Outdoor autonomous robots carrying sensors have the potential to provide transformative data for science, industry, and safety. However, limited battery life is a factor that restricts their wide deployment. To fully realize the potential of these robots, this restriction must be loosened. We improve battery life by designing algorithms that allow outdoor sensors to intelligently plan their movements to minimize power consumption while taking the necessary measurements for their tasks. This dissertation focuses on three main problems: The first problem we address is the problem of planning energy-minimal trajectories for solar-powered robots in a cluttered environment that contains shadows. We make contributions to this problem in both planning and estimation. We present a novel planning algorithm that efficiently computes an optimal trajectory on a discretized grid, given a solar map and a target position. We also present three methods of estimating the solar map that relates position and time with the amount of solar energy that can be collected if the robot is in that position at that time, using only previous measurements of solar power tagged with positions and times. The first method uses Gaussian Process regression to directly predict the solar power at a position based on neighboring measurements. The second method uses spatiotemporal Gaussian Process classification to estimate the likelihood that a particular position will be in the sun. The third method uses the measurements of sun and shade to infer shadow-casting objects, and performs raytracing through the uncertain height map to estimate the likely solar map for any given sun angle. These methods are justified through field experiments and in simulation. The second problem we address is the problem of exploring unknown environments in a manner that guarantees bounded motion cost. We consider two variants of this problem. First we consider a boat, equipped with a sonar sensor, that encounters an obstacle while it is attempting to reach a destination. Second we consider a solar-powered vehicle attempting to map the shadows in an environment. For both variants, we present online algorithms and examine their performance using competitive analysis. In competitive analysis, the performance of an online algorithm is compared against the optimal offline algorithm. For obstacle avoidance, the offline algorithm knows the shape of the obstacle. For solar exploration, the offline algorithm knows the geometry of the shadow casting objects. We obtain an O(1) competitive ratio for obstacle avoidance, and an O(log n) competitive ratio for solar exploration, where n is the number of critical points to observe. The strategies for obstacle avoidance are validated through extensive field experiments, and the strategies for exploration are validated with simulations. The third problem we address is a novel coverage problem that arises in aerial surveying applications. The goal is to compute a shortest path that visits a given set of cones--this is the path that maximizes the coverage ability for a UAV with a limited battery size. The apex of each cone is restricted to lie on the ground plane. The common angle alpha of the cones represent the field of view of the onboard camera. The cone heights, which can be varying, correspond with the desired observation quality (e.g. resolution). This problem is a novel variant of the Traveling Salesman Problem with Neighborhoods (TSPN), and we call it Cone-TSPN. Our main contribution is a polynomial time approximation algorithm for Cone-TPSN. We analyze its theoretical performance and show that it returns a solution whose length is at most O(1 + log(hmax/hmin)) times the length of the optimal solution where hmax and hmin are the heights of the tallest and shortest input cones, respectively. We demonstrate the use of our algorithm in a representative precision agriculture application. We further study its performance in simulation using randomly generated cone sets. Our results indicate that the performance of our algorithm is superior to standard solutions. The results in this dissertation have advanced the state of the art in planning energy-minimizing trajectories for outdoor vehicles, by presenting algorithms with strong theoretical guarantees, justified in field experiments and simulations.