This study presents an integrated framework for enhancing the safety and operational efficiency of robotic arms in laparoscopic surgery by addressing minimum distance estimation between multi-arm manipulators and the associated collision-aware warning. By combining analytical modeling, real time simulation, and machine learning, the framework offers a robust solution for ensuring safe robotic operations. An analytical model was developed to estimate the minimum distances between robotic arms based on their joint configurations, offering theoretical calculations that serve as both a validation tool and a benchmark. To complement this, a 3D simulation environment was created to model two 7 DOF Kinova robotic arms (Kinova inc., Boisbriand, QC, Canada), generating a diverse dataset of configurations for distance estimation and collision warning. Using these insights, a deep residual neural network model was trained with joint configurations as inputs. On the held out validation set, the model achieves R2 = 0.940, RMSE = 42.0 mm, MAE = 28.7 mm, and a near zero mean bias, demonstrating strong predictive accuracy and consistent generalization across the workspace. The framework is intended as an early collision warning layer, where a warning is triggered when the predicted inter-arm distance falls below a 0.2 m threshold, which corresponds to a surface to surface clearance of approximately 50 mm given the Kinova Gen3 (Kinova inc., Boisbriand, QC, Canada) cross sectional radius. This work demonstrates the effectiveness of combining analytical modeling with machine learning to enhance the precision and reliability of multi-arm robotic systems.