This article addresses the rigidity recovery problem of a formation controlled over an undirected sensing framework after a link is broken. In the formation control problem, mobile agents are controlled individually such that all interagent distances remain unchanged. The control law used by each agent is designed based on the relative positions of that agent to its neighbors, sensed in its local coordinate system. When the sensing graph is rigid, it has been shown that one can design a distributed control law to guarantee the stability of the formation. However, obtaining sensing measurements is a challenge in the formation control, since these systems are always subjected to sensing constraints, such as line-of-sight requirements and power limitations. This clearly affects the rigidity of the sensing graph, thus affecting the formation. This article proposes an online distributed algorithm based on the lattice of configurations to recover the distance-based controlled formation when a failure in a sensing link causes the network to lose its rigidity. The approach is to recover the rigidity of subframeworks of the formation, i.e., the subframework established by each node and its neighbors within the sensing range, by adding new sensing links locally so as to ensure a rigid formation. This article also proposes a multilayer rigidity recovery technique using a combination of sensing and communication networks, when the lattice of configurations fails due to a lack of neighbors for an agent. An upper bound is established on the delay in this indirect distance-measuring approach, which guarantees the tolerance of the formation against link failures. Simulations on a sample formation support the theoretical results.