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This article studies the optimal control policy learning for underactuated vertical take-off and landing (VTOL) aerial vehicles subject to the unknown mass and inertia matrix. A novel off-policy integral reinforcement learning (IRL) scheme is presented for simultaneously unknown parameter identification and optimal trajectory tracking. In the outer loop of the VTOL vehicles, a novel off-policy IRL scheme is proposed, where the fixed control policy for data generation is chosen to be different from the iterated control policy and the feedforward term with an unknown mass can be learned along with the optimal control policy. In the inner loop, a hybrid off-policy IRL algorithm is developed to tackle the optimal attitude control policy learning and inertia matrix identification under the hybrid control scheme introduced by the employed inner-outer loop control strategy. A simulation study is finally provided to demonstrate the effectiveness of the proposed algorithm.

In this article, the problem of fault-tolerant attitude-tracking control of spacecraft with quantized control torque is addressed. Actuator faults/failures, an uncertain inertia matrix and unknown disturbances are considered in the attitude controller design of the spacecraft. A dynamical quantization strategy is developed to quantize the signals of the control torque, which can reduce the data transmission rate. An adaptive fault-tolerant controller based on sliding mode techniques is constructed to address the impacts of the actuator faults/failures, quantization errors, inertia matrix uncertainties and unknown disturbances. The developed control strategy with a quantizer can ensure that the entire closed-loop system is asymptotically convergent and achieves satisfactory attitude-tracking performance. Finally, simulation results are provided to show the effectiveness of the proposed method.

International audience; Control of robots with kinematic constraints like loop-closure constraints or interactions with the environment requires solving the underlying constrained dynamics equations of motion. Several approaches have been proposed so far in the literature to solve these constrained optimization problems, for instance by either taking advantage in part of the sparsity of the kinematic tree or by considering an explicit formulation of the constraints in the problem resolution. Yet, not all the constraints allow an explicit formulation and in general, approaches of the state of the art suffer from singularity issues, especially in the context of redundant or singular constraints. In this paper, we propose a unified approach to solve forward dynamics equations involving constraints in an efficient, generic and robust manner. To this aim, we first (i) propose a proximal formulation of the constrained dynamics which converges to an optimal solution in the least-square sense even in the presence of singularities. Based on this proximal formulation, we introduce (ii) a sparse Cholesky factorization of the underlying Karush-Kuhn-Tucker matrix related to the constrained dynamics, which exploits at best the sparsity of the kinematic structure of the robot. We also show (iii) that it is possible to extract from this factorization the Cholesky decomposition associated to the so-called Operational Space Inertia Matrix, inherent to task-based control frameworks or physic simulations. These new formulation and factorization, implemented within the Pinocchio library, are benchmark on various robotic platforms, ranging from classic robotic arms or quadrupeds to humanoid robots with closed kinematic chains, and show how they significantly outperform alternative solutions of the state of the art by a factor 2 or more.

The article considers a method of examining the influence of dynamic couplings contained in the underwater vehicle model on the movement of this vehicle. The method uses the inertia matrix decomposition and a velocity transformation if the fully actuated vehicle is described in the earth-frame representation. Based on transformed equations of motion, a controller including dynamic couplings in the gain matrices is designed. In the proposed method, the control algorithm is used for the test vehicle dynamics model taking into account disturbances. The approach is useful for simulating the model of an underwater vehicle and improving it, thus avoiding unnecessary experiments or planning them better. The procedure is shown for a full model of an underwater vehicle, and its usefulness is verified by simulation.

Whenrobots physically interact with the environment, compliant behaviors should be imposed to prevent damages to all entities involved in the interaction. Moreover, during physical interactions, appropriate pose controllers are usually based on the robot dynamics, in which the ill-conditioning of the joint-space inertia matrix may lead to poor performance or even instability. When the control is not precise, large interaction forces may appear due to disturbed end-effector poses, resulting in unsafe interactions. To overcome these problems, we propose a task-space admittance controller in which the inertia matrix conditioning is adapted online. To this end, the control architecture consists of an admittance controller in the outer loop, which changes the reference trajectory to the robot end-effector to achieve a desired compliant behavior; and an adaptive inertia matrix conditioning controller in the inner loop to track this trajectory and improve the closed-loop performance. We evaluated the proposed architecture on a KUKA LWR4+ robot and compared it, via rigorous statistical analyses, to an architecture in which the proposed inner motion controller was replaced by two widely used ones. The admittance controller with adaptive inertia conditioning presents better performance than with a controller based on the inverse dynamics with feedback linearization, and similar results when compared to the PID controller with gravity compensation in the inner loop.

In order to solve the joint chattering problem of the manipulator in the process of motion, a novel dynamics model is established based on the dynamics model of the manipulator, by fitting parameters of the neural network and combining with the estimated value of the inertia matrix. We proposed a neural network adaptive control method with a time-varying constraint state based on the dynamics model of estimation. We design the control law, establish the Lyapunov function equation and the asymmetric term, and derive the convergence of the control law. According to the joint state tracking results of the manipulator, the angular displacement, angular velocity, angular acceleration, input torque, and disturbance fitting of the manipulator are analyzed by using the Simulink and Gazebo. The simulation results show that the proposed method can effectively suppress the chattering amplitude under the environment disturbances.

Interconnection and damping assignment passivity-based control scheme has been used to stabilize many physical systems such as underactuated mechanical systems through total energy shaping. In this method, some partial differential equations (PDEs) related to kinetic and potential energy shaping shall be solved analytically. Finding a suitable desired inertia matrix as the solution of nonlinear PDEs relevant to kinetic energy shaping is a challenging problem. In this paper, a systematic approach to solving this matching equation for systems with one degree of underactuation is proposed. A special structure for desired inertia matrix is proposed to simplify the solution of the corresponding PDE. It is shown that the proposed method is more general than that of some reported methods in the literature. In order to derive a suitable desired inertia matrix, a necessary condition is also derived. The proposed method is applied to three examples, including pendubot, VTOL aircraft, and 2D SpiderCrane.

This paper addresses the problem of position control or waypoint tracking for an underactuated quadrotor. The proposed control law is what is known as reaction based in the way that the attitude system reacts to errors in the translational motion. This methodology requires no generation of desired attitude or angular velocity and the resulting control law is model-independent in that it does not require the knowledge of the inertia matrix. In addition the controller has a very simple structure making it suitable for small quadrotor platforms. Simulation results are provided and discussed to demonstrate the proposed method.

In this note we identify a class of underactuated mechanical systems whose desired constant equilibrium position can be globally stabilised with the ubiquitous PID controller. The class is characterised via some easily verifiable conditions on the systems inertia matrix and potential energy function, which are satisfied by many benchmark examples. The design proceeds in two main steps, first, the definition of two new passive outputs whose weighted sum defines the signal around which the PID is added. Second, the observation that it is possible to construct a Lyapunov function for the desired equilibrium via a suitable choice of the aforementioned weights and the PID gains. The results reported here follow the same research line as (Donaire et al., 2016a) and (Romero et al., 2016a) bridging the gap between the Hamiltonian and the Lagrangian formulations used, correspondingly, in these papers. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Magnetic control has been used for decades for spacecraft detumbling, i.e., to bring a spacecraft to a final condition with a sufficiently small angular momentum after separation from the launcher. This task is typically achieved by controllers based on the so-called b-dot principle, which stands out thanks to its simplicity, reliability and ease of on-board implementation. In this paper, we first review existing control methods and study their convergence properties with tools borrowed from general averaging theory, which allows addressing in an accurate manner the time-varying nature of magnetic actuation. Then, some effort is devoted to showing the performance limitations of existing controllers for which increasing the gain too much deteriorates the convergence rate. To overcome this issue, a novel projection-based control law with a state-dependent time-varying gain is presented. By means of Lyapunov arguments for non-autonomous systems, we prove that the proposed controller guarantees that the spacecraft angular momentum converges exponentially fast to zero for all initial conditions, robustly with respect to sufficiently small uncertainties in the inertia matrix, and that the closed-loop solutions are globally uniformly ultimately bounded in the presence of exogenous bounded disturbances. Several numerical simulations have been carried out by referring to realistic detumbling scenarios to show the performance improvement with respect to existing controllers.

With the goal of moving towards implementation of increasingly dynamic behaviors on underactuated systems, this paper presents an optimization-based approach for solving full-body dynamics based controllers on underactuated bipedal robots. The primary focus of this paper is on the development of an alternative approach to the implementation of controllers utilizing control Lyapunov function based quadratic programs. This approach utilizes many of the desirable aspects from successful inverse dynamics based controllers in the literature, while also incorporating a variant of control Lyapunov functions that renders better convergence in the context of tracking outputs. The principal benefits of this formulation include a greater ability to add costs which regulate the resulting behavior of the robot. In addition, the model error-prone inertia matrix is used only once, in a non-inverted form. The result is a successful demonstration of the controller for walking in simulation, and applied on hardware in real-time for dynamic crouching., Accepted to the American Control Conference, 2020

The aim of this article is modifying the well-known homotopy perturbation method (HPM) to yield the solution of Sylvester matrix equation. Moreover, conditions are deduced to check the convergence of the homotopy series. The numerical implementations will be conducted to measure the accuracy and speed of the homotopy series. Finally, some applications of this linear matrix equation are given.

Nowadays, one of the most active research fields in space engineering is autonomous relative navigation around uncooperative objects. A common approach used to tackle this problem is through vision-based pose determination techniques. This paper investigates the possibility of using non-linear filtering techniques to improve the attitude estimation performance of vision-based methods. Furthermore, a simulation study is presented to compare the proposed nonlinear techniques with the multiplicative extended Kalman filter for attitude estimation. First-order and second-order nonlinear filters are adapted, implemented and tested for relative attitude estimation. Finally, the consequences of uncertainty in the knowledge of the target inertia matrix are investigated.

The moment of inertia parameters play a critical role in assuring the spacecraft mission throughout its lifetime. However, determination of the moment of inertia is a key challenge in operating satellites. During satellite mission, those parameters can change in orbit for many reasons such as sloshing, fuel consumption, etc. Therefore, the inertia matrix should be estimated in orbit to enhance the attitude estimation and control accuracy. This paper investigates the use of gyroscope to estimate the attitude rate and inertia matrix for low earth orbit satellite via extended Kalman filter. Simulation results show the effectiveness and advantages of the proposed algorithm in estimating these parameters without knowing the nominal inertia. The robustness of the proposed algorithm has been validated using the Monte-Carlo method. The obtained results demonstrate that the accuracy of the estimated inertia and angular velocity parameters is satisfactory for satellite with coarse accuracy mission requirements. The proposed method can be used for different types of satellites.

The article of record as published may be found at http://dx.doi.org/10.3389/frobt.2018.00041 The paper provides a step-by-step tutorial on the Generalized Jacobian Matrix (GJM) approach for modeling and simulation of spacecraft-manipulator systems. The General Jacobian Matrix approach describes the motion of the end-effector of an underactuated manipulator system solely by the manipulator joint rotations, with the attitude and position of the base-spacecraft resulting from the manipulator motion. The coupling of the manipulator motion with the base-spacecraft are thus expressed in a generalized inertia matrix and a GJM. The focus of the paper lies on the complete analytic derivation of the generalized equations of motion of a free-floating spacecraft-manipulator system. This includes symbolic analytic expressions for all inertia property matrices of the system, including their time derivatives and joint-angle derivatives, as well as an expression for the generalized Jacobian of a generic point on any link of the spacecraft-manipulator system. The kinematics structure of the spacecraft-manipulator system is described both in terms of direction-cosine matrices and unit quaternions. An additional important contribution of this paper is to propose a new and more detailed definition for the modes of maneuvering of a spacecraft-manipulator. In particular, the two commonly used categories free-flying and free-floating are expanded by the introduction of five categories, namely floating, rotation-floating, rotation-flying, translation-flying, and flying. A fully-symbolic and a partially-symbolic option for the implementation of a numerical simulation model based on the proposed analytic approach are introduced and exemplary simulation results for a planar four-link spacecraft-manipulator system and a spatial six-link spacecraft manipulator system are presented.

In this paper we present an energy shaping control law for set-point regulation of the Chaplygin sleigh. It is well known that nonholonomic mechanical systems cannot be asymptotically stabilised using smooth control laws as they do no satisfy Brockett's necessary condition for smooth stabilisation. Here, we propose a discontinuous control law that can be seen as a potential energy shaping and damping injection controller. The proposed controller is shown to be robust against the parameters of both the inertia matrix and the damping structure of the open-loop system., 7 pages, 5 figures, To be presented at 6th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control (LHMNC 2018)

We provide necessary and sufficient conditions for the generalized $\star$-Sylvester matrix equation, $AXB + CX^\star D = E$, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices $A, B, C, D$ (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the existence and uniqueness of solution for generalized Sylvester and $\star$-Sylvester equations., This new version corrects some inaccuracies in corollaries 7 and 9

© 2017 Informa UK Limited, trading as Taylor & Francis Group In many areas of applied linear algebra, it is necessary to work with matrix approximations. A usual situation occurs when a matrix obtained from experimental or simulated data is needed to be approximated by a matrix that lies in a corresponding statistical model and satisfies some specific properties. In this short note, we focus on symmetric and positive-semidefinite approximations and we show that the positive and negative indices of inertia of the symmetric approximation and the rank of the positive-semidefinite approximation are always bounded from above by the rank of the original matrix.

International audience; Stabilization of mechanical systems by shaping their energy function is a well-established technique whose roots date back to the work of Lagrange and Dirichlet. Ortega and Spong in 1989 proved that passivity is the key property underlying the stabilization mechanism of energy shaping designs and the, now widely popular, term of passivity-based control (PBC) was coined. In this chapter, we briefly recall the history of PBC of mechanical systems and summarize its main recent developments. The latter includes: (i) an explicit formula for one of the free tuning gains that simplifies the computations, (ii) addition of PID controllers to robustify and make constructive the PBC design and to track ramp references, (iii) use of PBC to solve the position feedback global tracking problem, and (iv) design of robust and adaptive speed observers.

This paper summarizes current scientific knowledge and the results of a study focused on the determination of a powertrain’s inertia ellipsoid. The work deals with the application of methods for the experimental determination of the inertia matrix and summarizes their basic potential. The work describes a proposed computational algorithm by means of which the inertia ellipsoid can be determined. The experimental section of the work presents the results of measurements for internal combustion engines (powertrains).

In this paper, for a significant class of uncertain MIMO systems, in particular for robots, new fundamental results are stated. These allow to design robust control laws without chattering, of pseudo-PID type, in order to track sufficiently regular trajectories with a preassigned maximum error. The proposed pseudo-PID controllers are easy to design and implement because these laws can also be decentralized and they are based on two design parameters. The first parameter is related to the minimum eigenvalue of a suitable matrix, depending on the inertia matrix in the case of a mechanical system, from which the practical stability depends on, and the second parameter is related to the practical region of asymptotic stability, to the precision of the tracking error and to the time constant of an appropriate linearized majorant system of the closed-loop control system. The utility and the efficiency of the main results proposed in this paper are illustrated with a significant example.

International audience; Control of underactuated mechanical systems via energy shaping is a well-established, robust design technique. Unfortunately, its application is often stymied by the need to solve partial differential equations (PDEs). In this paper a new, fully constructive, procedure to shape the energy for a class of mechanical systems that obviates the solution of PDEs is proposed. The control law consists of a first stage of partial feedback linearization followed by a simple proportional plus integral controller acting on two new passive outputs. The class of systems for which the procedure is applicable is identified imposing some (directly verifiable) conditions on the systems inertia matrix and its potential energy function. It is shown that these conditions are satisfied by three benchmark examples.

This paper presents the explicit dynamic equations of multibody mechanical systems. This is the second paper on this topic. In the first paper the dynamics of a single rigid body from the Boltzmann--Hamel equations were derived. In this paper these results are extended to also include multibody systems. We show that when quasi-velocities are used, the part of the dynamic equations that appear from the partial derivatives of the system kinematics are identical to the single rigid body case, but in addition we get terms that come from the partial derivatives of the inertia matrix, which are not present in the single rigid body case. We present for the first time the complete and correct derivation of multibody systems based on the Boltzmann--Hamel formulation of the dynamics in Lagrangian form where local position and velocity variables are used in the derivation to obtain the singularity-free dynamic equations. The final equations are written in global variables for both position and velocity. The main motivation of these papers is to allow practitioners not familiar with differential geometry to implement the dynamic equations of rigid bodies without the presence of singularities. Presenting the explicit dynamic equations also allows for more insight into the dynamic structure of the system. Another motivation is to correct some errors commonly found in the literature. Unfortunately, the formulation of the Boltzmann-Hamel equations used here are presented incorrectly. This has been corrected by the authors, but we present here, for the first time, the detailed mathematical details on how to arrive at the correct equations. We also show through examples that using the equations presented here, the dynamics of a single rigid body is reduced to the standard equations on a Lagrangian form, for example Euler's equations for rotational motion and Euler--Lagrange equations for free motion.

It is proved that the range of a Sylvester map defined by two matrices of sizes p× p and q × q, respectively, plus matrices whose ranks are bounded above, cover all p × q matrices. The best possible upper bound on the ranks is found in many cases. An application is made to a minimal rank problem that is motivated by the theory of minimal factorizations of rational matrix functions.

International audience; We address and solve some long-standing yet well– documented open problems on output feedback tracking control of Euler-Lagrange systems with arbitrarily high relative degree; this includes underactuated systems. Our main contribution is to establish a theoretical foundation for the use of so-called dirty derivatives, a common " ad-hoc " replacement of unavailable state measurements such as generalized velocities, whence obviating the the use of observers for the purpose of position-feedback tracking control. Reminiscent of passivity-based control for robot manipulators, our control law is globally Lipschitz and the controller dynamics is linear. For relative-degree-two fully-actuated Lagrangian systems without dissipative forces (friction) and with bounded inertia matrix we establish uniform global asymptotic stability in closed loop. Furthermore, we show that our control approach applies to Lagrangian systems augmented by a chain of integrators (relative degree m + 2 systems). The design method, which is based on a recursive procedure in the spirit of backstep-ping control, is intuitive as it exploits structural properties such as passivity and inherent input-output stability. As a corollary, we solve an output feedback global-tracking control problem for flexible-joint robots but also for systems coupled with output-feedback linearizable actuator dynamics. In addition, we discuss remaining open problems of fairly general interest in the realm of analysis and design of robust nonlinear systems.

The Sylvester matrix was first defined by JJ Sylvester. Some authors have studied the relationships between certain orthogonal polynomials and the determinant of the Sylvester matrix. Chu studied a generalization of the Sylvester matrix. In this paper, we introduce its 2-periodic generalization. Then we compute its spectrum by left eigenvectors with a similarity trick.

We develop a method to simplify the partial differential equations (PDEs) associated to the potential energy for interconnection and damping assignment passivity based control (IDA-PBC) of a class of underactuated mechanical systems (UMSs). Solving the PDEs, also called the matching equations, is the main difficulty in the construction and application of the IDA-PBC. We propose a simplification to the potential energy PDEs through a particular parametrization of the closed-loop inertia matrix that appears as a coupling term with the inverse of the original inertia matrix. The parametrization accounts for kinetic energy shaping, which is then used to simplify the potential energy PDEs and their solution that is used for the potential energy shaping. This energy shaping procedure results in a closed-loop UMS with a modified energy function. This approach avoids the cancellation of nonlinearities, and extends the application of this method to a larger class of systems, including separable and non-separable port-controlled Hamiltonian (PCH) systems. Applications to the inertia wheel pendulum and the rotary inverted pendulum are presented, and some realistic simulations are presented which validate the proposed control design method and prove that global stabilization of these systems can be achieved. Experimental validation of the proposed method is demonstrated using a laboratory set-up of the rotary pendulum. The robustness of the closed-loop system with respect to external disturbances is also experimentally verified.

International audience; Total energy shaping is a controller design methodology that achieves (asymptotic) stabilisation of mechanical systems endowing the closed-loop system with a Lagrangian or Hamiltonian structure with a desired energy function. The success of the method relies on the possibility of solving two partial differential equations (PDEs) which identify the kinetic and potential energy functions that can be assigned to the closed loop. Particularly troublesome is the PDE associated to the kinetic energy (KE) which is quasi-linear and non-homogeneous, and the solution that defines the desired inertia matrix must be positive definite. This task is simplified by the inclusion of gyroscopic forces in the target dynamics, which translates into the presence of a free skew-symmetric matrix in the KE matching equation that reduces the number of PDEs to be solved. Recently, it has been claimed that considering a more general form for the target dynamic forces that relax the skew-symmetry condition further reduces the number of KE PDEs. The purpose of this paper is to prove that this claim is wrong.

Although adaptive control schemes for spacecraft attitude tracking are abundant in controls literature, very few are designed to guarantee consistent performance for a spacecraft with both rigid and nonrigid (time-varying) inertia components. Because inertia matrix changes are a common occurrence due to phenomena like fuel depletion or mass displacement in a deployable spacecraft, an adaptive control algorithm that takes explicit account of such information is of significant interest. In this paper, a novel adaptive attitude control scheme is presented for a spacecraft with inertia matrix parameters that have both unknown rigid components and only partially determined variable components. The proposed controller directly compensates for inertia variations that occur as either pure functions of the control input, or as functions of time. For the particular case of an input-dependent inertia matrix, a bounded control solution is ensured by placing some mild restrictions on the initial conditions and by employing a smooth projection scheme that confines the parameter estimates to a well-dMefined convex set. Detailed derivations of the control law are provided, along with a thorough analysis for the associated stability and error convergence properties. In addition, numerical simulations are presented to highlight the performance benefits when compared with an adaptive control scheme that does not account for inertia variations.

In this survey paper, a summary of results which are to be found in a series of papers, is presented. The subject of interest is focused on matrix algebraic properties of the Fisher information matrix (FIM) of stationary processes. The FIM is an ingredient of the Cram´er-Rao inequality, and belongs to the basics of asymptotic estimation theory in mathematical statistics. The FIM is interconnected with the Sylvester, Bezout and tensor Sylvester matrices. Through these interconnections it is shown that the FIM of scalar and multiple stationary processes fulfill the resultant matrix property. A statistical distance measure involving entries of the FIM is presented. In quantum information, a different statistical distance measure is set forth. It is related to the Fisher information but where the information about one parameter in a particular measurement procedure is considered. The FIM of scalar stationary processes is also interconnected to the solutions of appropriate Stein equations, conditions for the FIM to verify certain Stein equations are formulated. The presence of Vandermonde matrices is also emphasized.

An approach is presented for the model identifi- cation of the so-called link dynamics used by the KUKA LWR- IV, a lightweight manipulator with elastic joints that is very popular in robotics research but for which a complete and reliable dynamic model is not yet publicly available. The control software interface of this robot provides numerical values of the link inertia matrix and the gravity vector at each configuration, together with link position and joint torque sensor data. Taking advantage of this information, a general procedure is set up for determining the structure and identifying the value of the relevant dynamic coefficients used by the manufacturer in the evaluation of these robot model terms. We call this a reverse engineering approach, because our main goal is to match the numerical data provided by the software interface, using a suitable symbolic model of the robot dynamics and the inertial and gravity coefficients that are being estimated. Only configuration-dependent terms are involved in this process, and thus static experiments are sufficient for this task. The main issues of dynamic model identification for robots with elastic joints are discussed in general, highlighting the pros and cons of the approach taken for this class of KUKA lightweight manip- ulators. The main identification results, including training and validation tests, are reported together with additional dynamic validation experiments that use the complete identified model and joint torque sensor data.

53rd IEEE Annual Conference on Decision and Control, CDC 2014; JW Marriott Hotel Los Angeles L.A.; United States; 15 December 2014 through 17 December 2014, This work concentrates on tracking control of dynamically positioned surface vessels with asymmetric added mass terms affecting the system model at the acceleration level. Specifically, we propose a novel continuous robust controller for surface vessels that, in addition to asymmetric added mass in its inertia matrix, contains unstructured uncertainties in all its system matrices. The proposed controller compensates the overall system uncertainties and ensures asymptotic tracking, while requiring only the knowledge of the sign of the leading principle minors of the input gain matrix. Lyapunov based approaches are applied in order to prove the stability of the closed-loop system and asymptotic convergence of the tracking error signal. © 2014 IEEE.

A robust control allocation scheme is developed for rigid spacecraft attitude stabilization in the presence of actuator partial loss fault, actuator failure, and actuator misalignment. First, a neural network fault detection scheme is proposed, Second, an adaptive attitude tracking strategy is employed which can realize fault tolerance control under the actuator partial loss and actuator failure withinλmin⁡=0.5. The attitude tracking and faults detection are always here during the procedure. Once the fault occurred which could not guaranteed the attitude stable for 30 s, the robust control allocation strategy is generated automatically. The robust control allocation compensates the control effectiveness uncertainty which caused the actuator misalignment. The unknown disturbances, uncertain inertia matrix, and even actuator error with limited actuators are all considered in the controller design process. All are achieved with inexpensive online computations. Numerical results are also presented that not only highlight the closed-loop performance benefits of the control law derived here but also illustrate its great robustness.

This paper presents a novel approach to simplify the dynamic model of industrial robot by means of interval method. Due to strong nonlinearities and complexities, some components of robot dynamic model such as the inertia matrix and the vector of centrifugal, Coriolis and gravitational torques, are very complicated for real-time control of industrial manipulators. Thus, an interval-based simplification algorithm is proposed in this study in order to reduce the computation time and the memory occupation. Besides, it also contributes to finding negligible inertia parameters, which is useful for robot model identification. More importantly, the proposed approach is suitable for arbitrary trajectories in the whole robot workspace. Simulation studies have been carried out on a test trajectory using a 6-DOF industrial robot model, showing good performance and effectiveness of the simplification.

Published version of an article in the journal: Abstract and Applied Analysis. Also available from the publisher at: http://dx.doi.org/10.1155/2014/302982 Open Access The problem of finite-time control for attitude tracking maneuver of a rigid spacecraft is investigated. External disturbance, unknown inertia parameters are addressed. As stepping stone, a sliding mode controller is designed. It requires the upper bound of the lumped uncertainty including disturbance and inertia matrix. However, this upper bound may not be easily obtained. Therefore, an adaptive sliding mode control law is then proposed to release that drawback. Adaptive technique is applied to estimate that bound. It is proved that the closed-loop attitude tracking system is finite-time stable. The tracking errors of the attitude and the angular velocity are asymptotically stabilized. Moreover, the upper bound on the lumped uncertainty can be exactly estimated in finite time. The attitude tracking performance with application of the control scheme is evaluated through a numerical example.

International audience; The problem of robustness improvement, vis à vis external disturbances, of energy shaping controllers for mechanical systems is addressed in this paper. First, it is shown that, if the inertia matrix is constant, constant disturbances (both, matched and unmatched) can be rejected simply adding a suitable integral action--interestingly, not at the passive output. For systems with non-constant inertia matrix, additional damping and gyroscopic forces terms must be added to reject matched disturbances and, moreover, enforce the property of integral input-to-state stability with respect to matched disturbances. The stronger property of input-to-state stability, this time with respect to matched and unmatched disturbances, is ensured with further addition of nonlinear damping. Finally, it is shown that including a partial change of coordinates, the controller can be significantly simplified, preserving input-to-state stability with respect to matched disturbances.

Discussion on equations of motion of planar flexible mechanisms is presented in this paper. The finite element method (FEM) is used for obtaining vibrational analysis of links. In derivation of dynamic equations it is commonly assumed that the shape function of elastic motion can represent rigid-body motion. In this paper, in contrast to this assumption, a model of the shape function specifically dedicated to the rigid-body motion is presented, and its influence on elastic motion is included in equations of motion; the inertia matrix related to the rigid-body acceleration vector depends on both shape functions of the elastic and rigid elements. The numerical calculations are conducted in order to determine the influence of the assumed shape function for rigid-body motion on the vibration of links in the case of closed-loop and open-loop mechanisms. The results of numerical simulation show that for transient analysis and for some specific conditions (e.g., starting range, open-loop mechanisms) the influence of assumed shape functions on vibration response can be quite significant.

Optimum design is a pivotal approach to fulfill the potential advantages of the parallel manipulator in practical applications. This paper concerns the optimum design issue of the 3-3 Stewart platform considering the inertia property, in addition to the kinematic performance. On the basis of spherical usable workspace, global conditioning index (GCI) is analyzed. Atlases of the workspace and GCI are deduced with the established nondimensional design space. Further, after dynamic modeling, the global inertia index (GII) is deduced from the joint-space inertia matrix, and corresponding GII atlases are drawn. In particular, an example is presented to illustrate the process of obtaining the practical optimum results based on these non-dimensional atlases. Since both kinematic and dynamic properties are considered, the optimum result will possess comprehensive performance improvements.

This paper is concerned with the numerical solution of the Sylvester tensor equation, which includes the Sylvester matrix equation as special case. By applying hierarchical identification principle proposed by Ding and Chen, 2005, and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective.

In this paper we present the Sherman-Morrison-Woodbury-type formula for the solution of the Sylvester equation of the form \[(A_0+U_1V_1)X+X(B_0+U_2V_2)=E, \] as well as for the solution of the $T$-Sylvester equation of the form \[ (A_0+U_1V_1)X+X^T(B_0+U_2V_2)=E, \] where $U_1, U_2, V_1, V_2$ are low-rank matrices. Although the matrix version of this formula for the Sylvester equation has been used in several different applications (but not for the case of a $T$-Sylvester equation), we present a novel approach using a proper operator representation. This novel approach allows us to derive a matrix version of the Sherman-Morrison-Woodbury-type formula for the Sylvester equation, as well as for the $T$-Sylvester equation which seems to be new. We also present algorithms for the efficient calculation of the solution of Sylvester and $T$-Sylvester equations by using these formulas and illustrate their application in several examples.

Indefinite symmetric matrices occur in many applications, such as optimization, least squares problems, partial differential equations, and variational problems. In these applications one is often interested in computing a factorization of the indefinite matrix that puts into evidence the inertia of the matrix or possibly provides an estimate of its eigenvalues. In this paper we propose an algorithm that provides this information for any symmetric indefinite matrix by transforming it to a block antitriangular form using orthogonal similarity transformations. We also show that the algorithm is backward stable and has a complexity that is comparable to existing matrix decompositions for dense indefinite matrices.

Roth's similarity theorem on the consistency of Sylvester's matrix equation AX − XA = C can be extended to a theorem on rank minimization if the common eigenvalues of A and B are nonderogatory or semisimple.

International audience; The problem of robustness improvement, vis 'a vis external disturbances, of energy shaping controllers for mechanical systems was addressed by the authors in a previous paper. It was shown that-if the inertia matrix is constant-- constant disturbances (both, matched and unmatched) can be rejected simply adding a suitable integral action. For systems with non–constant inertia matrix and non-constant disturbances the controller, that adds nonlinear damping and gyroscopic forces terms, is quite complicated. The purpose of this paper is to show that, including a partial change of coordinates, the controller can be significantly simplified, achieving the same robustness property of input--to-state stability with respect to matched and unmatched disturbances of the previous controller.

The active force control (AFC) method is known as a robust control scheme that dramatically enhances the performance of a robot arm particularly in compensating the disturbance effects. The main task of the AFC method is to estimate the inertia matrix in the feedback loop to provide the correct (motor) torque required to cancel out these disturbances. Several intelligent control schemes have already been introduced to enhance the estimation methods of acquiring the inertia matrix such as those using neural network, iterative learning and fuzzy logic. In this paper, we propose an alternative scheme called Knowledge-Based Trajectory Error Pattern Method (KBTEPM) to suppress the trajectory track error of the AFC scheme. The knowledge is developed from the trajectory track error characteristic based on the previous experimental results of the crude approximation method. It produces a unique, new and desirable error pattern when a trajectory command is forced. An experimental study was performed using simulation work on the AFC scheme with KBTEPM applied to a two-planar manipulator in which a set of rule-based algorithm is derived. A number of previous AFC schemes are also reviewed as benchmark. The simulation results show that the AFC-KBTEPM scheme successfully reduces the trajectory track error significantly even in the presence of the introduced disturbances.Key Words: Active force control, estimated inertia matrix, robot arm, trajectory error pattern, knowledge-based.

International audience; This paper deals with Newmark time stepping methods and finite rotations for nonlinear finite element analysis of flexible beam structures. The corotational method is used to develop expressions of the internal forces and the corresponding tangent stiffness matrices. For the dynamic part, four formulations based on different parameterizations of rotations are compared. The first three are classic formulations taken from the literature with some modifications for two of them. The last one is new and uses three of the four Euler parameters (quaternion) as rotational variables. For all these approaches, theoretical derivations as well as practical implementations are given in detail. The similarities and differences between the formulations are pointed out. Six numerical examples are studied in order to compare these four formulations in terms of numerical accuracy and computational efficiency. Regarding efficiency, the choice of the predictor at each time step and the possibility to simplify the tangent inertia matrix are carefully investigated. The numerical results show that these four formulations have the same numerical accuracy, but that the computational efficiency depends on the choice of the tangent inertia matrix. Besides, the new formulation proposed in this paper turns out to be the fastest one. (C) 2012 Elsevier B.V. All rights reserved.