In an unstable configuration, a mechanism loses controllability and gains at least one unwanted degree of freedom (DOF) instantaneously. Based on the concept of direct kinematics singularity, we develop an analytical formulation to express unstable configurations of planar N-link mechanisms with single or multiple loops. The dependence of unstable configurations, on the choice of input links, is illustrated through a 7-link 2-DOF mechanism. The unstable configurations are derived for fifteen input link combinations, which are chosen from two different active inputs in six movable links. To gain mechanism controllability at unstable configurations, actuators at passive joints can be introduced. By changing inputs at the unstable configuration and actively actuating one of the passive joints, we gain controllability. It is shown that for a mechanism with F degrees of freedom and L independent closed-loops, the minimum number of actuators needed to fully control the mechanism is F + L.