## 1 Introduction  
Variant constraint systems, such as structurally reconfigurable systems, soft body locomotion systems, and contact or collision systems and so on, are familiar in real world applications. The variant constraint systems (VCS) are generally heterogeneous multi-mode systems and contain state jumps in the dynamics of the systems. Thus, the VCS is essentially a class of hybrid dynamical systems (HDS) [1, 2]. Although the investigations about the HDS have been ongoing for more than thirty years [3], the dynamics modeling of general HDS has not been thoroughly discussed so far. For the VCS, a reasonable model can be used to analyze the dynamical behaviors of the systems, which can provide useful information for improving the mechanisms/structure designs and simultaneously reducing the complexity of synthesizing the control laws. Unfortunately, to date, a systemic approach of modeling the general VCS has not been presented. As shown in [1], many examples of HDS were presented, but the issues of modeling hybrid systems were discussed case by case. [PERSON] and [PERSON] [4] briefly presented the modeling approach of the hybrid dynamics of a passive one-legged running robot, and did not mention the theoretical basis of the approach. In a monograph [5], [PERSON] et al. introduced the impact model of several kinds of dynamical legged robots in brief and the main contents of the monograph are motion planning and hybrid control issues of the HDS. In literature [6, 7], [PERSON] et al. addressed the modeling and control approaches of the HDS with the background of the biped robots. Following the modeling approach presented by [PERSON] [6], [PERSON] et al. [8, 9] stablized the HDS of biped robots with the finite time controllers. From a viewpoint of nonlinear control [10], the VCS are generally a certain kind of nonholonomic system since a VCS usually has relatively fewer constraints in one mode while has more constraints in another mode, and the additional constraints themselves are possible nonholonomic constraints [11]. [PERSON]'s necessary condition [12] points out that the nonholonomic systems can not be stabilized by any smooth time invariant pure state feedback; thereafter, the presented controllers for nonholonomic systems commonly employ non-smooth feedback [11; 13; 14; 15] or time varying feedback [16; 17; 11; 18]. However, it is interesting that many theoretical or experimental investigations, such as the legged running robot systems [4; 5; 9], have shown that the underactuated nonholonomic systems [19] could be stabilized to certain time varying trajectories by switched linear controllers. The switched linear controllers [20] are a class of discontinuous controllers that do not violate the Brockett's necessary condition. Therefore, as it is revealed in this paper, from a standpoint of switching control, under certain conditions, the complex VCS can be stabilized to certain periodic orbits by simple switched linear controllers with only partial state feedback in a time-sharing manner. This point is appealing in practice and supports the innovations of the VCS in a broader field well. The remainder of this paper is organized as follows. In Section 2, modeling the hybrid dynamics of underactuated systems with variant constraints is discussed. It is shown that the impulse caused by the instantaneously variant constraints of the HDS can be explicitly calculated with the help of the concept of differential inclusions [21], so that searching for a time varying trajectory with zero impulse is possible, and the special time varying trajectory can be utilized to improve the energy efficient of a locomotion system. In Section 3, the switched linear control approaches for stabilizing the VCS are proposed. We show that the closed-loop stability of tracking trajectory for the HDS can be systematically investigated under the frame of switching control [22].