an alternative proof for the recursive formulae for computing the moore penrose m inverse of a matrix in this paper we provide an alternative proof for the recursive formulae for the determination of the generalized moore penrose m inverse of a matrix through the direct verification of the four properties of moore penrose m inverse

eliminating constraint drift in the numerical simulation of constrained dynamical systems by means of the udwadia kalaba approach we propose an explicit equation of constrained motion developed to simulate constrained dynamical systems without error accumulation due to constraint drift the basic idea is to embed a small virtual force and a small virtual impulse to the equation of motion in order to avoid the drift typically experienced in constrained multibody simulations the embedded correction terms are selected to minimally alter the dynamics in an acceleration and kinetic energy norm sense the formulation allows one to use a standard ode solver avoiding the need for iterative constraint stabilization the equation is based on the pseudoinverse of a constraint matrix such that it can be used under redundant constraints and kinematic singularities the proposed method takes into account the finite word length of the computational environment and also accommodates possibly inconsistent initial conditions

inequality constraints in the process of jumping abstract this work presents a study of jumping through a biomechanical model of a leg which is subjected to an inequality constraint the activation and deactivation of an equality constraint reproduce the inequality constraint within the framework of the generalized inverse equations of motion it is shown that the activation of this additional constraint can be implemented by greville s formulae

optimal tracking control of nonlinear dynamical systems this paper presents a simple methodology for obtaining the entire set of continuous controllers that cause a nonlinear dynamical system to exactly track a given trajectory the trajectory is provid

inversion of e simple block matrices we recall previous results on inverting matrices the digraph of which is e simple i e such that every edge is contained in at most one simple cycle we present and analyze a finite algorithm for the inversion applications to m matrices are included

a new perspective on the tracking control of nonlinear structural and mechanical systems based on recent results from analytical dynamics this paper develops a class of tracking controllers for controlling general nonlinear structural and mechanical systems unlike most control methods that perform some kind of linearization and or nonlinear cancellation the methodology developed herein views the nonlinear control problem from a different perspective this leads to a simple and new control methodology that is capable of exactly maintaining the nonlinear system along a certain trajectory which in general may be described by a set of differential equations in the observations measurements the approach requires very little computation compared with standard approaches it is therefore useful for online real time control of nonlinear systems the methodology is illustrated with two examples

on constrained motion the general explicit equations of motion for constrained discrete dynamical systems are obtained these new equations lead to a simple and new fundamental view of constrained motion where the forces of constraint may be ideal and or non ideal

the inverse of a singular linear transformation and some applications in constrained mechanics we extend the notion of the moore penrose inverse of a matrix to the case of a morphism between vector bundles and a more general concept is used to develop the geometric theory of udwadia and kalaba s approach to constrained lagrangian mechanics

a unified approach for the recursive determination of generalized inverses abstract using a unified approach simple derivations for the recursive determination of different types of generalized inverses of a matrix are presented these include results for the generalized inverse the least squares generalized inverse the minimum norm generalized inverse and the moore penrose inverse of a matrix

on a decomposition of generalized constraint forces an additive decomposition of generalized constraint forces into a component determined by the motion of the system and a constitutively prescribed component is discussed in this paper the former component is shown to be necessary and sufficient to ensure that the motion of the system satisfies the constraint while the prescription of the latter is shown to be at the discretion of the dynamicist the decomposition is also related to previous works on constraint forces in the dynamics of discrete mechanical systems

the bellman gauss principle for constrained motion gauss principle of least constraint is solved in a sequential fashion via dynamic programming in this paper the solution itself constitutes a new principle for constrained motion which we may name the bellman gauss principle for constrained motion

the weighted generalized inverses of a partitioned matrix a constructive proof for the recursive representation of the weighted moore penrose inverse of a partitioned matrix a uv is given using the same thread of reasoning similar results for a 1 3 m and a 1 4 n are derived and all of these results can be represented by an unified form

efficient algorithms for simulating complex mechanical systems using constraint dynamics the constrained lagrangian and constrained hamiltonian equations of motion for a general non relativistic classical mechanical system subject to rheonomous holonomic constraints are derived in an easy and straightforward manner the numerical integration of the constrained equations of motion are discussed it is shown how constraint errors introduced by the numerical integration can be avoided by introducing simple constraint correction schemes as an example the developed constrained methods are applied to the periodically driven inverted n linked pendulum it is demonstrated how the constrained methods leads to very efficient numerical algorithms in the case of the n linked pendulum the computational complexity using the constrained methods is o n compared to o n 3 using the conventional unconstrained approach

recursive modular modelling methodology for lumped parameter dynamic systems this paper proposes a novel approach to the modelling of lumped parameter dynamic systems based on representing them by hierarchies of mathematical models of increasing complexity instead of a single complex model exploring the multilevel modularity that these systems typically exhibit a general recursive modelling methodology is proposed in order to conciliate the use of the already existing modelling techniques the general algorithm is based on a fundamental theorem that states the conditions for computing projection operators recursively three procedures for these computations are discussed orthonormalization use of orthogonal complements and use of generalized inverses the novel methodology is also applied for the development of a recursive algorithm based on the udwadia kalaba equation which proves to be identical to the one of a kalman filter for estimating the state of a static process given a sequence of noiseless measurements representing the constraints that must be satisfied by the system

conformal energy conformal laplacian and energy measures on the sierpinski gasket on the sierpinski gasket sg and related fractals we define a notion of conformal energy e and conformal laplacian for a given conformal factor based on the corresponding notions in riemannian geometry in dimension n 2 we derive a differential equation that describes the dependence of the effective resistances of on we show that the spectrum of dirichlet or neumann has similar asymptotics compared to the spectrum of the standard laplacian and also has similar spectral gaps provided the function does not vary too much we illustrate these results with numerical approximations we give a linear extension algorithm to compute the energy measures of harmonic functions with respect to the standard energy and as an application we show how to compute the l p dimensions of these measures for integer values of p 2 we derive analogous linear extension algorithms for energy measures on related fractals

a unified approach to rigid body rotational dynamics and control this paper develops a unified methodology for obtaining both the general equations of motion describing the rotational dynamics of a rigid body using quaternions as well as its control this is achieved in a simple systematic manner using the so called fundamental equation of constrained motion that permits both the dynamics and the control to be placed within a common framework it is shown that a first application of this equation yields in closed form the equations of rotational dynamics whereas a second application of the self same equation yields two new methods for explicitly determining in closed form the nonlinear control torque needed to change the orientation of a rigid body the stability of the controllers developed is analysed and numerical examples showing the ease and efficacy of the unified methodology are provided

explicit poincare equations of motion for general constrained systems part i analytical results this paper gives the general constrained poincareequations of motion for mechanical systems subjected to holonomic and or nonholonomic constraints that may or may not satisfy d alembert s principle at each instant of time it also extends gauss s principle of least constraint to include quasi accelerations when the constraints are ideal thereby expanding the compass of this principle considerably the new equations provide deeper insights into the dynamics of multi body systems and point to new ways for controlling them

efficient numerical solution of constrained multibody dynamics systems abstract a numerical algorithm for conducting coupled system dynamical simulation is presented the interconnected system comprising numerous modules is treated as a constrained multibody dynamics system of particular focus is the efficient solution of coupled system simulation without sacrificing the independence of the separate dynamical modules the proposed algorithm maggi s equations with perturbed iteration mepi emanates from numerical methods for differential algebraic equations separate treatment of the constraint equations from the resolution of subsystem dynamical responses marks mepi s main characteristic

energy control of inhomogeneous nonlinear lattices this paper considers energy control of an n d f inhomogeneous nonlinear lattice with fixedfixed and fixedfree ends the lattice consists of dissimilar masses wherein each mass is connected to its

classical r matrices and the feigin odesskii algebra via hamiltonian and poisson reductions we present a formula for a classical r matrix of an integrable system obtained by hamiltonian reduction of some free field theories using pure gauge symmetries the framework of the reduction is restricted only by the assumption that the respective gauge transformations are lie group ones our formula is in terms of dirac brackets and some new observations on these brackets are made we apply our method to derive a classical r matrix for the elliptic calogero moser system with spin starting from the higgs bundle over an elliptic curve with marked points in the paper we also derive a classical feigin odesskii algebra by a poisson reduction of some modification of the higgs bundle over an elliptic curve this allows us to include integrable lattice models in a hitchin type construction

fundamental principles of lagrangian dynamics mechanical systems with non ideal holonomic and nonholonomic constraints this paper deals with the foundations of analytical dynamics it obtains the explicit equations of motion for mechanical systems that are subjected to non ideal holonomic and nonholonomic equality constraints it provides an easy incorporation of such non ideal constraints into the framework of lagrangian dynamics it bases its approach on a fundamental principle that includes non ideal constraints and that reduces to d alembert s principle in the special case when all the constraints become ideal based on this the problem of determining the equations of motion for the constrained system is reformulated as a constrained minimization problem this yields a new fundamental minimum principle of analytical dynamics that reduces to gauss s principle when the constraints become ideal the solution of this minimization problem then yields the explicit equations of motion for systems with non ideal constraints an illustrative example showing the use of this general equation for a system with sliding coulomb friction is given

an alternative proof for the explicit equations of motion for mechanical systems with independent non ideal constraints an alternative proof for the explicit equations of motion for mechanical systems with independent non ideal constraints is proposed in this paper these equations are derived without the use of generalized inverse matrices and based on the decomposition of the constraint forces along the acceleration constrained flat surface and its normal plane

on lagrangian dynamics and its control formulations this paper exploits the equivalence between constrained dynamics problems and a class of tracking control problems an alternative derivation of the modern formulation of parczewski and blajer j appl math 56 1989 676 684 is presented including some new analysis on the almost singular case followed by a critique of the classical lagrangian formulation in addition a new formulation which allows the constraints to be satisfied approximately is presented a general closed loop control law which depends on a user specified threshold of constraint errors e is proposed transforming the default dae problem into a stiff ode initial value problem which can be routinely solved on computers it is shown that certain explosively unstable open loop systems may be stabilized by the use of closed loop control using a small but finite e an example is worked out using both the open loop and the closed loop control formulations

symbolic and recursive computation of different types of generalized inverses abstract we propose a method and algorithm for recursive computation of different classes of generalized inverses of a given one variable rational matrix and corresponding algorithm for polynomial matrix these methods and algorithms are generalizations of the method for computing the generalized inverses for constant matrices originated in f e udwadia r e kalaba a unified approach for the recursive determination of generalized inverses comp math appl 37 1999 125 130 and the partitioning method for computing the generalized inverses of rational and polynomial matrices introduced in p s stanimirovic m b tasic partitioning method for rational and polynomial matrices appl math comput 155 2004 137 163 m b tasic p s stanimirovic m d petkovic symbolic computation of weighted moore penrose inverse using partitioning method appl math comput 189 2007 615 640 algorithms are implemented in the symbolic computational package mathematica

on singularity of energy measures on self similar sets we provide general criteria for energy measures of regular dirichlet forms on self similar sets to be singular to bernoulli type measures in particular every energy measure is proved to be singular to the hausdorff measure for canonical dirichlet forms on 2 dimensional sierpinski carpets

identifiability analysis of local oscillator phase self calibration based on hybrid cramer rao bound in mimo radar the identifiability of self calibration of local oscillator phase perturbation in multiple input multiple output mimo radar is analyzed in this paper four cases are considered the absolute phase perturbation app and phase perturbation differences ppds in mimo radar with widely separated antennas smimor and the app and ppds in co located mimo radar cmimor the tightness of the hybrid cramer rao bound hcrb is proved in all four cases it is interesting that the problem in this paper is a special case in which hcrb is asymptotically tight for nonlinear observation models containing unknown deterministic and random parameters for each case a signal model for multiple stationary targets is constructed the closed form hcrb of phase perturbation is derived and used to analyze the identifiability of phase perturbation it is proven that only the ppds in cmimor in the above mentioned four cases are identifiable numerical simulations verify the conclusion

the moore penrose inverses of m n block matrices and their applications abstract this article gives the expressions for the moore penrose inverses of m n block matrices when they satisfy rank additivity conditions and presents some of their special cases and applications

energy measures on p c f self similar sets abstract on connected post critically finite p c f self similar sets we give a linear extension method to compute the energy measures of harmonic functions with respect to the standard energy and as an application we also compute the l 2 dimensions of these measures on some p c f self similar sets

adaptive neural asymptotic control for uncertain nonlinear multiagent systems with a fuzzy dead zone constraint abstract a consensus control scheme is developed for a class of leader follower multiagent systems with a fuzzy dead zone constraint differently from previous work the method developed implements a predefined convergence of tracking errors for multiagent systems especially the dynamic model for each follower has a fuzzy dead zone input to solve the problem not only a set of smooth functions are used within the control design to increase the stability of the systems but also the unfuzziness and center of gravity method are used to analyze and process the actuator constraint model where each slope of the dead zone is uncertain and fuzzy it is verified that the method developed effectively guarantees that the tracking errors of multiagent systems can converge to a predefined interval that is the problem of better asymptotic consensus performance for nonlinear multiagent systems is solved simulation illustrates the results obtained

harmonic calculus on p c f self similar sets the main object of this paper is the laplace operator on a class of fractals first we establish the concept of the renormalization of difference operators on post critically finite p c f for short self similar sets which are large enough to include finitely ramified self similar sets and extend the results for sierpinski gasket given in 10 to this class under each invariant operator for renormalization the laplace operator green function dirichlet form and neumann derivatives are explicitly constructed as the natural limits of those on finite pre self similar sets which approximate the p c f self similar sets also harmonic functions are shown to be finite dimensional and they are characterized by the solution of an infinite system of finite difference equations

complete automation of the generalized inverse method for constrained mechanical systems of particles a completely automated method for simulating constrained mechanical systems of particles is presented a compiler generates matlab r program files that are specific to the problem presented by the user these automatically generated programs call several subroutines that perform numerical differentiation acceleration computation and numerical integration providing the accelerations velocities and coordinates of all the particles of the specific mechanical system

using the fundamental equation of constrained motion with inequality constraints in this paper we investigate a computational approach to keeping a moving particle within a predefined annulus or a predefined bounded space formed by two concentric spheres with radii l m i n and l m a x respectively assuming that said particle cannot maintain a perfectly circular trajectory the study develops an algorithm for dealing with a system in which constraints are expressed as inequalities the proposed approach expresses the trajectory in terms of winding unwinding logarithmic spirals with transitions expressed as damped vibrations between them these transitions are necessary to resolve incompatibility between initial conditions for winding unwinding spirals equations of motion for the particle are obtained by using the fundamental equation of constrained motion the obtained simulation results show that such an approach produces the desired pseudo periodic type of motion and the particle stays within the predefined region of space for a long duration although no cycle of its trajectory is repeated

should model based inverse inputs be used as feedforward under plant uncertainty bounds on the size of the plant uncertainties are found such that the use of the inversion based feedforward input improves the output tracking performance when compared to the use of feedback alone the output tracking error is normalized by the size of the desired output and used as a measure of the output tracking performance the worst case performance is compared for two cases 1 with the use of feedback alone and 2 with the addition of the feedforward input it is shown that inversion based feedforward controllers can lead to performance improvements at frequencies w where the uncertainty spl delta jw in the nominal plant is smaller than the size of the nominal plant g sub 0 jw divided by its condition number k sub g0 jw i e spl par spl delta jw spl par sub 2 spl par g sub 0 jw spl par sub 2 k sub g0 jw a modified feedforward input is proposed that only uses the model information in frequency regions where plant uncertainty is sufficiently small the use of this modified inverse with any feedback results in improvement of the output tracking performance when compared to the use of the feedback alone

dynamical computation of constrained flexible systems using a modal udwadia kalaba formulation application to musical instruments most musical instruments consist of dynamical subsystems connected at a number of constraining points through which energy flows for physical sound synthesis one important difficulty deals with enforcing these coupling constraints while standard techniques include the use of lagrange multipliers or penalty methods in this paper a different approach is explored the udwadia kalaba u k formulation which is rooted on analytical dynamics but avoids the use of lagrange multipliers this general and elegant formulation has been nearly exclusively used for conceptual systems of discrete masses or articulated rigid bodies namely in robotics however its natural extension to deal with continuous flexible systems is surprisingly absent from the literature here such a modeling strategy is developed and the potential of combining the u k equation for constrained systems with the modal description is shown in particular to simulate musical instruments objectives are twofold 1 develop the u k equation for constrained flexible systems with subsystems modelled through unconstrained modes and 2 apply this framework to compute string body coupled dynamics this example complements previous work debut antunes marques and carvalho appl acoust 108 3 18 2016 on guitar modeling using penalty methods simulations show that the proposed technique provides similar results with a significant improvement in computational efficiency

an order recursive generalized least squares algorithm for system identification in system identification an order recursive algorithm is of practical interest however such an algorithm for obtaining asymptotically unbiased and consistent parameter estimates in a noisy linear system is lacking an order recursive generalized least squares gls algorithm is presented this is based on a new recursive method for computing the least squares inverse developed in section ii and is an extension of hsia s gls algorithm 2

a new perspective on constrained motion the explicit general equations of motion for constrained discrete dynamical systems are obtained these new equations lead to a simple and new fundamental view of lagrangian mechanics

generalized lm inverse of a matrix augmented by a column vector this paper presents a direct proof of the recursive formulae for the generalized lm inverse of a matrix augmented by a column vector the recursive relations are proved by direct verification of the four conditions of the generalized lm inverse several auxiliary results pertaining to generalized inverses are also provided

least squares solution of equations of motion under inconsistent constraints abstract udwadia and kalaba have obtained explicit equations for the motion of discrete mechanical systems under consistent holonomic or nonholonomic constraints using the extremal property of the moore penrose pseudoinverse a direct proof now shows that the uk equations also determine the least squares solution for motion under inconsistent constraints

comment on penalized weighted residual method for the initial value problem n a recent paper dealing with the use of the hamiltonian law of varying action for the calculation of dynamic systems kim and cho 1 propose to introduce the initial values of the problem by using a penalty function the justi cation for the application of such a function is given in the paper 1 as follows a because the initial displacement u t0 is speci ed the variation of the initial displacement d u t0 must be zero for kinematically admissible variations therefore the initial velocity cannot be considered appropriately in m c ud u j t0 0 thus the theories using hamilton s lawof varying action must consider tentatively d u t0 as free variation independent of u t0 and impose u t0 c u t0 at the approximate stage o there is a basic misconception in the paper 1 the idea 2 i 4 of using a variation d u independent of u has a deeper importance than the one assumed in the paper 1 the independent d u is applied to stabilize the calculation process 2 i 4 for example in ref 3 the second derivative of u was used as d u in this way the calculation process has been stabilized it is interesting that gauss 5 in his principle treated e

is analytical dynamics a theoretical or an experimental science when a mechanical system is subjected to equality constraints use of the chain rule of differentiation and of generalized inverses of matrices enables us to write the most general possible equation of motion no use being made of any physical principles eq 8 then employment of standard physical principles enables us to further interpret the terms in this general equation of motion

new expressions of 2 times 2 block matrix inversion and their application a 2 times 2 block matrix inversion is a tool that is frequently used in areas of control estimation theory and signal processing however one of the two diagonal entries of the block matrix should be invertible to carry out a conventional block matrix inversion in this technical note we show that this assumption can be partially released with three new types of symbolic block matrix inversion also an application example of an inverse plant model of a multi inputs and multi output mimo plant which cancels plant noise and disturbance is suggested to show the effectiveness of these new types of matrix inversion

explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi body dynamics we present the new general explicit form of the equations of motion for constrained mechanical systems applicable to systems with singular mass matrices the systems may have holonomic and or non holonomic constraints which may or may not satisfy d alembert s principle at each instant of time the equation provides new insights into the behaviour of constrained motion and opens up new ways of modelling complex multi body systems examples are provided and applications of the equation to such systems are illustrated

about the generalized lm inverse and the weighted moore penrose inverse the recursive method for computing the generalized lm inverse of a constant rectangular matrix augmented by a column vector is proposed in udwadia and phohomsiri 2007 16 17 the corresponding algorithm for the sequential determination of the generalized lm inverse is established in the present paper we prove that the introduced algorithm for computing the generalized lm inverse and the algorithm for the computation of the weighted moore penrose inverse developed by wang and chen 1986 in 23 are equivalent algorithms both of the algorithms are implemented in the present paper using the package mathematica several rational test matrices and randomly generated constant matrices are tested and the cpu time is compared and discussed

conservative rigid body dynamics by convected base vectors with implicit constraints abstract a conservative time integration formulation is developed for rigid bodies based on a convected set of orthonormal base vectors the base vectors are represented in terms of their absolute coordinates and thus the formulation makes use of three translation components plus nine components of the base vectors orthogonality and unit length of the base vectors are imposed by constraining the equivalent green strain components and the kinetic energy is represented corresponding to rigid body motion the equations of motion are obtained via hamilton s equations including the zero strain conditions as well as external constraints via lagrange multipliers subsequently the lagrange multipliers associated with the internal zero strain constraints are eliminated by use of a set of orthogonality conditions between the generalized displacements and the momentum vector leaving a set of differential equations without additional algebraic constraints on the base vectors a discretized form of the equations of motion is obtained by starting from a finite time increment of the hamiltonian and retracing the steps of the continuous formulation in discrete form in terms of increments and mean values over each integration time increment in this discrete form the lagrange multipliers are given in terms of a representative value within the integration time interval and the equations of motion are recast into a conservative mean value and finite difference format the lagrange multipliers are eliminated explicitly within each integration interval leaving a projection operator expressed in terms of displacement component mean values hereby the number of variables is reduced by six for each rigid body in the problem and the difference equations lead to conservation of the orthonormality conditions for the local base vectors examples demonstrate the efficiency and accuracy of the procedure

an alternate proof for the equation of motion for constrained mechanical systems the newly developed equation of motion for constrained mechanical systems are derived using d alembert s principle