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Controller synthesis using Control Lyapunov and Barrier functions in a quadratic program has become increasingly popular due to its simplicity and ability to handle nonlinear systems. However, they suffer from local minima problems like standard potential field path planners. We augment the controller synthesis algorithm with RRT to re-plan dynamically to avoid the local minima.
A simulation of the planar double integrator using the controller synthesized from solving a quadratic program using Control Lyapunov and Barrier functions. The system (black dot) is tasked with following a moving trajectory (gray dot) while avoiding dynamic obstacles (green circles). The system is able to track the reference outside of obstacles and safely wait/avoid obstacles when it is unsafe to follow.
The CLF-CBF quadratic program controller synthesis algorithm also runs into local minima problems as shown above. The local minima occurs because the controller only satisfies the convergence property when it is safe to do so. In the above scenario, the attractor wants the system to move forward but the safety constraint prevents that motion. Thus the system is stuck.
To resolve the local minima problem, we instantaneously plan a temporary trajectory using RRT (red path) assuming the obstacles are stationary. Once the system starts following the temporary path, the CLF-CBF controller will account for any dynamic obstacles that were not considered by RRT.
After planning a temporary trajectory, the system follows the trajectory until it is safe to go directly to the goal. In this manner, the system is not bound to a trajectory that may no longer go to the goal. In this instance, the target position is stationary, but by leaving the temporary trajectory the controller is more optimal.
In the case where the goal position changes after the RRT planning, the condition to leave the temporary path allows the system to quickly converge to the moving target.
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