symbolic calculation of the base inertial parameters of ... - CiteSeerX

MULTIBODY DYNAMICS 2009, ECCOMAS Thematic Conference K. Arczewski, J. Fraczek, ˛ M. Wojtyra (eds.) Warsaw, Poland, 29 June–2 July 2009

SYMBOLIC CALCULATION OF THE BASE INERTIAL PARAMETERS OF ROBOTS THROUGH DIMENSIONAL ANALYSIS Xabier Iriarte , Javier Ros and Vicente Mata:

Mechanical, Energetic and Materials Engineering Department Public University of Navarra, Campus Arrosadia s/n, 31006 Pamplona, Navarra, Spain e-mails: [email protected],[email protected] web page: http://www.imac.unavarra.es

: Mechanical Engineering Department Polytechnic 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 in engineering. A very interesting property of these models is that they can be written in a linear form 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 columns of the matrix that represents the equations of the inverse dynamic system, may appear as a linear combinations of each other, making the systems dynamics dependent, not on single inertial parameters but on linear combinations of them. These combinations are called Base Inertial Parameters. Knowing the symbolic expressions of these base parameters is a very interesting information in some disciplines. That is the reason why much effort was made in the 90’s to calculate them for open- and closed-loop systems. There have been two approches for the calculation of these parameters, the numeric and the symbolic ones. The numeric approches happened to be easier to implement and more system independent, while the symbolic approaches were more involved and not easily applicable to closed-loop systems. In this paper a new approach is presented for the calculation of the symbolic expressions of the base inertial parameters of robots. This approach is based on Dimensional Analysis applied to the base parameters expressions obtained by a numeric method, and also on the underlying structure of the coefficients of the expressions. This makes much easier to guess the hidden symbolic expressions, and makes the method much easier and faster than the previous approaches, specially when dealing with closed-loop mechanisms.

1

Xabier Iriarte, Javier Ros and Vicente Mata

1

INTRODUCTION

It is well known that the Inverse Dynamics equations of a Multibody System can be written in 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 barycentric parameters. See also Ref. [3] for Parameter Linear (PL) model construction. This linearity is a great advantage in fields like dynamic parameter identification, design optimization, model reduction and others. When writing the equations for the external generalized forces and torques (τ ) in this linear form (W φ 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 fact that the joints couple the movement of the bodies, so that the inertias of the bodies also couple in linear combinations to give the base inertial parameters. This way, the dynamics of the multibody system does not depend on the individual inertial parameters of the bodies, but depends on linear combinations of them. Knowing the symbolic expressions of these dependences is a valuable 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: numeric and symbolic. The numeric methods developed by Gautier [4], gave the tool to obtain these relations in an easy and accurate manner, making use of the Singular Value Decomposition (SVD) or the QR decomposition [5]. However, the numeric methods gave the linear combinations so that the weigths were numbers (P R) and not symbolic expressions depending on the geometry and topology of the mechanical system at hand. In fact, these symbolic expressions are much more useful, and motivated the work of Gautier and Khalil [6, 7], Mayeda and Yoshida [8] and other authors in this field. However, the methods provided by these papers were only valid for open-loop systems or parallelogram closed-loops. It was not until 1995 that Khalil and Bennis [9] developed an algorithm to obtain the symbolic base parameters relations for any closed-loop mechanical system. However this method was certainly complex and much more involved than the methods developed for open-loop systems. Much later, Chen et al. [10] developed an easier method based on the mass and moment of inertia transfer concepts for planar mechanisms, and 3D mechanisms [11]. In this paper a new method is developed for obtaining the expressions of the base inertial parameters for open- and closed-loop robots. The method is much easier to apply than the one from Khalil and Bennis, and is general for any system provided that the equations of motion are written linear w.r.t. the inertial parameters, and some conditions are fulfiled in the geometric model. This method is based on the dimensional analysis [12] of the numeric solutions of Gautier [4], and the fact that the dynamic equations can be written with a certain structure that suggests to find the coefficients as products of the lengths and trigonometric functions of the angles of the geometric model. To show the existence of this structure for 2D systems, some demonstrations 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 the base parameters calculation is resumed. In Section 3, some of the properties of the Inverse Dynamic Model and base parameters structure are highlighted. In Section 4 the dimensional analysis is applied to the numerically obtained base parameters. In Section 5, the symbolic expressions searching algorithm is presented, and Sections 6 and 7 show the results and draw some conclusions. The demonstrations of the underlying structure are brought to the Appendix.

2


2 2.1

NUMERIC CALCULATION OF THE BASE INERTIAL PARAMETERS The inverse dynamic PL model

When writing the Inverse Dynamic Model (IDM) of a multibody system, it is often interesting to write it in a parametric linear form: W pq, q, 9 q:, λ, αqφ τ

(1)

where pq, q, 9 q:q are the generalized coordinates (and their derivatives) of the model, and λ and α represent, respectively, two vectors of the lengths and angles (their sines and cosines) that have been used for the geometric model of the multibody system. The length of the inertial parameters vector φ will be 10n 1, because ten inertial parameters are needed for each of the n bodies of the system. These parameters are the mass of the body (m), its three center of 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 other three inertial parameters tmx, my, mz u m tcgx , cgy , cgz u. This way the 10 inertial parameters 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 be defined in the center of gravity of the body, since its location depends on the unknown inertial parameters. • The only friction model that preserves the linearity of the IDM equations is any viscous friction model with the structure τvf

¸

fi pq9qφf r,i

(3)

for any velocity dependent function f . The vector φf r,i represents the friction parameters and 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 single inertial parameter. This is very easy to see for a system with a body that rotates w.r.t. the ground around an axis parallel to the gravity direction. In this case, the inertia of that body in that direction is the only of its 10 inertial parameters to have any influence on the equations of motion. Thus, the columns of W multiplying those parameters will be null vectors. In other cases, and depending on the topology of the mechanism, some columns of W can be written as linear combinations of each other, and this makes the dynamic equations dependent on linear combinations of the corresponding inertial parameters. These linear combinations will form a set of minimal knowledge of the inertia parameters for determining the dynamic model. They are 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 on the SVD [5] presented by Gautier [4]. After the decomposition (W U J SV ) the column 3


vectors of V related to the null singular values of S, V 2, are used to obtain the base parameters expressions. Σ 0 J V1 V2 W U (4) 0 0 Reordering the rows of V2 (and the inertial parameters (φ)) so that V22 is a regular matrix, P JV

2

V21 V22

φ P Jφ 1

(5)

φ2

where P is a permutation matrix, the expressions for the base inertial parameters (φb ) can be written as follows: φb φ1 V21 V22 1 φ2 φ1 βφ2 (6) where the base parameters appear to be linear combinations of the inertial parameters with one of the weights equal to 1 (the one which multiplys to φ1 ). Notice that matrix P is usually not univocally determined. Therefore, some different base parameters sets can be obtained, all of them equally valid. Since the βij coefficients are the weights of the linear combinations that define the base parameters, we will refere to them as the Base 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 a multibody systems are relatively easy to calculate, provided that we have the PL-IDM and a software to obtain the SVD of a matrix. However, in this paper, we are looking for the symbolic expressions for the base inertial parameters, i. e. symbolic expresions for the base parameters weights. Therefore, it seems to be interesting to investigate the underlying structure of W and the base parameters weights so that some information can be obtained to find their symbolic equivalent. Let us write a single term of the W matrix, Wij , as a product of two functions: Wij

hij pλ, αq gij pq, q,9 q:, λ, αq

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

W.j

$ ' h1j λ, α ' ' & h λ, α 2j

h.j pλ, αq d g.j pq, q,9 q:, λ, αq ' ' ' %

p p

(7) ,

q g1j pq, q,9 q:, λ, αq / / . q g2j pq, q,9 q:, λ, αq / .. .

hmj pλ, αq gmj pq, q, 9 q:, λ, αq

/ / / -

(8)

where d represents an element-by-element product of two vectors. Since gij depends not only on the coordinates pq, q, 9 q:q, but also on λ and α, the hij functions could always equal 1. However, let us build hij with the maximum available terms of λ and 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, q, 9 q:q)) their corresponding 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 only depend on λ and α, and this will reduce the amount of expressions to check. 4


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

Nλ ¹

n λk ijk

k 1

Nα ¹

n

αl ijl ,

nijk

P Z, nijl P N

(10)

l 1

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 of the 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 would still take a lot of time to check all the possible combinations of products that fulfil the structure of Eq. (10), and compare them with the values obtained through the numeric computation of the base parameters. However, this structure can still be shrunk to a narrower structure with the help of the dimensional analysis [12]. Taking into account that all the monomials that define the base parameters must have the same physical dimensions, the underlying dimensions of βij can be easily obtained. Let us show 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 has those dimensions and has no number multiplying to it. Remember that the method of Gautier [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 second moment of inertia dimensions. Notice that it appears twice multiplying two inertial parameters 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 inertia my3 . This way, since the dimensions of each βij are known, we can constraint the search to those combinations that fulfil: dij

Nλ ¸

nijk

(12)

k 1 dij

where the dimensions of βij are m . All the base inertial parameters can only have dimensions of m0 , m1 or m2 . Therefore, the βij coefficients only can have dimensions m2 , m1 , m0 , m1 or m2 . In order to simplify the search domain, if the dimensions of βij are m2 or m1 , 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 .

5


5

SYMBOLIC EXPRESSIONS SEARCHING ALGORITHM

In the previous two sections, we have pointed out some interesting properties that the base parameters weights have to fulfil, and they are going to be very useful for the algorithm to obtain their 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, find the 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 dimensional parameters that appear in the geometric model in a single vector. This way, all the (numeric values of the) lengths of the model will form the λ vector. Moreover, we will build another equivalent vector with the symbolic expressions related with the numbers we put in λ. This symbolic vector will be called λS and will be useful to tell the user the final symbolic expressions of the base parameters. In an equivalent manner, another two vectors will be built, with all the dimensionless expressions that appear in the geometric model (generally trigonometric functions 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 also be put in the vector, so that the search will only be done through products of components of α. The 1 number will also take part of the dimensionless vector, and the length of the α vector will be Nα 4q 1, with q the number of involved angles. Notice that if αi , αi and their supplementary and complementary angles appear in the geometric model, only αi needs to be taken into account for the α vector.

pL 1 , L 2 , L 3 , . . . , L N q 1 αS p1, sin α1 , cos α1 , . . . , sin αq , cos αq , . . . , sin α λS

1

5.2

(13a)

λ

,

1 1 1 ,..., , q cos α1 sin αq cos αq

(13b)

Dimensionless numbers search

In order to search for all the dimensionless numbers obtained through the second products of Eq. (10), we will find all the different possible combinations with repetitions available between the components of α. In this step it is necessary to decide how many numbers we are going to take into account for the products. If the unit number is the first of the values of α, increasing the number of products will not increase the computational cost, because the algorithm will stop searching each weight when the corresponding symbolic expression has been found. As an example, if the length of vector α p1, sin α1 , cos α1 , sin1 α1 , cos1 α1 q is Nα 5 and we want to involve the products of Np 2 numbers in the search, there will be 15 possible combinations: Number of Combinations with Repetitions CRNαp N

pNNp !pNNα11q!q! p

(14)

α

The first 8 combinations would be the following: Vα1 Vα2

p1, 1q p1, sin αq

Vα3 Vα4

p1, cos αq p1, sin1 αq

Vα5 Vα6

6

p1, cos1 αq psin α, sin αq

Vα7 Vα8

psin α, cos αq psin α, sin1 αq


Notice that even if the number of combinations is very high, the only computation that has to be done with each of them is a product of Np numbers and compare the result with another number. Thus, the computation is very fast. 5.3

Dimensional numbers search

Due to the dimensional analysis applied to the base parameters weights (β ), it shows us ° ° ij obvious 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 combinations formula, but with a variations with repetitions formula, since the order is important (see Eq. (16)). Number of Variations with Repetitions V RNαp N

Nα N

(15)

p

Therefore, if we want to involve 3 (must be an odd number) dimensional numbers for the βij -s with dimensions of m1 , and 4 (must be an even number) dimensional numbers for the βij -s with dimensions of m2 , the Vλ vectors for λ pL1 , L2 q would be the following: For 3 elements vectors, the 8 vectors would be: Vλ1 Vλ2

pL1, L1, L1q pL1, L1, L2q

Vλ3 Vλ4


Vλ5 Vλ6


For 4 elements vectors, the first 9 vectors would be: Vλ1 Vλ2 Vλ3

pL1, L1, L1, L1q pL1, L1, L1, L2q pL1, L1, L2, L1q

Vλ4 Vλ5 Vλ6


And the searching structures: For dimpβij q m1 For dimpβij q m2

ùñ ùñ

βijk βijk

Nα ¹

αnijl

l 1 Nα ¹

αnijl

Vλ7 Vλ8 Vλ9

Vλ7 Vλ8


VλkVp1qVp3λkq p2q

(16a)

λk

Vλk p1qVVλk pp24qqVλk p3q

l1 °Nα th where βijk is the k candidate to equal βij , and l1 nijl Np , nijl P N

5.4


(16b)

λk

Recursive implementation of the algorithm

The easiest manner to implement this algorithm in a computer program, would be to write some nested loops for the calculation of all possible combinations of dimensional and dimensionless parameters. The number of nested loops would be equal to Np . However, the number of loops would be fixed at code typing time, so that the code would not be useful for other number of products. The solution to this problem is the Recursive Implementation of the algorithm, making the number 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 a tolerance, the symbolic expression has been found, and the program writes it making use of the λS 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] . 7


6

RESULTS

This algorithm has shown to be valid for 2D open- and closed-loop mechanisms, because it 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. In the paper of Chen et al. [10] the symbolic base inertial parameters of 7 planar mechanisms are written, being those only examples of base parameters fulfilling the structure of Eq. (10). i 1 2 3 4 5 6 7 8 9

di q1 0 0 0 0 q6 0 q8 0

ai 0 0 lr 0 0 0 0 0 0

θi π {6 q2 q3 q4 q5 5π {6 q7 π { 2 q9

αi 0 π {2 0 π {2 π {2 0 π {2 0 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. Two tests have been made with the 3RPS and 3PRS parallel robots, with satisfactory results. The base parameters of the 3PRS with a Denavit-Hartenberg geometric model, see Table (1), are shown in Eq. (17).

φb

φ1

βφ2

$ , m ' / 1 ' / ' / ' / I ' / zz2 ' / ' / ' / mx ' / 2 ' / ' / ' / my ' 2/ ' / ' / ' / I ' / xx3 ' / ' / ' / I ' / xy3 ' / ' / ' / I ' xz3 / ' / ' / ' / I ' / yz3 ' / ' / ' & Izz3 / .

'my3 / ' mz3 / ' / ' / ' / ' / m ' 4 / ' / ' / ' / I ' / zz5 ' / ' 'mx5 / / ' / ' / ' / ' / my ' / 5 ' / ' / ' ' m6 / / ' / ' / ' / I ' zz7 / ' / ' / ' / ' / mx 7 ' / % -

my7

plm2 cos αq1 lm1 tan2 α lr2 {plm2 cos αq lr 2 tan2 α{lm plm2 {lr cos αq1 lr tan2 α{lm 0 2 tan α 1 tan α 0 0 sin α 1 sin α lm cos2 α 0 lm2 cos α 1 lr2 lm2 cos α lr lm2 cos α 0 lm2 cos2 α 1 lr2 lm cos α 2 lr lm2 cos2 α

p

q

p q {p

p

q

{p {p

p {p {p

q

q q

q

0

8

q

q

1 lr2 lr 0 0 lm tan2 α 0 lm tan α 0 0 0 0 0 , $ lm tan2 α 0 & Iyy3 . sin α{ cos2 α 0 %mx30 0 m3 1 lm 0 lr2{lm 0 lr{lm 0 0 0 plm cos2 αq1 0 lr2 {plm cos2 αq 0 lr{plm cos2 αq 0 0 0

(17)


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 in the base parameters of other robots. However, some of the base parameters solutions in the literature 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 would always be valid. Some examples are listed below: • The PUMA 560, in Ref. [4]: base parameters do not fit the structure because some sums of squared lengths appear. • A 5 DOF (1 closed-loop) robot. Example 1 in Ref. [9]: in one of the base parameters, the term sin2 γ cos2 γ appears. • Another 5 DOF (1 closed-loop) robot. Example 2 in Ref. [9]: sums of lengths and squared lengths 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 this robot do not fit the structure. φb8 φb9

IzzE szE

m3 m3

6 ¸

2 prBx

i

2 rBy q i

i 1 6 ¸

rBzi

(18a) (18b)

i 1

where rBxi , rByi and rBzi are geometric lengths, and szE is a first moment of inertia. However, this equations already include a symmetry condition that imposes that the masses of 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 polar 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: φ1

b8

φ1b9

IzzE szE

6 ¸

(19a)

m3i rBzi

(19b)

i 1 6 ¸

i 1

9

m3i L2i


7

CONCLUSIONS

A new method has been developed to obtain the symbolic expressions of the base intertial parameters of open- and closed-loop robots. It is much easier and faster to implement than the classic methods in the literature. The existence of an underlying structure has been demonstrated 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 been tested 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 be the one presented in Eq. (10) (if some conditions are fulfilled). The demonstrations presented here are based on the paper of Chen et al. [10]. In that paper the mass and inertia transfer concepts are used to obtain the base parameters of a planar mechanism. However, reader should notice that transferring mass (or inertia) from one body to another can be done without affecting the 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 when Coulomb friction (or another reaction dependent friction model) is present in the equations. In fact, 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 are literally taken from the paper of Chen et al. [10], and will not be demonstrated here. Making use of these 4 propositions, the symbolic expressions for the base inertial parameters of planar mechanisms can be obtained. The propositions from 5 to 9 have been developed in order to demonstrate that the base parameters obtained with this method preserve the structure of products highlighted in Eq. (10), and therefore show the validity of the approach presented in this paper for 2D mechanisms. The standard parameters of a 3D body have been introduced in Eq. (2). For a 2D body, the standard parameters are only the ones involved in the planar dynamics: φ2D

tm, mx, my, Izz uJ tm, mσx, mσy , J uJ

(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 be eliminated, and hence is not a base parameter. Proposition 2: If two links are jointed with a translational joint, their moments of inertia will be regrouped as 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 be estimated when the link has gravitational force. 10


(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 parameters preserve 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. The first moments of inertia, however, suffer from the translation, still preserving the structure:

pmσxq2 pmσxq1 L cos ϕ m pmσy q2 pmσy q1 L sin ϕ m 11

(21a) (21b)


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

J1 mpσx12 σy12 q J2 JC mppL cos ϕ σx1 q2 pL sin ϕ σy1 q2 q J2 J1 L2 m 2L cos ϕ mσx1 2L sin ϕ mσy1

JC

(22a) (22b) (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 preserve the structure. Once again the mass of a link is independent of the reference frame in which it has been defined. The second moment of inertia will also experiment no change because the origin of both references are located in the same point. The only inertial parameters that change will be the 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 mσx1

sin δ mσy1 cos δ mσy1

*

(23)

Proposition 7: If part of the mass of a link (a point mass) is transferred between two links jointed with a rotational 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, as depicted in Fig. (4), and transfering it to the point kj on link j (as shown in the revolution joint of 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 , σyi q and moment of inertia 1 , σ1 q, moment of inertia J 1 and a point mass m1 . Ji . And virtual link i with mass m1i , mass center in pσxi yi i 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 m1ik pmσxi q pmσxi q1 m1ik a1 pmσyi q pmσyi q1 m1ik b1 Ji Ji1 m1ik pa12 b12 q

(24a) (24b) (24c) (24d)

where the superscript (1 ) denotes the inertias and lengths of the virtual links after the mass (or moment of inertia transfer). After moving the point mass m1ik from link i to link j, the inertial parameters of the two virtual links can be written as follows: For the link i: m1i

mi m1ik pmσx q1 pmσx q m1ik a1 pmσy q1 pmσy q m1ik b1 Ji1 Ji m1ik pa12 b12 q i

i

i

i

(25a) (25b) (25c) (25d)

For the link j: m1j

mj m1ik (26a) 1 (26b) pmσx q pmσx q 1 pmσy q pmσy q (26c) 1 (26d) Jj Jj As can be seen in the equations, parameters ppmσx q1 ,pmσy q1 ,Jj1 q will suffer no modification j

j

j

j

from their original values, for any value of the transferred mass m1ik . However, Eq. (25a-26a) form a 5 linear equations system with 6 unknowns pm1i , pmσxi q1 , pmσyi q1 , Ji1 , m1j , m1ik q. 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 virtual parameters will be the so called base inertial parameters. It can be easily demonstrated that the base 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 b12 q in the equations. Instead of that, making use of polar coordinates (R, θq, the terms R cos φ a1 , R sin φ b1 and R2 a12 b12 will preserve the products structure. If we solved the equations (25a-26a) for some φ1 0, we would get a matrix that relates the φ1 and φ parameters. φ1 Aφ (27) j

where φ

j

tmi, mσxi, mσyi, Ji, mj uJ, and φ1 tm1i, mσxi1 , mσyi1 , Ji1, m1j uJ without one of its

13


components (the one that is chosen to equal zero). The matrices for the 5 possible choices are: $ 1 , mσ ' xi / ' / & .

R cos θ R sin θ R 2

1 mσyi φ1pm1i 0q Apm1i 0q φ 1 Ji / ' ' / % 1 mj $ 1 , m / ' ' . & i1 /

R cos1 θ tan θ cosR θ 1 R cos θ

1 mσxi 0 1 0q φ φ1pmσyi 1 0q Apmσyi 1 / J 0 ' ' % i1 / mj 0 $ 1 , m / ' ' & i1 / .

φ1pm1j 0q

xi Apm1 0qφ 'mσ mσ 1 / j

' %

yi /

Ji1

-

0 1 0 0

0 0 1 0

R sin1 θ tan1 θ sinR θ

0 1 0 0

1 mσxi 0 φ1pJi1 0q ApJi1 0q φ 1 / mσ 0 ' ' % 1yi / mj 0 $ , m1i / ' ' & . 1 /

0 1 0 0

1

1 mσ 0 yi 1 0q φ φ1pmσxi 1 0q Apmσxi 1 / 0 J ' ' % i1 / mj 0 $ 1 , m ' / ' & i1 / .

1 0 0 0

1 R sin θ

0 1 0 0

0 0 1 0

R1 cosR θ sinR θ

1 0 0 0

0 1 0 0 0 0 1 0

0 0 0 1

2

1 R2

$ , ' mi / ' / 0 0 ' / ' / mσ & . xi 0 0 mσyi 1 0 ' / ' Ji / ' / ' / 0 1 % mj $ , ' mi / ' / 0 ' / ' / mσ & . xi 0 mσ yi 0 ' / ' / J ' / i ' / 1 % -

(28a)

(28b)

mj

0 0 1 0

$ , ' mi / ' / 0 ' / ' / mσ & . xi 0 mσ yi 0 ' / ' / J ' / i ' / 1 % -

(28c)

mj

, $ ' mi / / ' 0 ' / / ' mσ . & xi 0 mσ yi 0 ' / / ' J ' i / / ' 1 % -

(28d)

mj

, $ ' mi / / ' 1 / ' / ' mσ . & xi R cos θ mσ yi R sin θ ' / / ' J ' i / 2 / ' R %

(28e)

mj

Notice that if φ1i 0 is chosen, the column multiplying to φi happens to be the one different from a canonical vector. Remembering the numeric method presented in Subsection (2.2) and the Eq. (6), it shows us obvious that the 4 4 identity matrices that appear in the previous equations are the matrices multiplying to φ1 in Eq. (6), and therefore, the remaining column vector of each matrix represents the matrix β. In fact, since V22 is a 1 1 matrix, we find that all the V2 vectors are proportional to each other. This means that all the choices selecting φ1i 0 give proportional kernel vectors for the observation matrix.

"

V V2 P J 21 V22

*

$ , 1 ' / ' / ' / ' & R cos θ/ .

$ ' ' ' ' &

,

R cos1 θ / / /

$ ' ' ' ' &

,

1 / . 9 ' R sin θ / 9 ' tan θ / 9 ' 1 R / R / ' ' ' R 2 / ' / ' sin ' cos θ / θ / ' / / / % - ' - ' % % 1

1 R cos θ

14

,

$ ' ' ' ' &

$

,

R1 / 1 / / ' ' / ' / / cosR θ / . ' &R cos θ/ . sin θ 9 R sin θ / 9 / ' R / '

R sin1 θ / / / tan1 θ / . 1 R sin θ

' ' ' %

2

1 / / /

1 R2

-

' R2 / ' / ' / % -

1

(29)


Proposition 8: If part of the second moment of inertia of a link is transferred between two links jointed with a 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 of the two bodies. The result is that the 2 moments of inertia couple in the inertia of a sigle body, preserving the products structure. "

1 Ji1 Ji Jik 1 1 Jj Jj Jik

*

ùñ

"

if Ji1 if Jj1

0 ñ Jik1 Ji ñ Jj1 Ji 0 ñ Jik1 Jj ñ Ji1 Ji

Jj Jj

*

(30)

Proposition 9: If two parts of the mass of a link (i) (two point masses) are transferred consecutively to two links (j and l) jointed to the first with rotational joints, the new inertial parameters of the three virtual links will preserve the structure if m1j and m1l are set to zero. Suppose that the points Oj and Ol are located at coordinates pRj cos θj , Rj sin θj q and coordinates pRl cos θl , Rl sin θl q respectively, with respect to the reference frame (Xi , Yi ).

Figure 4: Moving two consecutive point masses (m1ik ,m2ik ) on link i to links j and l.

If the mass of link j is transferred to the link i (m1j m1i mx1i myi1 J11

mi mxi myi J1

0), the new parameters of i will be:

mj Rj cos θj mj Rj sin θj mj Rj2 mj

(31a) (31b) (31c) (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 mx2i myi2 J12

mi mxi myi J1

mj ml Rj cos θj mj Rl cos θl ml Rj sin θj mj Rl sin θl ml Rj2 mj Rl2 ml

(32a) (32b) (32c) (32d)

The parameters denoted with the superscript (2 ) will be the base parameters, and they will fit the products structure. Nevertheless, this proposition will not be valid for any choice of the virtual parameters to be zero. Choosing m1j 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. REFERENCES [1] C.G. Atkeson, C.H. An and J.M. Hollerbach. Estimation of inertial parameters of manipulator loads and links. International Journal of Robotics Research, 5(3), 101–119, 1986. [2] P. Maes, J.-C. Samin and P.-Y. Willems. Linearity of multibody systems with respect to barycentric parameters - Dynamics and identification models obtained by symbolic generation. Mechanics of Structures and Machines, 17, 219–237, 1989. [3] S.S Siddhartha, D.G. Beale, and D. Wang. A general method for estimating dynamic parameters of spatial mechanisms. Nonlinear Dynamics, 16, 349–368, 1995. [4] M. Gautier. Numerical calculation of the base inertial parameters of robots. Journal of Robotic Systems, 8(4), 485–506, 1991. [5] G.H. Golub and C.F. Van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore and London, 1983. [6] M. Gautier and W. Khalil. A direct determination of minimum inertial parameters of robots. Proceedings of the IEEE International Conference on Robotics and Automation, 1682–1687, 1988. [7] M. Gautier and W. Khalil. Identification of the minimum inertial parameters of robots. Proceedings of the IEEE International Conference on Robotics and Automation, 1529– 1534, 1989. [8] H. Mayeda, K. Yoshida, and K. Osuka. Base parameters of manipulator dynamics models. Proceedings of the IEEE International Conference on Robotics and Automation, 6(3), 312–321, 1990. [9] W. Khalil, and F. Bennis. Symbolic calculation of the base inertial parameters of closedloop robots. The International Journal of Robotics Research, 14(2), 112–128, 1995. [10] K. Chen, D.G. Beale and D. Wang. A new method to determine the base inertial parameters of planar mechanisms. Mechanism and Machine Theory, 37, 971–984, 2002. 16


[11] K. Chen and D.G. Beale. A new symbolic method to determine base inertia parameters for general spatial mechanisms Proceedings of the DETC, Montreal, Canada, 731–735, 2002. [12] E. Buckingham. On Physically Similar Systems; Illustrations of the Use of Dimensional Equations. Phys. Rev., 4(4), 345–376, 1914. [13] N. Farhat, V. Mata, A. Page, F. Valero. Identification of dynamic parameters of a 3-DOF RPS parallel manipulator. Mechanism and Machine Theory, 43, 1–17, 2008. [14] X. Iriarte. Matlab/Octave code for the Calculation of the Symbolic Expressions of Base Inertial Parameters. www.imac.unavarra.es/xabiiriarte/ [15] M. M. Olsen, J. Swevers and W. Verdonck. Maximum likelihood identification of a dynamic robot model: implementation issues The International Journal of Robotics Research, 21(2), 89–96, 2002. [16] M. Grotjahn, M. Daemi and B. Heimann Friction and rigid body identication of robot dynamics International Journal of Solids and Structures, 38, 1889–1902, 2001. [17] H. Abdellatif, B. Heimann, 0. Hornungand and M. Grotjahn Identification and Appropriate Parametrization of Parallel Robot Dynamic Models by Using Estimation Statistical Properties Proceedings of the IEEE International Conference on Intelligent Robots and Systems, Alberta, Canada, 157–162, 2005.

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