In this paper, we consider flocking of multiple inertial agents with second-order dynamics. For a single agent, its state is therefore velocity and position. There exists both velocity coupling and position coupling between agents that are not necessarily equal. The nonequal couplings are distinguished here, instead of the common assumption of the equal velocity coupling and position coupling in most literature, since they play different roles in the system dynamics. The main contributions are in two aspects: first, we provide a general geometric decomposition approach for analyzing such systems of the agents. Then, we provide some stability results via Lyapunov analysis as a function of couplings and gains that generalize existing results in this area, which are especially useful when the couplings are partially known. Although the stability results are only sufficient, they serve as important complements of the eigenvalue analysis, which generally requires complete knowledge of the couplings in the eigenvalue analysis.

In this paper we study the problem of passive walking for a compass-gait biped with gait asymmetries. In particular, we identify and classify bifurcations leading to chaos caused by the gait asymmetries because of unequal leg masses. We present bifurcation diagrams showing step period versus the ratio of leg masses at various walking slopes. The cell mapping method is used to find stable limit cycles as the parameters are varied. It is found that a variety of bifurcation diagrams can be grouped into six stages that consist of three expanding and three contracting stages. The analysis of each stage shows that marginally stable limit cycles exhibit period-doubling, period-remerging, and saddle-node bifurcations. We also show qualitative changes regarding chaos, i.e., generation and extinction of chaos follow cyclic patterns in passive dynamic walking.