### Analytical Prediction Error in Controlled Lagrangian Particle Tracking

For several decades, autonomous underwater vehicles (AUVs), an instrument platform has been employed in oceanographic research. The small, mobile, and low-cost AUVs allow oceanographers to collect useful environmental data to characterize and model various oceanic environments such as open and coastal seas. However, ocean spatial and temporal variability significantly influence the paths of the vehicles because of their low velocities. Predicting the real trajectory of the vehicles becomes a challenging domain of research. To predict the route of the vehicles, many people have been using various ocean models that represent highly nonlinear time-varying dynamics of an ocean flow field. However, they have a variety of uncertainties: space and time sparseness of observational data sets and missing physics such as flow interaction. One approach to predicting the real trajectories of the vehicles with controlled velocity input in ocean flows is controlled Lagrangian particle tracking (CLPT) [1]. This concept includes the controlled input of the vehicles within the framework of Lagrangian particle tracking (LPT), an approach to finding the position of particles freely advected in ocean flows. To predict the error growth of the vehicles in a real flow field, the authors in [1] have defined CLPT error as position error between model-based simulated trajectory and real experimental trajectory of the vehicles. They performed an analysis of CLPT error based on probability theory, which assumes that the ocean field has spatial variability. In addition, they were able to predict the constant biased ocean flow field relatively accurately using linearized CLPT error dynamics [2]. Our works extend the previous results of [2], which assume that the ocean field has constant flow. We focus on predicting the real trajectory of the vehicles with waypoint controller, which guides a vehicle approach the waypoint by canceling out the normal component of a flow with respect to lines connected between the vehicle and the waypoint. To handle uncertain time-varying nonlinear systems that represent the vehicles on the uncertain oceanic environment, we use polar coordinate system and perturbed system theory. Unlike the previous research, our nominal and perturbed systems based on perturbed system theory are the key framework to have analytical solutions of the systems. In addition, range and angle channel of the systems provide us with physical insight of the real trajectory. To obtain the analytical solution of each channel, we develop three-step process to deal with two different perturbation terms. The first step is that we analytically solve the equation of angle channel by perturbation method when the perturbation term of the range channel equals zero. The second step has the same as the first step except that the perturbation term of the angel channel equals zero instead of the range channel. Final step is that we use superposition method to obtain the analytic solution of full perturbed systems from results of the first and second steps. Figure 1 and 2 show successful results obtained using analytical solutions in CLPT described here. (see summary)

#### References

[1] K. Szwaykowska and F. Zhang, “Trend and bounds for error growth in controlled lagrangian particle tracking,” IEEE Journal of Ocean Modelling, vol. 39, pp. 10–25, 2014.

[2] K. Szwaykowska and F. Zhang, “Controlled lagrangian particle tracking error under biased flow prediction,” in 2013 American Control Conference, Washington, DC, USA, 2013, pp. 92–98.