Koopman Operator Based Modeling and Control of Rigid Body Motion Represented by Dual Quaternions

In this paper, we systematically derive a finite set of Koopman based observables to construct a lifted linear state space model that describes the rigid body dynamics based on the dual quaternion representation. In general, the Koopman operator is a linear infinite dimensional operator, which means that the derived linear state space model of the rigid body dynamics will be infinite-dimensional, which is not suitable for modeling and control design purposes. Recently, finite approximations of the operator computed by means of methods like the Extended Dynamic Mode Decomposition (EDMD) have shown promising results for different classes of problems. However, without using an appropriate set of observables in the EDMD approach, there can be no guarantees that the computed approximation of the nonlinear dynamics is sufficiently accurate. The major challenge in using the Koopman operator for constructing a linear state space model is the choice of observables. State-of-the-art methods in the field compute the approximations of the observables by using neural networks, standard radial basis functions (RBFs), polynomials or heuristic approximations of these functions. However, these observables might not providea sufficiently accurate approximation or representation of the dynamics. In contrast, we first show the pointwise convergence of the derived observable functions to zero, thereby allowing us to choose a finite set of these observables. Next, we use the derived observables in EDMD to compute the lifted linear state and input matrices for the rigid body dynamics. Finally, we show that an LQR type (linear) controller, which is designed based on the truncated linear state space model, can steer the rigid body to a desired state while its performance is commensurate with that of a nonlinear controller. The efficacy of our approach is demonstrated through numerical simulations

The geometric structure of unit dual quaternion with application in kinematic control

AbstractIn this paper, the geometric structure, especially the Lie-group related properties, of unit dual quaternion is investigated. The exponential form of unit dual quaternion and its approximate logarithmic mapping are derived. Correspondingly, Lie-group and Lie-algebra on unit dual quaternions and the approximate logarithms are explored, respectively. Afterwards, error and metric based on unit dual quaternion are given, which naturally result in a new kinematic control model with unit dual quaternion descriptors. Finally, as a case study, a generalized proportional control law using unit dual quaternion is developed

Hybrid kinematic control for rigid body pose stabilization using dual quaternions

In this paper, we address the rigid body pose stabilization problem using dual quaternion formalism. We propose a hybrid control strategy to design a switching control law with hysteresis in such a way that the global asymptotic stability of the closed-loop system is guaranteed and such that the global attractivity of the stabilization pose does not exhibit chattering, a problem that is present in all discontinuous-based feedback controllers. Using numerical simulations, we illustrate the problems that arise from existing results in the literature—as unwinding and chattering—and verify the effectiveness of the proposed controller to solve the robust global pose stability problem

A convex-programming-based guidance algorithm to capture a tumbling object on orbit using a spacecraft equipped with a robotic manipulator

An algorithm to guide the capture of a tumbling resident space object by a spacecraft equipped with a robotic manipulator is presented. A solution to the guidance problem is found by solving a collection of convex programming problems. As convex programming offers deterministic convergence properties, this algorithm is suitable for onboard implementation and real-time use. A set of hardware-in-the-loop experiments substantiates this claim. To cast the guidance problem as a collection of convex programming problems, the capture maneuver is divided into two simultaneously occurring sub-maneuvers: a system-wide translation and an internal re-configuration. These two sub-maneuvers are optimized in two consecutive steps. A sequential convex programming procedure, overcoming the presence of non-convex constraints and nonlinear dynamics, is used on both optimization steps. A proof of convergence is offered for the system-wide translation, while a set of structured heuristics—trust regions—is used for the optimization of the internal re-configuration sub-maneuver. Videos of the numerically simulated and experimentally demonstrated maneuvers are included as supplementary material

Controle híbrido para estabilização de pose usando quaternions duais

Tese (doutorado)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2018.Motivado tanto pelas vantagens da representação em dual quatérnios duais e por problemas relativos à obstrução topológica de se ter um equilíbrio assintótico globalmente estável, esse trabalho visa usar o formalismo de quaternion dual e as ferramentas de sistemas dinâmicos híbridos para tratar o problema de estabilização de pose de corpos rígidos. O grupo de Lie dos quatérnios duais proporciona um modo eficiente de representar a cinemática linear e rotacional de um corpo rígido sem singularidades. Algumas estratégias híbridas são propostas para lidar com o problema de “chattering” presente em todos os controladores por realimentação descontínuos enquanto ao mesmo tempo garantindo atratividade global da pose de estabilização do corpo rígido.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) e Fundação de Apoio à Pesquisa do Distrito Federal (FAP-DF).Motivated both by the advantages of the dual quaternion representation and by the problems concerning the topological obstruction to global asymptotic stability, this work addresses the rigid body pose stabilization problem using dual quaternion formalism and dynamic hybrid systems tools. The Lie group of unit dual quaternions provides a computationally efficient way to represent coupled linear and rotational kinematics without singularities. Some hybrid control strategies are proposed to overcome the chattering problem present in all discontinuous-based feedback controllers while at same time also guaranteeing global attractivity of the stabilization pose of the rigid body

Quaternion-based Dual loop Nonlinear Trajectory Control of Quadrotors

Many multirotor controllers achieve globally stable attitude control through the usage of quaternions, as it does not inherent any singularities, unlike other alternatives such as Euler angles or the Direction Cosine Matrices (DCM). However, globally stable position or velocity control of quadrotors have rarely been achieved; most controllers limit their attitude within a certain range to avoid such singularities and lose stability. This thesis focuses on quadrotors and presents a globally stable quaternion-based dual loop nonlinear control scheme so that the quadrotor can achieve any attitude configuration during position or velocity control without losing stability. The two control loops are structured so that the outer loop controls the translational velocity or position, and the inner loop controls the attitude. The outer control loop uses a Proportional-Integral (PI) feedback structure. The proportional action is in terms of the translational position or velocity of the quadrotor, and the integral action is primarily used to eliminate steady-state error. The inner attitude control loop is a Proportion-Derivative (PD) feedback loop, where the proportional and derivative action is in terms of the vector component of the quaternion of the quadrotor attitude and quadrotor angular velocity, respectively. The two control loops are linked in a manner so that a globally exponentially stability can be achieved. Additional feedback and feedforward, which acts as compensations for the nonlinear dynamics of the quadrotor and gyroscopic torques, were further employed to guarantee the global stability of the quadrotor. The thesis also investigates different singularity and ambiguity issues, more specifically singularities that can arise during attitude control while trying to achieve globally stabilizing position control; these singularities, when undealt with, may cause the controller to lose control of the quadrotor leading to instability at different locations. The results section shows that once all solutions are employed robust and accurate control of the quadrotor can be achieved. The designed controller was both simulated and tested experimentally with a commercial quadrotor.M.S

Adaptive Position and Attitude Tracking Controller for Satellite Proximity Operations using Dual Quaternions

Presented at Astrodynamics Specialist Conference, Hilton Head, SC, August 11-15, 2013.In this paper, we propose a nonlinear adaptive position and attitude tracking controller for satellite proximity operations. This controller requires no information about the mass and inertia matrix of the satellite, and takes into account the gravitational force, the gravity-gradient torque, the perturbing force due to Earth’s oblateness, and other constant – but otherwise unknown – disturbance forces and torques. We give sufficient conditions on the reference motion for mass and inertia matrix identification. The controller is shown to be almost globally asymptotically stable and can handle large error angles and displacements. Unit dual quaternions are used to simultaneously represent the absolute and relative attitude and position of the satellites, resulting in a compact controller representation

Modeling and control of a tiltrotor unmanned aerial vehicle for path tracking

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia de Automação e Sistemas, Florianópolis, 2015.Abstract : This master thesis deals with the modeling and control of a small scale birotor tiltrotor unmanned aerial vehicle (UAV). A tiltrotor is characterized by a mechanism that tilts the aircraft's rotors in order to control the flight. An UAV with such characteristics is being developed by this author and other researchers in the scope of the project named ProVANT. The developed UAV prototype is used in this work to obtain experimental results. This kind of system can be characterized by its underactuated, highly nonlinear, and coupled dynamics. Instead of using a dynamic model available in literature, this work proposes a more accurate model considering the UAV as a multibody system. By doing so the tilting angles become generalized coordinates and the tilt mechanism dynamics are naturally added to the model, as well as the coupling between the bodies. The result is an eight degrees of freedom model obtained through Euler-Lagrange formulation. The path tracking problem is solved here with linear optimal controllers for the full model, instead of the classical approach of cascade control for the translation and rotation subsystems. The developed controllers are linear quadratic regulators, a H1 controller and a multiobjective H2=H1 controller, all with LMI formulation. A nonlinear backstepping controller taken from the literature is implemented in order to be compared with the designed controllers. In addition, controllers for the hovering problem are also designed to be used in experiments with ProVANT's tiltrotor. They reduce the complexity of the initial experimental flights, focusing not only in the validation of the control system, but the complete project, including its electronics, mechanical design, and additional software. Such experiments are presented and discussed in details along this work. The work also addresses how flight-related information are gathered and processed. This includes the design of a nonlinear complementary filter for the attitude estimation that works with data acquired from the UAV sensors.<br

Dynamics of Serial Manipulators using Dual Quaternion Algebra

This paper presents two approaches to obtain the dynamical equations of serial manipulators using dual quaternion algebra. The first one is based on the recursive Newton-Euler formulation and uses twists and wrenches instead of 3D vectors, which simplifies the classic procedure by removing the necessity of exhaustive geometrical analyses since wrenches and twists are propagated through high-level algebraic operations. Furthermore, the proposed formulation works for arbitrary types of joints and does not impose any particular convention for the propagation of twists. The second approach, based on Gauss's Principle of Least Constraint (GPLC), takes into account elements of the dual quaternion algebra and provides a linear relationship between twists derivatives and joint accelerations, which can be particularly useful in robot control. Differently from other approaches based on the GPLC, which have representational singularities or require constraints, our method does not have those drawbacks. We present a thorough methodology to obtain the computational cost of both algorithms and compared them with their classic counterparts. Although our current formulations are more computationally expensive, they are more general than their counterparts in the state of the art. Simulation results showed that both methods are as accurate as the classic recursive Newton-Euler algorithm.Comment: Submitted for publication (currently under review