The spherical parallel mechanism (SPM) offers several advantages such as high stiffness, precision, a large workspace, immunity to interference, and simple kinematic calculations. Consequently, SPM finds extensive applications in fields like surgical robots, exoskeleton robots, and others. This paper proposes a design principle based on the virtual middle-plane constraint method, which integrates the branch constraint of the mechanism into the intermediate virtual constraint plane. On the one side, a symmetric spherical 3R branch consisting of two spherical links is provided to offer 3 rotational degrees of freedom (DOFs). On the other side, a constraint force located on the middle plane constrains 1 rotational DOF, enabling the end effector link to achieve 2 DOFs. Several symmetrical SPMs are synthesized based on the constraint force provided by the branches. The mechanism can achieve continuous motion from an initial position to a final position by undergoing a single equivalent rotation around an axis on the virtual symmetric plane passing through the center. The forward and inverse kinematic solutions and the velocity Jacobian matrix of the symmetrical SPM are determined. The workspace of the mechanism is obtained by considering inverse kinematics and link interference conditions. The dexterity and force/torque transfer performance of the mechanism within a certain range are analyzed. The correctness of the kinematics of the symmetrical SPM is demonstrated through simulation analysis and prototype experiment. This research lays a foundation for motion planning and dynamic analysis of this kind of mechanism by providing a variety of configurations for practical applications.

The parallel mechanism has the advantages of high bearing capacity, high structural stiffness, and compact structure. The spherical parallel mechanism (SPM) is a special spatial parallel mechanism. Its structural feature is that all the axes of revolute joints intersect at one point, and the end effector can rotate freely around that point. The SPM plays an indispensable role in practical applications and has been widely used in many fields, such as the azimuth tracking system (Luo et al., 2014), the bionic robot (Kumar et al., 2017), and medical devices (Cao et al., 2019; Saafi et al., 2018). When applied in imaging devices, spherical parallel mechanisms excel in fulfilling a range of needs, from surveying surrounding environments and coarse positioning to subsequent autonomous fine adjustments for the optimal observation of target objects. In contrast to traditional imaging setups that are limited to adjusting angles in horizontal and vertical directions, spherical parallel mechanisms offer simultaneous 2-degree-of-freedom (DOF) adjustments on a spherical surface, providing superior imaging perspectives.

The current research on SPMs mainly focuses on 2-DOF SPMs and 3-DOF SPMs, and some research results have been achieved in configuration synthesis, kinematic analysis, and dynamic analysis. The theoretical research and practical engineering applications of 3-DOF SPMs are relatively mature – for example, the typical 3-RRR SPMs. Gosselin and Hamel (1994) developed the well-known agile eye device. In addition, various modifications of this mechanism have been proposed and its applications have been continuously expanded. In the field of bionic robots, Liu et al. (2002) used a spherical 3-RRR mechanism as the shoulder and wrist joint mechanisms to design a 7-DOF redundant humanoid arm based on a series–parallel structure. In the field of rehabilitation medicine, Li and Payandeh (2002) applied the spherical 3-RRR mechanism to the laparoscopic surgical robots and optimized its workspace. French scholar Saafi et al. (2015) proposed a medical device based on tactile feedback and put forward an optimal torque distribution method. The performance of the SPM was improved by adding redundant actuators, and the geometric parameters were optimized to eliminate singularities. Zhang (2016) used a decoupled 3-RRR spherical parallel mechanism for the design of a launch platform. Zhou and Ge (2017) added a 3R branch as the design basis and obtained various redundantly actuated spherical 3-DOF mechanisms for flight simulators. Detailed analyses and comparisons of their DOF properties and workspace motion properties were conducted. The development of other types of 3-DOF SPMs has also been very rapid. Enferadi and Nikrooz (2018) conducted kinematic and workspace analyses on the 3-UPS/S SPM and used genetic algorithms to optimize the dimensions of the manipulator for single-objective and multi-objective optimization. Chang et al. (2022) proposed a novel three-branch spherical parallel mechanism with semi-decoupled characteristics, which has 3 rotational DOFs around the

In many fields, the use of a 2-DOF SPM is sufficient to meet the requirements. Compared to 3-DOF SPMs, 2-DOF SPMs have lower manufacturing costs and are easier to control. For instance, pointing mechanisms (Yu et al., 2016) can be utilized for sphere-based carving machines, satellite-antenna azimuth tracking, and automatic tracking devices on the ground for different types of aircraft. The spherical 5R mechanism serves as a typical representative of a 2-DOF SPM. Saiki et al. (2021) designed a bidirectional oscillating 2-DOF SPM with an active arc slider. Based on the new design concept of a “linear input–rotational output” chain, 2-DOF decoupled parallel mechanisms with a spherical working space are designed (Yu et al., 2020). Unlike existing methods that exploit singularity in parallel mechanisms for synthesizing reconfigurable parallel mechanisms, we proposed a set of triangular decoupling conditions for spherical parallel mechanisms. These conditions facilitate the synthesis of a drivetrain-based reconfigurable parallel spherical joint (Hu and Liu, 2022) capable of achieving one-dimensional fixed-axis rotation, one-dimensional variable-axis rotation, and two-dimensional and three-dimensional rotations. By incorporating a passive US limb into a 3-RPS parallel mechanism, a novel 3-RPS/US parallel mechanism with 2 DOFs has been obtained that is composed of one revolute, one prismatic, and one spherical joint limb (Li et al., 2022) for the 3-RPS portion and universal joints and spherical joints for the US portion. This enhancement increases the payload capacity of the mechanism, making it suitable for potential applications such as dual-axis tracking photovoltaic supports. A basic design method for a 2-DOF SPM bending dual-axis oscillating mechanism with a circular arc slider (Naoto et al., 2021) as input is proposed. The swing area of the passive link is small, allowing for infinite rotation around a certain axis without collision or transitioning into a single posture. Dong et al. (2012) analyzed the kinematics, singularity, and workspace of a class of 2-DOF mechanisms. Kim and Oh (2014) deformed the spherical 5R mechanism; designed the spatial self-adaptive finger clamp; and conducted a constraint analysis, optimization design of the structure, and grasping experiment on it. Cao et al. (2019) obtained a three-rotation-and-one-translation (3R1T) manipulator for minimally invasive surgery by connecting the revolute pair and the prismatic pair to a 2-DOF SPM and analyzed its kinematics and singularity. Chablat et al. (2021) designed a 2-UPS-U 2-DOF SPM with a predetermined regular workspace shape to manipulate an endoscope for assisting in ear surgery.

In summary, many scholars have conducted comprehensive research on two typical SPMs. It can be seen that SPMs have great potential application value. However, most of the research on 2-DOF SPMs only analyses the proposed mechanisms and does not mention the design principles of their configurations. Therefore, we proposed a new design principle for a symmetrical 2-DOF SPM. Structurally symmetric parallel mechanisms with isotropic symmetry and fewer DOFs exhibit greater potential in applications. The symmetrical SPM can realize continuous rotation around any line on the middle plane that passes through the rotation center of the spherical mechanism. The axis of rotation is fixed during rotation, and the mechanism is symmetric at all times during movement, which means that any form of motion of the mechanism can be transformed into a rotation with a fixed axis. This study establishes the theoretical foundation for the design of parallel mechanisms and further enriches the research on parallel mechanism systems.

In this paper, the design principle of a novel 2-DOF SPM with a symmetrical structure is proposed, a series of SPMs without over-constrained force/torque are synthesized, and the kinematic and static characteristics of one of the mechanisms are analyzed in detail. The paper is organized as follows: Sect. 2 analyzes the design principle and structure design of the symmetric SPM. In Sect. 3, the models of forward kinematics and inverse kinematics are established and the Jacobian matrix of the mechanism is obtained. The performance of the symmetrical SPM is analyzed in Sect. 4. In Sect. 5, the kinematic simulation and prototype experiment of the SPM are conducted. Conclusions are presented in Sect. 6.

The motion of the spherical mechanism revolves around a fixed point which is called the rotation center of the mechanism. There are two requirements in order for the SPM to maintain symmetry during motion, as shown in Fig. 1. Firstly, the overall structure of the mechanism must be symmetrical about a plane which is called the middle plane of the mechanism. Secondly, the rotation axis of the end effector of the spherical mechanism must be located on the middle plane and pass through the rotation center of the mechanism.

Symmetric SPM.

In order to realize the symmetrical movement of the 2-DOF spherical mechanism, its design principle is to limit the end effector's 3 translational DOFs and 1 rotational DOF about the axis perpendicular to the middle plane, and the structure of the mechanism is symmetrical about the middle plane.

Therefore, the specific design principle of a 2-DOF spherical parallel mechanism forms a 3R mechanism for one side of the linkage, providing 3 rotational DOFs. To achieve 2 rotational DOFs, another branch is needed to provide a constraint force that does not pass through the center of the sphere and is located on the middle plane. This constraint can be equivalent to a constraint torque relative to the rotation center,

According to the design principle of the symmetrical SPM based on the branch constraint, three non-coplanar constraint forces intersecting at one point and a constraint torque that is perpendicular to the middle plane are provided by the branches connecting the end effector and the base, whose structure is symmetrical.

The simplest branch that can provide three non-coplanar constraint forces that intersect at one point is the spherical 3R branch. Assuming that one of the branches connecting the end effector and the base is the spherical 3R branch, its coordinate system is established as shown in Fig. 2; the

The coordinate system of the mechanism containing the spherical 3R branch.

The

The constraint screw system of the spherical 3R branch is

Based on screw theory, the constraint screws in Eq. (2) correspond to the constraint forces which pass through

In order to provide a mechanism without over-constrained force/torque, the branch providing only one constraint torque is considered first. However, Zhao et al. (2004) found that there are few branches suitable for the design of the symmetrical SPM, so the branches that provide only one constraint force are considered. The design requirement of the branches with a single constraint force is that the constraint force provided by the branch is located on the middle plane and does not pass through the rotation center. The constraint force can be equivalent to the constraint torque perpendicular to the middle plane. Chen et al. (2016) synthesizes a series of branches providing a constraint force, among which the branches satisfying structural symmetry are shown in Table 1.

Branches providing a constraint force and symmetry about the middle plane.

There exists a well-defined geometric correlation between the force or torque of the branch constraint and the motion pairs of said branch (Chen et al., 2016). In branches with symmetrical structures, the constraint force can effectively be confined to the middle plane by judicious arrangement of the motion pairs. To comply with design principles, this paper outlines the structural design of symmetrical SPMs utilizing the RSR, PSP, URU, and CRC branches, respectively.

The constraint force through the center of rotation of the spherical joint and the intersection of the axis of rotation is provided by the RSR branch, as shown in Fig. 3a. Since the intersection of the axis of rotation and the RSR branch is symmetric about the center of rotation of the spherical joint at point

Symmetrical SPM based on the RSR branch:

The symmetric SPM RSR branch (Fig. 3a) is utilized as an example, and the DOF property of the mechanism is analyzed based on screw theory. The coordinate system is established as shown in Fig. 1, where the rotation center

The SPM is then analyzed in detail. The specific structure of the end actuator shown in Fig. 1 is displayed in Fig. 4. The

End effector of symmetrical SPM.

With the exception of the varying radius, the configuration of the base and end effector are nearly identical, leading to a slight disparity in the architecture between rotation joints

The constraint screw system of the branch of the spherical 3R branch determined by rotation joints

The constraint screw system of the RSR branch is

It can be found from the direction number and the coordinate of the center point of rotation joint

As shown in Fig. 1, we assume that the intersection point of the

According to Eq. (4) and Eqs. (6), (7), and (8), it can be concluded that

Therefore, the constraint screw of the RSR branch is a constraint force along the direction of line

As shown in Fig. 5, the coordinate systems of the symmetrical SPM are established. The global coordinate system

Kinematic model of the symmetrical SPM.

The distance between

Because the mechanism has 2 DOFs, the configuration can be represented by two angles,

When the output parameters

The driving parameters

When the end effector and the base coincide, the coordinate systems

The coordinates of

The coordinates of

The coordinates of the rotation center of

The coordinate of the rotation center of spherical joint

The driving parameter,

The paraments

Four initial configurations of the symmetrical SPM.

Given the driving parameters

According to a reference (Zhang et al., 2006), the configuration parameters can be obtained as follows:

The end effector of the 2-DOF SPM has the ability to continuously rotate around the axis passing through the rotation center and lying on the middle plane throughout the entire motion. In addition, this mechanism possesses other noteworthy motion properties. Specifically, starting from the initial position and ending at the target position, the end link can achieve a pose transformation through a fixed-axis rotation, known as the equivalent rotation of the mechanism.

In Fig. 7a, the end effector travels from position I to position II, with middle planes

Equivalent rotation characteristics of the SPM:

Based on the structural characteristics of the mechanism, the fixed link and the end link remain symmetric about the middle plane during the motion process. Thus, we can obtain the following:

To provide a clearer illustration, a plane

The point

That is, if line

Therefore, point

In the global coordinate system,

The 3R branch has 3 DOFs; thus, the instantaneous motion screw of the moving link can be represented as a linear combination of the three screws as follows:

According to screw theory, within the branch, there are three screws that reciprocally commute with all the motion screws in that branch, denoted as

In matrix form, this can be expressed as

The RSR branch has 5 DOFs; thus, the instantaneous motion screw of the moving link can be represented as a linear combination of the five screws as follows:

In the same way, we obtain

In matrix form, this can be expressed as

The constraint screw of the mechanism is obtained by combining the constraint screws from the two branches:

The drivers of the mechanism are

The newly added constrained screw of the 3R branch is denoted as

The newly added constrained screw of the RSR branch is denoted as

Expressing this in matrix form, we have

By combining Eq. (30) and Eq. (32), we obtain

The

Due to mechanical interference, reference point

Workspace of the spherical mechanism.

The following conclusion can be drawn from Fig. 8: (1) as the angle

Limited configuration parameters of the spherical mechanism.

The dexterity primarily reflects the transmission accuracy of motion between input and output; Salisbury and Craig (1982) proposed using the condition number of the Jacobian matrix of a mechanism as a performance metric for the dexterity of the mechanism. The condition number of its inverse matrix,

The matrix norm,

The condition number of the Jacobian matrix of the institution considered in this paper can be defined as follows:

As defined by Eq. (34), the spectral norm of a matrix can be expressed as follows:

Since

In accordance with the structural parameters of the mechanism outlined in Sect. 4.2, the dexterity of the 2-DOF SPM presented in this article is expressed in Fig. 9.

Dexterity of the spherical mechanism.

When the condition number is large, the precision of the inverse matrix of the Jacobian matrix is relatively low, leading to significant distortion in the relationship between input and output velocities. Therefore, it is crucial to ensure a low condition number of the Jacobian matrix within the operational range of the mechanism during its design. The condition number of the Jacobian matrix is a value greater than or equal to 1. A condition number of 1 indicates optimal motion transmission performance, while an infinite condition number signifies a singular configuration.

Fang et al. (2014) introduced the dexterity index of the mechanism as

According to Fig. 9, it can be observed that the mechanism exhibits better transmission performance when 20°

The condition number of the mechanism Jacobian is used as the analysis index of torque transmission performance, reflecting the driving capacity of the spherical mechanism. The matrix spectral norm is used to calculate the condition number of the Jacobian matrix.

According to the kinematic principles, this can be expressed as

According to the properties of matrix norms, we have

The condition number of the Jacobian matrix can be defined as follows:

As the value of Eq. (42),

Transfer performance of the spherical mechanism.

From Fig. 10, we can see that the

After inputting two tiny input variables,

Verification of the Jacobian matrix.

The correctness of the kinematic and velocity analyses of the 2-DOF SPM studied in this paper is verified using this numerical validation method to validate the inverse model of the mechanism.

The kinematic simulation verification of the 2-DOF SPM mechanism was conducted using SolidWorks and RecurDyn software. In SolidWorks, the various components of the motion simulator were built and assembled into a three-dimensional model. After simplifying the model, it was imported into the RecurDyn software, where motion pairs and drivers were added to obtain a virtual prototype model, as shown in Fig. 11. During the RecurDyn simulation, the fixed link is kept stationary. Driving 1 is applied at the revolute joint

Importing to RecurDyn to add constraints and drivers.

The mathematical expression for the output motion of the moving link can be represented as follows:

Therefore, the simulation results of the 2-DOF SPM mechanism were obtained. The input angular displacement–time curves for each driving link are shown in Fig. 12, while the driving force–time curves are depicted in Fig. 13. From the figures, it can be observed that the required driving curves for each branch are remarkably smooth, indicating excellent kinematic performance.

Displacement–time curve.

Driving–time curve.

We observed that, when provided with a set of continuously varying driving forces, the motion of the end effector obtained through simulation remains continuous, with no sudden changes in torque occurring during the motion process. This demonstrates the continuity and stability of the mechanism's motion.

After the kinematic simulation, the kinematic performance of 2-DOF SPM is further verified. In this section, a prototype 2-DOF SPM is built, and its motion capability is verified by experiments. The four initial configurations with different arrangements of the driving links (

Four initial configurations.

Motion of the mechanism.

This paper proposes a design principle based on the virtual middle plane constraint method, which integrates the branch constraint of the mechanism into the middle virtual constraint plane. On the one side, two spherical 3R branches symmetrically provide 3 rotational DOFs, while on the other side, a constraint force located on the middle plane restricts one rotational DOF, allowing the end effector to achieve 2-rotational-DOF spherical motion. Specific requirements include (1) ensuring that the angle between the fixed link and the end effector remains constant with respect to the middle plane throughout the motion of the mechanism and symmetrically arranged. (2) The joint axes of the 3R branch on the left intersect at the center of the sphere. (3) The constraint force provided by the right constraint branch is located on the middle plane. Based on these principles, a series of symmetric SPMs is designed. The SPM can realize continuous rotation around any line on the middle plane which passes through the rotation center of the spherical mechanism, and the rotational axis can be fixed during the rotation process, which means that any form of motion of the mechanism can be transformed to a rotation with a fixed axis.

The kinematic solutions of the symmetrical SPM based on the RSR branch are given. The inverse and forward kinematic mechanisms are determined using screw theory. To obtain the inverse Jacobian matrix of the 2-DOF SPM, the constraint equation is differentiated. The workspace of the mechanism is obtained by considering inverse kinematics and link interference conditions. We find that when the orientation parameters

In the future, we will continue to carry out research work on 2-DOF SPMs. This will include research into the statics and dynamics of such mechanisms, and further research and development efforts should focus on fully harnessing the potential of this mechanism to advance the field of rehabilitation robots, ultimately benefiting patients in their recovery processes.

All code included in this study is available from the corresponding author on reasonable request.

The data generated during this study are available from the corresponding author on reasonable request.

XC conceived the idea and organized the paper structure. CX and ZZ prepared the figures and wrote this paper. YG and AY developed the simulation models. ZC was the project administrator.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors wish to thank Yanshan University for its support.

This research has been supported by the National Natural Science Foundation of China (grant no. 51775474).

This paper was edited by Med Amine Laribi and reviewed by six anonymous referees.