Reliability of cumulative navigation error propagation in multi-stage underwater delivery missions

In multi-stage underwater delivery missions, a mother platform, typically a large unmanned underwater vehicle (UUV) equipped with a high-precision inertial navigation system (INS), releases child platforms, such as medium or micro UUVs carrying low-cost inertial sensors, at preplanned areas to extend mission coverage. However, the estimation error of the mother's INS is inherited by the released child as its initial position error, so that navigation errors accumulate and propagate across stages and directly constrain the end-to-end mission success probability in GNSS-denied underwater environments. To address this issue, this paper proposes a reliability modeling method for cumulative navigation error propagation. An error transfer coefficient is defined to quantify how a child platform inherits the mother's position uncertainty at release; based on it, an inter-stage error transfer function and a closed-form expression for the cumulative terminal error of N-stage delivery are established, and closed-form expressions for the mission success probability of multi-stage delivery are further derived by unifying the navigation error physics of variance accumulation with the probability model of system reliability. A Monte Carlo simulation framework is constructed to verify the analytical model. Theoretical analysis reveals a leverage effect of the mother stage's INS accuracy on end-to-end reliability: given equal resources, improving the mother's accuracy yields significantly higher reliability gains than improving the child's. Sensitivity coefficients further quantify the influence of each stage's parameters, showing that the mother's operating time and error growth rate dominate the terminal reliability. Moreover, there exists a critical value of the error transfer coefficient; when the coefficient falls below this critical value (0.76 in a typical case), the end-to-end reliability converges as the number of stages increases, so delivery stages can be extended almost without reliability loss, whereas above the critical value reliability declines persistently, with an eight-stage delivery losing about 30% reliability under uncorrected conditions. Numerical simulations show that the analytical results agree well with Monte Carlo simulations, with a mean absolute error below 5% in all scenarios: a two-stage delivery achieves a reliability of 0.723; adding a micro third stage lowers it to 0.405 because the micro platform contributes 76.0% of the terminal error; and reducing the transfer coefficient from 1.0 to 0.5 through inter-stage correction raises the reliability from 0.723 to 0.933. Compared with traditional series reliability models that assume independence across stages, the proposed model explicitly characterizes the cross-stage error coupling and avoids overestimating the reliability by about 3.7%. The model provides a quantitative theoretical basis for stage number design, platform selection, inter-stage correction design, and mission reliability assessment of multi-stage underwater delivery systems.