Doctoral Degrees (Applied Mathematics)

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Browsing Doctoral Degrees (Applied Mathematics) by Subject "Mobile robots -- Mathematical models"

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    Path planning for wheeled mobile robots using an optimal control approach
    (Stellenbosch : Stellenbosch University, 2019-12) Matebese, Belinda Thembisa; Banda, Mapundi K.; Withey, Daniel; Brink, Willie; Stellenbosch University. Faculty of Science. Department of Mathematical Sciences (Applied Mathematics).
    ENGLISH ABSTRACT: The capability and practical use of wheeled mobile robots in real-world applications have resulted in them being a topic of recent interest. These systems are most prevalent because of their simple design and ease to control. In many cases, they also have an ability to move around in an environment without any human intervention. A main stream of research for wheeled mobile robots is that of planning motions of the robot under nonholonomic constraints. A typical motion planning problem is to find a feasible path in the configuration space of the mobile robot that starts at the given initial state and reaches the desired goal state while satisfying robot kinematic or dynamic constraints. A variety of methods have been used to solve various aspects of the motion planning problem. Depending on the desired quality of the solution, an optimal path is often sought. In this dissertation, optimal control is employed to obtain optimal collision-free paths for two-wheeled mobile robots and manipulators mounted on wheeled mobile platforms from an initial state to a goal state while avoiding obstacles. Obstacle avoidance is mathematically modelled using the potential field technique. The optimal control problem is then solved using an indirect method approach. This approach employs Pontryagin’s minimum principle where analytical solutions for optimality conditions are derived. Solving the optimality condition leads to two sets of differential equations that have to be solved simultaneously and whose conditions are given at different times. This set of equations is known as a two-point boundary value problem (TPBVP) and can be solved using numerical techniques. An indirect method, namely Leapfrog, is then implemented to solve the TPBVP. The Leapfrog method begins with a feasible trajectory, which is divided into smaller subdivisions where the local optimal controls are solved. The locally optimal trajectories are added and following a certain scheme of updating the number of subdivisions, the algorithm ends with the generation of an optimal trajectory along with the corresponding cost. An advantage of using the Leapfrog method is that it does not depend on the provision of good initial guesses along a path. In addition, the solution provided by the method satisfies both boundary conditions at every step. Moreover, in each iteration the paths generated are feasible and their cost decreases asymptotically. To illustrate the effectiveness of the algorithm numerically, a quadratic cost with the control objective of steering the mobile robot from an initial state to a final state while avoiding obstacles is minimized. Simulations and numerical results are presented for environments with and without obstacles. A comparison is made between the Leapfrog method and the BVP4C optimization algorithm, and also the kinodynamic-RRT algorithm. The Leapfrog method shows value for continued development as a path planning method since it initializes easily, finds kinematically feasible paths without the need of post processing and where other techniques may fail. To our knowledge the work presented here is the first application of the Leapfrog method to find optimal trajectories for motion planning on a two-wheeled mobile robot and mobile manipulator.