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Summary The concept of input-to-state stability (ISS) is important in robust control, as the state of an ISS system subject to disturbances can be stably regulated to a small region around the origin. In this study, the ISS property of the rigid-body attitude system with quaternion representation is thoroughly investigated. It has been known that the closed loop with continuous controllers is not ISS with respect to arbitrarily small external disturbances. To deal with this problem, hybrid proportional-derivative controllers with hysteresis are proposed to render the attitude system ISS. The controller is far from new, but it is investigated in a new aspect. To illustrate the applications of the results about ISS, 2 new robust hybrid controllers are designed. In the case of large bounded time-varying disturbances, the hybrid proportional-derivative controller is designed to incorporate a saturated high-gain feedback term, and arbitrarily small ultimate bounds of the state can be obtained; in the case of constant disturbances, a hybrid adaptive controller is proposed, which is robust against small estimate error of inertia matrix. Finally, simulations are conducted to illustrate the effectiveness of the proposed control strategies.

In this paper, we consider explicit and iterative methods for solving the Generalized Sylvester matrix equation AV + BW = EVF + C. Based on the use of Kronecker map and Sylvester sum some lemmas and theorems are stated and proved where explicit and iterative solutions are obtained. The proposed methods are illustrated by numerical example. The obtained results show that the methods are very neat and efficient.

It is well known that the least-squares QR-factorization (LSQR) algorithm is a powerful method for solving linear systems Ax = b and unconstrained least-squares problem minx | | Ax − b | | . In the paper, the LSQR approach is developed to obtain iterative algorithms for solving the generalized Sylvester-transpose matrix equation the minimum Frobenius norm residual problem and the periodic Sylvester matrix equation Numerical results are given to illustrate the effect of the proposed algorithms. Copyright © 2013 John Wiley & Sons, Ltd.

SUMMARY This paper investigates the quadratic stability and stabilization of a class of matrix second-order time-varying systems. All the system matrices including the second-order differential coefficient matrix are assumed to have the time-varying norm-bounded parameters. Necessary and sufficient conditions for the quadratic stability and stabilization of such time-varying systems are derived. All the results are obtained in terms of linear matrix inequalities. Two illustrative examples are given to show that our results are effective and less conservative than the results obtained by other researchers. Copyright © 2011 John Wiley & Sons, Ltd.

The aim of this paper is to develop an efficient method for decoupling joint and elastic accelerations, while maintaining the nonlinear inertia coupling between rigid body motion and elastic body deformation. Almost all of the existing recursive methods for analysis of flexible open-loop and closed-loop multibody mechanical systems lead to dense coefficient matrices in the equations of motion, and consequently there are strong dynamic couplings between the joint and elastic coordinates. When the number of elastic degrees of freedom increases, the size of the coefficient matrix in the equations of motion becomes large and unfortunately the use of these methods for solving for the joint and elastic accelerations becomes less efficient. This paper discusses the problems associated with the inertia projection schemes used in the existing recursive methods, and it is shown that decoupling the joint and elastic accelerations using these methods requires the factorization of nonlinear matrices whose dimensions depend on the number of elastic degrees of freedom of the system. An amalgamated method that can be used to decouple the elastic and joint accelerations is then proposed. In this amalgamated decoupling formulation, the relationships between the absolute, elastic, and joint variables and the generalized Newton–Euler equations are used to develop systems of loosely coupled equations that have sparse matrix structure. Utilizing the inertia matrix structure of flexible manufacturing systems and the fact that the joint reaction forces associated with the elastic coordinates do represent independent variables, a reduced system of equations whose dimension is dependent on the number of elastic degrees of freedom is obtained. This system can be solved for the joint accelerations as well as for the joint reaction forces. The application of the procedure developed in this paper is illustrated using flexible open-loop robotic manipulators and closed-loop crank-slider mechanical systems. Copyright © 2007 John Wiley & Sons, Ltd.

The aim of this paper is to develop non-linear recursive formulations for decoupling joint and elastic accelerations, while maintaining the non-linear inertia coupling between rigid body motion and elastic deformation in deformable mechanical systems. The inertia projection schemes used in most existing recursive formulations for the dynamic analysis of deformable mechanisms lead to dense coefficient matrices in the equations of motion. Consequently, there are strong dynamic couplings between the joint and elastic coordinates. When the number of elastic degrees of freedom increases, the size of the coefficient matrix in the equations of motion becomes large. Consequently, the use of these recursive formulations for solving the joint and elastic accelerations becomes less efficient. In this paper, the non-linear recursive formulations have been used to decouple the elastic and joint accelerations in deformable mechanical systems. The relationships between the absolute, elastic and joint variables and generalized Newton–Euler equations are used to develop systems of loosely coupled equations that have sparse matrix structure. By using the inertia matrix structure of deformable mechanical systems and the fact that joint reaction forces associated with elastic coordinates do represent independent variables, a reduced system of equations whose dimension is dependent of the number of elastic degrees of freedom is obtained. This system can be solved for the joint accelerations as well as for the joint reaction forces. The use of the approaches developed in this investigation is illustrated using deformable open-loop serial robot and closed-loop four-bar mechanical systems. Copyright © 2007 John Wiley & Sons, Ltd.

Interconnection and damping assignment passivity-based control is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed-loop system with a Hamiltonian structure with a desired energy function - that qualifies as Lyapunov function for the desired equilibrium. The assignable energy functions are characterized by a set of partial differential equations that must be solved to determine the control law. A class of underactuation degree one systems for which the partial differential equations can be explicitly solved-making the procedure truly constructive-was recently reported by the authors. In this brief note, largely motivated by the interesting Acrobot example, we pursue this investigation for two degrees-of-freedom systems where a constant inertia matrix can be assigned. We concentrate then our attention on potential energy shaping and give conditions under which an explicit solution of the associated partial differential equation can be obtained. Using these results we show that it is possible to swing-up the Acrobot from some configuration positions in the lower half plane, provided some conditions on the robot parameters are satisfied. Copyright � 2006 John Wiley & Sons, Ltd.

This paper investigates the constraint and coupling characteristics of underactuated manipulators by proposing an elastic model of the manipulator and examining the second order constraint equation. A dynamic model and a coupling constraint equation are developed from a Jacobian matrix and the Newton-Euler formulation. The inertia matrix and the Christoffel tensor are analyzed and decomposed into the part concerning actuated joints and the part concerning passive joints. This decomposition is further extended to the dynamic coupling equation and generates an actuation coupling matrix and a dynamic coupling tensor. Two new dynamic coupling indices are hence identified. One is related to an actuation input and the other is related to centrifugal and Coriolis forces. The former reveals the dynamic coupling between the input and the acceleration of passive joints and gives the actuation effect on the passive joints. The latter reveals the dynamic coupling between the centrifugal and Coriolis forces and the acceleration of passive joints and provides the centrifugal and Coriolis effect on the acceleration of passive joints. The study reveals the coupling characteristics of an underactuated manipulator. This is then demonstrated in a three-link manipulator and extended to a serial manipulator with passive prismatic joint. © 2003 Wiley Periodicals, Inc.

Summary Matrix equations of the kind A1X2+A0X+A−1=X, where both the matrix coefficients and the unknown are semi-infinite matrices belonging to a Banach algebra, are considered. These equations, where coefficients are quasi-Toeplitz matrices, are encountered in certain quasi-birth–death processes as the tandem Jackson queue or in any other processes that can be modeled as a reflecting random walk in the quarter plane. We provide a numerical framework for approximating the minimal nonnegative solution of these equations that relies on semi-infinite quasi-Toeplitz matrix arithmetic. In particular, we show that the algorithm of cyclic reduction can be effectively applied and can approximate the infinite-dimensional solutions with quadratic convergence at a cost that is comparable to that of the finite case. This way, we may compute a finite approximation of the sought solution and of the invariant probability measure of the associated quasi-birth–death process, within a given accuracy. Numerical experiments, performed on a collection of benchmarks, confirm the theoretical analysis.

The focus of this work is on a robust tracking control design for a 6 DOF parallel manipulator in the presence of nonlinearity and fast (or slowly) time-varying uncertainty. Two types of controllers are presented. The controls are based on the Lyapunov approach and guarantee a practical stability. The controls utilize the information of link displacements and its velocities. The first control scheme uses the quadratic Lyapunov function and other uses the geometry dependent Lyapunov function, which excludes the inverse matrix computation on the inertia matrix. Also, the hydraulic dynamics is considered in the control design and control performance. The control performances of the proposed algorithms are verified by simulations and experiments. © 2000 John Wiley & Sons, Inc.

This article summarizes four formulations of the composite body method for the inertia matrix of a manipulator in the earlier works and presents a new formulation. These five formulations all use the first moments and the inertia tensors of composite bodies about the origin of the local frame. This paper also presents an algorithm for computing these first moments and inertia tensors. This algorithm utilizes a set of minimal linear combinations of inertia parameters instead of the natural inertia parameters, so that a number of redundant computations are saved. It is found that the new algorithm for the first moments and the inertia tensors of composite bodies is computationally superior to the others in the literature. On the other hand, two among the five formulations for the inertia matrix are more efficient than the other three as well as the others in the literature. The new formulation is one of these two most efficient formulations, and is specially adequate to a manipulator with some translational joints. ©1999 John Wiley & Sons, Inc.

In this paper two non-linear control laws for ships are derived by using a non-linear ship model which includes the hydrodynamic effects due to time-varying speed and wave frequency. The non-linear ship model does not have the structural properties of symmetric inertia and positive damping at high speed, which are common assumptions, when designing non-linear ship tracking control systems. Backstepping control designs are used to circumvent the problem of a non-symmetrical inertia matrix. The first case is ship tracking with full actuation in surge, sway and yaw while the second case discusses an emergency situation where a ship is supposed to track a time-varying trajectory by using the bow thruster and one of the main propellers as actuation devices only. The application of speed- and wave-dependent models for ship control is a first step towards the design of a new generation of non-linear ship control systems that can operate under time-varying speed and wave frequency conditions. Previous work has often relied on the assumption of constant speed, and practical solutions have been based on switching between different control schemes at different speeds. Slowly varying environmental disturbances, such as ocean current, second-order wave-induced disturbances and wind forces, are compensated for by using adaptive backstepping. Global uniform asymptotic stability (GUAS) is proven for the fully actuated case by using a recent theorem for GUAS when backstepping with integral action, whereas asymptotic tracking of position and velocity are shown for the degraded control case. Computer simulations illustrate the performance and the robustness of the control laws for an offshore supply vessel. © 1998 John Wiley & Sons, Ltd.

The paper derives bounds on the distribution of the quadratic forms Z = yH(XΓXH)−1y and W = yH(σ2I + XΓXH)−1y, where the elements of the M× 1 vector y and the M×N matrix X are independent identically distributed (i.i.d.) complex zero mean Normal variables, Γ is some N×N diagonal matrix with positive diagonal elements, I, is the identity, σ2 is a constant and H denotes the Hermitian transpose. The bounds are convenient for numerical work and appear to be tight for small values of M. This work has applications in digital mobile radio for a specific channel where M antennas are used to receive a signal with N interferers. Some of these applications in radio communication systems are discussed.

In this article, a nonlinear dynamic model of a flexible manipulator is derived through finite element method associated with Lagrange approach. The flexible manipulator is modeled as an Euler‐Bernoulli beam driven by a motor at its base and with a point mass tip payload. The generalized coordinates of the system are selected to be the displacements and rotations of the nodes on the considered flexible beam, and such that a state space model is obtained with all the state variables having physical meanings. Based on this model, an effective nonlinear feedback controller is developed to control the tip position. Furthermore, an efficient algorithm is developed to calculate the inverse of the system’s inertia matrix for real-time implementation. Numerical simulation results are given to show the effectiveness of the controller and its robustness in handling payload variations. © 1997 John Wiley & Sons, Inc.

This article presents an efficient algorithm for computing the inertia matrix of rigid serial manipulators. The derivation of the algorithm is based on the closed-form formulation of the force and moment exerted on a link using a minimum set of dynamic parameters of the manipulator model. The minimum set of dynamic parameters can be derived completely from the original dynamic parameters using the recursive regrouping method before starting the simulation and the control. The proposed computation method is suitable for the control and the simulation based on parameter estimates because the minimum set of dynamic parameters is an identifiable parameter set. The computational efficiency of the proposed methods is compared with other published methods. It is shown that the proposed algorithm is the most efficient approach for serial manipulators. As an example, the number of computations for the inertia matrix of a manipulator with n rotational joints is 11n 2 +9 n - 35 multiplications and 7n 2 + 23 n - 57 additions by reformulating the dynamic model using the minimum set of dynamic parameters.

In this paper we discuss the problem of feedback linearization of rigid robot manipulators. We have shown previously that a necessary and sufficient condition for exact linearization of such systems under a nonlinear coordinate transformation in the state and input space is that the Riemannian curvature tensor associated with the robot inertia matrix vanish identically [17]. Moreover the coordinate transformation is given as the solution of an easily characterized system of partial differential equations. We first review the result from [17] and then discuss the idea of approximate feedback linearization, i.e. feedback linearization of a simplified model. Approximate feedback linearization is useful in this context since the class of robots that satisfy the conditions for exact linearizability is limited. We will show a connection between approximate feedback linearization based on Riemannian curvature and both the imaginary robot concept of Gu and Loh [12] and thepassive computed torque of Anderson [3].

This article deals with the minimal parameters of a manipulator in the least squares sense, so that the minimal parameters are equivalent to the identifiable parameters. The least squares concept is used to introduce terminology for the minimal linear combinations (MLCs) of the system parameters that define a set of linear combinations of the system parameters. The number of elements of the set is minimal, yet the set still completely determines the system. Furthermore, it is shown that the problem of finding a set of MLCs of a manipulator can be simplified to that of finding two individual sets of MLCs that determine the entries of the inertia matrix and the gravity load. Although the approach is applied to the inertia constants of composite bodies to obtain a set of MLCs identical to an earlier one, the result is newly interpreted in the least squares sense. The approach itself is a new method for finding the identifiable parameters of a manipulator, and it yields some new insight into the manipulator dynamics. The crucial feature is that a set of MLCs found by using the present approach is guaranteed to be identifiable. The earlier approaches always require an identification method to verify the results. An equivalence theorem is also presented that rigorously states the equivalence between the different sets of minimal parameters. © 1994 John Wiley & Sons, Inc.

In this article, the relation between the global optimization of joint velocity and local optimization of joint torque is investigated. The local minimization of the weighted joint torques can be matched to the global optimization of the corresponding weighted joint velocities when the weighted matrices satisfy certain sufficient conditions. A straightforward matching is obtained using the local optimization of the inertia inverse weighted dynamic torque and the global minimization of the kinetic energy. Another easy solution can be found, as will be shown later, if the inertia matrix is a constant and gravity is neglected. Based on that, it can be seen why a Cartesian robot, which has a constant inertia matrix, is always stable. © 1994 John Wiley & Sons, Inc.

This article establishes new goals for redundancy resolution based on manipulator dynamics and end-effector characteristics. These goals can be accomplished by employing the recently developed configuration control approach. Redundancy resolution is achieved by controlling the joint inertia matrix or the end-effector mass matrix that affect the inertial torques or by reducing the joint torques due to gravity loading and payload. The manipulator mechanical advantages and velocity ratio are also used as performance measurements to be improved by proper utilization of redundancy. Furthermore, end-effector compliance, sensitivity, and impulsive force at impact are introduced as redundancy-resolution criteria. The new goals for redundancy resolution presented in this article allow a more efficient utilization of the redundant joints based on the desired task requirements. Simple case studies using computer simulations are described for illustration.

This article presents a new composite body method for numerically forming the inertia matrix and the bias vector of manipulators, which is more efficient than the other two existing types of composite body methods. The main discrepancy of this one from the existing ones is that all points in a manipulator are observed from the origin of the base frame and the distances are all measured from this origin. The required computations of the present method for the inertia matrix and the bias vector of a manipulator with n rotational joints are (10.5n2 + 38.5n - 85)M + (6n2 + 39n - 70)A and (12.5n2 + 5.5n + 3)M + (9n2 + n)A, respectively, where “M” denotes multiplications, “A” does additions. In numerically forming the inertia matrix, the present method is more efficient than other methods in the literature for a manipulator with five or more joints; whereas this method is also superior to the recursive Newton-Euler formulation in computing the bias vector for a manipulator with six or less joints.

In this article an efficient algorithm for computation of the manipulator inertia matrix is presented. The algorithm is derived based on Newton's and Euler's laws governing the motion of rigid bodies. Using spatial notations, the algorithm leads to the definition of the composite rigid-body spatial inertia which is a spatial representation of the notion of augmented body. The equations resulting from this algorithm are derived in a coordinate-free form. The choice of the coordinate frame for projection of the coordinate-free equations, that is, the intrinsic equations, is discussed by analyzing the vectors and the tensors involved in the final equations. Previously proposed algorithms, the physical interpretations leading to their derivation, and the redundancy in their computations are analyzed. The developed algorithm achieves a greater efficiency by eliminating the redundancy in the intrinsic equations as well as by a suitable choice of coordinate frame for projection of the intrinsic equations.

The paper presents a method for reducing the size of a vibration analysis by generating an inertia matrix referred to a limited set of ‘master’ vibrational freedoms. In designing a computer program to perform this mass condensation process, the emphasis has been on providing a two way interface between structural analysis and vibration analysis programs. The transformation from the full set of degrees of freedom to the master set makes extensive vibration calculations economically feasible giving solutions in terms of the amplitudes of the master freedoms. However, by reversing the transformation after completion of a vibration analysis, it is possible to restore the full set of freedoms, allowing automatic plotting of mode shapes in full detail, and re-entry to the structural analysis program for the purpose of computing dynamic stresses. A worked example is given. This is a realistic machine tool structure rather than the simple plates or beams which are usually used for illustration purposes. Natural frequencies and mode shapes, calculated with various degrees of condensation, are compared with those obtained by using all possible freedoms. A comparison is also made with an intuitive selection of ‘lumped masses’.