Target-Tracking Forces Dynamics

• Target-tracking forces are dynamic control laws in multi-agent systems that use positional and velocity information to achieve finite-time target capture.
• The framework integrates nonlinear repulsion, self-regulation with stochasticity, and specialized tracking forces to maintain chaser synchronization.
• A four-group geometric configuration combined with predictive guidance (CP versus TDP) enhances capture stability and reduces mean capture time.

Target-tracking forces constitute a class of dynamics used in multi-agent systems where chasers are tasked with apprehending or confining a moving target by exploiting both positional and velocity information of the target. In the context of "A dynamical trap made of target-tracking chasers" (Gao, 27 Dec 2025), the system is formulated as a set of point masses—representing the target and multiple chasers—moving in a plane, with each chaser subjected to a combination of short-range repulsion, self-regulation plus stochasticity, and a uniquely defined target-tracking force. The design achieves robust target capture via critical interplay between moving-direction synchronization and geometric grouping of chasers around the target, yielding finite-time, provably stable encirclement.

1. System Formulation and Constituent Forces

Agents are treated as point masses $m$ confined to a planar domain. The equations of motion are formulated as:

$$m\,\mathbf{a}_i = \mathbf{F}_i^{\mathrm{rep}} + \mathbf{F}_i^{\mathrm{self}} + c \cdot \mathbf{F}_i^{\mathrm{track}}$$

where $c=0$ for the target (which does not track), and $c=1$ for each chaser. The forces are as follows:

• Repulsive Force: A nonlinear elastic repulsion ensures short-range collision avoidance,

$$\mathbf{F}_i^{\mathrm{rep}} = \sum_{j \neq i} \epsilon \left(\frac{\delta_{ij}}{d_{ij}}\right)^{3/2}\Theta(\delta_{ij})\hat{\mathbf{n}}_{ij}$$

where $\epsilon$ is force amplitude ($\epsilon^{\mathrm{t}}=25$, $\epsilon^{\mathrm{c}}=20$), $d_{ij}$ is the mean agent diameter, $\delta_{ij}=d_{ij}-r_{ij}$, $$m\,\mathbf{a}_i = \mathbf{F}_i^{\mathrm{rep}} + \mathbf{F}_i^{\mathrm{self}} + c \cdot \mathbf{F}_i^{\mathrm{track}}$$0 is center-to-center distance, $$m\,\mathbf{a}_i = \mathbf{F}_i^{\mathrm{rep}} + \mathbf{F}_i^{\mathrm{self}} + c \cdot \mathbf{F}_i^{\mathrm{track}}$$1 is the Heaviside function, and $$m\,\mathbf{a}_i = \mathbf{F}_i^{\mathrm{rep}} + \mathbf{F}_i^{\mathrm{self}} + c \cdot \mathbf{F}_i^{\mathrm{track}}$$2 is the unit vector from $$m\,\mathbf{a}_i = \mathbf{F}_i^{\mathrm{rep}} + \mathbf{F}_i^{\mathrm{self}} + c \cdot \mathbf{F}_i^{\mathrm{track}}$$3 to $$m\,\mathbf{a}_i = \mathbf{F}_i^{\mathrm{rep}} + \mathbf{F}_i^{\mathrm{self}} + c \cdot \mathbf{F}_i^{\mathrm{track}}$$4.

• Self–Regulation plus Noise: Implements frictional damping and stochasticity,

$$m\,\mathbf{a}_i = \mathbf{F}_i^{\mathrm{rep}} + \mathbf{F}_i^{\mathrm{self}} + c \cdot \mathbf{F}_i^{\mathrm{track}}$$5

with $$m\,\mathbf{a}_i = \mathbf{F}_i^{\mathrm{rep}} + \mathbf{F}_i^{\mathrm{self}} + c \cdot \mathbf{F}_i^{\mathrm{track}}$$6 (braking), $$m\,\mathbf{a}_i = \mathbf{F}_i^{\mathrm{rep}} + \mathbf{F}_i^{\mathrm{self}} + c \cdot \mathbf{F}_i^{\mathrm{track}}$$7, and $$m\,\mathbf{a}_i = \mathbf{F}_i^{\mathrm{rep}} + \mathbf{F}_i^{\mathrm{self}} + c \cdot \mathbf{F}_i^{\mathrm{track}}$$8 (random direction at each step).

• Target-Tracking Force (Chasers only):

$$m\,\mathbf{a}_i = \mathbf{F}_i^{\mathrm{rep}} + \mathbf{F}_i^{\mathrm{self}} + c \cdot \mathbf{F}_i^{\mathrm{track}}$$9

where $c=0$0, $c=0$1 is the activation radius, $c=0$2 is the target's instantaneous velocity direction, $c=0$3 (direction-alignment), $c=0$4 is the unit vector pointing to a designated point $c=0$5, and $c=0$6 (guidance strength).

2. Sensing, Actuation, and Integration

Chasers have real-time access to the target's position ($c=0$7) and velocity ($c=0$8), as well as their own positions. No explicit inter-chaser communication is required beyond a one-time group assignment—a configuration fixed throughout the chase. Time integration of equations is conducted via a velocity-Verlet scheme with a small timestep $c=0$9, and sensing/actuation are modeled as instantaneous and noise-free except for the intentional stochastic component $c=1$0.

3. Geometric Grouping and Domain Assignment

Effective capture necessitates dividing the $c=1$1 chasers into four equal groups, each assigned to an "arm" of a cross-shaped domain surrounding the target. Each domain is determined by:

• Minimal exclusion radius $c=1$2
• Arm thicknesses $c=1$3 (radial), $c=1$4 (tangential)
• Domain parameters: typically $c=1$5, $c=1$6, $c=1$7, $c=1$8

Each chaser within a group selects a random point $c=1$9 within its assigned domain arm and applies $$\mathbf{F}_i^{\mathrm{rep}} = \sum_{j \neq i} \epsilon \left(\frac{\delta_{ij}}{d_{ij}}\right)^{3/2}\Theta(\delta_{ij})\hat{\mathbf{n}}_{ij}$$0 towards it. The geometric configuration ensures simultaneous approach from multiple directions, essential for enforcing the dynamical trap.

4. Velocity Alignment and Stability of Capture

Synchronization of chaser motion to the target's heading is a critical stabilizing mechanism. The force term $$\mathbf{F}_i^{\mathrm{rep}} = \sum_{j \neq i} \epsilon \left(\frac{\delta_{ij}}{d_{ij}}\right)^{3/2}\Theta(\delta_{ij})\hat{\mathbf{n}}_{ij}$$1 in $$\mathbf{F}_i^{\mathrm{rep}} = \sum_{j \neq i} \epsilon \left(\frac{\delta_{ij}}{d_{ij}}\right)^{3/2}\Theta(\delta_{ij})\hat{\mathbf{n}}_{ij}$$2 ensures local velocity alignment for chasers within distance $$\mathbf{F}_i^{\mathrm{rep}} = \sum_{j \neq i} \epsilon \left(\frac{\delta_{ij}}{d_{ij}}\right)^{3/2}\Theta(\delta_{ij})\hat{\mathbf{n}}_{ij}$$3 of the target. For $$\mathbf{F}_i^{\mathrm{rep}} = \sum_{j \neq i} \epsilon \left(\frac{\delta_{ij}}{d_{ij}}\right)^{3/2}\Theta(\delta_{ij})\hat{\mathbf{n}}_{ij}$$4, directional deviations decay, resulting in collective motion that remains phase-locked with the target's direction. Empirical findings demonstrate that with $$\mathbf{F}_i^{\mathrm{rep}} = \sum_{j \neq i} \epsilon \left(\frac{\delta_{ij}}{d_{ij}}\right)^{3/2}\Theta(\delta_{ij})\hat{\mathbf{n}}_{ij}$$5 (i.e., without heading alignment), chasers cannot maintain proximity and $$\mathbf{F}_i^{\mathrm{rep}} = \sum_{j \neq i} \epsilon \left(\frac{\delta_{ij}}{d_{ij}}\right)^{3/2}\Theta(\delta_{ij})\hat{\mathbf{n}}_{ij}$$6 (the mean chaser-target separation) diverges, allowing the target to escape.

5. Predictive Guidance Algorithms

Two principal guidance paradigms are evaluated:

• Classical Pursuit (CP): Chasers track the target's current position ($$\mathbf{F}_i^{\mathrm{rep}} = \sum_{j \neq i} \epsilon \left(\frac{\delta_{ij}}{d_{ij}}\right)^{3/2}\Theta(\delta_{ij})\hat{\mathbf{n}}_{ij}$$7).
• Track–Direction Pursuit (TDP): Chasers aim for a predicted waypoint $$\mathbf{F}_i^{\mathrm{rep}} = \sum_{j \neq i} \epsilon \left(\frac{\delta_{ij}}{d_{ij}}\right)^{3/2}\Theta(\delta_{ij})\hat{\mathbf{n}}_{ij}$$8.

Simulations indicate TDP lowers mean capture time by approximately 20% relative to CP; however, eventual capture is achieved in both regimes as long as $$\mathbf{F}_i^{\mathrm{rep}} = \sum_{j \neq i} \epsilon \left(\frac{\delta_{ij}}{d_{ij}}\right)^{3/2}\Theta(\delta_{ij})\hat{\mathbf{n}}_{ij}$$9 and the four-group configuration is employed. Predictive guidance enhances efficiency but is not strictly required for finite-time capture.

6. Analytical Results: Robustness and Failure Modes

Key theoretical propositions established via simulation include:

Condition	Asymptotic Separation $\epsilon$0	Capture Outcome
All chasers in one group	$\epsilon$1 (plateau)	Target escapes
Four-group geometry, $\epsilon$2	$\epsilon$3 (finite $\epsilon$4)	Target is captured
Four-group, $\epsilon$5	Diverges	Target escapes
Four-group, $\epsilon$6 too small	Diverges	Target escapes

Proposition 1 asserts that non-grouped (single-domain) strategies are always unstable: $\epsilon$7 increases and does not vanish, resulting in target evasion. Theorem 1 demonstrates that for four-group configurations with appropriate parameters, finite-time capture occurs and $\epsilon$8 approaches zero at $\epsilon$9, after which the target is dynamically trapped at the chaser-induced potential minimum. Reducing $\epsilon^{\mathrm{t}}=25$0 below a critical threshold or disabling velocity-alignment again leads to target escape.

7. Applications, Limitations, and Prospects

Application domains include non-lethal wildlife control (e.g., deploying small UAV swarms for animal exclusion), multi-robot cooperative encirclement in search-and-rescue, and perimeter defense operations. Chasers can be real robots leveraging thermal or stereo-vision for sensing. Noted limitations are the restriction to planar (2D) simulation, idealized noise/actuation models, omission of communication/wind/terrain effects, and lack of explicit energy management or agent–agent collision avoidance. Additionally, animal behavioral responses may deviate significantly from model assumptions. Future extensions will address time-delayed feedback, 3D environments, and biologically realistic target behavior.

The target-tracking force framework represents an overview of (i) nonlinear repulsion, (ii) local velocity alignment ($\epsilon^{\mathrm{t}}=25$1), (iii) position-guided pursuit ($\epsilon^{\mathrm{t}}=25$2), and (iv) geometric four-group organization. The interdependency of these elements is essential; specifically, velocity-alignment and multi-group assignment are necessary for finite-time capture, while predictive guidance confers quantitative improvements in capture speed (Gao, 27 Dec 2025).