This thesis proposes new algorithms for a group of sensing robots to learn a parametric model for a dynamic spatio-temporal field, then based on the learned model trajectories are planned for sensing robots to best estimate the field. In this thesis we call these two parts learning and monitoring, respectively.;For the learning, we first introduce a parametric model for the spatio-temporal field. We then propose a family of motion strategies that can be used by a group of mobile sensing robots to collect point measurements about the field. Our motion strategies are designed to collect enough information from enough locations at enough different times for the robots to learn the dynamics of the field. In conjunction with these motion strategies, we propose a new learning algorithm based on subspace identification to learn the parameters of the dynamical model. We prove that as the number of data collected by the robots goes to infinity, the parameters learned by our algorithm will converge to the true parameters.;For the monitoring, based on the model learned from the learning part, three new informative trajectory planning algorithms are proposed for the robots to collect the most informative measurements for estimating the field. Kalman filter is used to calculate the estimate, and to compute the error covariance of the estimate. The goal is to find trajectories for sensing robots that minimize a cost metric on the error covariance matrix. We propose three algorithms to deal with this problem. First, we propose a new randomized path planning algorithm called Rapidly-exploring Random Cycles (RRC) and its variant RRC* to find periodic trajectories for the sensing robots that try to minimize the largest eigenvalue of the error covariance matrix over an infinite horizon. The algorithm is proven to find the minimum infinite horizon cost cycle in a graph, which grows by successively adding random points. Secondly, we apply kinodynamic RRT* to plan continuous trajectories to estimate the field. We formulate the evolution of the estimation error covariance matrix as a differential constraint and propose extended state space and task space sampling to fit this problem into classical RRT* setup. Thirdly, Pontryagin's Minimum Principle is used to find a set of necessary conditions that must be satisfied by the optimal trajectory to estimate the field.;We then consider a real physical spatio-temporal field, the surface water temperature in the Caribbean Sea. We first apply the learning algorithm to learn a linear dynamical model for the temperature. Then based on the learned model, RRC and RRC* are used to plan trajectories to estimate the temperature. The estimation performance of RRC and RRC* trajectories significantly outperform the trajectories planned by random search, greedy and receding horizon algorithms.
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