Academic literature on the topic '010101 Algebra and Number Theory'

Quantifiers occur in natural language as expressions using which we quantify a number of some objects in a given universe. A special class of them are intermediate quantifiers, for example, many, most, almost all, a few, a little and others. A detailed elaboration of the latter is provided in Peterson (2000). Their formal theory was established in (Novák, 2008). According to it, intermediate quantifiers form a special theory TIQ of higher-order fuzzy logic (fuzzy type theory) (FTT) with models based on the standard Łukasiewicz MVΔ-algebra.

Generic approach is one of the approaches to the study of algorithmic problems for almost all inputs, born at the intersection of computational algebra and computer science. Within the framework of this approach, algorithms are studied that solve a problem for almost all inputs, and for the remaining rare inputs give an undefined answer. This review reflects two areas of research of generic complexity of algorithmic problems in algebra, mathematical logic, number theory, and theoretical computer science. The first direction is devoted to the construction of generic algorithms for problems that are unsolvable and hard in the classical sense. In the second direction, algorithmic problems are sought that remain unsolvable or hard even in the generic sense. Such problems are important in cryptography.

Abstract A new, systematic approach to the synthesis of stiffness by springs is presented using screw (spatial vector) algebra. The space of solutions is fully characterized for all stiffnesses realizable by springs. The main result shows that a rank r stiffness can always be synthesized by r springs. Further, a stiffness can be synthesized by an arbitrarily large number of springs greater than r. Algorithms and numerical results support the theory.

Abstract The dimensional synthesis of spatial closed-loop linkages to match both force and position specifications is investigated. An efficient means of formulating force equations is introduced through the application of linear algebra to screw theory. The synthesis of spatial four-bar linkages is discussed in detail; it is shown that the maximum number of allowable design positions is not decreased after force constraints are imposed on classical position synthesis problems.

Recent results have shown that the application of group theory to the Euclidean group, E(3), and its subgroups yields a new and improved mobility criterion. Unlike the well known Kutzbach-Gru¨bler criterion, this improved mobility criterion yields correct results for both trivial and exceptional linkages. Unfortunately, this improved mobility criterion requires a little bit more than counting links and kinematic pairs. An important advance was made when it was proved that the improved mobility criterion, originally stated in a language of group theory and subsets and subgroups of the Euclidean group, E(3), can be translated into a language of the Lie algebra, e(3), of the Euclidean group, E(3), and its vector subspaces and its subalgebras. The language of the Lie algebra, e(3), is far simpler than the nonlinear language of the Euclidean group, E(3). Still, the computations required for the improved mobility criterion are more involved than those required for the Kutzbach-Gru¨bler criterion, and it might preclude the employment of the improved mobility criterion in prospective tasks such as the number synthesis of parallel and modular manipulators. This contribution dispels these doubts by showing that the improved criterion can be easily implemented by a simple computer program. Several examples are included.

The potential of In-Orbit Servicing (IOS) to extend the operational life of satellites and the need to implement Active Debris Removal (ADR) to effectively tackle the space debris problem are well known among the space community. Research on technical solutions to enable this class of missions is thriving, also pushed by the development of new control systems. Several solutions have been proposed over the years to safely capture orbital objects, the majority of which rely on robotic systems. A promising solution is the employment of an autonomous spacecraft (chaser) equipped with a highly dexterous robotic arm able to perform the berthing with a resident space object. In this respect, the design of an effective, reliable, and robust Guidance Navigation and Control (GNC) system, for which several architectures and hardware configurations are possible, plays a key role to ensure a safe mission execution. The proposed solution aims at the implementation of a combined control strategy wherein the spacecraft base and the robotic arm are controlled together. As shown by recent works, a combined architecture has several advantages over decoupled control strategies, from fuel efficiency improvement to performance improvement. Robust control methods are adopted to design control laws for the uncertain, nonlinear dynamics of the chaser and of the complete chaser-target stack after capture. Space robots are characterized by a high level of complexity due to the kinematic and dynamic coupling between its elements. The motion of a single body (be it the base or one of the links) is transmitted downstream (in the direction of the end-effector) according to the properties of the kinematic chain, while it dynamically affects the system also in the upstream direction (towards the satellite-base). From a practical standpoint this means that, unlike ground-fixed robots, the motion of the manipulator causes a motion of the base. Keeping these aspects in mind, the dynamic model of a space robot is mostly built upon the traditional theory of rigid multibody system. In this work a recursive method has been applied to the system of interconnected rigid bodies which, unlike the direct equivalent, models the interconnections in terms of forces and kinematic constraints acting at a single-body level. This results in a large set of equations which can be solved by exploiting recurrence relations descending from the tree-like structure of the system. The proposed recursive method leverages a floating-base version of both the Newton-Euler algorithm (RNEA) and the composite rigid-body algorithm (CRBA); in particular, spatial vector algebra has been used to increase compactness and efficiency of the algorithms. Following this approach, a rigid multi-body model of the system has been developed including the chaser platform and a 7 degrees-of-freedom redundant manipulator mounted on the spacecraft base. As for control design, the chosen architecture is based on a combined approach wherein base and manipulator states are controlled together, following ideas recently proposed in the literature. The specific approach developed in this work consists in using nonlinear control laws, based on extensions of the well–known computed torque controller to space robot, together with a systematic tuning procedure based on the H-Infinity framework. Indeed, while computed torque controllers deliver good tracking performance in a large domain of operating conditions, they suffer from modelling uncertainty (they are based feedback linearization) and no rule is given to tune the gains of the feedback component of the control law, which is typically based on a (nonlinear) Proportional Derivative (PD) law. Hence, trial and error procedures are often employed in practice to select the gains and achieve acceptable performance. Such an approach is made more challenging by the large number of states of space robots. Therefore, the following systematic tuning approach has been considered: first, both the plant and the control law are linearized about a nominal operating point and a linear uncertain description of the closed–loop system is derived; then, the gains of the control law are tuned by leveraging the structured H∞ framework. In this manner, the control law handles by design the rigid body nonlinearities while performance requirements can be imposed in the neighborhood of the desired configurations when tuning the gains. The proposed synthesis approach allows accounting for dynamics effects at synthesis time, such as sloshing, actuator dynamics, flexibility, orbital dynamics, which are neglected when deriving the nonlinear control law. The proposed robust control is designed in joint space and thus requires computing a reference trajectory in joint space. However, the capture of uncontrolled thumbling objects poses requirements to the trajectory generation in task space. A trajectory generation for the end effector in task space is proposed together with an inverse kinematic approach which exploits the manipulator redundancy to locally optimize the manipulability index. In addition, a feedback term is added in the inverse kinematic algorithm to make up for possible location and attitude errors, according to the Closed-Loop Inverse Kinematics (CLIK) algorithm. As the target is in an uncontrolled thumbling state, the reference trajectory generation is generated propagating forward in time the target motion, but continuously updating the propagation with information on the current state of the target coming from the Navigation System. After the controller synthesis, a robustness analysis with respect to rigid-body uncertainties (mass, moments of inertia (MoI), products of inertia (PoI) and center of mass (CoM) position) and sloshing is performed, while the performances of the proposed controller are evaluated on a representative scenario for the capturing of an uncontrolled thumbling object using a full nonlinear model of the dynamics.