A unified approach to the velocity-free consensus algorithms design for double integrator dynamics with input saturations Abdelkader Abdessameud and Abdelhamid Tayebi Abstract— This paper considers the consensus problem of double integrator multi-agent systems where each agent is subject to input saturations, and the velocity (second state) of each agent is not available for feedback. We present a unified approach to the consensus algorithms design that extends most of the existent consensus algorithms developed for double integrator multi-agents in ideal situations to handel these two problems simultaneously. To illustrate the effectiveness of the proposed approach, we present solutions to three different sec- ond order consensus problems and provide simulation results. I. I NTRODUCTION In contrast to multi-agents with first order dynamics, consensus algorithms for double integrators can be naturally extended to design cooperative control strategies for complex physical systems with applications to flocking [1], formation control of unmanned vehicles [2]-[3], rigid body attitude syn- chronization [4]-[5] and synchronization of networked Euler- Lagrange systems [6]-[7]. The consensus problem of double integrators involves the design of consensus algorithms such that agents can reach an agreement on their states, or on a common objective, using local information exchange. This information exchange is generally restricted to be directed, dynamically changing, and may be delayed. In the related literature to the second order consensus problem, tools from algebraic graph theory have been suc- cessfully applied to establish conditions under which second order consensus is reached. In directed networks, it has been shown that second order consensus will be reached if and only if the communication graph has a spanning tree and the control gains are carefully selected [8], [9]. Within a similar framework, several related problems to consensus have been considered such as the formation control problem, [10], consensus with group reference velocity, [11] and leader-follower problems, [12]. Also, the case of dynamically changing topologies have been discussed in [8] and [13]. The effects of communication delays that are inherently present in communication systems have also been considered in [14]- [16] and references therein. However, the above consensus algorithms are based on the assumption that the full state vector is available for feedback. In practice, it is sometimes desirable to design consensus algorithms that do not require full states information. If we This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors are with the Department of Electrical and Computer En- gineering, University of Western Ontario, London, Ontario, Canada. The second author is also with the Department of Electrical Engineering, Lake- head University, Thunder Bay, Ontario, Canada. aabdess@uwo.ca, tayebi@ieee.org consider, for example, a group of point masse agents, an important problem is to design consensus algorithms in the case where the velocity information (the second state) is not available for feedback, either because it is not precisely measured or agents are not equipped with velocity sensors. Another important problem arises when the input of each agent is subject to input saturations. Unfortunately, the papers dealing with these two problems are not numerous and only the simple case of fixed and undirected communication topology has been considered. The author in [11] proposed consensus algorithms that account for input saturations in the full state information case. In the same reference, the author presents a second order consensus algorithm that re- moves the requirement of velocity measurements. Consensus algorithms that take into account the two above problems simultaneously have been proposed in [17]. In this reference, a new approach, based on auxiliary systems, has been proposed to simplify the consensus algorithm design problem in this case. However, it is difficult to show that the results in [17] are applicable under more general communication topologies, that may be directed, time-varying and/or subject to communication delays. The main contribution of this paper is to provide a unified approach that extends most of the existent consensus algorithms, developed for double integrator dynamics with a certain communication topology, to account for input saturations and remove the requirement of velocity mea- surements. Instrumental to our approach is the introduction of two second order auxiliary systems that simplify the consensus algorithm design. The first auxiliary system is used to generate an intermediate reference trajectory for each agent, and its input is designed such that all agents reach an agreement on their reference trajectories. The input of the second auxiliary system is designed such that each agent tracks its corresponding intermediate reference trajectory without velocity measurements. With this setting, the control input of each agent is constructed using only the auxiliary states to account for input saturations. As a result, the consensus algorithm design problem with the above mentioned constraints is reduced to the design of a consensus algorithm in ideal situations, i.e., in the full state information case and without input saturations. To show the effectiveness of the obtained results, we consider three different problems related to second order consensus, and extend some consensus algorithms developed for double integrator dynamics to account for input saturations in the partial state feedback case. 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 2011 978-1-61284-799-3/11/$26.00 ©2011 IEEE 4903