Constraint Jacobian Matrix Research Articles

Projection continuation for minimal coordinate set formulation and singularity detection of redundantly constrained system dynamics

The formulation of (possibly redundantly) constrained system dynamics using coordinate projection onto a subspace locally tangent to the constraint manifold is revisited using the QR factorization of the constraint Jacobian matrix, using column pivoting to identify a suitable subspace, possibly detect any singular configurations that may arise, and extract it. The evolution of the QR factorization is integrated along with that of the constraint Jacobian matrix as the solution evolves, generalizing to redundant constraints a recently proposed true continuation algorithm that tracks the evolution of the subspace of independent coordinates. The resulting subspace does not visibly affect the quality of the solution, as it is merely a recombination of that resulting from the blind application of the QR factorization but avoids the artificial algorithmic irregularities or discontinuities in the generalized velocities that could otherwise result from arbitrary reparameterizations of the coordinate set, and identifies and discriminates any further possible motions that arise at singular configurations. The characteristics of the proposed subspace evolution approach are exemplified by solving simple problems with incremental levels of redundancy and singularity orders.

A projection continuation approach for minimal coordinate set constrained dynamics

The formulation of constrained system dynamics using coordinate projection onto a subspace locally tangent to the constraint manifold is revisited using the QR factorization of the constraint Jacobian matrix to extract a suitable subspace and integrating the evolution of the QR factorization along with that of the constraint Jacobian matrix, as the solution evolves. A true continuation algorithm is thus proposed to track the evolution of the subspace of independent coordinates. It does not visibly affect the quality of the solution but avoids the artificial algorithmic irregularities or discontinuities in the generalized velocities that could otherwise result from arbitrary reparameterizations of the coordinate set. The characteristics of the proposed subspace evolution approach are exemplified by solving simple single- and multi-degree-of-freedom problems.

Stiffness analysis of a 3-DOF parallel mechanism for engineering special machining

Abstract. There are considerably rigorous requirements for accuracy and stability of the mechanism to accomplish large-scale and complex surface machining tasks in the aerospace field. In order to improve the stiffness performance of the parallel mechanism, this paper proposes a novel three degrees of freedom (DOF) redundantly actuated 2RPU-2SPR (where R, P, U and S stand for revolute, prismatic, universal and spherical joints, respectively) parallel mechanism. Firstly, the kinematics position inverse solution is derived and a dimensionless generalized Jacobian matrix is established through the driving Jacobian matrix and constraint Jacobian matrix. Secondly, the stiffness model of the parallel mechanism is deduced and the accuracy of the stiffness model is verified through finite-element analysis. Using eigenscrew decomposition to illustrate the physical interpretation of the stiffness matrix, the stiffness matrix is equivalent to six simple screw springs. Finally, the simulation experiment results demonstrate that redundantly actuated parallel mechanism has better stiffness performance compared to the traditional 2RPU-SPR parallel mechanism.

Motion Sensitivity Analysis of Multibody Systems by Null Space Matrix Method of Differential Equation Type

One of the authors has proposed a motion analysis method for multibody systems called nullspace matrix method of differential equation type. In the proposed method, a nullspace matrix for the constraint Jacobian matrix is obtained by solving a differential equation. Then, the term which represents the constraint force in the equation of motion is eliminated with the nullspace matrix. The proposed method is applicable to systems with singular configurations. For efficient design of machines and structures, sensitivity of the behavior of the system with respect to design variables such as lengths of linkages and positions of joints is required. In this research, based on the procedure of the nullspace matrix method of differential equation type, a motion sensitivity analysis method of multibody systems is proposed. An example of motion sensitivity is presented.

Iterative refinement algorithm for efficient velocities and accelerations solutions in closed-loop multibody dynamics

To simulate closed-loop multibody systems in effective and accurate way, the semi-recursive formulations can be used together with optimal numerical integrators. Although numerous contributions have been reported in this field, there still exists a demand to improve the computation efficiency for the faster-than-real-time simulation and hardware-in-the-loop control applications. This paper presents an iterative refinement algorithm to accelerate the numerical integration process of a semi-recursive multibody formulation. By reusing the constraint Jacobian matrix factorization and generalized mass matrix factorization, respectively, the presented algorithm is introduced to enhance the solution of dependent relative velocities and independent relative accelerations. The introduced procedure consists of three steps; initial guess determination, iterative refinement process and termination criteria. The iterative refinement process is designed to find an accurate numerical solution, which contains residual computing, solution increment computing and solution updating. For higher efficiency, the initial guess can be re-determined to repeat the iterative refinement process. Three closed-loop multibody systems with increasing complexity are taken as numerical examples to verify the accuracy and efficiency of the presented iterative refinement algorithm. The results highlight more than 15% efficiency gain. The algorithm also can be implemented in other numerical integrators and multibody formulations.

Accuracy analysis and synthesis of asymmetric parallel mechanism based on Sobol-QMC

Aiming at the aircraft composite skin grinding, a new three degree-of-freedom (DOF) parallel mechanism with asymmetrical structure (TAM) is proposed to replace manual grinding. The TAM is achieved by integrating one of active limbs into the passive limb while keeping the required DOF unchanged, which is divided into two closed-loop chains: telescopic rod and parallelogram. The inverse kinematics models of the two chains are established according to closed-loop vector method. Thus, the actuation and the constraint Jacobian matrix are obtained. Based on the perturbation principle, the error modeling of the TAM is built. Adopting the constraint Jacobian matrix, 15 uncompensated errors are distinguished from the error model. In order to improve the working accuracy of the TAM, accuracy analysis and synthesis are necessary for all the uncompensated errors. The mapping function reflects the influence of uncompensated errors on the pose accuracy. The global sensitivity evaluation indexes are established by mapping function. Since Sobol sequences are superior in uniformity and convergence, the Quasi-Monte Carlo method based on Sobol sequences (Sobol-QMC) is introduced for sensitivity analysis. Taking the minimum manufacturing and installation costs as the optimization target, the objective function of accuracy synthesis is constructed. Ultimately, the reasonable tolerance zone of each uncompensated error is calculated by genetic algorithm. Simulation is performed by Sobol-QMC to verify the rationality of the optimization. The results show the probability is above 97% where most pose errors are in [[Formula: see text], [Formula: see text]] within the workspace. Therefore, accuracy synthesis is correct and practical.

Deformation basis and kinematic singularities of constrained systems

This paper demonstrates that arbitrarily introducing deformation degrees of freedom does not always solve the kinematic singularities of over-constrained systems. To address this fundamental problem, this paper proposes different methods that can be used to eliminate the initial-configuration singularity of over-constrained systems when modeled using multibody system formulations such as the general nonlinear finite element floating frame of reference (FFR) formulation. It is shown that the constraint Jacobian matrix of flexible-body models, which are not topologically over-constrained, can be singular. This fundamental problem, encountered in several heavily constrained flexible multibody system applications, such as vehicles, parallel mechanisms, and industrial robots, cannot be solved by automatic elimination of redundant constraints, a feature offered by several commercial multibody system computer programs. Automatic elimination of redundant contraints in systems which are not topologically over-constrained can lead to topological changes, resulting in different solutions with different frequency contents. A simple static-force benchmark problem, with known solution, is used in this paper to shed light on the effect of using different sets of deformation basis vectors, defined using the FFR reference conditions, on the rank of the constraint Jacobian matrix. Three methods for the systematic elimination of the initial-configuration rank deficiency of the constraint Jacobian matrix are proposed. The first method is the modal perturbation method in which selected elements of the modal transformation matrix are perturbed to preserve the system topology and the solution accuracy. The second method defines the deformation basis vectors using the new geometrically accurate absolute nodal coordinate formulation (ANCF)/FFR finite elements. These elements can lead to coupled axial and bending deformation basis vectors, which cannot be obtained using conventional finite elements. In the third method, an equivalent system that has the same solution as the original problem is developed by using a proper set of multibody system constraint equations. The results obtained in this investigation confirm the fact that the choice of the deformation basis vectors should not be restricted to one set, such as the free-free modes, to avoid obtaining incorrect results.

Bridging protein rigidity theory and normal modes using kino‐geometric analysis with hierarchical constraint relaxation

AbstractElastic‐Network Models (ENM) and pebble game based rigidity analysis are two distinct coarse‐grained approaches that have provided tremendous insight into conformational flexibility of proteins. However, the topological nature of the pebble game, and thereby the absence of motion modes have limited its applicability and a detailed comparison to ENM. Here, we present an alternative approach to rigidity analysis which eliminates these drawbacks through detailed kinematic analysis of dihedral degrees of freedom and non‐covalent hydrogen bond constraints, collected in the constraint Jacobian matrix J. Our new procedure reveals a spatial hierarchy of protein motions intrinsic to the hydrogen bonding network, ranked by the singular values of J. This spectrum of J yields a striking, fold‐specific signature, differentiating stiffer α‐helical from more collective β‐sheet proteins, which often goes undetected in similarly coarse‐grained methods. Overall, our results agree with experimental data and more detailed simulations, indicating that hydrogen bond networks have evolved with different protein folds to modulate structural dynamics and molecular mechanisms, with broad implications for protein and drug design.

Singular-Perturbation-Based Nonlinear Hybrid Control of Redundant Parallel Robot

High-speed robots are being applied more and more widely in the scenarios of aviation manufacturing, electronic packaging, and three-dimensional printing, etc. To achieve high speed, the robots are typically designed to be lightweight, hence suffer from undesired vibration, so that the dynamic performance would deteriorate. In light of this, this paper considers the dynamic modeling and active control of an emerging redundant parallel robot. First, the recursively dynamic model is modularly established by means of Lagrangian multipliers method. For completeness, the model of permanent magnet synchronous motor is further integrated with the model of mechanical system to formulate the electromechanically coupled dynamic model (ECDM). To facilitate dynamic control, the ECDM is cleverly projected into the null space of constraint Jacobian matrix eliminating the need to compute Lagrangian multipliers explicitly. Based on singular perturbation approach and Tikhonov theorem, the ECDM is decomposed into two subsystems in different timescales. On this basis, a nonlinear hybrid control scheme is proposed for both trajectory tracking and vibration suppression. Numerical simulations suggest that the nonlinear hybrid control exhibits overwhelming performance than the conventional joint-based proportional and differential feedback control.

A piston tribodynamic model with deterministic consideration of skirt surface grooves

The machining micro-grooves on the piston skirt surface are very common in internal combustion engines. To accurately investigate the influence of the grooves on piston tribodynamic performance, a coupling model of piston skirt-liner lubrication and crank-rod-piston multibody dynamics is built in this study. The lubrication model is solved by the Finite Element Method with deterministic treatment of the grooves. The multibody dynamics equations are formulated with Lagrange multipliers and constraint Jacobian matrix in the form of Differential Algebraic Equations (DAEs). The MEBDF approach is applied to a high efficient time integration of the DAEs. Based on the above model, the influences of grooves and some groove parameters on piston skirt-liner lubrication as well as piston slap are studied and revealed.

Accuracy analysis and design of A3 parallel spindle head

As functional components of machine tools, parallel mechanisms are widely used in high efficiency machining of aviation components, and accuracy is one of the critical technical indexes. Lots of researchers have focused on the accuracy problem of parallel mechanisms, but in terms of controlling the errors and improving the accuracy in the stage of design and manufacturing, further efforts are required. Aiming at the accuracy design of a 3-DOF parallel spindle head(A3 head), its error model, sensitivity analysis and tolerance allocation are investigated. Based on the inverse kinematic analysis, the error model of A3 head is established by using the first-order perturbation theory and vector chain method. According to the mapping property of motion and constraint Jacobian matrix, the compensatable and uncompensatable error sources which affect the accuracy in the end-effector are separated. Furthermore, sensitivity analysis is performed on the uncompensatable error sources. The sensitivity probabilistic model is established and the global sensitivity index is proposed to analyze the influence of the uncompensatable error sources on the accuracy in the end-effector of the mechanism. The results show that orientation error sources have bigger effect on the accuracy in the end-effector. Based upon the sensitivity analysis results, the tolerance design is converted into the issue of nonlinearly constrained optimization with the manufacturing cost minimum being the optimization objective. By utilizing the genetic algorithm, the allocation of the tolerances on each component is finally determined. According to the tolerance allocation results, the tolerance ranges of ten kinds of geometric error sources are obtained. These research achievements can provide fundamental guidelines for component manufacturing and assembly of this kind of parallel mechanisms.

A new numerical method for piston dynamics and lubrication analysis

A numerical model for piston dynamics and lubrication analysis is proposed by coupling the lubrication model of piston skirt–liner system and the dynamics model of multibody system consisting of crank, connecting rod and piston. The lubrication model is solved by the Finite Element Method, while the multibody dynamics equations are established with Lagrange multipliers and constraint Jacobian matrix. The validation of the method is verified through comparing with a well-known existing work. Apart from this, some design parameters of piston skirt were investigated so as to reveal their influence on the slap noise and lubrication performances of the piston.

Neural Network-Based Control of Networked Trilateral Teleoperation With Geometrically Unknown Constraints.

Most studies on bilateral teleoperation assume known system kinematics and only consider dynamical uncertainties. However, many practical applications involve tasks with both kinematics and dynamics uncertainties. In this paper, trilateral teleoperation systems with dual-master-single-slave framework are investigated, where a single robotic manipulator constrained by an unknown geometrical environment is controlled by dual masters. The network delay in the teleoperation system is modeled as Markov chain-based stochastic delay, then asymmetric stochastic time-varying delays, kinematics and dynamics uncertainties are all considered in the force-motion control design. First, a unified dynamical model is introduced by incorporating unknown environmental constraints. Then, by exact identification of constraint Jacobian matrix, adaptive neural network approximation method is employed, and the motion/force synchronization with time delays are achieved without persistency of excitation condition. The neural networks and parameter adaptive mechanism are combined to deal with the system uncertainties and unknown kinematics. It is shown that the system is stable with the strict linear matrix inequality-based controllers. Finally, the extensive simulation experiment studies are provided to demonstrate the performance of the proposed approach.

Real-Time Dynamic Simulations of Large Road Vehicles Using Dense, Sparse, and Parallelization Techniques

This article presents three multibody formulations with improved efficiency in order to achieve real-time simulations for the forward dynamic of two real-life road vehicles. The bigger is a semitrailer truck with 40 degrees of freedom (DOF). Two topological and semirecursive formulations are used as well as a global formulation based on the use of Euler parameters and flexible joints. The first semirecursive formulation carries out a double velocity transformation and the integration is done by means of the explicit fourth order Runge–Kutta method. The second semirecursive formulation and the global one use a penalty scheme at position level and orthogonal projections at velocity and acceleration levels. In both cases the integrator was the implicit Hilbert–Huges–Taylor (HHT) method. The double velocity transformation method involves the coordinate partitioning of the constraint Jacobian matrix which leads to the costly solution of a redundant but consistent with the constraints linear system of equations. The choice of a unique set of independent coordinates may not be valid for a complete simulation and additional repartitioning would be required. Based on previous experience and as the examples show in this article, a careful initial choice of the independent coordinates can remain valid for complete simulations involving common maneuvers. This represents a numerical advantage for dense matrix methods and can be further exploited if sparse matrix techniques are employed. This has been the case for both of the vehicles used, reaching real-time simulations even with the semitrailer truck. The implicit semirecursive formulation involves the numerical evaluation of the stiffness and damping matrices, which hamper obtaining real-time simulations. For the semitrailer truck, this computation represents the 76% of the total simulation time. The numerical computation of these matrices is carried out by columns and its algorithm is straightforwardly parallelizable. Using a quad-core processor and with a simple and efficient OpenMP implementation, it has been possible to achieve a speedup of 3.25 reducing the simulation times under the real-time limit. The sparse matrices of Euler parameters formulation show very different sparsity degrees, difference that grows with the size of the multibody model. This poses a challenge to sparse matrix implementations in order to be able to efficiently perform matrix operations without increasing fillings or handling zero entries. This has been successfully accomplished using a new sparse matrix representation. This one is not a feature of general purpose sparse software, requiring at some stages the implementation of our own algorithms. Reductions in time of three orders of magnitude have led to real-time simulations even with the semitrailer truck.

Automated independent coordinates’ switching for the solution of stiff DAEs with the linearly implicit Euler method

The real-time (RT) integration of the equations of motion describing complex mechanical systems requires the employment of proper integration schemes which must be able to handle the time integration at each time step in less than an a priori fixed sampling time interval. The linearly implicit Euler method has been successfully employed for the RT integration of large stiff differential algebraic systems of equations (DAEs) which typically arise from the complex mechanical systems of interest in practical applications. The current industry demand is to further increase the degree of complexity of multibody models employed in RT applications, pushing researchers to improve the efficiency of the currently available integration methods. In this paper, we investigate the improvements in the efficiency of the linearly implicit Euler method coming from the conversion of the equations of motion at each time step from a dependent to an independent coordinates’ formulation. The automatic switching from dependent to independent coordinates is achieved exploiting the properties of the matrix R whose columns represent a basis of the null-space of the constraint Jacobian matrix. A non-iterative projection method is also applied in order to avoid the drift-off from the constraint conditions.

DESIGN FOR OPTIMAL PERFORMANCE OF 3-RPS PARALLEL MANIPULATOR USING EVOLUTIONARY ALGORITHMS

This paper presents an optimal kinematic design for a general type of 3-RPS spatial parallel manipulator based on multi-objective optimization. The objective functions considered are Global Conditioning Index (GCI), Global stiffness Index (GSI) and Workspace volume. The objective functions are optimized simultaneously to improve the dexterity as well as the workspace volume which represents the working capacity of a parallel manipulator. A multi-objective Evolutionary Algorithm based on the control elitist non-dominated sorting genetic algorithm is adopted to find the true optimal Pareto front. A constraint Jacobian matrix is derived analytically and the manipulator workspace is generated by numerical search method. The static analysis of the manipulator is also carried out to determine the compliance of the end-effecter.

Model-based principal components of correlation matrices

A model for principal components of correlation matrices is proposed. The model satisfies the correlation constraint (i.e., unit valued diagonal elements) as well as optional constraints on eigenvalues and/or eigenvectors. The model yields simplified principal components that retain both orthogonality and variance maximization properties. Inference procedures for eigenvalues, eigenvectors, and loadings on rotated or raw principal components are given. Multivariate normality is not required. A major issue in the modeling process is that the eigen-structure of the population correlation matrix can induce rank deficiencies in the submatrix of the constraint Jacobian matrix that is associated with the correlation constraint. This rank deficiency is a property of the population constraint Jacobian matrix; it is not necessarily a property of the sample Jacobian matrix evaluated at the solution to the estimating equation. Furthermore, if degenerate constraints are eliminated, then the fitted correlation matrix need not satisfy the correlation constraint. Procedures are proposed for detecting rank deficiencies, eliminating degenerate constraints, and constructing auxiliary constraints that ensure that the correlation constraint is satisfied. The procedures are illustrated on two real data sets.

An adjoint-based SQP algorithm with quasi-Newton Jacobian updates for inequality constrained optimization

We present a sequential quadratic programming (SQP) type algorithm, based on quasi-Newton approximations of Hessian and Jacobian matrices, which is suitable for the solution of general nonlinear programming problems involving equality and inequality constraints. In contrast to most existing SQP methods, no evaluation of the exact constraint Jacobian matrix needs to be performed. Instead, in each SQP iteration only one evaluation of the constraint residuals and two evaluations of the gradient of the Lagrangian function are necessary, the latter of which can efficiently be performed by the reverse mode of automatic differentiation. Factorizations of the Hessian and of the constraint Jacobian are approximated by the recently proposed STR1 update procedure. Inequality constraints are treated by solving within each SQP iteration a quadratic program (QP), the dimension of which equals the number of degrees of freedom. A recently proposed gradient modification in these QPs takes account of Jacobian inexactness in the active set determination. Superlinear convergence of the procedure is shown under mild conditions. The convergence behaviour of the algorithm is analysed using several problems from the Hock–Schittkowski test library. Furthermore, we present numerical results for an optimization problem based on a small periodic adsorption process, where the Jacobian of the equality constraints is dense.

A constraint Jacobian based approach for static analysis of pantograph masts

This paper presents a constraint Jacobian matrix based approach to obtain the stiffness matrix of widely used deployable pantograph masts with scissor-like elements (SLE). The stiffness matrix is obtained in symbolic form and the results obtained agree with those obtained with the force and displacement methods available in literature. Additional advantages of this approach are that the mobility of a mast can be evaluated, redundant links and joints in the mast can be identified and practical masts with revolute joints can be analysed. Simulations for a hexagonal mast and an assembly with four hexagonal masts is presented as illustrations.

Hermite-Legendre-Gauss-Lobatto Direct Transcription in Trajectory Optimization

D IRECT methods have been widely applied for solving trajectory optimization problems [1–5]. Among the popular methods are the Hermite–Simpson method [6] and the Legendre pseudospectral (PS)method [7,8]. There has been some considerable interest in developing theory related to the Legendre PS method due to its high accuracy, although the Hermite–Simpson method continues to be applied to large-scale practical problems [9]. The main characteristic of the Hermite–Simpson method is the combination of reasonable accuracy with a highly sparse constraint Jacobian and Hessian matrix [10]. The PS method offers impressive accuracy (spectral accuracy) for smooth problems, but the constraint Jacobians are much denser than other methods. The sparsity of the constraint Jacobians can be increased in the PS method by using knots [11]. In addition to the Hermite–Simpson method, additional highorder methods have been proposed by Herman and Conway [2]. These methods are attractive from the point of view of accuracy, but in the framework proposed by Herman and Conway, they require detailed derivation when extended to arbitrary higher orders. For instance, in [2], the form of the constraints was derived via the symbolic manipulation software MAPLE. The purpose of this Note is to provide an alternative framework for arbitrary higher-order methods suitable for implementation on digital computers and in a reusable form. The optimal control problem is approximated by a discrete nonlinear programming problem (NLP) by expanding the state trajectories using local Hermite interpolating polynomials. For high accuracy, the collocation points are selected from the family of Gauss–Lobatto points. This also allows the integral performance index to be approximated viaGauss– Lobatto quadrature rules. For optimal control problems of the Bolza form, the natural choice of quadrature points are the Legendre– Gauss–Lobatto (LGL) points, because they are derived on the basis of a unity weight function, giving the highest accuracy for polynomial integrands. The generalization of the approach is referred to as the Hermite–Legendre–Gauss–Lobatto (HLGL) approach throughout the remainder of this Note.