Discrete motion systems move relative to a lattice, using bistable mechanisms to snap between lattice locations. A discrete motion system includes a lattice having a regular configuration of attachment points, one or more motion modules that move across the lattice in discrete increments, and controllers that direct the modules. A module includes a body, actuators, and feet having mechanisms for attaching and detaching the module from the lattice. The module may include actuated joints that cause movement of arm structures to engage and disengage the feet from the lattice. The module may be a digital inchworm, and may be a relative assembler having at least one assembler arm. A method for discrete extensible construction includes creating a lattice having a regular configuration of attachment points, causing a discrete motion relative assembler to systematically move across the lattice in discrete increments, and causing placement of materials by the assembler arm.