The potential for building smart, autonomous microrobots with on-board electronics like microprocessors, memory systems, and sensors has grown immensely with the continued miniaturization of semiconductor electronics. However, many of the standard mechanisms for motion at the microscale such as chemical reactions, external electromagnetic fields, and biological propulsion have no direct route towards integrating with these onboard electronics, limiting their feasibility for autonomous microrobots. In this thesis, we present an electronically-controlled electrokinetic motor that easily integrates with silicon electronics and can be built using only a single step of photolithography. Using on-board silicon photovoltaics to regulate electrical power, motors generate electric fields and produce electrokinetic flows that lead to propulsion. To demonstrate the capabilities of this approach, we build simple microrobots capable of turning, tracing out complex shapes, and swimming with other robots in unison under the influence of well-studied, macroscale control laws. Further, using this microrobot platform, we demonstrate a powerful correspondence between robot motion and geodesics of light in general relativity that is broadly applicable to both microscale and macroscale robots. Using this connection, we demonstrate how standard control problems such as maze traversal, convergence, and dispersion can be drastically simplified for robots that lack onboard computation and memory. In summary, these results have enabled the creation of the first reprogrammable microscopic robot and introduced a new framework for controlling resource-limited robots.