This paper proposes an improved RRT*-Connect algorithm (IRRT*-Connect) for robotic arm path planning in narrow environments with multiple obstacles. A heuristic sampling strategy is adopted with the integration of the ellipsoidal subset sampling and goal-biased sampling strategies, which can continuously compress the sampling space to enhance the sampling efficiency. During the node expansion process, an adaptive step-size method is introduced to dynamically adjust the step size based on the obstacle information, while a node rejection strategy is used to accelerate the search process so as to generate a near-optimal collision-free path. A pruning optimization strategy is also proposed to eliminate the redundant nodes from the path. Furthermore, a cubic non-uniform B-spline interpolation algorithm is applied to smooth the generated path. Finally, simulation experiments of the IRRT*-Connect algorithm are conducted in Python and ROS, and physical experiments are performed on a UR5 robotic arm. By comparing with the existing algorithms, it is demonstrated that the proposed method can achieve shorter planning times and lower path costs of the manipulator operation.