Symbolic Calculation of the Base Inertial Parameters of Closed-Loop Robots

MULTIBODY DYNAMICS 2009, ECCOMAS Thematic ConferenceK. Arczewski, J. Fraczek, M. Wojtyra (eds.)

Warsaw, Poland, 29 June–2 July 2009

SYMBOLIC CALCULATION OF THE BASE INERTIAL PARAMETERSOF ROBOTS THROUGH DIMENSIONAL ANALYSIS

Xabier Iriarte�, Javier Ros� and Vicente Mata:

�Mechanical, Energetic and Materials Engineering DepartmentPublic University of Navarra, Campus Arrosadia s/n, 31006 Pamplona, Navarra, Spain

e-mails: [email protected],[email protected] page: http://www.imac.unavarra.es

:Mechanical Engineering DepartmentPolytechnic University of Valencia, Camino de Vera s/n, 46022 Valencia, Spain

e-mail: [email protected]

Keywords: Base Parameters, Inertial Parameters, Robots, Symbolic, Dimensional Analysis.

Abstract. The inverse dynamic model of a multibody system is used for many applications inengineering. A very interesting property of these models is that they can be written in a linearform with respect to the inertial parameters of the solids, provided that some conditions hold.This property is very helpful in fields like dynamic parameter identification, design optimization,model reduction and others.

Due to the movement constraints that the joints impose to the bodies of a system, the columnsof the matrix that represents the equations of the inverse dynamic system, may appear as a linearcombinations of each other, making the systems dynamics dependent, not on single inertialparameters but on linear combinations of them. These combinations are called Base InertialParameters.

Knowing the symbolic expressions of these base parameters is a very interesting informationin some disciplines. That is the reason why much effort was made in the 90’s to calculate themfor open- and closed-loop systems. There have been two approches for the calculation of theseparameters, the numeric and the symbolic ones. The numeric approches happened to be easierto implement and more system independent, while the symbolic approaches were more involvedand not easily applicable to closed-loop systems.

In this paper a new approach is presented for the calculation of the symbolic expressionsof the base inertial parameters of robots. This approach is based on Dimensional Analysisapplied to the base parameters expressions obtained by a numeric method, and also on theunderlying structure of the coefficients of the expressions. This makes much easier to guess thehidden symbolic expressions, and makes the method much easier and faster than the previousapproaches, specially when dealing with closed-loop mechanisms.

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Xabier Iriarte, Javier Ros and Vicente Mata

1 INTRODUCTION

It is well known that the Inverse Dynamics equations of a Multibody System can be writtenin a linear form w.r.t. the inertial parameters φ. Atkeson et al. [1] were one of the first to do it,and Maes et al. [2] demonstrated the linearity of the equations w.r.t. the so called barycentricparameters. See also Ref. [3] for Parameter Linear (PL) model construction. This linearity is agreat advantage in fields like dynamic parameter identification, design optimization, model re-duction and others. When writing the equations for the external generalized forces and torques(τ ) in this linear form (Wφ � τ ) the observation matrix (W ) generally does not have full rank,so that some of its columns are linear combinations of each other. This property is due to factthat the joints couple the movement of the bodies, so that the inertias of the bodies also couple inlinear combinations to give the base inertial parameters. This way, the dynamics of the multi-body system does not depend on the individual inertial parameters of the bodies, but dependson linear combinations of them. Knowing the symbolic expressions of these dependences is avaluable information in design because it gives deep insight into the equations of motion.

There have been two approaches to obtain these relations between inertial parameters: nu-meric and symbolic. The numeric methods developed by Gautier [4], gave the tool to obtainthese relations in an easy and accurate manner, making use of the Singular Value Decompo-sition (SVD) or the QR decomposition [5]. However, the numeric methods gave the linearcombinations so that the weigths were numbers (P R) and not symbolic expressions dependingon the geometry and topology of the mechanical system at hand. In fact, these symbolic ex-pressions are much more useful, and motivated the work of Gautier and Khalil [6, 7], Mayedaand Yoshida [8] and other authors in this field. However, the methods provided by these paperswere only valid for open-loop systems or parallelogram closed-loops. It was not until 1995that Khalil and Bennis [9] developed an algorithm to obtain the symbolic base parameters rela-tions for any closed-loop mechanical system. However this method was certainly complex andmuch more involved than the methods developed for open-loop systems. Much later, Chen etal. [10] developed an easier method based on the mass and moment of inertia transfer conceptsfor planar mechanisms, and 3D mechanisms [11].

In this paper a new method is developed for obtaining the expressions of the base inertialparameters for open- and closed-loop robots. The method is much easier to apply than the onefrom Khalil and Bennis, and is general for any system provided that the equations of motion arewritten linear w.r.t. the inertial parameters, and some conditions are fulfiled in the geometricmodel. This method is based on the dimensional analysis [12] of the numeric solutions ofGautier [4], and the fact that the dynamic equations can be written with a certain structure thatsuggests to find the coefficients as products of the lengths and trigonometric functions of theangles of the geometric model. To show the existence of this structure for 2D systems, somedemonstrations are developed based on the symbolic approach of Chen et al. [10].

The paper is organized as follows. In Section 2, the most used numeric method for thebase parameters calculation is resumed. In Section 3, some of the properties of the InverseDynamic Model and base parameters structure are highlighted. In Section 4 the dimensionalanalysis is applied to the numerically obtained base parameters. In Section 5, the symbolicexpressions searching algorithm is presented, and Sections 6 and 7 show the results and drawsome conclusions. The demonstrations of the underlying structure are brought to the Appendix.

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2 NUMERIC CALCULATION OF THE BASE INERTIAL PARAMETERS

2.1 The inverse dynamic PL model

When writing the Inverse Dynamic Model (IDM) of a multibody system, it is often interest-ing to write it in a parametric linear form:

W pq, 9q, :q, λ, αqφ � τ (1)

where pq, 9q, :qq are the generalized coordinates (and their derivatives) of the model, and λ andα represent, respectively, two vectors of the lengths and angles (their sines and cosines) thathave been used for the geometric model of the multibody system. The length of the inertialparameters vector φ will be 10n � 1, because ten inertial parameters are needed for each ofthe n bodies of the system. These parameters are the mass of the body (m), its three centerof gravity location coordinates (cgx, cgy, cgz), and its six different inertia tensor components(Ixx, Ixy, Ixz, Iyy, Iyz, Izz).

It has to be pointed out that it is not always possible to write the IDM equations in a PL form.However, the following three conditions are sufficient to obtain a PL-IDM:

• The mass parameters and center of gravity location parameters have to merge in otherthree inertial parameters tmx,my,mzu � m � tcgx, cgy, cgzu. This way the 10 inertialparameters for the body i are:

φi � tmi,mxi,myi,mzi, Ixxi, Ixyi, Ixzi, Iyyi, Iyzi, IzziuJ (2)

• The inertia tensor of each body has to be defined in a known location, i. e., it can not bedefined in the center of gravity of the body, since its location depends on the unknowninertial parameters.

• The only friction model that preserves the linearity of the IDM equations is any viscousfriction model with the structure

τvf �¸fip 9qqφfr,i (3)

for any velocity dependent function f . The vector φfr,i represents the friction parametersand τvf represents the viscous friction force.

2.2 Base inertial parameters calculation

The equations of motion of a PL model are known not to depend on the value of every singleinertial parameter. This is very easy to see for a system with a body that rotates w.r.t. theground around an axis parallel to the gravity direction. In this case, the inertia of that body inthat direction is the only of its 10 inertial parameters to have any influence on the equations ofmotion. Thus, the columns of W multiplying those parameters will be null vectors. In othercases, and depending on the topology of the mechanism, some columns of W can be written aslinear combinations of each other, and this makes the dynamic equations dependent on linearcombinations of the corresponding inertial parameters. These linear combinations will form aset of minimal knowledge of the inertia parameters for determining the dynamic model. Theyare called Base Inertial Parameters and will be denoted by (φb).

One of the numerical methods to obtain the base inertial parameters, is the one based onthe SVD [5] presented by Gautier [4]. After the decomposition (W � UJSV ) the column

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vectors of V related to the null singular values of S, V 2, are used to obtain the base parametersexpressions.

W � UJ�Σ 00 0

� �V1 V2

�(4)

Reordering the rows of V2 (and the inertial parameters (φ)) so that V22 is a regular matrix,

PJV2 ��V21

V22

�PJφ �

�φ1

φ2

�(5)

where P is a permutation matrix, the expressions for the base inertial parameters (φb) can bewritten as follows:

φb � φ1 � V21V22�1φ2 � φ1 � βφ2 (6)

where the base parameters appear to be linear combinations of the inertial parameters with oneof the weights equal to 1 (the one which multiplys to φ1).

Notice that matrix P is usually not univocally determined. Therefore, some different baseparameters sets can be obtained, all of them equally valid. Since the βij coefficients are theweights of the linear combinations that define the base parameters, we will refere to them as theBase Parameters Weights.

3 OBSERVATION MATRIX AND BASE INERTIAL PARAMETERS STRUCTURE

As it has been shown in the previous section, the numeric base inertial parameters of amultibody systems are relatively easy to calculate, provided that we have the PL-IDM and asoftware to obtain the SVD of a matrix. However, in this paper, we are looking for the symbolicexpressions for the base inertial parameters, i. e. symbolic expresions for the base parametersweights. Therefore, it seems to be interesting to investigate the underlying structure of W andthe base parameters weights so that some information can be obtained to find their symbolicequivalent.

Let us write a single term of the W matrix, Wij , as a product of two functions:

Wij � hijpλ, αq � gijpq, 9q, :q, λ, αq (7)

and let us write a full column of W as follows:

W.j � h.jpλ, αq d g.jpq, 9q, :q, λ, αq �$'''&'''%

h1jpλ, αq � g1jpq, 9q, :q, λ, αqh2jpλ, αq � g2jpq, 9q, :q, λ, αq

...hmjpλ, αq � gmjpq, 9q, :q, λ, αq

,///.///- (8)

where d represents an element-by-element product of two vectors.Since gij depends not only on the coordinates pq, 9q, :qq, but also on λ and α, the hij func-

tions could always equal 1. However, let us build hij with the maximum available terms of λand α while preserving the structure. In this situation, if two columns of the full W matrix(W.i and W.j) are linearly dependent (for any extended state of the system (@pq, 9q, :qq)) theircorresponding g functions (g.i and g.j) must be the same. Or mathematically:

W.i � k �W.j ùñ g.i � g.j and h.i � k � h.j (9)

where k will generally depend on λ and α. This way, the base parameters weigths will also onlydepend on λ and α, and this will reduce the amount of expressions to check.

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Moreover, if we analyzed the structure of W or β obtained through a symbolic method, wewould notice that βij can be written as a product of the geometric parameters λ and α.

βij �Nλ¹k�1

λnijk

k � Nα¹l�1

αnijl

l , nijk P Z, nijl P N (10)

where Nλ and Nα are the lengths of vectors λ and α, respectively.This hypothesis has been demonstrated for the 2D case in the Appendix. The demonstration

is strongly based on [10], where the symbolic base parameters were obtained with the help ofthe mass and moment of inertia transfer concepts.

4 DIMENSIONAL ANALYSIS ON THE BASE INERTIAL PARAMETERS

Despite we have established a very simple structure for the base parameters weights, it wouldstill take a lot of time to check all the possible combinations of products that fulfil the structureof Eq. (10), and compare them with the values obtained through the numeric computation ofthe base parameters. However, this structure can still be shrunk to a narrower structure with thehelp of the dimensional analysis [12].

Taking into account that all the monomials that define the base parameters must have thesame physical dimensions, the underlying dimensions of βij can be easily obtained. Let usshow it through an example using one of the base parameters of a 3-RPS parallel robot [13]:

φb1 � Iyy3 � 0.3952 �my3 � 0.2082 �m2 � 0.2082 �m3 (11)

From this base parameter expression, one can derive the following information:

• All the monomials have second moment of inertia dimensions (kg � m2), since Iyy3 hasthose dimensions and has no number multiplying to it. Remember that the method ofGautier [4] ensures that one of the parameters has no coefficient. See Eq. (6).

• The weight 0.2082 has dimensions of m2, since it multiplys to a mass to give secondmoment of inertia dimensions. Notice that it appears twice multiplying two inertial pa-rameters with the same dimensions, showing that the approach is consistent.

• The weight 0.3952 has dimensions of m, since it multiplys to the first moment of inertiamy3.

This way, since the dimensions of each βij are known, we can constraint the search to thosecombinations that fulfil:

dij �Nλ

k�1

nijk (12)

where the dimensions of βij are mdij .All the base inertial parameters can only have dimensions of m0, m1 or m2. Therefore, the

βij coefficients only can have dimensions m�2, m�1, m0, m1 or m2.In order to simplify the search domain, if the dimensions of βij are m�2 or m�1, it will be

easier to find the symbolic expression of βij�1. This way, the symbolic expressions searching

algorithm will only have to look for coefficients with dimensions m0, m1 and m2.

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5 SYMBOLIC EXPRESSIONS SEARCHING ALGORITHM

In the previous two sections, we have pointed out some interesting properties that the baseparameters weights have to fulfil, and they are going to be very useful for the algorithm to obtaintheir symbolic expressions. The only thing that the algorithm will do is to take one of the βij

and compare its value with the many possible combinations of Eq. (10). Or mathematicaly, findthe nijk-s and nijl-s so that Eq. (10) holds.

5.1 Dimensional and dimensionless components vectors

The first to do in order to find the symbolic expressions is to collect all the dimensionalparameters that appear in the geometric model in a single vector. This way, all the (numericvalues of the) lengths of the model will form the λ vector. Moreover, we will build anotherequivalent vector with the symbolic expressions related with the numbers we put in λ. Thissymbolic vector will be called λS and will be useful to tell the user the final symbolic expres-sions of the base parameters. In an equivalent manner, another two vectors will be built, withall the dimensionless expressions that appear in the geometric model (generally trigonometricfunctions of the constant angles) in its numeric and symbolic versions, α and αS . However,for the simplicity of the algorithm, the inverse values of all the dimensionless values will alsobe put in the vector, so that the search will only be done through products of components ofα. The 1 number will also take part of the dimensionless vector, and the length of the α vectorwill be Nα � 4q � 1, with q the number of involved angles. Notice that if αi, �αi and theirsupplementary and complementary angles appear in the geometric model, only αi needs to betaken into account for the α vector.

λS � pL1, L2, L3, . . . , LNλq (13a)

αS � p1, sin α1, cos α1, . . . , sin αq, cos αq, . . . ,1

sin α1

,1

cos α1

, . . . ,1

sin αq

,1

cos αq

q (13b)

5.2 Dimensionless numbers search

In order to search for all the dimensionless numbers obtained through the second products ofEq. (10), we will find all the different possible combinations with repetitions available betweenthe components of α. In this step it is necessary to decide how many numbers we are going totake into account for the products. If the unit number is the first of the values of α, increasingthe number of products will not increase the computational cost, because the algorithm will stopsearching each weight when the corresponding symbolic expression has been found.

As an example, if the length of vector α � p1, sin α1, cos α1, sin�1 α1, cos�1 α1q is Nα � 5

and we want to involve the products of Np � 2 numbers in the search, there will be 15 possiblecombinations:

Number of Combinations with Repetitions � CRNp

Nα� pNp �Nα � 1q!

Np!pNα � 1q! (14)

The first 8 combinations would be the following:

Vα1 � p1, 1q Vα3 � p1, cos αq Vα5 � p1, cos�1 αq Vα7 � psin α, cos αqVα2 � p1, sin αq Vα4 � p1, sin�1 αq Vα6 � psin α, sin αq Vα8 � psin α, sin�1 αq

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Notice that even if the number of combinations is very high, the only computation that hasto be done with each of them is a product of Np numbers and compare the result with anothernumber. Thus, the computation is very fast.

5.3 Dimensional numbers search

Due to the dimensional analysis applied to the base parameters weights (βij), it shows usobvious that if its dimensions are meters1 (or meters2) then

°k nijk � 1 p°k nijk � 2q. This

time, the number of different Vλ vectors that can be generated will not be calculated with a com-binations formula, but with a variations with repetitions formula, since the order is important(see Eq. (16)).

Number of Variations with Repetitions � V RNp

Nα� Nα

Np (15)

Therefore, if we want to involve 3 (must be an odd number) dimensional numbers for theβij-s with dimensions of m1, and 4 (must be an even number) dimensional numbers for the βij-swith dimensions of m2, the Vλ vectors for λ � pL1, L2q would be the following:

For 3 elements vectors, the 8 vectors would be:

Vλ1 � pL1, L1, L1q Vλ3 � pL1, L2, L1q Vλ5 � pL2, L1, L1q Vλ7 � pL2, L2, L1qVλ2 � pL1, L1, L2q Vλ4 � pL1, L2, L2q Vλ6 � pL2, L1, L2q Vλ8 � pL2, L2, L2q

For 4 elements vectors, the first 9 vectors would be:

Vλ1 � pL1, L1, L1, L1q Vλ4 � pL1, L1, L2, L2q Vλ7 � pL1, L2, L2, L1qVλ2 � pL1, L1, L1, L2q Vλ5 � pL1, L2, L1, L1q Vλ8 � pL1, L2, L2, L2qVλ3 � pL1, L1, L2, L1q Vλ6 � pL1, L2, L1, L2q Vλ9 � pL2, L1, L1, L1q

And the searching structures:

For dimpβijq � m1 ùñ β�ijk ��

Nα¹l�1

αnijl

�� Vλkp1qVλkp2q

Vλkp3q (16a)

For dimpβijq � m2 ùñ β�ijk ��

Nα¹l�1

αnijl

�� Vλkp1qVλkp2qVλkp3q

Vλkp4q (16b)

where β�ijk is the kth candidate to equal βij , and°Nα

l�1 nijl � Np, nijl P N5.4 Recursive implementation of the algorithm

The easiest manner to implement this algorithm in a computer program, would be to writesome nested loops for the calculation of all possible combinations of dimensional and dimen-sionless parameters. The number of nested loops would be equal to Np. However, the numberof loops would be fixed at code typing time, so that the code would not be useful for othernumber of products.

The solution to this problem is the Recursive Implementation of the algorithm, making thenumber of nested loops a parameter for the program.

Every time the algorithm writes a new Vλ or Vα vector, it computes the new βij� candidate

in order to compare it with the searched βij . If the difference between them is smaller than atolerance, the symbolic expression has been found, and the program writes it making use of theλS and αS symbolic vectors, and the indexes with which βij

� was calculated.The Matlab/Octave code of this algorithm is freely available in the web [14] .

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6 RESULTS

This algorithm has shown to be valid for 2D open- and closed-loop mechanisms, becauseit has been demonstrated that the mathematical structure of the base parameters weights (βij)is the one shown in Eq. (10), provided some minimum conditions for the geometric model. Inthe paper of Chen et al. [10] the symbolic base inertial parameters of 7 planar mechanisms arewritten, being those only examples of base parameters fulfilling the structure of Eq. (10).

i di ai θi αi

1 q1 0 π{6 02 0 0 q2 π{23 0 lr q3 04 0 0 q4 π{25 0 0 q5 π{26 q6 0 5π{6 07 0 0 q7 π{28 q8 0 �π{2 09 0 0 q9 π{2

Table 1: Denavit-Hartenberg parameters of the 3PRS parallel robot.

However, the existence of this structure for 3D robots has not been demonstrated yet. Twotests have been made with the 3RPS and 3PRS parallel robots, with satisfactory results. Thebase parameters of the 3PRS with a Denavit-Hartenberg geometric model, see Table (1), areshown in Eq. (17).

φb � φ1 � βφ2 �

$''''''''''''''''''''''''''''''''&''''''''''''''''''''''''''''''''%

m1

Izz2

mx2

my2

Ixx3

Ixy3

Ixz3

Iyz3

Izz3

my3

mz3

m4

Izz5

mx5

my5

m6

Izz7

mx7

my7

,////////////////////////////////.////////////////////////////////-

�

��

plm2 cos αq�1 �lm�1 tan2 α 1lr2{plm2 cos αq �lr2 tan2 α{lm lr2

plm2{lr cos αq�1 �lr tan2 α{lm lr0 0 0ptan2αq�1 �lm tan2 α 0� tan α lm tan α 00 0 00 0 0psin αq�1 �lm tan2 α 0

sin α{plm cos2 αq � sin α{ cos2 α 00 0 0plm2 cos αq�1 �lm�1 0

lr2{plm2 cos αq �lr2{lm 0lr{plm2 cos αq �lr{lm 0

0 0 0�plm2 cos2 αq�1 plm cos2 αq�1 0�lr2{plm cos αq2 lr2{plm cos2 αq 0�lr{plm2 cos2 αq lr{plm cos2 αq 00 0 0

��

$&%Iyy3

mx3

m3

,.- (17)

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for α � π6, and (lm, lr) two characteristic lengths of the robot. It can be seen that all the βij-s fit

the products structure.It has also been found this structure in the base parameters of other parallel and serial 3D

robots in the literature. Besides the 3RPS and the 3PRS, it has been found this structure inthe base parameters of other robots. However, some of the base parameters solutions in theliterature do not fit the structure. The reason for that might be in the their geometric model,but it remains as a future work to know the conditions in which the proposed algorithm wouldalways be valid.

Some examples are listed below:

• The PUMA 560, in Ref. [4]: base parameters do not fit the structure because some sumsof squared lengths appear.

• A 5 DOF (1 closed-loop) robot. Example 1 in Ref. [9]: in one of the base parameters, theterm sin2 γ � cos2 γ appears.

• Another 5 DOF (1 closed-loop) robot. Example 2 in Ref. [9]: sums of lengths and squaredlengths appear.

• A one parallelogram closed-loop robot. Example 3 in Ref. [9]: this robot fits the structure.

• The KUKA IR 361 serial robot. Ref. [15]: this robot fits the structure.

• The Siemens manutec-r15 serial robot. Ref. [16]: this robot fits the structure.

• The Hexapod PaLiDA parallel robot. Ref. [17]: at first sight, the base parameters of thisrobot do not fit the structure.

φb8 � IzzE �m3

6

i�1

pr2Bxi� r2

Byiq (18a)

φb9 � szE �m3

6

i�1

rBzi(18b)

where rBxi, rByi

and rBziare geometric lengths, and szE is a first moment of inertia. How-

ever, this equations already include a symmetry condition that imposes that the massesof the 6 identical sliders are the same. Thus, if they were not supposed to be the same,each term in the summations in Eqs. (18a,18b) would multiply a different mass parameter.Moreover, if the end of sliders position w.r.t. the movable platform were expressed in po-lar coordinates (rBxi

� Li cos γ, rByi� Li sin γ) the sum of squares would be expressed

as L2i and the base parameters expressions would fit the structure:

φ1b8 � IzzE �6

i�1

m3iL2i (19a)

φ1b9 � szE �6

i�1

m3irBzi(19b)

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7 CONCLUSIONS

A new method has been developed to obtain the symbolic expressions of the base inter-tial parameters of open- and closed-loop robots. It is much easier and faster to implementthan the classic methods in the literature. The existence of an underlying structure has beendemonstrated for 2D mechanisms (provided that the geometric model fulfils some requirements)while the equivalent for 3D systems remains as a future work. However, the method has beentested with some 3D robots showing its validity for at least some mechanisms.

Moreover, the computation is very fast, leading the results in a few seconds of calculation.

APPENDIX

In this section, the underlying structure of the base parameters weigths will be shown to bethe one presented in Eq. (10) (if some conditions are fulfilled). The demonstrations presentedhere are based on the paper of Chen et al. [10]. In that paper the mass and inertia transferconcepts are used to obtain the base parameters of a planar mechanism. However, reader shouldnotice that transferring mass (or inertia) from one body to another can be done without affectingthe dynamics of the system, but will change the reaction forces in the joints linking the bodies.Therefore, the base parameters derived from the joint between two links, have no sense whenCoulomb friction (or another reaction dependent friction model) is present in the equations. Infact, the model with Coulomb friction would become nonlinear in the inertial parameters.

This Appendix is divided in 9 Propositions and their demonstrations. The first 4 of them areliterally taken from the paper of Chen et al. [10], and will not be demonstrated here. Makinguse of these 4 propositions, the symbolic expressions for the base inertial parameters of pla-nar mechanisms can be obtained. The propositions from 5 to 9 have been developed in orderto demonstrate that the base parameters obtained with this method preserve the structure ofproducts highlighted in Eq. (10), and therefore show the validity of the approach presented inthis paper for 2D mechanisms.

The standard parameters of a 3D body have been introduced in Eq. (2). For a 2D body, thestandard parameters are only the ones involved in the planar dynamics:

φ2D � tm,mx, my, IzzuJ � tm,mσx,mσy, JuJ (20)

The notation used in this section will be the one used in Ref. [10] (second vector in Eq. (20))to be coherent with that paper.

Proposition 1:

If two links are jointed with a rotational joint, one of their standard parameters can beeliminated, and hence is not a base parameter.

Proposition 2:

If two links are jointed with a translational joint, their moments of inertia will be regroupedas one linear combination.

Proposition 3:

If a link is jointed to ground by a rotational joint, only one parameter of the moving link [. . .]may be estimated when the link has no gravitational force, and three parameters [. . .] may beestimated when the link has gravitational force.

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(a) (b)

Figure 1: (a) Link i with a local frame. (b) Two links jointed with a rotational joint.

Proposition 4:

If a link is jointed to ground with a translational joint, only its mass may be estimated.

Proposition 5:

If the inertial parameters of a link are defined with respect to a new reference frame (R2)attached to it (built from a translation of the original one (R1)) the new inertial parameterspreserve the structure.

Figure 2: Definition of the inertial parameters of a link w.r.t. two different references.

The mass of a link is independent of the reference frame in which it has been defined. Thefirst moments of inertia, however, suffer from the translation, still preserving the structure:

pmσxq2 � pmσxq1 � L cos ϕ �m (21a)pmσyq2 � pmσyq1 � L sin ϕ �m (21b)

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To obtain the second moment of inertia w.r.t. the reference R2, see Fig. (2), the parallel axistheorem has to be applied twice.

JC � J1 �mpσ2x1 � σ2

y1q (22a)

J2 � JC �mppL cos ϕ� σx1q2 � pL sin ϕ� σy1q2q (22b)J2 � J1 � L2 �m� 2L cos ϕ �mσx1 � 2L sin ϕ �mσy1 (22c)

Notice that the number 2 (and 12) should also be put in the α vector (Eq. 13b) as part of the

dimensionless parameters vector.

Proposition 6:

If the inertial parameters of a link are defined with respect to a new reference system (R2)attached to it (built from a rotation of the original one (R1)) the new inertial parameters pre-serve the structure.

Once again the mass of a link is independent of the reference frame in which it has beendefined. The second moment of inertia will also experiment no change because the origin ofboth references are located in the same point. The only inertial parameters that change will bethe first moments of inertia, but they will preserve the structure."

mσx2

mσy2

*��

cos δ sin δ� sin δ cos δ

�"mσx1

mσy1

*�"

cos δ �mσx1 � sin δ �mσy1� sin δ �mσx1 � cos δ �mσy1

*(23)

Proposition 7:

If part of the mass of a link (a point mass) is transferred between two links jointed with arotational joint, the new inertial parameters of the two virtual links preserve the structure.

Moving part of the mass from mass center of link i to the jointing point ki on link i, asdepicted in Fig. (4), and transfering it to the point kj on link j (as shown in the revolution jointof Fig. (1b)), an inertial parameter dissapears from the equations.

Figure 3: Moving a point mass m1ik on link i. Link i with mass mi, mass center in pσxi, σyiq and moment of inertia

Ji. And virtual link i with mass m1i, mass center in pσ1xi, σ

1yiq, moment of inertia J 1i and a point mass m1

ik.

Let us write the inertial parameters of link i before (left hand side of the equations) and after

12


(right hand side) translating the mass m1ik:

mi � m1i �m1

ik (24a)pmσxi

q � pmσxiq1 �m1

ika1 (24b)

pmσyiq � pmσyi

q1 �m1ikb1 (24c)

Ji � J 1i �m1ikpa12 � b12q (24d)

where the superscript (1) denotes the inertias and lengths of the virtual links after the mass (ormoment of inertia transfer). After moving the point mass m1

ik from link i to link j, the inertialparameters of the two virtual links can be written as follows:

For the link i:

m1i � mi �m1

ik (25a)pmσxi

q1 � pmσxiq �m1

ika1 (25b)

pmσyiq1 � pmσyi

q �m1ikb1 (25c)

J 1i � Ji �m1ikpa12 � b12q (25d)

For the link j:

m1j � mj �m1

ik (26a)

pmσxjq1 � pmσxj

q (26b)pmσyj

q1 � pmσyjq (26c)

J 1j � Jj (26d)

As can be seen in the equations, parameters ppmσxjq1,pmσyj

q1,J 1jq will suffer no modificationfrom their original values, for any value of the transferred mass m1

ik. However, Eq. (25a-26a)form a 5 linear equations system with 6 unknowns pm1

i, pmσxiq1, pmσyi

q1, J 1i ,m1j,m

1ikq. If any

of these virtual parameters (except m1ik, because there would not be mass transfer) is chosen

to equal zero, the system of equations is determined and the expressions for the other virtualparameters will be the so called base inertial parameters. It can be easily demonstrated that thebase parameters will fit the products structure for any choice made.

Nevertheless, it will be necessary to use polar coordinates to refere to the m1ik mass location

in order to avoid the term pa12 � b12q in the equations. Instead of that, making use of polarcoordinates (R, θq, the terms R cos φ � a1, R sin φ � b1 and R2 � a12 � b12 will preserve theproducts structure.

If we solved the equations (25a-26a) for some φ1 � 0, we would get a matrix that relates theφ1 and φ parameters.

φ1 � Aφ (27)

where φ � tmi,mσxi,mσyi, Ji,mjuJ, and φ1 � tm1i,mσ1xi,mσ1yi, J

1i , m

1juJ without one of its

13


components (the one that is chosen to equal zero). The matrices for the 5 possible choices are:

φ1pm1i�0q � Apm1i�0qφ �$''&''%

mσ1xi

mσ1yi

J 1im1

j

,//.//- �� R cos θ 1 0 0 0�R sin θ 0 1 0 0�R2 0 0 1 0

1 0 0 0 1

��$''''&''''%

mi

mσxi

mσyi

Ji

mj

,////.////- (28a)

φ1pmσ1xi�0q � Apmσ1xi�0qφ �$''&''%

m1i

mσ1yi

J 1im1

j

,//.//- ��1 � 1

R cos θ0 0 0

0 � tan θ 1 0 00 � R

cos θ0 1 0

0 1R cos θ

0 0 1

��$''''&''''%

mi

mσxi

mσyi

Ji

mj

,////.////- (28b)

φ1pmσ1yi�0q � Apmσ1yi�0qφ �$''&''%

m1i

mσ1xi

J 1im1

j

,//.//- ��1 0 � 1

R sin θ0 0

0 1 � tan�1 θ 0 00 0 � R

sin θ1 0

0 0 1R sin θ

0 1

��$''''&''''%

mi

mσxi

mσyi

Ji

mj

,////.////- (28c)

φ1pJ 1i�0q � ApJ 1i�0qφ �$''&''%

m1i

mσ1xi

mσ1yi

m1j

,//.//- �� 1 0 0 � 1

R2 00 1 0 � cos θ

R0

0 0 1 � sin θR

00 0 0 1

R2 1

��$''''&''''%

mi

mσxi

mσyi

Ji

mj

,////.////- (28d)

φ1pm1j�0q � Apm1j�0qφ �$''&''%

m1i

mσ1xi

mσ1yi

J 1i

,//.//- �� 1 0 0 0 1

0 1 0 0 R cos θ0 0 1 0 R sin θ0 0 0 1 R2

��$''''&''''%

mi

mσxi

mσyi

Ji

mj

,////.////- (28e)

Notice that if φ1i � 0 is chosen, the column multiplying to φi happens to be the one differentfrom a canonical vector.

Remembering the numeric method presented in Subsection (2.2) and the Eq. (6), it showsus obvious that the 4 � 4 identity matrices that appear in the previous equations are the ma-trices multiplying to φ1 in Eq. (6), and therefore, the remaining column vector of each matrixrepresents the matrix β. In fact, since V22 is a 1 � 1 matrix, we find that all the V2 vectors areproportional to each other.

This means that all the choices selecting φ1i � 0 give proportional kernel vectors for theobservation matrix.

V2 � PJ"

V21

V22

*9

$''''&''''%1�R cos θ�R sin θ�R2

1

,////.////-9$''''&''''%� 1

R cos θ

1� tan θ� Rcos θ1

R cos θ

,////.////-9$''''&''''%� 1

R sin θ� 1tan θ

1� Rsin θ1

R sin θ

,////.////-9$''''&''''%� 1

R2� cos θR� sin θR

11

R2

,////.////-9$''''&''''%

1R cos θR sin θ

R2

1

,////.////-(29)

14


Proposition 8:

If part of the second moment of inertia of a link is transferred between two links jointed witha translational joint, the new inertial parameters of the two virtual links preserve the structure.

In this case the only inertial parameters that change will be the second moments of inertia ofthe two bodies. The result is that the 2 moments of inertia couple in the inertia of a sigle body,preserving the products structure."

J 1i � Ji � J 1ikJ 1j � Jj � J 1ik

*ùñ

"if J 1i � 0 ñ J 1ik � Ji ñ J 1j � Ji � Jj

if J 1j � 0 ñ J 1ik � �Jj ñ J 1i � Ji � Jj

*(30)

Proposition 9:

If two parts of the mass of a link (i) (two point masses) are transferred consecutively to twolinks (j and l) jointed to the first with rotational joints, the new inertial parameters of the threevirtual links will preserve the structure if m1

j and m1l are set to zero.

Suppose that the points Oj and Ol are located at coordinates pRj cos θj, Rj sin θjq and coor-dinates pRl cos θl, Rl sin θlq respectively, with respect to the reference frame (Xi, Yi).

Figure 4: Moving two consecutive point masses (m1ik,m2

ik) on link i to links j and l.

If the mass of link j is transferred to the link i (m1j � 0), the new parameters of i will be:

m1i � mi �mj (31a)

mx1i � mxi �Rj cos θj �mj (31b)my1i � myi �Rj sin θj �mj (31c)

J 11 � J1 �R2j �mj (31d)

And if the mass of link l is also transferred to the link i (m1l � 0), the new parameters of link

15


i will be:

m2i � mi �mj �ml (32a)

mx2i � mxi �Rj cos θj �mj �Rl cos θl �ml (32b)my2i � myi �Rj sin θj �mj �Rl sin θl �ml (32c)

J21 � J1 �R2j �mj �R2

l �ml (32d)

The parameters denoted with the superscript (2) will be the base parameters, and they will fitthe products structure.

Nevertheless, this proposition will not be valid for any choice of the virtual parameters to bezero. Choosing m1

j � 0 and m2i � 0, for example, the weigths of the resulting base parameters

will not fit the structure.Therefore, it will be generally necessary to correctly choose the inertial parameters to form

φ2 when selecting the P permutation matrix in Eq. (5) in order to ensure the products structure.

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[9] W. Khalil, and F. Bennis. Symbolic calculation of the base inertial parameters of closed-loop robots. The International Journal of Robotics Research, 14(2), 112–128, 1995.

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[11] K. Chen and D.G. Beale. A new symbolic method to determine base inertia parametersfor general spatial mechanisms Proceedings of the DETC, Montreal, Canada, 731–735,2002.

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[13] N. Farhat, V. Mata, A. Page, F. Valero. Identification of dynamic parameters of a 3-DOFRPS parallel manipulator. Mechanism and Machine Theory, 43, 1–17, 2008.

[14] X. Iriarte. Matlab/Octave code for the Calculation of the Symbolic Expressions of BaseInertial Parameters. www.imac.unavarra.es/xabiiriarte/

[15] M. M. Olsen, J. Swevers and W. Verdonck. Maximum likelihood identification of a dy-namic robot model: implementation issues The International Journal of Robotics Re-search, 21(2), 89–96, 2002.

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Symbolic Calculation of the Base Inertial Parameters of Closed-Loop Robots