A skew-symmetric property of rigid-body systems

The configuration space for the attitude of a vehicle can be modeled as SO(3), namely the rotation matrix group. This work investigates the globally valid dynamics of natural Lagrangian systems and gyroscopic systems with configuration space including SO(3). The dynamics is derived by using the global representations of jet bundles of SO(3). A skew-symmetric property associated with the systems can be then established. Such property can be used in many applications such as the adaptive controller design.     

Stable adaptive fuzzy control of nonlinear systems using small-gain theorem and LMI approach

A new design scheme of stable adaptive fuzzy control for a class of nonlinear systems is proposed in this paper. The T-S fuzzy model is employed to represent the systems. First, the concept of the so-called parallel distributed compensation (PDC) and linear matrix inequality (LMI) approach are employed to design the state feedback controller without considering the error caused by fuzzy modeling. Sufficient conditions with respect to decay rate α are derived in the sense of Lyapunov asymptotic stability. Finally, the error caused by fuzzy modeling is considered and the input-tostate stable (ISS) method is used to design the adaptive compensation term to reduce the effect of the modeling error. By the small-gain theorem, the resulting closed-loop system is proved to be input-to-state stable. Theoretical analysis verifies that the state converges to zero and all signals of the closed-loop systems are bounded. The effectiveness of the proposed controller design methodology is demonstrated through numerical simulation on the chaotic Henon system.     

摄影测量共线方程的单位四元数描述

为探讨四元数在摄影测量共线方程严密解算中的应用问题,从四元数的基本理论出发,详细推导以单位四元数矩阵表达的共线方程的严密线性化公式,该线性化公式无须对旋转矩阵进行微分。以单像后方交会和光束法平差为例采用模拟数据和实际数据实验。结果表明,推导的以四元数矩阵为基础的线性化共线方程具有形式简单、初值无关性和收敛速度快的优点。具有一定的实用价值。     

Adaptive fuzzy fault-tolerant attitude control of spacecraft

This paper investigates the attitude control of spacecraft in the presence of unknown mass moment of inertia matrix, external disturbances, actuator failures, and control input constraints. A robust adaptive controller is proposed with the utilization of fuzzy logic and backstepping techniques. The unit quaternion is employed to describe the attitude of spacecraft for global representation without singularities. The system uncertainty is estimated by introducing a fuzzy logic system. The adaptive mechanism has only two parameters to be adapted on-line because the adaptive law of the proposed controller is derived from the norm of the weight matrix. The stability of the closed-loop system is guaranteed by Lyapunov direct approach. Results of numerical simulations state that the proposed controller is successful in achieving high attitude performance in the presence of parametric uncertainties, external disturbances, actuator failures, and control input constraints.     

Coordinate independent adaptive attitude tracking control design for spacecraft robust to time-varying system uncertainties

In this paper, we consider the adaptive attitude control of spacecraft with time-varying inertial properties. We use a coordinate independent approach for the purpose of designing the control and estimation laws in terms of the rotation matrices representing the spacecraft body frame and reference tracking signals. This method helps to overcome the difficulties regarding the attitude representation on SO(3) such as ambiguities associated with quaternion representation and inherent singularities inside Euler parameters. We model the time variations in inertial parameters in two different ways, and design adaptive control schemes for each case. As the first uncertain dynamic model, we consider a setting of spacecraft with multiple moving appendages, and based on this model, design an adaptive control scheme with three different versions, where Frobenious norm is used in measuring the deviation of the estimated inertia tensors from their actual values. The proposed adaptive control scheme is later extended for the more direct model where the inertia tensor of the spacecraft has a nonlinear relation with the norm of the input moment. Further, we derive the allowable sets of initial conditions to ensure the convergence of the tracking error. Simulation results are provided to illustrate the effectiveness of our proposed approach.     

Quaternion‐based visual servo control in the presence of camera calibration error

Visual servo control systems use information from images along with knowledge of the optic parameters (i.e. camera calibration) to position the camera relative to some viewed object. If there are inaccuracies in the camera calibration, then performance degradation and potentially unpredictable response from the visual servo control system may occur. Motivated by the desire to incorporate robustness to the camera calibration, different control methods have been developed. Previous adaptive/robust controllers (especially for six degree‐of‐freedom camera motion) rely heavily on properties of the rotation parameterization to formulate state estimates and a measurable closed‐loop error system. All of these results are based on the singular axis–angle parameterization. Motivated by the desire to express the rotation by a non‐singular parameterization, efforts in this paper address the question: Can state estimates and a measurable closed‐loop error system be crafted in terms of the quaternion parameterization when the camera calibration parameters are unknown? To answer this question, a contribution of this paper is the development of a robust controller and closed‐loop error system based on a new quaternion‐based estimate of the rotation error. A Lyapunov‐based analysis is provided which indicates that the controller yields asymptotic regulation of the rotation and translation error signals given a sufficient approximate of the camera calibration parameters. Simulation results are provided that illustrate the performance of the controller for a range of calibration uncertainty. Copyright © 2009 John Wiley & Sons, Ltd.     

基于SO(3)的多四旋翼无人机编队协同控制

针对多四旋翼无人机系统的编队飞行问题,提出了基于特殊正交群SO(3)的协同控制设计方法.在给出编队空间队形和通信拓扑描述后,建立了多四旋翼无人机系统SO(3)控制模型.由于SO(3)与传统俯仰/偏航/滚转三通道模型具有不同的结构,文中进一步研究了SO(3)中无人机之间相对误差的表示方法,设计了适用于多飞行器的SO(3)控制器实现对编队和姿态的协同控制.推力控制器用于调节无人机的位置与速度,并在此基础上构造旋转矩阵形式的姿态协同指令.文中相应设计了SO(3)姿态控制器用于实现指令跟踪,最后从理论上对协同稳定性进行了分析.提出的控制方法能够使得多四旋翼无人机形成期望的队形,并且保持姿态一致进行稳定飞行.仿真结果验证了本文方法的有效性.     

Adaptive T-S fuzzy-neural modeling and control for general MIMO unknown nonaffine nonlinear systems using projection update laws

This paper describes a novel design of an on-line Takagi-Sugeno (T-S) fuzzy-neural controller for a class of general multiple input multiple output (MIMO) systems with unknown nonlinear functions and external disturbances. Instead of modeling the unknown systems directly, the T-S fuzzy-neural model approximates a virtual linearized system (VLS) of a real system with modeling errors and external disturbances. Compared with previous approaches, the main contribution of this paper is an investigation of more general MIMO unknown systems using on-line adaptive T-S fuzzy-neural controllers. In this paper, we also use projection update laws, which generalize the projection algorithm, to tune the adjustable parameters. This prevents parameter drift and ensures that the parameter matrix is bounded away from singularity. We prove that the closed-loop system controlled by the proposed controller is robust stable and the effect of all the modeling errors and external disturbances on the tracking error can be attenuated. Finally, two examples covering four cases are simulated in order to confirm the effectiveness and applicability of the proposed approach in this paper.     

Attitude guidance and tracking for spacecraft with two reaction wheels

This paper addresses the guidance and tracking problem for a rigid-spacecraft using two reaction wheels (RWs). The guidance problem is formulated as an optimal control problem on the special orthogonal group SO(3). The optimal motion is solved analytically as a function of time and is used to reduce the original guidance problem to one of computing the minimum of a nonlinear function. A tracking control using two RWs is developed that extends previous singular quaternion stabilisation controls to tracking controls on the rotation group. The controller is proved to locally asymptotically track the generated reference motions using Lyapunov's direct method. Simulations of a 3U CubeSat demonstrate that this tracking control is robust to initial rotation errors and angular velocity errors in the controlled axis. For initial angular velocity errors in the uncontrolled axis and under significant disturbances the control fails to track. However, the singular tracking control is combined with a nano-magnetic torquer which simply damps the angular velocity in the uncontrolled axis and is shown to provide a practical control method for tracking in the presence of disturbances and initial condition errors.     

Adaptive iterative learning control for switched nonlinear continuous-time systems

In this paper, an adaptive iterative learning control (ILC) method is proposed for switched nonlinear continuous-time systems with time-varying parametric uncertainties. First, an iterative learning controller is constructed with a state feedback term in the time domain and an adaptive learning term in the iteration domain. Then a switched nonlinear continuous-discrete two-dimensional (2D) system is built to describe the adaptive ILC system. Multiple 2D Lyapunov functions-based analysis ensures that the 2D system is exponentially stable, and the tracking error will converge to zero in the iteration domain. The design method of the iterative learning controller is obtained by solving a linear matrix inequality. Finally, the efficacy of the proposed controller is demonstrated by the simulation results.     

Adaptive Interconnection and Damping Assignment Passivity Based Control for Underactuated Mechanical Systems

In this paper, we present two adaptive control approaches to handle uncertainties caused by parametric and modeling errors in a class of nonlinear systems with uncertainties. The methods use the Port-controlled Hamiltonian (PCH) modelling framework and the interconnection and damping assignment passivity-based control (IDA-PBC) control design methodology being the most effectively applicable method to such models. The methods explore an extension on the classical IDA-PBC by adopting the state-transformation, yielding a dynamic state-feedback controller that asymptotically stabilizes a class of underactuated mechanical systems and preserves the PCH structure of the augmented closed-loop system. The results are applied to the underactuated mechanical systems that are a class of mechanical systems with broad applications and are more interesting as well as challenging control problems within this context. The results are illustrated with numerical simulations applied to two underactuated robotic systems; the Acrobot and non-prehensile planar rolling robotic (disk-on-disk) systems.

     

Robust adaptive control of synchronous generators with SMES unit via Hamiltonian function method†

Superconducting Magnetic Energy Storage (SMES) units can be used to enhance the stability of power systems. Using Hamiltonian function method, this article investigates robust adaptive control design for synchronous generators with such a unit, and proposes an energy-based robust adaptive controller for the systems with disturbances and unknown parameters. The generator used in this article is a 4th order model with both excitation and steam valve controls. It is shown that the generator with one SMES unit can be changed as a dissipative Hamiltonian system, with which the energy-based robust adaptive control law can be designed for the system by using the structural properties of dissipative Hamiltonian systems. Study on simulations shows that the controller proposed in this article works very well.     

Modeling, control, and stability analysis for time-delay TLP systems using the fuzzy Lyapunov method

In this study, we present a Takagi–Sugeno (T–S) fuzzy model for the modeling and stability analysis of oceanic structures. We design a nonlinear fuzzy controller based on a parallel distributed compensation (PDC) scheme and reformulate the controller design problem as a linear matrix inequalities (LMI) problem as derived from the fuzzy Lyapunov theory. The robustness design technique is adopted so as to overcome the modeling errors for nonlinear time-delay systems subject to external oceanic waves. The vibration of the oceanic structure, i.e., the mechanical motion caused by the force of the waves, is discussed analytically based on fuzzy logic theory and a mathematical framework. The end result is decay in the amplitude of the surge motion affecting the time-delay tension leg platform (TLP) system. The feedback gain of the fuzzy controller needed to stabilize the TLP system can be found using the Matlab LMI toolbox. This proposed method of fuzzy control is applicable to practical TLP systems. The simulation results show that not only can the proposed method stabilize the systems but that the controller design is also simplified. The effects of the amplitude damping of the surge motion on the structural response are obvious and work as expected due to the control force.     

Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations

A novel ODE time stepping scheme for solving rotational kinematics in terms of unit quaternions is presented in the paper. This scheme inherently respects the unit-length condition without including it explicitly as a constraint 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 (DAEs). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form a system of ordinary differential equations (ODEs) on the Lie algebra \(\mathit{so}(3)\) of the rotation group \(\mathit{SO}(3)\). This rotation vector defines an incremental rotation (and thus the associated incremental unit quaternion), and the rotation update is determined by the exponential mapping on the quaternion group. Since the kinematic ODE on \(\mathit{so}(3)\) can be solved by using any standard (possibly higher-order) ODE integration scheme, the proposed method yields a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions, avoiding integration of DAE equations. Besides being ‘more elegant’—in the opinion of the authors—this integration procedure also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation. As presented in the paper, the numerical integration of three non-linear ODEs in terms of the rotation vector as canonical coordinates achieves a higher accuracy compared to integrating the four (linear in ODE part) standard-quaternion DAE system. In summary, this paper solves the long-standing problem of the necessity of imposing the unit-length constraint equation during integration of quaternions, i.e. the need to deal with DAE’s in the context of such kinematical model, which has been a major drawback of using quaternions, and a numerical scheme is presented that also allows for longer integration steps during kinematic reconstruction of large three-dimensional rotations.     

C3连续的单位四元数插值样条曲线

目的 构造一类C³连续的单位四元数插值样条曲线,证明它的插值性和连续性,并把它应用于刚体关键帧动画设计中。方法 利用R³空间中插值样条曲线的5次多项式调配函数的累和形式构造了S³空间中单位四元数插值样条曲线,它不仅能精确通过一系列给定的方向,而且能生成C³连续的朝向曲线。结果 与Nielson的单位四元数均匀B样条插值曲线的迭代构造方法相比,所提方法避免了为获取四元数B样条曲线控制顶点对非线性方程组迭代求解的过程,提高了运算效率;与单位四元数代数三角混合插值样条曲线的构造方法（Su方法）相比,所提方法只用到多项式基,运算速度更快。本例中创建关键帧动画所需的时间与Nielson方法和Su方法相比平均下降了73%和33%。而且,相比前两种方法,所提方法产生的四元数曲线连续性更高,由C²连续提高到C³连续,这意味着动画中刚体的朝向变化更加自然。结论 仿真结果表明,本文方法对刚体关键帧动画设计是有效的,对实时性和流畅性要求高的动画设计场合尤为适用。     

基于状态估计的摩擦模糊建模与鲁棒自适应控制

针对一类多自由度机械系统, 研究了基于状态估计的摩擦模糊建模与鲁棒自适应控制问题. 提出用模糊状态估计器估计摩擦模型中的不可测变量, 并用严格正实和李雅普诺夫稳定性理论证明了状态估计误差的一致最终有界性. 运用模糊状态估计结果设计了多变量鲁棒自适应控制器, 其中摩擦模糊模型中的自适应参数是基于李雅普诺夫稳定性理论设计的, 并证明了闭环系统跟踪误差的一致最终有界性. 本文对多自由度质量、弹簧和摩擦阻尼系统进行的仿真, 结果表明所提出的状态估计算法和自适应控制策略是有效的.     

利用四元数实现三维地球场景的漫游

场景的交互性操作是地球三维可视化的重要研究内容。尽管人们通过直接滤波欧拉角来平滑方位数据,但是平滑欧拉角未必会产生平滑的运动,这是由旋转群和欧氏空间之间的度量单位的差异引起的。Hamilton于1853年发现的四元数提供了描述旋转的途径。文章首先讨论了四元数的一般原理籍优越性;接着论述了三维虚拟地球的建模方法,采用四元数技术实现了三维地球的任意旋转的问题;最后在数字地球原型系统上进行了实验研究,实验证明四元数技术是提供一种解决大规模地形场景实时漫游的有效途径。     

Synthesis of adaptive gain robust model-following/tracking controllers for a class of uncertain linear systems

This paper proposes a new adaptive gain robust model-following/tracking controller for a class of uncertain linear systems. The proposed adaptive gain robust controller is composed of state feedback laws with fixed gains, feedfoward inputs for the reference model and nonlinear compensation inputs with adjustable time-varying parameters. Moreover, the proposed control strategy can achieve good transient performance and avoid the excessive control input by means of the design parameter. In this paper, linear matrix inequality-based sufficient conditions for the existence of the proposed adaptive gain controller are given, and two design strategies are presented. Finally, simple illustrative examples are included to show the effectiveness of the proposed adaptive gain robust controller.     

Stability conditions of fuzzy systems and its application to structural and mechanical systems

This paper provided the stability conditions and controller design for a class of structural and mechanical systems represented by Takagi–Sugeno (T–S) fuzzy models. In the design procedure of controller, parallel-distributed compensation (PDC) scheme was utilized to construct a global fuzzy logic controller by blending all local state feedback controllers. A stability analysis was carried out not only for the fuzzy model but also for a real mechanical system. Furthermore, this control problem can be reduced to linear matrix inequalities (LMI) problems by the Schur complements and efficient interior-point algorithms are now available in Matlab toolbox to solve this problem. A simulation example was given to show the feasibility of the proposed fuzzy controller design method.     

一类多变量非线性动态系统的鲁棒自适应模糊控制

对一类非线性多变量未知动态系统,提出了一种自适应模糊控制策略.策略中采用 IF-THEN推理规则来构造模糊逻辑系统,实现对系统中未知函数的估计,在建模误差为零的 条件下设计状态反馈控制器及参数的自适应律.分析了当存在建模误差时,闭环系统的稳定 性和鲁棒性.