When operating in human environments, a robot should use predictable motions that allow humans to trust and anticipate its behavior. Heuristic search-based planning offers predictable motions and guarantees on completeness and sub-optimality of solutions. While search-based planning on motion primitive-based (lattice-based) graphs has been used extensively in navigation, application to high-dimensional state-spaces has, until recently, been thought impractical. This dissertation presents methods we have developed for applying these graphs to mobile manipulation, specifically for systems which contain closed chains. The formation of closed chains in tasks that involve contacts with the environment may reduce the number of available degrees-of-freedom but adds complexity in terms of constraints in the high-dimensional state-space. We exploit the dimensionality reduction inherent in closed kinematic chains to get efficient search-based planning. Our planner handles changing topologies (switching between open and closed-chains) in a single plan, including what transitions to include and when to include them. Thus, we can leverage existing results for search-based planning for open chains, combining open and closed chain manipulation planning into one framework. Proofs regarding the framework are introduced for the application to graph-search and its theoretical guarantees of optimality. The dimensionality-reduction is done in a manner that enables finding optimal solutions to low-dimensional problems which map to correspondingly optimal full-dimensional solutions. We apply this framework to planning for opening and navigating through non-spring and spring-loaded doors using a Willow Garage PR2. The framework motivates our approaches to the Atlas humanoid robot from Boston Dynamics for both stationary manipulation and quasi-static walking, as a closed chain is formed when both feet are on the ground.