Path defined directed graph vector (pgraph) method for multibody dynamics

The dynamic model of multibody systems plays an important role in a wide range of applications ranging from robotics, vehicle and machine design, computer animations and virtual reality applications. A significant number of studies have been conducted to develop algorithms that allow efficient computation of dynamic models of multibody systems in computer environment. However, fewer studies have been done to develop algorithms to reduce the configuration time of the system to be modeled and to minimize the most possible modeling errors. In the dynamic model of multibody systems, the main objective is to obtain the equation of motion of the system. Reduction of model configuration time is achieved by developing the modeling environments in which parameters including geometric and mass informations related to the system are taken as inputs and the equations of motion are obtained as output. For a system in serial topology, it is enough to take the information of joint degrees of freedom and axes of rotation in addition to geometric and mass informations of the system. For a system in a tree topology or closed kinematic-chain, the parameters that represent the connection between the bodies must also be taken as input. The connections between the bodies can be mathematically expressed by methods based on graph theory. What is important here is to include connections between the bodies into the formulation in such a way that equations of motion of the system can be generated automatically. While equations of motion are obtained, Euler-Lagrange and Newton-Euler based formulations are generally used. In Euler-Lagrange based modeling, while creating the Lagrangian function related to the kinetic and potential energy, the propagation of body velocities must be computed according to the global coordinate frame. On the other hand, in Newton-Euler based modeling, the propagation of body velocities and forces must be computed according to the global coordinate frame. The commonly used method in both approaches is to define the coordinate frames at the joint locations and rotation matrices between this coordinate frames, then computing velocity and forces related to bodies according to the global coordinate frame. From this point of view, for a system in serial topology, the above processes can be done without any graph-based method since the connections between the bodies are uniform. However, graph-based methods are crucial for complex systems involving tree topology or closed kinematic-chain. In the thesis, it is aimed to create a modeling environment which will allow to reduce the modeling errors and configuration time for modeling multibody systems. In this environment, a graph theory based approach which defines the connections between the bodies has been developed and the dynamic model of the systems including tree topology and closed kinematic-chain has been systematically obtained with this approach. Complex multibody systems may include multiple combinations of serial and tree topologies, as well as closed kinematic-chain systems. Simulations of such systems is a costly task not only for run-time but also for construction-time when model description and constraints are introduced. In this thesis, we present a systematic framework of a construction-time efficient modeling methodology for complex multibody systems by introducing path defined directed graph vector (Pgraph) and the associated methodology based on Linear Graph Theory (LGT). This method is especially useful for systems changing topology, such as walking mechanisms where bilateral and unilateral constraints are conditionally embedded. Although this is a general methodology to be combined with many dynamical modeling algorithms available in the literature, here we introduce methodology and applications using Spatial Operator Algebra (SOA) which is a recursive algorithm based on Newton-Euler formalism. We also introduced the implementation of Pgraph method to mostly known Denavit-Hartenberg Convention for updating body coordinate frames. Another focus of this thesis is analyzing external constraint forces, especially computation of unilateral constraint forces in wheeled ground vehicles. Our aim is to get a model to obtain traction forces between the wheels and ground. These forces are important for real time vehicle control systems to prevent the roll over. We developed a model that provides computation of traction forces for wheel dynamics with a modular approach that can be extended for two or more wheeled vehicles. Traction forces are computed by defining the point contact for the situation where the wheels are subjected to no-slip. Frictional forces which are dependent on the characteristic parameters of the road and wheel at the contact point can be included to the system. In this thesis, bicycle dynamics is taken as an example of a two-wheeled vehicle, the traction forces between the ground and the bicycle wheels are computed in the case of different actuating forces applied to a bicycle. Another example that is addressed within the context of ground vehicles is the truck and trailer system. The modeling of a system with both tree topology and many wheels in the developed environment is important to show the potential of the Pgraph method. Visualization of multibody systems in a 3D environment allows us to do better dynamic analysis of multibody systems. Both the 3D CAD drawings and the 3D shapes representing the bodies are supplied as input with Pgraph vector to the model configuration. Pgraph method is enabling the system to be automatically created in the 3D environment. In this thesis, although the "3D Animation Toolbox" which is included in the MATLAB software as an infrastructure for 3D environment is used, the developed algorithm can be integrated into a popular environment such as ROS (Robot Operating System) and so on. The thesis consists of five parts. In the first part, a summary of the existing studies in the literature about modeling multibody systems is given. Besides the aim and the motivation of the thesis are expressed. In the second part, the graphical representation of the systems is introduced, and the Spatial Operator Algebra (SOA) is given with Linear Graph Theory. Then, the Pgraph method is described by applying it to the SOA method. In the third chapter, motion equations under external constraint forces of multibody systems are analyzed. In the fourth chapter, modeling studies and simulation results of the example systems which will provide better understanding of the developed methods and algorithms are given. In the fifth section, the concluding remarks and future works are given.