Visual servoing (VS) is a control technique that employs visual features captured by a camera to guide robots toward desired targets. According to the retrieved visual features, VS is commonly divided into position-based VS (PBVS), image-based VS (IBVS) and homography-based VS (HBVS). Apart from the specified VS task, obstacle avoidance (OA) and joint-limit avoidance (JLA) are crucial for ensuring safety and reliability of the robot. This paper focuses on developing a quadratic programming (QP) framework that unifies the aforementioned different types of visual servoing methods with OA and JLA capabilities for joint-constrained redundant robots. Then, a gradient-dynamics based neurodynamic network (GDNN) is designed to serve as a QP solver. Experiments conducted using two Franka Emika Panda robots demonstrate the validity and practicality of the established QP framework for achieving VS tasks with OA and JLA considered.