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This paper lays a theoretical foundation for fault detection and tolerance in static walking of legged robots. Legged robots considered in this paper have symmetric structures and legs which have the form of an articulated arm with three revolute joints. A kind of fault event (locked joint failure) is defined, and its properties are closely investigated in the frame of gait study and robot kinematics. For the purpose of tolerating a locked joint failure, an algorithm of fault-tolerant gaits for a quadruped robot is proposed in which the robot can continue its walking after a locked failure occurs to a joint of a leg. In particular, a periodic gait is proposed as a special form of the proposed algorithm and its existence and efficiency are analytically proven. A case study on applying the proposed scheme to wave gaits verifies its applicability and capability.