In the field of structural analysis of mechanisms, it is very common to work with R (revolute) joints, P (prismatic) joints, C (cylindrical) joints, etc. which are mechanical connections between adjacent elements permitting some degrees of freedom in their relative motion. Nevertheless, the idea of kinematic joint or pair has always been a more general concept than a simple mechanical coupling (Angeles, 2005). As an example, we can think about the U (universal) joint, in which the pair itself contains an intermediate moving element, the cross shaft. This is the idea that underlies the kinematic joint named Pa joint (also known as Π joint; Hervé and Sparacino, 1992), and other more complex pairs, that can be included in a further generalized concept first termed by Hervé (1999) as liaison cinématique in French or mechanical bond in English. Later on, Yu et al. (2009) renamed this type of pairs, those that can be even formed by closed-loop kinematic chains, as complex joints (CJs), group in which the Pa joint is included. In Yu et al. (2009), the authors present an in-depth state of the art of these CJs and propose a new type classification and mobility analysis for these special joints.

Regarding the Pa joint, it consists of an articulated parallelogram which permits a translational degree of freedom (dof), and it is commonly used in the design of translational parallel manipulators (Gogú, 2004). One of the first practical application of the Pa joint was in the Delta translational manipulator (Clavel, 1988) designed by Reymond Clavel for Pick & Place operations. A similar architecture was the STAR parallel manipulator, proposed by Hervé and Sparacino (1992) in which this special kinematic pair was named as Π joint. The CAPAMAN robot designed by Ceccarelli and Ottaviano (2000) at the Laboratory of Robotics and Mechatronics (Cassino) is also another practical example of a three dof spatial parallel manipulator which incorporates an articulated parallelogram in each leg. A 2-D version of the Delta robot, named the Diamond robot, which also incorporated parallelogram linkages, was designed by Huang et al. (2004) for quality inspection of rechargeable batteries. The robot named Par4 (Pierrot et al., 2009), which is an enhanced design of the prior H4 (Pierrot and Company, 1999), incorporates the Pa joint not only in the legs of the manipulator but also in the special moving platform the authors termed as the traveling plate, allowing for a rotational dof in the end-effector without the need of a telescopic chain as occurs in the Delta design. Another robot with pure translational motion incorporating the Pa joint is the Micro Finger (Arai et al., 1996), a 3 dof parallel platform intended for manipulating micrometer size objects.

We can find in the scientific literature more architectures formed by the Pa joint, such as the two Schönflies designs, the 4-RRPaR and 4-PRPaR, proposed recently by Li et al. (2013) or the translational parallel manipulator presented by Affi et al. (2004), formed by three prismatic legs and two Pa joint based passive chains that eliminate the rotational capacity of the manipulator. However, it is quite notable the limited use of double Pa joints, termed as the Pa2 joint. The kinematic Pa2 joint is composed of two intertwined articulated parallelograms which connect in parallel the two elements that the joint links together. It possesses two translational dof that result in a translational plane varying with the position. A design with a double Pa joint, named by the author as the Π2 joint, was proposed by Angeles (2004). In the proposed design, the Π2 joint consist of four in-parallel bars connecting the fixed and moving platform with universal joints. This joint permits two translational dof, where the moving platform traces a spherical translational motion with respect to the fixed platform. Their kinematic characteristics are equivalent to those of the Pa2 joint.

The first clear proposal of designs including Pa2 joints is found in the Doctoral Thesis of Oscar Salgado (2008), where two 4 dof parallel manipulators based on Pa2 joints are investigated by means of Theory of Group of Displacements. Both robots are formed by four kinematic chains, each one generating 5 dof, the connection of all kinematic chains resulting in a Schönflies motion (3 translations and 1 rotation) in the moving platform. Some works related to this Thesis can be found in Salgado et al. (2007) where a parallelogram-based 4 dof manipulator intended for aeronautical industry is presented, and in Salgado et al. (2008) in which a new topology of 3T1R fully-parallel manipulator intended for Pick and Place operations is proposed.

With the purpose of evaluating the potential of a certain design of a parallel manipulator including this type of Pa joints, one of the main steps is to assess the operational workspace the manipulator can reach. This study is included in the last part of the present paper. As proposed in Macho et al. (2008, 2013), a complete analysis of the workspace requires the study of the singularity locus, as well as the determination of the working modes and the singularity-free regions associated with each working mode, so that an enlarged workspace can be found. A general systematic procedure to obtain all the singularity-free workspace regions in parallel manipulators, so that strategies to enlarge the accessible workspace can be planned, is presented in Macho et al. (2009).

The outline of the paper is the following. Firstly, a detailed description of the kinematic characteristics and current usage of the Pa joint will be introduced, and, based on that, the double Pa joint, that is, the Pa2 joint, will be described. Next, a complete kinematic analysis of a new manipulator designed using Pa2 joints will be presented, the 3-P¯Pa2 parallel manipulator. This study includes the mobility analysis, the solution of the direct and inverse position problems, the solution of the velocity problem, the singularity analysis and the workspace computation. Finally, the main conclusions will be presented.

The Pa joint is based on a planar four-bar linkage in a parallel configuration (or articulated parallelogram). As it is known, the four-bar linkage is a one degree of freedom mechanism and in a general configuration, the coupler element has an instantaneous rotation. However, in a parallel configuration, its instantaneous center of velocity goes to infinity. In this case, the coupler element moves following a permanent circumferential translation.

This is called the Pa joint, which has been successfully used in some translational parallel manipulators. Maybe, as pointed out in the introduction, the best-known example is the Delta robot, shown in Fig. 1, which is built from 3-RRPaR kinematic chains joining in a parallel architecture the fixed frame and the moving platform. The RRPaR kinematic chain has four degrees of freedom, three translations and one rotation, that is, it is a Schönflies motion generator. One translation comes from the Pa joint, while the remaining three parallel rotation axes (r1, r2, r3 in Fig. 1) provide the final rotation and the other two translations.

One of the main principles of the structural synthesis is the fact that in a parallel manipulator the mobility of the moving platform is the intersection of mobilities of the kinematic chains. This is the reason why in the Delta robot the rotational degree of freedom is lost. Each chain allows just a rotation around the direction of its parallel revolute joints. Since such a direction is not the same in the three chains, no rotation of the moving platform is possible when all chains are assembled.

More than one Pa joint can be combined in the same chain. As an example, the Micro Finger parallel manipulator, shown in Fig. 2, is built from three chains containing each one two Pa joints serially connected. This configuration provides at each chain two translational degrees of freedom (u1, u2), that is, a plane of translations. The kinematic chains of this example contain these double Pa joints and two parallel revolute joints (r1, r2), which provide the third translation and an extra rotational degree of freedom which is lost when all chains assembled in parallel.

Finally, several Pa joints can be connected in order to build the Pa2 joint, which will be used in the translational manipulator of this work. As it can be seen in Fig. 3a, it is constituted by two identical Pa joints (aIbIcIdI and aIIbIIcIIdII), which are parallel one to the other, and there is also a third Pa in a crossed orientation, eIfIeIIfII, connecting the previous ones. The axes of the revolute joints in the two parallel Pa have the same direction, while the axes in the cross-linked Pa are perpendicular to the previous ones. This assembly has 2 dof and 8 redundant constraints.

Taking into account the geometry of the resulting architecture, two imaginary planes appear, the fixed plane πf and the coupler plane πm, which are always parallel, Fig. 3b. This means that the two dof are translational, like in the previously shown double Pa, but in this case, instead of being planar, this translation is spherical. The two parallel Pa rotate an angle α and the crossed Pa rotates angle β, Fig. 3c. The equation relating the coordinates of reference points A and B is: xB-xA2+yB-yA2+zB-zA2=L2 Or alternatively, given a reference system as shown in the figure: xB-xA=Lsin⁡βyB-yA=-Lcos⁡βsin⁡αzB-zA=Lcos⁡βcos⁡α Next, the mobility analysis of two translational parallel manipulators proposed by the authors which include the Pa2 kinematic joint will be developed.

In references Hernandez et al. (2016a, b), a preliminary kinematic analysis of the manipulator under study was presented, showing the potential this manipulator has. As it will be shown in this section and subsequent ones, we present here a reformulated kinematic analysis based on an equivalent simplified manipulator. From this new formulation, a complete workspace and joint-space analysis is accomplished by means of obtaining the set of working modes and assembly modes, the singularity-free regions associated with each of the assembly modes and the enlarged operational workspace.

In the proposed 3-P¯Pa2 manipulator, shown in Fig. 4, one translation is provided by the prismatic P joint and the remaining two ones by the Pa2 joint. The slider goes along the direction of one Cartesian axis of the global reference frame and the home position of the Pa2 is set as the pose having all links perpendicular among them, as it is shown in Figs. 3a and in 4a. The BC link of the moving platform is normal to the coupler plane of the Pa2. The dimensional parameters are just the lengths L and l. The coordinates of point C are the output variables, while the distance s from the prismatic slider to the origin is the input variable. The closure loop equation relating all parameters can be obtained in a very simple way particularizing Eq. (1) for this case. Considering a vertical leg, as shown Fig. 4b:

xC-h-02+yC-h-02+zC-0-s2=L2xC-h2+yC-h2+zC-s2=L2 As it can be seen in Fig. 5, it has been chosen a configuration for the manipulator based on three identical chains with the P sliders along the three Cartesian axes of the global reference system (the three sliding directions intersecting at the origin O).

To solve any kinematic problem, the first step is to pose the position equations system of the assembled manipulator: ec1=(xC-s1)2+(yC-h)2+(zC-h)2-L2=0ec2=(xC-h)2+(yC-s2)2+(zC-h)2-L2=0ec3=(xC-h)2+(yC-h)2+(zC-s3)2-L2=0

The system of closure loop equations of the three chains contains the input and output variables and is used to solve the position problem.

The different solutions of the inverse position problem, which are also called working modes of the manipulator, are found in an easy way because they are decoupled. Given the coordinates of the coupler point C, each chain can have independently two different positions, or values of its input variable. s1=xC±(yC-h)2+(zC-h)2-L2s2=yC±(xC-h)2+(zC-h)2-L2s3=zC±(xC-h)2+(yC-h)2-L2 Combining these two solutions for the three chains the manipulator presents a total of eight working modes. In Fig. 6, p and n denote the positive or negative sign of the square root involved in Eq. (9).

On the contrary, the direct position problem is not so decoupled. To solve this problem, some algebraic manipulation on this Eq. (8) is required, to eliminate, for example, the output variables yC and zC, and achieve the univariate polynomial in the output variable xC. A quadratic polynomial is obtained, whose coefficients depend on the input variables, s1, s2 and s3: axC2-2ah+c1xC+ah2+c1h+s1-h2b1+s2-h2+s3-h2-4L2a232=0 Where: a12=(s1-h)(s2-h)a23=(s2-h)(s3-h)a31=(s3-h)(s1-h)b1=a122+a312b2=a232+a122b3=a312+a232a=2(b1+b2+b3)=4(a122+a232+a312)c1=4(s1-h)b1

In this process, the values of the output coordinates yC and zC as functions of inputs and xC are also obtained. Once xC is obtained, the remaining output variables yield: yC=h+(s2-h)2-(s1-h)2+2(s1-h)xC2(s2-h)zC=h+(s3-h)2-(s1-h)2+2(s1-h)xC2(s3-h) Obviously, for given values of the inputs there are two solutions of the direct problem, also called assembly modes. The final solutions of the direct position problem are:

xC=h+(s1-h)b1±a23Δsa/2yC=h+(s2-h)b2±a31Δsa/2zC=h+(s3-h)b3±a12Δsa/2 Being: Δs=s1-h2b1+s2-h2b2+s3-h2b3+2s1-h2s2-h2s3-h2-L2a Kinematic problems can be solved in an easier way making use of a simplified equivalent manipulator, shown in Fig. 7. To understand better the transformation, first an intermediate schematic representation is done, Fig. 7b, where each Pa2 joint is depicted as a single bar of the same length L with spherical joints. Next, the moving platform is merged into a single point D, making null the length l, resulting in the 3-P¯SS parallel manipulator, Fig. 7c. This process implies that each bar AB is translated to its new equivalent position A′D a magnitude h along the direction of its slider, as shown in Fig. 7d.

Since the directions of bars of the 3-P¯Pa2 manipulator and those of the 3-P¯SS are always parallel, from a kinematic point of view both are fully equivalent, this is, both have the same solutions and singularities. The relations between the input and output variables of the real and the simplified manipulator are: si=ri+h,i=1,2,3xC=xD+h,yC=yD+h,zC=zD+h Then, for example, for the kinematic chain with vertical sliding direction: ((xD+h)-h)+((yD+h)-h)2+((zD+h)-(r3+h))2=L2xD2+yD2+(zD-r3)2=L2 Using the equivalent manipulator, the solutions of the direct position problem are obtained from: (xD-ri)2+yD2+zD2=L2xD2+(yD-r2)2+zD2=L2xD2+yD2+(zD-r3)2=L2 Then, the quadratic univariate polynomial in xD and the values of yD and zD are: r12r22+r22r32+r32r122xD2-2r13r22+r322xD+r14r22+r32+r22+r32-4L2r22r32=02yD=r22-r12+r12xDr22zD=r32-r12+r12xDr3 And finally: 2xD=r12(r22+r32)±r2r3Δrr12r22+r22r32+r32r122yD=r23(r32+r12)±r331Δrr12r22+r22r32+r32r122zD=r33(r12+r22)±r1r2Δrr12r22+r22r32+r32r12 Where: Δr=r14r22+r32+r24r32+r12+r34r12+r22+2r12r22r32-4L2r12r22+r22r32+r32r12

To end with the position analysis the workspace of the manipulator is presented. Considering the position equations as they have been obtained, each chain theoretically generates a cylinder parallel to the sliding direction of the prismatic joint. Then, the theoretical workspace would be the intersection of three mutually perpendicular cylinders, as shown in Fig. 8.

Nevertheless, the Pa2 joint imposes a limit in the motion range. Since the two parallel Pa cannot lie in the same plane, Fig. 9a, the actual workspace of each chain is just a half cylinder, which produces a smaller real workspace of the whole manipulator, as depicted in Fig. 9b and c.

To carry out the singularity analysis, the position equations are derived with respect to time and the velocity equations are obtained. The problem is linear in the input and output velocities, so it can be expressed in a matrix form. In this approach, the Jacobian matrices are obtained. As it can be seen, the rows of the direct Jacobian matrix Jx are the vectors AB defining the directions of the Pa2 joints. The inverse Jacobian matrix Jq is simpler because it is diagonal. The velocity problem can be expressed as: Jx⋅x˙=Jq⋅q˙ Where: x˙=vC=x˙Cy˙Cz˙Cq˙=s˙1s˙2s˙3 The direct and inverse Jacobian matrices can be obtained from the constraint equations, Eq. (8): Jx=∂ec1∂xC∂ec1∂yC∂ec1∂zC∂ec1∂xC∂ec2∂yC∂ec2∂zC∂ec1∂xC∂ec3∂yC∂ec3∂zCJq=-∂ec1∂s1∂ec1∂s2∂ec1∂s3∂ec1∂s1∂ec2∂s2∂ec2∂s3∂ec1∂s1∂ec3∂s2∂ec3∂s3 By computing the terms inside Jx and Jq, it yields: xC-s1yC-hzC-hxC-hyC-s2zC-hxC-hyC-hzC-s3x˙Cy˙Cz˙C=-xC-s1000yC-s2000zC-s3s˙1s˙2s˙3 Direct singularities occur whenever the determinant of the direct Jacobian matrix vanishes. This means that a dependence relation among the input velocities is verified. It is essential to assess these positions because the controllability of the manipulator is lost, so they must be avoided.

The determinant of the direct Jacobian matrix can be obtained as: Jx=xCs2s3+yCs3s1+zCs1s2-s1s2s3-hxCs2+xCs3+yCs3+yCs1+zCs1+zCs2+h2xC+yC+zC+s1+s2+s3-2h3Jx=xD-r1yDzDxDyD-r2zDxDyDzD-r3 But the expression is much simpler in the variables of the equivalent manipulator: Jx=xDr2r3+yDr3r1+zDr1r2-r1r2r3 Direct singularity surface in the joint-space (input variables domain) is achieved by substituting in the equation Jx=0 the obtained solutions of the direct position problem, Eq. (20). On the one hand, doing this, firstly it is verified that: xDr2r3+yDr3r1+zDr1r2=r1r2r3±Δr/2 So: Jx=0⟶r1r2r3±Δr2-r1r2r3=0⟶Δr=0 This first condition, which is a surface in the 3-Dimensional joint space Δrr1,r2,r3=0, Eq. (21), can be also expressed in the input variables of the real manipulator Δss1,s2,s3=0, Eq. (14).

On the other hand, apart from the nullity of Δr, looking at Eq. (27), it is verified that Jx=0 when any of the following conditions is satisfied: r1=r2=0⟶s1=s2=hr2=r3=0⟶s2=s3=hr3=r1=0⟶s3=s1=h These are the expressions of three lines. As it is known, it is verified that direct singularities occur where the mechanism has double solutions of the direct position problem (where the two assembly modes merge). The surface and lines are shown in Fig. 10.

In case of the proposed manipulator, considering the structure of Jx, along with the mathematical analysis made, a geometrical interpretation is possible. Direct singularities occur when the three chains are parallel to a plane, as shown in Fig. 11, or when two chains have the same direction, as shown in Fig. 12a. Obviously, this second case includes a subcase that occurs when the three legs have the same direction, as depicted in Fig. 12b.

In the singularity locus obtained, the external continuous surface Δss1,s2,s3 corresponds to the situation of three legs parallel to a same plane (in the case of the equivalent manipulator, all postures where the three bars lie in the same plane). The three lines correspond to the three possible combinations of two parallel legs. For example, the line s1=s2=h corresponds to the case in which the chains 1 and 2 have the same direction. In the case of the equivalent manipulator, this situation (r1=r2=0) implies that two bars are coincident. The intersection point of the three lines, s1=s2=s3=h, corresponds to the subcase of three legs parallel.

Singular postures in the equivalent manipulator are shown next. In the cases of three legs parallel to a same plane, Fig. 13a, or just two legs parallel, Fig. 13b, the rank of Jx decreases in one (one dof appears with all inputs blocked). In the subcase of the three legs in the same direction, Fig. 13c, the rank of Jx decreases in two (two dof appear with all inputs blocked).

The direct singularity surface in the workspace (output variables domain) can be also obtained by eliminating input variables from Jx: Jx=0⟶ΓμxD,yD,zD⋅ΓπxD,yD,zD=0 In this case, two independent surfaces, Γμ and Γπ, are obtained. The first surface is: Γμ≡xD2+yD2+zD2-L2=0 This is a sphere centered at the origin and radius L. The second surface is more complex: Γπ≡L14-3L12σ+3L10σ2-L846τ+γ+32L6στ+8L4τδ+80L2τ2-16τ2σ=0 With: σ=xD2+yD2+zD2τ=xD2yD2yD2zD2γ=xD6+yD6+zD6+3(xD4yD2+xD2yD4+yD4zD2+yD2zD4+zD4xD2+zD2xD4)δ=xD4+yD4+zD4+6(xD2yD2+yD2zD2+zD2xD2) But once obtained for the simplified equivalent manipulator, both can be easily transformed to the output variables of the real manipulator, ΓμxC,yC,zC, Fig. 14a, and ΓπxC,yC,zC, Fig. 14b. In Fig. 14c both surfaces are represented overlapped and a cross section has been given to visualize the internal part (because Γπ is inside Γμ).

All figures in this and next sections have been obtained for L=10 length units and h=5 length units.

In the current domain (output variables), Γμ corresponds to the situation where two legs i and j are parallel, i∥j⟶si=sj=h, and Γπ where the three legs are parallel to a plane. In addition, Γμ is the same surface for the three possible combinations, s1=s2=h, s2=s3=h, and s3=s1=h, and also for the subcase where the three legs have the same direction, s1=s2=s3=h. An interesting way to understand how is this possible is to start supposing a posture where the three legs are all of them parallel, and from this position make a working mode change in one chain. Doing this, that leg is not any more parallel to the others, but the remaining two legs are still parallel. This phenomenon is shown in Fig. 12.

Finally, it is possible to depict the direct singularity locus for the real size workspace (that one in Fig. 9, where real motion limits are taken into account) and to obtain also the corresponding actual portions of the theoretical joint-space. In addition, it is possible to identify the specific geometrical conditions associated with each portion as well as the correspondence among the different portions in both domains, making use of the GIM software as explained in Macho et al. (2009). All this is shown in Fig. 15, where two different views are provided in order to properly show all portions and the existing associations between them.

Inverse singularities occur when the determinant of the inverse Jacobian matrix vanishes. This happens when any chain is in a completely extended position, that is, inverse singularities define the workspace boundaries. Taking into account that: Jq=xC-s1000yC-s2000zC-s3=xD-r1000yD-r2000zD-r3 Then: Jq=(xC-s1)(yC-s2)(zC-s3)=(xD-r1)(yD-r2)(zD-r3) So: Jq=0⟶xC=s1⟶(yC-h)2+(zC-h)2=L2yC=s2⟶(xC-h)2+(zC-h)2=L2zC=s3⟶(xC-h)2+(yC-h)2=L2⟶xD=r1⟶yD2+zD2=L2yD=r2⟶xD2+zD2=L2zD=r3⟶xD2+yD2=L2 These are the same surfaces depicted in Fig. 8. In Fig. 16 is shown the case of leg 2 totally extended, that is, a posture where the working modes ⊗p⊗ and ⊗n⊗ meet (where ⊗ means p or n).

Increased mobility singularities occur when the complete Jacobian matrix is rank deficient. The complete Jacobian is defined as:

Jx˙q˙=0⟶J=Jx⋮-Jq So, for the manipulator under study: J=xC-s1yC-hzC-h⋮-(xC-s1)00xC-hyC-s2zC-h⋮0-(yC-s2)0xC-hyC-hzC-s3⋮00-(zC-s3) Now, the specific conditions to verify the rank deficiency can be found. Here it will be shown an example. Firstly, note that when two legs are parallel, the locus of coupler point positions is a sphere centered at (h,h,h) and radius L. For legs 1 and 2 being parallel, s1=s2=h. Over this surface, defined in the three output variables, the values of two variables can be freely chosen. So, it is possible to set the posture where xC=yC=h. For these values, from the positions equations, Eq. (8), are obtained the values of zC and s3. Then, the posture of the manipulator, as well as the Jacobian matrix are shown in Fig. 17. It is obvious that the rank of this matrix decreases from 3 to 2.

The direct singularity locus for the real size workspace was depicted in Fig. 15. This surface comprises all existing working modes and this means that it cannot be directly used as it has been obtained. The problem is that each point of such a surface, which corresponds to a position of the coupler point C, is, in fact, compatible with as many different postures of the manipulator as existing working modes, but just one of those working modes will be in a real direct singularity. In other words, each working mode has its specific direct singularities, and the singularity locus that has been obtained analytically is the overlap of all of them.

Then, it is necessary to identify the portions of the singularity surface that are specifically associated with each working mode, that is, it is necessary to distribute the whole singularity locus among the eight existing working modes, evaluating the value of Jq at each point of the singularity surface for all existing working modes, as explained in Salgado et al. (2008). The result of this process is shown in Fig. 18, where in order to gain clarity, instead of the real manipulator, the schematic kinematic chains have been represented. The common situation for a parallel manipulator is having different singularities for each working mode, but in this case, due to the phenomenon previously described (the singularity surface when three or two legs are parallel is the same) there are four working modes sharing the same singularities.

Now, once the specific direct singularities of each working mode have been identified, the singularity-free regions in the workspace can be traced. For each working mode, the theoretical workspace is completely crossed by the singular surface and is divided into adjacent regions associated with the existing assembly modes. Both assembly modes can be identified as ⊕ and ⊖ because each one is associated with one sign of Jx (and separated by positions where Jx=0). For each working mode, these singularity-free regions are considered the operational workspaces. In Fig. 19, the safe regions for two working modes (ppp and npp) are shown, both for a same assembly mode (⊕).

Although inverse singularities are positions where the platform mobility is restrained, because a dependence relation among output velocities is satisfied, they are not a problem from the actuators control point of view. They can be reached during the motion and, in fact, communicate the regions associated with the different working modes (for the same assembly mode), so they can be used to enlarge the operational workspace. When a workspace boundary (where two solutions of the inverse position problem merge) is reached, the manipulator can change its working mode and make a transition from one workspace region to another, maintaining the same assembly mode, Macho et al. (2008, 2009, 2013).

Taking advantage of the possibility of carrying out this type of transitions, all regions associated with the same assembly mode, for all working modes, can be communicated and therefore the operational workspace can be enlarged. In Fig. 20 is shown the enlarged operational workspaces associated with one of the two existing assembly modes.

In this paper, the potentiality of the use of the Pa2 pair as an adequate kinematic chain to design translational parallel manipulators is studied. After a deep kinematic analysis, it is stated that the position and velocity equations are simpler than those of chains based on revolute joints to obtain translational motion patterns. Additionally, the architecture based on the Pa2 pair, is more robust than the designs based on two consecutive floating prismatic joints avoiding the problems of galling and blocking that can appear in the passive prismatic joints. Also, the Pa2 pair includes a structure based on four parallel bars structure that provides the manipulator with high stiffness and accuracy.

From a practical approach, the Pa2 pair is used in the design of a 3-P¯Pa2 translational parallel manipulator which is completely kinematically characterized by the solving of the position and velocity problem, the singularity analysis and the workspace computation. The results are obtained in terms of simplicity of the architecture and the position and velocity equations. The possibility of enlarging the operational workspace confirms the interest of the proposed manipulator design implementing the studied Pa2 pairs.

Given the potentiality of this manipulator, the design of 3-P¯Pa2 could be enhanced regarding the dimensional synthesis and optimization process, with the purpose of developing a real prototype in which also constructive considerations, reliability, stiffness and easiness of assembly should be considered. Additionally, by studying the possibilities of eliminating redundant constraints of the Pa2 pair, non-redundant designs could be synthesized to reduce as much as possible the influence of the manufacturing tolerances in the kinematic and dynamic behavior of the translational manipulator.