The study of autonomous systems such as drones is an “emerging field in network science and engineering.” In the future, these systems need to cooperatively operate without collision, communicating with each other “to achieve more sophisticated multi-agent system behavior.” Every agent not only controls its actions based on some predetermined spatial configuration, but also can solve problems in a distributed manner. Hence, the title of the book presents a systematic mathematical foundation for the formation of such autonomous systems in a distributed manner. Such formation control involves the study of distance sensing, bearing sensing, and unidirectional and bidirectional sensing to establish a distributed system.

This book, divided into four main parts, presents formation control in a well-structured, step-by-step manner. The first part defines distributed sensing, communications, computing, and control. It then explains the mathematical background required, including “graph theory, rigidity theory, key results in consensus, and basics of nonlinear control theory.”

Rigidity theory includes distance rigidity, persistence, bearing rigidity, and weak rigidity. In rigidity theory, the scalar or vector magnitude is calculated based on a unique framework between neighboring agents. The second part of the book explores a gradient-based approach, which is basically a “function for generating local controllers for distributed agents.” It creates a sound topology that “ensures a unique configuration when the desired distances are satisfied.” This approach basically has two solutions: stabilize the formation globally from any starting condition for a given graph, and stabilize the formation locally given general n-agents. Particularly, it considers a 2D or 3D space of three or four agents’ cases without any communication between agents.

The third part of the book presents formation control via orientation in which neighboring agents can communicate with each other: “the agents can sense each other” and the results can be exchanged between agents. Thus, by exchanging sensing messages between agents, “global convergence under more generalized initial conditions could be assured.” By exchanging frames between neighbors, misaligned coordinates can be corrected for strong global convergence. Until this stage, the frames are not aligned with respect to other agents. Hence, the next part covers alignment with respect to bearing vectors that can be normalized using the angles related to each agent.

Finally, the book explores some advanced topics, for example, the moving formation of three agents and how to control their angles and shape while moving. At the end, the book develops formation control for n-agents and resizing that addresses the problem of scaling.

The book presents a very sound mathematical solution for future autonomous systems in the advanced technological age. However, an applied solution to the above-mentioned mathematical foundation is not discussed.