#### Abstract

This paper investigates the mobility of a family of fully translational parallel manipulators based on screw system analysis by identifying the common constraint and redundant constraints, providing a case study of this approach. The paper presents the branch motion-screws for the 3-*R**C*-Y parallel manipulator, the 3-*RCC*-Y (or 3-*R**RC*-Y) parallel manipulator, and a newly proposed 3-*R**C*-T parallel manipulator. Then the paper determines the sets of platform constraint-screws for each of these three manipulators. The constraints exerted on the platforms of the 3-*R**C* architectures and the 3-*RCC*-Y manipulators are analyzed using the screw system approach and have been identified as couples. A similarity has been identified with the axes of couples: they are perpendicular to the *R* joint axes, but in the former the axes are coplanar with the base and in the latter the axes are perpendicular to the limb. The remaining couples act about the axis that is normal to the base. The motion-screw system and constraint-screw system analysis leads to the insightful understanding of the mobility of the platform that is then obtained by determining the reciprocal screws to the platform constraint screw sets, resulting in three independent instantaneous translational degrees-of-freedom. To validate the mobility analysis of the three parallel manipulators, the paper includes motion simulations which use a commercially available kinematics software.

#### 1. Introduction

The mobility, or degrees-of-freedom, of a mechanism is commonly defined as the number of independent coordinates that is required to define the configuration of a kinematic chain or mechanism [1]. The mobility of a particular mechanism is one of the most fundamental issues in the kinematic analysis and synthesis of mechanisms. One of the first contributions to the study of mobility was the formula presented by Chebychev [2] for calculating the independent variables in a mechanism. For a comprehensive historical survey of the studies of the mobility of mechanisms over the past 150 years see Gogu [3]. The most commonly used approach to determine the number of degrees-of-freedom of a mechanism is the Grübler-Kutzbach mobility criterion [4]. However, it is well known that this criterion does not provide the correct result for the mobility of a certain family of mechanisms (commonly referred to as overconstrained mechanisms). The primary reason for this is that the Grübler-Kutzbach mobility criterion does not consider the geometry of the mechanism [5–7] that redundant constraints are taken as effective constraints. An additional limitation is that the Grübler-Kutzbach criterion does not indicate if the identified degree-of-freedom (DOF) is a translational DOF or a rotational DOF.

The mobility of overconstrained mechanisms has been investigated by several different techniques; for example, Angeles and Gosselin [8] used the dimension of the nullspace of the Jacobian matrix. Angeles [9], Fanghella and Galletti [10, 11], and Hervé [12] used group theory to study the mobility properties of single-loop kinematic chains. Rico-Martinez and Ravani [13] have extended the work presented in [10–12] by developing a single equation that can be used to determine the mobility of kinematic chains using group theory. Then Rico, et al. [14] translated the mobility criterion presented in [13] from finite kinematics, based on group theory, into infinitesimal kinematics, based on Lie algebra. Screw theory has also been used by several researchers to investigate the mobility of mechanisms, in general, and parallel manipulators, in particular [15]. Huang and Li [16] used this geometric approach to determine the mobility and kinematic properties of lower mobility parallel manipulators using the constraint screw systems of the manipulators. Zhao et al. [17] proposed the concept of configuration degrees-of-freedom (CDOF) for a study of the mobility of spatial mechanisms. Then Zhao et al. [18] proposed a programmable algorithm to investigate the CDOF of a spatial parallel manipulator with reciprocal-screw theory. Dai et al. [19] made an important contribution to this field by investigating the screw systems of parallel manipulators and their interrelationships, relating the screw systems to motion and constraints. They identified the constraints as (i) platform, (ii) mechanism, and (iii) redundant constraints. This work resulted in a new approach to the mobility analysis of parallel manipulators based on the decomposition of motion- and constraint-screw systems.

This paper adopts the screw-based method proposed by Dai et al. [19], to study the mobility of three fully translational parallel manipulators, namely, (i) the 3-*R PC*-Y parallel manipulator (where

*R*denotes a revolute joint,

*P*denotes a prismatic joint,

*C*denotes a cylindric joint, the underline denotes the active joint, and Y denotes the configuration of the three identical legs equally distributed on the base, referred to here as a star configuration); (ii) the 3-

*RCC*-Y (or 3-

*R*-Y) parallel manipulator; and (iii) a newly proposed 3-

__P__RC*R*-T parallel manipulator (where T denotes the configuration of the legs on the base). The 3-

__P__C*R*-Y parallel manipulator was first proposed by Callegari and Tarantini [20] and the 3-

__P__C*RCC*-Y parallel manipulator was first proposed by Callegari and Marzetti [21]. However, the 3-

*RPC*-T parallel manipulator is a new architecture proposed by the authors of this paper. The three parallel manipulators are illustrated in Figures 1, 2, and 3. Callegari and Tarantini studied the mobility of the 3

*-R*-Y parallel manipulator using the general Grübler-Kutzbach mobility criterion and found that the platform is overconstrained. Hervé and Sparacino [22] showed that the topology of this manipulator complies with the conditions for full Cartesian motion. In other words, the platform has three translational degrees-of-freedom and cannot undergo a rotation [17]. From an inspection of the Jacobian matrix of the 3-

__P__C*RCC*-Y parallel manipulator, Hervé and Sparacino showed that this manipulator also complies with the conditions for full Cartesian motion (the platform has three translational degrees-of-freedom). The approach presented in this paper supports these results and is then used to study the mobility of the newly proposed 3-

*R*-T parallel manipulator. Finally, the results are validated by the use of motion simulations which make use of a commercially available kinematics software.

__P__CThe paper is arranged as follows. Sections 2 and 3 present the mobility analysis of the 3-*R PC*-Y manipulator and the 3-

*RCC*-Y manipulator, respectively. Then Section 4 presents the mobility analysis of a newly proposed 3-

*R*-T parallel manipulator. Section 5 includes the motion simulations of all three parallel manipulators. These motion simulations validate the results that were obtained from the mobility analysis presented in Sections 2, 3, and 4. Finally, Section 6 presents some important conclusions and suggestions for future research, in general, and on the newly proposed 3-

__P__C*R*-T parallel manipulator, in particular.

__P__C#### 2. Mobility Analysis of the 3-*R*__P__C-Y Parallel Manipulator

__P__C

The 3-*R PC*-Y parallel manipulator connects the platform to the base by means of three legs which are

*RPC*kinematic chains. The kinematic chain for leg

*i*where the subscript

*ij*denotes joint on the leg is shown in Figure 4

*.*Consider that the global Cartesian reference frame G (

*X*,

*Y*, and

*Z*) is placed at the origin

*O*, while a Cartesian reference frame H (

*U*,

*V*, and

*W*) is located at the centroid of the platform

*E*. The

*R*joint connected to the base is denoted ; the

*P*joint is denoted as ; and the

*C*joint is composed of the

*R*and

*P*joints , and , respectively. Consider that the

*R*joint axis

*s*_{i1}is set to be collinear with vector (representing the position of the

*R*joint), the

*P*joint axis

**s**

_{i2}is collinear with vector (denoting the position of the leg

*i*), and the

*C*joint axes

*s*_{i3}and

**s**

_{i4}are set to be collinear with vector (which represents the stroke of the

*C*joint). If the platform is initially assembled parallel to the base, then the joint axes

**s**

_{i1},

**s**

_{i3}, and

**s**

_{i4}are invariantly parallel to each other and perpendicular to

**s**

_{i2}. Then the 3-

*R*-Y parallel manipulator is obtained when the three identical legs are equally distributed on the base (, , and ), resulting in a star configuration (denoted here as Y) of the

__P__C*R*joints on the base. Likewise, the

*C*joints are equally distributed on the platform (where , , and ).

Since the *P* joints are selected as the active joints, then each leg is comprised of a system of four motion-screws which can be written as
where
Performing the vector operations in (1) and (2) results in the general branch motion-screw system for leg *i*; that is,
Then substituting , , and into (3), the general branch motion-screw systems for legs 1, 2, and 3, respectively, are

Since the general branch motion-screw system for leg is a four-screw system and their screws are independent (see (3), (4), (5), and (6)), then the branch constraint-screw system for leg *i* is formed by two independent reciprocal screws; that is,
Note that the reciprocal screws represent couples whose axes are coplanar and perpendicular to **s**_{i1}, while the reciprocal screws represent couples acting about the *Z*-axis. The branch constraint-screw systems for legs 1, 2, and 3 can be written, respectively, as

Note that (8) provide six constraints on the platform, leading to the conclusion that this manipulator will have zero DOF. This conclusion is consistent with the observation made by Callegari and Tarantini [20] and Callegari and Marzetti [21] concerning the overconstrained condition of this parallel manipulator. Also, note that the reciprocal screws , , and in these three equations form a common constraint that exerts the same constraint and reduces the order of the screw system in the platform.

The platform constraint-screw system can be expressed in terms of the three nonunique sets of independent reciprocal screws in (8); that is,
Therefore, the platform motion-screw system can be written from the reciprocal screws in (9); that is,
This equation shows that the *R PC*-Y parallel manipulator has three translational DOF which are along the

*X-*,

*Y-*, and

*Z*-axes. This result is consistent with the mobility predicted by Callegari and Marzetti [21].

#### 3. Mobility Analysis of the 3-*RCC*-Y Parallel Manipulator

When an additional rotational DOF (denoted as joint ) is given to the *P* joints of the *R PC*-Y parallel manipulator, the result is a

*RPRRP*kinematic chain. Consider that the joint axis of this chain is set to be collinear with vector , the and joint axes are collinear with vector , and the and joint axes are collinear with vector . For this architecture, three identical legs are equally distributed on the base (, , and ), while the

*C*joints (i.e., joints and ) are equally distributed on the platform (, , and ). The joints are selected as active joints. The resulting mechanism is a 3-

*R*-Y (or 3-

__P__RC*RCC*-Y) parallel manipulator. Leg

*i*of this manipulator, which is comprised of a system of five screws, is shown in Figure 5. For convenience, the joint screws are expressed in the global frame G (

*X*,

*Y*, and

*Z*) and will be expressed relative to the origin

*O*. For the sake of convenience, the platform is initially assembled parallel to the base. Note that for this architecture the joint axes

**s**

_{i1},

**s**

_{i2},

**s**

_{i3}, and

*s*_{i4}are identical to their corresponding joint axes of the 3-

*R*-Y parallel manipulator. Therefore, the joint screws , , , and for the 3-

__P__C*RCC*-Y parallel manipulator are as given by (1). The remaining joint screw can be written as Substituting (2) into this equation and performing the vector operations, the joint screw can be written as Therefore, the branch motion-screw systems for the manipulator can be written asSince each branch motion-screw system is composed of five independent screws, then the branch constraint-screw is given by the reciprocal screws The constraints exerted by these reciprocal screws represent couples whose axes are orthogonal to the vectors and .

The platform constraint-screw system consists of the set of three independent reciprocal screws (namely, the constraint-screws , , and ); that is,
The platform motion-screw system can be obtained by determining the reciprocal screws to platform constraint-screw system given by (15). The result is a screw system identical to (10). This indicates that the platform has three DOF which are translations along the *X-*, *Y-*, and *Z*-axes. It is also worth noting that this manipulator has singular configurations in the center of the workspace, leading to direct kinematics singularities when all three legs are perpendicular to the base. These conclusions and observations are consistent with the work of Callegari and Marzetti [21].

#### 4. Mobility Analysis of the New 3-*RPC*-T Parallel Manipulator

The *RPC* kinematic chain that was described in Section 2 for the 3-*RPC*-Y architecture (see Figure 4), is used in this section to obtain a new fully translational parallel manipulator, namely, the 3-*R PC*-T parallel manipulator. The design consists of three identical

*RPC*kinematic chains distributed on the base as follows: , , and . In a similar manner, the joints are distributed on the platform by setting , , and . The configuration of the three identical legs equally distributed on the base results in a T configuration of the joints, and hence the name, the 3-

*RPC*-T parallel manipulator. Note that the joint axes and are collinear and coplanar with the

*XY*plane, while the joint axes , , , and are collinear and coplanar with the

*UV*plane. Due to the geometry of the joint axes, legs 1 and 3 provide a synchronous motion. This section considers the selection of joints , , and as a nonunique combination of active joints defined for the manipulator.

Since the kinematic chain of the 3-*RPC*-T parallel manipulator is identical to the kinematic chain of the *R PC*-Y parallel manipulator, then the general branch motion-screw system for leg

*i*is given by (3). Substituting , , and into this equation, the branch motion-screw systems for legs 1, 2, and 3, respectively, are Also, the branch constraint-screw system for leg

*i*is given by (7). Substituting , , and into this equation, the branch constraint-screw system for legs 1, 2, and 3, respectively, are Equations (17) represent physical constraints exerted by each kinematic chain on the platform. The constraints can be interpreted geometrically as (i) couples whose axes are perpendicular to

*s*_{i1}and lie on the

*XY*plane are presented by reciprocal screws and that provide two constraints and by reciprocal screw that is considered as providing a redundant constraint and (ii) couples acting about the

*Z*-axis are represented by reciprocal screws , , and that provide a common constraint that any two of these screws can be considered redundant common constraints.

From the nonunique set of independent reciprocal screws , , and , the platform constraint-screw system can be written as
Therefore, the platform motion-screw system can be obtained from the reciprocal screws of in (18). The result is a screw system which is identical to (10), indicating that the platform of the 3-*RPC*-T parallel manipulator has three translational DOF, that is, translations along the *X-*, *Y-*, and *Z*-axes.

#### 5. Motion Simulations

To verify the results of the mobility analysis that was performed in Sections 2, 3, and 4, this section presents CAD models of the 3**-***R PC*-Y parallel manipulator, the 3-

*RCC*-Y parallel manipulator, and the newly proposed 3

**-**

*R*-T parallel manipulator. The three models are subjected to motion simulations. The parameters that are used in the simulations, such as the total simulation time, the velocity of the active joints, and the dimensional parameters of the three manipulators, are presented in Tables 1, 2, and 3. The tables include the simulation results of the initial and final poses of the platforms where , , and are the Cartesian coordinates of the platform centroid

__P__C*E*and , , and are the projection angles (that express the orientation of the platform) about the

*X-*,

*Y-*, and

*Z*-axes, respectively. Figures 6(a), 6(c), and 6(e) show the simulation results for the platform position, velocity, and acceleration of the 3-

*R*-Y parallel manipulator. Figures 6(b), 6(d), and 6(f) show the simulation results for the platform position, velocity, and acceleration of the 3-

__P__C*RCC*-Y parallel manipulator. Finally, Figures 7(a), 7(b), and 7(c) show the simulation results for the platform position, velocity, and acceleration of the 3-

*RPC*-T parallel manipulator. Figure 7(d) shows the results of the projected angles of the platform. Note that for all of these figures the

*X-*,

*Y-*, and

*Z*-axes components are shown in solid, dashed, and dotted plots, respectively.

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

**(a)**

**(b)**

**(c)**

**(d)**

According to (10), the platform of the 3-*R PC*-Y parallel manipulator has three translational degrees-of-freedom (that is, full Cartesian motion). Table 1 presents the simulation parameters used for obtaining the positioning profile of the platform. From the position, velocity, and acceleration of centroid (see Figures 6(a), 6(c), and 6(e)), the platform can occupy an arbitrary position on any axis of the global frame. Also, the results of the projected angles of the platform (see Figure 7(d)) indicate that there is no rotation in any of the axes of the global frame.

According to the platform motion-screw system that is identical to (10), the platform of the 3-*RCC*-Y parallel mechanism also has three translational degrees-of-freedom (that is, full Cartesian motion). Table 2 presents the simulation parameters used for obtaining the positioning profile of the platform as shown in Figure 6(b). The plots of the velocity and acceleration of the platform are presented in Figures 6(d) and 6(f). Note that the projected angles of the platforms of the 3-*R PC*-Y and 3-

*RCC*-Y parallel manipulators are identical; see Figure 7(d). The simulations prove that this parallel manipulator has fully translational motion.

According to the platform motion-screw system from (10), the newly proposed 3-*RPC*-T parallel manipulator (see Section 4) has three translational degrees-of-freedom. The mobility of this manipulator is verified with a motion simulation using the parameters listed in Table 3. Recall that for this architecture the joints , , and are selected as actuators. According to the simulation plots for the position, velocity, and acceleration shown in Figures 7(a)–7(c), the platform is able to position itself along any of the three Cartesian axes of the global frame. The simulation results for the projection angles, angular velocity, and angular acceleration of the platform are shown in Figure 7(d). This figure shows that the platform is parallel to the base for all positions of the manipulator.

#### 6. Conclusions

This paper presented a mobility analysis of the 3-*R PC*-Y parallel manipulator, the 3-

*RCC*-Y parallel manipulator, and a newly proposed 3-

*RPC*-T parallel manipulator. The analysis was based on screw theory which provided geometric insight into the investigation. The paper obtained the branch motion-screws for all three architectures and then identified the platform constraint-screw systems. The results showed that all

*RPC*-based architectures displayed the same constraints, namely, (i) couples whose axes are perpendicular to the

*R*joint axes and coplanar with the base and (ii) couples acting about the normal axis to the base. The constraints of the 3-

*RCC*-Y parallel manipulator are couples whose axes are orthogonal to the

*R*joint axis and the leg of each kinematic chain. The mobility of the platform is then obtained by determining the reciprocal screws to the platform constraint-screw sets and the platforms are identified to have three instantaneous independent translational DOF. The results of the mobility analysis were substantiated by using CAD models and motion simulations. The simulations showed that the platforms could translate along any axis of the global frame, while there was no rotation of the platform.

#### Acknowledgment

E. Rodriguez-Leal acknowledges the financial support from Tecnologico de Monterrey and CONACYT.