Geometrico-static Analysis of Under-constrained Cable-driven Parallel Robots

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1 Geoetrico-static Analysis of Under-constrained Cable-driven Parallel Robots M. Carricato and J.-P. Merlet 1 DIEM - Dept. of Mechanical Engineering, University of Bologna, Italy, e-ail: arco.carricato@ail.ing.unibo.it 2 INRIA Sophia Antipolis - Méditerranée, Sophia Antipolis, France, e-ail: jean-pierre.erlet@sophia.inria.fr Abstract. This paper studies the kineatics and statics of cable-driven parallel robots with less than six cables, in crane configuration. A geoetrico-static odel is provided, together with a general procedure aied at efficiently solving, in analytical for, the inverse and direct position probles. The stability of equilibriu is assessed within the fraework of a constrained optiization proble, for which a purely algebraic forulation is provided. A spatial robot with three cables is studied as an application exaple. Key words: Under-constrained cable-driven parallel robots, geoetric analysis, static analysis. 1 Introduction Cable-driven parallel robots (CDPRs) are ordinarily referred to as fully-constrained or under-constrained, depending on whether all six degrees of freedo of the oving platfor are controlled or not [6]. It is well known that, in the general case, fully-constrained CDPRs require at least seven cables, andt only six in crane configuration, naely when gravity acts as an additional cable. The distinction between the two aforeentioned categories of robots is soewhat fictitious, since a theoretically fully-constrained CDPR is, in considerable parts of its workspace, actually under-constrained, naely when a full restraint would require a negative tension in soe cable. Peranently under-constrained CDPRs with less than six cables are furtherore used in a nuber of applications, such as rescue operations, in which a liitation of dexterity is acceptable in order to decrease coplexity, set-up tie, likelihood of cable interference, etc. The above considerations otivate a careful study of under-constrained robots. However, while fully-constrained CDPRs have been extensively investigated [2], very few studies have been dedicated to under-constrained ones [4, 10]. A ajor challenge in the study of this kind of robots consists in the intrinsic coupling between kineatics and statics (or dynaics). Indeed, while in a fully-constrained CDPR the platfor posture is copletely deterined by assigning the cable lengths (provided that all cable tensions are positive), in an under-constrained robot the 1

2 2 M. Carricato and J.-P. Merlet pose depends on both cable lengths and equilibriu equations. As a consequence, kineatics and statics ust be dealt with siultaneously. Furtherore, since the platfor posture depends on the applied load, it ay change by the presence of external disturbances. It is therefore of great iportance to investigate equilibriu stability [1]. This paper presents a preliinary geoetrico-static study of under-constrained nn-cdprs, in crane configuration. An under-constrained nn-cdpr is controlled by n cables and it exhibits n distinct anchor points on both the base and platfor, with n < 6. A geoetrico-static odel is first presented, together with a general procedure aied at efficiently solving, in analytical for, the inverse and direct position probles. These consist in deterining the overall robot configuration (and cable tensions) once n configuration variables (e.g., platfor posture coordinates or cable lengths) are given. The proble of equilibriu stability is forulated as a constrained optiization proble, and a purely algebraic ethod, which rules out the need of differentiation, is provided. The geoetrico-static study of a general 33-CDPR is outlined as an application exaple. 2 Geoetrico-static odel Let a obile platfor be connected to a fixed base by n cables, with 2 n 5. A i and B i are, respectively, the anchor points of the ith cable on the frae and the platfor, and s i = B i A i. The set C of theoretical geoetrical constraints iposed on the platfor is deterined by the relations s i = s i s i =,...n, (1) where is the length of the ith-cable, which is assued, for apparent practical reasons, strictly positive (as a conseqeunce, B i A i ). Since only n geoetrical constraints are enforced, the platfor preserves 6 n degrees of freedo (dofs), with its posture being deterined by equilibriu laws. If Q$ e, Q>0, is an arbitrary external wrench acting on the platfor (including inertia forces, in case of dynaic conditions) and (τ i / )$ i is the force exerted by the ith cable ($ e and $ i / are assued to be unit screws), then with n τ i $ i + Q$ e = 0, (2) τ i 0,...n. (3) Equations (1)-(2) aount to 6+n scalar relations involving 6+2n variables, naely, the cable tensions and lengths, and the variables used to paraeterize the platfor posture. In general, a finite set of syste configurations ay be deterined if n of such variables are assigned.

3 Geoetrico-static Analysis of Under-constrained Cable-driven Parallel Robots 3 In this paper, only static equilibriu is considered and Q$ e is assued to be a constant force applied on a point G of the platfor (e.g., the platfor weight acting through its center of ass). Hence, Eqs. (1)-(2) are algebraic, or ay be easily rendered so. If Eq. (2) is written as (τ 1 /ρ 1 ) [ ] $1 $ n $ e } {{ } M. (τ n /ρ n ) Q = 0, (4) M is a 6 (n+1) atrix only depending on the platfor posture and equilibriu is possible only if rank(m) n. (5) Within the doain of rigid-body echanics, the proble is statically deterinate if the equality holds, indeterinate otherwise. In the forer case, it is always possible to replace Eq. (4) with 6 n scalar relations that do not contain the unknowns τ i,...n. A ost straightforward strategy consists in coputing cable tensions by way of n linearly independent relations chosen within Eq. (4), then substituting the back into the reaining ones. The resulting equations, however, exhibit a rearkable coplexity. A ore convenient strategy consists in setting all (n+1) (n+1) inors of M equal to zero, which aounts to ( 6 n+1) scalar relations, aong which 6 n linearly independent ones ay be suitably chosen. By such an approach, the resulting equations are significantly sipler. Furtherore, since they do not coprise cable lengths, they lead to a partial decoupling of the syste equations, with cable lengths only appearing in Eq. (1). Such an approach ay also be applied when the proble is statically indeterinate. Depending on the variables designated as input, one ay tackle an inverse geoetric proble (IGP), if n variables concerning the platfor posture are assigned, or a direct one (DGP), if cable lengths are given. The IGP takes particular advantage of the partial decoupling of syste equations, since the platfor configuration ay be coputed by siply solving the 6 n relations eerging fro Eq. (5). Cable lengths and tensions ay be subsequently (and straightforwardly) coputed by Eq. (1) and a suitable set of linear independent relations chosen within Eq. (4). The set of adissible solutions consists of all those for which cable tensions are non-negative (cf. Eq. (3)) and the platfor equilibriu is stable (cf. 3). The DGP is rearkably ore coplex, since in this case the platfor configuration ust be deterined by siultaneously solving both the 6 n relations eerging fro Eq. (5) and the n relations in Eq. (1). It ust be said that Eq. (1) represents a set of theoretical constraints, since the actual constraint iposed by a generic cable consists in that s i = s i s i. (6)

4 4 M. Carricato and J.-P. Merlet The above refineent causes no concern when the IGP is dealt with, since in this case the theoretical values of cable lengths are conveniently coputed by Eq. (1), after the platfor posture has been established. Conversely, when the DGP is tackled, cable lengths are assigned as inputs, and, a priori, nothing assures that all cables are called upon to sustain the load. Indeed, if a subset W of cable indexes exists such that card(w )<n and $ e span($ j, j W ), equilibriu configurations possibly exist such that s k < ρ k, for all k W, and thus τ k = 0. These are legitiate solutions of the proble at hand. It follows that the overall solution set is obtained by solving the DGP for all possible constraint sets { s j =ρ j, j W }, with W {1...n} and card(w ) n, and by retaining, for each corresponding solution set, the solutions for which s k <ρ k, for all k W. In ( n n h) DGPs. general, this aounts to solving h=0 n 1 A caveat is worth to be entioned. Equation (5) is only a necessary condition for equilibriu. In very special conditions, it ay happen that equilibriu is not possible, in spite of Eq. (5) being fulfilled and irrespective of the sign of cable tensions. In particular, this occurs if M loses its full rank because a subset of its n first coluns becoes linearly dependent, i.e. if the rank loss is concentrated aong the set of screws associated with the cable lines. In this case, the rank of the block 1 M 1...6,1...n is at ost equal to n 1 and Eq. (2) ay be satisfied only if rank(m) n 1. Cases like the one described here, however, are sufficiently unlikely to occur not to be, in practice, of particular concern. Nonetheless, a check of the rank of M 1...6,1...n is advisable before attepting to solve for cable tensions. Throughout the text, the following notation is adopted. Oxyz is a Cartesian coordinate frae fixed to the base, with i, j and k being unit vectors along the coordinate axes. Gx y z is a Cartesian frae attached to the platfor. e is a unit vector directed as $ e, x = G O, a i = A i O, r i = B i G, s i = B i A i = x+r i a i, and u i =(A i B i )/ = s i /, with...n. Without loss of generality, O is chosen to coincide with A 1 (so that a 1 = 0) and k = e. Furtherore, if b i is the projection of B i G on Gx y z, Φ is the array grouping the variables paraeterizing the platfor orientation with respect to the fixed frae, and R(Φ) is the corresponding rotation atrix, then r i = R(Φ)b i. Finally, the platfor posture is described by the array X=(x;Φ), with the coponents of x in Oxyz being denoted, for the sake of brevity, as x, y and z. If O is chosen as the reduction pole of oents, $ i and $ e ay be respectively expressed, in axis coordinates, as $ i =[s i ; a i s i ] and $ e =[e; x e]. Accordingly, M becoes M= [ ] x+r1 x+r i a i e, (7) 0 a i (x+r i ) x e or, equivalently, after subtracting the first colun fro the coluns 2 through n, [ ] M x+r1 r = i1 a i e, (8) 0 a i (x+r i ) x e 1 The notation M hi j,kl denotes, throughout the text, the block atrix obtained fro rows h, i and j, and coluns k, l and of M. When all coluns of M are used, the corresponding subscripts are oitted.

5 Geoetrico-static Analysis of Under-constrained Cable-driven Parallel Robots 5 7 z A 1 Ḡ 1 B 1 B 2 Ḡ 4 Ḡ 2 Ḡ 3 A x Fig. 1 Stationary configurations of a 22-CDPR with a 2 =(10,0, 2), b 1 =( 0.5,0, 0.5), b 2 = (3,0,0), and ρ 1 = ρ 2 = 7. where r i j = r i r j, i j. 3 The stability of equilibriu Let an equilibriu configuration ( X, ρ 1... ρ ) be considered, with being the nuber of active constraints (i.e. the nuber of cables contributing to supporting the platfor). By a convenient reordering of indexes, taut cables ay be assued to be the first, with n. Since the platfor preserves 6 dofs, it ay displace under the effect of a change in the external force acting on it, while cable lengths reain unvaried (for the sake of siplicity, it is assued that the nuber of cables in tension does not change because of the perturbation, which is reasonable, but not necessarily true). The proble of assessing equilibriu stability is in order. In particular, G ay generally ove within a closed region of R 3 (in soe cases, a surface or a curve). If g is the frontier of this region, the equilibriu is stable any tie the potential energy associated with the external force Q$ e, naely Qe x, is at a iniu on g. Loosely speaking, the platfor is at rest in all points Ḡ of g in which the variety tangent to g is perpendicular to e, with the equilibriu being stable if and only if a neighborhood WḠ of Ḡ exists such that (P Ḡ) e<0, for all P (g WḠ). In such a condition, when the platfor displaces under the effect of a perturbation, the original configuration is restored if the perturbation ceases. Figure 1 portrays the locus g of the positions that G ay assue for an exeplifying 22-CDPR, under the constraints (1), with =n=2. If the platfor is thought of as the coupler of a four-bar linkage whose grounded links are the cables (with as-

6 6 M. Carricato and J.-P. Merlet signed lengths), g is the coupler curve of G, naely a bicursal sextic. The stationary configurations of G are the points of g in which the tangent line is perpendicular to e, and they coincide with the real solutions of the DGP of the robot (which ay be proven to adit up to twelve coplex solutions). The potential energy is at a iniu in Ḡ 2 and Ḡ 4. Since cable tensions are negative in the configurations lying above the base, and positive otherwise, Ḡ 4 is the only feasible stable configuration for the exaple at hand. Finding the inia of a constrained function is a classical issue in optiization theory. An efficient algorithic foralization is presented hereafter. At equilibriu, the variation of the potential energy of the platfor due to a virtual displaceent of it ust be zero. Such a variation is the opposite of the work carried out by all forces acting on the platfor, naely δl= τ i u i δb i Qe δg=0. (9) If δx and δθ are, respectively, the virtual displaceent of G and the virtual rotation of the platfor, then δg=δx and δb i = δs i = δx+δθ r i, and thus ( ) ( ) δl= τ i u i + Qe δx i r i u i δθ = f δx+ δθ = 0. (10) τ Equation (10), fro which f and are inferred to be zero, is clearly equivalent to Eq. (2), by letting n=. Since, for =, δ( s i )=δ s i and δ s i = s i δs i δl ay be written as = s i δx+r i s i δθ = (u i δx+r i u i δθ), (11) δl= Qe δx+ i.e., as the virtual variation of the Lagrange function L= Qe x+ τ i δ( s i ), (12) τ i ( s i ). (13) The Lagrange ultipliers in L coincide with the cable tensions, naely, with the forces necessary to ipose the geoetrical constraints [7]. 2 The above observation is useful, since it allows the stability characteristics of the equilibriu to be assessed by evaluating the definiteness of the reduced Hessian H r of L, naely, the Hessian of 2 Notice that Eq. (9) plus the relations {τ i > 0, s i = } for... and {τ i = 0, s i < } for i =...n are equivalent to the Karush-Kuhn-Tucker conditions for the iniization of L under the constraints (6), provided that $ 1,...,$ are linearly independent.

7 Geoetrico-static Analysis of Under-constrained Cable-driven Parallel Robots 7 L taken with respect to the configuration variables, further projected on the tangent space of the constraints C [7]. An algebraic expression of H r is derived hereafter. The second-order variation of δl is given by with δ 2 L= Qe δ 2 x+ δs i δs i τ i + τ i s i δ 2 s i, (14) δ 2 s i = δ 2 x+δ 2 Θ r i + δθ (δθ r i ). (15) Substituting Eq. (15) in Eq. (14), and enforcing f==0, yields δ 2 L= τ i or, in atrix notation, {δx δx 2δx (r i δθ) (r i δθ) [(x a i ) δθ]} (16) δ 2 L= τ i [ δx T δx 2δx T r i δθ + δθ T r i ( x ã i )δθ ], (17) where ñ denotes, for a generic vector n, the skew-syetric atrix expressing the operator n. δ 2 L is a bilinear for in the twist space of the platfor. If the platfor virtual displaceent is expressed, in ray coordinates, as δt=[δx;δθ], and I 3 denotes the 3 3 identity atrix, the syetric atrix associated with this for is H p = τ i [ I3 r i r i 1 2 ( r i x r i ã i + x r i ã i r i ) ], (18) which represents the pseudo-hessian of L (H p is not a true and proper Hessian, since δθ is not generally integrable). The tangent space of C is obtained by setting Eq. (11) equal to zero for all values of i. In atrix notation, this aounts to s T 1 J p δt=. s T (r 1 s 1 ) T [ ] δx. = 0, (19) δθ (r s ) T where the ith row of J p coincides with $ i, expressed in axis coordinates and assuing G as the oent pole. J p is the pseudo-jacobian of the constraint equations. If N p is any 6 (6 ) atrix whose coluns generate the null space of J p, the reduced Hessian of C is the following (6 ) (6 ) atrix: H r = N T p H p N p. (20)

8 8 M. Carricato and J.-P. Merlet A sufficient condition for the equilibriu to be stable consists in H r being positive definite. It is worth rearking that, if E 3 is the atrix such that δθ = E 3 δφ, the Hessian and the Jacobian derived by differentiating L and C with respect to x and Φ are equal, respectively, to H=E T 6 H pe 6 and J=J p E 6, where E 6 =[I 3,0 3 ;0 3,E 3 ] and 0 3 is the 3 3 zero atrix. If the null space of J is generated by N=E 1 6 N p, then H r = N T p H p N p = N T HN. The advantage of the ethod described in this section consists in that H r is coputed in a purely algebraic way, with no need to perfor any differentiation. 4 Application exaple: the 33-CDPR Due to space liitations, only a brief outline of the IGP and the DGP of the 33- CDPR is sketched hereafter. Technical details and convenient discussions about epty and nonzero-diensional solution sets will be provided in future papers, as well as the study of the cases n=2,4,5. When n=3, Eq. (5) is satisfied and rank(m)=3 only if $ 1, $ 2, $ 3 and $ e belong to the sae tridiensional subspace of lines [3]. Letting all 4 4 inors of M vanish leads to 15 polynoial equations in x and Φ of the for p j = 0. We look for the variety V of the ideal generated by such equations. If three configuration variables are known (as in the IGP), any three p j, say p l, p h, p k, ay be chosen and a corresponding (generally zero-diensional) variety V lhk is obtained. V is the intersection of the five varieties that ay be generated this way. Clearly, a priary objective of the solving algorith consists in liiting the nuber of varieties to be coputed to the lowest possible value, possibly to just one. It is useful observing that letting B i A i causes the ith colun of M to vanish (since s i = a i s i = 0) and, hence, it causes all 4 4 inors of M (and thus of M ) to be zero. It follows that a configuration for which B i A i always belongs to V : we call it a trivial solution and we need to discard it (cf. 2). This observation is particularly iportant for the IGP with assigned orientation. In this case, in fact, it is always possible to displace the platfor (with a given orientation) so as to superipose B i onto A i. Consequently, all varieties V lhk necessarily contain the trivial solutions corresponding to B i A i, naely x i = a i r i,...3. Inverse geoetric proble. When the orientation is assigned, all vectors r i,...3, are known. If the equations p 1 := detm 1236 = 0, p 2 := detm 1235 = 0, p 3 := detm 1234 = 0, (21) are considered, it ay be proven that V V 123. Such equations coprise the lowestdegree polynoials aong all inors of M. In particular, p 1 is quadratic in x and y, whereas p 2 and p 3 are quadratic in x, y and z. By eliinating z and y fro Eq. (21), a 4th-degree polynoial equation in x ay be obtained, i.e. p 123 = 0. Since three roots of p 123 necessarily correspond to trivial solutions, the fourth root, i.e. x 4, is real

9 Geoetrico-static Analysis of Under-constrained Cable-driven Parallel Robots 9 and it ay be coputed by Vieta s forulas in closed for. The proble adits, thus, a single solution. Of course, it is adissible only if the corresponding cable tensions are nonnegative and the equilibriu is stable. When the position x is assigned, r 1, r 2 and r 3 are unknown. If the Rodrigues paraeters e 1, e 2 and e 3 are chosen to describe the platfor orientation, the relations in Eq. (21) assue a particularly favorable structure. Indeed, after letting r i = Rb i,...3, p 1, p 2 and p 3 becoe quartic polynoials in e 1, e 2 and e 3. The inors of M other than M 1234, M 1235 and M 1236 yield, instead, sextic equations in the Rodrigues paraeters. Another useful quartic is obtained as follows. By setting the inors M j456, j = 1...3, equal to zero, one has that (x+r 1)detM 456,234 = 0. The variety defined by the above equation coprises the trivial solution x= r 1 and the set of all configurations for which p 8 := detm 456,234 = 0. (22) Equation (22) is, indeed, of degree four in e 1, e 2 and e 3 (and it is quadratic in x, y, z). It is well known that three polynoial equations of the sae total degree always adit a Sylvester-type resultant free fro extraneous polynoial factors [8]. For the case of three quartics, such a resultant is a univariate 64th-degree polynoial in one of the unknowns, say e 3. However, if the resultant of p 1, p 2 and p 3 is attepted to be coputed, it appears to be identically nought. Conversely, if any one aong p 1, p 2 and p 3 is replaced by p 8, the expected 64th-degree polynoial is obtained. The proble adits, thus, at the ost 64 solutions. Direct geoetric proble. When cable lengths are assigned, the platfor configuration has to be deterined. Equation (5) provides up to 15 (non-independent) polynoial equations in the platfor posture variables. Aong the, Eqs. (21) and (22) are of degree four in the Rodrigues paraeters and degree two in the coponents of x. Equation (1) provides three further relations in the platfor posture variables. In particular, one ay conveniently consider q 1 := s 1 2 ρ 2 1 = 0, q 2 := s 2 2 ρ 2 2 s ρ 2 1 = 0, q 3 := s 2 3 ρ 2 3 s ρ 2 1 = 0, (23a) (23b) (23c) which, after clearing the denoinator 1+e e2 2 + e2 3, are quadratic in e 1, e 2 and e 3. q 1 is also quadratic in the eleents of x, whereas q 2 and q 3 are linear in these variables. The point-to-point distance relations in Eq. (23) represent the typical constraints governing the forward kineatics of parallel anipulators with telescoping legs connected to the base and the platfor by ball-and-socket joints. In particular, the DGP of the general Gough-Stewart anipulator depends on six equations of this sort, one of which is equivalent to Eq. (23a) and five ore to Eqs. (23b)-(23c). This proble is known to be very difficult and it has attracted the interest of researchers

10 10 M. Carricato and J.-P. Merlet for several years [5, 9]. The DGP of the 33-CDPR appears to be even ore coplex. In fact, three equations analogous to Eqs. (23b)-(23c), naely linear in the coponents of x and quadratic in the coponents of Φ, are replaced by relationships that are, at least, quadratic in the coponents of x and quartic in the coponents of Φ (cf. Eqs. (21) and (22)). Possible siplifications ay, indeed, arise fro the fact that soe power-products of x, y and z actually iss in the equations eerging fro Eq. (5). The redundancy of such equations ay also play a role. The proble appears to be a daunting task and it will be the subject of future research. 5 Conclusions This paper studied the kineatics and statics of under-constrained cable-driven parallel robots with less than six cables, in crane configuration. In these robots, kineatics and statics are intrinsically coupled and they ust be dealt with siultaneously. This poses ajor challenges. A geoetrico-static odel was presented, together with an original and general procedure aied at efficiently solving the inverse and direct position probles in analytical for. A purely algebraic ethod, based on a constrained optiization forulation, was also provided for the assessent of equilibriu stability. A spatial robot with three cables was considered as a case study, in order to show the feasibility of the presented approach. References 1. Bosscher, P., Ebert-Uphoff, I.: A stability easure for underconstrained cable-driven robots. In: IEEE Int. Conf. on Robotics and Autoation, pp , New Orleans, USA (2004). 2. Hiller, M., et al.: Design, analysis and realization of tendon-based parallel anipulators. Mechanis and Machine Theory, 40(4), (2005). 3. Hunt, K.H.: Kineatic Geoetry of Mechaniss. Clarendon Press, Oxford, UK (1978). 4. Maier, T., Woernle, C.: Inverse kineatics for an underconstrained cable suspension anipulator. In: Lenarčič, J., Husty, M.L. (eds.), Advances in Robot Kineatics: Analysis and Control, pp Kluwer Acadeic Publishers, Dordrecht, The Netherlands (1998). 5. Merlet, J.-P.: Parallel Robots. Springer, Dordrecht, The Netherlands (2006). 6. Ming, A., Higuchi, T.: Study on ultiple degree of freedo positioning echaniss using wires Part 1: Concept, Design and Control. Int. J. Japan Soc. Prec. Eng., 28(2), (1994). 7. Pagani, C.D., Salsa, S.: Analisi Mateatica Vol. 2, in Italian. Masson, Milan, Italy (1993). 8. Salon, G.: Lessons Introductory to the Modern Higher Algebra. Hodges, Figgis, and Co. Dublin, Ireland (1885). 9. Soese, A.J., Wapler, C.W.: The Nuerical Solution of Systes of Polynoials Arising in Engineering and Science. World Scientific Publishing, Singapore (2005). 10. Yaaoto, M., Mohri, A.: Inverse kineatic analysis for incopletely restrained parallel wire echaniss. In: IEEE/RSJ Int. Conf. on Intelligent Robots and Systes, pp , Takaatsu, Japan (2000).