High locomotion speed and energy efficiency are the most attractive characteristics of the jumping gait for small mobile robots. In this paper, the design and simulation of a lightweight jumping robot is described. We propose the use of shape memory alloy (SMA) for actuating the jumping mechanism. We also report that the robot design is inspired by vertebrates' lower musculoskeletal system, in particular the structure of legs and the functions of muscles. SMA wires are used as artificial muscles to realize the jumping motion. A parametric dynamics simulation study is conducted to develop a suitable pattern of muscle activation. The simulation results show that the robot can jump up half of its height and move forward by about 25% of its length.