The dynamics are

$\dot{\mathbf{x}}_I = \mathbf{u}_I,\qquad \dot{\mathbf{x}}_D = \mathbf{u}_D.$

Under the constant bearing law, the follower selects its heading $\psi_D$ to enforce a constant offset between $\mathbf{u}_D$ and the instantaneous LOS vector $\mathbf{x}_I - \mathbf{x}_D$ . The exact control policy in the planar case is given by

$\psi_D(t) = \sin^{-1}\left(\frac{v_I}{v_D}\sin\psi_I(t)\right),\quad \psi_D(t)\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right].$

This generalizes to higher dimensions using Frenet frame equations, with shape variables encoding the relative orientation between agents (Makkapati et al., 29 Nov 2025, Galloway et al., 2019).

2. Dependent Reachable Set Analysis

The primary reachability-theoretic object of interest is the Dependent Reachable Set (DRS): the set of all points $(x_D, y_D)$ the follower can occupy at time $t$ , given all admissible leader trajectories up to time $t$ . The DRS is characterized by

$y_D(t) = \int_0^t v_I\sin\psi_I(\tau)\,d\tau,\quad x_D(t) = \int_0^t \sqrt{v_D^2 - v_I^2\sin^2\psi_I(\tau)}\,d\tau,$

as $\psi_I(\cdot)$ ranges over feasible controls. Analytical results provide the DRS explicitly in two temporal regimes:

For $0 \le t \le t_2 = a/\sqrt{v_D^2 - v_I^2}$ ,

$\mathcal{D}(t) = \{(x, y)\mid x^2 + y^2 \leq (v_D t)^2,\; x \geq t\sqrt{v_D^2 - v_I^2}\}.$

The boundary is a vertical chord of the $v_D t$ -disk [(Makkapati et al., 29 Nov 2025), Thm. 3.3].

For $t_2 < t \le t_c$ ,

$\mathcal{D}(t) = \{(x, y)\mid x^2 + y^2 \leq (v_D t)^2,\, x \geq \frac{a^2 + v_D^2 t^2 - v_I^2 t^2}{2a}\}.$

The chord slides rightward; the DRS remains a minor segment of the disk.

Simulations with point-cloud propagation confirm the accuracy of these boundaries up to sampling error. The DRS is always a proper subset of the follower’s velocity-limited reach disk, revealing the geometric restriction imposed by constant bearing pursuit (Makkapati et al., 29 Nov 2025).

3. Optimization and Extremal Trajectories

To maximize (or minimize) the $x$ -coordinate of the follower at final time, for a fixed leader endpoint, the optimal leader trajectory is formulated as

$\max_{\mathbf{u}_I(\cdot)} \int_0^t \sqrt{v_D^2 - \dot y_I^2(\tau)} d\tau$

subject to $\dot x_I^2 + \dot y_I^2 = v_I^2$ and given boundary conditions. Euler–Lagrange analysis yields first-integrals, but closed-form solutions remain elusive. Simulation-based evidence supports that extremal leader paths are single-switch curves, partitioned into arcs with constant heading rates, and optimal switch points lie on an ellipse defined by kinematic constraints. The explicit structure of these extremal controls is central for worst-case analysis and pursuit-evasion optimality assessment (Makkapati et al., 29 Nov 2025).

4. Extensions to Multi-agent, Beacon, and Cyclic Frameworks

CB pursuit generalizes to multi-agent configurations and higher dimensions:

Cyclic pursuit: Each agent pursues its neighbor with prescribed CB offset, yielding formations such as parallel rectilinear motion, circling equilibria, and self-similar spirals. The set of possible global collective behaviors is fully classified by closure conditions on agent bearing offsets (Galloway et al., 2017).
Branching graphs: Weakly connected, outdegree-1 pursuit graphs permit agents to join or leave the core cycle without perturbing the global formation. A branch agent’s parameters and initial conditions do not affect the persistent trajectory of the main cycle, facilitating modular design (Galloway et al., 2017).
Beacon-referenced pursuit: Combining attention to both neighboring agents and static beacons enriches the spectrum of feasible collective behaviors. Feedback laws of the form

$u_i = (1-\lambda) u^i_{\text{CB}} + \lambda u^i_{B},\qquad \lambda\in(0,1)$

admit circling, stacked, or spiral equilibria where agents orchestrate around beacons with radii and angular separations tunable by control parameters. The conditions for existence and stability of these equilibria depend on gain, offset, and weight choices (Galloway et al., 2019, Galloway et al., 2017, Galloway et al., 2017, Galloway et al., 2015).

5. Three-dimensional Constant Bearing Pursuit

In three dimensions, agent states comprise position and Frenet frames $(\mathbf{r}, \mathbf{x}, \mathbf{y}, \mathbf{z})$ with unit-speed kinematics

$\dot{\mathbf{r}}_i = \mathbf{x}_i,\qquad \dot{\mathbf{x}}_i = u_i \mathbf{y}_i + v_i \mathbf{z}_i.$

The generalized steering law maintains the inner product $\mathbf{x}_i \cdot \widehat{\mathbf{r}_{ij}} = a_i$ (desired bearing-cosine to the target). Beacon-referenced modifications convexly combine pursuit and beacon objectives, enabling engineered circling equilibria in planes orthogonal to beacon axes, with explicit formulas for formation radii, vertical offsets, and agent separation (Galloway et al., 2019, Galloway et al., 2017). Trade-offs emerge: increasing beacon attention parameter $\lambda$ increases geometric anchoring but reduces mutual-tracking influence.

6. Stability and Collective Shape Dynamics

The stability of circling and spiral equilibria in CB pursuit frameworks is analytically tractable in select cases:

In the planar 2-agent case, stability reduces to explicit sine-sign conditions on CB and beacon offsets (Galloway et al., 2015).
For general $n$ -agent cyclic or branching graphs, stability of circling and spiral equilibria derives from eigenvalue analysis of block-circulant Jacobians and application of generalized Routh criteria (Galloway et al., 2017).
Beacon-referenced shape equilibria manifest as invariant manifolds with dimension reduction into scale (size) and pure-shape factors, revealing log-spiral dynamics or periodic orbits, depending on closure parameters and symmetries.

These characterizations permit rigorous parameter design for robust, scalable, and reconfigurable pursuit collectives.

7. Practical Implications and Applications

CB pursuit laws have direct application in motion planning for UAVs, marine robotics, collision-avoidance, surround coverage, and station-keeping in communication sensor swarms. The explicit reachability and constraint analysis in (Makkapati et al., 29 Nov 2025) affords guarantees for coverage, collision avoidance, and interception geometry, especially under adversarial or uncertain leader behavior. The structure of the DRS is fundamental for certification in safety-critical, adversarial, or resource-constrained scenarios. In multi-agent and formation control settings, modularity of the CB pursuit law enables decentralized, scalable algorithms with parametric tunability of collective geometry and convergence properties (Makkapati et al., 29 Nov 2025, Galloway et al., 2017, Galloway et al., 2015).

References:

(Makkapati et al., 29 Nov 2025) "Dependent Reachable Sets for the Constant Bearing Pursuit Strategy" (Galloway et al., 2019) "Beacon-referenced Pursuit for Collective Motions in Three Dimensions" (Galloway et al., 2017) "Collective Motion under Beacon-referenced Cyclic Pursuit" (Galloway et al., 2017) "Constant Bearing Pursuit on Branching Graphs" (Galloway et al., 2017) "Beacon-referenced Mutual Pursuit in Three Dimensions" (Galloway et al., 2015) "Station Keeping through Beacon-referenced Cyclic Pursuit"