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The traditional A∗ algorithm, applied to the motion planning of autonomous vehicles, easily causes high computational costs and excessive turning points generated in the planning path. In addition, the vehicle cannot track the path due to the unsmooth inflection point. To overcome these potential limitations, an improved A∗ algorithm-based motion planning algorithm and a tracking control strategy based on model predictive control theory were proposed in this work. The method of expanding the search neighborhood is adopted to improve the planning efficiency of A∗ algorithm. The artificial potential field method is also incorporated into the proposed A∗ algorithm. The resultant force generated by each potential field is further introduced into the evaluation function of A∗ algorithm to plan the driving path, which could be suitable for autonomous vehicles. The sharp nodes in the path are smoothed by cubic quasi-uniform B-spline curve. The tracking control strategy is designed based on model predictive control theory to realize the accurate tracking of the planned path. Typical obstacle avoidance conditions were selected for co-simulation test verification. The experimental results show that the proposed motion planning algorithm and tracking control strategy can effectively plan the obstacle avoidance path and accurately track the path in different environments.