The design of a knee joint is a key issue in robotics and biomechanics to improve the compatibility between prosthesis and human movements, and to improve the bipedal robot performances. We propose a novel design for the knee joint of a planar bipedal robot, based on a four-bar linkage. The dynamic model of the planar bipedal robot is calculated. Two kinds of cyclic walking gaits are considered. The first gait is composed of successive single support phases with stance flat-foot on the ground separated by impacts. The second gait is a succession of finite time double support phases, single support phases, and impacts. During the double support phase, both feet rotate. This phase is ended by an impact of the toe of the forward foot, while the rear foot is taking off. The single support phase is ended by an impact of the swing foot heel, the other foot keeping contact with the ground through its toe. For both gaits, the reference trajectories of the rotational joints are prescribed by cubic spline functions in time. A parametric optimization problem is presented for the determination of the parameters corresponding to the optimal cyclic walking gaits. The main contribution of this paper is the design of a dynamical stable walking gait with double support phases with feet rotation, impacts, and single support phases for this bipedal robot.