A Novel Reconfigurable 3-URU Parallel Platform Luca Carbonari(B) , David Corinaldi, Matteo Palpacelli, Giacomo Palmieri, and Massimo Callegari Polytechnic University of Marche, 60131 Ancona, AN, Italy {l.carbonari,d.corinaldi,m.palpacelli,g.palmieri,m.callegari}@univpm.it Abstract. A novel parallel robot stemming from the 3-SRU (sphericalrevolute-universal) under - actuated joints topology is presented in this paper. The conceptual design here proposed takes advantage of a reconﬁgurable universal joint obtained by locking, one at a time, diﬀerent rotations of a spherical pair by means of an automatized system. Such local reconﬁguration causes a slight modiﬁcation of the robot legs mobility which is enough to provide the end-eﬀector with diﬀerent kinds of motion. The ﬁrst part of the paper is dedicated at formally demonstrating the motion capabilities oﬀered by the 3-URU kinematics; in the second part, a mechanical solution for the realization of the reconﬁgurable joint is shown. Keywords: Parallel kinematics machine · Algebraic geometry · Study’s parameters · Lockable joint 1 Introduction Due to the increasing demand of ﬂexibility in industrial manufacturing, the research in the ﬁeld of robotics is focuest on versatile machines, which are able to readily adapt themselves to new functions. A possible answer to such quest of ﬂexibility is certainly given by the deﬁnition of smart workstations based on conventional industrial machines equipped with an on-board intelligence and with some degrees of adaptive capacity; a more recent answer is addressed by the development of novel mechanical solutions, thus the realization of reconﬁgurable manipulators. A reconfigurable or metamorphic manipulator is a machine that is able to change its end-eﬀector mobility according to a local change of kinematics. Kinematotropic linkages can be also mentioned when the change of mobility is originated by a gain of an inﬁnitesimal mobility [1–3], namely when a singular conﬁguration is reached. Modular manipulators with serial kinematics are reconﬁgurable when they are assembled from elemental building blocks, which can be combined to produce the desired sequence of joints [4,5]. Reconﬁgurable joints may have a great impact on parallel kinematics machines (PKMs). For this family of robots, in fact, a little modiﬁcation of the legs kinematics may lead to substantial modiﬁcations of their mobility. c Springer International Publishing AG 2018 C. Ferraresi and G. Quaglia (eds.), Advances in Service and Industrial Robotics, Mechanisms and Machine Science 49, DOI 10.1007/978-3-319-61276-8 8 64 L. Carbonari et al. This fact may help to overcome some of the drawbacks that are intrinsic in PKMs, such as their small workspace. Moreover, the reconﬁguration can be often performed by acting some speciﬁc joints without disassembly or assembly operations [6–9]: planar, pure translational, pure rotational or more complex motions result from a manual intervention. Some examples of lockable joints, instead, are provided in the literature [10,11]: the key idea in this kind of joints is that of realizing higher kinematics pair from the composition of elementary kinematic pairs and then locking them alternately. Nonetheless, the change of conﬁguration of the joints is often managed manually, without thinking about the behaviour of the manipulator during the transition. The researchers at the Robotics Lab of the Polytechnic University of Marche, in Ancona, studied the possibility to realize a reconﬁgurable parallel robot exploiting the peculiarities of the Cylindrical-Prismatic-Universal leg joints topology [12]. The researches yield the design of a reconﬁgurable tripod, able to perform motion of pure rotations and pure translations through the reconﬁguration of the universal joints performed by means of a special lockable device [13]. Notwithstanding the metamorphic capabilities shown, the 3-CPU architecture also owns some intrinsic drawbacks which pushed the research towards diﬀerent topology. In particular, the main issue that aﬀects the 3-CPU is that of needing a reconﬁguration of the passive joints which connect the end-eﬀector to the limbs structure: such characteristic implies that the lockable device must be part of the leg structure, with consequent complications in terms of mechanical design (for limitation of weight and size of the device), and in terms of actual payload reduction. Such drawbacks led authors to investigate diﬀerent topologies [14] capable of performance similar to those of the 3-CPU architecture. In particular this manuscript presents a novel concept for the design of a reconﬁgurable sphericaltranslational platform which exploits a 3-URU (universal - revolute - universal) joints architecture, stemming from the under-actuated 3-SRU (spherical - revolute - universal) topology. The key concept of reconﬁgurability is presented in the ﬁrst part of the paper. Later, the kinematics of the two proposed joints conﬁgurations is analysed with the aim of both demonstrating the actual mobility of the platform, and deriving the equations necessary for the solution of the position kinematics problems. The approach is that of the algebraic geometry which has proved to be successful in the case of the 3-CPU PKM. 2 Robot Description The main objective of this paper is to investigate the possibility to realize a reconﬁgurable multi-purpose parallel manipulator starting from the 3-SRU under-actuated topology. To this aim, a design that takes advantage of a reconﬁgurable actuated joint used to connect each leg to the chassis of the robot is proposed. Such joint is realized as a combination of revolute pairs; a locking system allows to alternatively lock one of the two revolute joints, giving the machine diﬀerent 3-URU kinematic conﬁgurations which correspond to diﬀerent types of mobility. A Novel Reconﬁgurable 3-URU Parallel Platform 65 Fig. 1. Reconﬁgurable 3-URU kinematic architecture. The kinematic architecture of each leg is composed of a Spherical-RevoluteUniversal joints chain. The spherical pair, which must be reduced to a universal joint in order to guarantee the functionality of the machine, connects the ﬁrst body of each leg to the chassis. The second link is connected to the ﬁrst one by a revolute joint. At last, a universal joint closes the kinematics chain connecting the end-eﬀector to the leg. The mutual arrangement of the joints, which is of paramount importance for determination of the mobility of the moving platform, is shown in Fig. 1. As visible, the three spherical pairs are located so that their centres lie on the axes of the ﬁxed reference frame {0} at the same distance, called a, from the origin. The three rotations that compose each pair is realized through three perpendicular revolute joints, whose axes concur in a point (namely the centre of the spherical couple). The three axes are disposed as follows: – the ﬁrst revolute is coaxial to the respective axis of the reference frame {0}. In the i-th leg local frame {l1,i } the unit vector describing such axis is {l1,i } r1,i = T 100 ; – the second revolute is perpendicular to the ﬁrst one and it is directed along T the ﬁrst body of the limb. In the frame {l2,i } it is {l2,i } r2,i = 0 0 1 ; – the third and last rotation is perpendicular to both the ﬁrst and the second T at home conﬁguration. With respect to frame {l3,i } it is {l3,i } r3,i = 0 1 0 . It is considered here that at home conﬁguration, frames {l1,i }, {l2,i }, and {l3,i } are coincident and for leg 1 they have the same orientation of the absolute frame {0}. The locking mechanism is used for this application to produce two diﬀerent conﬁgurations, here called Joints Configuration A and Joints Configuration B (see Fig. 1). In conﬁguration A, the ﬁrst rotation of the spherical pair is allowed and, in 66 L. Carbonari et al. particular, it represents the actuated degree of freedom of the i-th leg. The second rotation is locked, while the third one is free. In conﬁguration B the ﬁrst rotation is locked, the second one is actuated and the third one is free. As demonstrated in the following, such choice allows generating motions of pure rotations with Joint Configuration A and motions of pure translations with Joint configuration B without further modiﬁcation to the arrangement of the passive joints. The remainder of the kinematic chain of each leg is composed by a revolute joint, parallel to the last revolute of the reconﬁgurable spherical joint at a distance b, and by a universal joint that connects the second link of the leg to the end-eﬀector. As demonstrated by Palpacelli et al. in [14], the arrangement of the last joint of the chain is also crucial for deﬁnition of the end-eﬀector mobility. In particular, for the reconﬁgurable 3-URU parallel robot the terminal universal joint is disposed so that the ﬁrst rotation is parallel to previous revolute joint at a distance c, thus it is perpendicular the the leg plane πi (identiﬁed by point Ai , Bi , and Ci ). The second rotation lies on πi and is perpendicular to the previous one; on the end eﬀector, such axes are mutually perpendicular and concurrent at the origin of the moving reference frame {1}. Moreover, such axes coincide with the axes of {1}. With respect to such reference frame, the three attachment T T points Ci are identiﬁed by vectors {1} C1 = 0 0 −d , {1} C2 = −d 0 0 , and T {1} C3 = 0 −d 0 . The locking mechanism described in the last part of this manuscript is dedicated at alternatively allowing the motion of the ﬁrst and the second revolute joint. A similar solution was already proposed by authors in [12]. Nonetheless, the mechanical solution proposed for the 3-CPU robot was non negligibly diﬀerent since it was used to modify the mobility of a passive universal joint. In the present case, instead, the reconﬁguration implies that the actuated joints of the robots are to be changed. The mechanical design of the reconﬁgurable joint is obviously aﬀected by that. On the one hand, the need of connecting an actuator to diﬀerent joints relatively complicates the problem. On the other hand, the fact that the lockable device lies on the chassis and it is not moving together with the end eﬀector relaxes the design parameters in terms of weight and dimension, and consequently in terms of costs. 3 Mobility Analysis In order to fully represent the attitude of the mobile platform, the Study’s parameters are chosen for the parametrization of its kinematics. This well known notation [15,16] was introduced by Study who used a superabundant set of 8 parameters in the projective space of dimension 7 (P7 ) to describe the Euclidean transformations. The 8 parameters [x0 : x1 : x2 : x3 : y0 : y1 : y2 : y3 ] are related through a quadric polynomial equation (namely the Study quadric) and through a metric equation which ensures that the transformations actually represent rigid body motions. A Novel Reconﬁgurable 3-URU Parallel Platform 67 Thus, the key idea is to map the Euclidean space through a one to one representation x in the projective space P7 : x : 0 T1 ∈ SE (3) → x ∈ P7 (1) If the homogeneous notation is used, the transformation 0 T1 describing the pose of the robot platform looks like: 0 0 R1 p1 0 T1 = (2) 0 x20 + x21 + x22 + x23 where: ⎤ ⎡ 2 x0 + x21 − x22 − x23 2 (x1 x2 − x0 x3 ) 2 (x1 x3 + x0 x2 ) 0 R1 = ⎣ 2 (x1 x2 + x0 x3 ) x20 − x21 + x22 − x23 2 (x2 x3 − x0 x1 ) ⎦ 2 (x1 x3 − x0 x2 ) 2 (x2 x3 + x0 x1 ) x20 − x21 − x22 + x23 (3) ⎤ ⎡ 2 (−x0 y1 + x1 y0 − x2 y3 + x3 y2 ) 0 p1 = ⎣2 (−x0 y2 + x1 y3 + x2 y0 − x3 y1 )⎦ 2 (−x0 y3 − x1 y2 + x2 y1 + x3 y0 ) (4) The 8 parameters [x0 : x1 : x2 : x3 : y0 : y1 : y2 : y3 ] in (2)–(4) are called Study’s parameters; a set of such parameters corresponds to only one transformation in the Euclidean space and allows to uniquely deﬁne the conﬁguration of a rigid body, e.g. the mobile platform, when the Study’s quadric equations are satisﬁed: x0 y0 + x1 y1 + x2 y2 + x3 y3 = 0 x20 + x21 + x22 + x23 = 0 (5) If the inequality in (5) is assumed equal to an arbitrary constant, the expression becomes a normalizing equation that ensures that matrix (2) is eﬀectively a non singular transformation matrix; typically it is assumed x20 + x21 + x22 + x23 = 1. Thus, Eq. (5) yields the two polynomials: σ1 : x0 y0 + x1 y1 + x2 y2 + x3 y3 , σ1 = 0 σ2 = 0 σ2 : x20 + x21 + x22 + x23 − 1, (6) The vanishing sets of polynomials σ1 and σ2 relate the 8 Study parameters and represent an intrinsic characteristic that the parameters set must fulﬁl. It is then necessary to provide a complete algebraic description of the kinematics of the parallel platform taking advantage of the constraints imposed by the architecture of each joints arrangement. As shown in the following, a distinction can be made among the constraint equations that compose the kinematic model of the moving platform based on their dependency on the robot actuation parameters. Some equations, in fact, can be directly gathered by legs kinematic architecture. Other ones, instead, take into consideration the eﬀect of the motion of actuated joints on the whole robot. Subsequently, such sets of equations are deducted for both the translational and rotational 3-URU manipulators. 68 L. Carbonari et al. Fig. 2. Kinematics of the i-th leg for (a) spherical and (b) translational 3-URU architectures. 3.1 The Spherical 3-URU Robot In order to obtain a complete description of robot kinematics, Study quadric Eq. (5) must be put together to the algebraic relations which are characteristic of the speciﬁc platform. Firstly, the relations which are peculiar of the legs architecture are introduced. This procedure is also useful to formally demonstrate the mobility of the platform within its workspace. It is not trivial, in fact, to ﬁgure out what kind, or kinds, of motions a parallel manipulator might be capable of. As visible in Fig. 2, the conﬁguration A is built so as to constrain the axis of the last revolute joint on the plane πi . Equivalently, one can say that the line li , identiﬁed by the axis of such joint, has an intersection point with all the lines that lie on the plane πi , for example the axis of the ﬁrst revolute of the reconﬁgurable universal joint, here called mi . The two lines can be parametrized through their direction plus a passage point, as: li : αi 0 R1 {1} Ci + 0 p1 mi : βi {l1i } r1,i (7) with αi , βi ∈ R. It must be remarked that, in joints conﬁguration A, the direction of the unit vector {l1i } r1,i is ﬁxed. For the leg constraint to be fulﬁlled it might exist a set of αi and βi ∈ R so that: li = mi → αi 0 R1 {1} Ci + 0 p1 = βi {l1i } r1,i or, in matrix form: 0 R1 {1} Ci T {l1 } 0 − i r1,i p1 αi βi 1 = 0 (8) (9) A non trivial solution of this last homogeneous equation can be found only letting the determinant of the respective matrix vanish. Substituting (3) and (4) for 0 R1 A Novel Reconﬁgurable 3-URU Parallel Platform 69 and 0 p1 in (9), three polynomials in the Study parameters can be found, one for each leg. After simpliﬁcation [17], the three constraints can be expressed by the varieties of the following polynomials: g1 : x0 y2 + x1 y3 + x2 y0 + x3 y1 g1 = 0 g2 : x0 y3 + x1 y2 + x2 y1 + x3 y0 g2 = 0 g3 : x0 y1 + x1 y0 + x2 y3 + x3 y2 g3 = 0 (10) The vanishing set of polynomials g1 , g2 , and g3 , together with Study quadric Eq. (6) are still not suﬃcient to fully describe the kinematics of the 3-URU platform with joints conﬁguration A, since a set of actuation dependent equations are also needed. Nonetheless, such relations provide all the necessary information about the mobility of the end-eﬀector [16,18]. With the aim of performing a complete kinematics analysis of the robot it is possible to decompose the polynomial ideal in its primary components. Each primary components, then, is peculiar of a particular branch of robot mobility (the interested reader can approach authors’ previous work [12]). In any case, the main target of this section is to demonstrate the ability of the manipulator to perform motions of pure rotations and, to this aim, it is suﬃcient to ensure that the conditions provided by such mobility are compatible with the constraints provided by the three legs. In particular, it can be easily veriﬁed that the general transformation matrix 0 T1 describes a pure rotation when the four Study parameters y0 , y1 , y2 , and y3 are null. In this case the translation vector 0 p1 vanishes, denoting a pure rotational behaviour of the end-eﬀector. Also, it is as much easy to verify that such condition fulﬁls both the constraints expressed by Eq. (10), and the Study quadric σ1 = 0. Being demonstrated that the pure rotational behaviour is allowed by the Joints Conﬁguration A, the homogeneous transformation matrix can be simpliﬁed consequently as: 0 R1 03×1 0 T1 = (11) 01×3 x20 + x21 + x22 + x23 A set of simple equations dependent on the actuated variables can now be derived. The line li can be used again: given that li lies on plane πi , it follows that such line is perpendicular to the unit vector {l3,i } r3,i , whose expression in the ﬁxed reference frame is only function of the angle qi of rotation of the ﬁrst revolute joint. For the three legs it is: ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ 0 sin q2 cos q3 (12) r3,1 = ⎣cos q1 ⎦ r3,2 = ⎣ 0 ⎦ r3,3 = ⎣ sin q3 ⎦ sin q1 0 cos q2 The perpendicularity of vectors (12) with the lines li is respected when: rT3,i 0 Ri {1} Ci = 0 (13) 70 L. Carbonari et al. Substitution of the simpliﬁed homogeneous transformation matrix (11) in (13) yields the three polynomials: g4 : −2 cos q1 (x0 x1 − x2 x3 ) + sin q1 x20 − x21 − x22 + x23 d g4 = 0 g5 : −2 cos q2 (x0 x2 − x1 x3 ) + sin q2 x20 + x21 − x22 − x23 d g5 = 0 (14) g6 : −2 cos q3 (x0 x3 − x1 x2 ) + sin q3 x20 − x21 + x22 − x23 d g6 = 0 Now the kinematic problem of the 3-URU robot for motions of pure rotations, namely the joints conﬁguration A, is fully posed as the variety of the polynomial ideal HA = g4, g5, g6, σ2 . It is worth remarking that the solutions of such system of equations intrinsically belong to the Study projective space being veriﬁed the quadric σ1 . 3.2 The Translational 3-URU Robot At this point, the kinematic problem of the 3-URU spherical platform is completely deﬁned, and its mobility is veriﬁed. Similar steps should now be made for Joints Conﬁguration B so as to demonstrate its ability to perform motions of pure translation. To this aim, two sets of equations similar to (13) and (14) are to be found also for this joint arrangement. Even though they are very similar, joints conﬁgurations A and B oﬀer diﬀerent sets of polynomials because of the diﬀerent mobility that each one of the legs provide the manipulator with. Even in this case, equations that do not depend on actuated joints variables can be expressed through the co-planarity of two lines. As well visible in Fig. 2.b, lines li and ni both lie on the leg plane πi . In this case, the ni line equation is ni : γi l2i r2,i + Ai . As done for conﬁguration A, it is possible to state that the coplanarity constraint among lines li and ni is ful ﬁlled when the determinant of the matrix 0 R1 {1} Ci −{l2i } r2,i 0 p1 − Ai vanishes. The three polynomials, one for each leg, coming from the determinant of such matrix can be simpliﬁed as done for (10). The result is the following: h1 : a (x0 x1 − x2 x3 ) + 2 (x1 y1 + x2 y2 ) h1 = 0 h2 : a (x0 x2 − x1 x3 ) + 2 (x2 y2 + x3 y3 ) h2 = 0 h3 : a (x0 x3 − x1 x2 ) + 2 (x1 y1 + x3 y3 ) h3 = 0 (15) For the general transformation (2) to describe motions of pure translations, the Study parameters representing the rotation should assume the values x0 = 1, x1 = 0, x2 = 0 and x3 = 0. Under these conditions, the rotation matrix 0 R1 T is an identity matrix, and the translation vector becomes 0 p1 = −2 y1 y2 y3 . Moreover, the normalization equation σ2 = 0 is fulﬁlled, while the Study quadric σ1 = 0 is veriﬁed when y0 = 0. It is easy to verify that such values also belong to the vanishing sets of the constraint polynomials (15), demonstrating that the pure translational motion is actually allowed by the 3-URU platform with Joints Conﬁguration B. At last, three actuation dependent polynomial equations can be found as done for Conﬁguration A. To this aim, it is possible to observe that the vector A Novel Reconﬁgurable 3-URU Parallel Platform 71 connecting the reconﬁgurable universal joint and the origin of the moving frame {1}, namely the vector 0 p1 − Ai , lies on the leg plane πi . Therefore, it is also perpendicular to the unit vector r3,i whose coordinates only depend on the rotation of the i-th actuated joint. Being the second rotation of the reconﬁgurable joint active while the ﬁrst locked, the three r3,i vectors assume the values: ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ − sin q1 0 cos q3 r3,1 = ⎣ cos q1 ⎦ r3,2 = ⎣− sin q2 ⎦ r3,3 = ⎣ 0 ⎦ (16) 0 cos q2 − sin q3 Then, the three constraint conditions rT3,i 0 p1 − Ai = 0 yield the three polynomials: h4 : sin q1 (2y1 + a) + 2y2 cos q1 h4 = 0 h5 : sin q2 (2y2 + a) + 2y3 cos q2 h5 = 0 (17) h6 : sin q3 (2y3 + a) + 2y1 cos q3 h6 = 0 whose vanishing set provide a complete description of the 3-URU kinematics when Joints Conﬁguration B is used. 4 Reconfigurable Spherical Joint The functional design that is proposed here is based on a bevel gear coupling, as shown in Fig. 3. Such solution is similar to the one proposed by authors in [13]. Anyhow, in the present case the actuation of the locking mechanism has been re-conceived in order to allow the actuation of the bevel gear D. The motion of the sliding cursor C reconﬁgures the spherical joint alternatively locking one of the rotation of the spherical joint as described in the following. The cursor C is driven by an actuator in two diﬀerent positions, which provide the joint as many diﬀerent working modes. The cursor has a cylindrical shape with an external splined shaft and an internal splined hub at the top. In the Conﬁguration I (as in Fig. 3.a) the cursor C mechanically engages the splined hub of the mechanical fork B. In this case B, C and D globally behave like a rigid body. Due to the solid rotation of the fork B together with the bevel gear D, the second fork E does not rotate with respect to B. The body F remains free of rotating since it is not constrained anyhow. Considering the reference frame represented in Fig. 3.a, such conﬁguration turns to allow rotations about axes x and y as required by the 3-URU Joints Conﬁguration A. If the cursor is moved downward to Conﬁguration II (Fig. 3.b), the splined shaft engages with both bodies A and B. Being the body A ﬁxed to ground, the rotation of B remains locked as well. The rotation of the actuated bevel gear is transmitted instead to the second fork E which represents now the actuated degree o freedom of the leg. Thus, conﬁguration II allows an actuated rotation about axis z and a free rotation around y, exactly as required by the robot Joints Conﬁguration B. 72 L. Carbonari et al. Fig. 3. Functional scheme of the actuated reconﬁgurable joint for spherical motions (a) and translational motions (b) of the 3-URU. 5 Conclusions This manuscript introduced the mobility analysis of a novel reconﬁgurable parallel platform, based on the 3-URU kinematic architecture. The study has been performed by means of tools deriving from algebraic geometry, which allowed the demonstration of some interesting reconﬁgurability aspects of the robot. In particular, the paper shows that the 3-URU platform allows motions of both pure rotations and pure translations through a little reconﬁguration of the legs architecture performed by means of a spherical reconﬁgurable joint. To this aim, a set of equations for each joints conﬁguration was found exploiting the legs mobility. Such equations, written in polynomial form thanks to a particular projective approach, were simpliﬁed. Then, it was veriﬁed that the conditions necessary for the manipulator to perform particular motions (namely spherical in the one case, translational in the other) also fulﬁl the constraints imposed by legs mobility. The information obtained with such strategy allowed the simpliﬁcation of a general transformation matrix for the two cases of pure rotations and translations. This step brought to simple relations also for computation of the forward and inverse kinematics problems through two further sets of equations. 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