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We present designs, theory, and results of fabrication and testing for a novel parallel microrobotic assembly scheme using stress-engineered MEMS microrobots. The robots are 240-280 mum times 60 mum times 7-20 mum in size and can be controlled to dock compliantly together, forming planar structures several times this size. The devices are classified into species based on the design of their steering arm actuators, and the species are further classified as independent if they can be maneuvered independently using a single global control signal. In this paper, we show that microrobot species are independent if the two transition voltages of their steering arms, i.e., the voltages at which the arms are raised or lowered, form a unique pair. We present control algorithms that can be applied to groups of independent microrobot species to direct their motion from arbitrary nondead-lock configurations to desired planar microassemblies. We present designs and fabrication for four independent microrobot species, each with a unique transition voltage. The fabricated microrobots are used to demonstrate directed assembly of five types of planar structures from two classes of initial conditions. We demonstrate an average docking accuracy of 5 mum and use self-aligning compliant interaction between the microrobots to further align and stabilize the intermediate assemblies. The final assemblies match their target shapes on average 96%, by area.