Kinematic Differential Equations Research Articles

Motion of a two-wheeled solid along a flat curvilinear trajectory by considering tire and suspension elasticity

The unsteady motion of a solid in suspension by two elastic wheels placed one after another along a random curvilinear trajectory in vertical plane of gravity field is examined. The basic kinematical relationships and differential equations of motion for solid are presented. The relationships for the object’s effect to the trajectory are written. A method for calculating the trajectory of a solid by considering the elasticity of wheels and suspension elasticity is presented.

Precise trajectory optimization for articulated wheeled vehicles in cluttered environments

Trajectory planning refers to planning a time-dependent path connecting the initial and final configurations with some special constraints simultaneously considered. It is a critical aspect in autonomously driving an articulated vehicle. In this paper, trajectory planning is formulated as a dynamic optimization problem that contains kinematic differential equations, mechanical/environmental constraints, boundary conditions and an optimization objective. The prevailing numerical methods for solving the formulated dynamic optimization problem commonly disregard the constraint satisfactions between every two adjacent discretized mesh points, thus resulting in failure when the planned motions are actually implemented. As a remedy for this limitation, the concept of minute mesh grid is proposed, which improves the constraint satisfactions between adjacent rough mesh points. On the basis of accurate penalty functions, large-scale constraints are successfully incorporated into the optimization criterion, thus transforming the dynamic optimization problem into a static one with simple bounds on the decision variables. Simulation results verify that our proposed methodology can provide accurate results and can deal with various optimization objectives uniformly.

Co-simulation method for solver coupling with algebraic constraints incorporating relaxation techniques

Based on the stabilized index-2 formulation for multibody systems, a semi-implicit co-simulation approach for solver coupling with algebraic constraints has been presented by Schweizer and Lu (Multibody Syst. Dyn., 2014) for the case that constant approximation is used for extrapolating/interpolating the coupling variables. In the manuscript at hand, this method is generalized to the case that higher-order approximation is employed. Direct application of higher-order polynomials for extrapolating/interpolating the coupling variables fails. Using linear approximation polynomials, artificial oscillations in the Lagrange multipliers of the kinematical differential equations are observed. For quadratic and higher-order polynomials, the co-simulation becomes unstable. In this work, the key idea to obtain stable solutions without artificial oscillations is to apply a relaxation technique. A detailed stability and convergence analysis is presented in the paper for the case of higher-order approximation. In this context, the influence of the relaxation parameter on the stability and convergence behavior is investigated. Applicability and robustness of the stabilized index-2 co-simulation method incorporating higher-order approximation polynomials is demonstrated with different numerical examples. Using piecewise constant approximation polynomials for the coupling variables produces discontinuous accelerations and reaction forces in the subsystems at the macrotime points, which may entail problems for the subsystem integrator. With higher-order approximation polynomials, the coupling variables and in consequence the accelerations and reaction forces in the subsystems become continuous.

Determining the rotational motion of the Bion M-1 satellite with the GRAVITON instrument

Actual controlled rotational motion of the Bion M-1 satellite is reconstructed for the modes of the orbital and single-axis solar orientation. The reconstruction was performed using data of onboard measurements of the vectors of angular velocity and the Earth’s magnetic field (EMF) strength. The reconstruction procedure is based on the kinematic equations of the rotational motion of a solid body. In the framework of this procedure, measurement data for two types collected at a certain time interval are processed jointly. Measurements of angular velocity are interpolated by piecewise–linear functions, which are substituted in the kinematic differential equations for quaternion giving the transition from the satellite instrument coordinate system to the inertial (the second geoequatorial) coordinate system. Thus the obtained equations represent the kinematic model of the satellite rotational motion. The solution to these equations approximating the actual motion is derived from the condition of the best (in the sense of the least-square method) matching measurement data of the EMF strength vector with the calculated values. The described procedure allows us to reconstruct the actual rotational satellite motion using one solution to kinematic equations over time intervals with durations of more than 5 h. Found reconstructions were used to calculate the residual microaccelerations.

Lie-group integration method for constrained multibody systems in state space

Coordinate-free Lie-group integration method of arbitrary (and possibly higher) order of accuracy for constrained multibody systems (MBS) is proposed in the paper. Mathematical model of MBS dynamics is shaped as a DAE system of equations of index 1, whereas dynamics is evolving on the system state space modeled as a Lie-group. Since the formulated integration algorithm operates directly on the system manifold via MBS elements’ angular velocities and rotational matrices, no local rotational coordinates are necessary, and kinematical differential equations (that are prone to singularities in the case of three-parameter-based local description of the rotational kinematics) are completely avoided. Basis of the integration procedure is the Munthe–Kaas algorithm for ODE integration on Lie-groups, which is reformulated and expanded to be applicable for the integration of constrained MBS in the DAE-index-1 form. In order to eliminate numerical constraint violation for generalized positions and velocities during the integration procedure, a constraint stabilization projection method based on constrained least-square minimization algorithm is introduced. Two numerical examples, heavy top dynamics and satellite with mounted 5-DOF manipulator, are presented. The proposed Lie-group DAE-index-1 integration scheme is easy-to-use for an MBS with kinematical constraints of general type, and it is especially suitable for dynamics of mechanical systems with large 3D rotations where standard (vector space) formulations might be inefficient due to kinematical singularities (three-parameter-based rotational coordinates) or additional kinematical constraints (redundant quaternion formulations).

Numerical Solution of Stiff Multibody Dynamic Systems Based on Kinematic Derivatives

The Hermite–Obreshkov–Padé (HOP) method of numerical integration is applicable to stiff systems of differential equations, where the linearization has large range of eigenvalues. A practical implementation of HOP requires the ability to determine high-order time derivatives of the system variables. In the case of a constrained multibody dynamical system, the power series solution for the kinematic differential equation is the foundation for an algorithmic differentiation (AD) procedure determining those derivatives. The AD procedure is extended in this paper to determine rates of change in the time derivatives with respect to variation in the position and velocity state variables of the multibody system. The coefficients of this variation form the Jacobian matrix required for Newton–Raphson iteration. That procedure solves the implicit relations for the state variables at the end of each integration time step. The resulting numerical method is applied to the rotation of a dynamically unbalanced constant-velocity (CV) shaft coupling, where the deflection angle of the output shaft is constrained to low levels by springs of high rate and damping.

New algorithms for determining the inertial orientation of an object

Kinematic equations and algorithms for the operation of strapdown inertial navigation systems intended for the high-accuracy determination of the inertial orientation parameters (the Euler (Rodrigues–Hamilton) parameters) of a moving object are considered. Together with classical orientation equations, Hamilton's quaternions and new kinematic differential equations in four-dimensional (quaternion) skew-symmetric operators are used that are matched with the classical rotation quaternion and the quaternion rotation matrix using Cayley's formulae. New methods for solving the synthesized kinematic equations are considered: a one-step quaternion orientation algorithm of third-order accuracy and two-step algorithms of third- and fourth-order accuracy in four-dimensional skew-symmetric operators for calculating the parameters of the spatial position of an object. The algorithms were constructed using the Picard method of successive approximations and employ primary integral information from measurements of the absolute angular velocity of the object as the input information, and have advantages over existing algorithms of a similar order with respect to their accuracy and simplicity.

New kinematic parameters of finite rotation of a solid

A new family of kinematic (global and local) orientation parameters of a solid is presented and described. All kinematic parameters are obtained by the method of mapping the variables onto the corresponding oriented subspace (hyperplane). In particular, the method of stereographic projection of a point of a five-dimensional sphere S6 ⊂ R6 onto the oriented hyperplane R5 is presented for the classical direction cosines of the angles determining the orientation of two coordinate systems. A family of global kinematic parameters obtained by the method of mapping of five-dimensional kinematic Hopf parameters given in the space R5 onto the four-dimensional oriented subspace R4 is described. The theorem of the homeomorphism of two topological spaces (the four-dimensional sphere S4 ⊂R5 with one deleted point and the oriented hyperplane R4) is used to establish the correspondence between five- and four-dimensional kinematic parameters defined in the corresponding spaces. It is also shown what global four-dimensional orientation parameters, quaternions defined in the subspace R4 are associated with the classical local parameters, i.e., the Rodrigues and Gibbs three-dimensional finite rotation vectors. The projection method is used to obtain the kinematic differential equations (KDE) of rotation corresponding to the five- and four-dimensional orientation parameters. All above-introduced kinematic orientation parameters of a solid permit efficiently solving the classical Darboux problem by using the corresponding KDE, i.e., determining the body current angular position in the space R3 from the known (measured) angular velocity of the object rotation and its initial position in space.

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.

KED Modeling of PLS Mechanism

For the super-size and large flexibility of Pipe Lay-down System, considering the influence on the mechanism from elastic deformation and mechanical vibration during the movements, the kineto-elastodynamics model is established by using the KED theory which is based on the analysis of kinematics. The PLS mechanism is divided into several finite elements. Dynamic equations of beam element are established in the local coordinate by using Lagrange’s equation. In the process of changing from local coordinate into global coordinate, no longer considering the instantaneous structure assumes. In consideration of the first and second derivative of the coordinate transformation matrix versus time are not zero. The mass matrix, damping matrix and stiffness matrix of the final system kinematic differential equation are the function of time. It realizes the continuity of variable in the time domain. Derivation of the results in this paper lays a foundation for the next more accurate and efficient methods being applied to solve the KED equation of PLS mechanism.

Derivation of the Kinematic Differential Equation for Bending Characteristics of the Bimorph

The bimorphs in this paper are provided with electrodes in parallel manner. When an electric field is applied, one plate elongates and the other contracts. This results in a bending motion and used for actuators or sensors. According to plate theory assumptions, establish the strain deflection expression firstly. And then, derive the strain-stress relation of PZT layer from g-type piezoelectric equations. Finally, some general equations are derived for piezoelectric laminated plates and can be used in theoretical or simulation analysis. Mainly, a kind of deflection differential equation about 3-layer electrical parallel piezoelectric thin plate has been obtained. Furthermore, replacing static load by initial force per unit area, the kinematic differential equation has also been derived.

Series Solution for Finite Displacement of Single-Loop Spatial Linkages

The kinematic differential equation for a spatial point trajectory accepts the time-varying instantaneous screw of a rigid body as input, the time-zero coordinates of a point on that rigid body as the initial condition and generates the space curve traced by that point over time as the solution. Applying this equation to multiple points on a rigid body derives the kinematic differential equations for a displacement matrix and for a joint screw. The solution of these differential equations in turn expresses the trajectory over the course of a finite displacement taken by a coordinate frame in the case of the displacement matrix, by a joint axis line in the case of a screw. All of the kinematic differential equations are amenable to solution by power series owing to the expression for the product of two power series. The kinematic solution for finite displacement of a single-loop spatial linkage may, hence, be expressed either in terms of displacement matrices or in terms of screws. Each method determines coefficients for joint rates by a recursive procedure that solves a sequence of linear systems of equations, but that procedure requires only a single factorization of a 6 by 6 matrix for a given initial posture of the linkage. The inverse kinematics of an 8R nonseparable redundant-joint robot, represented by one of the multiple degrees of freedom of a 9R loop, provides a numerical example of the new analytical technique.

Solution of the Forward Dynamics of a Single-loop Linkage Using Power Series

The kinematic differential equations express the paths taken by points, lines, and coordinate frames attached to a rigid body in terms of the instantaneous screw for the motion of that body. Such differential equations are linear but with a time-varying coefficient and hence solvable by power series. A single-loop kinematic chain may be expressed by a system of such differential equations subject to a linear constraint. A single matrix factorization followed by a sequence of substitutions of linear-system right-hand-side terms determines successive orders of the joint rate coefficients in the kinematic solution for this mechanism. The present work extends this procedure to the forward dynamics problem, applying it to a Clemens constant-velocity coupling expressed as a spatial 9R closed kinematic chain.

Series Solution for Finite Displacement of Planar Four-Bar Linkages

A time-varying instantaneous screw characterizes the motion of a rigid body. The kinematic differential equation expresses the path taken by any point on that rigid body in terms of this screw. Therefore, when a revolute joint is attached to a moving link in a planar kinematic chain, the path taken by the center of that revolute joint is the solution to such an equation. The instantaneous screw of a link in that chain is in turn determined by the action of the joints connecting that link to ground, where the contribution of each joint to that instantaneous screw is determined by its actuation rate and center point. Substituting power series expansions for joint rates into the kinematic differential equations for joint centers, and expressing loop closure as a linear constraint on the instantaneous screws of the links, a recurrence relation is established that solves for the coefficients in those power series. The resulting solution is applied to determine the equilibrium pendulum tilt of the United Aircraft TurboTrain. Comparing that power series approximation with an exact kinematic analysis shows convergence properties of the series.

Hypersphere Stereographic Orientation Parameters

E ULER parameters (EPs), also referred to as quaternions, are a nonsingular set of four attitude coordinates that are constrained to a unit norm. The first analytical mapping from EPs to modified Rodrigues parameters (MRPs) is performed by Wiener in his 1962 dissertation [1], in which he discovered a singularity at the 360 deg rotation. In [2], Marandi and Modi exploit the nonuniqueness property of the MRPs by formulating a nonsingular minimal attitude description. Shuster also mentions the MRPs in his well-known survey of attitude parameterizations, and he gives the parameters the name modified Rodrigues parameters [3]. Tsiotras and Longuski point out that theMRPs can be viewed as the result of a stereographic projection of the EP constraint unit hypersphere onto a threedimensional (3-D) projection hyperplane [4,5]. He also discovers that the natural logarithm function forms an elegant attitude cost (Lyapunov) function in terms of MRPs, which leads to linear MRP feedback with nonlinear stability. Schaub and Junkins [6] further develop this work by showing that theMRP stereographic projection description discovered by Tsiotras can be expanded to describe general families of attitude parameters called the stereographic orientation parameters (SOPs). In particular, [6] presents the subgroup of symmetric SOPs and shows that the MRPs and classical Rodrigues parameters are a subset of this family. Later on, Southward et. al. [7] develop the full kinematic properties of the symmetric SOPs by allowing the projection point to be placed anywhere on the scalar EP coordinate axis within the EP constraint hypersphere. These symmetric SOPs are expressed algebraically in terms of scalar projection point coordinates and yield minimal set attitude coordinates, in which the singularity can occur at any desired orientation within 0 deg< < 360 deg. In contrast, the asymmetric stereographic attitude parameters (ASOPs) of [6] place the projection point at 1 along one of the vector EP coordinate axes. This leads to an interesting behavior, in which singularities are only encountered if a pure rotation about a particular principle body axis is performed. Further, a 180 deg rotation may lead to a singular attitude description, but a 270 deg rotation (exact same orientation) is nonsingular. Only 630 deg in the negative direction leads to a singular description. The nonsymmetric nature of the singular rotations and their dependency of the path to a particular orientation lead to the name of asymmetric SOPs. Other recent attitude coordinates that relate to the MRPs include the higher-order Rodrigues parameters [8]. Here, higher-order Cayley transforms are used to develop attitude coordinates that grow infinitely large a at multiples of 360 deg. These higher-order Rodrigues parameters are convenient to develop minimal sets of attitude coordinates, for which the differential equation can be made arbitrarily linear through the use of higher-order Cayley transformations. Hurtado uses the MRPs to create inner and outer parameters for attitude representations and presents new Cayley-like transformations [9]. This paper investigates a subfamily of attitude coordinates called the hypersphere SOPs (HSOPs), which contain both the previous MRPs (particular set of symmetric SOPs) and theASOP, allowing for all this work to be combined into a single, minimal attitude parameter description. HSOPs allow the projection point to lie at any point on the EP unit hypersphere constraint. Thus, depending on the choice of the project point, these attitude coordinates can display a singular behavior similar to that of the ASOP. The attitude of a spinning body can be described singularity-free, with a minimal three-parameter coordinate set as long as the body is not spinning about a particular combination of principal body axes. Or, the HSOP coordinates can be chosen, such that their singular behavior matches that of the MRPs, for which a particular 360 deg rotation about any body axis leads to a singular description. When different attitude coordinates are combined into a more general family of parameters, such as the joining of classical Rodrigues parameters and MRPs into symmetric SOPs in [7], the result is often a more complex set of algebraic equations. This paper investigates how a general projection point on the surface of the EP constraint hypersphere complicates the associated HSOP differential kinematic equations and their mapping to the shadow set. This paper is organized as follows. Section II describes the geometry and algebra of a general stereographic projection. Section III provides the analytical mapping between HSOPs from EPs, the DCM, as well as the derivation of the shadow sets, the kinematic differential equation, and the singularity condition. Section IV discusses how the HSOPs can be employed in attitude control strategies.

Transformation of amplitudes and frequencies of precession and nutation of the earth’s rotation vector to amplitudes and frequencies of diurnal polar motion

The temporal change of the rotation vector of a rotating body is, in the first order, identical in a space-fixed system and in a body-fixed system. Therefore, if the motion of the rotation axis of the earth relative to a space-fixed system is given as a function of time, it should be possible to compute its motion relative to an earth-fixed system, and vice versa. This paper presents such a transformation. Two models of motion of the rotation axis in the space-fixed system are considered: one consisting only of a regular (i.e., strictly conical) precession and one extended by circular nutation components, which are superimposed upon the regular precession. The Euler angles describing the orientation of the earth-fixed system with respect to the space-fixed system are derived by an analytical solution of the kinematical Eulerian differential equations. In the first case (precession only), this is directly possible, and in the second case (precession and nutation), a solution is achieved by a perturbation approach, where the result of the first case serves as an approximation and nutation is regarded as a small perturbation, which is treated in a linearized form. The transformation by means of these Euler angles shows that the rotation axis performs in the earth-fixed system retrograde conical revolutions with small amplitudes, namely one revolution with a period of one sidereal day corresponding to precession and one revolution with a period which is slightly smaller or larger than one sidereal day corresponding to each (prograde or retrograde) circular nutation component. The peculiar feature of the derivation presented here is the analytical solution of the Eulerian differential equations.

Exact analytic solution for the rotation of a rigid body having spherical ellipsoid of inertia and subjected to a constant torque

The exact analytic solution is introduced for the rotational motion of a rigid body having three equal principal moments of inertia and subjected to an external torque vector which is constant for an observer fixed with the body, and to arbitrary initial angular velocity. In the paper a parametrization of the rotation by three complex numbers is used. In particular, the rows of the rotation matrix are seen as elements of the unit sphere and projected, by stereographic projection, onto points on the complex plane. In this representation, the kinematic differential equation reduces to an equation of Riccati type, which is solved through appropriate choices of substitutions, thereby yielding an analytic solution in terms of confluent hypergeometric functions. The rotation matrix is recovered from the three complex rotation variables by inverse stereographic map. The results of a numerical experiment confirming the exactness of the analytic solution are reported. The newly found analytic solution is valid for any motion time length and rotation amplitude. The present paper adds a further element to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.

PID+ BACKSTEPPING CONTROL OF RELATIVE SPACECRAFT ATTITUDE

In this paper we present a PID+ backstepping controller, as a solution to the problem of coordinated attitude control in spacecraft formations. The control scheme is based on quaternions and modified Rodrigues parameters as attitude representation of the relative attitude error. Utilizing the invertibility of the modified Rodrigues parameter kinematic differential equation, a globally exponentially stable control law for the relative attitude error dynamics is obtained through the use of integrator augmentation and backstepping. Finally, simulation results are presented to show controller performance.

Fault-Tolerant Structured Adaptive Model Inversion Control

An adaptive dynamic inversion control formulation is presented that takes advantage of the inherent dynamic structure of the state-space description of a large class of systems. The formulations impose the exact kinematic differential equations, thereby restricting the adaptation process that compensates for model errors to the acceleration level. The utility of this formulation is demonstrated for the problem of fault tolerance to actuator failures on redundantly actuated systems. The approach incorporates an actuator failure model in the controller formulation, so that actuator failure can be identified as a change in the parameters of the failure model. Tracking of reference trajectories is imposed, and initial error conditions and structured parametric uncertainties are incorporated explicitly in both the plant parameters and the control influence matrix. A numerical example consisting of a nonlinear model of an F-16 type aircraft with thrust vectoring is presented. Simulation results show that the fault-tolerant adaptive controller is capable of simultaneously handling parametric uncertainties, large initial condition errors, and actuator failures while maintaining adequate tracking performance. N recent years, there has been much interest in the development of reconfigurable control systems that can accommodate actuator failures without compromising mission integrity. There has been substantial progress in the development of real-time failure detection and isolation algorithms, system identification after failure, and control reconfiguration techniques in aerospace applications. In Ref. 1, a survey of various reconfigurable flight control methodologies is presented and it is shown that most traditional reconfiguration flight control approaches rely on failure detection and isolation. The complexity of such a system with this feature grows with the increase in the number of failures, and there tends to be a significant possibility of false alarms. 1,2 A different approach to reconfigurable flight control is based on adaptive control theory, in which the adaptive control structure implicitly reconfigures the control law using adaptive estimates of the altered dynamics after failure. 3 In Ref. 3, an adaptive control scheme is presented that uses a linear approximation of the plant model to compute the control, and a neural network based adaptive control law for flight reconfiguration has been developed and successfully flight tested. 4−6 A robust fault-tolerant controller has also been developed to reject state-dependent disturbances. 7 The approach presented in this paper uses a structured nonlinear adaptive dynamic inversion control methodology. Instead of using an explicit failure detection and isolation algorithm, this methodology is based on the adaptive control theory where the controller is constantly updating itself. This methodology is applicable to a general class of nonlinear systems that are affine in the control with uncertain parameters appearing linearly. Fault-tolerance capability is introduced by incorporating a failure model in the controller so that a failure can be identified and compensated for by a change in the parameters of the failure model. First, model reference adaptive control, structured model reference adaptive control, and structured adaptive model inversion

The Conjugate Method for Solving Mono-Axial Frames

This paper discusses materials for a course on ‘Strength of materials and mechanics of solids’, addressed to students of second-year mechanical engineering, ocean engineering, civil engineering and aerospace engineering. In particular, the problem of the evaluation of the deflection and of the axial displacement of elastic mono-axial frames is considered. This problem is solved by the use of a generalization of the classical conjugate beam method, that here is defined conjugate mono-axial frame method. It lies on the similarity between all three kinematic differential equations governing the elastic deflection and axial displacement of the mono-axial frame and all three static differential equations governing its equilibrium. This implies some comparisons and theorems that are reported in the present paper. The mode of presentation of this material is by lecture, within a regular course, of 3–4 hours, with 2–3 revision hours. The material must be presented after the classical elastic beam and frame problem has been treated. The material helps students to analyse the important relationships between the kinematics and the statics of any elastic structure.