SUMMARY

The field of robotic path planning is rich and diverse. As more complicated systems have become automated, the need for simple methods that can navigate high dimensional spaces has increased. However, most path planning methods, such as Road Map methods and Search methods, increase exponentially with dimension, making them undesirable for complex robotics. Thus, the Secant and Traveling Artificial Potential Field (TAPF) approaches were developed. The Secant and TAPF approaches are modifications to the general Artificial Potential Field (APF) path planning algorithm with desirable properties which make them ideal for path planning in high dimensional space. All APF methods grow linearly with dimension; however, general APF methods are not guaranteed to converge given an arbitrary field of obstacles, significantly hindering the applicability of the APF algorithm. By specially tuning the artificial forces generated by the Secant and TAPF approaches, these methods can be shown to be globally asymptotically stable at the target location for a point robot in a field of point obstacles. To extend this theory for more practical applications, the concept of a boundary layer was introduced into the path planning algorithm. The boundary layer is a finite radius that encompasses an obstacle, such that the field is transformed within the boundary layer to account for the solid shape. By warping the landscape within the boundary layer, the system becomes mathematically equivalent to avoiding a point in space. From these advancements, the Secant and TAPF approaches were then demonstrated on planar robots and manipulators. These real-world systems were handled by selecting individual points on the robot that need to converge and threating them as separate systems coupled together by the defined constraints. For example, a planar robot is dynamically equivalent to two points constrained by a link. Similarly, a manipulator could be considered to be n-points jointed together. Thus, each point is converging to a target, but the motion is constrained by the limitations of the system. This method was successful in demonstrating convergence of robotic models. With the use of the Secant and TAPF approaches to the APF algorithm, robotic control and path planning could be drastically simplified, even for complex systems.