A new method for reverse force analysis of a spatial three-spring system

Mechanism and Machine Theory 36 (2001) 623±632

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A new method for reverse force analysis of a spatial three-spring system Limin Fan *, Qizheng Liao, Chonggao Liang Beijing University of Posts and Telecommunications, P.O. Box 108#, 100876 Beijing, People's Republic of China Received 14 August 1999; received in revised form 1 August 2000

Abstract Using the idea of Dixon's resultant, a new method for the reverse force analysis of a spatial three-spring system is presented. By this method a 6 by 6 matrix is derived, and the 22nd degree polynomial in a single spring length is obtained only by computing the determinant of the 6 by 6 matrix. The result has been veri®ed by a numerical example. The symbolic computation is carried out by computer algebraic system MATHEMATICA. Ó 2001 Published by Elsevier Science Ltd.

1. Introduction Recently some reverse force analysis of spring-compliant mechanisms has been performed in [1±4,6]. The compliant parallel mechanisms, which can be treated as spring systems, have attracted great interest of many mechanism researchers. Compared with rigid mechanisms, compliant parallel mechanisms posses many new prospects, which make them have a lot of potential applications in the areas of robotic mounting and robotic force control, etc. The signi®cance of this type of work has already been addressed in detail in [1,3]. The reverse force analysis of a spring system, i.e. computing the multiple equilibrium positions of the system for an applied external force, is a dicult analytical problem. It involves highly nonlinear equations. This paper will focus attention on the reverse force analysis of the spatial threespring system shown in Fig. 1. The structure of the spatial three-spring system comes from [3]. The system consists of three translational linear springs acting in parallel, each of which is attached to the ground via pivots (A, B, C) which form a triangular base. The three springs are joined together at a common pivot (P) at the other end. Each spring can be thought of as acting in the prismatic *

Corresponding author. E-mail address: [email protected] (L. Fan).

0094-114X/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd. PII: S 0 0 9 4 - 1 1 4 X ( 0 1 ) 0 0 0 1 6 - 7

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L. Fan et al. / Mechanism and Machine Theory 36 (2001) 623±632

joint of a spherical±prismatic±spherical serial chain. A known external static force ~ F ˆ Fx~i ‡ Fy~ j ‡ Fz~ k is applied to the spring system at P. The external force is in static equilibrium with the forces acting in the springs and the static equilibrium positions of the spring system are determined. The chief objective of the present work is to derive the resultant lengths of springs. Zhang et al. [3] ®rst solved this problem in 1994 by using the resultant of Sylvester, the 22nd degree polynomial obtained from the determinant of a 16 by 16 matrix. But the computation procedure is very complex and the extraneous roots are cancelled in several steps by factoring several formulae and the determinant of the 16 by 16 matrix. The new method presented here will prove to be more e€ective and simpler than that of [3]. The 22nd degree polynomial can be got only by computing the determinant of a 6 by 6 matrix and the determinant does not contain any extraneous root. 2. Constraint equations of the spatial three-spring system 2.1. Geometry constraint equations A Cartesian coordinate system is chosen with the joint center A of the base as the origin of the system and the positive x-axis passing through the joint center B. The y-axis in the base plane is perpendicular to the x-axis, and the z-axis is perpendicular to the base plane as shown in Fig. 1. The absolute coordinates of the joint centers A, B and C in base DABC are (0; 0; 0), (X2 ; 0; 0), and (X3 ; Y3 ; 0), respectively. We assume that the spring lengths in equilibrium positions are L1 ; L2 ; L3 , r2 ;~ r3 are the unit vectors specifying the orientations of the lines of action of the three and ~ r1 ;~ springs. The above six variables are all unknown system variables. The geometry constraint equations for the spring system can be written as follows:

Fig. 1. Spatial three-spring system.

L. Fan et al. / Mechanism and Machine Theory 36 (2001) 623±632

625

~ r1 ˆ 1; r1
~

…1†

~ r2 ˆ 1; r2
~

…2†

~ r3
~ r3 ˆ 1;

…3† !

L1~ r1 ˆ L2~ r2 ‡ BA;

…4†

!

L1~ r1 ˆ L3~ r3 ‡ CA:

…5†

2.2. Force balance equations The spring elasticity constants k1 ; k2 ; k3 and the spring free lengths L01 ; L02 ; L03 are considered known. The force balance equation can be expressed in the form k1 …L1

L01 †~ r1 ‡ k2 …L2

L02 †~ r2 ‡ k3 …L3

L03 †~ r3 ‡ ~ F ˆ 0:

…6†

3. Derivation of a univariate equation 3.1. The derivation of three fundamental equations Obviously Eqs. (4)±(6) are linear equations about variables ~ r1 ;~ r2 and ~ r3 . So we can obtain ~ r1 ;~ r2 and ~ r3 by solving Eqs. (4)±(6). Substituting ~ r1 ;~ r2 and ~ r3 into Eqs. (1)±(3), three equations can be obtained: P2

j;k;mˆ0;j‡k‡m P 2 n2

Eijkm Lj1 Lk2 Lm3

ˆ0

…i ˆ 1; 2; 3†;

…7†

where Eijkm …i ˆ 1; 2; 3† are rational functions of the known force and the structural parameters of the spring system and n ˆ k3 L03 L1 L2 ‡ k2 L02 L1 L3 ‡ k1 L01 L2 L3

k1 L1 L2 L3

k2 L1 L2 L3

k3 L1 L2 L3 :

…8†

Here n must not equal zero. Then multiplying (7) by n2 yield, fi …L1 ; L2 ; L3 † ˆ

2 X j;k;mˆ0;j‡k‡m P 2

Eijkm Lj1 Lk2 Lm3 ˆ 0

…i ˆ 1; 2; 3†:

…9†

In the set (9), the highest degree of L1 ; L2 , and L3 is 2. In fact (9) are the three fundamental equations of the system, from which the univariate equation in L3 without extraneous roots can be derived as follows.

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L. Fan et al. / Mechanism and Machine Theory 36 (2001) 623±632

3.2. The derivation of the basic 6 by 6 matrix Following Dixon's resultant principle [5], we construct the following matrix: f1 …L3 ; L1 ; L2 † f2 …L3 ; L1 ; L2 † f3 …L3 ; L1 ; L2 † D…L3 ; L1 ; L2 ; u; v† ˆ f1 …L3 ; u; L2 † f2 …L3 ; u; L2 † f3 …L3 ; u; L2 † : f1 …L3 ; u; v† f2 …L3 ; u; v† f2 …L3 ; u; v†

…10†

Expanding the above determinant, D is a polynomial of degree 4 in L2 , 2 in L1 , 4 in u, 2 in v, and 6 in L3 . D vanishes when u or v substitute for L1 or L2 , implying that …L1 u†…L2 v† is a factor of D. Therefore the expression d…L3 ; L1 ; L2 ; u; v† ˆ

D…L3 ; L1 ; L2 ; u; v† …L1 u†…L2 v†

…11†

is a polynomial of degree 3 in L2 , 1 in L1 , 3 in u, and 1 in v. d vanishes when fi have common zeros no matter what u and v are. The coecient of each power product ui vj …i ˆ 0; 1; 2; 3; j ˆ 0; 1† of d have common zeroes which are also the common zeroes of equations fi . This gives eight equations gi …L3 ; L1 ; L2 † in power product of L1 and L2 , whereas the number of the power product Li1 Lj2 …i ˆ 0; 1; j ˆ 0; 1; 2; 3† is also eight. Therefore, the coecients of each power product Li1 Lj2 in these eight equations form an 8 by 8 matrix D. All the above algorithms can be expressed as d…L3 ; L1 ; L2 ; u; v† ˆ ‰1; u; u2 ; u3 ; v; uv; u2 v; u3 vŠ
D
‰1; L2 ; L22 ; L32 ; L1 ; L1 L2 ; L1 L22 ; L1 L32 ŠT ;

…12†

where D is a matrix whose elements are polynomials in L3 . It can be easily checked that the rank of D is only 6. So the determinant of matrix D is always equal to zero. Thus we cannot obtain the univariate equation in L3 simply from the determinant of matrix D. So we must use a new method to treat with the matrix D. Let the coecients of power products u0 v0 ; u1 v0 ; u2 v0 ; u3 v0 ; u0 v1 and u1 v1 of d form six equations, which can be checked to be independent. The six equations can be rewritten as: H
‰L2 ; L1 L22 ; L1 L2 ; L32 ; L1 ; 1; L22 ; L1 L32 ŠT ˆ 0; where H is a 6 by 8 2 h11 h12 6 h21 h22 6 6 h31 h32 Hˆ6 6 h41 h42 6 4 h51 h52 h61 h62

…13†

matrix. In fact H is formed by the ®rst six rows of the matrix D: 3 h13 h14 h15 h16 h17 h18 h23 h24 h25 h26 h27 h28 7 7 h33 h34 h35 h36 h37 h38 7 7: h43 h44 h45 h46 h47 h48 7 7 h53 h54 h55 h56 h57 h58 5 h63 h64 h65 h66 h67 h68

We can rearrange Eq. (13) as follows: M
‰X Š ˆ 0;

…14†

L. Fan et al. / Mechanism and Machine Theory 36 (2001) 623±632

where

2

h11 6 h21 6 6 h31 Mˆ6 6 h41 6 4 h51 h61

h12 h22 h32 h42 h52 h62

h13 h23 h33 h43 h53 h63

h15 h25 h35 h45 h55 h65

h16 h26 h36 h46 h56 h66

3 h17 h27 7 7 h37 7 7; h47 7 7 h57 5 h67

627

3 L2 ‡ k1 L32 ‡ g1 L1 L32 6 L1 L2 ‡ k2 L3 ‡ g2 L1 L3 7 2 2 27 6 6 L1 L2 ‡ k3 L3 ‡ g3 L1 L3 7 2 27 ‰X Š ˆ 6 6 L1 ‡ k4 L3 ‡ g4 L1 L3 7: 2 2 7 6 4 1 ‡ k5 L3 ‡ g L1 L3 5 5 2 2 L22 ‡ k6 L32 ‡ g6 L1 L32 2

Each element of ‰X Š is considered as linear and homogeneous in the unknowns. ki and gi are constants determined by the equilibrium positions of the three-spring system, but we must state here that before we have worked out the values of L3 we still do not know what values they are. But this is not important and does not a€ect our following computation procedure, because it is already enough for us to just know that ki and gi always exist and can be found. Using ki and gi , Eq. (13) can always be rearranged as Eq. (14). This point is very important and must be clearly stated here. For the equation set (14) to have non-trivial solutions, the determinant of the coef®cient matrix M must vanish, i.e. det‰MŠ ˆ 0:

…15†

The left-hand side of (15) is the eliminant, which is an equation in L3 and known parameters. Because the highest degree of elements of matrix M is 6, the highest degree of Eq. (15) is 36. At the same time, elements of matrix M can be factored as follows: 3 2 3 L3 m11 L23 m12 L23 m13 L33 m15 L43 m16 L23 m17 6 L2 m21 L3 m22 L3 m23 L2 m25 L3 m26 L3 m27 7 3 3 7 6 3 6 L3 m31 m32 m33 L3 m35 L23 m36 m37 7 7: 6 Mˆ6 m42 m43 L3 m45 L23 m46 m47 7 7 6 L3 m41 4 L2 m51 L3 m52 L3 m53 L2 m55 L3 m56 L3 m57 5 3 3 3 L3 m61 m62 m63 L3 m65 L23 m66 m67 So L23 ; L3 ; L3 ; L3 ; L3 ; L23 can be factored out from the ®rst, second, and ®fth rows, and ®rst, fourth, and ®fth columns and L83 can be cancelled from the determinant of matrix M. Hence the degree of the univariate equation in L3 decreases to 28. In fact the univariate equation is determined by the following matrix: 3 2 m11 m12 m13 m15 m16 m17 6 m21 m22 m23 m25 m26 m27 7 7 6 6 m31 m32 m33 m35 m36 m37 7 7 Nˆ6 6 m41 m42 m43 m45 m46 m47 7 7 6 4 m51 m52 m53 m55 m56 m57 5 m61 m62 m63 m65 m66 m67 and the 28th order polynomial equation in L3 is obtained and can be rewritten as L63

22 X iˆ0

Ci Li3 ˆ 0;

…16†

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L. Fan et al. / Mechanism and Machine Theory 36 (2001) 623±632

where Ci are determined only by known parameters. Although from Eq. (16) the 22nd order polynomial equation can be obtained, Eq. (16) still contains six extraneous roots L63 ˆ 0. So the next step is to eliminate the extraneous roots. 3.3. The derivation of the 22nd degree polynomial In this part four steps are performed and the degree of Eq. (16) decreases from 28 to 22. The ®rst two steps make the degree decrease by 2 each step and the last two steps decrease 1 each step. 3.3.1. The ®rst step Let N0 ˆ ‰n1 n2 n3 n4 n5 n6 ŠT , where ni is the ith row of matrix N0 . k1i …i ˆ 1; 2; 3; 4† and g1i …i ˆ 1; 2; 3; 4† can always be found and using them we can construct a new ®fth and a new sixth row by elementary row operations and make the following equations: n5 ‡

4 X

k1i ni ˆ L3 n51 ;

…17†

g1i ni ˆ L3 n61 ;

…18†

iˆ1

n6 ‡

4 X iˆ1

where k1i …i ˆ 1; 2; 3; 4† and g1i …i ˆ 1; 2; 3; 4† are determined only by known parameters. Then L3 can be cancelled from the new ®fth and sixth rows, and the new matrix N1 ˆ ‰ n 1

n2

n3

n4

n51

n61 ŠT

is obtained. The order of the univariate equation from the determinant of matrix N1 decreases to 26. 3.3.2. The second step Repeat the above procedure of the ®rst step and ®nd k2i …i ˆ 1; 2; 3; 4† and g2i …i ˆ 1; 2; 3; 4†, then make n51 ‡

4 X

k2i ni ˆ L3 n52 ;

…19†

g2i ni ˆ L3 n62 :

…20†

iˆ1

n61 ‡

4 X iˆ1

Then L3 can be cancelled again from the new ®fth and sixth rows, and a new matrix N2 ˆ ‰n1 n2 n3 n4 n52 n62 ŠT is obtained. The order of the univariate equation from the determinant of matrix N2 decreases to 24. 3.3.3. The third step Using the similar procedures of ®rst and second steps, we can also ®nd g3i …i ˆ 1; 2; 3; 4; 5†, and construct a new sixth row and make the following equation:

L. Fan et al. / Mechanism and Machine Theory 36 (2001) 623±632

g35 n52 ‡ n62 ‡

4 X iˆ1

g3i ni ˆ L3 n63 ;

629

…21†

then L3 can be cancelled again from the new sixth row, and a new matrix N3 ˆ ‰ n1

n2

n3

n4

n52

n63 ŠT

is obtained. The order of the univariate equation from the determinant of matrix N3 decreases to 23. 3.3.4. The fourth step Repeat the above procedure again and ®nd g4i …i ˆ 1; 2; 3; 4; 5†, then construct a new sixth row and make the following equation, 4 X g4i ni ˆ L3 n64 ; …22† g45 n52 ‡ n63 ‡ iˆ1

then L3 can be cancelled again from the new sixth row, and a new matrix N4 ˆ ‰ n1

n2

n3

n4

n52

n64 ŠT

is obtained. The order of the univariate equation from the determinant of matrix N4 decreases to 22, and thus we get the ®nal 22nd order polynomial equation without any extraneous roots 22 X iˆ0

Ti Li3 ˆ 0;

…23†

where Ti is determined only by known parameters. 3.4. Determination of the remaining unknowns From Eqs. (4) and (5), we can construct the following two new equations: !

!

!

!

…L1~ r1 †
…L1~ r1 † ˆ …L2~ r2 ‡ BA†
…L2~ r2 ‡ BA†; r1 †
…L1~ r1 † ˆ …L3~ r3 ‡ CA†
…L3~ r3 ‡ CA†: …L1~ r2 and ~ r3 (~ r1 ;~ r2 and ~ r3 have been calculated out Expanding the two equations and substituting ~ r1 ;~ in Section 3.1) into them, the above two equations can be changed into: gi …L1 ; L2 † ˆ 0 …i ˆ 1; 2†; n where n is determined by Eq. (8). Then multiplying (24) by n yields gi …L1 ; L2 † ˆ 0

…i ˆ 1; 2†:

…24† …25†

Subtracting (24) from (25), one new equation is obtained as follows: L1 …C13 L32 ‡ C12 L22 ‡ C11 L2 ‡ C10 † ‡ …C03 L32 ‡ C02 L22 ‡ C01 L2 ‡ C00 † ˆ 0;

…26†

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L. Fan et al. / Mechanism and Machine Theory 36 (2001) 623±632

where Cij …i ˆ 0; 1; j ˆ 0; 1; 2; 3† are rational functions of the known force, the structural parameters of the spring system and L3 . Eq. (26) can be rewritten in the following two di€erent forms: ‰C13 ; C12 ; C11 ; C10 ; C03 ; …C02 L22 ‡ C01 L2 ‡ C00 †Š
‰L1 L32 ; L1 L22 ; L1 L2 ; L1 ; L32 ; 1ŠT ˆ 0; ‰C13 ; C12 ; C11 ; C10 ; …C03 L22 ‡ C02 L2 ‡ C01 †; C00 Š
‰L1 L32 ; L1 L22 ; L1 L2 ; L1 ; L2 ; 1ŠT ˆ 0: The equations in (13) all have the similar form as (26). Taking out arbitrarily ®ve equations from (13) and adding Eq. (26) into them, we can get a new equation set. The new equation set can also be rewritten in two di€erent forms like Eq. (26): M1
‰L1 L32 ; L1 L22 ; L1 L2 ; L1 ; L32 ; 1ŠT ˆ 0; M2
‰L1 L32 ; L1 L22 ; L1 L2 ; L1 ; L2 ; 1ŠT ˆ 0: For the new equation set to have non-trivial solutions, the determinants of the coecient matrix M1 ; M2 must vanish, i.e. det‰M1 Š ˆ 0; det‰M2 Š ˆ 0: Then two new equations in L2 can be obtained, A12 L22 ‡ A11 L2 ‡ A10 ˆ 0;

…27†

A22 L22 ‡ A21 L2 ‡ A20 ˆ 0;

…28†

where Aij …i ˆ 1; 2; j ˆ 0; 1; 2† are rational functions of the known force, the structural parameters of the spring system and L3 . Eliminating L22 from (27) and (28), the value of L2 is obtained, L2 ˆ

A20 A12 A11 A22

A10 A22 : A21 A12

…29†

For the given force and the structural parameters of the system, Eq. (23) can be solved for L3 . Substituting L3 into Eq. (29), the corresponding value of L2 for each L3 can be determined, and in the end substituting L2 ; L3 into Eq. (26), the corresponding value of L1 is determined. 4. Numerical veri®cation The example here comes from [3], and the known parameters of the spatial three-spring system are stated again as following. The structural parameters of the system are the spring free lengths, L01 ˆ L02 ˆ L03 ˆ 2; the spring elasticity constants, k1 ˆ k2 ˆ k3 ˆ 2; the coordinates of the three joint centers p (A, B, C) in the reference system shown in Fig. 1: A (0,0,0), B (X2 ˆ 2; 0; 0), C (X3 ˆ 1; Y3 ˆ 3; 0). The external force applied to the common pivot P is: Fx ˆ Fy ˆ Fz ˆ 1:

L. Fan et al. / Mechanism and Machine Theory 36 (2001) 623±632

631

Using the above values, the 22nd order polynomial was obtained: 91:953 . . . ‡ 932:075 . . . L3

3968:814 . . . L23 ‡ 8341:957 . . . L33

18308:105 . . . L53 ‡ 50913:480 . . . L63 ‡ 77350:113 . . . L93

50712:236 . . . L73

4926:336 . . . L43

7232:704 . . . L83

11 12 85733:132 . . . L10 3 ‡ 26876:955 . . . L3 ‡ 31767:585 . . . L3

14 41739:628 . . . L13 3 ‡ 19515:230 . . . L3 18 ‡ 733:962 . . . L17 3 ‡ 425:876 . . . L3

1178:184 . . . L15 3

2708:538 . . . L16 3

20 349:107 . . . L19 3 ‡ 106:598 . . . L3

16:095 . . . L21 3

‡ 1:L22 3 ˆ 0: The 22 roots of this polynomial equation are L3 ˆ

1:0704 . . . ; 0:9238 . . . ; 0:6462 . . . ; 0:6249 . . . ; 1:7194 . . . ; 1:9016 . . . ; 2:0200 . . . ;  2:4257 . . . ; 1:9772 . . .  1:0279 . . . ; 0:5609 . . .  0:3941 . . . ; 0:6460 . . .  0:4329 . . . ; 0:8980 . . .  0:1520 . . . ; 0:9544 . . .  0:2103 . . . ; 1:4045 . . .  0:1516 . . . ; 2:5354 . . .  0:1099 . . .

And k11 ˆ 0; k12 ˆ

1; k13 ˆ 1; k14 ˆ 0;

g11 ˆ 1; g12 ˆ

1; g13 ˆ 0; g14 ˆ 0;

k21 ˆ

1:06699; k22 ˆ 0; k23 ˆ

g21 ˆ 6:79247; g22 ˆ

6:40192; g23 ˆ 2:77837; g24 ˆ

g31 ˆ 110:734; g32 ˆ 59:2; g33 ˆ g41 ˆ

920466; g42 ˆ

2:55656; k24 ˆ 1:97258; 2:29537;

204:67; g34 ˆ 278:654; g35 ˆ

462080; g43 ˆ

383605; g44 ˆ

1;

2:2329  106 ; g45 ˆ

47662:5:

5. Conclusions This paper presents a new method for the reverse force analysis of the spatial three-spring system. Using the new method a 6 by 6 matrix is obtained and a 22nd degree polynomial in one unknown is derived only by computing the determinant of the 6 by 6 matrix. A numerical example from [3] veri®ed the new method. The procedure presented here proves to be e€ective and simple. Acknowledgements This paper is supported ®nancially by the 973 Plan of China and the Youth Foundation from the Ministry of Information Industry of China.

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