Autonomous driving has long grappled with the need for precise absolute localization, making full autonomy elusive and raising the capital entry barriers for startups. This study delves into the feasibility of local trajectory planning for level-2+ (L2+) semi-autonomous vehicles without the dependence on accurate absolute localization. Instead, we emphasize the estimation of the pose change between consecutive planning frames from motion sensors and integration of relative locations of traffic objects to the local planning problem under the ego car's local coordinate system, therefore eliminating the need for an absolute localization. Without the availability of absolute localization for correction, the measurement errors of speed and yaw rate greatly affect the estimation accuracy of the relative pose change between frames. We proved that the feasibility/stability of the continuous planning problem under such motion sensor errors can be guaranteed at certain defined conditions. This was achieved by formulating it as a Lyapunov-stability analysis problem. Moreover, a simulation pipeline was developed to further validate the proposed local planning method. Simulations were conducted at two traffic scenes with different error settings for speed and yaw rate measurements. The results substantiate the proposed framework's functionality even under relatively inferior sensor errors. We also experiment the stability limits of the planned results under abnormally larger motion sensor errors. The results provide a good match to the previous theoretical analysis. Our findings suggested that precise absolute localization may not be the sole path to achieving reliable trajectory planning, eliminating the necessity for high-accuracy dual-antenna GPS as well as the high-fidelity maps for SLAM localization.