| This dissertation has two primary goals which are to find an improved method to derive the Configuration space (C-space) associated with a manipulator in an obstacled environment, then to implement a novel approach to solve the minimum energy path planning problem in the newly formed C-space.; The Jacobian method of generating the C-space uses a uniform upper bound on the maximum displacement for any point on the manipulator given a defined input to each joint. This allows for blocks of C-space to be defined as "free", "obstacled", or "unknown" with the unknown blocks further refined into smaller blocks until they can be defined as free or obstacled. The process continues until the entire C-space is determined within a given precision. This research implements a non-uniform bound on the maximum displacement for points on the manipulator, making the maximum displacements a function of the current position of the manipulator. This non-uniform bound allows the Jacobian method to more quickly define areas of C-space as either obstacled or free, thus allowing a faster convergence of the entire C-space.; Once a given C-space has been determined, our path planning algorithm minimizes the energy required to perform any task where the manipulator moves from point A to point B. This begins with both the kinematic and dynamic modeling of the robot and subsequent derivation of the equations of motion and the joint torques of the dynamic system. The energy functional is built by adding the squares of the joint torques and integrating over the time allowed to perform the task. Obstacles are added to the energy integral using a penalty function technique which severely penalizes any path which attempts to cross into an obstacle-filled region.; The Euler-Poisson equation is applied to the integrand of the energy integral to produce two coupled, non-linear, ordinary differential equations. The solution to these ODE's gives the minimum energy path in the obstacled C-space environment. Finite-difference operators replace derivatives and the Newton-Raphson technique is applied to solve the coupled ODE's and provide the minimum energy path. The solution is only minimized in a local sense and therefore a series of initial paths are selected to check for multiple solutions. The results find minimum energy paths for even the more challenging C-space's which result from multiple obstacles within the work envelope. The algorithm proves robust by converging to solutions from initial paths which pass through both obstacle-free and obstacle-filled regions. |
