To date, most quadruped robots are either equipped with trunks that are rigid bodies or consist of blocks connected by passive joints. The kinematic performance of these quadruped robots is only determined by their legs. To release the mobility of trunks and enhance the performance of quadruped robots, this paper proposes a metamorphic quadruped robot with a moveable trunk (a planar six-bar closed-loop linkage), called MetaRobot I, which can implement active trunk motions. The robot can twist its trunk like natural quadrupeds. Through trunk twisting, the stability margin of the quadruped robot can be increased compared with that of a quadruped robot with a rigid trunk. The inner relationship between the stability margin and the twisting angle is analyzed in this paper. Finally, simulations are carried out to show the benefits facilitated by the twisting trunk to the quadruped robot.