Consensus of Networks of Nonidentical Robots with Flexible Joints, Variable Time–Delays and Unmeasurable Velocities Daniela Valle, Emmanuel Nu˜ no, Luis Basa˜ nez and Nancy Arana-Daniel Abstract— The present paper proposes two controllers for solving a consensus problem to a given desired position of networks composed of a class of under actuated mechanical systems: flexible joints robots. One of the controllers makes use of joint (motor) velocity signals while the other only uses joint positions. The only assumption on the directed and weighted interconnection graph is that it is connected. Further, the interconnection may induce variable time–delays. The paper presents some experiments, using three 3-Degrees of Freedom manipulators, which show the performance of the proposed approaches. I. I NTRODUCTION A wide range of applications in different areas are based on the consensus of networks of dynamic systems. The objective for the collective motion of the network is to reach some type of agreement between certain variables of interest of the interconnected systems. The literature that deals with the consensus of networks covers those composed of linear time invariant systems, which is relatively rich and large [1], [2], [3], [4], [5], [6], and those composed of nonlinear nodes, which is rapidly increasing [7], [8], [9], [10], [11]. Consensus of networks of Euler–Lagrange (EL) systems without time-delays has been considered in [12], [13] using simple proportional controllers together with filtered veloci- ties. However, in both papers the authors assume that time- delays in the agents communications are negligible. The work of Nu˜ no et al. [14] reports an adaptive controller for EL-systems that solves the consensus problem with constant time-delays. Further results are those by Liu and Chopra [15] and by Hatanaka et al. [16], which consider the consensus problem in Cartesian space with constant time-delays in the communications. Recently, in [17] it has been proved that networks composed by nonidentical EL-systems with variable time–delays can reach a consensus, using simple PD controllers, provided enough damping is injected. It should be underscored that, all these previous results deal with fully actuated EL-systems (fully actuated robots). However, in diverse applications, including space and surgical robots, the use of thin, lightweight and cable-driven manipulators is increasing. These systems exhibit joint or link flexibility and hence they are under actuated mechanical systems. It has been shown in [18] that the lumped (linear) dynamics of a Daniela Valle is with the Electronics Department at the University of Guadalajara (UdG). Guadalajara, Mexico (danny.valler@gmail.com). Emmanuel Nu˜ no and Nancy Arana-Daniel are with the Department of Computer Science at the University of Guadalajara (UdG). Guadalajara, Mexico ({emmanuel.nuno; nancy.arana}@cucei.udg.mx). Luis Basa˜ nez is with the Institute of Industrial and Control Engineering (IOC) at the Technical University of Catalonia (UPC). Barcelona, Spain (luis.basanez@upc.edu). flexible link is identical to the (linear) dynamics of a flexible joint. On the other hand, the literature on the control of net- works of under actuated EL-systems is scarce, with some nice exceptions [19] and, more recently, [20]. In [19] the Controlled–Lagrangian technique is employed to solve the consensus in networks without delays and in [20] the con- sensus problem is solved under the assumption that all the states are measurable, all the physical parameters are known and the time-delays are constant. In the present work, inspired by [21] and [22], two different controllers that are capable of solving a consensus problem to a given desired position in a network composed by nonidentical flexible joint robots are proposed. One of the controller makes use of joint (motor) velocity measurements while the other only needs joint position measurements. The robots are interconnected with a directed and weighted network topology and the only assumption on the inter- connection graph is that it is connected. Moreover, the interconnection can exhibit variable time–delays. It should be noted that since interconnection improves performance, as has been proved in [23], [24], the proposed controllers are, in principle, more robust to parameter uncertainties than the ones without the interconnection. Finally, using three 3-Degrees of Freedom (DOF) manipulators, the paper presents some experiments which show the performance of the controllers with and without the interconnection. A. Notation R := (−∞, ∞), R >0 := (0, ∞), R ≥0 := [0, ∞). λ m {A} and λ M {A} represent the minimum and maximum eigen- values of matrix A, respectively. ||A|| denotes the matrix- induced 2-norm. |x| stands for the standard Euclidean norm of vector x. I k and 0 k represent the identity and all-zeros matrices of size k × k. 1 k is a vector of all elements equal to one of size k. For any function f : R ≥0 → R n , the L ∞ - norm is defined as ‖f ‖ ∞ := sup t≥0 |f (t)|, and the square of the L 2 -norm as ‖f ‖ 2 2 :=  ∞ 0 |f (t)| 2 dt. The L ∞ and L 2 spaces are defined as the sets {f : R ≥0 → R n : ‖f ‖ ∞ < ∞} and {f : R ≥0 → R n : ‖f ‖ 2 < ∞}, respectively. II. MODELS OF THE ROBOT DYNAMICS AND THE NETWORK I NTERCONNECTION The complete system is composed by an interconnected network of different flexible–joint robots. This section presents the dynamics of the manipulators (nodes or agents) and the model of the interconnection of the network. 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) November 3-7, 2013. Tokyo, Japan 978-1-4673-6357-0/13/$31.00 ©2013 IEEE 5878