Spherical parallel manipulators (SPMs) have a great potential for industrial applications of robot wrists, camera-orientating devices, and even sensors because of their special structure. However, increasing with the number of links, the kinematics analysis of the complex SPMs is formidable. The main contribution of this paper is to present a kind of 2 degree-of-freedom (DOF) seven-bar SPM containing two five-bar spherical loops, which has the advantages of high reaction speed, accuracy rating, and rigidity. And based on the unusual actuated choices and symmetrical loop structure, an approach is provided to identify singularities and branches of this kind of 2 DOF seven-bar SPM according to three following steps. Firstly, loop equations of the two five-bar spherical loops, which include all the kinematic characteristics of this SPM, are established with joint rotation and side rotation. Secondly, branch graphs are obtained by Maple based on the discriminants of loop equations and the concept of joint rotation space (JRS). Then, singularities are directly determined by the singular boundaries of the branch graphs, and branches are easily identified by the overlapping areas of JRS of two five-bar spherical loops. Finally, this paper distinguishes two types of branches of this SPM according to whether branch points exist to decouple the kinematics, which can be used for different performance applications. The proposed method is visual and offers geometric insights into understanding the formation of mobility using branch graphs. At the end of this paper, two examples are employed to illustrate the proposed method.

Mobility identification is a basic problem in linkage analysis and synthesis, which includes branch analysis, sub-branch analysis, the full rotatability problem, and order of motion (Ting, 1989, 1993; Liu and Ting, 1991, 1992). Singularities are the core of the full rotatability problem, which represent the singular configurations of the manipulator. At these configurations, the manipulator may lose control momentarily and has zero mechanical advantage (Gosselin and Angeles, 1990; Yan and Wu, 1989; Hunt, 1978). Since singularities are commonly harmful and even damage the manipulator, it is necessary to identify all singularities of the spherical parallel manipulator (SPM) during the design stage.

Branches (or circuits) (Chase and Mirth, 1993; Davis and Chase, 1994; Mirth, 1995; Ting and Dou, 1996; Ting, 1994) are the most fundamental problem among the various aspects of mobility issues. Generally speaking, branches should be identified and rectified first. A branch is a gather of manipulator configurations, referring to the continuity motion of the manipulator. And a manipulator configuration cannot transform from one branch to another without disconnecting the manipulator. Branches are determined by the links' size of the spherical loops and are irrelevant to the input conditions (Ting and Dou, 1996; Ting, 1994; Ting et al., 2008, 2010).

A spherical manipulator has a special structure whose rotation axes of all links intersect at the center of the sphere. In some senses, a planar manipulator can also be considered to be a special spherical manipulator whose common center point of the rotation joints is at infinity. In comparison with the majority of planar parallel manipulators, spherical parallel manipulators have the advantages of large workspace, large payload, large stiffness, low inertia, and high dynamic performance when they have the same number of links. Therefore, they have attracted a considerable amount of research. According to the Burmester problem, Ruth and McCarthy (1999) presented a computer-aided design software system that is suitable for spherical four-bar linkages. Bai et al. (2009) used four-bar spherical chains to establish the input–output (I/O) equations of spherical four-bar linkages for the forward displacement relationship of the spherical parallel robots. Sun et al. (2018) presented a wavelet feature parameter method for the non-periodic design requirements. By establishing a mathematical model for the coupler curve of the spherical four-bar linkage, the geometrical characteristics of the open path are described to calculate installation positions of the desired linkage. For spherical six-bar path generation linkage, Yang and Xu (2005) proposed a synthesis method to make the linkage into some link groups, and the constraint conditions and object functions are given. An approach was proposed by Sancisi, Parenticastelli, and Baldisserri (2017) to analyze the human ankle joint with the spherical model. Di Gregorio (2018) developed an analytical method for the singularity analysis and exhaustive enumeration of the singularity conditions in single degree-of-freedom (DOF) spherical mechanisms. Owing to the analytical method, the generated systems of equations are used to find the singularities of a given mechanism and to synthesize mechanisms which have specific requirements for the singularities. Using the construction characteristics of the spherical six-bar linkage, Hernandez et al. (2006) designed a new gearless robotic pitch-roll wrist which consists of spherical cam rollers and spherical Stephenson linkages. Driven by roller-carrying disks, the whole system works as a differential mechanism. Wampler (2004) employed rotation matrices or equations to locate the displacement analysis of spherical linkages up to three loops. The solution method is a modification of Sylvester's elimination method. Nie and Wang (2018) solved the branch problem of the spherical six-bar linkage based on the loop equations. Duan et al. (2018) proposed a new 2 DOF spherical parallel manipulator which can be applied for an orientation device, or a torque sensor, and they also carried out a typical motion plan, simulations, and actuating torque needed for these motions by employing derived inverse dynamic equations. Danaei and Arian (2018) analyzed the dynamic modeling and inertial parameter determination of a general 5R 2 DOF spherical parallel manipulator. According to the proposed eliminating and grouping operation, the computation time is reduced by 37 %. For the tasks on a mini-/microscale, Palpacelli et al. (2018) proposed a sensitivity analysis for an error model of a mini 2 DOF five-bar spherical parallel manipulator to geometrically calibrate its kinematics. Wu and Bai (2019) redesigned a 3-RRR spherical parallel manipulator of co-axis input with reconfigurability and investigated the workspace, dexterity, and singularity of the new manipulator. Elgolli et al. (2019) presented a method for the dynamic problem of a 3-RRR spherical parallel manipulator using the Kane method and D'Alembert's principle. The obtained model can also be used for sensitivity analysis and a dimensional optimization to ensure a constant dynamic transmission over the entire desired workspace. Based on the concept of an equivalent spherical mechanism, Boscariol et al. (2017) proposed a method to investigate the direct kinematic problem of a tunnel digging machine, which reduces a lot of computation time compared to the traditional numerical solution. Nelson et al. (2018) provided a redundant serial spherical linkage to avoid collision with the patient and to produce good kinematics and workspace. Bai et al. (2019) reviewed the state of the art of spherical-motion-generator kinematics, dynamics, design optimization, and novel spherical mechanisms with emerging applications and discussed some new research problems and future developments. As can be seen from the discussion above, the research mainly focuses on simple SPMs such as spherical four-bar linkage (Ruth and McCarthy, 1999; Sun et al., 2018), spherical six-bar linkage (Yang and Xu, 2005), and 2 DOF five-bar SPMs (Duan et al., 2018). But for complex SPMs, such as the 2 DOF seven-bar SPMs, although they have the advantages of more types of topologies and higher rigidity compared to 2 DOF five-bar SPMs, there still is limited literature available. The reason for this is that the loop equations of complex SPMs are difficult to solve. With the number of links increasing, too many unknown variables are brought into the loop equations and cannot be eliminated, contrasting with simple SPMs, resulting in no solutions. However, in this study, the unusual actuated choices and symmetric loop structure of this kind of SPM make the loop equations solvable. And the roots can be obtained using the discriminant method. In other words, the kinematics characteristics of these SPMs can be studied. This kind of 2 DOF seven-bar SPM may provide another potential choice for industry application such as cardans (Kong and Gosselin, 2004), robot wrists (Gosselin and Hamel, 1994), camera-orientating devices (Li and Payandeh, 2002), and even sensors (Duan et al., 2018). The proposed method may also become a powerful design aid when the manipulation has singularity and branch requirements (Sect. 5) during the design stage. It gives a clear graphical interpretation and is also valid for existing simple SPMs.

The motivation of this paper is to study 2 DOF seven-bar SPMs to attract more research about complex SPMs. In addition, the identification of singularities and branches of the 2 DOF seven-bar SPM also provides a theoretical basis for the industrial application of this kind of SPM. Based on loop equations, singular configurations are located with the discriminants of two five-bar spherical loop equations. Then, using joint rotation space (JRS; Ting, 2008), the branch graph, which includes all the kinematic information of the 2 DOF seven-bar SPM, is obtained to identify the branches.

The first contribution of this paper is to provide a kind of 2 DOF seven-bar SPM with two spherical five-bar loops, which gives another potential choice for industry application. Compared to 2 DOF five-bar SPMs, this kind of SPM has higher rigidity. The second one is first to succeed in identifying singularities and branches of the 2 DOF seven-bar SPM, utilizing the unusual actuated choices and symmetric loop structure. And the beauty of it is that the unusual actuated choices and symmetric loop structure not only enable complex loop equations to be solved but also bring many other advantages to the SPM. Owing to the unusual actuated choices, the SPM has the advantage of high reaction speed. Meanwhile, only containing two five-bar spherical loops, this kind of 2 DOF seven-bar SPM, a symmetric loop manipulator, has several merits, such as easy control, compared to other planar linkages or SPMs. The third one is to use a branch graph to explain the kinetic characteristics of the SPM, which provides geometric insights into understanding the mobility information instead of a complex formula set. Branch graphs are not the workspace but the input joint rotation ranges which can be effectively utilized when the SPMs are required for a specific input range or branch choice (discussed in Sect. 5). The fourth one is to distinguish the branches of 2 DOF seven-bar SPMs as two types according to whether branch points exist in the corresponding branch graph, which offers convenience when the manipulator faces decoupling motion demand (discussed in Sect. 6).

This paper is organized as follows. In Sect. 2, the 2 DOF seven-bar SPM is presented, and the input condition is decided. In Sect. 3, the loop equations of the two five-bar spherical loops are established based on the concepts of joint rotation and side rotation. Then, the singularities are obtained by saving the discriminants of the two five-bar spherical loop equations in Sect. 4. Subsequently, two types of branches of the 2 DOF seven-bar SPM are identified according to joint rotation space and branch points in Sect. 5. Finally, two examples are illustrated for the validity of the proposed method. And conclusions are presented at the end of this paper.

Planar parallel manipulators (PPMs) are a special case of spherical parallel
manipulator. Similar to the 2 DOF seven-bar PPM, the 2 DOF seven-bar SPM has
three different topologies (Duffy, 1980). In this paper, this kind of 2 DOF
seven-bar SPM, which contains two spherical five-bar loops (loops
A

The 2 DOF seven-bar spherical parallel manipulator with only

As is known, different input conditions lead to different mechanism motion.
For 2 DOF SPM, the normal input choices are the two joints on the static
frame (such as the joints A

Two kinds of input joints and corresponding motion
simplification diagrams.

Loop equations are a common tool to analyze the kinematics of planar manipulators, which has been proved in Ting (1993), Ting and Dou (1996), and Wang et al. (2010). For a certain planar manipulator, the loop equations contain all the kinematic characteristic information, which includes the connections of the links, the input–output relationship, and the transient motion. As discussed in the Introduction, planar manipulators are special spherical manipulators. Therefore, loop equations can also be utilized to analyze the kinematics of SPMs. Two steps to establish corresponding loop equations are listed as follows. And the used symbols and corresponding explanations are shown in Table 1.

Joint rotation and side rotation established in a spherical
four-bar manipulator with only

According to the foregoing setting, considering

Symbols and explanations.

There are two spherical five-bar loops in the 2 DOF seven-bar SPM (Fig. 1): the spherical five-bar loop A

Loop A

Loop C

According to Eqs. (4) and (5), (6) and (7) can be rewritten as Eqs. (8) and (9):

Note that the exterior angles

Bringing the known parameters (i.e.,

Firstly, the equations cos

Secondly, since the SPM should be assembled, that is, the loop equations can
be established, the condition for the SPM to move is the discriminants

The discriminant method is valid for singularity analysis, which had been proved in Wang et al. (2010). For the 2 DOF seven-bar SPM (Fig. 1), the singularity problem can also be solved based on the discriminants of two spherical five-bar loop equations.

For Eqs. (10) and (11), if

Parameters of spherical parallel manipulation in Fig. 4.

Singular boundaries of two spherical five-bar loops.

As discussed above, singularities of the 2 DOF seven-bar SPM occur when
the SPM reaches valid singular boundaries. According to Eqs. (12) and
(13) (

Singularity points.

After the values of the variables

Several configurations of the SPM.

For convenience, the following steps are summarized for the singularity
analysis of this kind of 2 DOF seven-bar SPM.

A branch is an assembly configuration area in which the spherical manipulator can move continuously without meeting singularity points. The branch problem is strictly affected by the kinematic chain and is irrelevant to the input or output conditions (Ting, 1994, Ting and Dou, 1996), which means when the size of the links of the SPM is determined, the branches of this SPM are decided no matter where the input angles and output angles are chosen. Therefore, in order to meet the requirements of the input range and decoupling motion, the branches should be identified during the design stage. A branch usually contains three parts: a valid joint rotation space, a branch point, and a branch curve. The valid joint rotation space is the inner area of the branch. And the branch points and branch curves constitute the boundaries of the branch. The corresponding concepts are explained in detail in Sect. 5.1. Last but not least, the branches are not the workspace of the SPM but the effective joint rotation range which decides whether the SPM is assembled.

The joint rotation space (JRS) is a region that represents the maximum possible
configuration space of the manipulators (Ting, 1994, Ting and Dou, 1996). The JRS is not
the workspace and working modes of the robot but the input ranges, which is
useful when the manipulator has input and security requirements. It can be
obtained by discriminant equations such as Eqs. (12) and (13). The JRS is
composed of the singular boundaries (

The parameters for Example 1.

The parameters for Example 2.

To consider whether the branch points exist, the branches of the 2 DOF
seven-bar SPM can be classified into two types as follows.

Branch points in Example 2.

It is worth noting that Fig. 6a is also regarded as a circuit problem, and Fig. 6b is still a branch problem in Norton (2004).

As a result of the discussion above, the following steps for branch identification
of the 2 DOF seven-bar SPM can be summarized.

In short, different types of branches are suited to different performance applications. When the SPM needs a decoupled motion, the Type I branch is the right choice. When a coupling motion is required, the Type II branch is applicable. As for the input requirements, it is obvious to see that different branches have a different corresponding input range; suitable branches can be selected according to design demand.

Using the given parameters, the branches of the 2 DOF seven-bar SPM are
shown in Fig. 6a using Maple. It is obvious to find that the branches
are Type I. The process of identification is shown as follows.

Based on the given dimensions above, the branches of the 2 DOF seven-bar
SPM are shown in Fig. 6b with Maple. Since the branch points exist in this
case, the branches are Type II. The following steps are used to
identify the branches.

According to the concept of joint rotation space and the discriminants of two five-bar spherical loops, this paper utilizes the branch graph, which depends on loop equations to identify singularities and branches of a kind of 2 DOF seven-bar SPM. Compared to other literature, there are four advantages shown as follows.

The presented 2 DOF seven-bar SPM which only contains two spherical five-bar loops is the main contribution. The highlights of this SPM are the unusual active choices and the symmetric loop structure. On the one hand, they solve the variable problem of loop equations with the discriminant method; on the other hand, they lead to the advantages of high reaction speed, wide input range, easy control, and high rigidity, which are more often required in industry compared to other SPMs. Thus, it may offer a new structure for industry applications such as cardans, robot wrists, camera-orientating devices, and even sensors.

Singularities and branches of a kind of 2 DOF seven-bar SPM were identified first.

Based on the branch graph, the proposed method is succinct and visual. It offers a geometric insight into understanding the formation of mobility information and is especially useful when the SPM has a specific input range or branch choice.

Two types of branches of the 2 DOF seven-bar SPM are distinguished according to whether branch points exist. Different branches correspond to the different coupling motions of the spherical mechanism.

The data in this study can be requested from the corresponding author.

LN, HD, and AK conducted the numerical analyses and wrote the majority of the paper. HD and JG supervised the findings and organized and structured the paper. JW and KT reviewed the paper and gave constructive suggestions to improve the quality of the paper.

The authors declare that they have no conflict of interest.

The first author would like to thank those who have hurt him over the years; they have made him stronger. In addition, he thanks the topical editor, Giuseppe Carbone, and the reviewers Iosif Birlescu and two anonymous referees, whose hard work improved the quality of the paper.

This research has been supported by the Natural Science Foundation of China (nos. 51975544, 51805494, and 51675495), the Major Project of Hubei Province Technology Innovation (grant no. 2019AAA071), and the 111 Project (grant no. B17040).

This paper was edited by Giuseppe Carbone and reviewed by Iosif Birlescu and two anonymous referees.