On the unique solvability of a direct dynamics problem for mechanisms with redundant constraints and Coulomb friction in joints

Mechanism and Machine Theory 46 (2011) 312–334

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Mechanism and Machine Theory j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m e c h m t

On the unique solvability of a direct dynamics problem for mechanisms with redundant constraints and Coulomb friction in joints Janusz Frączek, Marek Wojtyra ⁎ Warsaw University of Technology, Institute of Aeronautics and Applied Mechanics, ul. Nowowiejska 24, 00-665 Warsaw, Poland

a r t i c l e

i n f o

Article history: Received 4 August 2010 Received in revised form 31 October 2010 Accepted 16 November 2010 Available online 15 December 2010 Keywords: Direct dynamics Dependent constraints Joint reactions Multibody system Coulomb friction

a b s t r a c t The uniqueness of simulated motion of an overconstrained rigid body mechanism with joint friction is studied. The investigated issue originates in the problem of joint reactions solvability. It is known that in case of redundant constraints existence the constraint reaction forces cannot be — in general — uniquely determined. It can be proved, however, that — under certain conditions — selected reactions can be specified uniquely. Analytical and numerical methods for reactions solvability analysis are available. It is shown in this paper that indeterminacy of normal reactions results in indeterminacy of friction forces, and moreover, non-uniqueness of friction forces results in non-uniqueness of simulated motion. A method of finding these joints, for which friction forces are unique, is presented. It is also proved that if only uniquely solvable friction effects are introduced, then simulated motion of the mechanism is unique, otherwise it is not. Finally, examples of dynamic analysis of overconstrained mechanisms with joint friction are presented; unique and non-unique results are obtained. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Direct dynamic analysis of mechanisms consists in calculating motion resulting from external or internal loads and from driving constraints imposed on the links or joints [1,2]. Time history of kinematic quantities, i.e., positions, velocities and accelerations of bodies, is determined during the analysis, and — in many cases — the joint reaction forces are calculated alongside the kinematic quantities. If mechanism motion is the only object of interest and all kinematic pairs are frictionless, it is not necessary to calculate joint reaction forces, since they can be eliminated from equations of motion [3]. Nevertheless, joint reaction forces are quite frequently calculated. The knowledge of joint reaction forces is, e.g., essential for various structural analyses of mechanisms bodies, like fatigue or static analysis, when stresses or strains are calculated. When Coulomb friction in joints is taken into account, calculation of reaction forces cannot be avoided, since friction forces depend directly on normal joint reactions [2,4]. In that case, both normal reactions and friction forces must be determined to solve for mechanism motion. It happens that for a given rigid body mechanism the problem of finding joint reactions does not have a unique solution (this issue is discussed in subsequent paragraphs). If the considered mechanism is frictionless, and thus joint reactions are not needed to calculate its motion, the simulated motion is unique even though joint reactions are indeterminate. On the contrary, if Coulomb friction in kinematic pairs is considered and reactions are non-unique, the simulated mechanism motion may not be unique as well. The normal joint reaction forces are necessary to calculate friction forces, thus non-uniqueness of normal reactions results in non-uniqueness of friction forces, and consequently in non-uniqueness of mechanism motion. In this paper the uniqueness of simulated mechanism motion and the problem of joint reactions solvability in the presence of Coulomb friction are studied in detail. Most often the joint reaction indeterminacy, as well as the non-uniqueness of simulated motion, is caused by the presence of redundant constraints; and only this cause is considered in our paper. ⁎ Corresponding author. Tel.: +48 22 234 5610; fax: +48 22 234 7513. E-mail addresses: [email protected] (J. Frączek), [email protected] (M. Wojtyra). 0094-114X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2010.11.003

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Redundant constraints are usually defined as constraints that can be removed without changing the kinematics of the system [2,5]. Redundant constraints appear when several kinematic pairs restrict the same degree of freedom (e.g., a door supported by three hinges is a simple example of an overconstrained system, since door kinematics would be the same with a single hinge). The joints that are responsible for redundant constraints are not necessary when kinematics is concerned, however, they are frequently introduced due to constructional reasons, in order to straighten or to simplify the construction. In engineering simulations complex spatial mechanisms are often modelled as systems of bodies interconnected by kinematic pairs, i.e., as multibody systems, and algorithms typical for multibody systems analysis are used for calculations. Various sets of coordinates are used to specify the configuration of multibody systems; most often absolute (Cartesian), natural or joint coordinates. Kinematic pairs are represented by constraint equations in mathematical model of multibody system; the constraint equations are formulated for all joints, when absolute or natural coordinates are used to describe the system, or only for selected loop-closing joints, in the case of relative coordinates. The problem of finding physical joint reactions in a multibody system is equivalent to the problem of finding generalized constraint reactions. Mathematically, it is clear that if all of the constraint equations are independent, the constraint reaction forces can be uniquely determined. Similarly, when the constraint equations are dependent and consistent, the constraint reaction forces, in general, cannot be uniquely determined. In this case, infinitely many sets of reaction forces fulfil the motion and constraint equations. If redundant constraints are present in the multibody system, the constraint equations are dependent [2,5,6]. For our further considerations it is important that, in the case of a redundantly constrained mechanism, despite the fact that all constraint reactions cannot be uniquely determined, selected single constraint reactions or selected groups of reactions can be specified uniquely [7]. An interesting approach to the problem of joint reactions solvability, proposed by Song and Gao, can be found in [8]. The authors of [8] adopt the Grübler–Kutzbach mobility equation to predict the indeterminacy of joint reaction forces. The method proposed in [8] requires prior information about the dependent constraints and is not suitable for computer applications, especially for automatic analyses using multibody software. The problem of joint reactions solvability was also addressed in our previously published works [7,9,10]. An algebraic criterion, allowing for detection of these joints for which reaction forces can be uniquely determined, was formulated. This criterion was followed by three numerical methods of finding such joints. These methods are based on constraint Jacobian investigation, and thus they can be easily included in any multibody code. It should be noted that results of papers [7,9,10] as well as [8] are limited to the case of frictionless joints. Although the problem of reaction forces uniqueness — crucial for results credibility — is often encountered in practical calculations, surprisingly small attention is paid to it. Quite often it is assumed that constraint equations are independent, e.g. [6,11,12], hence redundant constraints, and thus reactions solvability problems, are simply ignored. In multibody modelling the most popular method of dealing with overconstrained systems consists in redundant constraints elimination. In a given set of constraints, the redundant ones can be selected in many ways. Selection of the redundant constraints is usually based on the results of the constraint Jacobian matrix factorization [5,6,13]. It should be noted that the loads carried by the eliminated constraints are transferred to the constraints remaining in the model. The redundant constraints elimination method is frequently implemented in general purpose multibody packages. The software users are usually advised to replace overconstrained models with kinematically equivalent models without redundant constraints. If the software user fails to follow the advice, the redundant constraints detected in the model are automatically eliminated. The problem that model after redundant constraints elimination does not reflect the real system, and some of calculated reactions are non-unique, usually is not signalized to the multibody software user. Sometimes redundant constraint equations are preserved in a mathematical model of a mechanism [2]. In that case the system of equations used to calculate joint reaction forces is indeterminate. It is possible to use pseudo-inverse methods [14] or a penalty approach, often based on the augmented lagrangian formulation [15], to solve the indeterminate system of equations, however, this also results in choosing one of infinitely many possible solutions, and hence a non-unique solution is found. This article presents a direct continuation of works described in [7]. Redundantly constrained rigid body mechanisms with Coulomb friction in kinematic pairs are considered. It is shown that the methods (proposed in [7] and [10]) used to check whether particular joint reactions in an overconstrained mechanism can be uniquely determined, can also be used to check whether the simulated motion of the mechanism is unique (in terms of positions, velocities and accelerations of bodies). It is shown that — in general — the direct dynamic problem for an overconstrained rigid body mechanism with Coulomb friction in joints is not solvable. This is a direct consequence of the fact that due to constraint dependency the normal joint reaction forces are not unique, and thus the friction (tangent) forces are not unique as well. The main result of this paper is that, if friction forces appear only in these joints for which reaction solution is unique, then the simulated motion of mechanism is also unique, and thus — in this special case — the direct dynamic problem is solvable. This paper is divided into five sections. The following, i.e. second, section briefly recapitulates results presented in [7]. Mathematical description of constraints and constraint reaction forces is presented. The analytical conditions, under which some reaction forces are uniquely determined, despite the constraint dependency observed in a mechanism, are discussed. The third section is focused on mechanisms with joint friction. Firstly, mechanisms without redundant constraints are studied and methods of formulating and solving equations of motion are discussed. Then, redundantly constrained mechanisms with joint friction are investigated. Conditions that must be fulfilled to obtain unique mechanism motion (and unique Coulomb friction forces in joints) are formulated and proved. Two examples, showing details of calculations for a planar and a spatial mechanism, are presented in the fourth section. In the last section concluding remarks are formulated. The article is followed by two appendices. The first appendix presents the joint friction models used during exemplary simulations. The second appendix is devoted to the analysis of redundant constraints imposed on the exemplary mechanisms.

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2. Uniqueness of joint reactions — mechanisms without friction forces 2.1. Constraint reaction forces Let us assume that absolute (Cartesian) coordinates are chosen to describe a mechanism (or, more generally, a multibody h T system — MBS) consisting of b moving bodies. The generalized coordinates vector can be written as: q = qT1 ⋯ qTb
; where qi T denotes the vector of i-th body coordinates. For planar systems qi = rTi φi = ½ xi yi φi
T , and for spatial systems  T qi = rTi φTi = ½ xi yi zi αi βi γi
T . In both cases ri denotes position of the i-th body-fixed local frame origin with respect to the global frame. In the planar case φi is the angle of rotation of the local frame with respect to the global frame, and in the spatial case φi is a three-element-vector of Euler angles (z–x–z rotations) describing the i-th local frame orientation. The MBS equations of motion can be written in the following form [5,6]: Mnnh˙ n1 + f n1 = Q n1 :

ð1Þ

where n = 3b for planar and n = 6b for spatial systems, respectively. M denotes the mass matrix, f represents the generalised reaction forces, Q contains the external forces and all velocity dependent inertial terms. In the planar case h is simply the time derivative of generalized coordinates: ˙ h = q:

ð2Þ

In the spatial case, the vector h is built of linear velocities decomposed in the global frame and of angular velocities decomposed in the local frames: h T h = hT1 ⋯ hTb
;

h hi = r˙ Ti

i

ðiÞ T T

ωi

:

ð3Þ

The relationship between Euler angles derivatives φ˙ i and angular velocity ω(i) i can be written as: ðiÞ φ˙ i = Hi ωi :

ð4Þ

Note that explicit form of Hi(φi) is given in Appendix B. Kinematic constraint equations are formulated to represent joints interconnecting moving bodies. These holonomic constraints imposed on generalized coordinates form a set of m scalar algebraic equations: h T ΦðqÞ = Φ1 ðqÞ; ⋯ ; Φm ðqÞ
= 0m1 :

ð5Þ

We assume that constraints are consistent [2], i.e. no contradictory conditions are imposed on coordinates q. The constraint Jacobian matrix is defined as: 2 Φq = 4

3

∂Φ1 = ∂q1 ⋯ ∂Φ1 = ∂qn ⋮ ⋮ ∂Φm = ∂q1 ⋯ ∂Φm = ∂qn

 5 = Φr 1

Φφ1 ⋯ Φrb

 Φφb :

ð6Þ

In the case of spatial systems, during dynamics calculations, a modified Jacobian matrix is used, which is related to angular velocities rather than to Euler angles derivatives. The modified Jacobian matrix is defined as: Mod

Φq

 = Φr1

Φφ1 H1 ⋯ Φri

Φφi Hi ⋯ Φrb

 Φφb Hb :

ð7Þ

To simplify notation, in the following text symbol J will denote the transposed Jacobian matrix in the planar case (J = ΦTq), and the T transposed modified Jacobian matrix in the spatial case (J = (ΦMod q ) ). In both cases ji will denote the i-th column of matrix J: Jnm = ½ j1 ⋯ jm
:

ð8Þ

The generalized reaction forces are described by the following equation [5,6]: f n1 = J λm1 ;

ð9Þ

where λ is a vector of Lagrange multipliers. It is worth noting that if different set of angular coordinates was chosen in the spatial case (e.g., Euler parameters), the linear relationship (9) would be still valid, however, modified Jacobian matrix would be defined differently. Since our further considerations are valid for both Jacobian and modified Jacobian matrices, stressing the difference between these matrices is in most cases unnecessary. Therefore we will be using generic term ‘Jacobian matrix’ also for the modified Jacobian.

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The constraint reaction forces given by Eq. (9) have a clear physical interpretation. Let us assume that the first k columns of matrix J correspond to the constraints imposed by a kinematic pair X. In such a case the generalized force: f X = ½ j1

j2 ⋯ jk
λX = JX λX

ð10Þ

represents the joint reaction forces. After substituting Eq. (9) into Eq. (1) and appending twice differentiated Eq. (5), one obtains well-known formulation of MBS equations of motion [5,6]: 

M JT

J 0



   h˙ = Q ; Γ λ

ð11Þ

where the term Γ represents expression resulting from double differentiating constraint Eq. (5) and taking into account relationship (4). If the Jacobian matrix has full row rank (i.e. when the transposed Jacobian matrix has full column rank: rank(J) = m), Eq. (11), which is linear with respect to the Lagrange multipliers, has a unique solution for λ. Thus, constraint reaction forces are uniquely determined. If the Jacobian matrix is rank deficient, Eq. (11) has infinitely many solutions for λ. If the vector of Lagrange multipliers is not unique, the problem of joint reaction forces calculation does not have a unique solution. The rank deficiency is most often caused by the existence of redundant constraints (another reason can be a singular position of the mechanism). A typical procedure in rigid redundantly constrained MBS dynamic analysis (implemented in widely used multibody software) is to divide the analyzed constraint equations set into two separate subsets of independent and dependent equations. The subset of dependent equations is not unique (and thus, the subset of independent equations is not unique either), since it can be selected in many ways. Redundant constraints are usually removed from multibody system mathematical description, hence only the subset of independent equations is analyzed [5,6]. Most often the division into subsets is made automatically, according to the results of constraint Jacobian matrix factorization (another possibility is, e.g., to build a kinematically equivalent model without redundant constraints). It should be emphasized that, regardless of the used method of redundant constraints elimination, the reaction forces associated with eliminated constraints are arbitrarily set to zero. Obviously, in real mechanisms reactions associated with constraints neglected during analysis are not likely to be constantly equal zero. Moreover, setting the reactions of eliminated constraints to zero transfers their loads to the constraints that remain in the mathematical model. Consequently, redundant constraints elimination affects not only the reactions of eliminated constraints but the reactions of remaining constraints as well. As mentioned in the previous section, even when redundant constraint equations are preserved in the mathematical model (instead of being eliminated), only one of infinitely many possible, and thus non-unique, set of reactions is found. 2.2. Uniquely determined constraint reaction forces It can be shown [7] that despite the λ vector is not unique due to redundant constraints existence, some of the constraint reaction forces can be uniquely determined. Selected results, presented in greater detail in [7], are briefly recalled in this section and developed in the following sections. Linear algebra provides us with the concept of direct sum which is crucial for our further considerations. We will recall the appropriate definition [13]. Assume that Z is a linear vector space in Rn. Assume also that X and Y are the subspaces of Z. We say that Z is a direct sum of subspaces X and Y, and we denote it as Z = X ⊕ Y, when the following conditions are fulfilled: 1. Z is a sum of subspaces X and Y (we denote it as Z = X + Y), which means that any vector z ∈ Z can be represented as z = x + y, where x ∈ X and y ∈ Y. 2. If x1 + y1 = x2 + y2, provided that x1 ∈ X, y1 ∈ Y, x2 ∈ X and y2 ∈ Y, then x1 = x2 and y1 = y2. Three linear spaces need to be defined. Let Z be a linear space spanned by columns of transposed Jacobian matrix J: Z = spanðJÞ:

ð12Þ

We assume that the Jacobian matrix is rank deficient, thus: dimðZ Þ = r b m:

ð13Þ

Let X be a linear space spanned by columns of matrix J corresponding to a selected kinematic pair (for the sake of simplicity, let it be the first k columns). This can be written as: X = spanðj1 ; j2 ; :::; jk Þ = spanðJX Þ;

ð14Þ

dimðX Þ = rX ≤ k:

ð15Þ

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Let Y be a linear space spanned by the remaining columns of matrix J, that is:   Y = span jk+1 ; jk+2 ; :::; jm = spanðJY Þ:

ð16Þ

Eq. (13) shows that the Lagrange multipliers λm × 1 responsible for the constraint reaction forces are not unique (see previous section). Despite this fact, for a broad class of mechanisms with redundant constraints, selected components of generalised reaction force (e.g., fX defined by Eq. (10)), can be uniquely determined. It can be shown [7] that the following statement is true: If Z = X ⊕ Y, then the generalised reaction force vector fX = JXλX, which corresponds to the vectors spanning the linear space X, is uniquely determined. It should be stressed that the problem of finding Lagrange multipliers λX might not have a unique solution even when the product JXλX is uniquely determined. The Lagrange multipliers λX can be uniquely determined only if dim(X) = k (this condition may not be fulfilled — see Eq. (15)). The considered problem can be reformulated to address a set of kinematic pairs. • Let Z be a linear space spanned by all columns of matrix J. • Let K be an ordered set of kinematic pairs consisting of p elements. • Let Xi be a linear space spanned by these columns of matrix J which correspond to constraints imposed by i-th kinematic pair from the set K. • Let Y be a linear space spanned by these columns of matrix J which do not correspond to any kinematic pair from the set K. The previously discussed statement can be generalized and written it in the following form [7]: If the vector space Z can be represented by the following direct sum of subspaces: Z = X1 ⊕ X2 ⊕ ⋯ ⊕ Xp ⊕ Y;

ð17Þ

then constraint reaction forces corresponding to the kinematic pairs belonging to the set K are uniquely determined. It is worth noting that condition (17) is not fulfilled, when a joint, for which reaction forces cannot be uniquely determined, belongs to the set K. For some mechanisms condition (17) is not fulfilled for all kinematic pairs. On the other hand, if there are no redundant constraints, condition (17) is fulfilled for all pairs. The abovementioned conditions are purely mathematical, however, they form a basis for numerical methods that enable detection of kinematic pairs with uniquely solvable reactions. Three numerical methods of reaction solvability analysis are proposed in [7]. All of them are based on constraint Jacobian matrix investigation, which makes it possible to easily implement them in general purpose multibody software. The simplest, but the least effective, method consists in dividing the Jacobian matrix into submatrices and calculating their ranks. The most effective, but more complicated, method is based on QR decomposition of the Jacobian matrix. All necessary details are presented in [7], thus here they are omitted. 3. Joint reactions solvability and motion uniqueness — mechanisms with Coulomb friction in joints 3.1. Joint friction in mechanisms without redundant constraints To describe mechanisms with friction in joints we will apply slightly simplified model of Coulomb dry friction presented in [12]. Friction forces in joint determined by Coulomb's law are modelled as external forces, being known functions of normal reaction forces in this joint. The effect of stiction (modelled with constraint addition–deletion technique in [12]) can be incorporated into our model by modification of friction coefficient. Let us assume temporarily that constraint equations are independent. The equations of motion of a mechanism or a multibody system (11) can be extended for the case of joint friction and written in the form [2,5,12]: 

M T J

J 0



" #  f h˙ = Q + Q ðq; h; λÞ ; λ Γ

ð18Þ

where Q f is a vector of generalized friction forces, calculated for all joints where friction occurs. The vector Q f depends explicitly on Lagrange multipliers λ. We will discuss how the vector Q f(λ) is calculated. In this section equations are formulated in general form, details (for translational, revolute and spherical joints) are presented in Appendix A. Let us assume that friction occurs in joint labelled X, connecting bodies i and j. The columns of the transposed Jacobian matrix, corresponding to constraints imposed by this joint, are denoted by JX. Normal constraints reaction forces can be computed using

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Eq. (10). However, to calculate friction force in the joint, the physical reaction forces that act in the joint are required. The transformation of generalized reaction forces into physical reaction acting on body i can be expressed in the form [12]: p′i = Ai ðqÞðf X Þi = Ai ðqÞðJX Þi λX ;

ð19Þ

where (fX)i is a subvector of reaction forces (Eq. (10)) acting on body i (three-component in planar, and six-component in spatial systems), Ai(q) is a transformation matrix that depends on joint type and geometry, and sometimes on relative position of bodies i and j, p′i is a vector of physical reactions, which is expressed in an auxiliary joint frame. The vector of Coulomb friction forces t′ i acting on body i (with components in the auxiliary joint frame) can be expressed as:   t′i = μ ðq;hÞu p′i ;

ð20Þ

where uðp′i Þ is a function characteristic to joint type and geometry, which determines the operations performed upon physical reaction vector to evaluate friction forces and μ(q, h) is the coefficient of friction that depends on relative joint velocity (Fig. 1) and determines magnitude of the friction force. Finally, the generalized force vector for body i, due to friction in the joint, is: Q i = Ci ðqÞt′i ; f

ð21Þ

where Ci(q) is a transformation matrix. It is worth noting that Q fi is a subvector of Q f corresponding to body i. The friction force acting on body j is opposite to that acting on body i, i.e. t′j = −t′i (note that both vectors are expressed in the auxiliary joint frame), thus the generalized force vector for body j, due to friction in the joint, can be calculated as: Q j = −Cj ðqÞt′i : f

ð22Þ

The generalized friction forces Q if and Q jf are calculated for each joint, where friction occurs, to obtain Eq. (18). Note that the Lagrange multipliers vector λ appears on both the left and right hand-sides of Eq. (18). Recalling that λ determines magnitude of reaction forces in joints, one may interpret Eq. (18) as indicating that the friction force acting in a selected joint influences reaction forces in other joints of the system. h i T T T ˙ The numerical method used to solve Eq. (18) for h employs a fixed point iteration algorithm [12], in which the best λ estimate of λ that is available, usually the solution for λ at the preceding step time step, is substituted into the right-hand side of Eq. (18) and a new estimate of λ is calculated from the solution of Eq. (18). The process is repeated until the difference between two consecutive estimates is less than arbitrarily chosen small value. The practicality of this algorithm rests upon the fact, that the coefficient matrix on the left-hand side of Eq. (18) is non-singular in case constraints are independent and it is a sparse matrix [12]. Moreover, for given q and h, the coefficient matrix is constant during the whole iterative process. h i T T T ˙ It should be pointed out that in practical computations it is assumed in general, that Eq. (18) has only one solution h λ and the fixed point algorithm is guaranteed to converge to the solution in each step. 3.2. Mechanisms with redundant constraints and joint friction Let us assume now that constraints imposed on a multibody system are redundant and consistent. Eq. (18) is still valid in this case, but Jacobian matrix is rank deficient and coefficient matrix on the left-hand side of Eq. (18) is singular. Thus, Eq. (18) can have many solutions for Lagrange multipliers λ, and hence reactions forces are not uniquely determined. Since friction forces in ˙ joints are modelled as external forces and they depend on reaction forces, motion of the MBS, represented by acceleration vector h,

µ A B C v

Fig. 1. Friction coefficient vs. relative velocity in the zero velocity vicinity: A — Coulomb model (discontinuous), B — approximated Coulomb model (continuous), C — approximated Coulomb model with Stribeck effect.

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is not defined uniquely as well. If equations of motion of the MBS with consistent redundant constraints are solved using constraints elimination technique (Section 2.1) and fixed point iteration algorithm described above, then as a consequence of different selection of eliminated constraints, one can obtain different solutions for joint reactions and the MBS motion. Now we will show that, despite redundant constraints existence in the MBS, simulated motion of the MBS, obtained in the presence of friction in joints and after the dependent constraints elimination, is in some cases unique. Uniqueness is understood as the fact that redundant constraint elimination does not influence accelerations calculated form Eq. (18), i.e. the accelerations remain the same for any selection of independent constraints. Firstly, we will consider a MBS with friction occurring in only one joint. We will formulate sufficient conditions for uniqueness of acceleration calculated from Eq. (18). Afterwards similar conditions for a general MBS, with Coulomb friction occurring in many joints, will be formulated. Let friction forces occur only in the joint connecting bodies i and j. Let JX denote the columns of the transposed Jacobian matrix corresponding to constraints imposed by this joint, and let JY denote the remaining columns of the transposed Jacobian matrix. Friction forces in the considered joint can be determined using formulas (19)–(22) and incorporated into Eq. (18). Let us assume that dependent (redundant) constraints were eliminated from the system using one of the well-known methods (e.g., [5]). Eq. (18), after the elimination process, can be rewritten in the form (the asterisk symbol indicates the matrices after dependent constraints elimination): 2

JX

M

6  T 6 ðJ Þ 4 X

JY

h˙

2

3

6Q + Q 7 76 6λ 7 = 6 07 6 54 X 5 4 0 λY

0

ðJY ÞT

32

0

f

3

ððJ Þ λ Þ + Q ððJ Þ λ Þ 7  X i

f

X

 X j

X

7 7: 5

ΓX ΓY

ð23Þ

The symbol Q f((JX)iλX) is used to stress that generalized friction force acting on body i depends explicitly on joint reaction  T forces (see Eqs. (19)–(21)). h     iT T  T T Let us assume that exactly one solution of Eq. (23) exist, namely , where λ = λX T . Let Z, X and λY h˙ λ

ð Þ ð Þ

Y be the vector spaces spanned by columns of the transposed Jacobian matrix: X = spanðJX Þ ;

Z = spanðJÞ ;

Y = spanðJY Þ:

ð24Þ

The following two statements can be formulated and then justified: 

A. If vector spaces (24) satisfy relationship Z = X ⊕ Y, then vector h˙ is determined uniquely, i.e. it does not depend on selection of independent constraints. B. If vector spaces (24) satisfy relationship Z = X ⊕ Y, then the normal joint reaction forces and friction forces corresponding to constraints described by JX are determined uniquely, i.e. they do not depend on selection of independent constraints. To justify statement A let us assume that it is false, i.e. there exist a different selection of independent constraints, such that  Eq. (23) has a solution for acceleration not equal to h˙ . Let us denote this selection of independent constraints by the double asterisk symbol. Eq. (23) can be rewritten as: 2

M

6  T 6 ðJ Þ 4 X T ðJ Y Þ



J X

J Y

32

 h˙

3

2

ð

Þ

ð

f   f   6 Q + Q ðJX Þi λX + Q ðJX Þj λX 7 6  7 6 7 6 7 0 5 4 λX 5 = 6 Γ X 4  λY 0 Γ Y

0 0

3

Þ7

7 7; 5

ð25Þ



with h˙ ≠ h˙ . The theorem A assumptions yield:      X = spanðJX Þ = span JX = span JX :

ð26Þ

Eq. (26) indicates that there exist constant, square and non-singular transformation matrices AX and AY such that:











T



JX = JX AX ; JY = JY AY ; ΓX = AX ΓX :

ð27Þ

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Substituting Eq. (27) into Eq. (25), one obtains: 2

JX AX

M

6 T  T 6 A ðJ Þ 4 X X

JY AY

0

T  T AY ðJY Þ

0

32

 h˙

2

3

6 7 76 6 6  7 0 7 5 4 λX 5 = 6 4 0 λ Y

 3 + Q f ðJX Þj AX λ Q + Q f ðJX Þi AX λ X X 7 7 7: ATX ΓX 5

ð

Þ

ð28Þ

ATY ΓY

Let us substitute the following terms into Eq. (28): 



λX = AX λX ; λY = AY λY

ð29Þ

T T and A− and then premultiply the acceleration equations from Eq. (28) by matrices A− X Y , respectively. The following equation is obtained:

2

M

6  T 6 ðJ Þ 4 X ðJY ÞT

JX

 3 f  f  Q + Q ð J Þ λ + Q ð J Þ λ X X i j 6 X 7 X 7 76 6 7 6 7 07 7: 5 4 λX 5 = 6 ΓX 4 5  0 λY Γ

JY

0 0

32

 h˙

3

2

ð

Þ

ð30Þ

Y

It may be seen that coefficients of Eq. (30) and of Eq. (23) are identical. As a consequence, we obtain that solution of Eq. (30)    

h     iT  T T T T  T  T T λ = λ where λ = λX T . It proves that the statement A must satisfy Eq. (23), and thus h˙ λY h˙

ð Þð Þ

ð Þð Þ

is true. The justification of the statement B is now straightforward, since (see Eq. (27)): 













JX λX = JX AX λX = JX λX = JX λX ;

ð31Þ

and thus: Q

f      JX i λX

=Q

f      JX i λ X :

ð32Þ

Let us assume now that friction forces are present in many (namely p) joints of the overconstrained MBS. If the columns of transposed Jacobian, that correspond to constraints imposed by these p joints, span linear spaces that satisfy Eq. (17), then the acceleration vector of the whole system and reaction forces for the discussed joints are determined uniquely in the sense defined earlier. This statement can be justified by argumentation similar to that presented above for the case of friction in a single joint. Finally, we can formulate two additional remarks, which are conclusions of the presented considerations, provided that assumptions of statements A and B are still valid. • We have assumed that Eq. (23) has a unique solution. The computational practice shows that it is a reasonable assumption. ˙ and λ can exist. It is worth noting that in such a case a change of the set of Theoretically, however, many solutions for h independent constraints does not generate new solutions. • If a solution of Eq. (23) exists for a certain selection of independent constraints, then it also exists for other, different sets of independent constraints. 4. Examples The problem of joint reactions solvability and the problem of motion uniqueness in the presence of joint friction are illustrated in this section. Examples present overconstrained rigid body mechanisms; the first one is planar, and the second one is spatial. 4.1. Planar mechanism A planar rigid body mechanism (Fig. 2), consisting of 4 moving bodies interconnected by 3 translational and 3 revolute joints, is discussed. The mechanism is redundantly constrained. Even in such simple mechanism identification of joints for which reactions can be uniquely determined is not straightforward, both without and with Coulomb friction. The constraint equations will be formulated and analysed in order to identify joints for which reactions can be uniquely determined. Then, motion of the mechanism will be simulated, firstly without and then with Coulomb friction in selected joints.

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y0 y2 T2

y4

Joint 5 L

4

x2

2

M Joint 6

x4

Joint 2 Joint 3

y3

gravity

3

x3

v12 k

y1 Joint 4

0 O

Joint 1

1 T1

K

x1

x0

Fig. 2. Planar mechanism.

4.1.1. Constraint equations and reactions solvability analysis The global reference frame π0 (x0y0) is established on the basis 0 and the local reference frames πi (xiyi) are established on the other bodies (Fig. 2). The absolute coordinates describing the mechanism form the vector q: h iT  T T ð33Þ q = qT1 qT2 qT3 qT4 ; qi = rTi φi ; where ri = ½ xi yi
T represents the position of the local reference frame πi origin with respect to the global frame π0, and φi is the angle of the local frame πi rotation with respect to the global frame. Let us consider a translational joint, formed by bodies i and j, and assume that point A belongs to body i and point B as well as vector vij belong to body j (for example, in Fig. 2 joint 3 is formed by bodies 1 and 2, A = K, B = L). Points A and B lie on a line parallel to the axis of relative joint motion, and vector vij is perpendicular to this axis. The translational joint can be described by two scalar constraint equations. The first one represents the fact that vector BA is perpendicular to vector vij, and the second equation represents the fact that body i does not change its orientation with respect to body j: 2 3 ði Þ ð jÞ T ð jÞ  ri + Ri sA −rj −Rj sB Rj vij tra 5 = 021 ; qi ; qj ≡4 ð34Þ Φ φi −φj −ψij where s(k) P is the position vector of point P in the local reference frame πk, ψij is a constant value and Rk is the direction cosine matrix transforming quantities from πk to π0:   cos φk −sin φk : ð35Þ Rk ≡R k ðφk Þ≡ sin φk cos φk Let us consider a revolute joint formed by bodies i and j at point P (for example, in Fig. 2 joint 6 is formed by bodies 3 and 4, P= M). The constraint equations describing this joint can be derived by requiring that the point Pi on body i coincides with point Pj on body j:  rev ði Þ ð jÞ Φ qi ; qj ≡ri + Ri sP −rj −R j sP = 021 : ð36Þ There are three translational and three rotational joints in the analysed mechanism, thus twelve scalar constraint equations can be formulated (see Appendix B for details): ΦðqÞ≡½ Φ1 ðqÞ

T

… Φ12 ðqÞ
= 0121 :

ð37Þ

The mechanism is described by 12 coordinates and 12 scalar constraint equations. The mechanism has one degree of freedom (points K and L can simultaneously move along axes x0 and y0, respectively). Hence we can state that redundant constraints are imposed on the mechanisms. The methods presented in [7] were applied to find joints for which normal reactions can be uniquely determined. It was found that the reactions in the revolute joints (4, 5 and 6) can be determined uniquely, whereas the reactions in the translational joints (1, 2 and 3) are dependent and cannot be determined uniquely (see Appendix B for details).

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4.1.2. Simulated motion of the frictionless mechanism To observe the reaction forces during simulation, equations of motion for the mechanism have been formulated and solved. It was assumed that gravitational forces are present and constant torques T1 = 10 Nm and T2 = 30 Nm are applied to the sliders 1 and 2, respectively. Moreover, a spring with stiffness constant k = 30 N/m connects slider 2 with the base part 0. It was also assumed that, for each moving part, the centre of its mass coincides with the local reference frame origin. The following mass matrices characterize the inertial properties of the moving parts:

M = diagðM1 ; M2 ; M3 ; M4 Þ;

Mi = diagðmi ; mi ; Ji Þ;

mi = 3 kg;

2

Ji = 1 kgm ;

i = 1; 2; 3; 4:

ð38Þ

The equations of motion were integrated over time, with initial conditions q0 = ½ 2 0 0 00 2 0 02 1 0 01 2 0
T and 0 q˙ = 0121 . Two variants of redundant constraints elimination were studied (it should be noted that more than two variants of elimination are possible). Constraint equations Φ4 and Φ6 were eliminated in these two variants, respectively. The reaction forces in translational joint 1, applied to the slider, were investigated. The resultant moment of reaction forces about the local frame origin was calculated. It was found that in the first variant of redundant constraints elimination the investigated torque was constant and equal −30 Nm. In the second variant the torque was also constant, but equal −40 Nm. It is worth noting that the alternatively eliminated constraint equations Φ4 and Φ6 represent joints 2 and 3, nevertheless, their elimination affected reaction torque in joint 1. The same simulations showed that reaction forces in revolute joints 4, 5 and 6 were the same for both variants of redundant constraints elimination. Thus, the results of simulations corroborate correctness of detection of reaction forces that can be determined uniquely. It should be emphasized that, despite different joint reactions, for both variants of redundant constraints elimination simulated motion of the mechanism was the same. This result shows that when friction effects are neglected, simulated motion is unique, even when joint reactions are not unique. 4.1.3. Uniqueness of motion in the presence of Coulomb friction in joints The elimination of redundant constraint from Eq. (37) affects the joint reaction forces, and thus may affect the friction forces as well. As a result of different selection of redundant constraints, different motion of the mechanism may be obtained. In order to investigate this problem, three different cases of joint friction forces were considered. In each case two variants of redundant constraints elimination were studied, i.e. equation Φ4 or Φ6 was eliminated. In the first case it was assumed that friction effects are present only in the revolute joints 4 and 5 (see Appendix A for the details of friction forces calculation). The equations of motion were integrated over time. At each integration step the nonlinear algebraic equation set (18) was solved using a fixed point algorithm to find the unknown accelerations and Lagrange multipliers. Calculations were performed for shaft radius a = 0.15 m and for friction coefficient μR = 0.1. The approximated Coulomb model (see Fig. 1, variant B) was used, in all simulations, to establish relation between the friction coefficient and the joint relative velocity. The time histories of x1 coordinate (slider 1 displacement) obtained during simulations are presented in Fig. 3. Simulated motion of the mechanism obtained for the set of constraints without equation Φ4 was the same as motion obtained for the other variant of redundant constraint elimination (the magnitude of observed differences was comparable with the numerical precision). As it was discussed earlier, the reaction forces in the revolute joints 4 and 5 can be uniquely determined. Therefore, despite the friction effects in these joints, motion of the mechanism was uniquely determined. The calculated motion did not depend on the selection of eliminated constraints.

2.0

displacement (m)

without friction with friction (all cases of redundant constraints elimination)

1.5

1.0

0.5

0

1

2

3

4

time (s) Fig. 3. Coordinate x of the slider 1 vs. time, without and with Coulomb friction in the revolute joints 4 and 5.

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2.0

displacement (m)

1.8 without eq. 4 without eq. 6

1.6 1.4 1.2 1.0 0.8

0

1

2

3

4

time (s) Fig. 4. Coordinate x of the slider 1 vs. time, for two different constraints sets and Coulomb friction in the translational joint 1.

2.0

displacement (m)

1.8 without eq. 4 without eq. 6

1.6 1.4 1.2 1.0 0.8

0

1

2

3

4

time (s) Fig. 5. Coordinate x of the slider 1 vs. time, for two different constraints sets and Coulomb friction in the translational joint 1 and in the revolute joints 4 and 5.

In the second case it was assumed that friction forces are present only in the translational joint 1 (details of friction forces calculation are given in Appendix A). Calculations were performed for slider length l = 0.2 m and for friction coefficient μT = 0.01. The time histories of x1 coordinate obtained for both variants of redundant constraints elimination are presented in Fig. 4. It is clearly visible that simulated slider motion is different for different selections of eliminated redundant constraints. The obtained results are consistent with the considerations of Section 3. As it was shown earlier, the reaction forces calculated for translational joint 1 depend on the redundant constraints selection and cannot be uniquely determined. Thus, after applying friction forces in this joint, motion of the mechanism cannot be uniquely determined as well. In the third case it was assumed that friction effects are present in the translational joint 1 and in the revolute joints 4 and 5. The time histories of x1 coordinate obtained for both variants of redundant constraint elimination are presented in Fig. 5. It was found that the simulated slider motion depends on the redundant constraint selection. In that way, the obtained results are similar to the results discussed previously. As it was shown earlier, the reaction forces in the revolute joints 4 and 5 can be uniquely determined. On the contrary, the reaction forces in the translational joint 1 cannot be uniquely determined. Therefore, after applying friction forces in the translational joint, motion of the mechanism cannot be uniquely determined.

4.2. Spatial mechanism A spatial parallelogram mechanism (Fig. 6), consisting of the basis 0 and seven movable bodies 1–7 is discussed in this example. The bodies are connected by 12 spherical joints. The mechanism has seven degrees of freedom. The platform 7 can perform a translational motion (with constant orientation), moreover, links 1–6 can rotate along their longitudinal axes, not affecting the

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323

Fig. 6. Spatial parallelogram mechanism.

other bodies motion (the mechanism has 6 internal degrees of freedom). Redundant constraints are present in this mechanism. Absolute coordinates are chosen to describe position and orientation of bodies. The example starts with analysis of joint reaction solvability in a frictionless mechanism. Constraint equations are formulated, then Jacobian matrix is calculated, and finally the Jacobian matrix is analysed to verify which joint reaction forces can be uniquely determined. Joint friction and its influence on motion uniqueness are considered in the next part of the example. Motion of the mechanism is studied, and three different cases are examined. In the first case friction effects are introduced only in these joints for which reaction forces can be uniquely determined. In the second case friction is present only in joints with non-unique reactions. In the third case friction forces are applied in both categories of joints. In all three cases the results of different selections of eliminated redundant constraints are investigated. 4.2.1. Constraint equations and reactions solvability analysis The global reference frame π0 (x0y0z0) is established on the basis 0 and the local reference frames πi (xiyizi) are established on the other bodies (Fig. 6). The absolute coordinates of the mechanism form the vector q:  q = qT1

qT2

qT3

qT4

qT5

qT6

qT7

T

;

h qi = rTi

φTi

iT

;

ð39Þ

where ri = ½ xi yi zi
T is the position of πi origin with respect to the global frame π0, and φi = ½ αi βi γi
T are angles of z–x–z Euler rotations describing the orientation of πi with respect to π0. The direction cosine matrix transforming quantities from πi to π0 is given by: R i = Ri ðαi ; βi ; γi Þ≡R z ðαi Þ R x ðβi Þ R z ðγi Þ; 2

cosψ −sinψ 0 R z ðψÞ≡4 sinψ cosψ 0 0 0 1

3 5;

2

1 R x ðψÞ≡4 0 0

0 cosψ sinψ

3 0 −sinψ5: cosψ

ð40Þ

Consider a spherical joint formed by bodies i and j at point P. The constraint equations describing this joint can be derived by requiring that the point Pi on body i coincides with point Pj on body j:

P

Φ

 ðiÞ ð jÞ qi ; qj ≡ri + R i sP −rj −Rj sP = 031 ;

where s(k) P is the position vector of point P in the local reference frame πk.

ð41Þ

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The above equations can be formulated for all 12 spherical joints (A, B, C, D, E, F, G, H, K, L, M and N) in the mechanism (r0 = 0 and R0 = I should be substituted for the points belonging to the base 0). All constraint equations can be written together in the form: h T ΦðqÞ≡ ΦA

ΦB

T

ΦC

T

ΦD

T

ΦE

T

ΦF

T

ΦG

T

ΦH

T

ΦK

T

ΦL

T

ΦM

T

ΦN

T

iT

= 036×1 ;

ð42Þ

The above constraint equations were differentiated to obtain the Jacobian matrix. Details of calculations are presented in Appendix B. The mechanism is described by 42 coordinates, and 36 constraint equations are imposed on them. The rank of the Jacobian matrix equals 35, thus the mechanism has 7 degrees of freedom (six of them are associated with links 1–6 rotations along their longitudinal axes) and one constraint equation is redundant. The methods presented in [7] were applied to find joints for which normal reactions can be uniquely determined. It was found that the reactions in joints K, L, M and N can be determined uniquely, whereas reactions in the remaining joints are dependent and cannot be determined uniquely (see Appendix B for details). 4.2.2. Simulated motion of the frictionless mechanism To observe the reaction forces during simulation, equations of motion (11) for the mechanism have been formulated and solved. It was ð0Þ assumed that gravitational forces are present and external constant torque T7 = ½ 0 300 0
T ðNmÞ and force ð0Þ F7 = ½ 0 40 0
T ðNÞ are applied to the platform (part 7) centre. It was also assumed that, for each moving part, the centre of its mass coincides with the local reference frame origin. The following mass matrices characterize the inertial properties of the moving parts: M = diag ðM1 ;…; M7 Þ; Mi = diag ðmL ; mL ; mL ; JL ; JL ; JL = 100Þ;

i = 1; …; 6;

ð43Þ

M7 = diag ðmP ; mP ; mP ; JP ; 2JP ; JP Þ; mL = 1 ðkgÞ;

JL = 2 = 3 ðkgm2 Þ; mP = 10 ðkgÞ; JP = 10 = 3 ðkgm2 Þ:

The equations of motion were integrated over time, with initial conditions q0 = q⁎ (see Appendix B for the values of q⁎) and 0 ˙q = 0 421 . Three variants (out of several other possible) of the redundant constraints elimination were studied. In the first case constraint equation Φ5 (y component of ΦB constraints) was eliminated, in the second case — equation Φ9 (z component of ΦC), and equation Φ18 (z component of ΦF) in the third case. The reaction forces in joints G and N were investigated. It was found that joint N reaction forces were the same (to within numerical precision errors) in all variants of redundant constraint elimination. In the case of joint G, the calculated reaction forces differed depending on selections of eliminated constraints, as it is shown in Fig. 7. The alternatively eliminated constraint equations are associated with joints B, C and F. It is worth noting that elimination affected the reaction force in joint G. The same simulations showed that the reaction forces in joints K, L, M and N were not affected by the choice of eliminated redundant constraints. Thus, the results of simulations corroborate correctness of uniquely determined reaction forces detection. It should be emphasized that, despite calculating different joint reactions in different variants of redundant constraints elimination, the obtained mechanism motion was always the same. This confirms well-known fact that, in case of mechanism with frictionless joints, the individual reactions are not unique, but their resultant effect (when motion is concerned) is unique.

20

force (N)

0

without eq. 5 without eq. 9 without eq. 18

-20

-40

-60

-80

0

2

4

6

time (s) Fig. 7. Joint G reaction force (y component).

8

10

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325

2.5

displacement (m)

2 without eq. 5 without eq. 9 without eq. 18

1.5 1 0.5 0 -0.5

0

2

4

6

8

10

time (s) Fig. 8. Platform position (x coordinate) vs. time for three different constraints sets and friction effects in joints G and H.

4.2.3. Uniqueness of motion in the presence of Coulomb friction in joints The elimination of redundant constraint equation from the set (42) affects some of the calculated joint reaction forces, and thus may affect the friction forces as well. As a result of different selection of redundant constraints, different motion of the mechanism may be obtained. Exploiting the results of Section 3, it is possible to predict whether motion of the overconstrained MBS will be unique after introducing friction in selected joints. In order to investigate the problem of motion uniqueness, three variants of simulations were performed. In the first case friction effects were introduced in joints M and N only (see Appendix A for the details of friction forces calculation). The equations of motion were integrated over time, with initial conditions q0 = q⁎ and q˙ 0 = 0421 . At each integration step the nonlinear algebraic equation set (18) was solved using the fixed point algorithm to find the unknown accelerations and Lagrange multipliers. The calculations were performed for friction coefficient μ = 0.3 and ball-joint radius ρ = 0.1 m (the approximated Coulomb model — Fig. 1, variant B — was used to establish relation between the friction coefficient and the joint relative velocity). Various variants of the redundant constraints elimination were studied. The same motion of the mechanism was obtained (to within numerical precision errors) for all performed simulations. The results of simulations are consistent with the considerations of Section 3. In the second case, friction effects were introduced in joints G and H only. Various variants of the redundant constraints elimination were studied. Results obtained for three of them are presented in Fig. 8. In each case simulated motion was different. As it was shown earlier, the reaction forces calculated for joints G and H depend on the redundant constraints selection and cannot be uniquely determined. Thus, after applying friction forces in these joints, motion of the mechanism cannot be uniquely determined as well. It should be emphasized that constraint equations Φ5, Φ9 and Φ18 represent joints B, C and F, respectively. Nevertheless, their elimination affected friction in joints G and H. The obtained results confirm the considerations of Section 3. In the third case friction effects were introduced in joints G, H, M and N. Three exemplary variants of the redundant constraints elimination were studied again. The obtained results (x7 coordinate time histories) are presented in Fig. 9. It was found that simulated motion depends on the redundant constraints selection. In that way, the results of simulations are similar to those obtained for friction in joints G and H only. The reaction forces in joints M and N can be uniquely determined. On the contrary, the

2.5 without eq. 5 without eq. 9 without eq. 18

displacement (m)

2 1.5 1 0.5 0 -0.5

0

2

4

6

8

10

time (s) Fig. 9. Platform position (x coordinate) vs. time for three different constraints sets and friction effects in joints G, H, M and N.

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reaction forces in joints G and H cannot be uniquely determined. Therefore, after applying friction forces in the joints G and H, motion of the mechanism cannot be uniquely determined. Once again, the obtained results are consistent with the considerations of Section 3. 5. Conclusions and discussion • If redundant constraints exist in a multibody system, 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. • Methods presented here enable us to detect 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 loads on crucial joints and bodies. • In the case of MBS with friction, if the joint reaction force cannot be uniquely determined, then the joint fiction cannot be uniquely determined as well. As a result, motion of MBS may not be unique. Simulated motion of a MBS with joint friction may depend on the results of redundant constraints elimination. • Presented considerations enable us to formulate conditions sufficient for the uniqueness of MBS motion, despite the existence of redundant constraints and friction presence in selected joints. The simulated motion is unique when friction effects are considered only in these joints in which the joint reactions can be determined uniquely. • The methods presented here may be extended to analyse MBS with joint stiction, modelled by constraints addition–deletion, as proposed in [12]. • Uniqueness of joint reaction (and joint friction) depends only upon the structure of the multibody system, thus it is irrelevant which type of coordinates is used. It should be noted, however, that formulating presented considerations in other than absolute coordinates, e.g., in relative joint coordinates, is not straightforward and requires additional research. It should be emphasized that the paper is focused on rigid body mechanisms. The rigid body assumption is made in many technical problems, and thus developed methods can be helpful when dealing with this type of problems. However, it is worth noting that when redundant constraints are imposed on a rigid multibody system, some reaction forces cannot be uniquely determined and further analyses utilizing these forces, especially analyses involving joint friction, are pointless. Therefore, in many cases in order to find a unique set of all joint reaction forces in such overconstrained system, and then to find its motion uniquely, it is necessary to reject the assumption that all bodies are rigid. Flexible bodies introduce additional degrees of freedom to the mechanism, which usually makes the constraint equations independent (the previously redundant constraints are no longer applied to the same degree of freedom). Unfortunately, a flexible multibody model is much more complicated and numerically less effective than a rigid one, and additionally requires much more input data. Moreover, it can be shown that modelling some mechanism bodies as flexible does not guarantee that unique joint reactions can be found. The authors are preparing a separate paper to discuss these issues in detail. Acknowledgements The Project was co-financed by the European Regional Development Fund within the framework of the 1. Priority Axis of the Innovative Economy Operational Programme, 2007–2013, through grant PO IG 01.02.01-00-014/08-00, and by the Institute of Aeronautics and Applied Mechanics statutory funds. Appendix A. Joint friction models To illustrate problems addressed in the paper, models of Coulomb friction in joints, based on the ideas proposed in [12] and variant B of friction coefficient (presented in Fig. 1), were developed and used during simulations. The models are not very complicated, since some simplifications were introduced. However, to illustrate problems discussed in the article, any friction model in which joint friction forces depend on normal reactions may be used. A.1. Translational joint (planar) Consider a translational joint X connecting bodies i and j (Fig. 10). An auxiliary joint coordinate frame π′i(x′i y′i) is established on body i. The physical reaction force corresponding to the joint, calculated with respect to the auxiliary frame π′i origin, is given as:  p′i = Fx

Fy

T

T

" =

T

T

Bi ðβÞRi ðiÞT

−sA R Ti ΩT

0 1

# ðf X Þi = Ai ðf X Þi ;

ðA:1Þ

matrix where Bi(β) is a transformation matrix from the auxiliary joint frame π′i to the body frame πi (xi yi), Ri is a transformation  0 −1 (i) . Fx and Fy are the from the local frame πi to the global frame π0 (the vector sA is resolved in the frame πi) and Ω = 1 0 components of force applied at the auxiliary joint frame π′i origin, and T is the torque about this origin.

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327

Fig. 10. Translational joint geometry and reaction forces.

It was assumed that the joint friction force acts along AB edge of the slider body. The friction force, decomposed in the joint frame π′i, can be written as [12]: h  ti = μ vi; j d ðjN−FA j + jFB jÞ

0

0

iT

;

N = Fy ;

FA = FB = T = l;

ðA:2Þ

where vi, j is the relative sliding velocity of body j with respect to i, and l is the distance between A and B. The generalized Coulomb friction forces, that are applied to bodies i and j, can be expressed as: f

Q i = Ci ti

f

and Q j = −Cj ti ;

ðA:3Þ

with: 2

Ri Bi Ci = 4  ðiÞ T T sA Ω Bi

0 1

3 5

2 and

Cj = 4 

R i Bi ði Þ

ri + Ri sA −rj

T

0 T

Ω Ri Bi

1

3 5:

Fig. 11. Revolute joint geometry and reaction forces.

ðA:4Þ

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A.2. Revolute joint (planar) Consider a revolute joint X connecting bodies i and j (Fig. 11). An auxiliary joint coordinate frame π′i(x′i y′i) is established on body i. The physical reaction force corresponding to the joint, calculated with respect to the joint frame π′i origin, can be obtained using Eq. (19). The axes of the joint frame π′i are parallel to the body frame πi axes, thus the matrix Ai can be written as: " Ai =

RTi ðiÞT

−sA R Ti ΩT

0 1

# :

ðA:5Þ

For a revolute joint the friction effect is a torque in the joint [12] which can be expressed as:  ti = 0

 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 T 2 ; 0 aμ ωi;j Fx + Fy

ð Þ

ðA:6Þ

where a is the radius of the shaft and ωi,j denotes the relative angular velocity of body j with respect to i. The generalized Coulomb friction forces, that are applied to bodies i and j, can be expressed as: f

ðA:7Þ

f

ðA:8Þ

Q i = ti ; Q j = −ti ; hence Ci = Cj = I. A.3. Spherical joint (spatial)

Consider a spherical joint (labelled X) connecting bodies i and j, shown in Fig. 12. Local reference frames πi and πj are (j) established on bodies i and j, respectively. The vectors s(i) P and sP are constant in appropriate frames. Point P is the centre of the spherical joint. An auxiliary joint reference frame π′i(x′i y′i z′i) is established on body i, its origin coincides with point P and its orientation is the same as orientation of πi. The generalized normal joint reaction, applied to body i, can be calculated by multiplying the submatrix of the transposed Jacobian corresponding to joint X and body i by the Lagrange multipliers associated with joint X: 2 4

ð0Þ

Fi

ði Þ

Ti

3 5 = ðf X Þ = ðJX Þ λX : i i

ðA:9Þ

In our case the generalized reaction (fX)i can be interpreted as a six-element-vector consisting of force F(0) (with components i expressed in the global frame π0), acting along line passing through the frame πi origin, and of torque T(i) i (expressed the local

Fig. 12. Spherical joint normal and friction forces distribution.

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329

frame πi). The physical normal joint reaction force and torque, applied to body i, calculated with respect to the auxiliary frame π′i origin, are given as: "   F′ i ′ = pi= T′ i

# 2 ð0Þ 3 F 4 i 5 = Ai ðf X Þ ; i ði Þ I3×3 Ti

RTi

0

ðiÞ

−˜sP RTi

ðA:10Þ

ðiÞ

where ˜sP is a skew-symmetric matrix associated with vector s(i) P and Ri is a direction cosine matrix transforming quantities from πi to π0. It is worth noting that in case of spherical joint it is obvious that T′i = 0, thus Eq. (A.10) can be simplified. The joint radius equals ρ, thus coordinates of point C, which is the point of contact between bodies i and j, can be calculated as: s′C = −ρ

F′i F′i ffi: = −ρ qffiffiffiffiffiffiffiffiffiffiffi jF′i j F′ T F′

ðA:11Þ

i

i

The angular velocity of body j with respect to i (expressed in π′i) can be calculated as:

T ð jÞ ði Þ ω′i;j = Ri R j ωj −ωi :

ðA:12Þ

The linear relative velocity of sliding of body j with respect to body i (at point C) is given by the following formula: ˜ ′i; j s′ C : v′i; j = ω

ðA:13Þ

The Coulomb friction force F′i:fr is applied to body i at point C. Its magnitude depends on normal reaction F′i and on friction coefficient μ. The friction force F′i:fr acts along direction of relative velocity v′i;j . Thus, the following equation can be written:  v′  qffiffiffiffiffiffiffiffiffiffiffiffi v′i; j i; j F′i: fr = μ v′i; j ⋅ F′i ⋅



= μ v′i; j ⋅ F′i T F′i ⋅ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ′

v i; j v′i;Tj v′i; j

ðA:14Þ

The physical friction force and torque, calculated with respect to the auxiliary frame π′i origin (point P) and expressed in this frame, can be calculated as: t′i =



F′i:fr ′r ˜s′C Fi:f



 =

I33 ˜s′C



F′i:fr :

ðA:15Þ

The physical friction force and torque t′i can be transformed from the auxiliary frame π′i to the local frame πi, and then friction force can be expressed in the global frame π0. As a result, the generalized friction force is obtained: " f Qi

=

#

Ri

0

˜sðPiÞ

I33

t′i = Ci t′i :

ðA:16Þ

The physical friction force acting on body j is opposite to that acting on body i (this force, resolved in the frame πj, is denoted as t″j ). Thus, the following equation can be written: 2 ″ tj

=4

3

Rj Ri

T

0

0

R Tj Ri

  5 −t′i :

ðA:17Þ

Finally, the generalized friction force acting on body j can be calculated: " f Qj

=

Rj

0

˜sðPjÞ

I33

#

" ″ tj

=−

Ri

0

˜sðPjÞ RTj R i

RTj Ri

# t′i = −Cj t′i :

ðA:18Þ

Appendix B. Redundant constraints analysis In this appendix the Jacobian matrices are derived and then redundant constraint analyses are performed. Methods described in [7] are applied to find joints for which normal reaction forces can be uniquely determined. Both exemplary mechanisms are considered. Note that in the following text physical units are omitted, since all angular dimensions are expressed in radians and all linear dimensions are expressed in meters.

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J. Frączek, M. Wojtyra / Mechanism and Machine Theory 46 (2011) 312–334

B.1. Planar mechanism ðiÞ

ð1Þ

ð jÞ

ðiÞ

ð2Þ

ð jÞ

The constraint equations describing joint 1 formed by bodies 1 and 0 can be derived by substituting sA = sK = 0; sB = ð jÞ ð0Þ = 0; vij = v10 = ½ 0 1
T and ψij = ψ10 = 0 into Eq. (34), which yields (note that ri = r0 = 0 and φi = φ0 = 0 must be substituted):       y Φ ðqÞ 0 1 ≡ 1 = = 0: ðB:1Þ Φ ðqÞ≡ 1 φ1 Φ2 ðqÞ 0

ð0Þ sO

Similarly, the equations for joint 2, formed by bodies 2 and 0, can by derived by substituting sA = sL = 0; sB = ð0Þ ð jÞ ð0Þ sO = 0; vij = v20 = ½ 1 0
T and ψij = ψ20 = 0 into Eq. (34), which yields:       x Φ ðqÞ 0 2 Φ ðqÞ≡ 3 = 0: ≡ 2 = φ2 Φ4 ðqÞ 0 ðiÞ

ð1Þ

ð jÞ

ðB:2Þ

ð2Þ

ð jÞ

ð2Þ

1
T and ψij = ψ12 = 0 into Eq. (34), one obtains the constraint

Substituting sA = sK = 0; sB = sL = 0; vij = v12 = ½ 1 equations for joint 3, formed by bodies 1 and 2:

      ðx1 −x2 Þðcos φ2 −sin φ2 Þ + ðy1 −y2 Þðsin φ2 + cos φ2 Þ Φ ðqÞ 0 3 Φ ðqÞ≡ 5 ≡ = = 0: φ1 −φ2 Φ6 ðqÞ 0

ðB:3Þ

(1) The constraint equations describing joint 4, formed by bodies 1 and 3, can be derived by substituting s(i) P = sK = 0 and ð jÞ ð3 Þ sP = sK = ½ 0 −1
T into Eq. (36), which yields:

      x1 −x3 −sin φ3 Φ ðqÞ 0 4 Φ ðqÞ≡ 7 ≡ = = 0: Φ8 ðqÞ y1 −y3 + cos φ3 0

ð jÞ sP

ðB:4Þ

(2) Similarly, the equations for joint 5, formed by bodies 2 and 4, can be derived by substituting s(i) P = sL = 0 and ð4 Þ T = sL = ½ −1 0
into Eq. (36), which yields:

      x −x4 + cos φ4 Φ9 ðqÞ 0 5 Φ ðqÞ≡ ≡ 2 = = 0: y2 −y4 + sin φ4 Φ10 ðqÞ 0 ði Þ

ð3Þ

ð jÞ

ðB:5Þ ð4Þ

1
T and sP = sM = ½ 1

And finally, substituting sP = sM = ½ 0 joint 6, formed by bodies 3 and 4:

0
T into Eq. (36), one obtains the constraint equations for

      Φ ðqÞ x3 −sin φ3 −x4 −cos φ4 0 6 Φ ðqÞ≡ 11 ≡ = = 0: y3 + cos φ3 −y4 −sin φ4 Φ12 ðqÞ 0

ðB:6Þ

The constraints imposed on the mechanism can be written in the short form (see Eq. (37)): ΦðqÞ≡½ Φ1 ðqÞ

T

… Φ12 ðqÞ
= 0121 :

ðB:7Þ

Differentiating the constraint equations, one obtains the constraint Jacobian matrix: 2

0

6 60 6 60 6 6 6 7 6 2 60 7 6 Φ ðqÞ 7 6 6 q 6 7 6 6κ 7 6 6 1 7 6 3 6 7 6 Φq ðqÞ 7 60 6 6 7 T J ðqÞ = Φq ðqÞ = 6 7=6 6 4 61 7 6 Φq ðqÞ 7 6 6 6 7 6 60 7 6 5 6 7 6 Φq ðqÞ 7 6 6 60 7 4 6 5 6 6 60 Φq ðqÞ 6 60 4 2

3 Φ1q ðqÞ

0 si = sin φi ;

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

κ2

0

−κ1

−κ2

0

0

0

0

0

0

0

1

0

0

−1

0

0

0

0

0

0

0

0

0

0

−1

0

−c3

0

0

1

0

0

0

0

0

−1

−s3

0

0

0

0

1

0

0

0

0

0

−1

0

0

0

0

1

0

0

0

0

0

−1

0

0

0

0

0

1

0

−c3

−1

0

0

0

0

0

0

0

1

−s3

0

−1

ci = cos φi ; κ1 = c2 −s2 ;

κ2 = c2 + s2 :

0

3

7 0 7 7 0 7 7 7 0 7 7 7 0 7 7 7 0 7 7 7; 0 7 7 7 0 7 7 7 −s4 7 7 7 c4 7 7 s4 7 5 −c4

ðB:8Þ

J. Frączek, M. Wojtyra / Mechanism and Machine Theory 46 (2011) 312–334

331

The redundant constraint analysis can be performed in any non-singular configuration of the mechanism. For example, in the position described by the vector q = ½ 2 0 0 00 2 0 02 1 0 01 2 0
T (which is presented in Fig. 2) the Jacobian matrix is: 2

0 60 6 60 6 60 6 61 6 60 T   J q =6 61 6 60 6 60 6 60 6 40 0

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

0 0 1 0 0 1 0 0 0 −1 1 0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 0 0 −1 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0 0 0 −1 0 0 −1 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 −1 0 0 0 0 0 −1 0 0 0 0 −1 0 −1 −1 0 1 0 0 −1

3 0 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7: 0 7 7 0 7 7 0 7 7 1 7 7 0 5 −1

ðB:9Þ

Rank of the Jacobian matrix equals 11, thus among 12 equations (B.7) one is redundant. Since redundant constraints are imposed on the mechanism, some or all constraint reaction forces cannot be uniquely determined (holding the rigid body assumption). Using any of the three methods of the Jacobian matrix analysis (described in [7]), it was possible to verify which joint reactions are uniquely solvable. Firstly, the rank comparison method was used. The calculations were performed for each joint separately. In each case a matrix JP was built of the columns of the transposed Jacobian matrix J associated with joint P, and a matrix J−P was built of the remaining columns. All calculations were made for the position described by the vector q⁎. The following results were obtained: r = rankðJÞ = 11; for joints 4; 5 and 6; rankðJP Þ + rankðJ−P Þ−r = 0 rankðJP Þ + rankðJ−P Þ−r = 1 N 0 for joints 1; 2 and 3:

ðB:10Þ

The above results show that the reactions in revolute joints 4, 5 and 6 can be determined uniquely. The reactions in translational joints 1, 2 and 3 are dependent and cannot be determined uniquely. Next, the SVD based method [7] was used. The matrix V, resulting from singular value decomposition of the transposed Jacobian matrix, was divided into six submatrices (VP)2 × 12 that correspond to the joints. The transposed Jacobian matrix J was divided into submatrices (JP)12 × 2 in the same way. Then matrices BP were calculated ( JPVP = [(DP)12 × 11 (BP)12 × 1]):

BJoint 1 ≈½ 012

0:58 019
T ; BJoint 2 ≈½ 015

−0:58 016
T ;

T

BJoint 3 ≈½ 012 −0:58 012 0:58 016
; BJoint 4 = 0121 ; BJoint 5 = 0121 ; BJoint 6 = 0121 :

ðB:11Þ

The matrices BP for joints 1, 2 and 3 have nonzero elements, thus the resultant constraint reaction forces in these joints cannot be uniquely determined. The other matrices BP consist of zeros only, thus for joints 4, 5 and 6 it is possible to uniquely determine the constraint reaction forces. Finally, the QR based method [7] was used. The matrix Q , obtained during QR decomposition of the Jacobian matrix JT, was divided into six submatrices (Q P)2 × 12 that correspond to the joints. Similarly, the transposed Jacobian matrix J was divided into submatrices (JP)12 × 2. Then matrices BP were calculated ( JPQ P = [(DP)12 × 11 (BP)12 × 1]): BJoint 1 ≠0; BJoint 2 ≠0;

BJoint 3 ≠0;

BJoint 4 = BJoint 5 = BJoint 6 = 0121 :

ðB:12Þ

The obtained conclusions, concerning the joint reaction uniqueness, coincide with the conclusions derived previously. B.2. Spatial mechanism The Jacobian matrix is obtained by differentiation of the constraint Eq. (42). The partial derivative of constraint equation ΦP (see Eq. (41)) with respect to the vector of absolute coordinates q (see Eq. (39)) provides three rows to the Jacobian matrix. Since

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J. Frączek, M. Wojtyra / Mechanism and Machine Theory 46 (2011) 312–334

ΦP is a function of only qi and qj, ΦPq may have nonzero elements only in the columns associated with qi and qj. The possible nonzero entries are: ΦPri ≡I33 ;

ΦPrj ≡−I33 ;

h ΦPφi ≡ ΦPα i

ΦPβi

i h ΦPγi ≡ Ωz Rz ðα i ÞRx ðβi ÞR z ðγi ÞsðPiÞ

h P ΦPφj ≡ Φαj

ΦPβj

i h    ð jÞ ΦPγj ≡− Ωz R z α j Rx βj Rz γ j sP

2

0 0

6 Ωx ≡4 0 0

0

3

2

7 −1 5;

0 1

0

6 Ωz ≡4 1

0

0

−1

0

ðiÞ

Rz ðα i ÞΩx R x ðβi ÞRz ðγi ÞsP

ðiÞ

R z ðα i ÞRx ðβi ÞΩz Rz ðγ i ÞsP

   ð jÞ R z α j Ωx Rx βj Rz γ j sP

i 3×3

;

   i ð jÞ Rz α j Rx βj Ωz Rz γj sP

3×3

;

3

0

7 0 5:

0

0

ðB:13Þ

The above formulae and Eq. (7) are used to calculate the modified constraint Jacobian matrix, which can be written in a form: 2

I 6 6 6 6I 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 JT ðqÞ = ΦMod q ðqÞ = 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

ΦAφ1 H1

0

ΦBφ1 H1

−I I

ΦCφ2 H2

I

ΦDφ2 H2

0 −I

0 E

I

Φφ3 H3

0

I

ΦFφ3 H3

−I I

ΦGφ4 H4

I

Φφ4 H4

0

H

0

−I I

ΦKφ5 H5

I

Φφ5 H5

0

L

−I I ΦM φ6 H6

0

I ΦNφ6 H6

−I

0

3

7 7 7 7 7 7 7 0 7 7 7 7 D Φφ7 H7 7 7 7 7 7 0 7 7 7 7 F Φφ7 H7 7 7 7 7 0 7 7 7 7 7 H Φφ7 H7 7 7 7 7 0 7 7 7 7 L Φφ7 H7 7 7 7 7 0 7 7 7 5 N Φφ7 H7 ΦBφ7 H7

;

ðB:14Þ

3642

where: 2

sin γ i = sin βi Hi ðφi Þ = 4 cos γ i −cos βi sin γi = sin βi

cos γ i = sin βi −sin γ i −cos βi cos γi = sin βi

3 0 0 5: 1

ðB:15Þ

The coordinates of points A, B, C, D, E, F, G, H, K, L, M and N (Fig. 6), which must be substituted to Eq. (B.14), are constant in appropriate frames (indicated by superscripts): ð0Þ

sA = 0; ð0Þ

0
;

ð0Þ

0
T ;

ð0Þ

0
;

ð0Þ

4
T ;

ð0Þ

4
;

sC = ½ 2 0 sE = ½ 2 2 sG = ½ 0 2 sK = ½ 0 1 sM = ½ 2 1

T

T

T

 ð1Þ sA = 0  ð2Þ sC = 0  ð3Þ sE = 0  ð4Þ sG = 0  ð5Þ sK = 0  ð4Þ sM = 0

pffiffiffi T 0 − 2 ; pffiffiffi T 0 − 2 ; pffiffiffi T 0 − 2 ; pffiffiffi T 0 − 2 ; pffiffiffi T 0 2 ; pffiffiffi T 0 2 ;

 ð1Þ sB = 0  ð2Þ sD = 0  ð3Þ sF = 0  ð4Þ sH = 0  ð5Þ sL = 0  ð6Þ sN = 0

0 0 0 0 0 0

pffiffiffi T 2 ; pffiffiffi T 2 ; pffiffiffi T 2 ; pffiffiffi T 2 ; pffiffiffi T − 2 ; pffiffiffi T − 2 ;

ð7Þ

sB = ½ −1 ð7Þ

sD = ½ −1 ð7Þ

sF = ½ 1 ð7Þ

sH = ½ 1 ð7Þ

sL = ½ 0 ð7Þ

0

−1
T ;

0

1
;

T

0

1
T ;

0

−1
;

0

−1
T ;

sN = ½ 0 0

T

T

1
:

ðB:16Þ

J. Frączek, M. Wojtyra / Mechanism and Machine Theory 46 (2011) 312–334

333

The mechanism can be described by the following set of absolute coordinates:

q1 q3 q5 q7

T

= ½ −sin θ cos θ 1 θ −π= 4 0
; = ½ 2−sin θ 2 + cos θ 1 θ −π =4 0
T ; = ½ −sin θ 1 + cos θ 3 θ π =4 0
T ; T = ½ 1−2 sin θ 1 + 2 cos θ 2 π =2 π =2 0
;

T q2 = ½ 2−sin θ cos θ 1 θ −π =4 0
; q4 = ½ −sin θ 2 + cos θ 1 θ−π =4 0
T ; q6 = ½ 2−sin θ 2 + cos θ 3 θ π =4 0
T ;

ðB:17Þ

where θ is an arbitrarily chosen angle depicted in Fig. 6. The redundant constraint analysis can be performed in any non-singular configuration of the mechanism. For example, in the position described by the vector q⁎, obtained for θ = −π/4, the submatrices of the Jacobian matrix are calculated as: 2 pffiffiffi 3 2 = 2 −1 0 6 7 p ffiffiffi 7 ΦAφ1 H1 ðq Þ = ΦCφ2 H2 ðq Þ = ΦEφ3 H3 ðq Þ = ΦGφ4 H4 ðq Þ = 6 4 2 = 2 1 0 5; −1 0 0 2 pffiffiffi 3 − 2= 2 1 0 6 pffiffiffi 7 B  D  F  H  7 Φφ1 H1 ðq Þ = Φφ2 H2 ðq Þ = Φφ3 H3 ðq Þ = Φφ4 H4 ðq Þ = 6 4 − 2 = 2 −1 0 5; 1 0 0 2 pffiffiffi 3 2 pffiffiffi 3 2 = 2 −1 0 − 2=2 1 0 6 7 6 7 pffiffiffi pffiffiffi  L  N  6 6 7 ΦKφ5 H5 ðq Þ = ΦM 1 07 φ6 H6 ðq Þ = 4 − 2 = 2 −1 0 5 ; Φφ5 H5 ðq Þ = Φφ6 H6 ðq Þ = 4 2 = 2 5 1 0 0 −1 0 0 2 3 2 3 0 −1 0 0 −1 0 6 7 6 7 B  F  D  H  Φφ7 H7 ðq Þ = −Φφ7 H7 ðq Þ = 4 0 1 0 5; Φφ7 H7 ðq Þ = −Φφ7 H7 ðq Þ = 4 0 −1 0 5; 2

−1 0

6 L  N  Φφ7 H7 ðq Þ = −Φφ7 H7 ðq Þ = 4 0 1

0 0 −1 0

0

1 3

1

0

ðB:18Þ

1

7 0 5: 0

The mechanism is described by 42 coordinates and the rank of the transposed Jacobian matrix J(q) equals 35, thus the mechanism has 7 degrees of freedom (although six of them are associated with links 1–6 rotations along their longitudinal axes). Since redundant constraints are imposed on the mechanism, some or all constraint reaction forces cannot be uniquely determined (holding the rigid body assumption). Using any of the three methods of the Jacobian matrix analysis (described in [7]), it was possible to verify which joint reactions can be uniquely determined. Firstly, the rank comparison method was used. The calculations were performed for all 12 joints separately. In each case a matrix JP was built of the columns of the transposed Jacobian matrix J associated with joint P, and a matrix J−P was built of the remaining columns. All calculations were made for the position described by the vector q⁎. The following results were obtained: r = rankðJÞ = 35; rankðJP Þ + rankðJ−P Þ−r = 0 rankðJP Þ + rankðJ−P Þ−r = 1 N 0

for joints K;L;M and N; for the remaining joints:

ðB:19Þ

The above results show that the reactions in joints K, L, M and N can be determined uniquely. Reactions in the remaining joints are dependent and cannot be determined uniquely. Next, the SVD based method [7] was used. The matrix V, resulting from singular value decomposition of the transposed Jacobian matrix, was divided into twelve submatrices (VP)3 × 36 that correspond to the joints. The transposed Jacobian matrix J was divided into submatrices (JP)42 × 3 in the same way. Then matrices BP were calculated ( JPVP = [(DP)42 × 35 (BP)42 × 1]): BA ≈½ 0:18 0:18 0:25 0139
T ; BB ≠0; BC ≠0; BD ≠0; BE ≠0; BF ≠0; BK = BL = BM = BN = 0421 :

BG ≠0; BH ≠0;

ðB:20Þ

The matrices BA–BH have nonzero elements, thus the resultant constraint reaction forces in joints A–H cannot be uniquely determined. The matrices BK–BN consist of zeros only, thus for joints K, L, M and N it is possible to uniquely determine the constraint reaction forces.

334

J. Frączek, M. Wojtyra / Mechanism and Machine Theory 46 (2011) 312–334

Finally, the QR based method [7] was used. The matrix Q, obtained during QR decomposition of the Jacobian matrix JT, was divided into twelve submatrices (QP)3 × 36 that correspond to the joints. Similarly, the transposed Jacobian matrix J was divided into submatrices (JP)42 × 3. Then matrices BP were calculated ( JPQP = [(DP)42 × 35 (BP)42 × 1]): BA ≠0; BB ≠0; BC ≠0; BD ≠0; BE ≠0; BF ≠0; BG ≠0; BH ≈½ 0118 0:18 0:18 0:25 0115 −0:18 −0:18 −0:25 −0:25 BK = BL = BM = BN = 0421 :

0:35 −0:25
T ;

ðB:21Þ

The obtained conclusions, concerning the joint reaction uniqueness, coincide with the conclusions derived previously. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

E.J. Haug, Intermediate Dynamics, Prentice Hall, 1992. J. Garcia de Jalon, E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems: the Real-Time Challenge, Springer, 1994. A. Shabana, Computational Dynamics, 2nd editionJohn Wiley & Sons, New York, 2001. K. Arczewski, J. Frączek, Friction models and stress recovery methods in vehicle dynamics modelling, Multibody System Dynamics 14 (2005) 205–224. E.J. Haug, Computer aided kinematics and dynamics of mechanical systems, Allyn and Bacon, 1989. P.E. Nikravesh, Computer-Aided Analysis of Mechanical Systems, Prentice Hall, 1988. M. Wojtyra, Joint reactions in rigid body mechanisms with dependent constraints, Mechanism and Machine Theory 44 (2009) 2265–2278. S.-M. Song, X. Gao, The mobility equation and the solvability of joint forces/torques in dynamic analysis, ASME Journal of Mechanical Design 114 (1992) 257–262. M. Wojtyra, Joint reaction forces in multibody systems with redundant constraints, Multibody System Dynamics 14 (2005) 23–46. M. Wojtyra, J. Frączek, Reactions of redundant or singular constraints in mechanisms with rigid links, Proceedings of the 12th IFToMM World Congress, Besançon, France, June 18–21 2007. W. Blajer, On the determination of joint reactions in multibody mechanisms, Transactions of the ASME, Journal of Mechanical Design 126 (2004) 341–350. E.J. Haug, S.C. Wu, S.M. Yang, Dynamics of mechanical systems with coulomb friction, stiction, impact, and constraints addition, deletion. Part I, II, III, Mechanism and Machine Theory 21 (1986) 401–425. L.J. Corwin, R.H. Szczarba, Multivariable Calculus, Marcel Dekker, 1982. F.E. Udwadia, R.E. Kalaba, Analytical Dynamics: A New Approach, Cambridge University Press, 1996. E. Bayo, R. Ledesma, Augmented Lagrangian and mass-orthogonal projection methods for constrained multibody dynamics, Nonlinear Dynamics 9 (1996) 113–130.