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A necessary attribute of a mobile robot planning algorithm is the ability to accurately predict the consequences of robot actions to make informed decisions about where and how to drive. It is also important that such methods are efficient, as onboard computational resources are typically limited and fast planning rates are often required. In this article, we present several practical mobile robot motion planning algorithms for local and global search, developed with a common underlying trajectory generation framework for use in model-predictive control. These techniques all center on the idea of generating informed, feasible graphs at scales and resolutions that respect computational and temporal constraints of the application. Connectivity in these graphs is provided by a trajectory generator that searches in a parameterized space of robot inputs subject to an arbitrary predictive motion model. Local search graphs connect the currently observed state-to-states at or near the planning or perception horizon. Global search graphs repeatedly expand a precomputed trajectory library in a uniformly distributed state lattice to form a recombinant search space that respects differential constraints. In this article, we discuss the trajectory generation algorithm, methods for online or offline calibration of predictive motion models, sampling strategies for local search graphs that exploit global guidance and environmental information for real-time obstacle avoidance and navigation, and methods for efficient design of global search graphs with attention to optimality, feasibility, and computational complexity of heuristic search. The model-invariant nature of our approach to local and global motions planning has enabled a rapid and successful application of these techniques to a variety of platforms. Throughout the article, we also review experiments performed on planetary rovers, field robots, mobile manipulators, and autonomous automobiles and discuss future directions o- the article.