Spinorial Method of Terminal Control of Spatial Rotations

In this paper, we introduce the notion of three-dimensional generalized rotations. We obtain relations between the parameters of the spinor representation of the group of three-dimensional generalized rotations and the coordinates of the initial and terminal points of rotation. Simple relations between elements of a three-dimensional orthogonal matrix of the basic representation and the Euler angles, on the one hand, and the coordinates of the initial and terminal points of rotation, on the other hand, were derived. The spinor method of solution of the inverse kinematic problem for spatial mechanisms with spherical pairs is developed and the corresponding algorithm is proposed. The obtained results allow one to reduce the three-dimensional problem of spatial motion control to the one-dimensional problem. Simple adaptive algorithms are suggested, by means of which various partial problems on the terminal control are solved under various terminal conditions. New algorithms of control of spatial rotations of manipulating robots are studied.     

Parametric optimization of a four-link close-chain manipulator with active and passive actuators

We investigate the problem of optimization of motion laws and design parameters of a four-link manipulator with a closed-chain kinematic structure. The manipulator performs cyclic transfer operations in a horizontal plane under the action of active and passive (springs and dampers) actuators. As a minimization criterion, we take a quadratic (with respect to control moments of forces) functional. An algorithm is proposed for constructing a suboptimal solution of the formulated problem based on parametrization of the generalized coordinates of the manipulator with a family of given functions and on the use of numerical procedures of mathematical programming.     

Computer system for kinematic and dynamic analysis synthesis of rigid and flexible multibody systems

In the paper an overview of a general numerical algorithm and program system library for deriving the kinematic constraint equations and dynamic equations of motion, as well as, computation of their first and second order partial derivatives with respect to kinematic parameters of motion, design parameters and mass and inertia characteristics for rigid and flexible multibody systems is presented. These are the main basic computational modules for implementation of kinematic and dynamic synthesis, optimization and design. The main theoretical basis consists in matrix methods for deriving the kinematic constraints and dynamic equations, as well as, the generalized Newton – Euler dynamic equations for rigid and flexible bodies, and finite element discretization in relative coordinates. Block–scheme of the computational procedures and problem oriented program compilation is presented. An example of kinematic synthesis of six–link path generating mechanism with singular points is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computed torque control of an under-actuated service robot platform modeled by natural coordinates

The paper investigates the motion planning of a suspended service robot platform equipped with ducted fan actuators. The platform consists of an RRT robot and a cable suspended swinging actuator that form a subsequent parallel kinematic chain and it is equipped with ducted fan actuators. In spite of the complementary ducted fan actuators, the system is under-actuated. The method of computed torques is applied to control the motion of the robot.The under-actuated systems have less control inputs than degrees of freedom. We assume that the investigated under-actuated system has desired outputs of the same number as inputs. In spite of the fact that the inverse dynamical calculation leads to the solution of a system of differential–algebraic equations (DAE), the desired control inputs can be determined uniquely by the method of computed torques.We use natural (Cartesian) coordinates to describe the configuration of the robot, while a set of algebraic equations represents the geometric constraints. In this modeling approach the mathematical model of the dynamical system itself is also a DAE.The paper discusses the inverse dynamics problem of the complex hybrid robotic system. The results include the desired actuator forces as well as the nominal coordinates corresponding to the desired motion of the carried payload. The method of computed torque control with a PD controller is applied to under-actuated systems described by natural coordinates, while the inverse dynamics is solved via the backward Euler discretization of the DAE system for which a general formalism is proposed. The results are compared with the closed form results obtained by simplified models of the system. Numerical simulation and experiments demonstrate the applicability of the presented concepts.     

Three-Dimensional Maneuvering and Control of Flexible Robots

This paper develops a general approach to the three-dimensional maneuver and vibration control of a robot in the form of a chain of flexible links. The equations for the rigid-body maneuvering motions are derived by means of Lagrange equations in terms of quasi-coordinates and the equations for the elastic deformations by means of ordinary Lagrange equations. The equations of motion are derived for the full system simultaneously, using recursive equations to relate the motions of a given link to the motions of the preceding links in the chain. The maneuver is carried out by means of joint torques and the vibration is suppressed by means of point actuators dispersed throughout the links. The controls are designed by the Liapunov direct method. A numerical example demonstrates the theoretical developments.     

Mathematical modeling and trajectory planning of mobile manipulators with flexible links and joints

This paper is concerned with mathematical modeling and optimal motion designing of flexible mobile manipulators. The system is composed of a multiple flexible links and flexible revolute joints manipulator mounted on a mobile platform. First, analyzing on kinematics and dynamics of the model is carried out then; open-loop optimal control approach is presented for optimal motion designing of the system. The problem is known to be complex since combined motion of the base and manipulator, non-holonomic constraint of the base and highly non-linear and complicated dynamic equations as a result of the flexible nature of both links and joints are taken into account. In the proposed method, the generalized coordinates and additional kinematic constraints are selected in such a way that the base motion coordination along the predefined path is guaranteed while the optimal motion trajectory of the end-effector is generated. This method by using Pontryagin’s minimum principle and deriving the optimality conditions converts the optimal control problem into a two point boundary value problem. A comparative assessment of the dynamic model is validated through computer simulations, and then additional simulations are done for trajectory planning of a two-link flexible mobile manipulator to demonstrate effectiveness and capability of the proposed approach.     

The problem of the time-optimal control of spacecraft reorientation

The use of Pontryagin's maximum principle to solve spacecraft motion control problems is demonstrated. The problem of the optimal control of the spatial reorientation of a spacecraft (as a rigid body) from an arbitrary initial angular position to an assigned final angular position in the minimum rotation time is investigated in detail. The case in which velocity parameters of the motion are constrained is considered. An analytical solution of the problem is obtained in closed form using the method of quaternions, and mathematical expressions for synthesizing the optimal control programme are given. The kinematic problem of spacecraft reorientation is solved completely. A design scheme for solving the maximum principle boundary-value problem for arbitrary turning conditions and inertial characteristics of the spacecraft is given. A solution of the problem of the optimal control of spatial reorientation for a dynamically symmetrical spacecraft is presented in analytical form (to expressions in elementary functions). The results of mathematical modelling of the motion of a spacecraft under optimal control, which confirm the practical feasibility of the control algorithm developed, are given. Estimates have shown that the turn time of modern spacecraft with a constrained magnitude of the angular momentum can be reduced by 15–25% compared with conventional reorientation methods. The greatest effect is achieved for turns through large angles (90° or more) when the final rotation vector is equidistant from the longitudinal axis and the transverse plane of the spacecraft.     

On the theory of the gyropendulum

Precession equations of motion of the gyropendulum relative to the accompanying Darboux trihedron /1/ and, also, precession equations of the gyropendulum motion relative to the geographic trihedron, considered in /2, 3/, are given a kinematic interpretation. Linear differential equations that define the gyropendulum behavior at finite deflection angles of the rotor axis from the vertical are established for arbitrary motions of its suspension point over the surface of the Earth. These equations have the form of kinematic equations of a solid body spherical motion in terms of Rodrigues-Hamilton parameters, and in the case of stationary base they are in agreement with equations established in /4/. The Liapunov stability ot the gyropendulum equations in both the finite Euler—Krylov angles and in the Rodrigues — Hamilton parameters is proved. Particular cases of integrability in quadratures of the gyropendulum precession equations at finite angles are indicated.     

On the stability of uniformly rotating liquid in a weak magnetic field

We analyze the simplest free boundary problem of magnetohydrodynamics governing the evolution of an isolated mass of a viscous incompressible liquid in the presence of the magnetic field. The motion of the liquid is governed by the Navier–Stokes equations, and for the magnetic field we have the Maxwell equations with an excluded displacement current. The magnetic field should be determined not only in the domain filled with the liquid, but also in the surrounding vacuum region. On the free boundary of the liquid standard jump conditions for the magnetic field are prescribed, as well as kinematic and dynamic boundary conditions, where the magnetic stress tensor is taken into account. We prove that the solution corresponding to a rigid rotation of the fluid and to zero magnetic field is stable if the functional of potential energy has a positive second variation. Bibliography: 11 titles.     

Stabilization of Multiple Constraints in Multibody Dynamics Using Optimization and a Pseudo-inverse Matrix

An approach for solving the forward dynamics problem for mechanical systems with many closed kinematic chains is presented. The dynamic model takes the form of Differential-Algebraic Equations. An optimization method for stabilization of kinematic constraints using the pseudo-inverse mass matrix of the dynamic equations is suggested. The stabilization algorithm provides minimal deviations of the parameters and their velocities with respect to the solution of the differential equations. Estimation of independent coordinates is not required. The forward and inverse dynamic problems of a spatial mechanism and a spatial moving platform with many closed chains are solved. The effectiveness of the algorithm is analyzed.     

Large-scale geodetic least-squares adjustment by dissection and orthogonal decomposition

Very large-scale matrix problems currently arise in the context of accurately computing the coordinates of points on the surface of the earth. Here geodesists adjust the approximate values of these coordinates by computing least-squares solutions to large sparse systems of equations which result from relating the coordinates to certain observations such as distances or angles between points. The purpose of this paper is to suggest an alternative to the formation and solution of the normal equations for these least-squares adjustment problems. In particular, it is shown how a block-orthogonal decomposition method can be used in conjunction with a nested dissection scheme to produce an algorithm for solving such problems which combines efficient data management with numerical stability. The approach given here parallels somewhat the development of the natural factor formulation, by Argyris et al., for the use of orthogonal decomposition procedures in the finite-element analysis of structures. As an indication of the magnitude that these least-squares adjustment problems can sometimes attain, the forthcoming readjustment of the North American Datum in 1983 by the National Geodetic Survey is discussed. Here it becomes necessary to linearize and solve an overdetermined system of approximately 6,000,000 equations in 400,000 unknowns—a truly large scale matrix problem.     

A New approach to finding the control that transports a system from one phase state to another

In their previous papers, the authors have considered the possibility of applying the theory of motion for nonholonomic systems with high-order constraints to solving one of the main problems of the control theory. This is a problem of transporting a mechanical system with a finite number of degrees of freedom from a given phase state to another given phase state during a fixed time. It was shown that, when solving such a problem using the Pontryagin maximum principle with minimization of the integral of the control force squared, a nonholonomic high-order constraint is realized continuously during the motion of the system. However, in this case, one can also apply a generalized Gauss principle, which is commonly used in the motion of nonholonomic systems with high-order constraints. It is essential that the latter principle makes it possible to find the control as a polynomial, while the use of the Pontryagin maximum principle yields the control containing harmonics with natural frequencies of the system. The latter fact determines increasing the amplitude of oscillation of the system if the time of motion is long. Besides this, a generalized Gauss principle allows us to formulate and solve extended boundary problems in which along with the conditions for generalized coordinates and velocities at the beginning and at the end of motion, the values of any-order derivatives of the coordinates are introduced at the same time instants. This makes it possible to find the control without jumps at the beginning and at the end of motion. The theory presented has been demonstrated when solving the problem of the control of horizontal motion of a trolley with pendulums. A similar problem can be considered as a model, since when the parameters are chosen correspondingly it becomes equivalent to the problem of suppression of oscillations of a given elastic body some cross-section of which should move by a given distance in a fixed time. The equivalence of these problems significantly widens the range of possible applications of the problem of a trolley with pendulums. The previous solution of the problem has been reduced to the selection of a horizontal force that is a solution to the formulated problem. In the present paper, it is offered to seek an acceleration of a trolley with which it moves by a given distance in a fixed time, as a time function but not a force applied to the trolley, while the velocities and accelerations are equal to zero at the beginning and end of motion. In this new problem, the rotation angles of pendulums are the principal coordinates. This makes it possible to find a sought acceleration of a trolley on the basis of a generalized Gauss principle according to the technique developed before. Knowing the motion of a trolley and pendulums it is easy to determine the required control force. The results of numerical calculations are presented.     

Perturbation analysis of short-crested waves in Lagrangian coordinates

A new second-order asymptotic solution that describes short-crested waves is derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid satisfies the irrotational condition and there being zero pressure at the free surface, in contrast with the Eulerian solution, in which there is residual pressure at the free surface. The explicit parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water level. The mass transport velocity and Lagrangian mean level associated with particle displacement can also be obtained directly. In particular, the mean level of the particle motion in a Lagrangian form differs that of the Eulerian form. The new formulation reduces to second-order standing or progressive wave solutions in Lagrangian coordinates at the limiting angles of approach. Expressions for kinematic quantities are also presented.     

Systematic dynamic modelling of mechanical systems containing kinematic loops

In this tutorial paper a systematic procedure is presented to obtain the dynamic models of mechanical systems containing kinematic loops, with a main emphasis on efficiency and with particular regard to robotic systems. The procedure retains a minimal set of generalized coordinates for the corresponding open loop structure, obtained by removing some additional constraints closing loops in the original structure, while adopting an efficient Newton-Euler formulation of the equations of motion. Two methods for including the loop closure equations are then discussed: the Lagrange multipliers method and the method based on an explicit solution of the constraint equations. In the first case the dynamic model is given in the form of an implicit Differential Algebraic Equations (DAE) system, while in the second case an Ordinary Differential Equations (ODE) system could be obtained.     

Modeling the angular capability of the ball joints in a complex mechanism with two degrees of mobility

The paper puts forward a complex linkage mechanism with two degrees of mobility and three kinematic loops, which is used for the guiding (suspension and steering) system of the vehicles. The geometric parameters and the coordinates frames that define the mechanical system are presented, as well as the specific kinematic functions. For this complex mechanical system, the angular capability of the ball (spherical) joints is defined by two angles. The equations for these angles have been determined by matrix algebra tools, considering the transformation matrices between the bodies reference frames. The diagrams of the angular capability of the ball joints, which are represented in angular coordinates, describe the form, orientation and size of the sockets from the spherical casings. Wears, shocks, functional locks or the compromising of the joint strength can occur if scarce sockets are implemented. The risk points, in which the angular parameters have maximum values, have been determined, the simulation being performed for a real system (vehicle).     

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.     

Controllability and stabilization of programmed motions of an automobile-type transport robot

A non-linear model of the motion of an automobile-type transport robot (TR) with absolutely rigid wheels, a steering device and actuators based on DC motors, is considered. Such a model for TR motion is a non-holonomic electromechanical system and, if the dynamics of the actuators and the steering device (forces of elasticity and attenuation in its elements) is ignored, corresponds to the model of automobile motion devised by Lineikin [1]. Non-linear canonical transformations of the state and control space coordinates are constructed which reduce the initial equations of motion of the TR to a simpler canonical form, convenient for the analysis and synthesis of control systems for the TR. These transformations are used to find the conditions for the controllability of the TR as a controlled object. Algorithms are given for constructing programmed controls and programmed motions of the TR. Stabilizing control laws are synthesized that make the programmed motions of the TR asymptotically stable and guarantee that the transients will have preassigned properties     

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.     

轴向拉伸细长杆的非线性力学模型

聚酯系泊缆是深海工程中具备一定抗弯刚度、易拉伸变形的细长杆件结构.聚酯缆的轴向变形属大拉伸范畴,分析中应当区分变形前后状态,特别是缆索长度的改变使得基于小拉伸假设的细长杆模型需要予以改进.因此,基于Garrett细长杆模型,应用总体坐标和斜率取代Euler-Bernoulli(欧拉 伯努利)梁元的转角,解决缆索在空间中大转动变形的几何非线性问题;使用轴向拉伸变形前后物质点对应的方法,借助单元两个节点和一个中点,以及3个二次多项式形函数描述轴向拉伸变形下细长杆元的运动微分方程.通过与轴向拉伸悬臂梁的对比分析,验证了该拉伸杆元的收敛性和准确性.