This paper proposes a method to navigate a mobile robot by estimating its state over a number of distributed sensor networks (DSNs) such that it can successively accomplish a sequence of tasks, i.e., its state enters each targeted set and stays inside no less than the desired time, under a resource-aware, time-efficient, and computation-and communication-constrained setting. We propose a new robot state estimation and navigation architecture, which integrates an event-triggered task-switching feedback controller for the robot and a two-time-scale distributed state estimator for each sensor. With the controller, the robot is able to accomplish a task by following a reference trajectory and switch to the next task when an event-triggered condition is fulfilled. With the estimator, each active sensor is able to estimate the robot state. We provide conditions to ensure that the state estimation error and the trajectory tracking deviation are upper bounded by two time-varying sequences, respectively. Furthermore, we find a sufficient condition for accomplishing a task and provide an upper bound of running time for the task. Numerical simulations of an indoor robot's localization and navigation are provided to validate the proposed architecture.