We present an optimization-based design process that can generate the best insertion-based joints with respect to different errors, including manipulation error, manufacturing error, and sensing error. We separate the analysis into two stages, the insertion and the after-insertion stability. Each sub-process is discretized into different modes of contacts. The transitions among the contact modes form a directed graph and the connectivity of the graph is achieved and maintained through the manipulation of the socket edge-angle and peg contact-point locations. The analysis starts in 2D with the assumption of point-edge contacts. During the optimization, the edges of the socket are rotated and the points on the peg are moved along the edges to ensure the successful insertion and the stability after insertion. We show in simulation that our proposed method can generate insertion-based joints that are tolerant to the given errors. and we present a few simple 3D projections to show that the analysis is still effective beyond 2D cases.