Velocity Transformation Matrix Research Articles

Practical Position Tracking for 3D Unicycle Model

This paper considers practical position tracking for the 3D unicycle model. In contrast to the existing hand position coordinate transformation, a novel modified hand position coordinate transformation is proposed for the 3D unicycle model which can guarantee that the induced velocity transformation matrix is nonsingular for all the possible orientations. By employing the proposed modified hand position coordinate transformation, the nonlinear 3D unicycle position kinematics can be converted into the single integrator form, which enables arbitrary position reference trajectory tracking given any prespecified upper bound for the steady-state tracking error. The control performance is examined by numerical examples.

Strain-Based Geometrically Nonlinear Beam Formulation for Rigid–Flexible Multibody Dynamic Analysis

Geometrically nonlinear strain-based beam formulation has the potential to analyze flexible slender components in multibody systems efficiently owing to the minimum number of variables and constant stiffness matrix. The objective of this study is to develop a multibody dynamic analysis framework based on the strain-based beam formulation. To this end, we describe the constraint equation using the vector variables of the absolute nodal coordinate formulation that exhibits a velocity-transformation relationship with the strain-based formulation. Subsequently, we divide the Jacobian of the constraint equation into two terms. One term is equivalent to the velocity transformation matrix that is currently implemented in the existing strain-based analysis framework. The other term is a simple constant or linear Jacobian defined based on the orthonormal vectors of the absolute nodal coordinate formulation. This simple Jacobian description integrates the strain-based beam formulation with multibody dynamic theories to handle open- and closed-loop joints. The obtained numerical results validate that the proposed strain-based multibody dynamic analysis method concurs well with and exhibits a better convergence than the conventional flexible multibody dynamic analysis method.

Convergence Characteristics of Geometrically Accurate Spatial Finite Elements

Abstract The convergence characteristics of three geometrically accurate spatial finite elements (FEs) are examined in this study using an eigenvalue analysis. The spatial beam, plate, and solid elements considered in this investigation are suited for both structural and multibody system (MBS) applications. These spatial elements are based on geometry derived from the kinematic description of the absolute nodal coordinate formulation (ANCF). In order to allow for an accurate reference-configuration geometry description, the element shape functions are formulated using constant geometry coefficients defined using the position-vector gradients in the reference configuration. The change in the position-vector gradients is used to define a velocity transformation matrix that leads to constant element inertia and stiffness matrices in the case of infinitesimal rotations. In contrast to conventional structural finite elements, the elements considered in this study can be used to describe the initial geometry with the same degree of accuracy as B-spline and nonuniform rational B-spline (NURBS) representations, widely used in the computer-aided design (CAD). An eigenvalue analysis is performed to evaluate the element convergence characteristics in the case of different geometries, including straight, tapered, and curved configurations. The frequencies obtained are compared with those obtained using a commercial FE software and analytical solutions. The stiffness matrix is obtained using both the general continuum mechanics (GCM) approach and the newly proposed strain split method (SSM) in order to investigate its effectiveness as a locking alleviation technique.

Spatial ANCF/CRBF beam elements

In this paper, the consistent rotation-based formulation (CRBF) is used to develop new three-dimensional beam elements starting with the absolute nodal coordinate formulation (ANCF) kinematic description. While the proposed elements employ orientation parameters as nodal coordinates, independent rotation interpolation is avoided, leading to unique displacement and rotation fields. Furthermore, the proposed spatial ANCF/CRBF-based beam elements adhere to the noncommutative nature of the rotation parameters, allow for arbitrarily large three-dimensional rotation, and eliminate the need for using co-rotational or incremental solution procedures. Because the proposed elements have a general geometric description consistent with computational geometry methods, accurate definitions of the shear and bending deformations can be developed and evaluated, and curved structures and complex geometries can be systematically modeled. Three new spatial ANCF/CRBF beam elements, which use absolute positions and rotation parameters as nodal coordinates, are proposed. The time derivatives of the ANCF transverse position vector gradients at the nodes are expressed in terms of the time derivatives of rotation parameters using a nonlinear velocity transformation matrix. The velocity transformation leads to lower-dimensional elements that ensure the continuity of stresses and rotations at the element nodal points. The numerical results obtained from the proposed ANCF/CRBF elements are compared with the more general ANCF beam elements and with elements implemented in a commercial FE software.

An Efficient Dynamic Formulation for Solving Rigid and Flexible Multibody Systems Based on Semirecursive Method and Implicit Integration

This paper presents a method for solving the dynamic equations of multibody systems containing both rigid and flexible bodies. The proposed method uses independent coordinates and projects the dynamic equations on the constraint tangent manifold by means of a velocity transformation matrix. It can be used with a wide variety of integration formulae, considering both fixed and variable stepsizes. Topological semirecursive methods are used to take advantage of the relatively small number of parameters needed. An in depth implementation analysis is performed in order to evaluate the terms involved in the integration process. Numerical and stability issues are also discussed.

Inverse dynamics of a 3-prismatic—revolute—revolute planar parallel manipulator using natural orthogonal complement

The performance of robotic systems with parallel kinematics can be evaluated by their kinematic, static, and dynamic properties. These properties are directly used in model-based controllers which potentially offer higher accuracy for robotic systems. Inverse dynamic solution is an essential part of these controllers. In the present work, the inverse dynamics model of a 3-PRR (prismatic—revolute—revolute) planar parallel manipulator based on the natural orthogonal complement (NOC) method is developed. To drive the NOC for the 3-PRR closed-loop systems, the explicit expressions of the loop constraints equations and the associated Jacobian matrices are first obtained. Next the NOC matrix, which is a velocity transformation matrix relating the Cartesian angular/translational velocities of various bodies to the motor joint rates, is calculated. Finally results of the NOC method are compared with simulation of a 3-PRR planar parallel manipulator using two commercial softwares: SimMechanics toolbox of Matlab and COSMOSMotion of SolidWorks. In order to verify the theoretical results, two different configurations for the robot are considered: a horizontal and a vertical. Results of the NOC method as well as the two simulations are compared for the two robot configurations.

Kinematics Analysis of a New 5 DOF Parallel-Serial Machine Tool Based on the Vector Method

A 5-axis parallel-serial machine tool is proposed, which consists of a 3-DOF parallel mechanism and a 2-DOF serial mechanism, and the kinematics analysis is discussed in detail based on the vector method. The size of the velocity transformation matrix is decreased. The problem of parameter coupling is resolved by adopting identity matrix and four-element vector. The relationship between the velocity and acceleration of the movable platform and the input parameters of the parallel mechanism is established. These algorithms are verified correct through ADAMS (Automatic Dynamic Analysis of Mechanical System) and can be referenced by kinematics analysis, dynamics analysis of parallel or parallel-serial machine tools, which have coupling kinematics parameters.

Two-step generation of the equations of motion of closed-chains

This paper presents a two-step generation of the equations of motion of planar mechanisms using point and joint coordinates. First, the formulation replaces a rigid body by a dynamically equivalent constrained system of particles and uses Newton’s second law to study the motion of the particles without introducing any rotational coordinates. Then, the equations of motion are transformed to a reduced set in terms of selected relative joint variables using a velocity transformation matrix. For an open-chain, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For a closed-chain, suitable joints should be cut and few cut-joints constraint equations are included. An example of a closed-chain is used to demonstrate the generality and efficiency of the proposed method.

A matrix formulation for the dynamic analysis of spatial mechanisms using point coordinates and velocity transformation

This paper presents a matrix formulation for the dynamic analysis of spatial mechanisms with common types of kinematic joints. The formulation is derived in two steps. Initially an equivalent constrained system of particles that replaces the rigid bodies is constructed and used to define the configuration of the mechanical system. This results in a simple and straightforward procedure for generating the equations of motion in terms of the rectangular Cartesian coordinates of the particles without introducing any rotational coordinates. The equations of motion are then derived in terms of relative joint coordinates through the use of a velocity transformation matrix. The velocity transformation matrix relates the relative joint velocities to the Cartesian velocities. For the open loop case, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For the closed loop case, suitable joints should be cut and few cut-joints constraint equations should be included for each closed loop. An example is used to demonstrate the generality and efficiency of the proposed method.

Massless links with external forces and bushing effect for multibody dynamic analysis

When the contribution of lightweight components to the total energy of a system is small, the inertia effects are sometimes ignored by replacing them to massless links. For example, a revolute-spherical massless link generates two kinematic constraint equations between adjacent bodies and allows four relative degrees of freedom. In this paper, to implement a massless link systematically in a computer program using the velocity transformation technique, the velocity transformation matrix of massless links is derived and numerically implemented. The velocity transformation matrix for a revolute-spherical massless link and a revolute-universal massless link are appeared as a 6X4 matrix and a 6X3 matrix, respectively. A massless link model in a suspension composite joint transmitting external forces is also developed and the numerical efficiency of the proposed model is compared to a conventional multibody model. For a massless link transmitting external forces, forces acting on links are resolved and transmitted to the attached points with a quasi-static assumption. Numerical examples are presented to verify the formulation.

A matrix formulation for the dynamic analysis of planar mechanisms using point coordinates and velocity transformation

This paper presents a matrix formulation for the dynamic analysis of planar mechanisms consisting of interconnected rigid bodies. The formulation initially uses the rectangular Cartesian coordinates of an equivalent constrained system of particles to define the configuration of the mechanical system. This results in a simple and straightforward procedure for generating the equations of motion. The equations of motion are then derived in terms of relative joint coordinates through the use of a velocity transformation matrix. The velocity transformation matrix relates the relative joint velocities to the Cartesian velocities. For the open-loop case, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For the closed-loop case, suitable joints should be cut and few cut-joints constraint equations should be included for each closed loop. Two examples are used to demonstrate the generality and efficiency of the proposed method.

A generalized recursive formulation for constrained flexible multibody dynamics

AbstractThis research develops a relative co‐ordinate formulation for the multibody flexible dynamics. The velocity transformation method is notationally compact, because the Cartesian generalized velocities are simultaneously transformed to the relative generalized velocities in a matrix form. However, inherent computational efficiency in the recursive kinematics between two adjacent bodies has not been exploited. This research presents a recursive formulation which is both notationally compact and computationally efficient. The velocity transformation method is used to derive the equations of motion and their derivatives. Matrix operations associated with the velocity transformation matrix in the resulting equations of motion and their derivatives are classified into several categories. A joint library of the generalized recursive formulas is developed for each category. When one category is encountered in implementing the equations of motion and their derivatives, the corresponding recursive formulas in the category are invoked. When a new force or joint module is added to a general purpose programme in the relative co‐ordinate formulation, the modules for the rigid body are not reusable for the flexible body. Since the flexible body dynamics handles additional generalized co‐ordinates associated with deformation, implementation of the flexible dynamics is generally complicated and prone to coding mistakes. A virtual rigid body is introduced at every joint and force reference frames. A virtual flexible body joint is introduced between two body reference frames of the virtual and original bodies. This makes a flexible body subjected to only the kinematic admissibility condition for the virtual flexible body joint. As a result, the only extra work to handle the flexible bodies is to add the virtual flexible body joint modules in all recursive formulas. Since computation time in a relative co‐ordinate formulation is approximately proportional to the number of relative co‐ordinates, computational overhead due to the additional virtual bodies and joints are minor. Meanwhile, implementation convenience is dramatically improved. Copyright © 2001 John Wiley & Sons, Ltd.

A Formulation for Automated Computer-Aided Design of Machine Parts.

A mathematical formulation for the automated design of parts and assemblies is presented as a numerical algorithm and its accompanying experimental code. A mechanical part is modeled as an assembly of links connected via joints. Constraints are imposed by the designer such that some dimensions are specified as variables while others are allowed to propagate due to a design change. Graph theory is used to model the assembly where the method of cut-joint constraints is used to obtain a spanning tree. Joint constrains between the various links are introduced and their variations are computed. A velocity transformation matrix is introduced and the Jacobian of for each constraint is developed. The Jacobian of the constraint is transformed to joint coordinate space. Finally, because the resulting system of equations has a non-square Jacobian, the Moore-Penrose inverse is used to compute an assembled configuration, hence, redesigning the assembly. The formulation is extended to include various methods for computing an assembled configuration. While the accompanying numerical code is only experimental, the formulation presented herein is believed to form the basis of a new methodology for automated computer-aided design of parts and systems. The significance of this work is characterized by the formulation’s ability to automatically design and redesign parts.

Use of Cholesky coordinates and the absolute nodal coordinate formulation in the computer simulation of flexible multibody systems

In a previous publication, procedures that can be used with the absolute nodal coordinate formulation to solve the dynamic problems of flexible multibody systems were proposed. One of these procedures is based on the Cholesky decomposition. By utilizing the fact that the absolute nodal coordinate formulation leads to a constant mass matrix, a Cholesky decomposition is used to obtain a constant velocity transformation matrix. This velocity transformation is used to express the absolute nodal coordinates in terms of the generalized Cholesky coordinates. The inertia matrix associated with the Cholesky coordinates is the identity matrix, and therefore, an optimum sparse matrix structure can be obtained for the augmented multibody equations of motion. The implementation of a computer procedure based on the absolute nodal coordinate formulation and Cholesky coordinates is discussed in this paper. Numerical examples are presented in order to demonstrate the use of Cholesky coordinates in the simulation of the large deformations in flexible multibody applications.

Computer Implementation of the Absolute Nodal Coordinate Formulation for Flexible Multibody Dynamics

Deformable components in multibody systems are subject to kinematic constraints that represent mechanical joints and specified motion trajectories. These constraints can, in general, be described using a set of nonlinear algebraic equations that depend on the system generalized coordinates and time. When the kinematic constraints are augmented to the differential equations of motion of the system, it is desirable to have a formulation that leads to a minimum number of non-zero coefficients for the unknown accelerations and constraint forces in order to be able to exploit efficient sparse matrix algorithms. This paper describes procedures for the computer implementation of the absolute nodal coordinate formulation' for flexible multibody applications. In the absolute nodal coordinate formulation, no infinitesimal or finite rotations are used as nodal coordinates. The configuration of the finite element is defined using global displacement coordinates and slopes. By using this mixed set of coordinates, beam and plate elements can be treated as isoparametric elements. As a consequence, the dynamic formulation of these widely used elements using the absolute nodal coordinate formulation leads to a constant mass matrix. It is the objective of this study to develop computational procedures that exploit this feature. In one of these procedures, an optimum sparse matrix structure is obtained for the deformable bodies using the QR decomposition. Using the fact that the element mass matrix is constant, a QR decomposition of a modified constant connectivity Jacobian matrix is obtained for the deformable body. A constant velocity transformation is used to obtain an identity generalized inertia matrix associated with the second derivatives of the generalized coordinates, thereby minimizing the number of non-zero entries of the coefficient matrix that appears in the augmented Lagrangian formulation of the equations of motion of the flexible multibody systems. An alternate computational procedure based on Cholesky decomposition is also presented in this paper. This alternate procedure, which has the same computational advantages as the one based on the QR decomposition, leads to a square velocity transformation matrix. The computational procedures proposed in this investigation can be used for the treatment of large deformation problems in flexible multibody systems. They have also the advantages of the algorithms based on the floating frame of reference formulations since they allow for easy addition of general nonlinear constraint and force functions.

Inverse Dynamic Analysis of Mechanical Systems Using the Velocity Transformation Technique

This paper presents a method for the inverse dynamic anlaysis of mechanical systems. Actuating forces(or torques) depending on the driving constraints are analyzed in the relative coordinate space using the velocity transformation technique. A systematic method to compose the inverse velocity transformation matrix, which is used to determine the joint reaction forces, is proposed. Two examples are taken to verify the method developed here.

A simple and highly parallelizable method for real-time dynamic simulation based on velocity transformations

A semi-recursive and easy-to-parallelize algorithm for real-time dynamic simulation of open- and closed-loop multi-rigid-body systems is presented. The equations of motion are obtained in terms of a minimal set of relative joint coordinates using an efficient implementation of the velocity transformation method. The open-loop velocity transformation matrix, which relates body translational and rotational velocities to joint relative velocities, is computed in parallel using an extremely simple and intuitive idea. Similarly, the open-loop projected mass matrix is computed in parallel. Fine grain parallelization and optimum use of the cache memory are achieved by using a body-by-body procedure for the computation of vectors and matrices. Closed-loop systems are transformed into open-loop systems through the penalty formulation. The performance of the method is tested through an arithmetic operation count of a 45 degree of freedom open-loop model of a human body and an 18 degree of freedom closed-loop model of a heavy truck.

Systematic Construction of the Equations of Motion for Multibody Systems Containing Closed Kinematic Loops

This papers presents a systematic method for deriving the minimum number of equations of motion for multibody system containing closed kinematic loops. A set of joint or natural coordinates is used to describe the configuration of the system. The constraint equations associated with the closed kinematic loops are found systematically in terms of the joint coordinates. These constraints and their corresponding elements are constructed from known block matrices representing different kinematic joints. The Jacobian matrix associated with these constraints is further used to find a velocity transformation matrix. The equations of motions are initially written in terms of the dependent joint coordinates using the Lagrange multiplier technique. Then the velocity transformation matrix is used to derive a minimum number of equations of motion in terms of a set of independent joint coordinates. An illustrative example and numerical results are presented, and the advantages and disadvantages of the method are discussed.

Joint coordinate method for analysis and design of multibody systems: Part 1. System equations

This paper presents a method, known as the joint coordinate method, for the computer aided design process of multibody systems. The relationship between joint and absolute velocities is described by a linear velocity transformation matrix representing the system kinematics. Joint coordinate method can generate a minimal set of equations of motion in terms of the generalized joint accelerations. This method can also yield reaction and actuator forces acting at the kinematic joints that are necessary for forward and inverse dynamics analyses. Applications of this method to static equilibrium and design sensitivity analyses are also studied. Comparisons between the absolute and joint coordinate formulations are given in terms of their computational efficiency.

Dynamic analysis of spatial flexible multibody systems using joint co‐ordinates

AbstractThis paper presents a systematic method for deriving the minimum number of equations of motion for spatial flexible multibody systems. Relative kinematics are developed using relative joint co‐ordinates and relative joint velocities to formulate a minimum number of equations of motion. The present method takes advantage of the simplicity of the absolute co‐ordinate formulation and computational efficiency of the relative joint co‐ordinate formulation. The system equations of motion are first formulated in terms of a coupled set of absolute reference co‐ordinates and elastic modal co‐ordinates. These equations are transformed to the joint co‐ordinate space by use of a velocity transformation matrix including the elastic modal velocities. A computer algorithm is proposed and extended to closed loop mechanisms. One example of a flexible vehicle is presented and the results are discussed to illustrate the computational efficiency of the method.