Extended analysis of the 3CPU reconfigurable class of parallel manipulators

During past years, two different minor mobility parallel robots have been developed and prototyped at the Laboratory of Robotics of the Polytechnic University of Marche in Ancona: the first one is a pure translational machine, called I.Ca.Ro., characterised by a Cartesian achitecture driven by linear modules while the second one, Sphe.I.Ro., provides the mobile platform with a spherical motion, driven by 3 linear induction motors. Notwithstanding the quite different kinematic performances of the machines, they are both based on the same 3-CPU architecture, with of course a different setting of the joints. The present work was aimed at investigating whether a common mechanical architecture would be able to provide both motions by a simple reconfiguration or even if the same machine could yield the two different kinds of motions by meeting some “switching configuration”, i.e. whether it could show a kinematotropic behaviour. The key idea was to realize a reconfigurable universal joint that allows to vary robot kinematics without changes in legs structure. A simple solution to this problem is using a spherical joint made of three consecutive revolute pairs: in this way, three different universal joints are obtained by locking, one at a time, the three rotations of the spherical joint. It is worth to remark that both I.Ca.Ro. and Sphe.I.Ro. are comprehended within these joints configurations. The extended analysis carried on pointed out that the only chance to yield different kinds of motions with the 3-CPU configuration is to lock the first rotation of spherical joint, thus obtaining the kinematics scheme of the robot Sphe.I.Ro. which is able to provide spherical motions. The study of mechanism kinematics requires to use a parametrization possibly free of representation singularities. To this aim the transformation matrix between denoting end-effector configuration is expressed as a function of Study’s parameters, which are 8 parameters defining a point in the prjective space which correspond a unique Euclidean transformation. In order to obtain a complete algebraic description of robot kinematics, Study quadric equations, that are intrinsic of the used parametrization, must be juxtaposed to relations characteristic of the specific legs architecture. Among these constraint equations, a distinction can be made based on their dependence on robot actuation. Indeed, each leg is made of a serial kinematic chain whose joints constrain the manipulator mobility regardless of the actuation parameters. On the other hand some equations are needed to describe the influence of actuation displacements on end effector pose. In the following a geometric interpretation of legs mobility will express actuation independent constraints while loop closure equations will provide actuation influence. The robotic system is then fully algebraically described by a set of polynomial constraint equations, that can be collected in a polynomial ideal. As suggested by relevant past works, a simpler formulation of the constraint equations is fundamental for a deeper understanding of robot kinematic behaviour. With this aim, a specialized study is required on that portion of the polynomial ideal collecting those constraints equations purely dependent on kinematic architecture. Therefore, the sub-ideal is analysed through the computation of its primary decomposition; this method consists of splitting the ideal into several sub-ideals such that the union of their vanishing sets correspond to the vanishing set of the starting ideal. It is worth to remark that the zero set, or vanishing set, of a polynomial ideal is the set of all points that simultaneously satisfy the homogeneous equations composing the ideal. Each one of the sub-ideals does not complete by its own the characterization of the kinematics problem since its vanishing set just refers to a portion of the feasible solutions of the starting ideal. In particular, sub-ideals denote the different types of mobility that the robotic platform is able to perform. Such conclusion turns evident if homogeneous equations that they contain are substituted into end-effector transformation matrix, which assumes a different shape for each one of the sub-ideals. A deeper investigation on every sub-ideal revealed that several types of motions are allowed by the 3-CPU architecture: 1 pure rotational working mode, already widely studied by several past works. 4 ideals pointed out the same type of mobility, characterized by 3 spurious DOFs: the end-effector can change its orientation rotating about two distinct axes and it can translate along a direction which rotates solidly with the moving platform. 6 not usable working modes: the mobilities deriving from these ideals can not be exploited by the actuation chosen for the 3-CPU manipulator, since for each one of them the end-effector is only allowed to move on a plane perpendicular to one of the cylindrical joints passing through the origin of the absolute reference frame. 4 ideals denoted the ability to move without varying its own configuration with respect to the reference frame, i.e. to perform pure translational motions. The different modes are distinguished for the different orientations of the manipulator, which are fixed in these cases. This mathematical process allows to demonstrate that it is possible to obtain a multifunctional robot by using the same 3-CPU architecture, since it proved capable of both pure rotational and pure translational motions: therefore it is relevant to investigate the possibility to switch from a type of mobility to the other. In order to be a feasible configuration for a switch between two working modes, a particular pose must belong to both the characteristic vanishing sets. As a matter of fact, such configurations should be common solutions of the forward kinematics problems of both working modes. Solutions that are common to a couple of vanishing sets are solutions of the intersection of the sets, i.e. a solution is common to a couple of ideals if it satisfies all the homogeneous polynomial equations collected in their zero sets. The number of solutions of each ideal intersection is equal to the dimension of the vanishing set of the respective basis; thus, computation of such dimensions directly provides the dimension of the space of solutions for each operational modes intersection. A particular interest is aroused by the transitions involving the pure rotational working mode: ideal denoting such mobility shares solutions with all other ideals. Therefore, an intermediate passage through this working mode makes possible, although not directly, all other transitions. This thesis also presents a workspace analysis of the spherical 3-CPU parallel machine, with particular attention to the spherical working mode since, as said, its presence effectively allows the switch between all working modes. It is known that transitions between two operational behaviours take place on singular configurations, i.e. when robot limbs can move independently from platform displacement/rotation and assume a configuration that allows a different kind of motion. Therefore, the subspaces of actuated joints space allowing such transitions must represent singularity surfaces of the manipulator and they should match the singularity regions detection carried on by means of Jacobian matrix analysis. Thus, results achieved in the first section are compared with the workspace analysis performed through the study of the manipulator Jacobian matrix. To do this, the forward position kinematics of the manipulator is briefly faced to provide the solution which is needed for the formulation of the differential kinematics. Theory of screws is then exploited for the formalization of velocity kinematics problem. Analysis of determinant of Jacobian matrix is probably the most used tool for detection of singularity surfaces. Nevertheless, it is often difficult to find the actual mathematical equation describing the borders of robot workspace in terms of actuation variables; in the specific field of PKM, this can represent an impossible task since in most cases the solution of the forward kinematics is not achievable in closed form. In the specific case of the rotational 3-CPU platform, forward kinematics solution yields 8 different configurations for each set of actuators displacements, deriving from zeroing of a polynomial of same degree; therefore, a closed form symbolical solution is actually. not achievable and it is not possible to explicitly express the singularity surfaces in terms of actuators displacements. The Jacobian analysis pointed out that no singular points are present within the tetrahedral space enclosed into the singularity surface. Hence it is reasonable to deduce that singularities denoted by switching conditions coincide with the points of workspace borders, i.e. the 4 planes, the 6 lines and the 4 points where transitions are possible respectively correspond to the faces, the edges and the vertices of the tetrahedral workspace. Therefore, a mathematical explicit description of singularity surfaces in terms of actuated joints displacements is given.

Negli ultimi anni il gruppo di ricerca in Robotica dell’Università Politecnica delle Marche, in Ancona, ha sviluppato due robot paralleli a mobilità ridotta: una delle macchine, chiamata I.Ca.Ro. è capace di eseguire moti di pura translazione, mentre la seconda, detta Sphe.I.Ro., dispone di una piattaforma mobile capace di moti di pura rotazione. Nonostante il loro opposto comportamento cinematico, entrambi i robot sono basati sulla stessa architettura di giunto 3-CPU, a meno di una inevitabile differenza nella disposizione reciproca dei giunti all’interno delle singole gambe. Il presente lavoro è indirizzato all’indagine di questo particolare comportamento manifestato dell’architettura parallela 3-CPU. Si cercherà dunque di approfondire le motivazioni per cui due tipi di mobilità diametralmente opposti possono essere eseguiti con una stessa architettura. In particolare l’attenzione viene focalizzata sulla possibilità di passaggio da una modalità operativa all’altra, che può avvenire attraverso un vera e propria riconfigurazione dell’assetto dei giunti all’interno della gamba, oppure attraverso il passaggio per particolari configurazioni, dette di transizione, o di switch, della piattaforma mobile. Un’analisi di questo tipo richiede lo studio approfondito delle equazioni di vincolo che regolano il sistema meccanico. Necessariamente, tali equazioni devono essere scritte con il fine di ridurre al minimo il numero di singolarità di rappresentazione e per questo motivo viene utilizzata una notazione propria della geometria algebrica, i.e. la rappresentazione di Study che consente di associare ad 8 parametri complessi una, ed una soltanto, posa di un corpo rigido nello spazio. Inoltre la rappresentazione di Study permette la scrittura delle equazioni di vincolo in forma polinomiale, caratteristica che può essere sfruttata per lo studio del sistema attraverso la sua suddivisione in sotto-problemi, più semplici da trattare, e descriventi ciascuno una particolare mobilità della macchina. Così facendo, viene dimostrata la possibilità dell’architettura 3-CPU ad operare secondo diversi tipi di mobilità, in funzione dell’attuale condizione di montaggio delle gambe. Anche il passaggio da una modalità operativa all’altra può essere investigata facendo ricorso alla natura polinomiale delle equazioni di vincolo. Infatti lo studio delle soluzioni comuni ai sotto-problemi sopra accennati permette di identificare delle condizioni, sotto forma di equazioni, che gli spostamenti dei giunti attuati devono rispettare affinchè la transizione avvenga. Il passaggio da un modo di lavoro all’altro non può che passare attraverso configurazioni singolari della macchina dato che le gambe devono potersi opportunamente riconfigurare. Questo consente di affermare che le condizioni matematiche di passaggio trovate coincidono con le superfici dette di singolarità del robot e che, storicamente, vengono ndividuate attraverso l’azzeramento del determinante della matrice Jacobiana, entità geometrica definita dalla cinematica di velocità della piattaforma. Al fine di operare un confronto tra i due metodi di individuazione delle superfici di singolarità, la cinematica differenziale della macchina viene formulata attraverso un approccio basato sulla teoria dei torsori cinematici, o screw theory. Il valore del determinante della matrice Jacobiana consente poi di quantificare le prestazioni della macchina all’interno dello spazio di lavoro delimitato dalle superfici in cui questo parametro si annulla. Il confronto diretto del determinante del Jacobiano con le condizioni di switch precedentemente individuate permette la formulazione matematica esatta di questi luoghi di punti. Va sottolineato che per la stessa natura del manipolatore, tale operazione non sarebbe possibile tramite il solo studio della cinematica differenziale. La cinematica di posizione del robot, infatti, non è computabile in forma chiusa nonostante tutte le 8 soluzioni ammissibili per ogni configurazione degli attuatori siano note. Ciò impedisce la formulazione matematica esatta delle superfici singolari che, invece, viene fornita in maniera semplice e completa dallo studio delle condizioni di passaggio tra mobilità diverse. In ultimo, viene formulata la dinamica di una delle possibili configurazioni studiate nel corso della trattazione. Le equazioni che costituiscono il modello vengono scritte sfruttando il principio dei lavori virtuali. Tale approccio si rivela di notevole efficienza computazionale nel caso specifico della robotica dato che un campo di spostamenti virtuali di un qualsiasi membro della macchina può essere correlato allo spostamento compiuto dai manipolatori grazie alla cinematica differenziale. Una breve analisi, dopo un’opportuna verifica, viene successivamente effettuata sulle equazioni scritte.