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This paper addresses the leader-follower flocking problem with a movingleader for networked Lagrange systems with parametric uncertainties under aproximity graph. Here a group of followers move cohesively with the movingleader to maintain connectivity and avoid collisions for all time and alsoeventually achieve velocity matching. In the proximity graph, the neighborrelationship is defined according to the relative distance between each pair ofagents. Each follower is able to obtain information from only the neighbors inits proximity, involving only local interaction. We consider two cases: i) theleader moves with a constant velocity, and ii) the leader moves with a varyingvelocity. In the first case, a distributed continuous adaptive controlalgorithm accounting for unknown parameters is proposed in combination with adistributed continuous estimator for each follower. In the second case, adistributed discontinuous adaptive control algorithm and estimator areproposed. Then the algorithm is extended to be fully distributed with theintroduction of gain adaptation laws. In all proposed algorithms, only one-hopneighbors' information (e.g., the relative position and velocity measurementsbetween the neighbors and the absolute position and velocity measurements) isrequired, and flocking is achieved as long as the connectivity and collisionavoidance are ensured at the initial time and the control gains are designedproperly. Numerical simulations are presented to illustrate the theoreticalresults.

Fully Distributed Flocking with a Moving Leader for Lagrange Networks with Parametric Uncertainties