Optimal control of rigid body motion with the help of rotors using stereographic coordinates

This paper considers the problem of optimal controlling the rotational motion of a rigid body using three independent control torques developed by three rotors attached with the principal axes of inertia of the body and rotate with the help of electric motors rigidly mounted on the body. The optimal control law is given as non-linear function of new parameterizations of the rotation group derived by using the stereographic projection of the Euler parameters. Given a cost function we seek for a stabilizing feedback control law that minimizes this cost and asymptotically stabilizes the rotational motion of the body. The stabilizing properties of the proposed controllers are proved by using the optimal Liapunov function. Numerical examples and simulation study are presented.     

Matrix formulation of the motion equations of a rigid body and identification of the ten inertia characteristics

The motion equations of a rigid body involve ten inertial characteristics: the mass, the mass center position and the inertia matrix. In order to identify these ten inertia characteristics, we propose an approach unifying them in a 4 × 4 positive definite symmetric matrix. The translation vector and the rotation matrix of the rigid body are also gathered in a 4 × 4 matrix. Therefore the motion equations are formulated as an equality between 4 × 4 skew–symmetric matrices: one representing the sum of external forces and torques, the second representing the dynamic force and torque. The identification is performed by a projected conjugate gradient algorithm developped in the 10–dimensional linear space of 4 × 4 symmetric matrices. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

推导刚体动力学方程的张量方法

本文提出了基于连续介质力学概念推导刚体动力学方程的张量方法,运用具有零共旋率的惯性张量的时间导数公式,证明了Lagrange方程、Nielsen方程、Gibbs-Appell方程、Kane方程和广义动量式Kane方程等五种方法的等价性,给出了角速度、角加速度之间的一些微分关系式.     

Control of the motion of a helical body in a fluid using rotors

This paper is concerned with the motion of a helical body in an ideal fluid, which is controlled by rotating three internal rotors. It is proved that the motion of the body is always controllable by means of three rotors with noncoplanar axes of rotation. A condition whose satisfaction prevents controllability by means of two rotors is found. Control actions that allow the implementation of unbounded motion in an arbitrary direction are constructed. Conditions under which the motion of the body along an arbitrary smooth curve can be implemented by rotating the rotors are presented. For the optimal control problem, equations of sub-Riemannian geodesics on SE(3) are obtained.     

Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod

A rigid–flexible coupling dynamic analysis is presented where a mass is attached to a massless flexible rod which rotates about an axis. The rod is limited to small deformation so that the mass is constrained to move in the plane of rotation. A strongly nonlinear model of the system is established based on the couplings between the elastic deflections of the mass and rigid rotation, in which the mass deflection and rigid rotation are both treated as unknown variables. The additional inertia forces on the mass and coupling mechanism are elucidated in the system model. In the case of varied but prescribed rigid rotation, a set of time-varying differential equations governing mass motion is obtained. The trajectories of mass motion are examined for the spin-up and spin-down rotation. Under constant rigid rotation, a set of ordinary differential equations is further attained, and the issues with dynamic frequencies and critical angular velocity of the system are analyzed. The effects of the centrifugal, Coriolis and tangential inertia forces on the dynamic responses are discussed.     

Dynamics and chaos control of gyrostat satellite

We consider the chaotic motion of the free gyrostat consisting of a platform with a triaxial inertia ellipsoid and a rotor with a small asymmetry with respect to the axis of rotation. Dimensionless equations of motion of the system with perturbations caused by small asymmetries of the rotor are written in Andoyer-Deprit variables. These perturbations lead to separatrix chaos. For gyrostats with different ratios of moments of inertia heteroclinic and homoclinic trajectories are written in closed-form. These trajectories are used for constructing modified Melnikov function, which is used for determine the control that eliminates separatrix chaos. Melnikov function and phase space trajectory are built to show the effectiveness of the control.     

Dynamisches Verhalten einer einfach besetzten rotierenden Welle mit Kardangelenken

Summary If a rotating, massless, elastic shaft carrying a disk is supported at the ends by Cardan links, the motion of the disk depends on the angles at the joints and the torques transmitted by the joints. The system is considered for constant angular velocity and constant torques of the driving shafts. The investigation of this nonstationary system leads to two second order differential equations with periodic coefficients. In order to establish conditions for instability the characteristics exponents are calculated by means of generalized Hills determinants. It is found that there exist critical intervals for the angular velocity.     

Identification of the ten inertia parameters of a rigid body

A special antisymmetric 4 × 4 matrix form of the equation of motion of a rigid body is proposed. This form depends linearly on the symmetric (4 × 4)-matrix of the Fayet global inertia tensor, containing the ten inertia parameters of a rigid body (the mass, the three coordinates of the centre of mass and the six components of the classical inertia tensor). For identifying the global inertia tensor, an algorithm is proposed which is based on the method of least squares and the method of conjugate gradients and tested using the example of a rigid body, the motion of which is obtained by computer modelling.     

Sufficient Global Stability Condition for a Model of the Synchronous Electric Motor under Nonlinear Load Moment

We study a model of the synchronous electric motor, which is described by a system of ordinary differential equations, including equations for electric currents in the windings of the rotor. The load moment is assumed to be a nonlinear function of the angular velocity of the rotor, allowing a linear estimate. The system of differential equations under consideration has a countable number of stationary solutions corresponding to the operating mode of uniform rotation of the rotor with the angular velocity equal to the angular velocity of rotation of the magnetic field in the stator. An effective sufficient condition is derived under which any motion of the rotor of the synchronous electric motor tends with time to uniform rotation.     

New kinematic parameters of the finite rotation of a rigid body

A new family of kinematic parameters for the orientation of a rigid body (global and local) is presented and described. All the kinematic parameters are obtained by mapping the variables onto a corresponding orientated subspace (hyperplane). In particular, a method of stereographically projecting a point belonging to a five-dimensional sphere S⁵ ⊂ R⁶ onto an orientated hyperplane R⁵ is demonstrated in the case of the classical direction cosines of the angles specifying the orientation of two systems of coordinates. A family of global kinematic parameters is described, obtained by mapping the Hopf five-dimensional kinematic parameters defined in the space R⁵ onto a four-dimensional orientated subspace R⁴. A correspondence between the five-dimensional and four-dimensional kinematic parameters defined in the corresponding spaces is established on the basis of a theorem on the homeomorphism of two topological spaces (a four-dimensional sphere S⁴ ⊂ R⁵ with one deleted point and an orientated hyperplane in R⁴). It is also shown to which global four-dimensional orientation parameters–quaternions defined in the space R⁴ the classical local parameters, that is, the three-dimensional Rodrigues and Gibbs finite rotation vectors, correspond. The kinematic differential rotational equations corresponding to the five-dimensional and four-dimensional orientation parameters are obtained by the projection method. All the rigid body kinematic orientation parameters enable one, using the kinematic equations corresponding to them, to solve the classical Darboux problem, that is, to determine the actual angular position of a body in a three-dimensional space using the known (measured) angular velocity of rotation of the object and its specified initial position.     

A kinematic interpretation of the motion of a heavy rigid body with a fixed point

A kinematic interpretation of the motion of a rigid body with a fixed point is proposed using the rolling of a mobile hodograph, which describes, on the ellipsoid of inertia, a vector collinear with the vector of the angular velocity of the body, with respect to a fixed vector. On the basis of this, an interpretation of the motion of the body in the Steklov, Grioli, Dokshevich and Bobylev – Steklov solutions is obtained. A new formula is derived indicating the connection between the angle of precession and the polar angle of the equations of the fixed hodograph, indicated by Kharlamov.     

On a periodic motion of a rigid body carrying a material point in the presence of impacts with a horizontal plane

A material system consisting of an outer rigid body (a shell) and an inner body (a material point) is considered. The system moves in a uniform field of gravity over a fixed absolutely smooth horizontal plane. The central ellipsoid of inertia of the shell is an ellipsoid of rotation. The material point moves according to the harmonic law along a straight-line segment rigidly attached to the shell and lying on its axis of dynamical symmetry. During its motion, the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. The periodic motion of the shell is found when its symmetry axis is situated along a fixed vertical, and the shell rotates around this vertical with an arbitrary constant angular velocity. The conditions for existence of this periodic motion are obtained, and its linear stability is studied.     

A Study of the Motions of an Autonomous Hamiltonian System at a 1:1 Resonance

We examine the motions of an autonomous Hamiltonian system with two degrees of freedom in a neighborhood of an equilibrium point at a 1:1 resonance. It is assumed that the matrix of linearized equations of perturbed motion is reduced to diagonal form and the equilibrium is linearly stable. As an illustration, we consider the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on a circular orbit in a neighborhood of cylindrical precession. The abovementioned resonance case takes place for parameter values corresponding to the spherical symmetry of the body, for which the angular velocity of proper rotation has the same value and direction as the angular velocity of orbital motion of the radius vector of the center of mass. For parameter values close to the resonance point, the problem of the existence, bifurcations and orbital stability of periodic rigid body motions arising from a corresponding relative equilibrium of the reduced system is solved and issues concerning the existence of conditionally periodic motions are discussed.     

Integrable cases of the problem of the free motion of a gyrostat

The free spatial motion of a gyrostat in the form of a carrier body with a triaxial ellipsoid of inertia and an axisymmetric rotor is considered. The bodies have a common axis of rotation, which coincides with one of the principal axes of inertia of the carrier. In the Andoyer–Deprit variables the equations of motion reduce to a system with one degree of freedom. Stationary solutions of this system are found, and their stability is analysed. Cases in which the longitudinal moment of inertia of the carrier is greater than the largest of the transverse moments of inertia of the system of bodies, is smaller than the smallest or belongs to a range composed of the moments of inertia indicated, are investigated. General analytical solutions that describe the motion on separatrices and in regions corresponding to oscillations and rotation on the phase portrait are obtained for each case. The results can be interpreted as a development of the Euler case of the motion of a rigid body about a fixed point when one degree of freedom, namely, relative rotation of the bodies, is added.     

Partial stability and stabilization of dynamical systems

We prove the theorem on necessary and sufficient conditions of partial instability and the theorem on partial stabilization of nonlinear dynamical systems. We obtain sufficient conditions of controllability for systems linear with respect to control. We also study the problem of control and stabilization of an angular motion of a solid body by rotors.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 186–193, February, 1995.     

Classification of complete integrability cases in four-dimensional symmetric rigid-body dynamics in a nonconservative field

This work is a relatively final result in studying the equations of motion of a dynamically symmetric, four-dimensional rigid body in a nonconservative force field in two logically possible cases of its tensor of inertia. The form of the force field considered is taken from the dynamics of real three-dimensional rigid bodies interacting with a medium.     

Optimal control of the motion of a multilink system in a resistive medium

The optimal control of the motion of a system consisting of a main body and one or two links joined to it by cylindrical joints in a resistive medium is investigated. The resistance force of the medium acting on the moving body is assumed to depend on their velocity. The control is accomplished through high-frequency angular oscillations of the links. The equations of motion are analysed, and the mean velocity of translational motion of the system is estimated under certain assumptions. Optimal control problems are formulated and solved, and the laws of control of the oscillations of the links for which the maximum mean velocity of motion is obtained are found as a result. The data obtained are in qualitative agreement with observations of the swimming of fish and animals. The results of this study can be used in developing mobile robots that move in a liquid.     

Properties of solutions for the generalized Euler‐Poisson equations

The present paper deals with equations, which generalize the known Euler‐Poisson equations for the motion of a heavy rigid body about a fixed point. These equations arise in dynamics of systems of coupled rigid bodies. In these equations the generalized inertia tensor depends upon components of vertical vector, i.e. it is not constant. Our aim is to analyze Lyapunov stability of stationary solutions and orbital stability of periodic solutions of the equations under study.     

The problem of the time-optimal turning of a manipulator

Two types of manipulator that perform three-dimensional motions are considered, and the control problem in which the manipulator rotation is performed in minimum time is studied. The rate of rotation of a rigid body about an axis rises as the moment of inertia about this axis falls. Manipulator control amounts to a problem of the rotation of a system of rigid bodies about an axis. In addition to the angle of rotation, there is a further controlled coordinate, whose variation can vary the moment of inertia about the axis. Assuming that the moment of inertia can be stantaneously “frozen” (that pulse control signals are possible), the in-time-optimal control modes were found in /1, 2/, (see also Akulenko, L.D. et al., “Optimization of the control modes of manipulation robots”, Preprint 218, In-t. Problem Mekhaniki Akad. Nauk SSSR, Moscow 1983). In these modes, the rotation, occurs in the entire time interval with minimum moment of inertia about the axis of rotation. The rotation when there are constraints on the control (pulse control signals are not permitted) was considered in /3/. Numerical studies there led to the false conclusion that, in the optimal motion, with a finite number of control switchings, the moment of inertia is also a minimum throughout the time interval. Below, for a set of extreme configurations, a control is constructed for the two types of manipulator, which satisfies the Pontryagin maximum principle, when there are constraints on the control signals. During its rotation the manipulator section then performs oscillations about a position corresponding to minimum moment of inertia about the axis of rotation. It is shown that the motion considered in /3/, which contains a singular mode with minimum moment of inertia, is not optimal. The motion which satisfies the maximum principle is compared with it. There can be a singular mode in the optimal motion /4/ only when the number of control switchings is infinite.     

Active vibration control of unbalanced flexible rotor–shaft systems parametrically excited due to base motion

Rotor vibrations caused by large time-varying base motion are of considerable importance as there are a good number of rotors, e.g., the ship and aircraft turbine rotors, which are often subject to excitations, as the rotor base, i.e. the vehicle, undergoes large time varying linear and angular displacements as a result of different maneuvers. Due to such motions of the base, the equations of vibratory motion of a flexible rotor–shaft relative to the base (which forms a non-inertial reference frame) contains terms due to Coriolis effect as well as inertial excitations (generally asynchronous to rotor spin) generated by different system parameters. Such equations of motion are linear but time-varying in nature, invoking the possibility of parametric instability under certain frequency–amplitude combinations of the base motion. An investigation of active vibration control of an unbalanced rotor–shaft system on moving bases is attempted in this work with electromagnetic control force provided by an actuator consisting of four electromagnetic exciters, placed on the stator in a suitable plane around the rotor–shaft. The actuator does not levitate the rotor or facilitate any bearing action, which is provided by the conventional suspension system. The equations of motion of the rotor–shaft continuum are first written with respect to the non-inertial reference frame (the moving base in this case) including the effect of rotor internal damping. A conventional model for the electromagnetic exciter is used. Numerical simulations performed on the flexible rotor–shaft modelled using beam finite elements shows that the control action is successful in avoiding the parametric instability, postponing the instability due to internal material damping and reducing the rotor response relative to the rigid base significantly, with sufficiently low demand of control current in comparison with the bias current in the actuator coils.