Constrained rigid body stability and control

Stability and control of a single or three‐body constrained system are considered. Several different types of constrained motion are among them: the impact phase of a free body colliding with the ground, contact with a stationary or moving platform, movement on a frictionless surface or multiple rigid bodies connected by holonomic constraints, and moving as in the human arm. The single body constrained system is controlled by sliding mode control. The stability of the three‐link arm at arbitrary equilibrium points and Lyapunov stability in the vicinity of the equilibrium point are formulated. The formulation and derivations are by computational tools, that is, state space analysis and matrices. The approach can easily be extended to larger systems with many rigid bodies such as skeletal systems. The formulation minimizes human labor in formulations and simulations. The sliding mode behavior of the model on a frictionless surface and the three link arm stability are demonstrated via simulation. Challenges for application to natural systems are outlined. Copyright © 2014 John Wiley & Sons, Ltd.     

Modelling and monitoring of passive control structures in human movement

Passive tissues, ligaments and cartilage are vital to human movement. Their contribution to stability, joint function and joint integrity is essential. The articulation of their functions and quantitative assessment of what they do in a healthy or injured state are important in athletics, orthopaedics, medicine and health. In this paper, the role of cartilage and ligaments in stability of natural contacts, connections and joints is articulated by including them in two very simple skeletal systems: one- and three-link rigid body systems. Based on the Newton–Euler equations, a state space presentation of the dynamics is discussed that allows inclusion of ligament and cartilage structures in the model, and allows for Lyapunov stability studies for the original and reduced systems. The connection constraints may be holonomic and non-holonomic depending on the structure of the passive elements. The development is pertinent to the eventual design of a computational framework for the study of human movement that involves computer models of all the relevant skeletal, neural and physiological elements of the central nervous system (CNS). Such a structure also permits testing of different hypotheses about the functional neuroanatomy of the CNS, and the study of the effects and dynamics of disease, deterioration, aging and injuries. The formulation here is applied to one- and three-link systems. Digital computer simulations of a two rigid body system are presented to demonstrate the feasibility and effectiveness of the approach and the methods.     

Use of penalty formulations in dynamic simulation and analysis of redundantly constrained multibody systems

The determination of particular reaction forces in the analysis of redundantly constrained multibody systems requires the consideration of the stiffness distribution in the system. This can be achieved by modeling the components of the mechanical system as flexible bodies. An alternative to this, which we will discuss in this paper, is the use of penalty factors already present in augmented Lagrangian formulations as a way of introducing the structural properties of the physical system into the model. Natural coordinates and the kinematic constraints required to ensure rigid body behavior are particularly convenient for this. In this paper, scaled penalty factors in an index-3 augmented Lagrangian formulation are employed, together with modeling in natural coordinates, to represent the structural properties of redundantly constrained multibody systems. Forward dynamic simulations for two examples are used to illustrate the material. Results showed that scaled penalty factors can be used as a simple and efficient way to accurately determine the constraint forces in the presence of redundant constraints.     

Stability,positive invariance and design of constrained regulators for networked control systems

In this article, the stability analysis, the positive invariance of polyhedral sets and the design of state-feedback regulators for networked control systems (NCS) with bounded transmission delays, constant and unknown or time-varying, are investigated. The dynamics of the NCS is described by autoregressive-moving-average (ARMA) models. Contrary to former approaches based on quadratic Lyapunov functions, in this article polyhedral Lyapunov functions are used for both stability and positive invariance analysis and state-feedback synthesis. Then, based on the property that the exponential of a matrix can be expressed as a weighted sum of its constituent matrices, it is proven that the problems of determination of stability margins or the design of stabilising controllers can be reduced to linear programming optimisation problems. The use of ARMA models allows the development of methods for the design of state-feedback controllers satisfying state constraints or convergence rate specifications defined on the NCS state space and not on the state of an augmented state space representation.     

Analysis and control of parabolic PDE systems with input constraints

This paper develops a general framework for the analysis and control of parabolic partial differential equations (PDE) systems with input constraints. Initially, Galerkin's method is used for the derivation of ordinary differential equation (ODE) system that capture the dominant dynamics of the PDE system. This ODE systems are then used as the basis for the synthesis, via Lyapunov techniques, of stabilizing bounded nonlinear state and output feedback control laws that provide an explicit characterization of the sets of admissible initial conditions and admissible control actuator locations that can be used to guarantee closed-loop stability in the presence of constraints. Precise conditions that guarantee stability of the constrained closed-loop parabolic PDE system are provided in terms of the separation between the fast and slow eigenmodes of the spatial differential operator. The theoretical results are used to stabilize an unstable steady-state of a diffusion-reaction process using constrained control action.     

Analysis of Rigid Body Interactions for Compliant Motion Tasks Using the Grassmann–Cayley Algebra

This paper studies the statics and the instantaneous kinematics of a rigid body constrained by one to six contacts with a rigid static environment. These properties are analyzed under the frictionless assumption by modeling each contact with a kinematic chain that, instantaneously, is statically and kinematically equivalent to the contact and studying the resulting parallel chain using the Grassmann-Cayley algebra. This algebra provides a complete interpretation of screw theory, in which twist and wrench spaces are expressed by means of the concept of extensor and its inherent duality reflects the reciprocity condition between possible twists and admissible wrenches of partially constrained rigid bodies. Moreover, its join and meet operators are used to compute sum and intersections of the twist and wrench spaces resulting from serial and parallel composition of motion constraints. In particular, it has an explicit formula for the meet operator that gives closed-form expressions of twist and wrench spaces of rigid bodies in contact. The Grassmann-Cayley algebra permits us to work at the symbolic level, that is, in a coordinate-free manner and therefore provides a deeper insight into the kinestatics of rigid body interactions.     

Regulator problem for linear systems with constraints on control and its increment or rate

This paper discusses the problem of constraints on both control and its rate or increment, for linear systems in state space form, in both the continuous and discrete-time domains. Necessary and sufficient conditions are derived for autonomous linear systems with constrained state increment or rate (for the continuous-time case), such that the system evolves respecting incremental or rate constraints. A pole assignment technique is then used to solve the inverse problem, giving stabilizing state feedback controllers that respect non-symmetrical constraints on both control and its increment or rate. An illustrative example shows the application of the method on the double integrator problem.     

State estimation for linear systems with state equality constraints

This paper deals with the state estimation problem for linear systems with linear state equality constraints. Using noisy measurements which are available from the observable system, we construct the optimal estimate which also satisfies linear equality constraints. For this purpose, after reviewing modeling problems in linear stochastic systems with state equality constraints, we formulate a projected system representation. By using the constrained Kalman filter for the projected system and comparing its filter Riccati equation with those of the unconstrained and the projected Kalman filters, we clearly show, without using optimality, that the constrained estimator outperforms the other filters for estimating the constrained system state. Finally, a numerical example is presented, which demonstrates performance differences among those filters.     

State representations of linear systems with output constraints

We derive state space representations for linear systems that are described by input/state/output equations and that are subjected to a number of constant linear constraints on the outputs. In the case of a general linear system, the state representation of the constrained system is shown to be essentially nonunique. For linear Hamiltonian systems satisfying a nondegeneracy condition, there is a natural and unique choice of the representation which preserves the Hamiltonian structure. In the linear systems setting we give an algebraic proof that a system withn degrees of freedom underk constraints becomes a system withn−k degree of freedom. Similar results are obtained for linear gradient systems.     

航天器全状态约束输出反馈控制

针对无角速度测量的刚性航天器姿态跟踪问题,提出一种全状态约束输出反馈控制方法.建立修正罗德里格参数描述的系统模型,提出能够适用于约束与非约束情况的改进型障碍李雅普诺夫函数(MBLF),拓展传统对数型障碍李雅普诺夫函数的适用范围.构造二阶辅助系统,将控制输入和饱和输入之间的差作为构造系统的输入,进而产生信号以补偿饱和的影响.设计状态观测器估计未知状态量,并结合反步法设计输出反馈控制律,保证系统全状态约束性能和姿态跟踪精度.通过李雅普诺夫稳定性分析证明姿态观测误差和跟踪误差能够达到一致最终有界.仿真结果验证所提方法的有效性.     

Task-level approaches for the control of constrained multibody systems

This paper presents a task-level control methodology for the general class of holonomically constrained multibody systems. As a point of departure, the general formulation of constrained dynamical systems is reviewed with respect to multiplier and minimization approaches. Subsequently, the operational space framework is considered and the underlying symmetry between constrained dynamics and operational space control is discussed. Motivated by this symmetry, approaches for constrained task-level control are presented which cast the general formulation of constrained multibody systems into a task space setting using the operational space framework. This provides a means of exploiting task-level control structures, native to operational space control, within the context of constrained systems. This allows us to naturally synthesize dynamic compensation for a multibody system, that properly accounts for the system constraints while performing a control task. A set of examples illustrate this control implementation. Additionally, the inclusion of flexible bodies in this approach is addressed.     

Virtual control with hard constraints

Practical control problems are always subject to plant state and/or input constraints, which make designing an effective controller a challenging task. This paper introduces a novel virtual control approach to handling the presence of hard constraints in control systems by utilizing virtual mechanisms in the form of nonlinear springs and dampers. The augmented virtual mechanisms are to assist in better shaping the closed‐loop responses, especially when operating near the constrained boundary. A linear quadratic regulator based model predictive control method is utilized to develop stabilizing controllers that not only achieve desired system performance, but also meet the imposed hard constraints. The basic idea is to dramatically increase control penalty by way of tuning the spring and damper effect when the constrained state/input response is close to its hard constraint. The proposed method is applied to a balancing ball problem to demonstrate its applicability and effectiveness, and the simulation results validate the proposed concept.     

Hybrid threshold strategy‐based adaptive tracking control for nonlinear stochastic systems with full‐state constraints

This article focuses on the adaptive tracking control problem for a class of interconnected nonlinear stochastic systems under full‐state constraints based on the hybrid threshold strategy. Different from the existing works, we propose a novel pre‐constrained tracking control algorithm to deal with the full‐state constraint problem. First, a novel nonlinear transformation function and a new coordinate transformation are developed to constrain state variables, which can directly cope with asymmetric state constraints. Second, the hybrid threshold strategy is constructed to provide a reasonable way in balancing system performance and communication constraints. By the use of dynamic surface control technique and neural network approximate technique, a smooth pre‐constrained tracking controller with adaptive laws is designed for the interconnected nonlinear stochastic systems. Moreover, based on the Lyapunov stability theory, it is proved that all state variables are successfully pre‐constrained within asymmetric boundaries. Finally, a simulation example is presented to verify the effectiveness of proposed control algorithm.     

Estimation of asymptotic stability regions via homogeneous polynomial Lyapunov functions

Asymptotic stability regions of non-linear dynamical systems are estimated by using homogeneous polynomial Lyapunov functions, which include quadratic Lyapunov functions as a special case. The non-linear system is represented as a convex hull of some linear systems in a specified region of the state space. It is shown that the estimation can be formulated as an LMI optimization problem in an augmented state space.     

Optimal Synchronization Control of Heterogeneous Asymmetric Input-Constrained Unknown Nonlinear MASs via Reinforcement Learning

The asymmetric input-constrained optimal synchronization problem of heterogeneous unknown nonlinear multiagent systems(MASs)is considered in the paper.Intuitively,a state-space transformation is performed such that satisfaction of symmetric input constraints for the transformed system guarantees satisfaction of asymmetric input constraints for the original system.Then,considering that the leader’s information is not available to every follower,a novel distributed observer is designed to estimate the leader’s state using only exchange of information among neighboring followers.After that,a network of augmented systems is constructed by combining observers and followers dynamics.A nonquadratic cost function is then leveraged for each augmented system(agent)for which its optimization satisfies input constraints and its corresponding constrained Hamilton-Jacobi-Bellman(HJB)equation is solved in a data-based fashion.More specifically,a data-based off-policy reinforcement learning(RL)algorithm is presented to learn the solution to the constrained HJB equation without requiring the complete knowledge of the agents’dynamics.Convergence of the improved RL algorithm to the solution to the constrained HJB equation is also demonstrated.Finally,the correctness and validity of the theoretical results are demonstrated by a simulation example.     

Design of networked control systems with bounded arbitrary time delays

This paper is mainly concerned with the model predictive control (MPC) of networked control systems (NCSs) with uncertain time delay and data packets disorder. The network-induced time delay is described as bounded and arbitrary process. For the usual state feedback controller, by considering all the possibilities of delays, an augmented state space model of the closed-loop system, which characterizes all the delay cases, is obtained. The stability conditions are given according to the Lyapunov method based on this augmented model. The stability property is inherited in MPC which explicitly considers the physical constraints. A numerical example is given to demonstrate the effectiveness of the proposed MPC.     

On Cartesian stiffness matrices in rigid body dynamics: an energetic perspective

Several Cartesian stiffness matrices for a single rigid body subject to a conservative force field are developed in this paper. The treatment is based on energetic arguments and an Euler angle parameterization of the rotation of the rigid body is employed. Several new representations for the stiffness matrix are obtained and the relation to other works on Cartesian stiffness matrices and Hessians is illuminated. Additional details are presented with respect to determining the Cartesian stiffness matrix for a pair of rigid bodies, as well as for a system of rigid bodies constrained to a plane.     

Principal Geodesic Analysis in the Space of Discrete Shells

Important sources of shape variability, such as articulated motion of body models or soft tissue dynamics, are highly nonlinear and are usually superposed on top of rigid body motion which must be factored out. We propose a novel, nonlinear, rigid body motion invariant Principal Geodesic Analysis (PGA) that allows us to analyse this variability, compress large variations based on statistical shape analysis and fit a model to measurements. For given input shape data sets we show how to compute a low dimensional approximating submanifold on the space of discrete shells, making our approach a hybrid between a physical and statistical model. General discrete shells can be projected onto the submanifold and sparsely represented by a small set of coefficients. We demonstrate two specific applications: model‐constrained mesh editing and reconstruction of a dense animated mesh from sparse motion capture markers using the statistical knowledge as a prior.     

On Stability of Constrained Receding Horizon Control with Finite Terminal Weighting Matrix

In this paper, a new stabilizing receding horizon control, based on a finite input and state horizon cost with a finite terminal weighting matrix, is proposed for time-varying discrete linear systems with constraints. We propose matrix inequality conditions on the terminal weighting matrix under which closed-loop stability is guaranteed for both cases of unconstrained and constrained systems with input and state constraints. We show that such a terminal weighting matrix can be obtained by solving a linear matrix inequality (LMI). In the case of constrained time-invariant systems, an artificial invariant ellipsoid constraint is introduced in order to relax the conventional terminal equality constraint and to handle constraints. Using the invariant ellipsoid constraints, a feasibility condition of the optimization problem is presented and a region of attraction is characterized for constrained systems with the proposed receding horizon control.     

Constrained output feedback model predictive control for nonlinear systems

A constrained output feedback model predictive control approach for nonlinear systems is presented in this paper. The state variables are observed using an unscented Kalman filter, which offers some advantages over an extended Kalman filter. A nonlinear dynamic model of the system, considered in this investigation, is developed considering all possible effective elements. The model is then adaptively linearized along the prediction horizon using a state-dependent state space representation. In order to improve the performance of the control system as many linearized models as the number of prediction horizons are obtained at each sample time. The optimum results of the previous sample time are utilized for linearization at the current sample time. Subsequently, a linear quadratic objective function with constraints is formulated using the developed governing equations of the plant. The performance and effectiveness of the proposed control approach is validated both in simulation and through real-time experimentation using a constrained highly nonlinear aerodynamic test rig, a twin rotor MIMO system (TRMS).