Comprehensive closed-form solution for the reachable workspace of 2-RPR planar parallel mechanisms

In the design of both planar and spatial parallel mechanisms, structural parameter selection for a desired workspace generally employs the use of numerical methods. However, using a closed-form solution facilitates workspace-based design and workspace optimization. This paper presents a general and comprehensive closed-form solution for the reachable workspace of 2-RPR planar parallel mechanisms. The workspace of the mechanism is analyzed in detail and its boundary is analytically derived to provide a closed-form solution for the reachable workspace. A discussion of the methodology is provided which highlights metrics for rapid determination of desired workspace parameters based on given structural parameters. An example is provided to validate the presented closed-form solution against a numerical method solution. The example is extended to demonstrate the use of the closed-form solution for workspace-based design and workspace optimization. The presented closed-form solution can be used to derive the closed form solution to the workspace of spatial multi-legged parallel mechanisms such as axially symmetric hexapod robots.