Zero-Effort Miss (ZEM)

Author: Tianjiang Says
This article is based on Zhang Chengming (2021) Research on Guidance Strategies for Spacecraft Pursuit-Evasion Games.

Definition

Zero-Effort Miss (ZEM) is the closest approach distance between the pursuer and the target at a future time, assuming both the pursuer and the target maintain their current states (no additional control inputs). When ZEM is zero, it indicates that the pursuer will successfully capture the target.

ZEM is a core parameter in pursuit-evasion games and guidance law design, used for:

Assessing the feasibility of the current interception/pursuit scheme
Designing feedback guidance laws
Real-time threat assessment for the pursuing party

Mathematical Definition

In an inertial coordinate frame, the zero-effort miss at time $t$ is defined as:

\text{ZEM}(t) = \boldsymbol{r}_p(t_f) - \boldsymbol{r}_e(t_f | \boldsymbol{r}_p(t), \boldsymbol{r}_e(t))

i.e., the position difference when both parties are propagated to the terminal time $t_f$ assuming they maintain their current states.

Simplified Form

For short-range pursuit-evasion problems, the CW equation-based ZEM can be approximated as:

|\text{ZEM}| \approx \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}

where $\Delta x, \Delta y, \Delta z$ are the current relative position components.

Application in Pursuit-Evasion Games

Pursuit-Evasion-Defense Problem

Zhang Chengming (2021) combined ZEM with fuzzy comprehensive evaluation methods in spacecraft pursuit-evasion-defense problems:

Real-time ZEM computation: The pursuing party computes ZEM with the evading party in real time
Threat assessment: Assess the pursuing party's threat level from the defending party based on ZEM
Guidance decision: Determine the pursuing party's guidance strategy based on threat level

Fuzzy Comprehensive Evaluation

Steps of the ZEM-based fuzzy comprehensive evaluation method:

Establish membership functions between ZEM and threat levels
Consider multiple factors comprehensively (ZEM, relative velocity, remaining fuel, etc.)
Determine the fuzzy threat level for the pursuing party
Select the corresponding guidance strategy based on the threat level

Relationship with Guidance Laws

The guidance command of proportional navigation is closely related to the ZEM direction:

a_c \propto \dot{r} \cdot \text{ZEM}

where $\dot{r}$ is the closing velocity.

Optimal Guidance Law

In optimal guidance law design considering zero-effort miss, the objective is to minimize the terminal ZEM.

Typical Values

ZEM Range	Tactical Meaning
ZEM approx 0	Interception/capture about to succeed
ZEM > 0	Possible miss, course correction needed
ZEM >> intercept radius	Significant miss, replanning required

Application Scenarios

Missile interception: Guidance law design for air-to-air and surface-to-air missiles
Spacecraft rendezvous and docking: Collision avoidance during the approach phase
Space debris avoidance: Timing decisions for evasive maneuvers
Pursuit-evasion games: Real-time threat assessment and strategy selection

References

Zhang Chengming. Research on Guidance Strategies for Spacecraft Pursuit-Evasion Games[D]. National University of Defense Technology, 2021. [in Chinese]
Shinar J, Shima T. Non-Unique Solutions in Missile Interception Guidance[J]. Journal of Guidance, Control, and Dynamics, 1998.
Zarch P. Guided Weapons[M]. Elsevier, 2013.