Differential-algebraic equations (DAEs) consist of mixed systems of ordinary differential equations (ODEs) coupled with linear or nonlinear equations. Such systems may be viewed as ODEs with integral curves lying in a manifold. DAEs appear frequently in applications such as classical mechanics and electrical circuits. This thesis concentrates on systems of index 2, originally index 3, and mixed index 2 and 3. Fast and efficient numerical solvers for DAEs are highly desirable for finding solutions. We focus primarily on the class of Gauss-Lobatto SPARK methods. However, we also introduce an extension to methods proposed by Murua for solving index 2 systems to systems of mixed index 2 and 3. An analysis of these methods is also presented in this thesis. We examine the existence and uniqueness of the proposed numerical solutions, the influence of perturbations, and the local error and global convergence of the methods. When applied to index 2 DAEs, SPARK methods are shown to be equivalent to a class of collocation type methods. When applied to originally index 3 and mixed index 2 and 3 DAEs, they are equivalent to a class of discontinuous collocation methods. Using these equivalences, (s,s)--Gauss-Lobatto SPARK methods can be shown to be superconvergent of order 2s. Symplectic SPARK methods applied to Hamiltonian systems with holonomic constraints preserve well the total energy of the system. This follows from a backward error analysis approach. SPARK methods and our proposed EMPRK methods are shown to be Lagrange-d'Alembert integrators. This thesis also presents some numerical results for Gauss-Lobatto SPARK and EMPRK methods. A few problems from mechanics are considered.

In recent years, there has been a significant increase in the interest for snake like modular robots due to their superior locomotion capabilities in terms of versatility, adaptability and scalability. Passive wheeled planar snake like robots are a major category and they are being actively researched. Due to the nonholonomic constraints imposed on them, certain configurations lead to the singularity which must be avoided at all costs. Furthermore, it is vital to generate a locomotion pattern such that they can track a wide range of trajectories. All of these objectives must be accomplished smoothly and in an energy efficient manner. Studies indicate that meeting all of these requirements is a challenging problem. In this study, a novel form of the serpenoid curve is proposed in order to make the robot track arbitrary paths. A controller has been designed using the feedback linearization method. Afterwards, a new performance measure, considering both the efficiency and sustainability of the locomotion, has been proposed to evaluate the locomotion. Optimal parameters for the proposed serpenoid curve and the linear controller have been determined for efficient locomotion by running series of simulations. Relations between the locomotion performance, locomotion speed and eigenvalues of the linear controller have been demonstrated. Simulation results show striking differences between the locomotion by using the proposed serpenoid curve with optimal parameters and the locomotion by purely tracking a given path. Obtained results also indicate that the aforementioned requirements are met successfully and confirm the validity and consistency of the proposed performance measure.