Abstract
The distributed formation control can be studied from various approaches. In distance-based formation control, which uses relative displacements under misaligned orientations as sensing variables and distances as control variables, the most intuitive approach is the gradient-based approach . The gradient-based formation control laws use a potential function for generating local controllers for distributed agents. If the potential function is a function of distance errors that can be sensed in each agent, then the agent can implement a control law, which attempts to reduce the potential function, via a local coordinate frame. Thus, if the underlying topology ensures a unique configuration when the desired distances are satisfied, the desired formation can be considered as achieved. Since a gradient of the potential function has to lead a distributed formation controller, it is important to select an appropriate potential function. There are two solutions in gradient control laws. The first one is to stabilize the formation globally from any initial condition for some specific graphs. The second one is to stabilize the formation locally when extending to general n-agents, under a general rigidity topology. It is not possible to stabilize the formation to a desired one from any initial condition under gradient control laws. That is why the existing works consider specific formations for a global convergence. This chapter considers three-agent cases in 2-dimensional space, four-agent cases in 3-dimensional space, and polygon graphs for a global convergence. Although the control laws of this chapter use relative displacements as sensing variables and distances as control variables, there is no communication between neighboring agents.
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Notes
- 1.
Let a matrix A have row vectors as \(v_1, v_2\) in the 2-D case, and as \(v_1, v_2, v_3\) in the 3-D case. Let the area determined by the paralellogram by the vectors \(v_1, v_2\) in 2-D be \(\xi _2\), and the volume determined by the skewed cube by the vectors \(v_1, v_2, v_3\) in 3-D be \(\xi _3\). Then, the determinant of A, i.e.,\(\text {det}(A)\), is equal to the area or volume of A as \(\text {det}(A) = \xi _2\) in 2-D, and \(\text {det}(A) = \xi _3\) in 3-D. Thus, the area decided by the triangular defined by the vertex points of the two vectors \(v_1, v_2\) is equal to \(\frac{1}{2} \text {det}(A) = \frac{1}{2} \xi _2\), and the tetrahedral defined by the vertex points of the three vectors \(v_1, v_2, v_3\) in 3-D is equal to \(\frac{1}{6} \text {det}(A) = \frac{1}{6} \xi _3\).
- 2.
The saddle points could be one of equilibrium points when the linearized dynamics has positive and negative eigenvalues. When a trajectory approaches toward the eigenvectors corresponding to the positive eigenvalues, it will converge to the origin asymptotically; but when it reaches to the eigenspace spanned by the eigenvectors corresponding to the negative eigenvalues, it will escape from the origin. Figure 3.5 depicts a saddle point, \(x_o\).
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Ahn, HS. (2020). Global Stabilization. In: Formation Control. Studies in Systems, Decision and Control, vol 205. Springer, Cham. https://doi.org/10.1007/978-3-030-15187-4_3
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