Generation of under-actuated manipulators with non-holonomic joints from ordinary manipulators

1 Patrick Grosch Institut de Robòtica i Informàtica Industrial (CSIC-UPC) Llorens Artigas 4-6, 2 planta 08028 Barcelona – Spain E-mail: [email protected] Raffaele Di Gregorio Department of Engineering University of Ferrara Via Saragat, 1 44100 FERRARA - Italy [email protected] Federico Thomas Institut de Robòtica i Informàtica Industrial (CSIC-UPC) Llorens Artigas 4-6, 2 planta 08028 Barcelona – Spain E-mail: fthom[email protected] Generation of Under-Actuated Manipulators with Non-Holonomic Joints from Ordinary Manipulators This paper shows how to generate under-actuated manipulators by substituting nonholonomic spherical pairs (nS pairs) for (holonomic) spherical pairs (S pairs) in ordinary (i.e. not under-actuated) manipulators. As a case study, an under-actuated manipulator, previously proposed by one of the authors, is demonstrated to be generated, through this pair substitution, from an inversion of the 6-3 fully-parallel manipulator (FPM). Moreover, the kinetostatic analysis of this under-actuated manipulator is reconsidered, and a simple and compact formulation is obtained. The results of this kinetostatic analysis can be used both in the design of the under-actuated manipulator, and in its control. Keywords: kinetostatics, non-holonomic constraint, ordinary manipulator, underactuated manipulator, parallel manipulator. NOMENCLATURE P = prismatic pair U = universal joint S = spherical pair nS = non-holonomic spherical pair SPU = spherical-prismatic-universal kinematic chain (limb) nSPU = (non-holonomic spherical)-prismatic-universal kinematic chain (limb) FPM = fully parallel manipulator DPA = direct position analysis dof = degrees-of-freedom RB = rigid body 1 Introduction Non-holonomic constraints arise in many different areas of robotics [1-4] such as motion planning and control of mobile robots, reorientation of free-flying space robots, rolling contacts of multi-fingered hands, etc. In all these cases, the non-holonomic constraints are inherent to the problem, but there are some cases in which the artificial introduction of this kind of constraints can provide important advantages. In pick-and-place applications of manipulators, only the initial and the final poses (position and orientation) of the end effector are assigned by the task, whereas the end-effector path between them is free. The ideal manipulator for these applications should be able to make the end effector reach any pose in the sixdimensional operational space, and, by exploiting the free fly of the end effector, it should be able to satisfy additional design conditions that reduce its hardware complexity. Joints with nonholonomic constraints do not reduce the reachable relative poses of the links connected by the joint since non-holonomic constraints have the only effect of reducing the set of paths that can be covered for moving between two reachable relative poses. This reduction of practicable paths is accompanied by the rising of new reaction forces in the joint which can be usefully exploited to eliminate actuators. Thus, designing a manipulator with fewer actuators than the degrees-of-freedom (dof)1 of its configuration space ---to reduce bulk, weight and expense--- becomes feasible by introducing mechanical elements that lead to non-holonomic constraints. The literature on the use of non-holonomic devices in the design of manipulators is limited to few examples. In [7], Stammers et al. presents a robot wrist that can attain any orientation with two motors only. This is achieved by means of a friction drive, using rollers on a spherical ball to which the end effector is fixed, and by fixing the two motors to the arm. In [8], Peshkin et al. present a passive spherical robot which can display programmable constraints. The device is based on a nonholonomic element involving a sphere and three reorientable rollers. In [9], Nakamura et al. describe an n-joint serial robot which can reach any pose in its n-dimensional configuration space with only two actuators. The joints of this manipulator are coupled by (n-1) non-holonomic devices, based on spheres and rollers, so that it reaches a desired pose by following a path whose computation is algorithmically equivalent to maneuvering a car with n-trailers. More recently, in [10], Ben-Horin and Thomas proposed a three-legged parallel robot where each leg is connected to the base through a sphere whose motion is constrained by a roller. This latter parallel architecture permits to attain any 1 The degrees-of-freedom (dof) of the configuration space, also called configuration (or finite) dof [5], are the minimum number of geometric parameters necessary to uniquely identify the configuration of the mechanical system [6]. They may be different from the instantaneous dof, also called velocity dof [5], of the same mechanical system. 2 position and orientation for the platform using only three prismatic actuators. Despite the difference of purpose, all mentioned examples include at least one sphere whose motion is constrained by a roller that can freely roll in contact with the sphere without slipping laterally. This no-slip constraint is a non-holonomic constraint, a constraint relating the velocities of the sphere and the roller. The kinematics of this sphere-roller assembly is equivalent to that of a unicycle on a sphere whose equations of motion can be represented by first-order differential equations [11]. Many research efforts have been made to clarify different aspects of non-holonomic mechanical systems including its controllability, stability, feedback stabilization, time-periodic control, chained form transformation, etc. but, in any case, achieving a formulation for the kinematics of the system, as compact and simple as possible, is essential to explore the applicability of all these results available in the literature. Herein, how under-actuated manipulators can be generated through the substitution of a spherical pair (S pair) by a particular non-holonomic pair, named non-holonomic spherical pair (nS pair), in ordinary (i.e., not under-actuated) manipulators is shown. As a case study, the under-actuated parallel architecture presented in [10] is demonstrated to be generable from an inversion of the 63 fully-parallel manipulator (FPM). Its kinetostatic analysis is reformulated, and a simple and compact formulation, useful for its design and control, is obtained. This paper is structured as follows. Next section describes how to generate under-actuated manipulators through substitution of suitable kinematic pairs in ordinary manipulators. Section 3 is devoted to the case study: a compact formulation for its instantaneous kinematics and statics is obtained, and some clues for the characterization of its singularities are provided. Eventually, section 4 offers the conclusions. 2 Generation of Under-Actuated Manipulators Two rigid bodies (RBs) connected by a spherical pair (S pair) can assume any relative orientation, and can move from one relative orientation to another by covering any spherical-motion path that joins the two relative orientations. Actually, the possibility of freely orientating two RBs with respect to one another is not related to the possibility of performing relative rotations around axes which pass through the center of spherical motion and have any direction. In fact, a suitable sequence (at least three) of finite rotations around coplanar axes that pass through the spherical-motion center can freely orientate one RB with respect to another. Thus, if the only free relative orientation of two RBs is required, the use of an S pair will be redundant. The use of a kinematic pair that allows only rotations around coplanar axes that pass through a fixed point would be sufficient. Due to frictional forces, the rolling contact between a sphere and a roller forbids the sphere rotations around the axis through the sphere center, and perpendicular to the plane defined by the roller axis and the sphere center. By combining such a nonholonomic constraint with other constraints that forbid the relative translation between the sphere center and the roller axis, a nonholonomic joint will result. This joint constrains two RBs: one fixed to the sphere and the other fixed to the plane, defined by the roller axis and the sphere center. So that the resulting constrained motion permits only relative rotations around axes lying on the above-mentioned plane and passing through the sphere center. Hereafter, this type of joint will be called non-holonomic spherical pair (nS pair). Figure 1. 3D CAD model of one out of many different manufacturing schemes for the nS pair: (1) sphere, (2) roller (carried out through a roller bearing), (3) spherical bearing [there are three spherical bearing (only two are visible) that forbids the translation of the sphere], (4) pre-load adjust screw, and (5) parallelism adjust screws. The constraint forces, which two RBs, joined by a nS pair, exert on one another through the joint, can be reduced to a resultant force applied on the sphere center and a torque perpendicular to the plane defined by the roller axis and the sphere center. The torque is the static effect of the non-holonomic constraint, whereas the resultant force on the sphere center is the same static effect that an S pair would have generated. An nS pair could be manufactured as shown in Fig. 1. From a manufacturing point of view, it is worth noting that, in a nS pair, the presence of any number of roller-sphere contacts does not alter the kinetostatics of the nS pair, provided that all the roller axes lie on a same plane passing through the sphere center2. Moreover, the maximum torque transmitted through the nS pair, due to its frictional origin, can be fixed by suitably choosing the number of roller-sphere contacts together with the normal force transmitted through each contact. The above discussion brings to the proposition: i) the substitution of a number of nS pairs for as many S pairs in a kinematic chain does not change the configuration space of that chain (i.e., neither the degrees-of-freedom (dof)3 of the configuration space nor the reachable configurations change), it only reduces the practicable paths for moving that chain from one configuration to another. Moreover, due to the torque that arises in a nS pair and to proposition i), the following proposition holds, too: ii) in a manipulator, the substitution of a number of nS pairs for as many S pairs does not change its workspace and allows the elimination of a number of actuators equal to the number of introduced nS pairs (i.e., generates an under-actuated manipulator). S pairs are extensively employed in spatial parallel manipulators (see, for instance, [12-15]). In particular, fully2 In general, two rollers whose axes locate with the sphere center two different planes constrains the sphere to rotate around the intersection line between the two planes, whereas three rollers whose axes locate with the sphere center three different planes lock the sphere. 3 The presence of non-holonomic constraints does not change the configuration dof [5, 6]. It only affects the instantaneous dof of the mechanism. Hereafter, the acronym dof used alone will mean configuration dof. 3 parallel manipulators (FPMs) feature two platforms, one mobile (end effector) and the other fixed (frame), connected to each other by means of six universal(U)-prismatic(P)-spherical(S) kinematic chains (UPS limbs) where the prismatic pairs are the only actuated pairs. In each limb, the centers of the universal joint and of the spherical pair (limb's attachment points) are points, fixed either to the end effector or to the frame, whose distance (limb length) is controlled by the actuated prismatic pair. Two or more attachment points, either in the end effector or in the frame, can coalesce into a unique point. According to the number of attachment points (no matter if they are multiple or not) in the end effector, say p, and in the frame, say q, different FPM architectures, named p-q FPM, are distinguished [15]. Due to the high number of S pairs appearing in FPMs, the substitutions of nS pairs for S pairs, accompanied by as many eliminations of actuators in the prismatic pairs, can be operated in many ways in all the FPM architectures. By exploiting all the possible substitutions, a lot of new under-actuated parallel manipulators can be generated. It is worth noting that a passive UPS limb only affects the workspace borders since it has connectivity six, and, if this effect is not necessary, the elimination of the actuator in a prismatic pair could be accompanied by the elimination of the whole resulting passive UPS limb. 3 Case Study In this section, an under-actuated parallel manipulator generated from the 6-3 FPM (Fig. 2) is studied. The 6-3 FPM architecture features three couples of UPS limbs with coalesced S pairs in the end effector. This architecture was proposed first by Stewart [16], in the 1965, for a flight simulator. Successively, with the renewed interest for the parallel architectures, started at the end of the eighties, it was diffusely studied. In particular, regarding the direct position analysis of the 6-3 FPM, Innocenti and Parenti-Castelli [17] demonstrated that at most sixteen end-effector poses correspond to a given set of limb lengths. Then, Parenti-Castelli and Di Gregorio [18] demonstrated that the end-effector pose is uniquely determined when the value of one passive joint variable is measured besides the six limb lengths. The direct position analysis of this FPM can also be used for spatial parallel manipulators that become 3-RS structures when the actuators are locked (see, for instance, [19-22]). Starting from the 6-3 architecture, each couple of UPS limbs with coalesced S pairs (Fig. 3(left)) can be transformed into an UPnS limb, as shown in Fig 3(right), without affecting the workspace of the manipulator (see proposition i)). By operating this substitution in all the three couples of UPS limbs together with the inversion of the end effector with the frame, the underactuated manipulator with topology 3-nSPU, shown in Fig. 4, is obtained. This under-actuated manipulator is able to move the end effector in a six-dof workspace by changing only the three limb lengths. The 3-nSPU has been proposed first in [10]. Regarding the direct position analysis (DPA) of the 3-nSPU, since its configuration space has six dof, a number of closure equations equals to the number of unknowns can be written if, and only if, over the three limb lengths, three more passive joint variables are assigned (measured). By assigning (measuring) the three joint variables of the three revolute pairs4 not adjacent to the end effector, the closure equation system coincides with the one of the 6-3 FPM for assigned limb lengths [18], and admits at most 4 Each U joint is constituted by two revolute pairs: one adjacent to, and the other not adjacent to the end effector. sixteen solution for the end-effector pose. Moreover, if the joint variable of a revolute pair adjacent to the end effector is measured (or coherently assigned) too, only one end-effector pose satisfies the closure equations [18]. Figure 2. 6-3 FPM Figure 3. UPnS limb (right) generated by substituting an nS pair for the S pair in the two UPS limbs with coalesced S pairs (left). 3.1 Instantaneous Kinematics: Fig. 5 shows the ith limb, i = 1, 2, 3, together with the notation that will be used. w1i and w2i are two any mutually orthogonal unit vectors fixed to the frame and lying on the plane defined by the roller axis and the center, Ai, of the sphere, in the roller-sphere contact. w3i and w4i are the two mutually orthogonal unit vectors of the axes of the two revolute pairs constituting the U joint. Bi is the center of the U joint. ai and bi are the two position vectors which locate the points Ai and Bi, respectively, in a generic Cartesian reference fixed to the frame, whereas p is the position vector of an end-effector point, P, in the same Cartesian reference. θji, for j =1, …, 4, is a joint variable denoting a rotation angle around the joint-axis defined by wji, for j=1, …, 4, and positive if counterclockwise with respect to wji. The length of the ith limb is equal to ||bi − ai||, and it will be denoted li. Moreover, the limb-axis' unit vector, g i, and the unit vector, hi (ri) normal to the plane located by the U-joint's revolutepair axes (by the roller axis together with the sphere center in the nS joint) satisfy the following relationships: li g i = bi − ai, hi = w3i × w4i, ri = w1i × w2i. (1) The time differentiation of the first of the relationships (1) yields 4 Figure 4. Under-actuated manipulator with topology 3-nSPU: (a) kinematic model, (b) 3D CAD model of a manufacturing scheme. +=ggb ii ii i ll   (2) Since 11 22 (θθ)×=+gwwg iiiiii  , and = + ×( ) − bpωbp ii , where ω denotes the end-effector angular velocity, Eq. (2) can be rewritten as 11 22 + [θ(×) + θ(×)]=+×( )−gwgwgpωbp ii i i i i i i i i ll . (3) The dot products of (3) by w1i and w2i yield the following two scalar equations: 122 1 1 1 () + θ(× ) = +×( )⋅⋅⋅−⋅gw w gw pw ωbpw ii i ii i i i i i i ll , (4) 211 2 2 2 () + θ(× ) = +×( )⋅⋅⋅−⋅gw w gw pw ωbpw ii i ii i i i i i i ll . (5) Figure 5. ith limb of type nSPU: notations. On the other hand (see Fig. 5), the end-effector angular velocity is equal to 1,4 θ = ∑ w ji ji j , whose dot product by hi gives the following expression 11 22 = θ() + θ() ⋅ ⋅⋅ωhwh wh iiii iii  . Solving (4) and (5) for 2 θi  and 1 θi , respectively, and replacing the result in the above equation, yields: 222 1 12 111 2 21 + ×( ) = ( ) (× ) + ×( ) + ( ). (× ) ⎡⎤ ⋅−⋅−⋅ ⋅⋅ ⎢⎥ ⋅ ⎢⎥ ⎣⎦ ⎡⎤ ⋅−⋅−⋅ ⋅ ⎢⎥ ⋅ ⎢⎥ ⎣⎦ pw ωbpw gw ωhwh wgw pw ωbpw gw wh wgw ii iiii iii iii i iiiiii ii iii i l l l l     (6) Taking into account the identities 12 2 1 1 2 × = × ( × ) = ( ) ( ) ⋅ −⋅hr h w w whw whw ii i i i ii i ii i , 12 1 2 2 1 = × = × = × ⋅ ⋅− ⋅ ⋅rg w w g w gw w g w ii i ii i i i i i i , relationship (6) can be rewritten as: ( × ) = ( × ) + [( )×( × ) ( ) ]⋅⋅⋅− −⋅ghr phr ωbp hr rgh ii ii ii i ii ii i i ll . (7) Since i l can also be obtained as the projection of bi  on g i [see Eq. (2)], the following expression holds = = + [( )× ]⋅⋅⋅−bg pg ωbpg iii i i i l. (8) Replacing expression (8) for i l in (7), gives 5 + [( )× ( ) ] = 0⋅⋅− −⋅ps ωbps rgh iiiiiii l . (9) where = × [ ( × )] −⋅shr ghrg iii iiii . (10) is the component of hi×ri perpendicular to g i. Eventually, rewriting Eqs. (8) and (9), for i = 1, 2, 3, in matrix form yields 33 33 33 33 33 33 = ××× ××× ⎛⎞⎛ ⎞ ⎛⎞ ⎜⎟⎜ ⎟ ⎜⎟ ⎝⎠ ⎝⎠⎝ ⎠ 1GK p l 0SJ ω   (11) where 13×3 and 03×3 are the 3×3 identity and zero matrix, respectively, 123 = ( , , )llll  T is the vector collecting the joint rates of the actuated joints, and KT [i,:] = (bi − p) × g i (12) GT [i,:] = g i (13) JT [i,:] = (bi − p) × s i − li (ri ⋅ g i) h i (14) ST [i,:] = s i (15) with the notation AT [i,:] to mean the i-th column of matrix 3×3 AT. Matrix relationship (11) is the sought-after input-output instantaneous relationship necessary to implement the control algorithms of the 3-nSPU. 3.2 Static Analysis: The only input-output static relationship can be immediately deduced from (11) through the principle of virtual work. Nevertheless, in order to highlight how the loads act upon the limbs and are transmitted through the joints, the complete static analysis of the 3-nSPU will be developed here independently of (11). Figure 6 shows the free-body diagram of the ith limb. With reference to Fig. 6, the force fbi (fai), applied on Bi (Ai), together with the torque mhi hi (mri ri) are the resultants of constraint forces exerted by the end effector (frame) on the ith limb through the U joint (the nS joint). Moreover, the force −fext, applied on the endeffector point P, together with the torque −mext will denote the resultants of the interaction forces exerted on the end effector. The force τi gi will denote the axial force exerted on the upper part of the ith limb by the actuator in the prismatic pair. It is worth noting that the force equilibrium, along the limb axis, of the upper part of the ith limb yields the following relationship τi = −fbi ⋅ gi. With these notations, the equilibrium of the forces applied on the ith limb yields fbi + fai = 0; whereas, taking Ai as reference point, the equilibrium of the moments applied on the same limb is: mhi hi + mri ri + li gi × fbi = 0. (16) The dot product of Eq. (16) by gi, yields the relationship = ⋅ −⋅ hg rg ii ri hi ii mm . (17) whose substitution for mri in (16) leads to Figure 6. ith limb of type nSPU: free-body diagram. [ × ( × )] + × = ⋅ghr gf0 rg hi iiiiibi ii ml. (18) where the vector identity gi × (hi × ri) = (ri ⋅ gi) hi − (hi ⋅ gi) ri has been used. The dot product of Eq. (18) by hi × ri yields the relationship: (gi × fbi) ⋅ (hi × ri) = 0. Such a relationship is satisfied if, and only if, fbi is a linear combination of gi and hi × ri. Subtracting from hi × ri its component along gi, the vector si is obtained. Since gi and si are two orthogonal vectors that span the same subspace as gi and hi × ri, fbi can be expressed as follows: fbi = −τi gi − τ ⊥ isi. (19) Equation (19) can be interpreted as the equilibrium of the forces applied on the upper part of the ith limb. In fact, the two forces τi gi and τ ⊥ isi are, respectively, the active axial and the passive shear forces applied through the actuated prismatic pair. Replacing expression (19) for fbi in (18), and taking into account that gi ×(hi × ri) = gi × si yields τ( × ) = ⊥ ⎛⎞ − ⎜⎟ ⎜⎟ ⋅ ⎝⎠ gs 0 rg hi ii i i ii ml which is satisfied if mhi = li τ ⊥ i (ri ⋅ gi). (20) Using (17), (20) can be rewritten as mri = −li τ ⊥ i (hi ⋅ gi). (21) 6 Regarding the end-effector equilibrium, the equilibrium of the forces is: 33 3 11 1 τ+τ⊥ == = =− = ∑∑∑ ffgs ext bi i i i i ii i , (22) and, taking the end-effector point P as reference point, the equilibrium of the moments is: 33 11 ()× == =− − − ∑∑ mhbpf ext hi i i bi ii m. (23) The substitution of fbi, according to (19), and of mhi, according to (20), into (23) yields 33 11 τ()×+τ[( )× ( ) ] ⊥ == =− −−⋅ ∑∑ mbpgbpsrgh ext i i i i i i i i i i ii l. (24) Finally, Eqs. (22) and (24) can be rewritten in matrix form as follows: 33 33 33 33 = ×× ⊥ ×× ⎛⎞⎛ ⎞ ⎛⎞ ⎜⎟⎜ ⎟ ⎜⎟ ⎝⎠ ⎝⎠⎝ ⎠ fGK τ mSJ τ T ext ext (25) where τ = (τ1, τ2, τ3)T is a vector collecting the signed magnitudes of the forces applied by the actuators in the prismatic pairs, and τ⊥ = ( 123 τ,τ,τ ⊥⊥⊥ )T. Matrix relationship (25) is the input-output static relationship of the 3-nSPU. It is worth noting that, as expected, (11) and (25) satisfy the instantaneous power balance: fext ⋅p  + mext ⋅ ω = τ ⋅ l . 3.3 Singularity Analysis: Singularities are manipulator configurations where the relationship (input-output instantaneous relationship) between the rates of the actuated-joint variables and the end-effector twist, ( p , ω), fails [23−25]. Three types of singularities can be distinguished [23]: (I) singularities of the inverse kinematic problem, (II) singularities of the direct kinematic problem, and (III) singularities both of the inverse and of the direct kinematic problems. Type-I singularities occur when the actuated joint rates cannot be uniquely computed for an assigned end-effector twist. Vice versa, type-II singularities occur when the end-effector twist cannot be uniquely determined for assigned actuated joint rates. For the 3-nSPU, the input-output instantaneous relationship is (11) where the actuated joint rates are collected in the vector l . This relationship highlights that the 3-nSPU has only three instantaneous dof. Therefore, its singularity analysis can be addressed by using the scheme proposed in [26]. Regarding type-I singularities, provided that the assigned twist, ( p , ω), satisfies the last three equation of system (11), the inverse kinematic problem can be always solved5. Regarding type-II singularities, the equation of the singularity locus is 5 System (11) does not model the mobility limitations due to the physical constitution of the real joints, and to the real sizes of the links. Such limitations bound the workspace and, when correctly modeled, yield type-I singularities. 33 33 33 33 det = 0 ×× ×× ⎛⎞ ⎜⎟ ⎝⎠ GK SJ (26) The geometric interpretation of the above algebraic condition is not straightforward. Nevertheless, the last three equations of system (11) allows the elimination of p  provided that det(S3×3) = s1⋅s2×s3 is different from zero. In this case, system (11) becomes l  = Q ω (27) where Q is the 3×3 matrix (K3×3 − G3×31 33 − × SJ 3×3). Thus, the analytic expression of the singularity locus becomes det (Q) = q1 ⋅ q2 × q3 (28) where the vectors qi, for i = 1, 2, 3, are the column vectors of matrix Q. In conclusion, if the mixed product s1 ⋅ s2 × s3 is different from zero (i.e., the three vectors si, for i = 1, 2, 3, are neither coplanar nor null vectors), the type-II singularities are geometrically identified by either the coplanarity of the three vectors qi, for i = 1, 2, 3, or by the fact that at least one of the qi vectors is a null vector. If the mixed product s1 ⋅ s2 × s3 is zero, the determinant of the whole 6×6 matrix appearing in (26) must be considered, and geometric interpretations of (26) are much more difficult to provide. The zeroing of s1 ⋅ s2 × s3 can be geometrically identified since it occurs when either (a) at least one of the si vectors is a null vector, or (b) the three si vectors are coplanar. Vector si (see definition (10)) is related to the configuration of the ith limb, and it is the component of hi × ri perpendicular to gi (i.e., to the limb axis). As a consequence, condition (a) occurs when, in at least one limb, either (a.1) the two unit vectors hi and ri are parallel (i.e., when, in a limb, the revolute pair axes in the U joint are both parallel to the plane defined by the roller axis and the sphere center in the nS pair), or (a.2) the limb axis is the intersection line between the plane, defined by the roller axis and the sphere center in the nS pair, and the plane, defined by the revolute pair axes of the U joint. Condition (a.2) is forbidden in practice by the actual sizes of joints and links. Regarding condition (a.1), a very special case occurs when hi and ri are parallel in all the limbs. This occurrence makes the matrix S3×3 a null matrix and allows the determinant at the left-hand side of (26) to be factorized as det(G3×3) det(J3×3) where det(G3×3) is equal to g1 ⋅ g2 × g3, whereas, in this case, det(J3×3) is equal to −l1l2l3 (r1 ⋅ g1) (r2 ⋅ g2) (r3 ⋅ g3) h1 ⋅ h2 × h3. Thus, in this case, a type-II singularity occurs when either the limb axes are all parallel to a unique plane, or the hi vectors are coplanar, or, finally, in at least one limb, the limb axis lies on the plane defined by the roller axis and the sphere center of the nS pair. Moreover, it is worth noting that, in this case, the end effector performs an instantaneous translation, if neither det(G3×3) nor det(J3×3) are equal to zero (i.e., out of singularity). Regarding condition (b) (i.e., the coplanarity of the si vectors), it occurs when the limb axes are all parallel, and in other configurations more difficult to visualize. 3.4 Local and Global Controllability: Each end-effector configuration (pose) can be modeled as a point in R3×SO(3) 7 which is locally diffeomorphic to R6 equipped with a proper set of local coordinates: x ≡ (pT, ηT)T where η is a three-dimensional vector collecting the values of the three orientation parameters chosen to locate end-effector’s orientation. By using the orientation parameters’ rates, η , the endeffector’s angular velocity, ω, can be expressed as: ω = H3×3 η  (29). Relationship (29) allows system (11) to be rewritten in the form 6x6 3x1 ⎛⎞ =⎜⎟ ⎜⎟ ⎝⎠ l xV 0   (30) with 1 33 33 33 33 66 123456 1 33 33 33 33 (,,,,,) = − ×× × × − ××× ⎛⎞⎛ ⎞ ⎜⎟⎜ ⎟ ⎝⎠⎝ ⎠ 10GK V vvvvvv 0H S J x x  (31) where vi is the i-th 6-dimensional column vector of matrix V6x6. The vectors vi depend only on x; so, they are vector fields defined on end-effector’s configuration space. Definition (31) allows (30) to be further simplified as follows: 11 2 2 33 =+ +xv v vlll   (32). Relationship (32) states that, in the neighborhood of a generic configuration, x0, all the configurations, x, reachable without maneuvering (i.e., without a sequence of coordinated actions of the actuators) are so located that (x−x0) ∈ Span(v1, v2, v3). (32) is the relationship to be considered for discussing endeffector’s ability to reach any configuration in the neighborhood of a generic configuration, x0 [27, 28]. The presence of nonholonomic constraints in the 3-nSPU manipulator might allow all neighboring configurations be reachable possibly by maneuvering. If this happened, the system would be “locally controllable” [27] at the configuration x0. It can be shown (see [28] pp 323-324) that, if a system, satisfying (32) and at the configuration x0, first follows vi, i∈{1,2,3}, for a small time ε, then follows vj, j∈{1,2,3⎪j≠i}, for the same time ε, then −vi for ε, and finally −vj for ε, it will reach the following configuration of x0’s neighborhood: 0 2 0ij 0 lim (4ε)ε[,] = ε→ =+ xx xxvv (33) where [vi, vj] is the 6-dimensional vector field named Lie product of vi and vj, defined as follows ji ij i j [,] ∂∂ =− ∂∂ vv vv v v xx (34), and the trailing subscript, x = x0, indicates the point the two vector fields, vi and vj, are evaluated at. By reiterating the same reasoning (first, on pairs of vector fields of type vi and [vj, v k], i,j,k∈{1,2,3⎮i≠j≠k}, and, successively, on pairs of vector fields belonging to the set which collects all the vector fields that, in the previous iterations, were demonstrated to point from x0 toward reachable configurations), it can be demonstrated that all the vector fields obtained through Lie products of any degree of elements of the set {v1, v2, v3} point toward configurations that are reachable by maneuvering from x0 [27]. In other word, for any reachable configuration, say x, the vector (x−x0) belongs to the Lie algebra6 of {v1, v2, v3}. In our case, demonstrating that the dimension of the linear space Span(v1, v2, v3, [v1, v2], [v1, v3], [v1, v2]) is six7 is sufficient for concluding that the manipulator is locally controllable at a given configuration since end-effector’s configuration space is 6dimensional (Chow’s theorem [28]). Moreover, showing that the set of configurations where the system is locally controllable is a simply connected region is sufficient to demonstrate the existence of finite regions of end-effector’s configuration space where, for any two configurations belonging to that region, at least one path exists, which the system can follows, for moving from one configuration to the other (i.e., the system is “globally controllable” in that region). According to the above discussion, the configurations where the local controllability of our manipulator is not guaranteed are the geometric locus of the roots of the following equation det(L6×6) = 0 (35) where L6×6  (v1, v2, v3, [v1,v2], [v1,v3], [v2,v3]) (36). The locus of the roots of (35) is in general a 5-dimensional variety; thus, a finite region where our manipulator is globally controllable in general exists. This statement will be verified through the numerical example reported below. 3.5 Numerical Example: With reference to the notation defined in the Figs. 5 and 6, a 3-nSPU is considered where the points Ai (Bi) for i=1,2,3 are at the vertices of an equilateral triangle fixed to the frame (to the end effector). The Cartesian reference system fixed to the frame (to the end effector) has the origin O (P) at the centroid of the equilateral triangle, z axis perpendicular to the plane of the triangle, x axis passing through A3 (B3) with direction from A3 (B3) toward O (P), and y axis accordingly chosen. In an arbitrary unit of length (aul), the distance of the triangle vertices Ai (Bi) from its centroid is 39.7 aul (11.76 aul). In the same unit, the geometry of the frame and of the end effector are defined by the following data (the vector without any leading superscript are measured in the frame reference, whereas the vectors with the leading superscript e(⋅) are measured in the end-effector reference): a1=(19.85, −34.3812, 0)T, a2=(19.85, 34.3812, 0)T, a3=(39.7, 0, 0)T, r1=(0.7887, −0.2113, 0.5774)T, r2=(−0.2113, 0.7887, 0.5774)T, r3=(−0.5774, 0.5774, 0.5774)T, e(b1−p)=(5.88, −10.1845, 0)T, e(b2−p)=(5.88, 10.1845, 0)T, e(b3−p)=(11.76, 0, 0)T, ew41=(0.3536, −0.6124, 0.7071)T, ew42=(0.3536, 0.6124, 0.7071)T, ew43=(−0.7071, 0, 0.7071)T. With this manipulator geometry, the singularity locus defined by Eq. (26) and the root locus of Eq. (35) have been computed for a fixed orientation of the end effector with respect to the frame. 6 The “Lie algebra” of a set of vector fields is the linear span of all Lie products, of all degrees, of vector fields belonging to that set [27]. 7 It is worth noting that, if the dimension of Span(v1, v2, v3, [v1,v2], [v1,v3], [v2,v3]) is six, all the Lie products of any degree in {v1, v2, v3} must belong to Span(v1, v2, v3, [v1,v2], [v1,v3], [v2,v3]); thus, all the reachable configurations, x, satisfy the condition (x−x0)∈ Span(v1, v2, v3, [v1,v2], [v1,v3], [v2,v3]). 8 Figure 7. Singularity locus [Eq. (26)] for the 3-nSPU geometry of the numerical example, and end-effector orientation fixed to XZX Euler angles’ values (0, 1, 0) radians: (a) 3D view, (b) top view The results of these two computations are shown in the Figs. 7 and 8, respectively. By comparing the singularity locus (Fig. 7) and the root locus of Eq. (35) (Fig. 8), a wide free-from-singularity region that is globally controllable can be easily identified. Figure 8. Locus of the Eq. (35) roots for the 3-nSPU geometry of the numerical example, and end-effector orientation fixed to XZX Euler angles’ values (0, 1, 0) radians: (a) 3D view, (b) top view 4 Conclusions If a number of non-holonomic spherical pairs replaces as many spherical pairs in a manipulator, the same number of actuators can be eliminated. The resulting manipulator will keep 9 the same workspace of the generating manipulator, but it will be under-actuated. This technique for generating under-actuated manipulators can be applied to fully-parallel manipulators, where many spherical pairs are present, and the elimination of an actuator in an UPS limb can be accompanied with the elimination of the whole UPS limb. Through this pair substitution, an under-actuated manipulator, previously proposed by one of the authors, has been generated from an inversion of the 6-3 FPM. The kinetostatic analysis of this manipulator has been reconsidered to obtain a simple and compact formulation. This reformulated analysis can be used both in the design of the under-actuated manipulator, and in its control. Further works on this manipulator will present an exhaustive singularity analysis, and will provide design criteria for increasing its useful workspace. Acknowledgements This work has been partially supported by the Spanish Ministry of Education and Science, under the I+D project DPI2007-60858, and by Italian MIUR funds. We also like to thank our colleagues Patricia Ben-Horin, from Technion, Israel, and Krzysztof Tchon, from Wroclaw University of Technology, Poland, for their helpful comments. 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