| Sažetak (engleski) | The research focuses on the adaptation and application of a newly designed geometric integration algorithms on numerical tasks of flight vehicle dynamics and evaluation of their benefits and quality of adaptation in the framework of numerical simulation procedures for calculating the dynamic response of aircrafts and spacecrafts. With regard to the standard numerical methods for integrating ordinary differential equations that operate on vector spaces, targeted geometric algorithms are designed on manifolds and Lie groups. Starting geometric methods are shaped in the form that algorithmically satisfies kinematic and dynamic constraints that come from the differential geometric structure of system. Upon completion of description of the individual derivation of mathematical models of the selected algorithms and their adaptation to the target tasks of integration of flight vehicle dynamics, the examples of applying geometric integration methods are presented and compared with classical procedures of linear vector spaces. This thesis is organized in seven chapters, as follows: Chapter 1: Introduction. This chapter presents the relevance of the research. It gives an overview of the relevant literature related to the topic of the thesis and explains the scope of the research. The first chapter is divided into five sections which includes: motivation, overview of the previous research, research goals and hypotheses, methodology of the research, expected scientific contributions and structure of the thesis. Chapter 2: Manifolds. In this chapter manifolds are introduced and explained. Firstly manifolds are explained and then vector fields, vector spaces, curves and vector fields commutation are shortly described. The mathematics and terminology described here enables a better understanding of the following sections. Chapter 3: Rotations and the SO(3) group. In this chapter rotations and their mathematical description in terms of groups and manifolds is presented. Until now no formal mathematical mention of the rotations was made and the dynamics was mostly focused on particles and body translations. After the detailed presentation of rotations of the following chapter the description of the individual derivation of mathematical models of the selected algorithms is presented. Chapter 4: Lie-group integration method for constrained multibody systems in state space. Coordinate-free Lie-group integration method of arbitrary (and possibly higher) order of accuracy for constrained multibody systems (MBS) is described, adapted and applied to the helicopter forward dynamics in this chapter. Mathematical model of MBS dynamics is shaped as DAE system of equations of index 1, while dynamics is evolving on the system state space modelled as a Lie-group. Since formulated integration algorithm operates directly on the system manifold via MBS elements’ angular velocities and rotational matrices, no local rotational coordinates are necessary and kinematical differential equations (that are prone to singularities in the case of 3-parameters-based local description of the rotational kinematics) are completely avoided. Basis of the integration procedure is the Munthe-Kaas algorithm for ODE integration on Lie-groups, which is reformulated and expanded to be applicable for the integration of constrained MBS in the DAE-index-1 form. Dynamic simulation procedures of aircraft 3D motion need robust and efficient integration methods in order to allow for reliable (and possibly real-time) simulation missions. To this end, derivation of such integration schemes in coordinate-free Lie-group setting could be a promising start, since Lie-group dynamical models operate directly on SO(3) rotational matrices and angular velocities, avoiding local rotation parameters and additional algebraic constraints as well as kinematical differential equations. These integration characteristics should be especially beneficial for the flight vehicle simulation missions since a realization of the aircraft complex 3D maneuvers often requires numerical forward dynamics that includes complete 3D rotation domain. In such cases, the utilization of the 'standard' vector-space-based modeling procedures (with the local rotation parameters) leads toward kinematical singularities and re-parameterization of the rotation domain, which requires further computational burden. Along this line, a numerical integration scheme in Lie-group settings for the helicopter forward dynamics is presented and discussed in this chapter. The presented formulation provides a compact integration platform for smooth flight vehicle dynamics numerical integration, independently of the character of the 3D rotations involved and without introducing additional 'artificial' algebraic constraints. For the initial case study a simple maneuver of a helicopter is selected with generic transport helicopter modeled as a single 6DOF rigid body problem. The proposed Lie-group DAE-index-1 integration scheme is easy-to-use for a MBS with kinematical constraints of general type and it is especially suitable for dynamics of mechanical systems with large 3D rotations where standard (vector space) formulations might be inefficient due to kinematical singularities (3-parameters-based rotational coordinates) or additional kinematical constraints (redundant quaternion formulations). Chapter 5: Singularity-free time integration of rotational quaternions using non redundant ordinary differential equations. In this chapter a novel time stepping scheme for solving rotational kinematics, formulated as an ODE in terms of unit quaternions, is described, adapted and applied to the numerical forward dynamics of a fixed-wing aircraft. This scheme inherently respects the unit-length condition without including it explicitly as a further equation, as it is common practice. In the standard algorithms, the unit-length condition is included as an additional equation leading to kinematical equations in the form of a system of differential-algebraic equations (DAE). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form an ordinary differential-equations (ODE) on Lie-algebra so(3) of the rotation group SO(3). This rotation vector then defines an incremental rotation (and thus the associated unit-quaternion), and the rotation update is determined by projection of the incremental vector on the quaternion group via pertinent exponential mapping. Since the kinematic ODE on so(3) can be solved by using any standard (possibly higher-order) ODE integration scheme, the result of the procedure is a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions that allows for application of general ODE integration schemes, and thus avoids integration of DAE equations. This solves a long-standing problem of necessity to deal with DAE’s during integration of rotational kinematics, which has been a major drawback of using quaternions. As a numerical example, a 3D motion of a general aviation airplane, modeled as a flat-earth 6DOF single rigid body problem, is presented. Chapter 6: An angular momentum and energy conserving Lie-group integration schemes for rigid body rotational dynamics originating from Störmer-Verlet algorithm. This chapter describes two novel 2nd order conservative Lie-group geometric methods for integration of rigid body rotational dynamics. Firstly described algorithm is a fully explicit scheme that exactly conserves spatial angular momentum of a free spinning body. The method is inspired by the Störmer-Verlet integration algorithm for solving ordinary differential equations, which is also momentum conservative when dealing with ODEs in linear spaces but loses its conservative properties in a non-linear regime, such as non-linear SO(3) rotational group. Then, is described an algorithm that is an implicit integration scheme with a direct update in SO(3). The method is algorithmically designed to conserve exactly both of the two 'main' motion integrals of a rotational rigid body, i.e. spatial angular momentum of a torque-free body as well as its kinetic energy. As it is shown in the thesis, both methods also preserve Lagrangian top integrals of motion in a very good manner, and generally better than some of the most successful conservative schemes to which the proposed methods were compared within the presented numerical examples. The proposed schemes can be easily applied within the integration algorithms of the dynamics of general rigid body systems. The system under consideration is a satellite modeled as a single rigid body problem. Chapter 7: Conclusion. This chapter summarises the main contributions of the thesis and gives several recommendations for future research. |
