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The use of random sampling in decision-making and control has become popularwith the ease of access to graphic processing units that can generate andcalculate multiple random trajectories for real-time robotic applications. Incontrast to sequential optimization, the sampling-based method can takeadvantage of parallel computing to maintain constant control loop frequencies.Inspired by its wide applicability in robotic applications, we calculate asampling complexity result applicable to general nonlinear systems consideredin the path integral method, which is a sampling-based method. The resultdetermines the required number of samples to satisfy the given error bounds ofthe estimated control signal from the optimal value with the predefined riskprobability. The sampling complexity result shows that the variance of theestimated control value is upper-bounded in terms of the expectation of thecost. Then we apply the result to a linear time-varying dynamical system withquadratic cost and an indicator function cost to avoid constraint sets.

Sampling Complexity of Path Integral Methods for Trajectory Optimization