Dynamic Maneuvering Cue (DMC)

• Dynamic Maneuvering Cue (DMC) is a closed-form metric that quantifies the minimal heading change required for a vehicle to evade adversarial threat zones.
• It leverages current geometry and kinematics to compute safe boundaries in real-time, facilitating both lightweight feedback and MPC-based controllers.
• DMC's integration into vehicle control systems offers a balance between rapid goal achievement and robust collision avoidance in dynamic environments.

Dynamic Maneuvering Cue (DMC) is a state-dependent, closed-form metric quantifying the instantaneous heading change required by a controlled vehicle (“Agent”) to guarantee it lies outside the Basic Engagement Zone (BEZ) of one or more adversarial “Threat” vehicles. DMC directly captures the minimum evasive maneuver necessary, given current geometry and kinematics, and serves as a conservative risk metric for capture or collision in dynamic threat regions. Its analytic computability and suitability for real-time, onboard application enable feedback-based guidance and constraint enforcement for both lightweight and optimization-based vehicle controllers (Moll et al., 9 Dec 2025).

1. Mathematical Formulation and Definition

Let the Agent $A = (x_A, y_A) \in \mathbb{R}^2$ travel at constant speed $v_A$ with heading $\theta \in (-\pi, \pi]$, and let a single Threat $T = (x_T, y_T)$ travel at speed $v_T > v_A$. Define the speed ratio $p \equiv v_A/v_T \in (0,1)$, a reaction time $t_r$, the Threat’s maximum reach $R = v_T t_r$, and safety radius $r$ around $T$. The BEZ consists of $A$-positions from which $T$, on optimal interception, can reach within $r$ of $A$ in time $t_r$, corresponding to a circle of radius $R + r$ centered at

$B = T - p\,R\, [\cos\psi;\, \sin\psi],$

where $\psi = \operatorname{atan2}(y_A - y_T, x_A - x_T)$ is the LOS angle from $T$ to $A$.

Given the Agent's current range $d = \|A - T\|$ and aspect angle $\psi_{err} = \theta - \psi$, the DMC is defined as the minimum signed heading change needed so that, moving at velocity $v_A$ along the new heading, $A$ lies on or outside the BEZ boundary and never enters it (assuming instantaneous turn and constant speed). Specifically,

$$\phi^* = \arccos\left(\frac{d^2 + p^2R^2 - (R + r)^2}{2\,p\,R\,d}\right),$$

$$\mathrm{DMC}(\theta, d) = \begin{cases} 0 & |\theta - \psi| \geq \phi^* \ \operatorname{sign}(\theta - \psi) \min\{|\theta - \psi - \phi^*|, |\theta - \psi + \phi^*|\} & |\theta - \psi| < \phi^* \end{cases}$$

A DMC of zero indicates a safe (non-threatening) configuration; higher values denote increased immediate risk (Moll et al., 9 Dec 2025).

2. Extension to Multiple Simultaneous Threats

With $N$ threats $T_i$, each parameterized by $p_i, R_i, r_i$, and LOS angle $\psi_i$, the unsafe region defined by each threat is the cone $[\psi_i - \phi^*_i, \psi_i + \phi^*_i]$. The set of headings safe with respect to all threats is

$C = \bigcap_{i=1}^N C_i,$

where $C_i$ is the complement of the unsafe band for threat $i$. The aggregate DMC is the smallest heading correction required to project the current heading $\theta$ onto the nearest boundary of the (possibly nonconvex) safe set $C$. Computationally, this requires $O(N)$ comparisons, yielding only linear scaling in the number of dynamic threats (Moll et al., 9 Dec 2025).

3. Integration of DMC into Vehicle Controllers

DMC constraints can be incorporated into two primary control architectures:

• Simple Feedback (Aim-at-Goal with DMC Limiting): A nominal heading $$\theta_{nom} = \operatorname{atan2}(y_G - y_A, x_G - x_A)$$ is selected, with a DMC-threshold $\varepsilon \geq 0$ enforced:

$$\theta_s = \begin{cases} \theta_{nom} & |\mathrm{DMC}(\theta_{nom}, d)| \leq \varepsilon \ \theta_{nom} + \mathrm{DMC}(\theta_{nom}, d) - \operatorname{sign}(\mathrm{DMC})\,\varepsilon & \text{otherwise} \end{cases}$$

If the nominal heading violates the threshold, it is clamped to the boundary of the safe set via a minimal correction. This approach requires only a handful of trigonometric and min–max operations per cycle, suitable for real-time, onboard control.

• Model Predictive Control (MPC): The finite-horizon optimization enforces DMC-constraints at each step:

$$\begin{aligned} &\min_{\{\theta_0 \dots \theta_H\}} \|x_H - G\|^2 \ &\text{subject to}\;\; x_{k+1} = f(x_k, \theta_k),\quad |\mathrm{DMC}(\theta_k, d_k)| \leq \varepsilon, \quad k=0\dots H \end{aligned}$$

$f(\cdot)$ specifies kinematic updates. Decision variables are the future headings; the cost function favors rapid arrival at the goal without DMC violation. Solvers such as SQP or Interior–Point methods enable online solution, with computational cost scaling with planning horizon $H$ (Moll et al., 9 Dec 2025).

4. Performance Comparison and Computational Characteristics

Empirical evaluation demonstrates key performance trade-offs:

• In a single-threat scenario ($v_A=1$, $p=0.9$, $r=0.2$, $R=0.8$), simple feedback reaches the goal in $6.62$ s, while the numerically optimal trajectory achieves $6.54$ s, a $1.2\%$ improvement.
• In dual-threat cases ($\varepsilon>0$), the simple controller may exhibit suboptimal winding direction or oscillations at DMC switching surfaces, while MPC (with $H=25$, $t_s=0.07$ s) globally optimizes and shortens paths by up to $10\%$.

Computationally, simple feedback incurs sub-millisecond per-update cost (“zero-planning”), whereas MPC requires solving nonlinear programs with $O(10^2)$ variables per cycle (tens to hundreds of ms per update) (Moll et al., 9 Dec 2025).

5. Assumptions and Key Limitations

The DMC methodology, as currently formulated, relies on the following primary assumptions:

• Agent kinematics are restricted to constant-speed, planar motion with instantaneous heading changes (no explicit turn-rate or acceleration bounds).
• Threats are assumed adversarial and possess instantaneous arbitrary heading change—this yields conservative capture zones; extensions to non-adversarial or stochastic elements would be less conservative.
• Engagement is purely two-dimensional; no explicit vertical maneuvering or high-fidelity dynamic constraints are modeled (Moll et al., 9 Dec 2025).

6. Directions for Extension and Application

A range of extensions to the DMC framework are identified as promising:

• Incorporation of Agent turn rate and acceleration bounds into the analytical DMC construction.
• Allowing for speed change maneuvers (e.g., climb or acceleration) within the risk cue framework.
• Integration of DMC with Control Barrier Functions to formally guarantee safety.
• Robust or stochastic variants of DMC to address threat motion uncertainty.

DMC’s analytical real-time computability and extensibility position it as a fundamental component in next-generation reactive guidance, especially for deconfliction and collision avoidance in adversarial and dynamic environments (Moll et al., 9 Dec 2025).