Design of the dynamic stability properties of the collective behavior of articulated bipeds Albert Mukovskiy, Jean-Jacques Slotine and Martin A. Giese, Member, IEEE Abstract— The control of the collective behavior of multiple interacting agents is a challenging problem in robotics and autonomous systems design. Such behaviors can be character- ized by the dynamic interaction between multiple locomoting bipeds with highly nonlinear articulation dynamics. The anal- ysis and design of the stability properties of such complex multi-component systems is a largely unsolved problem. We discuss a first approach to this problem exploiting concepts from Contraction Theory, a recent framework for the analysis of the stability of complex nonlinear dynamical systems. We demonstrate the application of this framework to groups of humanoid agents interacting collectively in different ways, requiring different types of control rules for their propaga- tion in space and their articulation dynamics. We illustrate the framework based on a learning-based realtime-capable architecture for simulation of the kinematics of propagating bipeds, suitable for the reproduction of natural locomotion trajectories and walking styles. Exploiting central theorems from Contraction Theory and nonlinear control, we derive conditions guaranteeing the global exponential stability of the formation of the coordinated multiagent behavior. In addition, we demonstrate that the same approach permits to derive bounds that guarantee minimum convergence speeds for the formation of ordered states for collective behaviors of multiple humanoid agents. Index Terms— walking bipeds, crowd steering, coordination, distributed control, self-organization, stability. I. I NTRODUCTION Human movements and the collective behavior of inter- acting characters in crowds can be described by nonlinear dynamical systems, e.g.: [1], [2]. The design of stability properties of multiagents systems is a challenging task in control theory and robotics. Especially for humanoids agents, the reason is the complexity of the dynamical systems that are required for the accurate modelling of human body move- ments, and even more for the interaction between multiple interacting agents. Path planning for multi-agent systems has been studied extensively in robotics, primarily for cooperative tasks of multiple robots with relatively simple platform dynamics. The approach of the centralized planners, which is designing the motion of all agents through space-time [3], has expo- nential computational complexity in the number of agents. And it is not appropriate for large groups, where agents A. Mukovskiy and M.A. Giese are at the the Section Computational Sensomotorics, Department of Cognitive Neurology, Hertie Institute for Clinical Brain Research, Center for Integrative Neuroscience, University of T¨ ubingen, D-72070 T¨ ubingen, Germany. albert.mukovskiy@medizin.uni-tuebingen.de, martin.giese@uni-tuebingen.de J.J. Slotine at Nonlinear Systems Laboratory, Massachusetts Institute of Technology, MA 02139, USA. jjs@mit.edu optimize for personal goals (like mutual avoidance behavior) in a presence of global common navigation goal. In contrast to this approach, decoupled planning strategies, where agents plan their behaviors individually, require priority schemes to fix conflicts between the different plans [4]. Another alternative is to use local planners for obstacle avoidance and goal-directed navigation, but to control glob- ally the homogeneous interaction forces among agents in order to re-coordinate them by self-organization after the accomplishment of the individual subgoals [5], [6]. Such re- coordination is required to realize controllable and steerable crowds that pursue common global navigational tasks. Some works on self-organization in systems of dynam- ically coupled agents have been inspired by observations in biology. These works show that coordinated behavior of large groups of agents, such as flocks of birds, can be modelled as emergent behavior arising from the dynamical coupling between interacting agents, without requiring an external central mechanism ensuring coordination [7], [8], [9], [10]. These biological observations have inspired a variety of approaches in robotics. Group coordination and cooperative control have been studied in the context of the navigation of groups of vehicles [11], and also with the objective goal to generate collective behavior by self- organization, including spontaneous adaptation to perturba- tions or changes in the number of agents [12]. In addition, the dynamics of interactive group behavior has been extensively studied in the field of computer animation [13], [14], [15], [16]. Specifically, some recent studies have tried to learn interaction rules from the behavior of real human crowds [17], [18], [19]. Other recent work has tried to optimize interaction behavior in crowds by exhaustive search of the parameter space exploiting computer simulations by defini- tion of appropriate cost functions (e.g. [20]). However, most of the existing approaches for the control of group motion in computer graphics have not taken into account the effects of articulation during locomotion on the control dynamics [21], [22], [23]. Distributed control theory has started to study the temporal and spatial self-organization of crowds of agents by design of appropriate dynamic interactions, typically assuming rather simple and often even linear agent models (e.g. [24], [25], [26]). However, humanoid agents are characterized by highly complex kinematic and even dynamic properties, c.f. [27], [5]. This raises the question how approaches for the stability design of such complex dynamical systems can be developed. This paper presents some first attempts to address this problem for simplified, but highly nonlinear dynamical 2010 IEEE-RAS International Conference on Humanoid Robots Nashville, TN, USA, December 6-8, 2010 978-1-4244-8689-2/10/$26.00 ©2010 IEEE 66