Due to the high demand for aerial robots, it is essential to guarantee stable systems for the tasks assigned to these robots. Which is linked to control and stability; therefore, we talk about the design of optimal control algorithms. In this work, Lyapunov stability theory is used for trajectory tracking and three-dimensional potential field theory for obstacle avoidance. The Lyapunov candidate function was chosen in compliance with the requirements for the necessary stability, being necessary in the tracking of trajectories to saturate the speeds of the aerial robot and in the avoidance of obstacles, the theory of potential fields is applied, which builds a field potential with gradient therefore rejects obstacles. To demonstrate that there is an optimal algorithm that allows the aerial robot to follow trajectories in a stable way and avoid obstacles, we have compared the results with solutions implemented with controllers using numerical methods and implemented in reality and in simulation, seeing that the errors tend to zero from one quickly and their speeds are consistent with the reality of these robots. We have worked different test trajectories and we have had speeds in different ranges such as 5 m/s, -3.8 m/s and 7 m/s or 1.8 m/s and 2.2 m/s these speeds depend on the type of trajectory, as well as if it has obstacles , we can see all this in the figures of the work simulations, in the same way we can see the errors that tend to 0 m at different times 2 s, 6 s. The results of this research can be applied in the design of controllers for aerial robots, offering stable systems in the task assigned to the aerial robot.

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