In unstructured environments, picking robots could collide with branches, thus reducing the success rate of picking. In this study, collision-free motion planning is performed in two steps based on the binocular stereo vision information of the environment. First, according to the 3D information collected in the stereo vision environment, the improved adaptive weight particle swarm optimization (APSO) algorithm is adopted to solve the inverse kinematics problem of robots and obtain the collision-free picking posture. Second, to address the limitations of the Bi-RRT algorithm in a high-dimensional environment, such as randomness and slow convergence speed, the target gravity concept and adaptive coefficient adjustment method are introduced to the Bi-RRT algorithm, which is called the AtBi-RRT algorithm. The AtBi-RRT algorithm is used to determine the collision-free path of the robot. Simulation results indicate that the APSO algorithm can swiftly obtain the proper collision-free picking posture, and the average path-determination time of the AtBi-RRT algorithm is 4.24 s. The success rate of path determination is 100% for the laboratory’s picking scene. Experimental results verify that the proposed collision-free motion-planning method allows the picking robot to avoid obstacles in the workspace and efficiently complete the picking task.