##### Abstract

This thesis proposes new algorithms for a group of sensing robots to learn a para-
metric 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 temper-
ature 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.