### Environmental Uncertainty

#### Controlled Lagrangian Particle Tracking

The goal of our research in Controlled Lagrangian Particle Tracking (CLPT) is the understand the motion of mobile robotic agents in complex ocean flow fields. Our focus is on using ocean models to inform navigation decisions for these autonomous agents, and using in-situ flow measurement data to update ocean model flow predictions. To compare the real-life performance of mobile underwater agents with simulations, we define the CLPT error as the difference in position between a mobile agent in the ocean, and a simulated agent moving in a flow field derived from an ocean flow model. The CLPT error limits the performance of ocean model-based control laws and can be implemented on the agent. Our investigation of the CLPT error is based on data collected in the 2006 Adaptive Sampling and Prediction (ASAP) experiment which was performed with underwater gliders in Monterey Bay, CA. The gliders were deployed to take measurements of ocean states along preset trajectories. A simultaneous virtual experiment was run, in which simulated gliders followed the same trajectories under a flow field derived from an ocean model (ROMS, NCOM, or HOPS). We observe that the CLPT error in this experiment exhibits similar growth for each ocean model used; the error increases exponentially until it reaches approximately twice the grid size used in the ocean model, and linearly thereafter. We have derived a lower bound on the steady-state CLPT error for an autonomous agent moving in a two-dimensional flow field. This lower bound is reached in finite time, and accounts for the change in the error growth rate. A dynamic programming approach has been implemented to generate optimal paths for an underwater glider to maintain minimal distance from a set goal point under the influence of flow. This approach uses a cost function that integrates the glider's distance from the goal over a finite time horizon. The domain of operation is discretized in both space and time, and the cost-to-go and associated optimal control action are computed at each point in the discretized domain, starting at the final time (see Figure 2). The glider's position is then integrated forward using a simple particle model for the glider dynamics. At each time step in the integration, the glider's control action (e.g. the choice of heading angle) is taken to be a bilinear interpolation of the optimal control actions at the nearest states in the discretized domain. The glider's total velocity is taken as a sum of the glider's through-water velocity and the predicted flow velocity, which is obtained from an ocean model. This gives a near-optimal trajectory that can be converted to a waypoint list to be passed to the glider. The dynamic programming path-planning algorithm has been tested in the simulation module of GCCS, which uses ocean-model flow data and an approximated model of glider dynamics, as well as an implementation of the glider's on-board control algorithms, to simulate glider motion given waypoint lists passed to the glider. We use controlled Lagrangian particle tracking to evaluate the accuracy of the simulated glider position. Errors in glider position simulation are due to limited resolution of ocean models, missing physics in the models, and sparseness of available ocean measurements used to drive the model. Using a modified Langevin equation to model the growth of the expected glider position error (termed CLPT error), we have shown that the magnitude of the expected error in simulated position grows exponentially until reaching a lower bound equal to twice the gridsize of the ocean model used (see Figure 4). The error growth then slows to a polynomial function of time. (see summary)