Determination of Optimal Path Replanning Overview
One of the principals worked with the US Navy in developing energy efficient optimal control-based path replanners for an autonomous generic surface vehicle in ocean current. The optimal control algorithm, which was constrained by the vehicle dynamics, determined the optimal strategy for the thrust and steering for the vehicle to first reorient itself and then determine the optimal trajectory to follow the initial prescribed nominal path.
US Navy Increased Autonomous Capabilities
IDM Solutions provided a detailed dynamic mathematical model that represented a generic autonomous surface vessel that enables any vessel to be used in the model. A simulation environment was provided to the US Navy that allowed for the determination of an energy efficient optimal control strategy for thrust and steering for the vehicle to first reorient itself and then determine the optimal trajectory to follow the initial prescribed nominal path.
Autonomous Systems Planning Requirements
A critical component of all autonomous mechanical systems is the ability to follow prescribed motion trajectories. For autonomous vehicles, this motion trajectory is given by the determination of the motion path for the vehicle to follow to get from one point to another. Determining the best path to take and performing the mechanical functions to follow that path (that is, setting a thrust and steering) are basic functions of all driving and piloting that are often taken for granted when the vehicle is a maimed (or remotely operated) system. For autonomous military systems, however, optimal path planning and optimal vehicle control are not simple, routine functions; they are complex operations affected by many parameters. As more and more tasks are transferred to autonomous agents, determining an optimal control strategy becomes increasingly crucial to system performance to assure platform safety, platform efficiency, and successful mission completion.
Description of Generic Surface Vehicle
IDM Solutions used a three degree-of-freedom surface vehicle model (Fig. 1) that considers surge, sway, and yaw dynamics and neglects heave, roll, and pitch dynamics. The vehicle-fixed coordinate system is positioned at the center of mass with linear translational velocities {u, v} and rotational velocity {r = θ}. Similarly, the inertial (earth-fixed) reference frame {X,Y} is positioned at the center of mass of the vehicle with linear translational velocities {x,y} and rotational velocity { θ}. The equations of motion for the vehicle whose state variables are linear and rotational velocities are depicted in Fig. 2.
Figure 1: Schematic of generic surface vehicle
Figure 2: Full state equations of motion with forcing and external ocean current forcing
Optimal Control to Follow Nominal Path
In practice, an autonomous vehicle may not be in the specific location or orientation when that vehicle must execute a prescribed nominal path trajectory to complete its mission. This is especially true for autonomous vehicles in ocean current that may have just completed a mission or arrived at a location and has been waiting for its next task to start.
IDM Solutions considered an initial prescribed nominal path trajectory with an initial orientation. Next, IDM Solutions developed a two-part methodology where an optimal control strategy was developed to first reorient the vehicle from its current orientation followed by a nominal path replanner to allow for the vehicle to follow the nominal path trajectory using the least amount of energy. The optimal control strategy for the vehicle was found using dynamic programming. The constraints were introduced to the optimize framework by the vehicle’s dynamic state equations are show in Fig. 2. The optimal goal was to minimize the following objective function:
Figure 3 depicts the results for a controlled and no control case. Figure 3 (left) depicts the nominal path that was initially prescribed for the vehicle to follow. Due to the perturbed initial conditions on the vehicle’s orientation, the trajectory of the vehicle almost goes in the complete opposite direction. The controlled solution is obtained, and the optimal trajectory first reorients the vehicle and then follows the nominal path. The figure in the middle shows the results of the reorientation. The figure on the right depicts the applied thrust that is determined from the optimal control strategy where there is a significant amount of energy first taken and then minimal energy is used for the remainder of time.
Figure 3: (Left) Optimal constrained control applied to the perturbed initial conditions, (middle) difference between the nominal and actual trajectories, and (right) applied thrust needed for control