Parallel manipulators are increasingly utilized in extensive industrial applications due to their high accuracy, compact structure, and significant stiffness characteristics. However, most of the time, massive actuators are involved in constructing and controlling a parallel manipulator, which burdens the structure design and controller development. In this paper, a novel underactuated positioning system been built by different sets of linear motion units (defined as the positioning lines) is proposed, enabling to actuate the multiple degree-of-freedom manipulators with one motor. To achieve this, a smart shape memory alloy (SMA) clutch is presented to obtain the positioning function of each positioning line. Further, to get the decoupled motion regulation of the positioning lines, a new thermal kinematic model of the SMA clutch, which considers the heat dissipation influence on the metal components, was built and validated by the physical prototypes. The experimental results show that the constitutive model of the SMA clutch developed in this paper can be validated within the error of 5.3%. It can also be found that the heat dissipation of the metal component has a significant influence on the model accuracy of the SMA clutch (i.e., 2.6% of the model accuracy). The experiments on the underactuated positioning system produce the following results: the single positioning line can achieve high positioning (i.e., average error: 1.01%) and tracking (i.e., average error $\leq$ 1 mm) abilities; the underactuated positioning system can perform decoupled motions in the three positioning lines with high accuracy (i.e., ±2 mm within the stroke of 180 mm).