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Most rigid formation controllers reported in the literature aim to only stabilize a rigid formation shape, while the formation orientation is not controlled. This paper studies the problem of controlling rigid formations with prescribed orientations in both 2-D and 3-D spaces. The proposed controllers involve the commonly-used gradient descent control for shape stabilization, and an additional term to control the directions of certain relative position vectors associated with certain chosen agents. In this control framework, we show the minimal number of agents which should have knowledge of a global coordinate system (2 agents for a 2-D rigid formation and 3 agents for a 3-D rigid formation), while all other agents do not require any global coordinate knowledge or any coordinate frame alignment to implement the proposed control. The exponential convergence to the desired rigid shape and formation orientation is also proved. Typical simulation examples are shown to support the analysis and performance of the proposed formation controllers.

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We propose a control strategy that could steer the group of mobile agents in the plane to achieve a specified formation. The control law could be implemented in a fully decentralized manner so that each agent moves on their own local reference frame. Under the acyclic minimally persistent graph topology, each agent measures the relative displacements of neighboring agents and then adjusts the distances between them to achieve the desired formation. As well as achieving a fixed formation, we could resize the formation only by changing the leader edge, which connects the leader with the first-follower in acyclic minimally persistent graph, without changing the structures of the control law.A directed graph, which is denoted by G D .V; E/, is a pair of two sets, where V D ¹1; 2; : : : ; N º is a set of vertices, and E Â V V is a set of edges. Thus, we have jVj D N . We assume that there is no self edge, that is, .i; i/ … E for all i 2 V, and there are no multiple edges between By stacking the relative displacements, we have § The derivative of g is the sum of products consisting of e 12 ; e 13 ; e 23 ; O e 13 ; O e 23 ; z 12 ; z 13 , and z 23 terms that are bounded. ProofFrom (17), if .z 0 / D 0, then .z.t // D 0 for all t > 0 along the solution trajectory of (9). Corollary 3.2 means that if the agents are initially collinear, then they are collinear ever after. This result is the same as the cases that could be found in the literature dealing with distance-based formation control problems [6-8, 13, 15].