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Fully Distributed Flocking with a Moving Leader for Lagrange Networks with Parametric Uncertainties
Sheida Ghapani, Jie Mei, Wei Ren, Yongduan Song
TL;DR
The paper addresses leader-follower flocking for uncertain networked Lagrange systems when followers receive only local information under a proximity graph. It develops adaptive distributed controllers and estimators for constant- and varying-velocity leaders, including a fully distributed gain-adaptive extension. The proposed algorithms achieve connectivity maintenance, collision avoidance, and velocity matching under stated initial conditions and gain choices, with simulations illustrating the results.
Problem
Linear-agent flocking algorithms do not directly apply to nonlinear Lagrange systems with parametric uncertainties, especially when leader information is available only to nearby followers.
Method
The paper proposes distributed adaptive controllers and estimators for constant- and varying-velocity leaders, extending the latter with gain adaptation for fully distributed operation.
Results
The proposed algorithms achieve connectivity maintenance, collision avoidance, and velocity matching with a moving leader using only one-hop neighbor information, subject to initial conditions and properly designed gains.
Takeaways & Limitations
Flocking can be achieved for uncertain networked Lagrange systems under proximity-based local interaction without requiring every follower to access the leader directly.
Takeaways & Limitations
The varying-velocity algorithm uses a signum function, whose discontinuity may cause chattering; the continuous algorithm is preferred for constant-velocity leaders.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper motivates distributed flocking for nonlinear Lagrange systems, where uncertain dynamics make linear-model algorithms inapplicable. It proposes local, adaptive leader-follower control under proximity-based information constraints for constant- and varying-velocity leaders.
- Multi-agent systems coordinate global tasks cooperatively through distributed interactions using only local information from neighbors.
- Prior flocking studies largely address linear single- or double-integrator agents, whereas many practical systems have nonlinear Lagrange dynamics.Examples include autonomous vehicles, walking robots, and spacecraft formations.
- Existing leader-follower results either require the leader to neighbor every follower under varying velocity or do not address uncertain Lagrange dynamics.The paper identifies the all-followers leader-information requirement as unrealistic for large distributed networks.
- The paper studies moving-leader flocking with unknown parameters under a proximity graph, covering constant and varying leader velocities.It combines adaptive control and distributed estimation, using continuous algorithms for constant velocity and discontinuous algorithms with gain adaptation for varying velocity.
- The proposed algorithms use one-hop neighbor information and target connectivity maintenance, collision avoidance, and eventual velocity matching.The guarantees require initial connectivity and collision avoidance, together with properly designed control gains.
2 Background
The background models followers as uncertain Euler-Lagrange systems and represents local interactions with graph-theoretic and proximity-based structures. The leader connects to only nearby followers, while topology matrices encode leader-follower reachability.
- 2.1 Lagrange Dynamics: Each follower is modeled by Euler-Lagrange dynamics with generalized coordinates, inertia, Coriolis and centrifugal forces, gravity, and control input.The model is Mi(qi)¨qi + Ci(qi, ˙qi) ˙qi + gi(qi) = ui.
- 2.1 Lagrange Dynamics: The dynamics assume bounded inertia, Coriolis and centrifugal terms, and gravity, with a skew-symmetric ˙Mi(qi) − 2Ci(qi, ˙qi) property.
- 2.1 Lagrange Dynamics: The Lagrange dynamics are linearly parameterized by a regression matrix and an unknown but constant parameter vector.
- 2.2 Graph Theory: Directed edges specify which node can provide information, while adjacency and Laplacian matrices encode graph connectivity and interaction structure.The leader-follower topology matrix is H = LF + Λ; it is positive definite when the leader has directed paths to all followers.
- 2.2 Graph Theory: A proximity graph makes followers i and j neighbors when ||qi − qj|| < R, and makes the leader a neighbor of follower i when ||qi − q0|| < R.The leader has no neighbors and its motion is not necessarily dependent on the followers.
3 Main Results
The paper develops distributed adaptive flocking controllers for uncertain networked Lagrange systems with constant or varying leader velocity. The results preserve initial connectivity, avoid collisions, and achieve asymptotic velocity matching using local information.
- Problem formulation: The flocking objective combines connectivity maintenance, collision avoidance, and asymptotic velocity matching with a moving leader under unknown parameters.Followers interact through a proximity graph and aim to achieve ||˙q_i(t) − ˙q_0(t)|| → 0.
- Constant-velocity leader: For a constant-velocity leader, the proposed adaptive controller is continuous and uses local positions, velocities, and neighbor-relative measurements.The controller includes potential-function terms for collision avoidance and connectivity, velocity-matching terms, and parameter adaptation.
- Constant-velocity leader: Initial directed paths from the leader to all followers and collision-free initialization suffice for leader-follower flocking under the constant-velocity algorithm.The Lyapunov analysis establishes boundedness, preserves the initial connectivity pattern, prevents collisions, and yields asymptotic velocity matching.
- Varying-velocity leader: For a varying-velocity leader, a discontinuous adaptive controller addresses tracking when the leader is directly connected to only a subset of followers.The resulting algorithm achieves flocking with unknown parameters under suitable gain conditions and the same initial connectivity and collision assumptions.
- Controller trade-offs: The signum-based varying-velocity controller has discontinuous closed-loop dynamics and may cause chattering, while replacing signum with tanh is reported to retain satisfactory performance.For a constant-velocity leader, the continuous controller is more favorable because the discontinuous alternative may cause chattering.
- Fully distributed extension: The fully distributed extension introduces adaptive gain schemes and achieves flocking without requiring global information.The theorem assumes initial directed paths to all followers and no initial collisions.
4 Simulation
Simulations model four spacecraft with uncertain masses and test constant- and varying-velocity leader cases. The results show cohesive, collision-free flocking with velocity matching and preserved or expanded connectivity.
- Simulation setup: The spacecraft are modeled in chief-fixed LVLH coordinates as Lagrange systems with unknown mass parameters θi = mi.The relative translation dynamics have the form of the paper’s general Lagrange model.
- Simulation setup: The simulations use four spacecraft with masses mi = 30 + 5i kg, interaction range R = 200 m, and initial velocities equal to zero.The leader and spacecraft initial positions are specified, and the potential minima are set to 80 m.
- Constant-velocity leader: In the constant-velocity case, followers move cohesively without collisions, converge to the leader’s velocity, and add two graph edges without losing any.The leader velocity is [0.1, 0.1, 0.2]T, with γ = 0.04 and Γi = 5I3.
- Varying-velocity leader: In the varying-velocity case, agents maintain initial connectivity, avoid collisions, and eventually match the leader’s velocity.The experiment uses control algorithm (17)-(20), α = 0.04, Γi = 5I3, and tanh(1000·) in place of sgn(·).
- Fully distributed control: The fully distributed varying-velocity simulation achieves leader-following flocking with no edge added or lost.This case uses control algorithm (24)-(31), Γi = 5I3, and γ1i = γ2i = 0.003.
5 CONCLUSIONS
The paper studies leader-follower flocking for uncertain Lagrange systems under proximity-based local interactions, covering constant- and varying-velocity leaders. Its algorithms achieve connectivity maintenance, collision avoidance, and velocity matching using one-hop information.
- The study considers Lagrange agents with unknown but constant parameters, a moving leader, and follower interactions defined by a proximity graph.
- It treats both constant-velocity and varying-velocity leaders, with the leader directly neighboring only a group of followers.
- In the third simulation case, leader-following flocking is achieved under the fully distributed control algorithm without any edge being added or lost.
- All proposed control algorithms require only one-hop neighbors’ information while achieving connectivity maintenance, collision avoidance, and velocity matching with the moving leader.