Uncertainties are an inherent element of the manufacturing and control of mechanisms. Generally, uncertainties are neglected in the analysis and synthesis of mechanisms, as they can be difficult to account for. However, the performance of real mechanisms depends on the uncertainties;​ thus, there is a need for reliable methods which are able to provide bounds on the true performance of mechanisms.

Parallel mechanisms are considered in this work, where the design of a parallel mechanism accounting for uncertainties is termed an appropriate design. Reliable methods, based on interval analysis, are developed to analyze and synthesize appropriately designed parallel mechanisms, fittingly termed appropriate analysis and appropriate synthesis respectively. These appropriate methods are applied to the four-bar linkage and the 3-R̲RR planar parallel mechanism. Appropriate analysis methods compute the associated workspaces of mechanisms with an appropriate design. These workspaces consider the reachability, self-collisions, singularities, and wrench-capabilities of the mechanisms, and are used to guarantee whether the desired task is able to be accomplished by the mechanism. Appropriate synthesis methods are able to completely explore a continuous design parameter space to determine the complete set of appropriate designs of a mechanism which accomplish the desired task. Any design selected from this set is guaranteed to accomplish the desired task. The development of appropriate analysis and appropriate synthesis introduces a powerful new tool to mechanisms designers.