In this paper, we present a mathematical approach to synthesize multistable compliant mechanisms by combining multiple bistable equilibrium mechanisms. More specifically, we identify and categorize various types of bistabilities by characterizing the essential elements of their complicated deformation pattern. The behavior of a bistable compliant mechanism, in general, is highly nonlinear. Using combinations of such nonlinearities to capture the behavior of multistable (more than two stable positions) mechanisms can be quite challenging. To determine multistable behavior, our simplified mathematical scheme captures the essential parameters of bistability, such as the load-thresholds that cause the jump to the next stable position. This mathematical simplification enables us to characterize bistable mechanisms by using piecewise lower-order polynomials and, in turn, synthesize multistable mechanisms. Three case studies involving combinations of two, three, and four bistable behaviors are presented for the purpose of generating multistable mechanisms with up to 16 stable positions. The methodology enables us to design a compliant mechanism with a desired number of stable positions. A design example of a quadristable equilibrium rotational compliant mechanism consisting of two bistable submechanisms is presented to demonstrate the effectiveness of the approach.

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