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Identifiable

Parameters

of

a

Manipulator

* * *

Shir-Kuan Lin

Institute of Control Engineering National Chiao Tung University Hsinchu 30050, Taiwan

Received April 22, 1993; revised December 10, 1993; accepted March 4, 1994

This article deals with the minimal parameters of a manipulator in the least squares sense, so that the minimal parameters are equivalent to the identifiable parameters. The least squares concept is used to introduce terminology for the minimal linear combinations (MLCs) of the system parameters that define a set of linear combinations of the system parameters. The number of elements of the set is minimal, yet the set still completely determines the system. Furthermore, it is shown that the problem of finding a set of MLCs of a manipulator can be simplified to that of finding two individual sets of MLCs that determine the entries of the inertia matrix and the gravity load. Although the approach is applied to the inertia constants of composite bodies to obtain a set of MLCs identical to an earlier one, the result is newly interpreted in the least squares sense. The approach itself is a new method for finding the identifiable parameters of a manipulator, and it yields some new insight into the manipulator dynamics. The crucial feature is that a set of MLCs found by using the present approach is guaranteed to be identifiable. The earlier approaches always require an identification method to verify the results. An equivalence theorem is also presented that rigorously states the equivalence between the different sets of minimal parameters. 0 1994 John

W i l q G. Sons, Znc.

Journal of Robotic Systems 11(7), 641-656 (1994)

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642 Journal of Robotic Systems-1994

1. INTRODUCTION

System analysis involves investigating the dynamic behavior of a system to understand the system or to design regulators to control it. Two methodologies used in system analysis are mathematical modeling and system identification. However, the compound approach is always recommended for a deterministic system. An analytic approach can provide a dynamic model with wide-range validity and some physical insights, while parameter identification improves the accuracy of the model parameters. If the model of a nonlinear system such as a manipulator can be written in the form of linear equations with respect to the parameters, the parameter identification is then a least squares problem. The difficulty of this parameter identification is that not all parameters are identifiable, because some parameters determine the system not independently, but in combination. Essentially, the system is uniquely determined by a set of minimal parameters that are linear combina- tions of the modeling parameters and are linearly independent. These parameters are termed the mini- mal linear combinations (MLCs) of the system parame- ters in the context. Finding a set of MLCs will facili- tate solving the parameter identification problem.

The dynamic model of a manipulator is now well-known. It is highly nonlinear and requires knowledge of the kinematic parameters (relations between two adjacent links) and the inertia parame- ters (mass, center of mass and inertia tensor of each link). The kinematic parameters are always provided by the manufacturer or can be precisely calibrated, whereas the inertia parameters of industrial robots are almost all unavailable from the manufacturer be- cause these values are not needed for commercial controllers. However, most modern precision con- trol schemes for manipulators incorporate the in- verse dynamics of the manipulator, which requires the values of the inertia parameters. To evaluate the inertia parameters of the manipulator dynamics, Armstrong et a1.l disassembled a PUMA 560 robot and used a mechanical method to measure the inertia

parameters. This approach is tedious and does not yield precise results. Fortunately, Atkeson et a1.2 have found that the actuator forces of a manipulator are linear functions of the inertia parameters, i.e., the dynamics of a manipulator can be expressed as linear equations with respect to the inertia parame- ters. Previous attempts to identify the inertia param- eters have tried to formulate the linear equations either e ~ p l i c i t l y ~ - ~ or i m p l i ~ i t l y . ~ , ~ - ' ~ As mentioned above, without knowledge of the MLCs of the inertia parameters, identifying the parameters is difficult. Khosla and Kanade7 intuitively regrouped the closed-form dynamic equations, and other research- e r ~ ~ - ~ developed regrouping rules to minimize the number of inertia parameters appearing in the linear equations. These approaches are not practical for a manipulator with six or more joints because the closed-form dynamic equations of a six-joint manip- ulator are too large and too complicated to analyze. The MLCs of the inertia parameters of manipula- tors have drawn the attention of many researchers. Some researcher^^,'^-'^ have presented numerical ap- proaches such as the singular value decomposition method and the Q R method. Gautier et a1.16-lS devel- oped a regrouping rule to symbolically form a set of MLCs. Mayeda et al.19-22 found an explicit set of MLCs of the inertia parameters. Although these two sets of results are substantially equivalent, the ex- plicit form is more attractive.

It should be remarked that, instead of MLCs, minimum inertia parameters,'6-'8 base parame- t e r ~ , ' ~ - ~ ~ and the basis set of the essential parameter ~ p a c e ' ~ , ' ~ have all been used in the literature. This article will show that the identifiable parameters of a linear deterministic system belong to a set of MLCs of the system parameters in the least squares sense. Of course, a set of MLCs is also the necessary (i.e., minimal) parameters required to determine the sys- tem dynamics. The name of MLCs illuminates the role of the system parameters in the dynamic model and in the problem of parameter identification. In this article, the problem of finding the MLCs for the full manipulator dynamics is divided into a search

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Lin: Identifiable Parameters of a Manipulator 643

for two individual sets of MLCs, one for determining the inertia matrix and the other for determining the gravity load.

In an earlier article,23 we showed that some iner- tia constants of composite bodies form the minimal knowledge of the inertia parameters needed to deter- mine the manipulator dynamics. This article applies a new approach to the inertia constants of composite bodies and shows directly that the minimal knowl- edge of the inertia parameters in the earlier paper23 is a set of MLCs. The advantage of the present ap- proach is that it is not necessary to verify the identi- fiability of MLCs by an identification method. The approach also appears to provide a systematic method for analyzing the identifiable parameters of other mechanical systems. At the end of the article, an equivalence theorem is presented that rigorously states the equivalence between the different sets of minimal linear combinations of parameters. This the- orem could also be a tool for exploring other possible sets of MLCs in the future.

This article is organized as follows. Section 2 introduces the concept of minimal linear combina- tions (MLCs) of the system parameters. Several theo- rems are established to simplify the problem of find- ing the MLCs for a manipulator. To make the idea easier to comprehend, all proofs are presented in the Appendix. In section 3, we review the inertia constants of composite bodies, which are used to construct a set of MLCs in section 4. The rigorous derivation of the set of MLCs in section 4 is the main effort of this article. The approach is systematic and comprehensible, although some of the proofs are tediously long.

2. PRELIMINARIES

Motivated by the fact that the dynamics of a manipu- lator can be formulated as linear equations with re- spect to the inertia parameters,2 we consider a dy- namic system to be identified that has the linear deterministic form of

where y E R", 8 E R"' are observable signals, x E RP consists of the system parameters, p

>

n, and A( 8 ) : R"' -+ R'"P. We are concerned with the identfi- ability of the system parameters.

Definition 1. A set of columns a,( 8): R" -+ R" is said to be linearly dependent over Rln if tkere exist constants

a,, i = 1,

. . .

, n, not all zero such tkat

J I

2

ala,(0) = 0, V 8 E R"'. (2) i = l

I f a, are all zero, the set is said to be linearly independent

over R"'. w

Theorem 1. The number of linearly independent columns

of A(0) over R'" is k 5 p if and only if there exist A( 0): R"' Rnx' whose columns are linearly independent over R"' and w(x): RP + Rk whose components are linear combinations of x and are linearly independent over RI', such that

-

A( 0)x =

A(

8)w(x), V 8 E R"' and x E Rp. (3) w

The proof of this theorem is presented in the Appendix. According to the least squares theory,24 not all system parameters x are identifiable if the columns of A(8) are linearly dependent over R". However, the linear combinations w(x) in Theorem 1 are identifiable because the matrix

(xTx)

in the normal equation of the least squares problem is non- singular for a persistently exciting trajectory. A( O)x fully determines y, as does

A(

0)w(x). If not all values of x are available, the knowledge of w(x) is a neces- sary condition for determining y. In the sense of identification, we are interested in finding w(x) for the system in Eq. (1). Therefore, we introduce the following definition.

Definition 2. A set w(x) is a set of minimal linear combinations of the system parameters for the system in

E9. (1) if the elements of the set are linear combinations of x and linearly independent over tke domain of x and there exists

A(

8 ) whose columns are linearly independent Before turning to the dynamics of manipulators, we briefly review the literature. Ha et aL6 showed that the dynamic model of a manipulator can be formulated in a form like Eq. (3) by using some intu- itive regrouping rules. Theorem 1, however, gives a necessary condition for the number of linearly inde- pendent columns of A(B) and rigorously shows that

w(x) in Eq. (3) is a set of minimal linear combinations of the system parameters for the system in Eq. (1). Because there are numerous methods for selecting A( 0) from A( 8 ) , the set of minimal linear combina- tions is not unique.

We recognize that the dynamic equations of a manipulator with n joints are

over the domain suck that €9. (3) kolds. w

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644 Journal of Robotic Systems--1994

where q E R" consists of the joint displacements, x E RP consists of the inertia parameters, T E Rn con- sists of the actuator forces, H(q, x): R"+P + RfIxn is the symmetric inertia matrix, Tg(q, x): Ril+F' -+ R" consists of the gravitational forces, T E R" consists of the actuator forces, and &(q, q, x): R2"+p- R"'" consists of the Coriolis and centrifugal forces, which can also be related to the inertia matrix with Christof- fel symbols ( c ~ ~ ) ~ ~ ' ~ ~

(5)

(7)

where T ? is the ith element of ',T qi is that of q, and

h, is the ( i , j)th entry of H.

Our present problem is to find a set of MLCs of the inertia parameters (simply MLCs in the following context) for determining T, provided that the geo- metrical parameters of the manipulator are known. The strategy is to separate Eq. (4) into two parts:

7 8 and

Once the MLCs for 7'and are obtained, we can exclude the linear dependent elements and then ob- tain a set of MLCs for T. This strategy is supported by the following two theorems.

Theorem 2. A set is a set of MLCs for determining T if

it is the set consisting of the linearly independent elements of [wJT(x), wgT(x)IT, where wJ and wg are two sets of MLCs for determining T I and 78, respectively. Further-

more, when wJ is partitioned as

(9)

where PI and P, are some constant matrices and the compo- nents of wl are linearly independent of one another and those of wg, then the components of

d

and wg form

a set of MLCs for determining T. rn

Theorem 3. A set is a set of MLCs for determining T' if

and only if it is also a set of MLCs for determining the

Theorem 3 follows from Christoffel symbols in Eqs. (5)-(7). The detailed proofs of these theorems can also be found in the Appendix. By Theorems 2 and 3, the problem of finding the MLCs for T turns out to be a search for two individual sets of MLCs that determine the entries of the inertia matrix and the gravity load, respectively.

entries of H(q, x).

3. INERTIA CONSTANTS OF COMPOSITE BODIES

This section revisits the inertia constants of compos- ite bodiesz3 and Renauds formulation for the inertia m a t r i ~ . ~ ~ , ' ~ The structure of the inertia constants in Renaud's formulation will reveal that these constants are consistent with the MLCs of the inertia param- eters.

We consider a manipulator with n low-pair joints, which are labeled joint 1 to n outward from the base. Assign a body-fixed frame on each joint (Lee, frame El is fixed on joint i) in accord with the normal driving-axis coordinate s y ~ t e m ~ ~ , ~ ~ (known also as modified Denavit-Hartenberg notation). The distance from the origin of E l to that of E, is desig- nated asis, and that to the center of mass of link i

as c,.

In the normal driving-axis coordinate system (see Fig. l), the z-axis of a body-fixed frame is the driving axis of the corresponding link, i.e., the unit vector along joint i is uJ(') = [0, 0,

1IT,

where super- script "( i)" denotes the representation of a vector with respect to frame E l . The distance from the origin of frame EJ-* to frame E , is

where SOi = sin Oil COi

=

cos Oil and bi, d i , and Oi are the geometrical parameters of the coordinate system. The coordinate transformation matrix from Ei-l to

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Lin: Zdentifiable Parameters of a Manipulator 645

i"

U

Figure 1. Illustration of the normal driving-axis coordi- nate system.

The composite body i is defined as the union of link i to link n. Let the mass of the composite body

i

and the first moment of the composite body about the origin of Ei be denoted by hi and ti, respectively, to obtain

n

where mi is the mass of link j. The inertia tensor of the composite body about the origin of frame

Ei

(denoted by J i ) results from using the Huygeno- Steiner formula30 to obtain

where I? is the representation of the inertia tensor of link ,Iwith respect to frame E, and [ax] denotes

a skew-symmetric matrix representing vector multi- plication, i.e., [axlb = a x b. In the context, the overhead symbol " A " is used to denote the inertia

parameters (mass, first moment and inertia tensor) of a composite body.

I

We introduce the notation of

1, for rotational joint i,

0, for translational joint i. (15)

K? f (1 - Ki) sz

Renaud's f o r m u l a t i ~ n ~ ~ - ~ ~ , ~ * for the entries of the in- ertia matrix is

+

K,,K~ hi(u!i))z, m 5

i.

(16) where (.)ii denotes the (i, j)th entry of a matrix, the x-component of a vector, and

a(i) (i) x i (i)

(17) r,m Um m s

which is, physically, the part of the acceleration of the origin of frame E j due to a unit angular accelera- tion of joint m(i.e., Ij', = 1). We can also obtain the gravity term of the actuator force applied on joint i

in the form of23

where r f is the ith element of d ( q , x) and g is the gravitational acceleration.

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646 Journal of Robotic Systems-2994

By the principle of mathematical induction, it has been s h o ~ n ~ ~ , ~ ~ that the first moment and the inertia tensor of the composite body i can be formed as the sum of a constant vector (ki or Ui) and a varying vector

(ti

or UJ, such as

el')

= ki

+

t i

p

=

u.

+

v i

(19) (20) where k, = m,cF),

e,,

= 0, U,, = IF) - m,[ck')x]

If joint i

+

1 is a rotational joint, then

whereas for translational joint i

+

1

Note that '+!Rh in Eq. (23) is the third column of '+!R, i.e.,

which is a constant vector, and

dj?:) =

[

(i+ls(i+

:

1) ) ] =

[

i,]

(29) We shall call hi, the components of ki, and the entries of Ui, i = 1,

. .

.

, n, inertia constants of compos-

ite bodies. The salient feature of these constants is that the varying terms in Cji) and

i!')

can be calculated with only some (not all) of the inertia constants of composite bodies. Namely,

ti

in Eq. (22) are calcu- lated with some Y?Z~ and the x- and y-compounds of kj, j

> i,

and the recursive form for computing Vi in Eqs. (24) and (26) requires only the (1, 2)th, (1, 3)th, (2, 3)th entries and the difference of the (1, 1)th and (2,Z)th entries of Ui+l and some components of ki+l.

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Lin: Identifiable Parameters of a Manipulator 647

This property is consistent with the minimal linear combinations of the inertia parameters. Indeed, some elements of k, and

U,

and

mi

constitute a set of MLCs for determining the actuator forces of a ma- nipulator.

For instance, the gravity load 7 8 in Eq. (18) re- quires KTtji) and Kimi. Because

ye(')

can be expressed in terms of Kjmj and the

x-

and y-components of K;Ckj, j

>

i (see Eqs. (19) and (22)), we obtain the fol- lowing theorem.

Theorem 4. Consider a manipulator with n low-pair joints in which joint r is the first rotational joint counting from the base and joint s is the nearest rotational joint not parallel to joint r . A set of MLCs for determin- ing the gravity load 7 8 is the set 9 8 = {6,K(kJX, 6,K?(ki),, uiKimi, i = 1,

. .

. ,

n } - (0).

and IT, are either 1 or 0 to denote the

redundancy of the parameters, which are defined as follows.

ai

= 0 for r 5 i

<

s and ur//g, otherwise 6 , = 1. On the other hand, ui = 0 for ui i g, Vq E R" (if r

< i <

s

foy translational joint

i,

ui is always perpendicular to g

only when ur//g or when u, I g and ui//ur; while this can happen for i

>

s only if ur//g, us I u, and

u,//ui/bs

for any rotational joint j , s I

j < i),

otherwise ui = 1.

The proof of this theorem is similar to that for Theorem 2 in a previous and is thus omitted. According to Theorems 2 and 3, the rest of the MLCs for determining the actuator forces Tare those for determining h = [h,,,

.

. . ,

hln, h22,

. .

. ,

h J ,

which contains all upper triangular entries of the inertia matrix H(q, x) in Eq. (4) because the inertia matrix is symmetric. By Theorem 2, the problem can be temporarily formulated as the problem of finding

-

the components of v such that there is a matrix H(q) with linearly independent columns satisfying

Note that

h = R(q)v(q, x)

+

T(q)vg = Wq)v(q, x)

+

m w g (30) where T and

T

are some matrices, v is composed of

mi,

tji)

and Ui, i = 1,

.

,

.

,

n, other than those in

v8, and

V '

8 , w*=

in which ui is defined as in Theorem 4. Note that the components of wg belong to

Yg

in Theorem 4, and v8 = Wgwg, where W8 is a nonsingular upper triangular matrix with the diagonal entries of 1 (which can be seen from Eqs. (19) and (22)). After v in Eq. (30) is found, the terms associated with Eli)

can be rewritten in terms of the

x-

and y-components of kj and mj, j

> i,

i.e., the component of w8. The set of MLCs for determining h then has a form like Eq. (9).

4. MINIMAL LINEAR COMBINATIONS OF THE INERTIA PARAMETERS

To deduce the components of v in Eq. (30), we re- quire the following property, which directly follows from the definition.

Property

I.

If all nonzero elements of a row in

FT

are linearly independent over R", the columns of

n

containing the nonzero elements of this row are linearly independent of one another and of the other columns H We treat hnli of Renauds formulation given in Eq. (16) separately for the different types of joint i. First, suppose joint i is a translational joint; it follows from Eq. (16) that

over R". h j i =

[O.

* . 0 1 0 .

.

. O ] f

t

(32) m; h,, = [0

. . .

0

( ~ ( n ) ) ~

0

. . .

012 for K,,, = 1, m

<

i

t

(33) mi

hMi = [0

. . .

0 (a$$)z 0

. . .

0 - (u:))~ (u:))~ 0

. . .

O]f,

t

t

(tp)x (t$+)y

t

mj

for K: = 1, m

<

i (34) where f consists of the masses, the first moments, and the inertia tensors of the composite bodies. By Property 1, Eq. (32) shows that the column of

E

corresponding to K@zi is linearly independent of the other columns. This also reveals that the columns of

n

corresponding to the

x-

and y-components of Kit$') must be linearly independent of that corresponding to Kimi regardless of whether (a(:&)z in Eq. (34) is independent of - (u(n)), and (u:)),. For translational joint i, s

<

i 5 n, there are at least two elements of h having the form of Eq. (34), e.g.,

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648 Journal of Robotic Systems-1994

Because us is not parallel to ur, the x- and y-compo- nents of u$) vary independently with

q S ,

so that the second and third columns in Eq. (35) are linearly

-

independent. This implies that the columns of H corresponding to Ki(i?ji)), and Ki(Cji))y for s

<

i 5 n are linearly independent of the other columns. For the case of r

<

i

<

s, ugj is constant and u,//u, for any rotational joint m, m

<

i, so that the combination of ( - (uf)),,(tj')),

+

( ~ y ) ) ~ ( t j ~ ) ) ~ )

is constant and is a component of v in Eq. (30) if it is nonzero. It is zero when ui//ur.

It follows from Eq. (33) that Ui and

tji)

for i

<

r

are unnecessary for determining the inertia matrix. This result is summarized as follows.

Property 2. v in Eq. (30) has the following components:

1.

Ki@,

i = 1,

. . .

, n,

2. ~ ~ ( t j j ) ) ~ and Ki(tj'))Y, s

<

i 5 n, and 3. K,Rli for r

<

i

<

s and u,)((u,, where

kIi = -(up) Y I

(tv))x

+

(u;))x(ej'j)Y (36)

However, K~U, and

ti'),

i

<

r, are not components

of v. w

If joint i is a rotational joint, we obtain from Eq. (16) that

h i ; = [ O .

.

. O 1 0 .

.

.O]a

t

(37)

( J 9 3 3

h,,; = [0

. . .

0 (u:))~ -(u;')~

0

. .

.

O]k,

t

t

(tji)),

(ey)Y

for K,, = 1, m

<

i (38) hnli = [0

.

,

.

0 (a:$), -(a?))

0

. . .

0 (u:))' 0

. . .

01%

t

tnl

t

J!"

(tji))x (tji))y

Third Column of for Ktz = 1, m

<

i (39) Note that

Jj')

= Ui

+

Vi. To obtain a general form for Vi, we suppose that joints i and j , i

<

j , are rotational joints, but the joints between them are all translational joints. It follows from Eq. (22) that

Applying Eqs. (24), (26), and

(40),

we obtain

where is*(') is constant in the form of

i - 1

(43) For convenience of analysis, we relate frames Ei and Ej directly through a common normal of ui and

ui, which is in accord with the normal driving-axis coordinate ~ y s t e m ~ ~ , ~ ~ (see Fig. 2). The x-axis of frame

E i

will be in alignment with the common normal by a rotation of angle +q about ui (i.e., the z-axis of frame E;). Then, the rotation of /3: about the normal followed by the rotation of 0; about the z-axis of frame

Ei

transforms frame E , so that it is in alignment with frame E j . In mathematical form, the coordinate transformation matrix from frame E , to E j can be written as28,29

Note that +ii and

/3;

are constants since joints i

+

1,

. .

. ,

j - 1 are translational joints, whereas

e;

is a variable because it is the sum of

qi

and some con- stant. It can be shown that

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Lin: Identifiable Parameters of a Manipulator 649

U

Figure 2. Geometrical relation of any two joints.

(45)

This equation also reveals that ui//uj if and only if We are concerned with the third column of Vi.

sp;=

0.

Expanding Eq. (42) yields

where Vl') = 4R V j 4RT,

fk

are some appropriate vec- tor functions and

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650 Journal of Robotic Systems-1994

It should be remarked that V!',) is a function of qi, q j t l , ,

.

.,

q,, and B(p;,O;) and

IR

in the second and

third terms of Eq. (46) are functions of

qj

only, while

D(p,*, 6;) in the last term is a constant matrix. The terms in Eq. (46) can be divided into three groups: (1) the first term, (2) the second and third terms, and (3) the last two terms. The columns of the coefficient matrices in any one group are linearly independent of those in the other groups because of the differ- ent variables.

KkUk, k = 1,

. .

.

n do not appear in Eq. (46), so they are not needed to compute KTV,, i = 1,

. . . ,

n, and are then redundant for determining h. By Property 2, the columns of

R

corresponding to f i k for translational joint k are linearly independent of the other columns, and the role of the terms associ- ated with f i k for computing J!') in Eqs. (37) and (39) can be ignored, as can those with

(tik)),

and

(tik))y,

k

>

s. In Eq. (46), only the coefficient matrices in front of

tik)

(i.e., the last term) vary with

d k , the displacements of the translational joints. In

row rT of hii = rrv, as in Eq. (37), the component

corresponding to ( I $ ~ ) ) ~ is then linearly independent of the components other than those corresponding to

(tik))x,

(tik$,,

and f i k if it is nonzero. It is zero when ui//uk (i.e.,

Sp,*

= 0), because it is 2S2p,*dk according

to Eqs. (46) and (48). For k

>

s, uk may be parallel to uj, i 2 s, but it is never parallel to u,. Because Eq. (42) is in a recursive form, there must be a nonzero term associated with

(tik)),

for computing k,,, which is also one of the terms varying with d, (the other terms are those associated with

(tik$

and f i k ) . We then conclude that Kk(e(l;))z, k

>

s, is a component of v in Eq. (30).

For the case of i

<

k

<

s, kmi in Eq. (39) requires only the (3, 3)th entry of fj')because (u:) = [0, 0, +1IT. Combining Eqs. (37), (39), (46), and (48) yields

where p is some appropriate vector that is not a function of

tik)

and d k , and

which follows from Eq. (45) and ug) = ulk) for r 5 m

<

k

<

s. If uj/,bk, then

Sp,*

is zero, as is R2k. It should be remarked that Rlk in (36) is also zero when

Sp,*

= 0. This entails that

Kktik),

r

<

k

<

s, is redun- dant for computing h if u,//uk. Applying Property 1 to Eq. (49), we may state the following property. Property 3. v in Eq. (30) has the following components:

1.

Ki(tli))Z,

s

<

i I n, and

2. KiRZi (see E9. (50)) for r

<

i

<

s and u, x u i . However, Kit!') for r

<

i

<

s and u, x u i are not compo- nents of v, nor are KiUi, i = 1,

. . .

,

n.

Let us return again to Eqs. (37)-(39). It is now apparent that K;fijandK;(i.i')),, j = 1,

.

.

. ,

n, are redundant for computing h.

We still assume that joints i and j are rotational joints and joints k , i

<

k

<

j, are translational joints. Because the components of Vi are all functions of

qi+l or qj, Eq. (37) indicates that the column of

R

in Eq. (30) corresponding to K 3 . J i ) 3 3 , i = 1,

. . .

, n, is linearly independent of the other columns. On the other hand, the rotational joints in front of joint s are parallel to one another, so SPY in Eq. (44) for j

<

6

is zero and Eq. (47) is reduced to

0 0 0 0

0 0

cp;co; -cp;so;

o o c2p;so; czp;ce;

j < s (51) Therefore, the contribution of

Ul

to (V,),,, i

<

j

<

s, is zero. By Eqs. (37) and (39) and the parallelism of the rotational joints, the entries (other than the (3, 3)th) of KfV,, i

<

s, are redundant for computing h. This entails that the entries, other than the (3, 3)th, of KfU, for r I i

<

s are not components of v in Eq. (30).

Consider rotational joint j, j 2 s, and reformulate

Eq. (39) as

h, = (a$y(q% - (a$x(~f))Y

+

[@))I3 @')23

ij:I))33~u?,

j 2 s (52)

Because the terms associated with K*(tf)), and K*(L?!))~, j 2 s, for computing h in Eq. (30) are parti- tioned into Tvg, the third term associated with

K*S') in Eq. (46) and the first two terms in Eq. (52) for j 2 s can be ignored in the analysis of v. Keeping in mind that the

x-

and y-components of ul]), j 2 s, vary independently (because joint s is not parallel to joint r ) , and U, is constant while V, varies, it follows

(11)

Lin: Identifiable Parameters of a Manipulator 651

from Eq. (52) that (Uj)13 and (Uj)z3,

j

2 s, in addition to (U,),,, are components of v in Eq. (30).

Now let i

<

j = s. hii in Eq. (37) requires (Vi)33. The contribution of Us to (Vi)33 is the first row of

B(P,*, 0:) in Eq. (47), whose components are either zero or linearly independent of one another and of the other terms in (VJ3, because the terms associated with t f ) are ignored. The fact that rotational joint i

is not parallel to joint s implies that the first two components of the first row of B must be nonzero. Thus, (Us)ll - (Us)22 and (Us)lz are components of v. Fix joint s such that rotational joint

j ,

j

>

s, is not parallel to joint r. Let

Py

and 0; be the transformation parameters between joint r and joint

j

when joint s is held stationary; then the above result applies to

(U,)z3 for rotational joint j , I 2 s, as well as

&+(Ui),,, i = 1,

. .

.

,

n, are components of v in

Eq. (30).

The x- and y-components of KTti') for r 5 i

<

s are still left to be considered.

For any two rotational joints m and

i,

m

<

i

<

s, we define Case 1 to be Lls = Oor ;,s//u, (i.e., the origins of frames Ei and En, are coincident or the distance between them is collinear with urn), and Case 2 to be all other situations. In Case 1, = 0

according to Eq. (17). In addition, if there are no translational joints in front of joint i that are not parallel to joint i, the x- and y-components of Kkuf) for all translational joints k , k

<

i, are zero. Thus, Eqs. (38) and (39) are, respectively, reduced to Kkhki = 0, k

<

i, and h,,, = + ( J ; i ) ) 3 3 , because u;) = [0, 0, +1]? In addition,

(LI~*('n))x

= = 0 in Case 1, so that

t(i)

has no contribution to (Vm)33 according to Eq. (46). We then conclude that (tii))x and (t!i))y of rotational joint i, Y

< i <

s, are redundant for comput- ing h if there are no translational joints in front of joint i that are not parallel to joint i and if the distance from any rotational joint in front of joint

i

to joint i

is either zero or parallel to joint

i.

On the other hand, in Case 2, ai,nr is not zero and perpendicular to u,,,

.

Although a!,;) may be con- stant (when $("') is constant), the x- and y-compo-

nents of a;:; vary with the rotation of joint i. They are linearly independent of each other and of the nonzero component of u;) (i.e., (u$))~, which is k l ) .

Applying Property 1 to Eq. (39) yields the result that

(?ii))x

and

(t$i))y

of rotational joint i, r

<

i

<

s, in Case

2, are components of v in Eq. (30).

In front of joint r, if there are no translational joints not parallel to joint r, then (I?:))~ and (tl'))y are redundant for computing h. Otherwise, they are components of v in Eq. (30) according to Eq. (38). This also applies to rotational joint i, r

<

i

<

s. j

>

s. In summary, (uj)11 - (Uj).z, (uj)12, (uj)13, and

Property 4. v in Eq. (30) has the following components:

K?((ui)ll - (ui)22)1 K?(ui)lZ, K?(Ui)13r K?(Ui)23r s s i s n ,

2. K1(UJ3,, r 5 i 5 n,

3. K ; ( I ? ! ~ ) ) ~ , K(t!')),,for r 5 i

<

s if there is a transla-

tional joint k , k

<

i, such that ukx(ui or if there is

a

rotational joint m , m

< i,

such that L,s #

0 and )((ur (otherwise, they are not).

However, K:r?ziand

&+(~?j~))~,

Y I

i

I n, are redundant for computing h, as are all entries, other than the (3,3)th,

All components of v in Eq. (30) are included in Properties 2 to 4. Let V, consist of all Kikli, Kiri2,,

y(t(i))x

and KT(tii))y, i

<

s, in v, v B consist of the three

components of all Kjty),

j >

s, and vc consist of all other components of v. Furthermore, the compo- nents of v,, vB, and vc are all assumed to be pre- sented in the order from joint 1 to joint n. On the other hand, W, and w B are, respectively, defined the

same as V, and v B except that

tj'),

kli and k2, are,

respectively, replaced with ki and

of KYU,, r 5 i

<

s.

K~~ = -(up))y(ki)x

+

(up))x(ki)y (53)

The last two equations are different from Eqs. (36) and (50) only in the terms of k,.

Suppose that there are translational joint k and rotational joint i, k

<

i

<

s. By Properties 2 to 4, if

uk )((ui, then klk, k2k, and

(tji))y

are all compo-

nents of v in Eq. (30). This means that if Kkrilk and KkkZk are components of v, then so are

&+(t!i))x

and K;(tji)& for all i, k

<

i

<

s. In the other situation, when there are only parallel translational joints in front of joint i, if there is a rotational joint m in front of joint i such that ~ l ~ ) ( ( ~ i , then for any rotational

joint j behind joint i, i

<

j

<

s, neither ',s nor

LS

is parallel to

u,

because ',s = i s

+

is. Applying Prop- erty 4 yields that if KT(t;i))x and K?(tji))y are compo- nents of v, then so are K*(t!j)), and

K;(t?))y

for all j , i

<

j

<

s. Consequently, if either Kikl, and Kik2, or

K?(tji))x

and

&+(tji))y,

i

<

s, are in v, then K;(ty))x and K;(t!)&, i

< j <

s, are also in v. Note that Ali and k2i

are linear combinations of the components of

tji),

and that

tji)

for translational or rotational joint i can be expressed in terms of K . m . and the x- and

y-

components of Kykj, j

>

i. Thus it can be shown that

I . 1

I . 1

(12)

652 Journal of Robotic Systems-1994

V B = IwB

+

PBwg (56) where I is the identity matrix, WA is an upper trian- gular square matrix with the diagonal entries of 1,

PA and PB are some appropriate matrices, and wg is defined as in Eq. (31).

Therefore, Eq. (30) can be rewritten as

(57)

where

Because the columns of

RAT

RBI

and

Hc

are linearly independent to one another and WA is nonsingular, the columns of C are also linearly independent ac- cording to Lemma A2. Note that some columns of D may be linearly dependent on those of C. In this case, there exists a matrix

GD

whose columns are linearly independent to one another and to those of C, such that the second term in Eq. (57) can be decoupled into

DWg = CPlWg

+

HDP2Wg (60)

where P, and P, are constant matrices. Finally, we conclude that a set of MLCs for determining h is

w ' =

[;;j

+P,wg

(61)

...

P2wg

which is also the MLCs for determining 7' according

to Theorem 3. By Theorem 2, w,, wB, vc, and wg constitute a set of MLCs for determining r, as stated in the following theorem.

the set 1. 2. 3. 4. 5.

Y consisting of all nonzero elements of Kj+(Uj)33, GjKT(kj),, SjKj+(kj), for r 5 j

<

s,

Ky(Uj)23, Kj+(kj),, K;(kj), for s 5

j

5 n, Kifii for i = 1,

. . .

,

n,

Kj(kJX, Ki(ki),, Ki(ki), for s

<

i I n, and

uiKiKlir ~ , K , ~ , , f u r r

<

i

<

s,

Ky((Ujh1 - (Uj),2>/ K;(Uj)33/ Ky(Ujht q(Ujhr

where tij = 0 for the case that ur//uk//g, Vk 5 j

<

S ,

and ',s (when j > r ) is zero or parallel to u, for every rotational joint m , r 5 m

< j ,

otherwise Si = 1, and where ui = 0 for the case of ui//ur, r

<

i

<

s, otherwise

uj = 1.

An example for a set of MLCs of the Stanford arm can be found in previous We are now ready to state and prove the equivalence theorem. Corollary 6. Suppose that two sets Y , and YPb, formed by the linear combinations of the inertia parameters of a

manipulator, have the same number of elements and Y , is a set of MLCs for determining r. Then 9 , is also a set of

MLCs for 7 if and only if the values of the elements of Y , can be obtained from those of Y , and vice versa, i.e., there is a nonsingular constant matrix M such that

b = Ma (62)

where the components of a and b are the elements of 9,

and Y b , respectively.

Proof: According to the assumption, there are matrix A(q, q, q) with linearly independent columns and matrix P with linearly independent rows, such that r = Aa and a = Px, where x is the vector of all inertia parameters.

When Eq. (62) holds, applying Lemma A2 to PTMT yields the result that the rows of (MP) are linearly independent, so the components of b are also linearly independent. Moreover, we also obtain T = AM-lb, where the columns of (AM-I) are lin- early independent by Lemma A2. This completes the proof of the sufficiency.

If YPb is also a set of MLCs for T, there are matrix

B(q, q, q) with linearly independent columns and matrix Q with linearly independent rows, such that

r = Bb and b = Qx. Suppose that x E RP and a and b are k-tuples. There are (

p

-

k)

linear combinations (denoted by c) of x, such that the components of a

and c are linearly independent and form a basis of the domain of x. Thus x can be expressed as Theorem 5. For the manipulator considered in Theorem

4, a set of MLCs for determining the actuator forces T is

(13)

Lin: Identifiable Parameters of a Manipulator 653

which yields the results that

(64)

b = QW,a

+

QW2c

7 = BQW,a

+

BQW,c (65)

Note that Y a is a set of MLCs. By Theorem 1, (BQW,) must be zero and the columns of (BQW,) are linearly independent. This leads to the conclusion that QW, = 0 because the columns of B are linearly inde- pendent, and that (QW,) is a nonsingular square matrix according to Lemma A2. The claim of neces-

sity then holds. Q.E.D. rn

5. CONCLUSION

The minimal linear combinations of the inertia pa- rameters (MLCs) introduced in this article illuminate the role of the identifiable parameters in the manipu- lator dynamics. This article has presented a system- atic approach to finding a set of MLCs. The problem is first divided into two searches for two individual sets of MLCs that determine the entries of the inertia matrix and the gravity load. The MLCs for the gravity load are easier to find and have been addressed in an earlier To find a set of MLCs for the inertia matrix, we begin by decoupling the inertia parame- ters of composite bodies into two parts, so that the varying part of each parameter can be calculated from the constant parts of the other parameters. The rest of our task is then to carefully inspect the roles of the constant parts of the parameters in the inertia matrix. The technique for this is presented in section 4, where it is used to formulate Properties 2 to 4. Applying Theorems 2 and 3 converts the elements in Properties 2 to 4 to those of the MLCs for the full manipulator dynamics. This step-by-step approach is different from that in the author's earlier

although the results are identical. However, this set of MLCs is slightly different from others in the litera- tUre16,17,19-22 in some minor terms because a set of MLCs is not unique. The equivalence of these differ- ent results can be proved using Corollary 6 in the last section. The crucial feature of the present approach is that all derivations are in accordance with the least squares theory, so the identifiability of the present set of MLCs is assured.

The present set of MLCs also has some other advantages: a systematic off-line identification method for it has already been proposed,23 and a recursive formulation of the manipulator inverse dy- namics in terms of the set of MLCs has been derived

and shown to be more efficient than most other for- mulations of the inverse dynamics in the l i t e r a t ~ r e . ~ ~ The main emphasis of the present article is that the minimal parameters of a manipulator should be treated in the least squares sense, so that the minimal parameters are equivalent to the identifiable param- eters.

The author gratefully acknowledges the reviewers for their constructive comments on the presentation of this article. This article was supported by the National Science Council, Taiwan, under Grant NSC80-0404- E-009-31.

APPENDIX

Lemma A l . Suppose that

where x E RP are the system parameters, q E R", a..R"X1lXnXPjR" b . * R l l x P + R n l , c. :RnxP+Rfll+ Then,

' I ' lk

a set is a set of MLCs for determining a if and only if it is also a set of MLCs for determining Lemma A2. Suppose thnt the columns of matrix A(8) :

R" + Rnxk are linearly independent over R"'. The constant

square matrix B E Rkxk is nonsingular if and only if the columns of the product A( 8)B (or BA( 8)) w e also linearly

Lemmas A1 and A2 are used in the proofs of Theorem 3 and Corollary 6, respectively. The proofs of these two lemmas are very simple and can be found in a previous In the following, we prove the theorems in section 2.

Proof of Theorem 1: Sufficiency (+): According to the assumption that the elements of w are linear combi- nations of x, there exists a constant matrix B E RkxP such that rn T T T T ~ 1 1 2 , ~ 2 1 ,

. . .

r cnJ

.

[b:,

. . .

b,T,

c T ~ ,

.

*

.

independent over R". rn

The rank of B is k since the rows of B are linearly independent. It follows from Eqs. (3) and (A2) that

ai(8) = x(8)bi (A3)

where ai and bi are the ith columns of A and B, re- spectively.

(14)

654 Journal of Robotic Systems-1994

We select k linearly independent columns bi and then reorder them and the corresponding ai( 8 ) to be the first to kth columns of B and A( O), respectively. If we let

a,a,(8)

+

. .

+

akak(8) =

-

A(8)(blal

+

* * *

+

bkak) = 0 (A4)

then a, = ,

. .

= (Yk = 0 because the linear indepen- dence of the columns of x ( 8 ) over R" implies (b,al

+

.

+

bkaJ = 0. We conclude that al(8),

. . . ,

ak(8) are linearly independent over R". Be- cause the rank of B is k, there exist not all zero ai

such that (bla,

+

. . .

+

bk+,ak+,) = 0. Because

c

ai( 8)ai =

x(

8 )

,

V 8 E Rm (A5)

I

any k

+

1 or more columns of A( 8 ) are then linearly dependent over R". Consequently, the number of linearly independent columns of A is k.

Necessity (4): Because the number of linearly

independent columns of A( 8 ) is k, we choose k lin- early independent columns to construct

x(

e)

: R" + Rnxk, and let the other ( m - k) columns form A(0) : R" --.) Rnx(m-k). We can partition x into X and 2

such that

A(e)x =

A(e)X

+

A(e)n

(-46)

Every column

(ai)

of A can be expressed as a linear combination of the columns

(ai)

of in the form of

where aij are some constants. Substituting Eq. (A7) into Eq. (A6), we get

where

a l k l

. . . . . .

pll

.

Lu;,

. . . . . .

Proof of Theorem 2: Suppose that there are H(q, q, q) = [H, : H,] and G(q) whose columns are linearly independent over R3" and R", respectively, such that

_

_ -

Thus,

where

and I is the identity matrix. The fact that each entry of

H

is associated with q and/or q (see Eqs. (4) and (5)) implies that the columns of [H i GI are linearly independent over R3". This completes the proof of the first part. Let

([HI

i

H,

i G]P)a = 0. Because the columns of [H i GI are linearly independent, Pa =

0. This is only possible when a = 0. The claim of the second part is then true. Q.E.D. Proof of Theorem 3: Let h = [h,,,

. . .

,

hln, h,,,

. . . ,

Suppose that w(x):RP + Rk is a set of MLCs for determining h, and there exists H(q) : R" RnZxk

such that

- _

- _

L I T

and c = [ciiir ~ 1 1 2 ,

.

* * c i i n r ~ 1 2 2 , * *

.

cnnnI'*

h = H(q)w(x) 6414)

By Christoffel symbols in Eqs. (6) and (7), we get

c = T(q)w(x) (A151

where c ( q ) is some appropriate matrix, whose col- umns may be linearly dependent over R". Therefore, w(x) is also a set of MLCs for [h', cTIT because the columns of

[HT,

CTlT

are linearly independent over R"

.

Conversely, we assume that the columns of whose elements are linearly independent over

[nT,

CTlT

are linearly independent, but the columns

(15)

Lin: Identifiable Parameters of a Manipulator 655

combine the dependent columns of

k

to form H, whose columns are linearly independent over R”, such that

h = H(q)w(x) (A16)

where the elements of w(x) are linear combinations of w(x) and are linearly independent over RP. Using Christoffel symbols again, we obtain the result that w(x) is also a set of MLCs for [h’, $1’. This contra- dicts Theorem 1 since the dimensions of w and w are not the same. Consequently, the columns of

H

are linearly independent over R” if and only if the columns of

[HT,

c’]’

are linearly independent over R”. This implies that w(x) is a set of MLCs for h if and only if it is also a set of MLCs for [h’, cTIT.

The rest of the proof consists of showing that a set is a set of MLCs for T I if and only if it is also one for [hT,

$IT,

which follows directly from Lemma

A1 above. Q.E.D. H

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數據

Figure  1.  Illustration  of  the  normal  driving-axis coordi-  nate  system.
Figure 2.  Geometrical relation  of  any two joints.

參考文獻

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