This paper presents the forward and inverse displacement analysis of a novel three-legged walking robot Self-excited Tripedal Dynamic Experimental Robot (STriDER) in its triple stance phase. STriDER utilizes the concept of passive dynamic locomotion to walk, but when all three feet of the robot are on the ground, the kinematic configuration of the robot behaves like an in-parallel manipulator. To plan and control its change in posture, the kinematics of its forward and inverse displacement must be analyzed. First, the concept of this novel walking robot and its unique tripedal gait are discussed, followed by the overall kinematic configuration and definitions of its coordinate frames. When all three feet of the robot are on the ground, by assuming there is no slipping at the feet, each foot contact point is treated as a passive spherical joint. Kinematic analysis methods for in-parallel manipulators are briefly reviewed and adopted for the forward and inverse displacement analysis for this mobile robot. Both loop-closure equations based on geometric constraints and the intersection of the loci of the feet are utilized to solve the forward displacement problem. Analytical solutions are identified and discussed in the cases of redundant sensing with displacement information from nine, eight, and seven joint angle sensors. For the nonredundant sensing case using information from six joint angle sensors, it is shown that analytical solutions can only be obtained when the displacement information is available from unequally distributed joint angle sensors among the three legs. As for the case when joint angle sensors are equally distributed among the three legs, it will result in a 16th-order polynomial with respect to a single variable, and closed-form forward displacement solutions can be obtained. Finally, results from the simulations are presented for both inverse displacement analysis and the nonredundant sensing case with equally distributed joint angle sensors. It is found that at most 16 forward displacement solutions exist if displacement information from two joint angle sensors per leg are used and one is not used.