Quasi-Optimal Braking of Rotations of a Body with a Moving Mass Coupled to It through a Quadratic Friction Damper in a Resisting Medium

This paper addresses the problem of the time-optimal braking of rotations of a dynamically symmetric rigid body under a small control moment in the ellipsoidal range with close unequal values of the ellipsoid’s semiaxes. This problem is considered a problem of quasi-optimal control. The body is assumed to have a moving mass connected to it through elastic coupling with quadratic dissipation. In addition, the body is exposed to a small braking moment of the linear resistance of the medium. The problem of synthesizing the quasi-optimal braking of the rotations of a dynamically symmetric body in a resisting medium is investigated analytically and numerically. An approximate solution is found by the phase-averaging of the processional motion. The qualitative properties of quasi-optimal motion are analyzed and the corresponding graphs are presented.     

Attitude determination and stabilization of a spherically symmetric rigid body in a magnetic field

Observation and stabilization problems for a rigid body rotating about its mass centre in a time-periodic magnetic field is considered. The moment of the applied forces about the mass centre is a vector product of the field and the vector of control. The only measurement is of the field and its time derivatives in the body co-ordinates. The model describes the angular motion of a satellite in the geomagnetic field with the control carried out by a magnetic moment of current coils (magnetorquers) mounted on the satellite body. It is shown that generically the knowledge of the field and its derivatives is sufficient for the attitude determination. A problem of stabilization along one direction is considered. The stabilizer constructed in this work drives the system to a given motion from all initial points that do not belong to an unstable manifold. It is shown that all stabilizers have such a manifold of initial conditions.     

Substructure synthesis methods for dynamic analysis of multi-body systems

The formulation for the dynamic analysis of flexible multi-body systems that undergo large rigid body motion, leads to geometrically non-linear inertia properties due to large rotations. These inertia non-linearities that represent the coupling between gross rigid body motion and small elastic deformation, are dependent on the assumed displacement field. As alternatives to the finite element methods, deformable body shape functions and shape vectors are commonly employed to describe elastic deformation of linear structures. In this paper, substructure shape functions and shape vectors are used to describe elastic deformation of non-linear inertia-variant multi-body systems. This leads to two different representations of inertia nonlinearities; one is based on a consistent mass formulation, while the other is a lumped mass technique. The multi-body systems considered are collections of interconnected rigid and flexible bodies. Open and closed loop systems are permitted.     

Analytical Solution of the Minimum Time Slew Maneuver Problem for an Axially Symmetric Spacecraft in the Class of Conical Motions

The conventional problem of the time-optimal slew of a spacecraft considered as a solid body with a single symmetry axis subject to arbitrary boundary conditions for the attitude and angular velocity is considered in the quaternion statement. By making certain changes of variables, the original dynamic Euler equations are simplified, and the problem turns into the optimal slew problem for a solid body with a spherical distribution of mass containing one additional scalar differential equation. For this problem, a new analytical solution in the class of conical motions is found; in this solution, the initial and terminal attitudes of the space vehicle belong to the same cone realized under a bounded control. A modification of the optimal slew problem in the class of generalized conical motions is made that makes it possible to obtain its analytical solution under arbitrary boundary conditions for the attitude and angular velocity of the spacecraft. A numerical example of a spacecraft’s conical motion and examples demonstrating the proximity of the solutions of the conventional and modified optimal slew problems of an axially symmetric spacecraft are discussed.     

Analytical solution of the optimal slew problem for an axisymmetric spacecraft in the class of conical motions

The traditional problem is discussed of an optimal spacecraft slew in terms of minimum energy costs. The spacecraft is considered as a rigid body with one symmetry axis under arbitrary boundary conditions for the angular position and angular velocity of the spacecraft in the quaternion formulation. Using substitutions of variables, the original problem is simplified (in terms of dynamic Euler equations) to the optimal slew problem for a rigid body with a spherical mass distribution. The simplified problem contains one additional scalar differential equation. A new analytical solution is presented for this problem in the class of conical motions, leading to constraints on the initial and final values of the angular velocity vector. In addition, the optimal slew problem is modified in the class of conical motions to derive an analytical solution under arbitrary boundary conditions for the angular position and angular velocity of the spacecraft. A numerical example is given for the conical motion of the spacecraft, as well as examples showing the closeness of the solutions of the traditional and modified optimal slew problems for an axisymmetric spacecraft.     

Analytical and numerical study of the effects of an elastically-linked body on sloshing

In order to investigate the effects of an elastically-linked moving body on liquid sloshing inside a tank, an analytical formulation and a numerical approach were proposed to assess hydrodynamic loads in a partially filled rectangular tank with a body connected to the tank by springs. The analytical approach was developed based on the potential theory to calculate fluid velocity field, and the dynamics of the liquid sloshing coupled to the moving body are described as a mechanical system with two degrees of freedom. The coupling between the fluid and the moving body is given by a damping force calculated based on the body geometry and the fluid velocity field. The proposed numerical approach is based on the Moving Particle Semi-implicit (MPS) method, which is a Lagrangian particle-based method and very effective to model nonlinear hydrodynamics due to fluid–structure interaction. In the numerical approach, the rigid body is modeled as a cluster of particles and the motions are calculated considering its mass, moment of inertia, hydrodynamic loads and springs restoring forces. The elastic link between the body and tank is modeled by applying Hooke’s law. Simple cases of floating body motion were used to validate the numerical method. Finally, analytical and numerical results were compared. Despite its simplicity, the analytical approach proposed in the present work is an efficient approach to provide qualitative understanding and a first estimate of the moving body effects on the sloshing inside the tank. On the other hand, the numerical approach can provide more detailed information about the coupling phenomena, and it is an effective mean for the assessment of the reduction of the sloshing loads due to the moving body with elastic link. Finally, the effectiveness of the concept as a sloshing suppressing device is also investigated.     

Asymptotic output tracking with a two level hierarchical system

In this paper a two level hierarchical system is considered. This system consists of two discs that have limited rotational movement. It is assumed that one disc has large inertia and low angular acceleration and the other disc has low inertia and high angular acceleration. The disc with large angular acceleration is mounted on the periphery of the disc with low acceleration. The angular velocity and orientation of both discs is controllable. It is shown how the theory of asymptotic output tracking for linear systems, combined with some easily programmable mathematical optimization problems leads to very good results when designing a controller for asymptotically tracking a moving object, in particular meeting the angular acceleration limits imposed on the two discs. The paradigm for this system is the human eye-head-neck system and the tracking of a moving object at a distance from the eye.     

Use of Principal Axes as the Floating Reference Frame for a Moving Deformable Body

In this paper a method to adopt the principal axes of a deformable body as its body-attached frame is presented. The deformable body in a multibody setting is allowed to deform while it undergoes rigid-body motion. The fundamental concepts of imposing the principal axes as a moving reference frame are that the origin of the frame must remain at the instantaneous mass center and that the three products of inertia must remain zero as the body deforms. These conditions require the construction of several auxiliary matrices that are used in the constraint equations at the position, velocity, and acceleration levels. These auxiliary matrices are constructed only once and remain unchanged through the motion of the deformable body. The presented formulation does not depend on the type of finite element and multibody formulations or any associated assumptions.     

Experimental determination of the hydrodynamic coefficients of an underwater manipulator

A single degree‐of‐freedom (DOF) manipulator arm with a rectangular cross section is built to investigate the hydrodynamic effects, including drag force, added mass, and the moments induced by these forces. The drag–velocity relationships (linear and angular) are experimentally established and the drag force/moment coefficients are deduced. The added mass and the added moment of inertia are determined for the first time through the relationship between the added mass of the manipulator and its acceleration. These data are very useful for developing the dynamic model and therefore the optimum control of underwater manipulators. ©1999 John Wiley & Sons, Inc.     

飞轮安装偏差的过驱动航天器有限时间姿态控制

针对反作用飞轮安装存在偏差的过驱动航天器姿态跟踪问题, 提出一种有限时间姿态补偿控制策略. 通过设计自适应滑模控制器保证实现对不确定性转动惯量与外部干扰的鲁棒控制, 同时实现对飞轮安装偏差的补偿控制, 并应用Lyapunov 稳定性理论证明了该控制器能够在有限时间内实现姿态跟踪控制. 最后, 将该控制器应用于某型航天器的姿态跟踪控制, 仿真结果验证了所提出方法的有效性.     

Quasi-optimal deceleration of rotations of a rigid body with a moving mass in a resistive medium

The problem of time quasi-optimal deceleration of the rotations of a dynamically symmetric rigid body is studied. It is assumed that the body contains a point mass connected to it with a strong viscoelastic element (damper). The body is acted on by a small linear resistance torque of the medium that is proportional to the angular momentum and a small control torque bounded by an ellipsoidal domain. An approximate synthesis of control is proposed, and an asymptotic solution based on a procedure of averaging the precession motion over the phase is obtained; numerical integration is performed. The main properties of the quasi-optimal motion are determined.     

A non-linear parameter identification approach with applications†

This article is concerned with the identification of unknown parameters for a proposed non-linear time-variant, multivnriable model, that is observed in an additive statistically known white Gaussian noisy environment. The estimation problem is investigated by partitioning the system into subsequent sub-systems to yield a computationally more pragmatical solution. An optimal predictor-corrector maximum-likelihood state estimator and a stochastic hill-climbing procedure are delineated for solving the identification problem. Two particular applications are furnished to illustrate the suggested technique. In the first application, the approach presented is carried out to determine the inertia constant and damping coefficient of a synchronous machine in an unsteady operation, The second application is devoted to tho evaluation of the moments of inertia for a rigid body rotating about its principal axes and subject to external random disturbances and control torques. It is highlighted for the two preceding applications that, though the equations of motion are non-linear in the unknown quantities and/or tho state variables, the solutions obtained via partitioning are expressible in a linear form.     

Application of computer algebra methods for investigation of stationary motions of a gyrostat satellite

With the help of computer algebra methods, properties of a non-linear algebraic system that determines the equilibrium orientations of a gyrostat satellite moving along a circular orbit were investigated. The main attention is paid to the study of equilibrium orientations of the satellite with given principal central moments of inertia and given gyrostatic torque in special cases, when the projection of the vector of gyrostatic torque is located in one of the principal central planes of inertia of the satellite. Computer algebra method based on the algorithm for the construction of a Gröbner basis were applied to reduce the satellite stationary motion system of nine algebraic equations with nine variables to a single algebraic equation in one variable that determines all the equilibrium orientations of the satellite. Bifurcation curves in the space of system parameters that determine boundaries of domains with a fixed number of equilibria of the satellite were obtained symbolically. A comparative analysis of the effectiveness of different algorithms of Gröbner bases computation in solving the problem was carried out.     

Control of the angular orientation of the platform of a uniaxial wheeled module moving without slippage over an underlying surface

A method for controlling the angular orientation with respect to the horizon plane of the platform of a uniaxial wheeled module moving without slippage over the underlying surface is considered. For a module involving a one-axis gyroscope, a compensating flywheel, and a stabilizing weight, equations of motion are derived, which describe the module as a nonholonomic system due to the absence of slippage of its wheels over the underlying surface. The conditions for the “force of undisturbability” of the platform by the inertial forces upon its arbitrary translational-rotational motion and the conditions for “informational undisturbability” by the inertial forces of the accelometric sensor of the deviation of the platform from the horizon plane are determined. The analytical relationships determining the conditions for the absence of slippage of the wheels are found. By numerical simulation, rational values of the coefficient in the laws governing the moments of forces developed by the control elements of the structure are obtained. The effectiveness of these laws is confirmed by the results of simulating the control processes over the angular motion of the platform of the module along a typical trajectory of its motion over the underlying surface.     

Control of Motion of an Actively Bending Body in Channels with Dry Friction

Mechanics is described and equations are given of the motion of an actively bending homogeneous body of a finite thickness in a channel with dry friction under conditions of a weak bending. Control is suggested of the motion of a body in a sinusoidal channel, which retains a continuous first derivative of the controlling moment and a bounded second derivative. A process is described of the maximization of the speed of motion by control and then by the choice of the best parameters of a channel. Control is also considered of other modes of the motion of a body in the same channel.     

Optimal Control of Rotation of a Rigid Body by a Movable Internal Mass

Journal of Computer and Systems Sciences International - The two-dimensional problem on the fastest turn of a rigid body by moving an internal mass is considered. It is assumed that the rigid body...     

Experimental study of a momentum-based method for identifying the inertia barycentric parameters of a human body

Physical modeling, simulation and analysis of an individual human body require inertia properties of the body segments of the human. Such subject-specific inertia data can be obtained only by measuring the individual human body as opposed to be derived from statistically generated anthropometric database. This paper presents experimental validation of a momentum-based approach for identifying the barycentric parameters of an individual human body which fully describes the inertia properties of the human. The identification algorithm is derived from the impulse–momentum equations of the human body which is assumed to be a multibody system with tree-type topology. Since the impulse–momentum equations are linear in terms of the unknown barycentric parameters, these parameters can be solved from the equations using a least-squares method. The approach does not require measuring or estimating accelerations and joint forces/torques because they do not appear in the impulse–momentum equations, and thus, the resulting identification procedure is less demanding on measurement data than the methods derived from the equations of motion. In this paper the test results of the identification method are validated by comparing the identified inertia parameters against the statistically established anthropometric data. Additionally, the identification results are also confirmed by comparing the contact forces using inverse dynamics to those obtained by forces plates.