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Inverse Geometrico-Static Analysis of

Under-Constrained Cable-Driven Parallel

Robots with Four Cables

Marco Carricato, Ghasem Abbasnejad and Dominic Walter

Abstract This paper presents the inverse geometrico-static analysis of under-

constrained cable-driven parallel robots with 4 cables. The problem consists in find-

ing all equilibrium configurations of the end-effector when either its orientation or

the center-of-masss position is assigned. In both cases, a further point of the end-

effector is constrained to lie on a given plane. A major challenge is posed by the

intrinsic coupling between kinematics and statics, which must be tackled simulta-

neously. The problems at hand are solved by analytical elimination procedures, thus

leading to univariate polynomials free of spurious factors. All potential solutions

may be real.

Key words: Cable-driven parallel robots, under-constrained robots, kinematic anal-

ysis, static analysis

1 Introduction

Cable-driven parallel robots (CDPRs) employ cables in place of rigid-body exten-sible legs in order to control the end-effector (EE) pose, thus strengthening classic

advantages characterizing closed-chain architectures versus serial ones. A CDPR is

under-constrained if the EE preserves some freedoms once actuators are locked.

Typically, this occurs when the EE is controlled by a number of cables n smaller

than the number of degrees of freedom (dofs) that the EE possesses with respect to

the base [2].

Marco Carricato Ghasem AbbasnejadDepartment of Mechanical Engineering, University of Bologna, Bologna, Italy, e-mail: marco.

[email protected]

Dominic Walter

Institute for Basic Sciences in Engineering, University of Innsbruck, Innsbruck, Austria

J. Lenarcic, M. Husty (eds.), Latest Advances in Robot Kinematics,

DOI 10.1007/978-94-007-4620-6 46,

Springer Science+Business Media Dordrecht 2012

365
http://[email protected]/http://[email protected]/http://dx.doi.org/10.1007/978-94-007-4620-6_46http://dx.doi.org/10.1007/978-94-007-4620-6_46http://dx.doi.org/10.1007/978-94-007-4620-6_46http://[email protected]/http://[email protected]/

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366 M. Carricato et al.

Fig. 1 A cable-driven parallel robot with 4 cables: (a) model for the IGP with assigned orientation,

(b) model for the IGP with assigned position.

A major challenge in the kinematic study of under-constrained CDPRs emerges

from the fact that the EE configuration depends on both the cable lengths and the

applied forces (e.g. gravity). Accordingly, loop-closure and mechanical-equilibrium

equations must be solved simultaneously and displacement-analysis problems be-

come significantly more complex than analogous tasks concerning rigid-link fully-

constrained manipulators [7, 9, 10]. Recently, Carricato and Merlet [24] proposed

a general methodology for the kinematic, static and stability analysis of under-

constrained CDPRs equipped with n 5 cables. By properly formulating the math-ematical model, the method allows one to find the entire set of equilibrium config-

urations when either n EE-pose coordinates (inverse problem) or n cable lengths

(direct problem) are assigned.

In this paper, the inverse geometrico-static problem of CDPRs with 4 cables is

solved. Two instances are considered, depending on whether the EE orientation or

the EEs center-of-mass position is assigned. Since 4 dofs of the EE are to be con-

trolled, an additional constraint must be set on the EE-pose coordinates: when the

EE orientation is assigned, the EEs center-of-mass is constrained to lie on a given

plane (this condition may be useful, for instance, to set the center-of-mass at a given

height); when the EE orientation is assigned, an additional point of the EE is set to

lie on a given plane (this condition may be useful for obstacle avoidance, in order

to guide the EE tilt). In both cases, the overall robot configuration, the cable lengthsand the cable tensions are to be computed.

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Inverse Geometrico-Static Analysis of Under-Constrained CDPRs with 4 Cables 367

2 Geometrico-Static Model

The EE is connected to the fixed base by 4 cables (Fig. 1). The ith cable exits

from the base at point Ai and it is connected to the EE at point Bi . The geometric

parameters of both the EE and the base are arbitrary. A is a fixed Cartesian frame

attached at A1, whereas B is a Cartesian frame attached to the EE at the platforms

center of mass G. The platform pose is described by XT = [gT;T], where gT =[x , y , z]T is the position of G in A and T = [e1, e2, e3]T is the array groupingthe Rodrigues variables parameterizing the EE orientation with respect to A. The

EE is acted upon by a constant force of magnitude Q, applied at G. This force is

described as a 0-pitch wrench QLe, where Le is the normalized Plucker vector of

the line of action. The normalized Plucker vector of the line associated with the ith

cable is Li /i , where i is the cable length, pi is any vector from the reduction

pole of moments O to the cable line and LTi = [(Ai Bi )T; {pi (Ai Bi )}T].Accordingly, the wrench exerted by the ith cable on the EE is (i /i )Li , with ibeing a positive scalar representing the intensity of the cable tensile force. Without

loss of generality, one may assume O A1.When all cables of the robot are in tension, the set of geometrical constraints

imposed on the EE is

||Ai Bi ||2 = 2i , i = 1 . . . 4, (1)

with the overall pose being determined by the EE static equilibrium, i.e.

4i=1

i

iLi + QLe =

L1 L2 L3 L4 Le

M(O)

(1/1)

(2/2)

(3/3)

(4/4)

Q

= 0, (2)

with i 0, i = 1 . . . 4.Equations (1)(2) amount to 10 scalar relations in 14 variables, namely g, , i

and i , i = 1 . . . 4.Generally, a finite set of system configurations may be determined if 4 additional

constraints are assigned on the variables. When these constraints concern the EE-

pose coordinates, an inverse geometrico-static problem (IGP) must be solved. Two

relevant cases may be considered, depending on whether: (i) the orientation is as-

signed and G is constrained to lie on a given plane (IGP with assigned orientation);

or (ii) the position ofG is known and a further point B5 of the EE is required to lie

on a given plane (IGP with assigned position). In both cases, the pose coordinates

are subject to 4 linear constraints, i.e. qi (X) = 0, i = 1 . . . 4.By following the method presented in [2], cable tensions may be eliminated from

the set of unknowns by observing that Eq. (2) holds only if

rank[M(O)] 4, (3)

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which is a purely geometrical condition, since M(O) is a 6 5 matrix only de-pending on X. By setting all 5 5 minors of M(O) equal to zero, 6 polynomialrelations that do not contain cable tensions may be obtained, 5 of which are linearly

independent, namely pj(X)

=0, j

=1 . . . 5. The 0-dimensional variety V of the

ideal generated by {q1, . . . , q4, p1, . . . , p5} provides the equilibrium configurationsof the EE that satisfy the imposed constraints.

Once X is known, cable lengths may be computed by Eq. (1) and cable tensions

may be obtained by 4 linearly-independent relations chosen within Eq. (2). Stability

may be assessed as in [2]. Once a configuration is found, it proves feasible only if it

is stable and therein cable tensions are positive.

3 IGP with Assigned Orientation

In this case, is known and G is constrained to lie on a plane (Fig. 1(a)). The

constraints qi (X) = 0, i = 1 . . . 4, are

e1 = e1, e2 = e2, e3 = e3, g n d = xn1 + yn2 + zn3 d = 0, (4)

where e1, e2 and e3 are known scalars, n = [n1, n2, n3]T is a unit vector perpen-dicular to and |d| is the distance of from A1. Two subcases may be identified,depending on whether n3

=0 or n3

=0.

3.1 n3 = 0

When n3 = 0, is a nonvertical plane and z may be expressed, from the last rela-tionship in Eq. (4), as z = (n1/n3)x (n2/n3)y + d/n3. By taking advantage ofthis expression and by imposing the first three constraints in Eq. (4), the 5 relations

pj(X)

=0, j

=1 . . . 5, emerging from Eq. (3) become cubic relations in x and y

comprising 10 monomials, i.e. [y3, y2x , y x2, x3, y2, y x , x2, y , x , 1].The problem may be efficiently solved by implementing a Sylvester dialytic

method, namely by rewriting the relations pj = 0 as linear equations in all mono-mials involving the original unknowns except one, which is hidden in the equation

coefficients. If these monomials are treated as linear unknowns, a square homo-

geneous system is obtained and the determinant of the coefficient matrix provides

a resultant in the hidden variable. In the case at hand, by hiding y, 4 monomi-

als in x emerge and, thus, four relations pj = 0, j = 1 . . . 4, may be used tobuild up a square Sylvester matrix. However, the corresponding resultant exhibits

a spurious solution. In order to get rid of the extraneous factor, all five relationspj = 0, j = 1 . . . 5, may be linearized in the 5 monomials contained in the array1 = [y3, x3, x2, x, 1]T, namely

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Table 1 A 4-4 CDPR whose IGP with assigned orientation admits 5 real potential solutions (d =3/

6).

[A2]A [A3]A [A4]A [B1 G]A [B2 G]A [B3 G]A [B4 G]A

6n

8 9 1 3 3 2 3 20 7 8 4 2 1 1 15 6 4 3 1 0 2 1

S1(y)1 = 0, (5)

where S1(y) is a 5 5 matrix whose entries are known polynomial functions of y.Letting the determinant of S1(y) vanish yields a 5th-degree univariate equation in

y. This was obtained in symbolic form and it is devoid of spurious roots. For eachroot, a unique value for x may be obtained by solving the linear system (5).

Solutions may be either complexor real, with only the latter ones having phys-

ical interest. By varying the robots geometry, the count of real roots may change.

Table 1 reports an example for which the IGP with assigned orientation admits 5

real solutions (not all of them necessarily feasible). Examples of this kind may be

obtained by using the algorithms developed in [1], namely a continuation procedure

adapted from a routine originally proposed by Dietmaier [6] and two evolutionary

techniques based on a genetic algorithm and particle swarm optimization.

3.2 n3 = 0

When n3 = 0 (without loss of generality, n2 = 0), is vertical and y may beexpressed, from the last relation in Eq. (4), as y = (n1/n2)x + (d/n2).

By substituting this expression in the relations pj(X) = 0, j = 1 . . . 5, oneobtains 5 cubics in the monomials [z2x,zx2, x3, z2, z x , x2, z , x , 1]. By a proceduresimilar to that described in Sec. 3.1, a least-degree univariate equation free from

extraneous polynomial factors may be obtained by linearizing all five relations pj =0, j = 1 . . . 5, in the 5 monomials contained in the array 2 = [z2, x3, x2, x, 1]T,and by writing them in the form

S2(z)2 = 0, (6)

where S2(z) is a 5 5 matrix whose entries are known polynomial functions of z.Letting the determinant ofS2(z) vanish yields a 4th-degree univariate polynomial in

z, which is available in symbolic form and devoid of spurious roots. For each root,

a unique value for x may be obtained by solving the linear system (6). Even in thiscase, all roots may possibly be real.

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4 IGP with Assigned Position

In this case, the position of G is known and a further point B5 of the EE is con-

strained to lie on an assigned plane (Fig. 1(b)). The constraints qi (X)

=0,

i = 1 . . . 4, become

x = x, y = y, z = z, r5 n d = 0, (7)

where x, y and z are known scalars, r5 is the position vector of B5 in A, n is a unit

vector perpendicular to and |d| is the distance of from A1.If R() is the rotation matrix between B and A, the position vector of Bi , i =

1 . . . 5, in A may be expressed as ri = [Bi]A = g + R()[Bi]B, where the positionofBi in B is known. By substituting these expressions in the 5 relations pj(X) = 0,j = 1 . . . 5, and by imposing the first three constraints in Eq. (7), one obtains 5sextics in e1, e2 and e3. By a similar expansion, the fourth relationship in Eq. (7),

i.e. q4 = 0, becomes a quadratic equation in the Rodrigues parameters.By denoting the ideal generated by the set J = {p1, . . . , p5, q4} as J, the solu-

tions of the IGP with assigned position form the variety V of J. The high orderof the polynomials in J suggests applying elimination procedures based on Groeb-

ner bases in order to solve the problem. Even though the lexicographic monomial

order is, in general, particularly suitable to solve polynomial systems, for it pro-

vides equation sets whose variables may be eliminated successively, it is highly

inefficient in terms of computation time and memory requirements. A Groebner ba-sis G[J] of J with respect to a graded reverse lexicographic order, instead, i.e.grevlex(e1, e2, e3), may be computed in a very expedited way (tenths of seconds,

for the case at hand, on a PC with a 2.67GHz Intel Xeon processor and 4GB of

RAM). Once G[J] is known, the FGLM algorithm [8], converting a Groebner basisfrom one monomial order to another, may be called upon to compute a univariate

polynomial in J. However, a more efficient method is provided by the Groebner-Sylvester hybrid approach proposed in [5]. The method is based on the observation

that G[J] comprises 12 polynomials and these contain 12 monomials in e1 and e2,i.e. 3

= [e1e

4

2

, e5

2

, e1e3

2

, e4

2

, e1e2

2

, e3

2

, e2

1

, e1e2, e2

2

, e1, e2, 1]

T. Accordingly, G[J

]may be set up as a square system of homogeneous linear equations in the form

S3(e3)3 = 0, (8)

where S3(e3) is a 1212 matrix polynomial in e3. Letting the determinant ofS3(e3)vanish yields a spurious-root-free univariate polynomial of degree 32 in e3. For each

root, unique values for e1 and e2 may be obtained by solving the linear system (8).

An alternative procedure is based on the properties of the normal set N[J],which is the array grouping all monomials which are not multiples of any lead-

ing monomial in G[J] [11]. N[J] contains 32 monomials in , i.e. N[J] =[1, . . . , 32]T. If rh is the remainder on division of e3h by G[J], rh is a linearcombination of the monomials of N[J], i.e. rh =

32k=1 ahkk , with ahk being a

constant coefficient. Since rh e3h belongs to J, it must vanish on the variety

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Table 2 A 4-4 CDPR whose IGP with assigned position admits 32 real potential solutions (d =0.0543588).

[A2]A [A3]A [A4]A [B1]B [B2]B [B3]B [B4]B [B5]B n (x, y, z)

0.0096715 0.1602038 0.3227272 0.8585338 0.3187879 0.6598471 0.6273182 0.4579794 0.8894538 0.1339193

0 0.0649423 0.7151215 0 0.4888859 0.6661870 0.9610494 0.5558744 0.4392272 0.2021438

0.5484151 0.6597958 0.5378416 0 0 0.8744523 0.0797477 0.8222827 .1262981 0.1180386

V, for any h. Thus, one may assemble all equations of this kind in the form

(A[J, e3] e3I32) N[J] = 0, (9)

where A[J, e3] = [ahk] is a 32 32 numeric matrix (called multiplication matrixfor e3) and I32 is the 32 32 identity matrix. Equation (9) is a linear eigenvalueproblem, whose 32nd-degree characteristic polynomial is the desired resultant in

e3. Equation (9) provides an efficient way to numerically compute all solutions of

the problem at hand as the eigenvalues of A[J, e3].By taking advantage of the algorithms developed in [1], several sets of robot

parameters were found proving that all 32 solutions may be real. Table 2 reports an

example.

5 Conclusions

This study solved the inverse geometrico-static problem (IGP) of under-constrained

cable-driven parallel robots with 4 cables. The problem consists in finding all equi-

librium configurations of the robot when a subset of the end-effector pose coordi-

nates is assigned. Two relevant cases were considered.

In the former, the orientation of the end-effector is assigned and the center of

mass is constrained to lie on a plane (IGP with assigned orientation). The prob-

lem was solved by an elimination procedure based on a Sylvester dialytic method.

A least-degree univariate polynomial was obtained in symbolic form. This has de-

gree 4 or 5, depending on the orientation of the plane constraining the center of mass

being vertical or not, and all roots may be real.

In the latter case, the position of the center of mass is assigned and a further point

of the end-effector is required to lie on a known plane (IGP with assigned position).

The problem was solved by exact-arithmetic procedures based on Groebner-basis

computation. In this case, at the most 32 solutions (and a corresponding univariate

polynomial) were obtained. By the algorithms developed in [1], a numerical exam-

ple was found that proves that all potential solutions may be real.

It is worth observing that all solution counts reported above refer to potential

solutions of the problems at hand, since they do not take into account the con-

straints imposed by the sign of cable tensions and the stability of equilibrium. Once

such constraints are imposed and solutions are sifted, the number of feasible con-

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figurations reduces. When multiple feasible configurations exist, the EE may switch

across them, due to inertia forces or external disturbances. Accordingly, the compu-

tation of the entire set of equilibrium configurations is essential for robust trajectory

planning. This motivates the relevance of the presented algorithms, even when they

are not applicable to real-time computation (as for the IGP with assigned position).For real-time applications, these authors are exploring interval-analysis-based ap-

proaches, in collaboration with Dr. Merlets team at INRIA.

References

1. Abbasnejad, G., Carricato, M.: Real solutions of the direct geometrico-static analysis of under-

constrained cable-driven parallel robots with three cables. In: Proc. of AIMETA 2011, the 20th

Congress of the Italian Association of Theoretical and Applied Mechanics, pp. 115. Bologna,

Italy, (2011)

2. Carricato, M., Merlet, J.P.: Geometrico-static analysis of under-constrained cable-driven par-

allel robots. In: Lenarcic J., Stanisic M.M. (eds.) Advances in Robot Kinematics: Motion in

Man and Machine, pp. 309319. Springer, Dordrecht (2010)

3. Carricato, M., Merlet, J.P.: Direct geometrico-static problem of under-constrained cable-

driven parallel robots with three cables. In: Proc. of the 2011 IEEE Int. Conf. on Robotics

and Automation, pp. 30113017. Shanghai, China, (2011)

4. Carricato, M., Merlet, J.P.: Inverse geometrico-static problem of under-constrained cable-

driven parallel robots with three cables. In: Proc. of the 13th World Congress in Mechanism

and Machine Science, pp. 110, Paper No. A7 283. Guanajuato, Mexico (2011)

5. Dhingra, A.K., Almadi, A.N., Kohli, D.: A GrobnerSylvester hybrid method for closed-form

displacement analysis of mechanisms. ASME J. Mech. Des. 122(4), 431438 (2000)

6. Dietmaier, P.: The StewartGough platform of general geometry can have 40 real postures.

In: Lenarcic J., Husty M.L. (eds.) Advances in Robot Kinematics: Analysis and Control, pp.

716. Kluwer Academic Publishers, Dordrecht (1998)

7. Fattah, A., Agrawal, S.K.: On the design of cable-suspended planar parallel robots. ASME J.

Mech. Des. 127(5), 10211028 (2006)

8. Faugere, J.C., Gianni, P., Lazard, D., Mora, T.: Efficient computation of zero-dimensional

Grobner bases by change of ordering. J. Symb. Comput. 16(4), 329344 (1993)

9. Jiang, Q., Kumar, V.: The direct kinematics of objects suspended from cables. In: Proc. of the

ASME 2010 Int. Design Engineering Technical Conferences, pp. 110, Paper no. DETC2010

28,036. Montreal, Canada, (2010)

10. McCarthy, J.M.: 21st century kinematics: Synthesis, compliance, and tensegrity. ASME J.

Mech. Robot. 3(2), 020201 (2011)

11. Stetter, H.J.: Matrix eigenproblems are at the heart of polynomial system solving. ACM

SIGSAM Bull. 30(4), 2225 (1996)

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