  This paper addresses the leader-follower flocking problem with a moving
leader for networked Lagrange systems with parametric uncertainties under a
proximity graph. Here a group of followers move cohesively with the moving
leader to maintain connectivity and avoid collisions for all time and also
eventually achieve velocity matching. In the proximity graph, the neighbor
relationship is defined according to the relative distance between each pair of
agents. Each follower is able to obtain information from only the neighbors in
its proximity, involving only local interaction. We consider two cases: i) the
leader moves with a constant velocity, and ii) the leader moves with a varying
velocity. In the first case, a distributed continuous adaptive control
algorithm accounting for unknown parameters is proposed in combination with a
distributed continuous estimator for each follower. In the second case, a
distributed discontinuous adaptive control algorithm and estimator are
proposed. Then the algorithm is extended to be fully distributed with the
introduction of gain adaptation laws. In all proposed algorithms, only one-hop
neighbors' information (e.g., the relative position and velocity measurements
between the neighbors and the absolute position and velocity measurements) is
required, and flocking is achieved as long as the connectivity and collision
avoidance are ensured at the initial time and the control gains are designed
properly. Numerical simulations are presented to illustrate the theoretical
results.
