Application of quasi-Monte Carlo in Mine Countermeasure Simulations with a Stochastic Optimal Control Framework

TL;DR

The paper tackles CPP-like mine countermeasure missions by formulating a stochastic optimal control problem where residual detection risk is an expected-value integral $\mathbb{E}[q(T_F)]$. It introduces an adaptive relaxation strategy that enlarges the search domain until the residual risk constraint is satisfied and evaluates the integral using MC and quasi-Monte Carlo methods, notably showing a significant speedup with Rank-1 Lattice qMC. It extends the framework to convex quadrilateral domains by decomposing into triangles and applying triangular qMC points. The results demonstrate reliable risk satisfaction and up to a 2× reduction in computation time with qMC, supporting practical deployments for autonomous MCM search planning and enabling more flexible domain shapes.

Abstract

Modelling and simulating mine countermeasures search missions performed by autonomous vehicles equipped with a sensor capable of detecting mines at sea is a challenging endeavour. The output of our stochastic optimal control implementation consists of an optimal trajectory in a square domain for the autonomous vehicle such that the total mission time is minimized for a given residual risk of not detecting sea mines. We model this risk as an expected value integral. We found that upon completion of the simulation, the user requested residual risk is usually not satisfied. We solved this by implementing a relaxation strategy which consists of incrementally increasing the square search domain. We then combined this strategy with different quasi-Monte Carlo schemes used for solving the integral. We found that using a Rank-1 Lattice scheme yields a speedup up to a factor two with respect to the Monte Carlo scheme. We also present an implementation which allows us to compute a trajectory in a convex quadrilateral domain, as opposed to a square domain, and combine it with our relaxation strategy.

Figures (12)

• Figure 1: A typical solution produced by our stochastic optimal control implementation of the MCM problem when considering an autonomous vehicle in the $\left[5,25\right]^2$ domain with a requested residual risk of maximally $10\,\%$. The black dotted line represent the trajectory, the red area is the part of the domain that has been 'seen' by the onboard sensor, and the blue area is the part that has not been 'seen' by the sensor.
• Figure 2: Boxplot and individual results of the residual MCM risk in % of 1000 simulations. The green line indicates the (maximal allowable) 5% user requested residual MCM risk which all individual simulations should satisfy.
• Figure 3: Using qMC points to compute a trajectory in a quadrilateral convex domain: (Left) Division of the convex quadrilateral domain into two triangles with the qMC points generated on each triangle in orange. (Center) The partitioned domain with the generated qMC points and the computed trajectory in blue. (Right) The domain delimited by the black lines, the ensonification of the domain in red and the trajectory represented by the black lines.
• Figure 4: The boxplots and individual results for the residual risk of 1000 independent simulations for different qMC and MC point sets. The green line at $5\,\%$ is the maximal user requested tolerance that all simulations should satisfy.
• Figure 5: The boxplots and individual results for the computation time of 1000 independent simulations for different qMC and MC point sets.