The research consists of two separate projects, in discrete robotic system design and discrete motion planning. First, the kinematics, motion planning, and design of robotic mechanisms with binary actuators are investigated. Robots with binary (on-off) actuators have a finite number of states. Major benefits of binary-actuated manipulators are that they can be operated without extensive feedback control, task repeatability is very high, and two-state actuators are generally very inexpensive, thus resulting in reliable low cost robots for industrial use. Second, the design and coordination of`metamorphic' robotic systems for use in unknown environments are investigated. A metamorphic robot is a collection of independent modules each of which occupies a space in a lattice, and has the ability to locomote over adjacent modules. The morphology of the collection of modules changes as a function of the collective motion. Such self-reconfigurability gives metamorphic robots unique abilities. For instance, modules can `flow' into crevasses unreachable by fixed morphology robots, or grow to form structures which envelop objects and form stable grasps.