Optimal body fixed coordinate systems in Newton/Euler modelling

modelling

Citation for published version (APA):

Lucassen, F. H. R., & Ven, van de, H. H. (1988). Optimal body fixed coordinate systems in Newton/Euler modelling. (EUT report. E, Fac. of Electrical Engineering; Vol. 88-E-210). Eindhoven University of Technology.

Document status and date: Published: 01/01/1988 Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

Fixed Coordinate Systems

in Newton/Euler Modelling

by F.H.R. Lucassen and H.H. van de Ven

EUT Report 88-E-210 ISBN 90-6144-210-9 Novem ber 1988

ISSN 0167- 9708

Faculty of Electrical Engineering Eindhoven The Netherlands

OPTIMAL BODY FIXED COORDINATE SYSTEMS IN NEWTON/EULER MODELLING

by

F.H.R. Lucassen and

H.H. van de Ven

EUT Report 88-E-210 ISBN 90-6144-210-9

Eindhoven November 1988

Lucassen, F.H~R.

Optimal body fixed coordinate systems in Newton/Euler

modelling / by F.H.R. Lucassen and H.H. van de Ven.

-Eindhoven: Eindhoven University of Technology, Faculty of Electrical Engineering. - Fig. - (EUT report, ISSN

0167-9708; 88-E-210)

Met lit. opg., reg.

ISBN 90-6144-210-9

SISO 527.7 UDC 007.52 NUGI832

Summary

In Newton/Euler modelling the multi body mechanical systems are con-sidered to be an open kinematic chain consisting of n arbitrary bodies without branching. In every link a so-called body fixed cartesian co-ordinate system is introduced.

In real time applications the number of multiplications necessary for the computation of the transition matrices is of major importance. In this report, an optimal body fixed coordinate system with respect to the number of multiplications is derived.

Starting with the collection of all possible body fixed coordinate systems and their corresponding transition matrices, the number of multiplications necessary for the computation of the transition mat-rices is counted.

Several possible coordinate systems are considered and we shall prove that for real time applications the Denavit-Hartenburg convention is the optimal choice. It contains only 4 multiplications.

Lucassen, F.H.R. and H.H. van de Ven

OPTIMAL BODY FIXED COORDINATE SYSTEMS IN NEWTON/EULER MODELLING.

Faculty of Electrical Engineering, Eindhoven University of Technology,

The Netherlands, 1988. EUT Report 88-E-210

Addresses of the authors: ir. F.H.R. Lucassen. Lintronics B. V. ,

P.O. Box 99, 5640 AB VEGHEL, The Netherlands

ir. H.H. van de Ven,

Measurement and Control Group, Faculty of Electrical Engineering, Eindhoven University of Technology,

P.O. Box 513, 5600 MB EINDHOVEN,

Contents Sununary 1. 2. 3. 4. 5. Introduction

Collection of all possible coordinate systems The transition matrix

The choice of an optimal coordinate system Conclusion References i i i 1 2 3 8 13 14

1 . Introduction

In Newton/Euler modelling a robot mechanism is considered to be an open (kinematic)

branching [1,2). tional degree of

chain consisting of n arbitrary bodies without The joints connecting the segments have one rota-freedom.

A joint allows a relative rotation around an axis determined by a unit vector, eU or e

1 .

-

1

-where: ~i is the unit vector of rotation of the i-th body with the u

~i

(i-l)th body;

is the unit vector of rotation of the i-th body with the (i+l)th body

[5).

In every link a body fixed cartesian coordinate system is introduced. A fixed external cartesian coordinate system with a vertical z-axis is also defined (fig. 1.1).

Zo

fixed coordlnat. aya'em _Z.

Z,

-

-Z,

,

'aegment

'1 '

Fig. 1.1 Coordinate systems

In Newton/Euler modelling it is important to compute the different representations of vectors in the different coordinate systems. The transformation of one coordinate system to another is possible with transition matrices.

There are three possible orientations for the body fixed coordinate system:

10 _{parallell to the principle axis of the segment,}

20 _{the Denavit-Hartenburg [3) convention,}

30 _arbitrarily

For real time applications (e.g. control) the number of multiplication necessary for the computation of the transition matrices is of major importance.

In this paper a method is described to determine an optimal body fixed coordinate system.

2 . Co11ection of &11 poaaib1e systems

Consider the i-th body with its two rotational axes

e~

_-1 and

e~

_-1. (fig. 2.1).

--e·-t

-.

(

.

_"1 )

,.

_bOdy

/

Xi . th 1- body

~~---Fig. 2.1 Configuration of the i-th body

Only rotations around _-1.

e~

and _{- l .}

e~

are possible; so i t is clear that the body fixed cartesian coordinate system should be referred to these two

Assuming e.

-~

*

-~

e~

and (e .. e. ) -~ -~

*

0, the following choice is rather an obvious one.

xi-axis --? i _{'Y ll~i} u + i _'Y12~i J.. + i _{'Y 13~i} u x -i eJ..

Yi- axis --? i _{'Y 21~i} u + i _{'Y 2 2~i} J.. + i _'Y23~i u x _-~

e~

z.-axis --? i _{'Y 31~i} u + i _{'Y 32~i} J.. + i _'Y33~i u x eJ..

~ - i

where x denotes the cross product.

Call this the i-th coordinate system with coordinate matrix

Of course, there are some restrictions on i 'Yk,j (e.g. det i

r

=

The choice for the (i+1)th coordinate system gives: x_i +

1-axis

Yi+1- axis

zi+1- axis

3 . The transition matrix

In this section we determine the transition matrix needed for the transformation of the (i+1)th body fixed cartesian coordinate system into the i-th body fixed cartesian coordinate system; this matrix is

(2.1)

called

A rotational joint axis determined by Hence:

(see fig. 3.1) allows a relative a unit vector

e~

(or

e~+l).

rotation around an e~ - ] .

1

+- ~i+l - ] . - ] .

where "+-,, is a symbolic notation for: "is transformed in".

___ Zj

Yj

Fig. 3.1 Rotational joint A vector in a plain

(fig. 3.2) and will

perpendicular to e~ _{- ] .} will remain in this plain be transformed according to Rodrique's formula [4

J.

The transformations of the other two cases are:

1

1

u

+cos CPi (~ i+l x .(~i+l x ~i+l)) _{/sin lli +1}

(3.1)

(3.2)

with u 1. (e .• e . ) -1. -1. I

I

}-/ / / / eU

_-.

Fig. 3.2 Rotation around the angle ~i Furthermore and 1 !1+1 I I I I cos 0. ~u

The components of the vectors

~u,

e1. and

~u

x

~1.

are the elements of the inverse matrix

r,

so:

(3.4)

(3. Sa)

~u

~u

x

~1.

With equations (3.4), (3.5) and (3.6) a matrix formulation of equa-tions (3.1), (3.2) and (3.3) yields:

i5 11 cot a_i . i511-1/sin _{a i}i_{5 21 i531/sin a i} i+15 21 i5 12 a i · i 5 12 -1/sin aii5 22 i532/sin i H15 cot _{a i} Ai+l" 22

i5l3 cot a_{i . i5l3-1/sin a i}i_{5 23} i533/sin _{a i} H15 ₂₃

i+1 H1 H1

- sin 'Pi/sin a_i+1 _{5 31} + cos 'P/ sin ai+1(cos a i +1 . 521- 5 11

i+1 H1 H1

sin _{'Pi/sin a H1 5 32+ cos 'Pi/sin} (li+l(cOS a i +1 5 22- 5 12

sin 'Pi/sin _{a i +1} i+1 _{5 33}+ cos 'Pi/sin ai+1(cos _{a i}+₁ H1t; 23-H15 13

i+1 i+1 i+1

- cos 'Pi/sin a i+_{1 5 31 - sin 'Pi/sin ai+1 (cos} a_i+_{1 5 21- 5 11}

H1 H1

- cos 'Pi/sin ai+1i+1532 - _{sin 'Pi/sin ai+1 (cos a i +1 5 22- 5 12}

i+1 i+1 i+1

- cos 'Pi/sin a_i+_{1 5 33 - sin 'Pi/sin ai+1 (cos} a_i+_{1 523- 5 13}

Equation (3.7) can be written as:

(3.6)

(3.7)

with:

v

_{(Vk ,j)3x3}

W

and the rotation matrix

1 0

R 0 cos

0 -sin

0 <Pi sin <Pi <Pi cos <Pi

cot (Xi -l/sin (Xi

o

1

o

o

o

o

-sin

o

o

l/sin (Xi

].

Both the matrices V and W can be interpreted as transition matrices; translation of the origin of the (i+1)th coordinate system to the

joint and from the joint to the origin of the i-th coordinate system respectively.

(3.9)

(3.10)

(3.11 )

(3.12)

With equations (3.9), (3.10), (3.11) and (3.12) the elements of matrix i Ai+1 can be calculated.

i

a k . - v k 1w1 . + cos <po _{(vk 2w2 . + v k 3w3 .) +}

, ] , , ] J . , , ] , , ]

(3.13) (k and j 1, 2, 3)

The terms Vk,jWk,j can be calculated. beforehand, so the number of mul-tiplications for the computation of 1Ai+1 equals 18.

4 • The choice of an OPtimal. coordinate system

In this section a coordinate system will be chosen in such a way that the computation of the transition matrix will be as simple as possible

(~ minimal number of multiplications). Throughout this section we will assume Euler modelling not only the transition

u

£.

~i if ~i· Because in Newton/ matrix is necessary but also its inverse, only orthonormal coordinate systems are considered. The following method is propo~ed:

1. Prescribe a matrix ir.

Start with a matrix with six zeros and try all possibilities un-til r has become a full matrix.

2. Check the orthogonality of ir.

3. Transform ir into an orthonormal matrix. 4.

5.

I.

i

Compute the elem

₇

_{nts of Ai +l .}

Choose the free LYk .'s in such a way that

_,

)

simple as possible.

Assume 3 elements iYk . unequal to zero.

_,

)

i

Ai+l becomes as

r _

[lg

The only relevant choice of r is: -

o

1

o

~

]

which vio-II I I , l 1 0 0

lates the orthogonal condition.

i

Assume 4 elements Yk . unequal to zero.

,

)

The following choices are possible (C i is a constant which has to be determined) . Ci 0 1 0 C_i 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 _Ci 1 0 0 1 C_i 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 C_i 0 1 0 C_i a b c d e f 0 0 1

11,2 A check of the orthognality of the various matrices gives: a: _Ci

= -

cos (Xi

b: impossible to choose C_i in such a way that the coor-dinate system becomes orthogonal.

c: identical to a.

d-f: impossible to achieve orthogonality.

11,3 Cases II,2,a and c give the possible bases with four elements i

Yk . unequal to zero. , J

In accordance with II,2,a:

xi-axis ~ _.§.i u

Yi- axis ~ (e.

1.

--~ cos (Xi e':')/sin -~ (Xi (4.1)

zi- axis ~ e':'

-~ x

~~/sin

(Xi

From (4.1) follows for (2.2)

[

:

cot (Xi

a

₁

i r _{l/sin (Xi}

_a

a

l/sin (Xi

The matching transition matrix (3.8) is:

cos (Xi+l sin (Xi+1

a

i

Ai+1 cos 'Pi sin (Xi+l cos 'PicOS (Xi+l sin 'Pi (4.2)

sin 'Pi sin _(Xi+l - _{sin 'PicOS (Xi+l}

-

_{cos 'Pi}

In accordance with II,2,c:

Yi- axis ~

e~

-~

zi- axis ~ u

e~/sin

_Il

i ~i x -~ From (4.3) follows:

[ ,t.>o

Ili

a

a

1

ir -cot Ili 1

a

a

a

l/sin Il i The matching transition matrix is:

- sin 'Pi cos Il

i

a

The matrices (4.2) and (4.4) contain only four multiplications. The bases are permutations of the so-called Denavit-Hartenberg [3] convention.

III Assume 5 elements·Yk . unequal to zero. , J

111,1 The following choices are possible (C

i and Bi are constants which have to be determined).

1 C_i B. 1 ~ Ci 0 1 Ci 0 1 Ci 0 1 Ci 0 1 0 1 0 _Bi 1 0 0 1 B. ~ 0 1 0 0 1 0 Bi 0 0 1 0 0 1 0 0 1 _Bi 0 1 0 _Bi 1 0 a b c d e (4.3) (4.4) 0 _Ci 1 0 0 1 f

III,2 III,3 with From 1 0 _Ci 1 0 C_i 1 0 C_i 1 0 0 1 0 0 1 0 1 _Bi 0 1 0 0 1 0 C_i 1 _{Bi Ci} 1 0 C i 0 0 1 _Bi 0 1 0 _Bi 1 0 0 1 _Bi 0 1 0 g h i j k 1 0 0 1 0 0 1 0 0 0 1 C_i 0 1 C i 0 1 0 B. ~ 0 1 0 B. ~ 1 Ci Bi 1 m n 0

A check of orthogonality gives:

a: impossible to achieve orthogonality b: C. =

-~ (Bi + cos ai)/(l + cos ail

c-o: impossible to achieve orthogonality. The only possible

(III,2,b) :

base with five

~k

_,

.'s unequal to zero is ) xi-axis -+ Yi- axis ~ Pi (1 + 2Bi cos a i + B~) ~ _1. qi = (1 + 2C_{i cos a i} + C~) ~ 1. (4.6) follows:

[ >lp

Ci/qi 0

1

i r Bi/P: l/qi 0 0 0 Ilsin a_i 0 0 1 0 B. 1 ~ 1 (4.5) (4.6) (4.7a) (4.7b) i

The matching transition matrix Ai +1 can be calculated with equations (3.9) to (3.13).

V

llWl2 - cos CPi(cOS a i + Ci ) (1+Bi cos ai)/(pi sin ail . (CHl sin aHl/qi+l) v_2lw_l2 + _{cos CPi (1+Bi cos ail (1+Ci cos ai)/(qi sin ail . (CHl} sin _aHl/qHl)

sin CPi(Ci+l sin _aHl/qHl)

sin CPi (cos 0i + _{Ci ) (1+Bi cos ai)/(pi sin ail}

1

sin CPi(1+Bi cos ail (1+Ci cos ai)/(qi sin ail

- cos _CPi

(4.8)

The matrix (4.8) contains eight multiplications. To reduce the number of multiplications, the following attempts are possible:

1· _{cos a i + Ci} ~ 0 ~

Ci ~

-cos a i

From equation (4.5) _{follows Bi} 0 ~ impossible

2· _{1 + Bi cos a.} D ~ _Bi - _{l/cos a i}

l.

From (4.5) _{follows Ci} ~ _~ impossible

3· 1 + C_i cos a_i ~ 0 _~ C

i - l/cos ai

and with equation (4.5) _Bi ~ _~ impossible

4· sin _{a i +1 /Pi+1} ± 1

50

From equation (4.7a) follows Bi+1 - - cos a i +1 and from (4.5) _Ci+₁

=

a

± 1

With equation (4.7b) _{Ci +1 = - l/cos} a_{i +1}

and with (4.5) 00

~ impossible

~ impossible

Hence it is impossible to simplify the expression for the transition matrix with five Yk,j'S unequal to zero.

IV _{Assume 6 to 9 Yk .'s unequal to zero.}

_,

)

The expressions for Vk,j and Wk,j are so complicated that it is impossible to reconstruct a transition matrix with less than four multiplications.

5. CONCLUSXON

An arbitrarily chosen body fixed coordinate system can be extremely inconvenient for real time applications. The number of multiplica-tions, needed for the calculation of the transition matrix, can rise to 18.

The Denavit-Hartenberg convention minimizes the number of mUltiplica-tions in the transition matrix.

REFERENCES

[lJ Vukobratovic, M. and V. Potkonjak

SCIENTIFIC FUNDAMENTALS OF ROBOTICS 1. DYNAMICS OF MANIPULATION ROBOTS: Theory and application.

Berlin: Springer, 1982.

Communications and control engineering series

[2J Paul, R.P.

ROBOT MANIPULATORS: Mathematics, programming, and control. The computer control of robot manipulators.

Cambridge, Mass.: MIT Press, 1981.

The MIT Press series in artificial intelligence [3] Denavit, J. and R.S. Hartenberg

A KINEMATIC NOTATION FOR LOWER-PAIR MECHANISMS BASED ON MATRICES.

Trans. ASME Ser. E, J. Appl. Mech., Vol. 22(1955), p. 215-221.

[4J Frazer, R.A. and W.J. Duncan, A.R. Collar

ELEMENTARY MATRICES AND SOME APPLICATIONS TO DYNAMICS AND DIFFERENTIAL EQUATIONS.

Cambridge University Press, 1938. Reprinted 1957.

[5] Lucassen, F.H.R. and H.H. van de Yen

A NOTATION CONVENTION IN RIGID ROBOT MODELLING.

Faculty of Electrical Engineering, Eindhoven University of Technology, 1988.

(189)

(190)

( 191 )

(192 )

(193)

EUT Report 88-E-188. 1988. ISBN 90-6144-188-9

Pineda de Gyvez, J.

ALWAys: A system for wafer yield analysis.

EUT Report 88-E-189. 1988. ISBN 90-6144-189-7

Siuzdak, J.

OPTICAL COUPLERS FOR COHERENT OPTICAL PHASE DIVERSITY SYSTEMS. EUT Report 88-E-190. 1988. ISBN 90-6144-190-0

Bastiaans, M.J.

LOCAL-FREQUENCY DESCRIPTION OF OPTICAL SIGNALS AND SYSTEMS. EUT Report 88-E-191. 1988. ISBN 90-6144-191-9

Worm, S.C.J.

AiMOLTI-FREQUENCY ANTENNA SYSTEM FOR PROPAGATION EXPERIMENTS WITH THE OLYMPUS SATELLITE.

EUT Report 88-E-192. 1988. ISBN 90-6144-192-7

Kersten, W.F.J. and G.A.P. Jacobs

ANALOG AND DIGITAL SIMULATI~LINE-ENERGIZING OVERVOLTAGES AND COMPARISON WITH MEASUREMENTS IN A 400 kV NETWORK.

EUT Report 88-E-193. 1988. ISBN 90-6144-193-5

(194) Hosselet, L.M.L.F.

MARTINU5 VAN MARUM: A Dutch scientist in a revolutionary time.

EUT Report 88-E-194. 1988. ISBN 90-6144-194-3

(195) Bondarev, V.N.

ON SYSfEM IDENTIFICATION USING PULSE-FREQUENCY MODULATED SIGNALS. EUT Report 88-E-195. 1988. ISBN 90-6144-195-1

(196) Liu Wen-Jiang, Zhu Yu-Cai and Cal Da-Wei

MODEL BUILDING toR AN INGOT HEAifNG PROCESS: Physical

identification approach.

EUT Report 88-E-196. 1988. ISBN 90-6144-196-X

modelling approach and (197) Liu Wen-Jiang and Ye Dau-Hua

x-REW METHOD FOR DYNAMIC HUNTING EXTREMUM CONTROL, BASED ON COMPARISON OF MEASURED AND ESTIMATED VALUE.

EUT Report 88-E-197. 1988. ISBN 90-6144-197-8

(198) Liu Wen-Jiang

ANrEXTREMUM HUNTING METHOD USING PSEUDO RANDOM BINARY SIGNAL. EUT Report 88-E-198. 1988. ISBN 90-6144-198-6

(199) Jozwiak, L.

THE FULL DECOMPOSITION OF SEQUENTIAL MACHINES WITH THE OUTPUT BEHAVIOUR REALI ZATJON.

EUT Report 88-E-199. 1988. ISBN 90-6144-199-4

(200) Huis in It Veld, R.J.

A FORMALISM TO DESCRIBE CONCURRENT NON-DETERMINISTIC SYSTEMS AND AN APPLICATION OF IT BY ANALYSING SYSTEMS FOR DANGER OF DEADLOCK. EUT Report 88-E-200. 1988. ISBN 90-6144-200-1

(201) j~~~~YNIH~i.iE~van and R. van den Born

~ IS WITH THE AID OF-oYNAMIC PROGRAMMING. EUT Report 88-E-201. 1988. ISBN 90-6144-201-X

(202) En~elshoven, R.J. van and R. van den Born

CO I CALCULATION FOR INCREMENTAL HARDWARE SYNTHESIS. EUT Report 88-E-202. 1988. ISBN 90-6144-202-8

(203) Delissen, J.G.M.

THE LINEAR REGRESSION MODEL: Model structure selection and biased estimators.

EUT Report 88-E-203. 1988. ISBN 90-6144-203-6 (204) Kalasek, V.K.I.

COMPARISON OF AN ANALYTICAL STUDY AND EMTP IMPLEMENTATION OF COMPLICATED THREE-PHASE SCHEMES FOR REACTOR INTERRUPTION.

(205) Butterweck, H.J. and J.H.F. Ritzerfeld, M.J. Werter

FINII£ WORDLENGTH EFFECTS IN DIGITAL FILTERS:~iew.

EUT Report 88-E-205. 1988. ISBN 90-6144-205-2

(206) Bollen, M.H.J. and G.A.P. Jacobs

~IVE TESTING OF AN AL~ FOR TRAVELLING-WAVE-BASED DIRECTIONAL DETECTION AND PHASE-SELECTION BY USING TWONFIL AND EMTP.

(207)

(208)

(209)

(210)

(211 )

EUT Report 88-E-206. 1988. ISBN 90-6144-206-0

Schuurman, W. and M.P.H. Weenink

STABILITY OF A TAYLOR-RELAXED CYLINDRICAL PLASMA SEPARATED FROM THE WALL BY A VACUUM LAYER.

EUT Report 88-E-207. 1988. ISBN 90-6144-207-9

Lucassen, F.H.R. and H.H. van de Ven

A NolAtloN CONVENTION IN RIGID ROBOT MODELLING. EUT Report 88-E-208. 1988. ISBN 90-6144-208-7

Jozwiak, L.

MINIMAL REALIZATION OF SEQUENTIAL MACHINES: The method of maximal

adjacencies.

EUT Report 88-E-209. 1988. ISBN 90-6144-209-5

Lucassen, F.H.R. and H.H. van de Ven

OPTIMAL BODY FIXED COORDINATE SYSTtMS IN NEWTON/EULER MODELLING. EUT Report 88-E-210. 1988. ISBN 90-6144-210-9

Boom, A.J.J. van den

~ONTROL: An exploratory

EUT Report 88-E-211. 1988.

study.