Dynamic equations of controlled Mechanica l system with redundant holonomic constraints Текст научной статьи по специальности «Математика»

УДК 519.711.3, 531

R. G. Mukharlyamov, C. T. Deressa

DYNAMIC EQUATIONS OF CONTROLLED MECHANICA L SYSTEM WITH REDUNDANT HOLONOMIC CONSTRAINTS

Ключевые слова: Обобщенная обратная матрица, голономная связь, положительно определённая матрица, знакопостоянная положительная матрица, ранг матрицы, избыточная связь, сингулярная матрица, разложение по сингулярным числам

матрицы.

Dynamic equations of motion for mechanical .systems with regard to Lagrange's equation of motion are discussed. The case of redundant constraints and positive semidefinite mass matrix in obtaining the Lagrangian multipliers and the constraint forces corresponding to a constraint equation based on a generalized inverse of a rectangular matrix and Singular Value Decomposition (SV D) is detailed. Different ways of applying the generalized inverse of a matrix in relation to obtaining an acceleration of a mechanical system on which a holonomic constraint is imposed is investigated. It is established that there could be an infinite set of solutions for an acceleration and Lagrangian multipliers of a dynamic system. The conditions that need to be satisfied for an equation of a mechanical system to have a unique solution is also spelled out and ascertained by proof. Method of obtaining the generalized inverse of a matrix is also discussed. A numerical example is given to show the application of the methods.

Key Words: Generalized Inverse, Holonomic Constraint, Positive definite matrix, Positive Semidefinite matrix, Rank Deficient Matrix, Redundant Constraints, Singular Mass matrix, Singular Value Decomposition.

Рассматривается задача построения уравнений динамики управляемой механической системы в обобщенных координатах в случае голономных связей и знакопостоянной положительной матрицы масс. Получены множители Лагранжа и реакций связи, соответствующие уравнению связей. Предлагается алгоритм, основанный на построении обобщенной обратной прямоугольной матрицы и сингулярного разложения (SVD). Исследованы различные способы использования обобщенной обратной матрицы для приведения уравнений динамики к виду, разрешенному относительно ускорений. Установлено, что решение уравнений относительно ускорений и множителей Лагранжа динамической системы не является единственным. Определены условия единственности решения. Предложен метод построения обобщенной обратной матрицы. Приводится числовой пример, иллюстрирующий применение предлагаемых методов.

1. Introduction

Modeling motion of mechanical system can be made in several equivalent ways. For instance, Lagrange multipliers method, Null space method and Magi's method are some of them [1,2,3]. In each of this methods a set of Differential-Algebraic-Equations (DAE) results. If these set of DAE of motion does not use explicitly the position and velocity equations associated to the constraints, then it leads to the problem of stability at the position and velocity level of the constraints. The strategies generally used to overcome this problem are the Coordinate Partitioning Method [2], the Baumgarte's Stabilization Method [4], the Augmented Lagrangian formulation or Mass-orthogonal projections of position and velocity vectors [5].

Moreover, in addition to the stability problems that may happen while simulating constrained mechanical systems, problems related to the presence of redundant constraints is also unavoidable in practice. In the presence of more equations than strictly needed the Jacobian matrix becomes rank deficient. This can be observed when some of the equations are dependent in the remaining ones. This makes the leading matrix, for example in Lagrangian Index one equation, singular. The leading matrix can also be singular when the mass matrix is singular. A Singular Mass matrices may appear when more than six coordinates are used to

define the position of a rigid body in R3 [2].The Jacobian matrix can also be rank deficient when the mechanical system reaches a configuration in which there is a sudden change in the number of degree of

freedom. For instance, a slider crank mechanism [5] reaches a singular configuration when both the two links are at vertical position. In this position both links overlap and the mechanism has not one but two degree of freedom that corresponds to two possible motions that the mechanism can undergo.

This paper presents a brief discussion on modeling a mechanical system with Index one Lagrange's equation of motion in the presences of redundant holonomic constraints. As the Index one Lagrange's equation of motion does not include the position and velocity constraint equations explicitly, it does not provide any solution for the constraint violation problem. Therefore, a technique useable to minimize the constraint violation errors is still required. To this end Baumgarte's method of stabilization is included in the discussion.

The main purpose of this paper is to show the application of Generalized Inverse of a matrix in finding the acceleration, the Lagrangian multipliers and as a result, the constraint forces of a mechanical system with a holonomic constraint imposed. This is discussed based on properties of Generalized Inverse of a Matrix and Singular Value Decomposition. The advantage of this method is that, it can handle singular mass matrices and redundant constraints. That is, the mass matrix can in general be assumed to be square symmetric and positive semidefinite. The positive definiteness of the mass matrix is not required in the application of this method. Redundant constraints are handled in the solution of system of equations of motion and the problems that involve singular configurations and redundant

constraints, and other problems associated with changing the number of degrees of freedom can be managed using the method developed in this paper.

2. Constructing Equation of Motion for a Holonomic Constrained Mechanical Systems

In this section we will discuss a general approach of constructing equation of motion of a mechanical system in which a holonomic constraint is involved based on the literature.

Let q = [q1,q2,...,qnfbe n generalized coordinate of a system and suppose that the system is subjected to m < n holonomic constraints given by:

0(q,t) = 0. (1)

Let L = l-(q, q) = T(q, q)- v(q) be the Lagrangian

of the system where T = T(q,q) and V = V(q) are respectively the kinetic and potential energy of the system. Then the Lagrange's equation of motion of the system can be given by [3, 6, 7]:

—---+ cp'A = QB

8q, 8q q

(2)

dt

Where QBX generalized external is force, and X is

Lagrange's multiplier. The kinetic energy of a multibody

system can be written in the form [1,2,7]:

T(q, q )4

qTM

(q) q

(3)

where M = M (q ] is the mass matrix of the system which is assumed to be symmetric and positive definite square matrix. Since L = T-V we have:

dfdO dfd(T-V))

dt

8 q

dt

8 q

Then equation (2) can be written as:

Mq + <pjA = Lq+QBX - Mq 8L

= M q + Mq

(4)

(5)

where L„ =

8

Putting Q = Lq + QBX - Mq equation (5) becomes:

Mq + cpjA = Q. (6)

The position, velocity and acceleration vectors in Equation (5) must satisfy the corresponding constraint equations:

cp (q,t)=0, (7a)

cp = cpqq + cp, = 0 (7b)

cp = cpqq + cpqq + cp, =0 (7c)

Equation (7a) - (7c) and (6) together constitutes n + m DAE of Index 3 [7] with q and X as unknowns. However, if only equations (7c) and (6) are considered, the following index 1 DAE system equivalent to an ODE system is obtained: This system is

~M (cpq)T_ "q" Q"

0 _ A G

said to be [2] Lagrange's Index one system of dynamic equations.

(8),

whereG = -cp, - cpqq

The system of differential equations (8) presents a constraint stabilization problem. As only the acceleration constraint equations have been imposed, the positions and velocities provided by the integrator suffer from a drift phenomenon. Some solutions to this problem are the Baumgarte's stabilization method [4] and the mass orthogonal projections of position and velocity vectors [2,5]. We will discuss Baumgarte's method of constraint stabilization. To secure Baumgartes stabilization of the constraint equation we

replace [4] ( in equation 7c by:

cp + a CP + Pep = 0, (9)

Where a and ¡3 are appropriately chosen constants. After replacing we obtain:

cpqq = -a (cpqq + cpt hP<p- cpqq - cp,

Substituting

G = -a (cpqq + cp, )-pcp-cpqq-cp, ,

we obtain:

<paq = G

(10)

"M (cpq)T_ "q" Q"

_<Pq 0 _ A G

Considering equations (6) and (10) we obtain the equation of the dynamic system to be:

(11)

Equation (11) constitutes m + n index one Ordinary Differential Equations, with the required values q and X .But still after Baumgarte's stabilization method is applied to the system, there could be a problem of redundant constraints and singular mass matrices. Let us see the following example that describes how singular mass matrix may happen. A Singular Mass matrices may appear when more than six coordinates are used to define the position

of a rigid body in R3 . When Euler parameters or natural coordinates are used this is always the case. With natural coordinates [2,9] the constant inertia matrix of a rigid body requires that the body be defined with two points and two unit vectors (or a similar configuration, for instance with four non-coplanar points). If this body has additional points and unit vectors, the corresponding rows and columns of the inertia matrix have null values, making this matrix positive semidefinite. In the case of redundant constraints in equation (1) the Jacobian matrix does not have full rank. In this situation we can obtain the

resultant reaction force of the constraints (j)TqX but not

[2,9,10] each of the X , ¡u =1,2,... m .Let us consider the following two cases:

1. Assuming that M is positive definite and the Jacobean Matrix (q has full rank, that is:

M = MT e Rnxn, ( e Rmxn, ran(M) = n, ran((q ) = m. Then the value of q from equation (6) can be obtained to be:

q = M-1Q-M -1cpJ A, (12)

and substituting(12) into (10) we obtain

(<p„M -1(pi)A = (pqM-1Q-G, (13)

which yields:

A = MqM> jn^qM-1Q-G]. (14)

The value of X can be obtained from (14) and then putting its value in (12) the corresponding value of q can be obtained. That is:

q = [M-1-M -1cpJ (cpqM-1cpJ -1cpqM-1] -M^ (cpqM-1(pJ -1G

2. Assume that M is positive definite and the Jacobian Matrix is rank Deficient.

As it is discussed above, redundant constraints in equation (1) can be reflected by the fact that some of the equations are dependent in the remaining ones. This lack of equation independence in the system (1) may

lead to a rank deficiency in the Jacobian matrix ( q and

an over constrained system of linear equations (more equations than unknowns) which will not have a solution that satisfy all the equations. In such situations, suppose we need to find each of the Lagrangian Multipliers Xp, ¡U =1,2,... m andthen each of the constraint forces of a mechanical system.

(15)

That is, assuming that

< er-

ran ((pJ )= r < m and for any convenient vector, let

<A = H

The general solution, by the Minimum Norm Solution method using Generalized Inverse of a matrix, of[8]:

<PjA = H, (16)

is given by:

a= (cpJ )h + [- (< j (< h, (17)

where I is n by n identity matrix and 7] e Rnx1 is arbitrary Lagrangian multiplier vector, I - ((pJ j ((pJ ] represents the orthogonal projection

matrix in the null space of (( .That is Equation (17) can be decomposed [7,8] as:

a = a0+nr| (18)

where A0 = ((pl JH is the minimum norm solution

that minimizes

cplA-H and N = 1- (c^q ) (cp

t > q >

is a vector in the kernel of (f)Tg .

With regard to (17) we have the following two cases:

a) If B = $ has a full rank then, b+ = bt (bbt -and hence equation (17) reduces to a = a0 = b+h = bt (bbt -1h, since I - BJB = 0. Therefore in this case we have a unique solution a = a0 which is called the pseudoinverse solution [8].Note that, if B is a square matrix and has a full rank then BJ = B-1 and in this case the nullspace of (( contains only the zero vector.

b) If B is rank deficient we apply the method of

in which case Bjcan be calculated as b = usvt where U and V are n by n and m by m square orthogonal matrices respectively and S has the same size as B and is n by m matrix. The non-square matrix S :has non-zero elements only on its diagonal and therefore, the calculation of its generalized inverse

SJ is trivial [8]. Note that, svd is based on a theorem from linear algebra which says that a rectangular matrix B can be broken down into the product of three matrices - an orthogonal matrix U , a diagonal matrix S , and the transpose of an orthogonal matrix V . The theorem is usually presented something like this: Bmn = UmmSVT ,where

mn mm mn nn '

UTU = I ;VTV = I the columns of U are orthonormal eigenvectors of BBT , the columns of V

are orthonormal eigenvectors of BBT , and S is a diagonal matrix containing the square roots of eigenvalues from U or V in descending order. The generalized inverse of B is [7] b+ = vs ut where the relation uT = u-1; vt = v- valid for orthogonal matrices have been used and:

S- 0

S).„ =

0 0

The sub-matrix ^ contains the reciprocal of the non-zero singular values along the principal diagonal. In this case, B is rank deficient, then equation (16) has an infinite number of solutions given by equation (17). In summary:

i).When we apply the method discussed above to Equation (13) assuming that (( is rank deficient we obtain X to be:

A= (cpqM-1 (cpq J ) (cpqM-1q-g>[i- (<Pqm-1Q-G) ] where r) is an arbitrary Lagrangian multiplier. By pressing on r we can obtain infinite solutions for X .Substituting ) = 0 we obtain the minimum norm solution:

S is rectangular diagonal matrix, which is an m-by-n matrix with only the entries of the form di t possibly non-zero.

A= (<pqM-1 (cpq J + (cpqM-1Q-G) that minimizes ((pqM-1(pJ ) A - [(pqM-1Q - G ^

ii).We can obtain q from equation (15) as follows. When we closely look at equation (15) we observe that (<PqM-1 (<Pq j - should exist for the values of q be

obtained. The existence may fail in case the Jacobian matrix is rank deficient. In this instance one method to obtain q is to make use of the assumption that M is positive definite that grantees the diagonalizability of M.

Let the mass matrix M resulting from unconstrained mechanical system be positive definite and the acceleration of the unconstrained system be denoted by

q . Then we have q c = M-1Q and referring to

equation (15) we have:

q = qnc + M VJ (cpqM-1cp; - (G - cpqqnc) (19)

From the fact that M is positive definite we obtain

M = M1/2M1/2 and hence M-1 = M-1/2M-1/2 . Then equation (19) becomes:

q = qnc +M-1/2M-1/2cpl -1(G-(Pqqnc)(20)

Putting f = (pqm equation (20) becomes

q = qnc+M-1/2FT (FFT /g-cp^T) (21) From the properties of generalized inverse of a matrix we have FT (FFT - = F+. Hence equation (21) becomes:

q = qnc + M-1/2F+(G-(Pqqnc) (22)

It can be seen in (22) that the values of q is independent of whether the coefficient matrix in equation (8) or (11) is rank deficient or not. Moreover,

F + can be calculated as explained above using SVD method.

The next point will be on how to obtain M 12: Since M is assumed to be symmetric and positive definite [8] it has a unique square root B , such that M = BB. In order to find matrix B first we need to diagonalize matrix M . That is, we need to find a matrix K which consists of the orthonormal eigenvectors of M and a diagonal matrix D with its diagonal elements the corresponding eigenvectors of M such that M = KDKT .

Next, since all the eigenvalues of M are positive we can write d=dv2d1/2 where

D1/2

is obtained from D by replacing all the diagonal elements with its square

root. Finally we calculate b = kd1/2kt to obtain the

square root of M. [Using MAT LAB, B = sqrtm (M ].

Indeed, b.b = K d1/2 (KTK )d1/2Kt =

= KD1/2ID1/2Kt = KDKT = M .The inverse of the square

root of M, denoted by M 12 is obtained to be:

¡-I)

m 1/2 = b 1 = kd 1/2kT, Where for orthonormal

matrix K, K 1 = KT is used.

iii).For the case of positive semidefinite mass matrix,

from Equation (6) we obtain:

q = M+(Q-cpqTA )+(l-M+M )q0,

where q is an arbitrary acceleration vector of the

system. As in the above choosing q0 = 0 gives the

minimum norm solution for the acceleration of the system to be:

q^+(Q- Tq.

It is to be noted that the resultant constraint force (TqX can always be obtained irrespective of the rank of matrix ((.

3. Acceleration and Lagrangian Multipliers of Mechanical System in the case of Redundant Constraints and Positive Semidefinite Mass Matrices

We can also apply the properties of generalized inverse of a matrix to obtain the acceleration and Lagrange's multipliers, at the same time, from equation of a constrained mechanical system given by (8) or (11).

Let as assume the mass matrix M, resulting from the unconstrained mechanical system is symmetric positive semidefinite nxn square matrix.

We dropped the assumption that M is positive

definite. M is in general considered to be positive semidefinite. Consider equation (11) and let:

(23)

Then equation (11) can be written in the form:

AX=Y (24)

With the same method used above the General Solution of equation (24) becomes:

X = A+Y + [l-A+Ajj (25) where I is an Identity matrix of (m+n) x(m+n) and ¡1 is an arbitrary vector of size (m+n) x1.The Minimum Norm solution that minimizes (AX-^iS X = A+Y. The main advantage of this

formulation lies in the fact that A+ ,the generalized inverse of A, always exists provided that Y is in the range of AX .In other words, the

formulation is applicable even if the mass matrix M is singular and the Jacobian matrix ((is rank deficient. Let us write this result as follows.

Result 1:

The General Solution of equation of motion of a constrained mechanical system described by equation (11), whether the matrix M that arises in the unconstrained system is singular, whether the constraint is redundant is given by:

"M < ,X = ,Y = Q"

0 A g

"q" "M < + "Q + f I - "M < + "M N

0 G_ V 0 _<Pq 0_ /

M

(26)

It is clear that, because of the arbitrary vector / in equation (26) the solution of q and 1 is not necessarily unique. However, if A in equation (23) has a full rank then A+A = 1 and hence the solution becomes unique.

We can state this result as follows: Result 2:

IVI

When the matrix A =

m <p;

Pa 0

has a full rank then

"q" "m < + q" "m < -1 q"

a 0 g" 0 g_

to have a full rank so that we can

the General Solution of the mechanical system given by equation (11) becomes unique and is given by:

(27)

The next logical question could be what are the necessary and sufficient conditions for

"m cp; _<pq o

have a unique solution? Let us investigate it as follows:

a).Assume that M is symmetric positive semidefinite and (q has full rank. If M is positive definite on the

kernel of (q (i.e. ( x = 0 and x ^ 0 implies that

xTMx > 0), then A is nonsingular.

Proof: We show that Au ^ 0 for u ^ 0. Indeed let

u = [x, yf, ((x = 0, x ^ 0. Then Au = 0 implies:

[Mx + cp^y = 0 I pax = 0

(28)

It then follows that XTMx + X^y = 0 yielding

XTMX = 0 since X^y = (p^xy = 0 . From

XTMX = 0 we obtain X = 0 since XTMx > 0 for (pqX = 0 and X ^ 0. Substituting x = 0 and nothing

that (has a full rank one can obtain from equation

(28) that y = 0 .Hence we showed that

Au = 0 implies that u = 0. This shows that A is

nonsingular. (Sufficient condition for A to be nonsingular)

b).On the other hand suppose A is nonsingular, ( x = 0 and x ^ 0.

Proof. We want to show thatXTMX^0 . On the contrary let = then since M is positive

semidefinite we obtain = yielding

X e kernel (M ; letting U = [x,0f and nothing that X ^ 0 we can see that Au = 0 for a non zero vector

u. This contradicts the supposition that A is

nonsingular. (Necessary condition)

The result obtained from a) and b) can be written as

follows:

Result 3:

Let M be symmetric positive semidefinite and (q has

full rank. Matrix A =

M < Pc 0

has a full rank if and

only if M is positive definite on the kernel of (f)q .

Remark:

1. In Result 3 above the condition that M is positive definite on the kernel of (can be relaxed to "the mass

matrix is definite on the kernel of (q ". This is because

in the proof we used XTMx only. If M is indefinite the following example shows that A is singular even though the Jacobian matrix (has a full rank.

f1 0 -fl

A = 0 -1 1

1-1 1 0

f M

°J

From the above matrix it can be seen that the Jacobiam

matrix has a full rank and M is indefinite but A is singular.

2. In equation (26) the Minimum Norm Solution is always unique and is given by

"m < + q"

a 0 g

obtained by putting / = 0 . Example:

This problem is adapted from the exercises given in chapter 2 of [11].A uniform hoop of mass m and radius r rolls without slipping on a fixed cylinder of radius R as shown in figure 1. The only external force is that of gravity. If the smaller cylinder starts rolling from rest on top of the bigger cylinder, find the acceleration and each of the constraint forces before the hoop falls off the cylinder.

Fig. 1 Solution:

Two equations of constraints:

p = r + R, r (0-0)=Re.

The generalized coordinates are 0 and 0. The first equation is from the fact that as long as the hoop is touching the cylinder the center of mass of the hoop is exactly R + r away from the center of the cylinder. The second one comes from no slipping:

r0- (r + R) 0 = 0.

Where 0 is the angle r makes with the vertical and 0 is the angle p makes with the vertical. The kinetic energy is the sum of the kinetic energy of the center of mass of the hoop and the kinetic energy of the hoop about the cylinder given by:

The potential energy is the height above the center of the cylinder and is given by:

V = mgpgpcos .

The Lagrangian is given by L = T - V , and from the Lagrange's

dtl 5Q,

equations: —f^i j - — = 0 and

59

A f'^L'L.^o we obtain:

dtl 50J 50

fmp2

0

0 mr2

V Q j

v Q )

f-mgpgpsij 0

(29)

The constraint equations at the acceleration level are given by: 0

(0 oY Q j fOj

v Q )

v0y

(30)

Now from equation (13) and using (29) and (30) we obtain:

fo o Y1 fo j

(31)

gsin0

v0 2/rriy

Note that I ® ^ I is a singular matrix. Now using the

Vo 2/mJ

method developed on equation (17) we have the General Solution of the Lagrangian multipliers given by:

fK..... 1 +

VA2)

(

Where I I is arbitrary non-zero vector. It can easily

V^J

be shown that

0 0 YfO 0 2/mJf-gsin0

1 oj fo o Yfo o 0 lj~{ 0 Zmllo 2/m

(32)

f0 0 j o 2/m

f 1 Yo o j

(2/m)2

0 2/m

V" -----J VV-'"7 J\~ -----J

Hence the General Solution becomes:

f1 f

f0 0 j o m/2

VA2)

V

Mi

(sine)

(33)

Note that: The Minimum Norm Solution is unique and is given by = j. On the other hand

combining the equations of unconstrained mechanical system, Equation (29), the constraint equation (30) and using equation (11) ,(In fact in this example no constraint stabilization method is used) together we obtain the equation of the constrained system as:

f —2 0 0 -(R + r)jf Q j f-mnr,ns

2

mp

0 mrz 0 r

o o o o

V-(R + r) r 0 0

From which the General Solution of the system becomes

0

mgpgpsi 0 0 0

f Q j

r

tet(4)+

mp 0 0

-(R + r) f mp2

o o

0

mr2 0 r

0 -pj+f-mgpgpsij

0

mr2

o

r

r 0 0 ^

0 -p 0 r 0 0 0 0

0 +

0

0 J

mp2 o 0

o mr2 0

o o 0

-(R + r)

l-(R + r)

Numerically, let us suppose that /?=1m, /=0.2m, m=2kg, g=9.8m/s2, r = (rx,T2,T3,T4) is arbitrary non-ze/o vector then

{Q j '-4.O831sin0j

0 -4.0831sin9

K r

VA2J v 9.8008sin9 y

From which it can be seen that:

9.8008sin9

where T3 is an arbitrary constant.

One can verify that the values we obtained for the Lagrangian multipliers here and in (33) are the same for

m=2kg

1. Each of the constraint forces is given by:

0 -pYMfo -1.2Y r3 0 r J{A2J {0 0.2 j{9.8008sin6y -117609s^j 1.96O2sin0

2. The acceleration of the system is given by: fQj = f - 4.O831sin0 j

v0 j = V-24.5008sineJ

From which it can be solved that: 9 (t^^^lsine + C1 + C2and0 (t)=

= -24.50o8t2sne + C,

where Cx, C2 and C are arbitrary constants that can be determined based on initial conditions. If the hoop starts from rest then 0 = 0, 0 = 0 From these initial conditions we obtain:

+

e (t)= 4.0831 (sin9-t£nd0 (t)= -24.5008t2sin9

4. Conclusions

The application of the methods we used here can equivalently be applied to other methods of modeling mechanical systems mentioned in the introductory part of this paper including nonholonomic constraint systems. It must also be noted that, even though we may come up with infinite number of equations of motion and infinite number of Lagrangian multipliers of a mechanical system, in all practical purposes, we make use of the minimum norm solutions which is always unique. The calculation of generalized inverse of a matrix, especially with one or more variable entries seems to be expensive, but if all the entries are scalars, obtaining the generalized inverse is not that expensive. In the second case one can also use MATLAB and other software.

Pабота выполнена при финансовой поддержке РФФИ, проект 10-08-00535.

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© R. G. Mukharlyamov - Professor. Doctor of Physical-Mathematical Sciences. Head department of Theoretical Mechanics. Peoples' Friendship University of Russia, rmuharliamov@sci.pfu.edu.ru; C. T. Deressa - Lecturer at Department of Mathematics. Jimma University. Jimma. Ethiopia. chernettuge@ymail.com.

© Р. Г. Мухарлямов - д-р физ.-мат. наук, проф. каф. теоретической механики Российского университета дружбы народов, rmuharliamov@sci.pfu.edu.ru; С. Т. Дересса - доцент каф. математики Jimma University. Jimma. Ethiopia. chernettuge@ymail.com.