Leader–follower cooperative attitude control of multiple rigid bodies
Introduction
Cooperative distributed control strategies for multiple vehicles have gained increased attention in recent years in the control community, owing to the fact that such strategies provide attractive solutions to large-scale multi-agent problems, both in terms of complexity in the formulation of the problem, as well as in terms of the computational load required for its solution.
A typical control objective for a team of agents is the state-agreement or consensus problem. This control objective has been extensively pursued in recent years. Several results are based on treating the vehicle as a single integrator [1], [2], [3], [4] or double integrator [5], [6], [7]. A recent review of the various approaches for solving the consensus problem when the underlying dynamics are linear can be found in [8]. A common analysis tool that is used to model these distributed systems is algebraic graph theory [9].
Extending the previous results to systems whose dynamics are nonlinear is, in general, a nontrivial task. A large and important class (in terms of applications) of systems whose dynamics are nonlinear are systems of rotating rigid bodies. Motivated by the fact that–despite the nonlinear dynamics–linear controllers can stabilize a single rigid body [10], in this paper we propose a control strategy that exploits graph theoretic tools for cooperative control of multiple rigid bodies. We extend our previous work in this area [11] to address the case of teams with heterogeneous agents. For some applications (i.e., earth monitoring or stellar observation using a satellite cluster with a large baseline) it may be necessary for some satellites to acquire and maintain a certain (perhaps nonzero) relative orientation among themselves. A primary control objective is therefore to stabilize a subgroup of the team (leaders) to certain relative orientations. The orientations of the rest of the team (followers) are to remain within a certain orientation boundary, determined–in this case–by the convex hull of the leaders’ orientations. At the same time, each agent is allowed to communicate its state (orientation and angular velocity) only with certain members of the team. These constraints limit the information exchange between the agents. The control laws for each agent proposed in this paper respect this limited information each rigid body has with respect to the rest of the team (leader or followers). A preliminary version of the paper appeared in [12].
We should mention that cooperative control of multiple rigid bodies has been addressed recently by many authors, notably [13], [14], [15], [16]. While these papers use distributed consensus algorithms to achieve the desired objective, the specific algebraic graph theoretic framework (that is, the use of graph Laplacians) encountered in this work has not been considered in these papers. Recall that the Laplacian matrix encodes the limited communication capabilities between team members. Similarly to the linear case, the convergence of the multi-agent system relies on the connectivity of the communication graph.
Section snippets
System and problem definition
We consider a team of rigid bodies (henceforth called agents) indexed by the set . The dynamics of the th agent are given by [10]: where is the angular velocity vector, is the external torque vector, and is the symmetric inertia matrix of the th agent, all expressed in the th agent’s body-fixed frame. The matrix denotes a skew-symmetric matrix representing the cross product between two vectors, i.e., .
In this paper,
Tools from algebraic graph theory
In this subsection we review some tools from algebraic graph theory [9] that we will use in what follows.
For an undirected graph with vertices, the adjacency matrix is the symmetric matrix given by , if and , otherwise. If there is an edge connecting two vertices , that is, , then are adjacent. A path of length from a vertex to a vertex is a sequence of distinct vertices starting from and ending at , such that consecutive
Leader relative orientation control design
In this section we present a control algorithm that drives the team of leaders to the desired relative orientations. This is a problem that resembles the formation control problem in multi-vehicle systems. The relative orientation for each pair of leaders may be different, and is dictated by the mission requirements. We impose the condition that, for each pair , there exists a desired relative orientation , to which the corresponding pair of leaders must converge (see
The case of lack of a global objective
In this section we assume that no global objective is imposed by the team of leaders. In particular, we assume that . The objective is to build distributed algorithms that drive the team of multiple rigid bodies to a common constant orientation with zero angular velocities.
In order to ensure that all agents converge to the same constant orientation, in this section we show that it is sufficient that one agent has a damping element on the angular velocity. Without loss of generality, we
Numerical example
In this section we present a numerical example that supports the theoretical developments.
The simulation involves four rigid bodies indexed from 1 to 4. We assume that there are two leaders and two followers . We further assume that leader 1 is the reference point, and according to (11) we have , and . The reference point was randomly produced for this example. We also have and . The control law of leader 2 is given by (12).
Conclusions
We propose distributed control strategies that exploit graph theoretic tools for cooperative rotational control of multiple rigid bodies. We assume that the agents are divided into leaders and followers. The leaders must maintain certain relative orientations with respect to each other, while the followers’ orientations are to remain within a certain region that is dictated by the orientations of the leaders. Similarly to the case with linear agent dynamics, the convergence of the system was
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