For safe and efficient navigation of heterogeneous multiple mobile robots (HMRs), it is essential to incorporate dynamics (mass and inertia) in motion control algorithms. Many methods rely only on kinematics or point-mass models, resulting in conservative results or occasionally failure. This is especially true for robots with different masses. In this article, we develop a novel navigation methodology for a distributed scheme by incorporating the robots' dynamics through calculating the time to collision (TTC) and designing a new controller accordingly that avoids collisions. We first propose a new predictive collision term by TTC that will be used to quantify imminent collisions among HMRs. Subsequently, using this term, we develop a novel nonlinear controller that explicitly incorporates TTC in the design and guarantees collision-free motion. Simulations and experiments were performed to demonstrate the effectiveness of the developed methods. We first compared the results of our proposed approach with controllers that only consider the robots' kinematics. It was shown that the proposed control strategy (a TTC-based controller) proves to be less conservative when determining safe motions. Specifically, for environments with limited space, it was demonstrated that using robots' kinematics may result in a collision, while our strategy results in safe motion. We also performed experiments that proved collision-free navigation of HMRs with this approach. The outcomes of this work provide more reliable motion control for HMRs, especially when the robots' masses or inertias are significantly different, for example, warehouses. The developments in this work are also applicable to vehicles and can therefore be beneficial in automated collision avoidance in autonomous driving and intelligent transportation.