**1. Introduction**

In recent years, multiagent robotic systems have been firmly established as an important topic of research owing to the continued emergence of potential applications. Cooperating teams of robots remotely operated or capable of autonomous navigation and sensing can be used to enhance or extend the functions of single-agent systems in areas such as airborne [1–3] and terrestrial-distributed mobile sensing [4–6], search and rescue [7–9], and intelligent transpor‐ tation [10–12].

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For mobile systems, one of the key technical considerations is the development of a technique to coordinate the motions of individual vehicles. The mutual goal of each agent is typically to establish and maintain a certain spatial configuration or to perform complicated geometrically time-varying maneuvers, a use case of particular interest to this investigation. These desired behaviors lead to a variety of formation control problems, for which a wide range of solutions have been and continue to be explored.

Notable work in this area includes the development of leader-follower strategies in which follower agents control their position relative to a designated leader to meet formation requirements [13–15]. Artificial potential fields have similarly been shown effective as a construct to establish formation-keeping forces between robots within a group [16, 17]. Cluster space, an approach that allows for intuitive specification of formation characteristics and implements control directly on these variables, has also been demonstrated successfully for a number of robotic systems [18]. Current trends in research, however, indicate a focus on the incorporation of formation requirements into the framework of optimal control, and will be discussed in this chapter.

Due to the inherent physical distribution of agents and the potential for limited information exchange, decentralized control protocols for multirobot systems are popular. However, centralized architectures, which exploit global information, are more amenable to executing specific time-varying formation trajectories. The latter pertains to a class of similar methods highly relevant to this chapter that delegate path design to an earlier step, relying on motion planning to avoid inter-agent collisions and achieve varying degrees of coordination [19–24]. Distinguished among these is [25], which frames motion coordination as a velocity-optimiza‐ tion problem. Formations are defined by robot-pair relative geometries as a function of distance along their respective paths, forming a constraint net. Optimal trajectories for each robot are then generated such that these formation errors are minimized. Alternatively, [26] represents the set of robot positions at a particular time as a 2D planar curve, and develops a synchronization controller that regulates robot motions to simultaneously track the prescribed formation boundaries as well as their individual paths. Although an optimal velocity signal is not derived, as in [25], the model allows for complete specification of any formation over time, provided the potentially complicated individual robot paths are attainable. For a number of conceivable formations, this is a non-trivial step and can prove prohibitive. The methods proposed in this chapter, which build from the cluster space approach to multirobot system specification, address this issue.

In contrast to the investigations discussed above, much of current research in multirobot systems is dedicated to the use of distributed optimal control techniques. These are appropriate for applications that permit simple formation specification where robots must operate with communication restrictions and local information such as relative positioning. Consensus algorithms, which draw from concepts in distributed computing and graph theory, are present in many of these approaches [27–29]. In the context of a cooperative multivehicle system, information consensus refers to the convergence of agents in a networked system to a common task or variable such as the center of a formation shape, the rendezvous time, or the direction of formation translation [30]. Also of note are distributed control-based methods to handle swarms, or multiscale dynamical systems, which are comprised of many agents [31–34]. In general, these techniques are only capable of assigning coarse-grained dynamics; specific paths and distributions are not determined a priori. Each unit is held subject to local objectives and constraints, giving rise to certain coherent macroscopic behaviors. For example, Ferrari et al. [35] model global behavior of a multiagent system using PDFs and optimizes subject to coupled local agent dynamics in such a manner that cohesion is achieved and the result requires far less computation than classical optimal control.

For mobile systems, one of the key technical considerations is the development of a technique to coordinate the motions of individual vehicles. The mutual goal of each agent is typically to establish and maintain a certain spatial configuration or to perform complicated geometrically time-varying maneuvers, a use case of particular interest to this investigation. These desired behaviors lead to a variety of formation control problems, for which a wide range of solutions

Notable work in this area includes the development of leader-follower strategies in which follower agents control their position relative to a designated leader to meet formation requirements [13–15]. Artificial potential fields have similarly been shown effective as a construct to establish formation-keeping forces between robots within a group [16, 17]. Cluster space, an approach that allows for intuitive specification of formation characteristics and implements control directly on these variables, has also been demonstrated successfully for a number of robotic systems [18]. Current trends in research, however, indicate a focus on the incorporation of formation requirements into the framework of optimal control, and will be

Due to the inherent physical distribution of agents and the potential for limited information exchange, decentralized control protocols for multirobot systems are popular. However, centralized architectures, which exploit global information, are more amenable to executing specific time-varying formation trajectories. The latter pertains to a class of similar methods highly relevant to this chapter that delegate path design to an earlier step, relying on motion planning to avoid inter-agent collisions and achieve varying degrees of coordination [19–24]. Distinguished among these is [25], which frames motion coordination as a velocity-optimiza‐ tion problem. Formations are defined by robot-pair relative geometries as a function of distance along their respective paths, forming a constraint net. Optimal trajectories for each robot are then generated such that these formation errors are minimized. Alternatively, [26] represents the set of robot positions at a particular time as a 2D planar curve, and develops a synchronization controller that regulates robot motions to simultaneously track the prescribed formation boundaries as well as their individual paths. Although an optimal velocity signal is not derived, as in [25], the model allows for complete specification of any formation over time, provided the potentially complicated individual robot paths are attainable. For a number of conceivable formations, this is a non-trivial step and can prove prohibitive. The methods proposed in this chapter, which build from the cluster space approach to multirobot system

In contrast to the investigations discussed above, much of current research in multirobot systems is dedicated to the use of distributed optimal control techniques. These are appropriate for applications that permit simple formation specification where robots must operate with communication restrictions and local information such as relative positioning. Consensus algorithms, which draw from concepts in distributed computing and graph theory, are present in many of these approaches [27–29]. In the context of a cooperative multivehicle system, information consensus refers to the convergence of agents in a networked system to a common task or variable such as the center of a formation shape, the rendezvous time, or the direction of formation translation [30]. Also of note are distributed control-based methods to handle

have been and continue to be explored.

discussed in this chapter.

142 Recent Advances in Robotic Systems

specification, address this issue.

Energy efficiency in mobile robotics systems is of great concern given that energy sources are often carried, and practical applications require extended remote operation wherein limited resources must be conserved. Many investigations seek to address this issue through the use of motion planning techniques to reduce the energy consumption in system components. Typically, proposed energy models are incorporated into optimal control frameworks to derive trajectories that minimize energy expenditure [36–40]. For example, in a recent work, Liu and Sun [41] decompose energy consumption into three categories: kinetic energy transformation, overcoming traction resistance, and maintenance of electrical sources for the operation of sensors, on board PCs, and control circuits. This analysis is then used to generate both an energy optimal path and a velocity trajectory, which further conserves energy. While a complete and detailed energy model can be beneficial, it is well known that preventing large torque variations in motors by smoothing velocity profiles is most fundamental to energy conservation [42, 43]. This chapter concentrates on minimizing energy expenditure for a formation of mobile robots in this regard.

The investigations referenced above are geared toward individual robots and therefore do not adequately address the optimal planning of energy-efficient trajectories for multiagent systems where a holistic approach is appropriate. There are, however, a number of methods that include provisions for minimizing aggregate energy consumption. For example, Sieber et al. [44] formulate an LQR-like optimal control problem designed to move a formation of mobile robots to a goal while minimizing input energy and incorporating a provision for formation rigidity into the cost functional. Similarly, Wigstrom and Lennartson [45] generated trajectories that reduce energy consumption using pseudo-spectral optimal control, though paths are considered free. Coverage algorithms such as [40, 46] also implement energy conservation constraints while maximizing the reach of mobile robot networks and sensing. A few swarmlike methods have accounted for power consumption as well, but they minimize global energy in order to achieve stability and cohesion for the group, or to affect the size and internal coverage of the swarm [47, 48]. While these examples address the conservation of group energy, they do not deal with the generation of smooth energy-efficient velocity trajectories along predefined paths, an objective of this chapter.

The concepts of time and energy optimality in the context of trajectory generation for multi‐ robot systems are well represented in the literature. However, they are largely accounted for alongside formation constraints, thus the derived control signals are suboptimal with regard to time and/or energy exclusively. To our knowledge, there are no methods that achieve high precision time-varying maneuvers and successfully separate the generation of optimal trajectories for each robot in a multiagent group from formation requirements. The techniques proposed in this chapter address this shortcoming.

The contribution of this investigation is a method to generate continuous force, acceleration, and velocity profiles for a formation of mobile robots with predetermined paths, while time and energy are minimized in chosen proportions. Previously proposed methods with similar objectives have employed optimization techniques with robot level dynamic constraints and formation level cost functions. In contrast, our treatment uniquely optimizes and imposes constraints at the cluster or formation level, with favorable results. Further advantages of our technique are explained in Section 4.

We first introduce the cluster space control framework, which presents a group of mobile robots as a virtual articulating mechanism in order to facilitate characterization and to implement coordinated control of the system. A parameterization of the cluster dynamic equations is next proposed which results in a reduced second-order state space model. The structure and numerical solution of the optimization are then shown. Finally, a simulation of a three-robot cluster controller is used to verify and validate the solution trajectory, after which analysis and results are presented.
