Pseudospectral Motion Planning for Autonomous Vehicles

Abstract

A COMMON task for autonomous vehicles is motion planning. Discipline-based design of motion planning algorithms have led to the development and evolution of different techniques to solve specific problems. For instance, the artificial–potential–function technique [1] is a popular method for the motion planning of unmanned ground vehicles (UGV) and robotic manipulators [1–3]. Although this technique has been used for over 30 years, it suffers from the possibility of the vehicle not achieving its goal and difficulties in accommodating various environmental constraints [4]. To overcome such issues, recent developments in motion planning algorithms place heavy emphasis on so-called sampling-based planning techniques such as probabilistic road maps, rapidly exploring random trees, and expansive space trees to name a few [5– 9]. These methods use probabilistic means of connecting the initial configuration to thefinal configuration thereby enabling an improved capacity to achieve the goal and a capability to generate initial feasible paths. In all these techniques, the initial feasible paths do not automatically incorporate vehicle dynamics; hence, path-following control techniques are needed to satisfy the physics of themotion [9]. This is one reason why nonholonomic constraints play such a crucial role in constrained control techniques that are designed to serve pathfollowing systems. There is no doubt that optimal control theory is the most natural framework for solving motion planning problems; however, solving optimal control problems has historically been considered difficult due to the twin curses of dimensionality and complexity. These difficulties are exacerbated in the presence of state constraints; hence, an obstacle-cluttered environment becomes a substantially more difficult problem under the framework of optimal control theory [10,11]. Nonetheless, it is possible to solve simplified motion planning problems wherein the cost function is quadratic or the environment is obstacle free. Because the motion planning techniques developed in robotics applications are unsuitable for flight vehicles, such as launch and reentry, for example, aerospace problems have motivated the development of efficient optimal control algorithms. In aerospace applications, satisfaction of the dynamical constraints is exceedingly important; hence, trajectory and control become fundamentally intertwined. On the other hand, aerospace problems do not have the vast number of path constraints that are common in an obstacle-cluttered environment. In recent years, paradigm-changing advancements have taken place in computational optimal control that challenge conventional wisdom. For instance, onboard the International Space Station, Bedrossian et al. [12] discovered and implemented a revolutionary momentum-dumping approach that they call a zero-propellant maneuver. This discovery was made possible by an application of pseudospectral methods to solve a real-life challenging optimal control problem [13]. Other applications of such advancements are discussed in [14,15] and include the ground test of a revolutionary attitude control concept for the NPSAT1 spacecraft. Motivated by these advancements, we apply pseudospectral methods to develop motion planning algorithms for autonomous vehicles characterized by nonlinear dynamical constraints, an obstacle-cluttered environment, and a need to generate solutions in real time. We consider the problem of generating optimal trajectories for generic autonomous vehicles. Shapes of arbitrary number, size, and configuration are modeled in the form of path constraints in the resulting optimal control problem. The method is tested under various obstacle environments on different platforms such as sea surface vehicles, ground vehicles, and aerial vehicles. The optimality of the computed trajectories is verified by way of the necessary conditions. We show that it is possible to do motion planning for different problems under the unified framework of optimal control and pseudospectral methods [16–19].