This paper investigates the stability principle of a planar, underactuated, four-linked limit cycle walker which is strictly controlled to follow the desired-time trajectories. We first linearize the equation of motion of the robot and analytically derive the transition equations of the state error for the stance and heel-strike phases. Second, we investigate the limit-cycle stability, especially the relation between the heel-strike posture and the gait convergence, through numerical simulations. Throughout the gait analysis, it is shown that the gait convergence changes from the speed mode to the totter mode through the deadbeat mode with respect to the asymmetry of the heel-strike posture.