Gangfeng Yan

Leader-follower formation via complex Laplacian

The paper introduces complex-valued Laplacians for graphs whose edges are attributed with complex weights and studies the leader-follower formation problem based on complex Laplacians. The main goal is to control the shape of a planar formation of point agents in the plane using simple and linear interaction rules related to complex Laplacians. We present a characterization of complex Laplacians that preserve a specific planar formation as an equilibrium solution for both single integrator kinematics and double integrator dynamics. Planar formations under study are subject to translation, rotation, and scaling in the plane, but can be determined by two co-leaders in leader-follower networks. Furthermore, when a complex Laplacian does not result in an asymptotically stable behavior of the multi-agent system, we show that a stabilizing matrix, which updates the complex weights, exists to asymptotically stabilize the system while preserving the equilibrium formation. Also, algorithms are provided to find stabilizing matrices. Finally, simulations are presented to illustrate our results.

Local control strategy for moving-target-enclosing under dynamically changing network topology

Abstract This paper studies the moving-target-enclosing problem for a group of autonomous mobile robots, which can be seen as the requirement of achieving a formation surrounding a moving target whose movement is not known a priori. A local information control law is proposed to solve the problem utilizing only the relative position information from the target and its neighbors. The derivation of the controller is based on the idea of decoupling the task of target tracking and the task of inter-robot coordination. It is shown that the group of autonomous mobile robots collaboratively estimates the moving velocity of the target and asymptotically reaches a regular polygon formation to keep the moving target as its centroid. The neighbor topology may dynamically change as the system evolves. Hence, a non-smooth version of LaSalle’s invariance principle is used to show the convergence.

Distributed control of cooperative target enclosing based on reachability and invariance analysis

The paper presents a hybrid control approach to the problem of steering a group of unicycle-type mobile robots to reach desired relative positions and orientations with respect to a specific target and other group-mates, which is referred to as the cooperative target enclosing problem. With the idea of having independent motion towards the target without inter-individual interactions in the further range and switching to coordinated motion control in the closer range to the target, reachability and invariance analysis is recalled to yield a hybrid control law using only local available information such that a group of unicycle-type mobile robots achieves a uniform circular motion around the target at equal angular distances from each other.

Brief paper: Collective motions and formations under pursuit strategies on directed acyclic graphs

A novel pursuit-based approach is presented to investigate collective motions and formations of a large number of agents with both single-integrator kinematics and double-integrator dynamics on directed acyclic graphs (DAGs). Each agent pursues its neighbors according to a directed acyclic graph, in which the agents without neighbors are leaders. Based on signal flow graph analysis and Masons rule, necessary and sufficient conditions are derived for BIBO stability of resulting pursuit systems. Moreover, achievable collective motions and formations are analyzed by adjusting a set of control parameters when leaders keep stationary, perform uniform rectilinear motions, and perform uniform circular motions. Finally, simulations are provided for achieving a static formation and mimicking several complex collective behaviors observed in nature, such as V-formation, vortex motions, and tornado motions.

Feedback Control of Planar Biped Robot With Regulable Step Length and Walking Speed

For a compass-like biped robot, the problem of achieving stable walking on both ideal inclined surfaces and complex environments is studied. For the case of walking on ideal inclined surfaces, a feedback-control law is obtained via feedback-linearization techniques so that the trajectory of the robot converges to a desired passive walking gait. Simulations show that it leads to a larger basin of attraction. For the case of walking in complex environments, a scheme is developed with regulable step length and walking speed. Different reference trajectories are constructed for different steps, and corresponding feedback-control laws are updated at the beginning of each step. Then, it is shown that the errors that measure the difference of the response trajectory and the reference one asymptotically converge to zero. An example of walking over stairs is given to numerically verify and demonstrate our approach.

Brief paper: Ring-coupled unicycles: Boundedness, convergence, and control

In cyclic pursuit a platoon of vehicles are coupled in a unidirectional ring at the interaction level according to some control scheme. In the paper, a new cyclic pursuit control law is proposed, where each vehicles linear speed and angular speed are proportional to the projection of its preys position on its forward direction and lateral direction respectively. Through these interactions a cooperative behavior emerges and vehicles in the platoon eventually move at a constant speed on a circle with constant inter-vehicle spacings. The control scheme ensures ultimate boundedness and leads to only two stable equilibrium polygons. This contrasts with other cyclic pursuit control schemes, where vehicles may diverge to infinity and there are more stable equilibrium polygons as the total number of vehicles increases. For this control scheme, ultimate boundedness is proved using the pseudo-linearization technique. Possible equilibrium polygons are analyzed and stability and convergence properties are established through root locus analysis of a complex characteristic polynomial. Design rules are discussed, showing how the radius of the circle they converge to is controlled by an appropriate choice of control parameters.

Synthesis of Distributed Control of Coordinated Path Following Based on Hybrid Approach

The coordinated path following problem, steering a fleet of unicycles along a path while achieving a desired inter-vehicle formation, is investigated. The path is a paved road in the world or a marked curve on the floor of work area, for which each vehicle can measure the distance to the path and the heading error as well as the curvature of the path segment in its sensing range. Moreover, the vehicles can measure the arc distances of their neighbors that lie in their sensing range. A hybrid control approach is proposed to solve the problem. Depending on the posture with respect to the partly observed path, each vehicle either interacts with one of the others or works at the single-agent level. It is shown that the path following error of each vehicle is eventually reduced to zero and vehicles of every weakly connected component asymptotically converge to formations with equal arc distances.

Adaptive leader-follower formation control for autonomous mobile robots

In this paper, adaptive formation control is addressed for a network of autonomous mobile robots in which there are only two leaders knowing the prescribed reference velocity while the others just play the role of followers. Assuming that each follower has only two neighbors to form a cascade interconnection, an adaptive formation control law is designed that allows each follower to achieve a specific triangular formation with its two neighbors without the need to know the velocity of its neighbors. With this scalable design approach, any expected geometric pattern of a group of n robots with two leaders can be realized by assigning an appropriate neighbor relationship and specifying a desired formation for each follower to reach. Both rigorous analysis and simulations are provided to demonstrate the effectiveness of the adaptive formation controller.

A distributed reconfigurable control law for escorting and patrolling missions using teams of unicycles

Recent years have seen rapidly growing interest in the development of networks of vehicles for which adaptive cooperation and autonomous execution become a necessity. In the paper, we develop a distributed reconfigurable control law to distribute unicycle-type vehicles evenly on a circle surrounding a moving target for the escorting and patrolling missions. The even distribution of the vehicles provides the best overall coverage of the target in its surroundings. It is shown that as the target moves, the group formation moves and rotates around the target to keep the target around the formation centroid. When some vehicles in the group are lost due to faults, the remaining vehicles recognize the loss and adaptively reconfigure themselves to a new evenly distributed formation.

Complex Laplacian and pattern formation in multi-agent systems

The paper studies the problem of pattern formation on spatial multi-agent systems. It is shown that complex-valued graph Laplacians in pattern formation are as important as the real one for consensus. First, formation patterns with four degrees of freedom (translation, rotation, and scaling) can be characterized by the null space of complex Laplacians associated with the sensing graph of networked agents. Second, formation patterns can be achieved via simple linear interaction rules related to complex Laplacians and the system can be stabilized by pre-multipling a stabilizing matrix. Several graphical and algebraic conditions are obtained for formation control of spatial multi-agent systems with their interaction topology modeled by an undirected graph.