The work aims to propose an improved numerical method to solve problems of dependence on the initial value and poor convergence of singular poses in the traditional numerical method in terms of complex inverse kinematics of the offset redundant manipulator. First, the nonlinear equations were transformed into homotopy equations. The introduction of homotopy continuation algorithm could effectively avoid the initial value dependence problem, and at the same time obtain the inverse kinematics solution space. Then taking the singular pose into account, the homotopy equations were transformed into a least squares problem, and the Levenberg Marquardt algorithm was used to track the path of the homotopy equations to obtain the inverse kinematics solution space. Finally, the joint limit avoidance problem was mapped to a solution space optimization problem, and the binary improved particle swarm optimization algorithm was introduced to obtain the optimal inverse kinematics solution. The experimental results showed that the numerical method proposed in this paper had a higher convergence rate and faster convergence speed for inverse kinematics than the traditional numerical method, and the binary improved particle swarm optimization algorithm could effectively avoid the joint limit problem. This numerical method has high accuracy in solving inverse kinematics and can meet real-time requirements, and has certain theoretical significance and engineering application value for the subsequent use of the manipulator in packaging operations.