joint reaction forces in multibody systems

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Multibody System Dynamics (2005) 14: 2346 C Springer 2005

Joint Reaction Forces in Multibody Systems

with Redundant Constraints

MAREK WOJTYRAWarsaw University of Technology, Institute of Aeronautics and Applied Mechanics, Nowowiejska

24, 00-665 Warsaw, Poland; E-mail: [email protected]

(Received 7 July 2003; accepted in revised form 29 September 2004)Abstract. Redundant constraints are defined as those constraints which can be removed without

changing the kinematics of the mechanism. They are usually eliminated from the mathematical model

of a multibody system. For a given mechanism the set of redundant constraints can be chosen in many

ways. Rigid body systems with redundant constraints do not have a unique solution to the problem of

joint reaction forces calculation. If redundant constraints are present in the mechanical system, then

thesystem is statically undetermined. If in the case of dynamics problem the constraints areconsistent,

all of them are frictionless and we are interested only in positions, velocities and accelerations of the

bodies, then the calculation of joint reaction forces is not necessary. In many cases, however, e.g. when

we want to take into accountfrictionin joints, thecalculation of joint reaction forcescannot be avoided.

In some rigid body systems, despite the redundant constraints existence, reaction forces in selected

joints can be uniquely determined. The paper presents three methods of finding the constraints for

which reaction forces can be uniquely determined using rigid body model. Three different techniques

of Jacobian matrix analysis are used.Keywords: redundant constraints, joint reaction force.

1. Introduction

In many mechanical systems the observed number of degrees of freedom (DOF)

is greater than the number of DOF calculated using Gruebler structural equation

(DOF = 6 [number of bodies] [number of constraints]). This happens when

some joints are constraining the same degrees of freedom as other joints. In such a

case redundant constraints are present in the multibody system [13].

Redundant constraints are usually defined as those constraints which can be

removed without changing the kinematics of the mechanism. After removing them,

their role in the kinematics of the system is played by the remaining constraints. Theredundant constraints are not necessary from the kinematics point of view, however

they are often introduced for constructional reasons. A door supported by three or

four hinges is a simple example of a system with redundant constraints. From the

kinematical point of view a single hinge removing five degrees of freedom would

be enough. Multiple hinges are usually used to strengthen the construction.

Redundant constraints are usually removed from the mathematical model of

a multibody system. In majority of cases the elimination is done explicitly [15],

sometimes it is assumed that there is no redundant constraints in the model [6]. For a


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24 MAREK WOJTYRA

given mechanism the set of redundant constraints can be chosen in many ways. The

choice of redundant constraints, which are removed from the mathematical model,

does not affect the results of kinematical analysis (however the computational

problem can be better or worse conditioned due to this choice). But in the case of a

kinetostatic or dynamic analysis the obtained results can be different for different

sets of removed redundant constraints.

Two bodies that are forming a kinematical pair are acting on each other by a

reaction force and torque. The same kinematics can be obtained for different sets

of constraints, so the same motion of a mechanism can be obtained for different

reaction forces and torques. Thus, rigid body systems with redundant constraints

do not have a unique solution to the problem of joint reaction forces calculation

[7]. If redundant constraints are present in the mechanical system, then the systemis statically undetermined the number of equilibrium equations is less than the

number of unknown joint reaction forces.

If in the case of dynamics problem the constraints are consistent, all of them are

frictionless and we are interested only in positions, velocities and accelerations of

the bodies, then the calculation of joint reaction forces is not necessary. In many

cases, however, e.g. when we want to take into account friction in joints, we cannot

avoid the calculation of joint reaction forces.

In order to find a unique set of all joint reaction forces in an overconstrained

system it is necessary to reject the assumption that all bodies are rigid. The analysis

of multibody system with deformable bodies needs more input data and much more

computation. So the process becomes more difficult and less effective.It should be noted that in some purely rigid body multibody systems, despite

the redundant constraints existence, the reaction forces in selected joints could

be uniquely determined. For example, let us consider two simple mechanisms

presented in Figure 1: a parallelogram with an additional link and a pendulum.

The parallelogram is a system with redundant constraints. The pendulum is a sys-

tem without redundant constraints. Both the mechanisms together can be formally

treated as one system with redundant constraints. It is obvious however, that the

reaction forces in the pendulums joint can still be uniquely determined.

The decision which reaction forces can be uniquely determined is not so straight-

forward in the case of more complicated multibody systems.When we are interested

in selected reactions only, the information whether we can determine them using

a rigid body model can be useful. The paper addresses the problem of finding theconstraints for which reaction forces can be uniquely determined.

Figure 1. Parallelogram and pendulum.


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JOINT REACTION FORCES IN MULTIBODY SYSTEMS WITH REDUNDANT CONSTRAINTS 25

Usually multibody software analyses the Jacobian matrix of constraint equa-

tions, subjectively determines which constraints are redundant and deletes them

from the set of equations, however, it does not provide us with the information

which constraint equations are independent upon any other equation in the original

set. The paper shows that reaction forces corresponding to the independent con-

straint equations can be determined uniquely. It should be emphasised that the fact

that we can or we cannot uniquely determine reaction in a selected joint depends

only upon the structure of the multibody system. The choice of coordinates in

mathematical model does not change this fact. The paper presents three methods of

finding the constraints for which reaction forces can be uniquely determined using

rigid body model. Three different techniques of Jacobian matrix analysis are used.

2. Constraint Equations and Constraint Reaction Forces

In our further consideration we assume that constraints are holonomic, consistent

and the multibody system position is non-singular. We also assume that absolute

coordinates [1, 2] are used to describe the multibody system. It is worth noting that

obtained results are valid for any system of coordinates.

A kinematic pair (joint) imposes certain conditions on the relative motion be-

tween the two bodies it comprises. Let q be the vector of coordinates of a multibody

system. The conditions on the relative motion between bodies imposed by the ith

pair can be expressed analytically as constraint equation:

i (q) = 0. (1)

This vector equation is equivalent to a set of scalar equations.

Constraint equations for all kinematic pairs can be treated jointly as one vector

equation. If the system is described by n coordinates (in the case of absolute coor-

dinates n = 3k for planar and n = 6k for spatial mechanisms, respectively, while

k is the number of bodies) and all kinematic pairs can be expressed by m scalar

equations, then we obtain:

(q) =1(q)

2(q)...

m(q)

=1(q1, q2, . . . ,qn)

2(q1, q2, . . . ,qn)...

m(q1, q2, . . . ,qn)

=

0

0...

0

= 0. (2)

In a multibody system some of the constraints can be redundant, i.e. some

kinematic pairs may repeat constraints imposed by other pairs. In a mathematical

model, redundant constraints arepresent in thesystem if some of thescalarequations

from the set (2) are dependent. If an equation from the set (2) is dependent, then it

is automatically fulfilled when the other equations are fulfilled.


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26 MAREK WOJTYRA

Letq denote the Jacobian matrix of constraint equations

q(q) =

1q1

1q2

1qn

2q1

2q2

2qn

......

...

mq1

mq2

mqn

=

(1)q

(2)q...

(m)q

. (3)

To check if all of the constraint equations are independent it is sufficient to check

the rank of the Jacobian matrix. If the rank of the Jacobian is equal to the number

of scalar equations, then constraint equations are independent; otherwise they aredependent [8].

Thus, if redundant constraints are present, then the rank of the Jacobian matrix

is less then m. That means that at least one of the rows ofq can be expressed as a

linear combination of other rows. If for example equation m is not independent,

then the row of the Jacobian matrix corresponding to this equation can be expressed

as:

(m)q = 1(1)q + 2(2)q + + m1(m1)q, (4)

where1, . . . , m1 are constant coefficients. At this point of consideration it is not

important how to calculate the coefficients in practice. One of the possible methods

is presented in Section 4.

Generalised forces and torques of constraint reaction can be calculated as follows

[1, 2]:

fn1 = (q)Tm1, (5)

where f is a generalised reaction, (q)T is the transposed Jacobian of constraint

equations, and is a vector of Lagrange multipliers.

For further considerations it is convenient to rewrite (5) in the following form:

fT = Tq. (6)

We can see that the generalised constraint reaction force can be expressed as a

linear combination of rows of the Jacobian matrix.

Let us consider the case of a multibody system with redundant constraints. Let

us assume that the last row of the Jacobian matrix can be expressed as a linear

combination (4) of the other rows. For an arbitrarily chosen scalar the following

equation is fulfilled:

m = + (m ). (7)


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By substituting Equation (4) to (6) and taking into account (7) we obtain:

fT = Tq

= 1(1)q + 2(2)q + + m1(m1)q + ( + m )(m )q

= 1(1)q + 2(2)q + + (1(1)q + 2(2)q

+ + m1(m1)q) + (m )(m)q (8)

= (1 + 1)(1)q + (2 + 2)(2)q +

+ (m1 + m1)(m1)q + (m )(m)q.

For different values of we obtain different coefficients of linear combinationof the Jacobian matrixs rows. These coefficients (Lagrange multipliers) determine

how the resultant load is divided between constraint reactions. In Equation (8) we

can choose the value of arbitrarily, so the vector of Lagrange multipliers cannot

be uniquely determined. In other words, if redundant constraints exist in the system,

then the problem of determining of all of the constraint reactions has an infinite

number of solutions. It will be shown however, that some of the constraint reactions

can be determined uniquely.

3. Elimination of Redundant Constraints and Results of This Procedure

In numerical calculations redundant constraints are usually removed from the math-ematical model of the multibody system. If the NewtonRaphson method is em-

ployed in the kinematic problem, the elimination of redundant constraints is nec-

essary to obtain a non-singular matrix of a linear equation set. In the dynamics

problem, the elimination of redundant constrains enables us to uniquely determine

Lagrange multipliers (corresponding to the remaining non-redundant constraints).

This is needed to obtain the numerical solution.

Redundant constraints cannot be uniquely identified. In the simplest case if the

role of constraint A can be played by the constraint B, then the role of constraint B

can also be played by constraint A. That means, that the set of redundant constraints

can be chosen in many ways.

Redundant constraints are detected by checking the rank of the Jacobian matrix.

Usually the selection of redundant constraints that are to be removed is based onGaussian elimination or one of its modifications. The elimination process is a purely

mathematical operation; the real mechanism remains unchanged. Let us see how

this procedure influences the obtained results of calculations.

Let us assume that after numerical analysis of the Jacobian matrix, the constraint

m was selected to be eliminated as a redundant one. Consequently, the term with

(m)q will not be present in Equation (8). Thus, the elimination of equation m is

equivalent to such a choice of parameter value,thatthetermwith(m )q disappears

from Equation (8), namely = m. In doing so we are arbitrarily choosing one of


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28 MAREK WOJTYRA

an infinite number of possibilities. By eliminating constraint m we are assuming

that the corresponding reaction equals zero. According to Equation (8), the choice

of = m influences also the reactions of the other constraints. Of course, there

is no reason to believe that in the real mechanism reaction of constraint m equals

zero.

In certain multibody systems, some of the reaction forces can be determined

uniquely, despite the existence of redundant constraints. Let us assume once again

that the constraint m was selected to be eliminated. Let us also assume that for

one of the other constraints, namely s , the coefficient s in Equation (4) equals

zero. In this case the choice of value does not influence the value of the coefficient

corresponding to the term (s )q in Equation (8). Thus, elimination ofm constraint

does not affect reaction corresponding to constraint s .Redundant constraints can be chosen in many ways. Instead of constraint m

any other constraint i can be eliminated, provided that in Equation (4) coefficient

i for this constraint is non-zero. It does not change the fact, that ifs equals zero,

then in Equation (8) the coefficient corresponding to (s )q is always the same.

Equations of type (4) can be written for all redundant constraints. If in all these

equations coefficients s corresponding to (s )q are equal to zero, then reaction

corresponding to constraint s can be determined uniquely. All coefficients sequal zero only if the (s )q row of the Jacobian matrix is not a linear combination

of the other rows. In such a case the constraint equation s is independent.

Results of above considerations can be concluded as follows: if specific con-

straints imposed on a multibody system are independent, that is if their role cannotbe played by the other constraints, then the reactions corresponding to these con-

straints can be determined uniquely, despite the existence of redundant constraints

in the system as a whole.

In numerical calculations detection of redundant constraints is based on Jacobian

matrix analysis. Usually the information which constraint reactions can be deter-

mined uniquely is not obtained. The methods of Jacobian matrix analysis can be

supplemented with procedures enabling to identify such constraints.

4. Identification of Constraints for Which Reaction Forces

Can be Determined Uniquely

4.1. METHOD A

Let us consider the Jacobian matrix q, given by Equation (3). Let us assume that

the rank of the Jacobian equals r(r m). Let \iq be the matrix q with crossed

out ith row:

\iq =

(1)

Tq (i 1)

Tq (i +1)

Tq (m)

Tq

T. (9)

If the ith row of the Jacobian matrix can be expressed as a linear combination

of other rows, then crossing it out does not change the rank of the matrix. In such


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a case rank\iq equals r. On the other hand, ifith row ofq is independent, then

rank of\iq matrix equals r 1.

It is possible to identify independent constraints by comparing the rank of the

Jacobian matrix q with ranks of matrices\iq , for i = 1, . . . ,m. Reaction forces

corresponding to detected independent constraints can be uniquely determined.

The described method is extremely simple, but requires a lot of numerical com-

putations. There is a possibility of using more efficient methods.

4.2. METHOD B

Let us assume that among m constraints (m r) were identified as redundant and

selected to be eliminated from the mathematical model. It is not important whichprocedure of redundant constraint identification was used. The important fact is that

rows of the Jacobian matrixcorrespondingto redundant constraints can be expressed

as linear combinations of other rows. Constraint equations can be reordered in such

a way that Jacobian matrix can be divided into two submatrices: Rq corresponding

to redundant constraints and Nq corresponding to other constraints:

q =

Nqrn

Rq(mr)n

mn

. (10)

The rankrof the Jacobian matrixq equals the rank of matrix Nq . Matrix

Nq

has full row rank. Each row of matrixRq can be expressed as a linear combinationof rows ofNq , thus we can write:

Rq = (mr)r

Nq . (11)

Equation (11) is a generalisation of Equation (4) for (m r) redundant con-

straints. Postmultiplying (11) with (Nq )T we obtain:

Rq

Nq

T= (mr)r

Nq

Nq

T. (12)

The matrix [Nq (Nq )

T]rr is invertible (since rank(Nq ) = r and rank(A) =

rank(AAT) for any matrix A), so can be easily calculated from Equation (12):

(mr)r = Rq

Nq

T

Nq

Nq

T1. (13)

As was stated in Section 3, if for a certain constraint equation (that remained in

the model after redundant constraint elimination) all coefficients in Equation (4) are

equal to zero, then corresponding reaction can be determined uniquely. In the case

of such a constraint, the appropriate column of matrix in Equation (11) consists

of zeros only.


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30 MAREK WOJTYRA

The method of identification of constraints for which reaction forces can be

determined uniquely consists of four steps. Firstly, redundant constraint equations

are being detected (this operation is always performed in multibody simulational

packages anyway). Then the Jacobian matrix is divided into submatrices Rq and

Nq . In the next step matrix is calculated using Equation (13). Finally, zero

columns are sought for in the matrix .

It is worth noting that in our considerations the Jacobian matrix elements are

treated as dimensionless. Usually some of the Jacobian elements refer to linear

quantities, whereas the others refer to angular quantities. As a result in operation

Nq (

Nq )

T we add components of different dimension. The physically consistent

formulation should be Nq M(q)(Nq )

T, where M(q) is a generalized mass matrix,

serving as a metric tensor [9, 10]. The reason why we are not using this formulation

is that our redundant constraint analysis is purely geometrical, and we are not

referring to masses of bodies. There is no point in introducing masses just to obtain

a different formulation. Neglecting the dimensions is a common practice [3].

4.3. METHOD C

The Jacobian matrix singular value decomposition can be computed [11] as

qmn = UmmmnVTnn. (14)

Matrices U and V are orthonormal and matrix is diagonal:

mn =

1 0

. . .

0 r

0

0 0

mn

, (15)

with 1 2 r > 0 and r = rank(q).

Let us consider Equation (6). Substituting (14) into (6) and postmultiplying with

V, we obtain a form equivalent to (6):

fTV = T, (16)

where:

T = TU. (17)

Only the first rdiagonal elements of are non-zero, thus Equation (16) uniquely

defines only the first r elements of vector . The remaining elements can be


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arbitrarily chosen, so vector that satisfies Equation (16) can be written in the

following form:

= + ,

= [1 r 0 0]T, = [ 0 0 r+1 m ]

T,(18)

where values 1, . . . ,r are uniquely defined by (16), and values r+1, . . . , m can

be arbitrarily chosen.

Postmultiplying (18) with UT and transposing the result we obtain:

= U = U + U. (19)

Equation (19) shows, that Lagrange multipliers vector depends on vector

, whose non-zero elements can be arbitrarily chosen. Thus, if r < m Lagrange

multipliers cannot be determined uniquely. This conclusion is consistent with con-

siderations presented in Section 3.

If in the s-th row of matrix U, elements from r + 1 to m are equal to zero, then

the Lagrange multiplier s always has the same value, for any vector :

s =[ Us1 Usr 0 0 ]

1...

r

0...

0

+

0

...

0

r+1...

m

=Us11 + + Usrr. (20)

The above conclusion can form the basis for a method of detecting those con-

straints for which reaction forces can be uniquely determined. In this method firstly

the SVD decomposition of the Jacobian matrix q is calculated and then appropri-

ate zero fragments of matrix U are sought.

5. Joint Reaction ForcesIn Sections 3 and 4 we were considering a reaction force corresponding to a scalar

constraint equation. Quite often a kinematic joint is described by more than one

scalar constraint equation. In a majority of cases we are interested in a unique

determination of all reaction forces acting in a joint, not only in selected components

corresponding to scalar equations of constraints.

In the previous sections we were checking if reactions corresponding to scalar

constraint equations could be determined uniquely. Let us now consider a slightly

different question: Is it possible to determine uniquely forces acting between two


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32 MAREK WOJTYRA

bodies, i.e. the resultant reaction in the kinematic pair? The difference between

these two problems can be explained using a simple example. Let us consider

a pendulum, forming with its basis a revolute joint, which removes five degrees

of freedom. Let us assume that a mathematical model of this joint consists of

more than five scalar constraint equations (for example the same constraints were

imposed twice). Unique determination of all reactions corresponding to constraints

is of course impossible. It is also obvious, that resultant force acting between the

pendulum and the basis can be uniquely determined.

Let us assume that each constraint equation belongs to exactly one kinematical

pair. In such a case constraint equations can be divided into separate sets i (q) =

[i1(q) ifi

(q)]T for all the joints. A generalised constraint reaction force,

given by (6), can be treated as a sum of reactions corresponding to kinematic pairs.If a multibody system consists ofp kinematic pairs and each pair is described by fiscalar constraint equations, then Equation (6) can be rewritten in the following form:

fT = Tq =

pi =1

(fi )T =

pi =1

(i )Tiq =

pi =1

fi

j =1

ijij

q

. (21)

If there are no redundant constraints in the multibody system, then each joint

reaction fi can be uniquely determined. The situation may change if redundant

constraints are present. Let us assume that, as a result of redundant constraints

existence, the row (pfp

)q of the Jacobian matrix can be expressed as a linear

combination of the other rows. Let i denote the coefficients of linear combinationcorresponding to joint p, and let ij denote the coefficients corresponding to other

joints. We obtain

pfp

q

=

p1i =1

fi

j =1

ijij

q

+

fp 1j =1

j

pj

q. (22)

For an arbitrarily chosen scalar value , Lagrange multiplier pfp

can be written

in the following form:

p

fp= + p

fp . (23)

Substituting (22) and (23) into (21), we obtain after some rearrangements:

fT = (fp)T +

p1i =1

(fi )T =

fp 1j =1

pj + j

pj

q

+

pfp

pfp

q

+

p1i =1

fi

j =1

ij +

ij

ij

q

(24)


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Similarly to the results from Sections 2 and 3, Equation (24) shows that the

choice of the parameter value influences the values of joint reaction forces.

Let us now consider a special case, when all theij coefficients are equal to zero.

In such a case, we can notice that joint reaction forces fi (for i = 1, . . . , p 1)

remain the same for any value of the parameter. As a result, since the generalized

constraint reaction f does not change, the resultant reaction fp in pth joint

must remain the same as well, for any value of the parameter (although this

resultant reaction cannot be uniquely distributed between the scalar constraints

reactions).

Our considerations show, that if constraint pfp

depends only on other con-

straints representing the same joint, then the resultant reaction fp in this joint can

be uniquely determined. Otherwise, if at least one non-zero coefficient ij exists,then a resultant joint reaction cannot be uniquely determined. This conclusion can

be generalised for more than one redundant constraint equation: if all scalar con-

straint equations describing a kinematic pair are independent, or if they depend

only on other equations describing the same pair, then the resultant joint reaction

force can be uniquely determined.

Methods of Jacobian matrix analysis can be supplemented with procedures

enabling for identification of such joints.

6. Identification of Joints for Which Resultant Reaction Forces

Can be Determined Uniquely

6.1. METHOD A+

Let us consider the Jacobian matrix q, given by Equation (3). Let us assume

that the rank of the Jacobian equals r (r m). Let iq be the matrix q with

crossed-out rows corresponding to ith kinematic pair and ri = rank(iq ). Let

iq be the matrix built of all rows ofq corresponding to ith kinematic pair and

ri = rank(iq).

If all scalar constraints describing the ith joint are independent upon scalar

constraints describing the other joints of multibody system, the following equation

is fulfilled:

r = ri + ri . (25)

Otherwise, if any row ofiq can be expressed as a linear combination of other

rows ofiq and at least one row ofiq , then the rank of the full Jacobian matrix

is less than the sum of the ranks of the matricesiq andiq : r < ri + ri .

Resultant reactions in joints for which Equation (25) is fulfilled can be uniquely

determined.


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34 MAREK WOJTYRA

6.2. METHOD B+

Let us consider Equation (11). Rows of matrix Rq can be expressed as linear

combinations of rows of matrix Nq . Matrix consists of coefficients of these

combinations. Rows of correspond to rows ofRq and columns of correspond

to rows ofNq .

As it was shown in Section 5, joint reaction forces can be uniquely determined

only if constraint equations describing this joint are independent upon equations

describing other joints. If a non-zero element of belongs to the row corresponding

to the ith joint, and to the column corresponding to the jth joint, then equations

describing both the joints are dependent. Thus, reaction forces in both joints cannot

be uniquely determined.Joints for which resultant reactions can be uniquely determined can be detected

using a four-step procedure. Three first steps are the same as in the method B,

described in the Section 4.2. The fourth step involves finding non-zero elements

in matrix and checking which joints these elements correspond to. If for all

scalar constraint equations describing the specified joint, non-zero elements of

correspond only to the constraint equations describing the same joint, then the

resultant reaction in the specified joint can be uniquely determined.

6.3. METHOD C+

Equation (21) shows, that generalised reaction in ith joint is given by:

(fi )T = (i )Tiq. (26)

The vector of all Lagrange multipliers can be written in the form (19). Let Ui

denote a matrix consisting only these rows ofU that correspond to ith joint. The La-

grange multipliers corresponding to ith joint can be expressed in the following form:

i = Ui = Ui + Ui . (27)

Substituting (27) into (26) we obtain:

(fi )T = T(Ui )Tiq

+ T(Ui )Tiq. (28)

Let matrices Bi and Ci be defined by:

(Ui )Tiq =

Cirn

Bi(mr)n

. (29)

The first r elements of vector are zeros, the other (m r) can be chosen

arbitrarily. Equation (28) shows that generalised reactions corresponding to ith

joint can be determined uniquely only if matrix B is zero.


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The above conclusion is useful in detection of joints for which resultant reactions

can be uniquely determined. Firstly, the SVD decomposition of the Jacobian matrix

q is calculated and then rows of U and q corresponding to the joint being

investigated are extracted. Finally, matrix Bi is calculated and checked to see if it

contains only zero elements.

7. Examples

7.1. SIMPLE PLANAR MECHANISM

In some mechanical systems it is obvious which joint reaction forces can be

uniquely determined. An example of such system is presented in Figure 1in the introductory section. The developed methods of the Jacobian matrix

analysis may be used in such case to confirm our intuitive guess. In the example

presented here we will analyse a bit more complicated mechanism. In this case the

identification of joints for which reactions can be uniquely determined is not so

straightforward.

The Figure 2 presents a simple planar mechanism, consisting of the basis 0

and four movable bodies 1, 2, 3 and 4. The absolute coordinates describing the

mechanism are formed in the vector q as

q = rT1 1 r

T2 2 r

T3 3 r

T4 4

T,

where ri = [xi yi ]T represents the position of the local reference frame xiyi origin

with respect to the global frame x0y0, and i is the angle of the local frame xiyirotation with respect to the global frame. The direction cosine matrix transforming

Figure 2. Planar mechanism.


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36 MAREK WOJTYRA

quantities from xiyi to x0y0 is given by

Ri =

cos i sinisin i cos i

.

The coordinates of points K, L and M(see Figure 2) are constant in appropriate

local frames:

s(1)K = 0, s

(3)K = 0, s

(2)L = 0, s

(4)L = 0, s

(3)M = [ 0 1 ]

T, s(4)M = [ 1 0 ]

T.

The translational joint 1, formed by the bodies 0 and 1, is described by two scalar

constraint equations. The first one represents the fact, that point Kmoves along x0axis and the second represent the fact, that body 1 does not change its orientation

with respect to body 0:

1(q)

1(q)

2(q)

y1

1

=

0

0

= 0.

The translational joint 2, formed by the bodies 0 and 2, can be described similarly

as

2(q)

3(q)

4(q)

x2

2

=

0

0

= 0.

The first constraint equation describing the translational joint 3, formed by

the bodies 1 and 2, represents the fact that the vector from point K to point L is

perpendicular to vector v = [1 1]T. The second constraint equation represents the

fact, that body 2 does not change its orientation with respect to body 1:

3(q)

5(q)

6(q)

[x2 x1 y2 y1 ]v

T

2 1

x2 x1 + y2 y1

2 1

=

0

0

= 0.

Bodies 1 and 3 form a rotational joint 4, which can be described by the following

constraint equation:

4(q)

7(q)

8(q)

r1 + R1s

(1)K r3 R3s

(3)K

x1 x3

y1 y3

=

0

0

= 0.

Similarly rotational joints 5 and 6 can be described as

5(q)

9(q)

10(q)

r2 + R2s

(2)L r4 R4s

(4)L

x2 x4

y2 y4

=

0

0

= 0,


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6(q)

11(q)

12(q)

r3 + R3s

(3)M r4 R4s

(4)M

x3 sin 3 x4 cos 4

y3 + cos 3 y4 sin 4

=

0

0

= 0,

At the time instant being considered, the mechanism is described by the follow-

ing coordinates: q = [ 1 0 0 | 0 1 0 | 1 0 0 | 0 1 0 ]T; this situation is presented

in Figure 2. By differentiating the constraint equations we obtain the constraint

Jacobian matrix:

q =

1q

2q

3q

4q

5q

6q

=

(1)q(2)q

(3)q

(4)q

(5)q

(6)q

(7)q

(8)q

(9)q

(10)q

(11)q

(12)q

=

0 1 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0

1 1 0 1 1 0 0 0 0 0 0 0

0 0 1 0 0 1 0 0 0 0 0 0

1 0 0 0 0 0 1 0 0 0 0 0

0 1 0 0 0 0 0 1 0 0 0 0

0 0 0 1 0 0 0 0 0 1 0 0

0 0 0 0 1 0 0 0 0 0 1 0

0 0 0 0 0 0 1 0 1 1 0 0

0 0 0 0 0 0 0 1 0 0 1 1

.

The mechanism is described by 12 coordinates and 12 scalar constraint equa-

tions.The mechanism hasone degree of freedom (points KandL can simultaneously


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38 MAREK WOJTYRA

move along axes x0 and y0, respectively). Hence, we can state that redundant con-

straints are imposed on the mechanisms. Let us check now which constraint reac-

tions can be uniquely determined. We will use all three methods.

7.1.1. Method A

Firstly, we must compute the rank of Jacobian matrix:

r = rank(q) = 11.

Then the rank of submatrix \1q should be computed:

rank

\1q

= rank

(2)q...

(12)q

= 10.

Similarly we can compute the ranks of other submatrices:

rank

\iq

= 11, for i {2, 4, 6},

rank

\iq

= 10, for i {1, 3, 5, 7, 8, 9, 10, 11, 12}.

The obtained results show, that reactions of constraints 2, 4 and 6 cannotbe uniquely determined, since the ranks of\2q ,

\4q and

\6q are equal the rank of

q. The other constraint reactions can be uniquely determined.

7.1.2. Method B

The Gaussian elimination procedure shows, that constraint equation 6 is redun-

dant, therefore the Jacobian matrix can be divided into two submatrices, as follows:

Nq =

(1)q...

(5)q(7)q

...

(12)q

, Rq = (6)q.

In the next step matrix is calculated as

= Rq

Nq

T

Nq

Nq

T1= [ 0 1 0 1 0 0 0 0 0 0 0 ].


17/24


There are nine zero columns in the obtained matrix. They are corresponding

to constraints 1, 3, 5, 7, 8, 9, 10, 11 and 12, so reactions of these

constraints can be uniquely determined.

7.1.3. Method C

Firstly, we compute singular value decomposition of Jacobian matrixq, to obtain

the matrix U. There are 12 scalar constraint equations and the rankr of Jacobian

matrixq equals 11, thus we are looking for zeros in rows in the 12th column only.

The 12th column of the matrix U is as follows:

U(12)

[ 0 0.5774 0 0.5774 0 0.5774 0 0 0 0 0 0]T.

There are nine zero elements in this column. They are corresponding to the set

of independent constraints previously detected using methods A and B.

All three analyses have shown, that the reactions of constraints 1, 3, 5, 7,

8,9,10,11 and12 are not influencedby the redundant constraintelimination.

Therefore, simulating the considered mechanism with multibody package, we will

obtain proper reactions of these nine constraints.

Let us check now in which joints the resultant reaction of all constraints defining

the joint can be uniquely determined. We will use all three methods.

7.1.4. Method A+

The rank of the Jacobian matrix has been already computed (r = 11), so now we

calculate the ranks of submatrices1q and1q :

r1 = rank

1q

= 2, r1 = rank

1q

= rank

2q

...

6q

= 10.

Similarly, we calculate the ranks of other submatrices:

r2 = rank2q = 2, r2 = rank2

q = 10,

r3 = rank

3q

= 2, r3 = rank

3q

= 10,

r4 = rank

4q

= 2, r4 = rank

4q

= 9,

r5 = rank

5q

= 2, r5 = rank

5q

= 9,

r6 = rank

6q

= 2, r6 = rank

6q

= 9.

For joints 1, 2 and 3 we observe that: r < ri + ri . Thus for these joints we

cannot uniquely determine the resultant constraints reaction. For joints 4, 5 and 6


18/24

40 MAREK WOJTYRA

we observe that: r = ri + ri . Thus, for these joints we can uniquely determine the

resultant constraints reaction.

7.1.5. Method B+

Firstly, the matrix must be calculated, as it is described in the method B above.

Then, we are looking fornon-zeroelements in the matrix. The first non-zero element

12 corresponds to scalar constraints 2 and 6. These scalar constraints belong

to joints 1 and 3, respectively. Thus, for both the joints resultant reactions cannot

be uniquely determined. The second non-zero element 14 corresponds to scalar

constraints4 and6. These scalar constraints belongto joints2 and3, respectively.

Thus, also for the joint 2, resultant reaction cannot be uniquely determined. None

of the non-zero elements corresponds to joints 4, 5 or 6, therefore the reactions in

these three joints can be determined uniquely.

7.1.6. Method C+

The matrix U (obtained during singular value decomposition of the Jacobian matrix)

is divided into six submatrices Ui212 that correspond to the joints.

Then matrices Bi are calculated as

B1 [ 0 0 0.5774 0 0 0 0 0 0 0 0 0],

B2 [ 0 0 0 0 0 0.5774 0 0 0 0 0 0],

B3 [ 0 0 0.5774 0 0 0.5774 0 0 0 0 0 0],

B4 = 0112,

B5 = 0112,

B6 = 0112.

Matrices B1, B2 and B3 have at least one non-zero element, thus the resultant

constraints reaction in joints 1, 2 and 3 cannot be uniquely determined. The matrices

B4, B5 and B6 consist of zeros only, thus for joints 4, 5 and 6 we can uniquely

determine the resultant constraints reaction.

The results of the constraints analysis might be very useful. For example, after

the analysis we are sure that in our rigid body model we can take into accountfriction in the revolute joints 4, 5 and 6. On the other hand, if we were intending to

consider friction in the translational joints 1, 2 and 3, then we would have to take

into account the elasticity of bodies.

7.2. SPATIAL MECHANISM

The Figure 3 presents a multibody system, consisting of the basis 0 and three

movable bodies 1, 2 and 3. Bodies 0 and 1 form a revolute joint 1; bodies 1 and

2 form a revolute joint 2. Body 3 forms spherical joints 3 and 4 with bodies 2


19/24


Figure 3. Spatial mechanism.

and 0, respectively. The global reference frame x0y0z0 is established on 0. The local

reference framesxiyizi are established on the other bodies. The absolute coordinates

of the multibody system are formed in the vector q:

q =

rT1 T1 r

T2

T2 r

T3

T3

T,

where ri = [xi yi zi ]T is the position of xiyizi origin with respect to the global

framex0y0z0, andi = [i i i ]T are angles ofz-x-z Euler rotations describing the

orientation ofxiyizi with respect tox0y0z0. The direction cosine matrix transforming

quantities from xiyizi to x0y0z0 is given by:

Ri (i , i , i )

= cos i sin i 0

sin i cos i 0

0 0 1

1 0 0

0 cos i sin i

0 sin i cos i

cos i sin i 0

sin i cos i 0

0 0 1

=

cos i cos i sin i cos i sin i sin i sin i cos i sin i cos i cos i sini

sin i cos i cos i cos i cos i cos i sin i cos i+ cos i sin i sin i sini

sin i sin i cos i sin i cos i

.


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42 MAREK WOJTYRA

The coordinates of points K, L, M, N, P and Q (see Figure 3) are constant

in appropriate local frames (dimensions are the following: d = 2 [m], l = 5

[m]):

s(0)K = [ 0 0 0 ]

T, s(1)K = [ 0 0 d]

T, s(0)L = [ 0 d 0 ]

T, s(1)L = [ 0 0 0 ]

T,

s(1)M = [ 0 l d]

T, s(2)M = [ 0 0 0 ]

T, s(1)N = [ 0 l 0 ]

T, s(2)N = [ 0 0 d]

T,

s(2)P = [ 0 l 0 ]

T, s(3)P = [ 0 0 0 ]

T, s(3)Q = [ d 0 l ]

T, s(0)Q = [ d l 2l ]

T.

Bodies 0 and 1 form a rotational joint 1, thus points K and L must remain intheir original positions on the axis of relative rotation. This can be described by the

following constraint equation:

1(q)

1

...

6

r1 + R1s

(1)K s

(0)K

r1 + R1s(1)L s

(0)L

=

0

0

.

The above vector equations are equivalent to six scalar equations. The revolute

joint removes only five degrees of freedom. It is obvious that in the mathematical

model some constraints are redundant.The rotational joint 2, formed by bodies 1 and 2, can be described in a similar

way:

2(q)

7

...

12

r2 + R2s

(2)M r1 R1s

(1)M

r2 + R2s(2)N r1 R1s

(1)N

=

0

0

.

The spherical joints 3 and 4 can be described by the following constraint equa-

tions:

3(q) [13 14 15 ]

T r2 + R2s(2)P r3 R3s

(3)P = 0,

4(q) [16 17 18 ]

T r3 + R3s(3)Q s

(0)Q = 0.

At the time instant being considered, the mechanism is described by the coordi-

nates: q = [ 0 d 0 0 /2 0|0 0 l 0 /2 0|0 0 2l 0 /2 0 ]T; this situation

is presented in Figure 3. Calculating the Jacobian matrix of constraint equations


21/24


we obtain:

q =

1q

T

2q

T

3q

T

4q

TT

=

1 0 0 d 0 0

0 1 0 0 0 0

0 0 1 0 d 0 066 066

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

1 0 0 d 0 l 1 0 0 0 0 00 1 0 0 l 0 0 1 0 0 0 0

0 0 1 0 d 0 0 0 1 0 0 0 066

1 0 0 0 0 l 1 0 0 d 0 0

0 1 0 0 l 0 0 1 0 0 0 0

0 0 1 0 0 0 0 0 1 0 d 0

1 0 0 0 0 l 1 0 0 0 0 0

036 0 1 0 0 l 0 0 1 0 0 0 0

0 0 1 0 0 0 0 0 1 0 0 0

1 0 0 l 0 0

036 036 0 1 0 d 0 0

0 0 1 0 l d

Let us check now which constraint reactions can be uniquely determined. We

will use all three methods.

7.2.1. Method A

Firstly, we must compute the rank of Jacobian matrix:

r = rank(q) = 16.

Then the ranks of submatrices\iq should be computed:

rank

\iq

= 16, for i {2, 5, 8, 11},

rank

\iq

= 15, for i {1, 3, 4, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18}.

The obtained results show, that reactions of constraints 2, 5, 8 and 11cannot be uniquely determined. The result is quite obvious, since the constraints

2 and 5 express no relative translation between the bodies 0 and 1 along axis


22/24

44 MAREK WOJTYRA

of rotation. The reactions of these constraints cannot be determined, as we do not

know how they are shared between the points K and L. Similar situation occurs

with the constraints 8 and 11.

7.2.2. Method B

The Gaussian elimination procedure shows, that constraint equation 5 and 11are redundant, thus the Jacobian matrix can be divided into two submatrices, as

follows:

Nq =

(1)

Tq (4)

Tq (6)

Tq (10)

Tq (12)

Tq (18)

TqT,

Rq =

(5)

Tq (11)

Tq

T.

In the next step matrix is calculated:

= Rq

Nq

T

Nq

Nq

T1=

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

.

The zero columns in matrix correspond to constraints 1, 3, 4, 6, 7,

9, 10, 12, 13, 14, 15, 16, 17 and 18, so reactions of these constraints

can be uniquely determined.

7.2.3. Method C

Firstly, we compute singular value decomposition of Jacobian matrix q.

There are 18 scalar constraint equations and the rank r of Jacobian matrix qequals 16, thus we are looking for zeros in rows only in the last two columns

of the matrix U. The 17th and the 18th columns of the matrix U are the

following:

U(17) U(18)

0 0.69 0 0 0.69 0 0 0.14 0 0 0.14 0 0 0 0 0 0 0

0 0.14 0 0 0.14 0 0 0.69 0 0 0.69 0 0 0 0 0 0 0

T

The obtained results show, that reactions of constraints 2, 5, 8 and

11 cannot be uniquely determined. The other reactions can be determined

uniquely.

Let us check now in which joints the resultant reaction of all constraints defining

the joint can be uniquely determined. We will use all three methods.


23/24


7.2.4. Method A+

The rank of the Jacobian matrix has been already computed (r = 16), thus we

calculate the ranks of submatricesiq andiq as

r1 = rank

1q

= 5, r1 = rank

1q

= 11,

r2 = rank

2q

= 5, r2 = rank

2q

= 11,

r3 = rank

3q

= 3, r3 = rank

3q

= 13,

r4 = rank

4q

= 3, r4 = rank

4q

= 13.

We observe that for each joint: r = ri + ri . Thus, for all the joints the resultant

reactions can be determined uniquely. Let us look at the joint 1. We do not knowhow the reaction force acting along the axis of rotation is shared between points K

and L, nevertheless we know the sum of these two components, i.e. the resultant

reaction. The situation in the joint 2 is similar.

7.2.5. Method B+

Firstly, the matrix must be calculated, as it is described in the method B above.

Then we are looking for non-zero elements in the matrix. The first non-zero element

12 corresponds to scalar constraints 2 and 5. Both these scalar constraints

belong to the same joint, namely joint 1. Thus, the constraint redundancy does not

affect other joints. The second non-zero element 27 corresponds to scalar con-

straints 8 and 11. Both these scalar constraints belong to the joint 2. Similarly,the constraint redundancy does not affect other joints. The final conclusion is, that

for all the joints the resultant reactions can be determined uniquely.

7.2.6. Method C+

The matrix U (obtained during singular value decomposition of the Jacobian matrix)

is divided into submatrices Ui that correspond to the joints.

Then matrices Bi are calculated as

(Ui )Tiq =

Ci1618

Bi218

.

We obtain:

B1 = 0218, B2 = 0218, B

3 = 0218, B4 = 0218.

All the matrices Bi consist of zeros only, thus for all the joints we can uniquely

determine the resultant constraints reaction.

In the example presented redundant constraints were imposed in a slightly artifi-

cial way, and the results obtained were quite obvious. However, in the case of more

complicated mechanisms, being modelled using professional multibody packages,


24/24

46 MAREK WOJTYRA

redundant constraints are natural and appear quite frequently. Intuitive analysis of

constraints is difficult in complex mechanical systems. In such cases, presented

here systematical methods of redundant constraint analysis become really useful.

8. Conclusions

If redundant constraints exist in a multibody system, then it is not possible to

determine uniquely all constraint reactions. In order to find a unique set of all

joint reaction forces in an overconstrained system it is necessary to abandon the

assumption that all bodies are rigid, which changes the class of problem being

analysed. Flexible mechanism modelling is much more difficult and requires a lot

of additional data. If the modelled mechanism is in the early stage of construction,then some necessary data are not available.

Methods presented here enable us to detect constraints and joints for which

reactions can be uniquely determined despite the existence of redundant constraints.

In many technical problems it is possible to avoid the flexibility analysis and to gain

information about loadson crucial joints and bodies. Redundant constraints analysis

is especially advised when we intend to consider the joint friction.

It should be emphasised that the fact that we can or we cannot uniquely determine

reaction in a selected joint depends only upon the structure of the multibody system.

The choice of coordinates in a mathematical model does not change this fact. It is

also important that the redundant constraints analysis can be performed only once,

at the beginning of a simulation. There is no reason for repeating it in the subsequent

steps of computation.

References

1. Haug, E. J., Computer Aided Kinematics and Dynamics of Mechanical Systems, Allyn and

Bacon, Boston, MA, 1989.

2. Nikravesh, P. E., Computer-Aided Analysis of Mechanical Systems, Prentice Hall, New York,

1988.

3. Garcia de Jalon, J. and Bayo, E., Kinematic and Dynamic Simulation of Multibody Systems: The

Real-Time Challenge, Springer, New York, 1994.

4. Park, T., Haug, E. J. and Yim, H. J., Automated kinematic feasibility evaluation and analysis of

mechanical systems, Mechanisms and Machine Theory 23(5), 1988, 383391.

5. Haug, E. J., Intermediate Dynamics, Prentice Hall, 1992.

6. Blajer, W., On the determination of joint reactions in multibody mechanisms, ASME Journalof Mechanical Design 126(2), 2004, 341350.

7. Udwadia, F. E. and Kalaba, R.E.,Analytical Dynamics: A New Approach, Cambridge University

Press, Cambridge, 1996.

8. Corwin, L. J. and Szczarba, R. H., Multivariable Calculus, Marcel Dekker, New York, 1982.

9. Jungnickel, U., Differential-algebraic equations in riemannianspacesand applications to multi-

body system dynamics, ZAMM74(9), 1994, 409415.

10. Blajer, W., A Geometrical interpretation and uniform matrix formulation of multibody system

dynamics, ZAMM81(4), 2001, 247259.

11. Strang, G., Linear Algebra and Its Applications, Academic Press, New York, 1980.

joint reaction forces in multibody systems

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