The transition from one equilibrium state to another via rapid snap-through can store elastic energy and release it as kinetic energy for rapid motion as seen in Venus flytrap and hummingbird to catch insects mid-flight. They are explored in soft robotics for repeated and autonomous motions. In this study, we synthesize curved liquid crystal elastomer (LCE) fibers as the building blocks that can undergo buckling instability upon heated on a hot surface, leading to autonomous snap-through and rolling behaviors. When they are connected into lobed loops, where each fiber is geometrically constrained by the neighboring ones, they demonstrate autonomous, self-regulated, and repeated synchronization with a frequency of ~1.8 Hz. By adding a rigid bead on the fiber, we can fine-tune the actuation direction and speed (up to ~2.4 mm/s). Last, we demonstrate various gait-like locomotion patterns using the loops as the robot's legs.