A unique droplet microfluidic technology enabling programmable deterministic control over complex droplet operations is presented. The platform provides software control over user-defined combinations of droplet generation, capture, ejection, sorting, splitting, and merging sequences to enable the design of flexible assays employing nanoliter-scale fluid volumes. The system integrates a computer vision system with an array of membrane displacement traps capable of performing selected unit operations with automated feedback control. Sequences of individual droplet handling steps are defined through a robust Python-based scripting language. Bidirectional flow control within the microfluidic chips is provided using an H-bridge channel topology, allowing droplets to be transported to arbitrary trap locations within the array for increased operational flexibility. By enabling automated software control over all droplet operations, the system significantly expands the potential of droplet microfluidics for diverse biological and biochemical applications by combining the functionality of robotic liquid handling with the advantages of droplet-based fluid manipulation.