We present an incremental configuration space (CS) construction algorithm for mechanisms described as collections of subassemblies of rigid parts. The inputs are the initial subassembly configurations and the subassembly CSs partitioned into uniform motion regions in which part contacts are constant and motions are monotonic. The output is a partition of the mechanism CS into uniform motion regions. The algorithm optimizes CS construction by incrementally enumerating and testing only the regions reachable from the initial configuration. We implement the algorithm for subassemblies whose uniform motion regions are polyhedral or are of dimension two or lower. The program constructs the exact CS when possible and an approximate CS otherwise. The approximate CS usually is qualitatively correct and in good quantitative agreement with the true CS. The program covers most mechanisms composed of linkages and fixed-axes kinematic pairs, two subassembly types for which CS construction programs are available.