There are considerably rigorous requirements for accuracy and stability of the mechanism to accomplish large-scale and complex surface machining tasks in the aerospace field. In order to improve the stiffness performance of the parallel mechanism, this paper proposes a novel three degrees of freedom (DOF) redundantly actuated 2RPU-2SPR (where R, P, U and S stand for revolute, prismatic, universal and spherical joints, respectively) parallel mechanism. Firstly, the kinematics position inverse solution is derived and a dimensionless generalized Jacobian matrix is established through the driving Jacobian matrix and constraint Jacobian matrix. Secondly, the stiffness model of the parallel mechanism is deduced and the accuracy of the stiffness model is verified through finite-element analysis. Using eigenscrew decomposition to illustrate the physical interpretation of the stiffness matrix, the stiffness matrix is equivalent to six simple screw springs. Finally, the simulation experiment results demonstrate that redundantly actuated parallel mechanism has better stiffness performance compared to the traditional 2RPU-SPR parallel mechanism.

In the field of industrial manufacturing, the processing equipment with the parallel mechanism has been widely used and researched because of its excellent performance. Tian et al. (2019) proposed a novel robot leg deformable parallel mechanism with the ability of reconstruction and motion change based on an innovative rotatable-axis revolute joint. Araujo-Gómez et al. (2019) carried out kinematic analysis and size optimization of 2R2T (where R and T stand for rotational and translational degrees of freedom, respectively) parallel mechanism. Wu et al. (2014) systematically introduced and discussed a series of 2-DOF parallel manipulators with equal–diameter spherical pure rotation. Three-DOF parallel mechanism is one of the most noticeable and widespread mechanisms and Li and Hervé (2010) synthesized a series of 3-DOF parallel mechanisms. The finite motions of 1T2R parallel mechanisms with parasitic motions were described and analyzed by Sun and Huo (2018) and their topologies were synthesized. Huang et al. (2019) proposed a simple and highly visual method for type synthesis of families of over-constrained parallel mechanisms with one translational and two rotational motion capabilities. In order to expand the application of parallel manipulators, Wu et al. (2018) created a 5-DOF hybrid machine and a mechatronics model was derived based on this parallel manipulator.

Stiffness design refers to optimization based on a certain stiffness analysis model with structural parameters as variables to improve the stiffness performance of the mechanism. A stiffness synthesis strategy was proposed for the desired elastic center of a 3-DOF parallel mechanism composed of 3RPR links (Wen et al., 2015). The Exechon parallel mechanism with over-constrained kinematic chain has attracted extensive attention in the research community and its stiffness has been studied by Zhang et al. (2016). Moosavian et al. (2016) introduced a method for designing variable geometry parallel mechanism with enhanced static properties. Generally, higher stiffness allows greater variable load and higher precision of the end–effector. There is no unified strategy to establish the stiffness model of the parallel mechanism and many experts have done relevant research on different types of parallel mechanisms. A new variable stiffness design method was proposed to optimize the geometric parameters of the parallel mechanism by Li et al. (2017). For the heavy-duty parallel robot manipulator, Wang et al. (2017) proposed a method to simultaneously optimize the size and structural parameters and introduced a stiffness performance evaluation index. This stiffness distribution index mainly focuses on the stiffness components of the manipulator. Moosavian et al. (2014) introduced a new method to enhance the stiffness of parallel mechanisms without redundant actuation. Chen et al. (2018) showed a negative stiffness criterion for active stiffness using the example of a redundantly actuated planar parallel mechanism. Ali et al. (2018) also studied the stiffness of the parallel mechanism. Moreover, the over-constrained parallel mechanism can also be more widely applied by combining with sliding rail such as the serial parallel hybrid 5-axis machining machine tool. In order to meet the requirements of the large workspace and high dexterity of aerospace industry processing equipment, Fang et al. (2020) and Jiang et al. (2021) proposed a new type of series–parallel hybrid processing robot mechanism. Portman (2020) presented a new method to define stiffness values for elastic support systems and demonstrated the effectiveness and generality of the method with examples of simulated applications. Cao et al. (2017) summarized a method for modeling stiffness of a 3R2T over-constrained parallel robot mechanism. The proposed method can be conveniently applied to build stiffness models with high accuracy for 3R2T over-constrained parallel mechanisms. Jiang et al. (2019) designed a 3T2R parallel mechanism with a large output rotation angle in terms of the Lie group theory and the integration of configuration, and analyzed various performance indices of the parallel mechanism.

The commonly used modeling methods at this stage can be roughly divided into two categories. One is the employ of some methods such as principle of virtual work, structural matrix, screw theory or a combination of them. Based on strain energy analysis and screw theory, Cao et al. (2017) proposed a systematic approach to a limb stiffness model that considers the coupling of constrained wrench stiffness and actuated wrench stiffness, and Sun et al. (2016) proposed a 2-DOF constrained rotating parallel mechanism with an articulated walking platform and established its stiffness model. Yan et al. (2016) performed a stiffness analysis using a strain energy method that considers compliance of the mobile platform. Wu et al. (2022) proposes a novel method to evaluate the dynamic performance of the robot along joint–space directions. Furthermore, a new stiffness index is proposed to evaluate stiffness properties. Fang et al. (2002) proposed a comprehensive method of 4-DOF parallel mechanism based on the screw theory and summarize a comprehensive method of 3R parallel mechanism. Gosselin and Schreiber (2018) considered two types of redundancy for parallel mechanisms, namely, actuation redundancy and motion redundancy. Then, each concept was mathematically formulated to clearly demonstrate their characteristics and properties. The other is with the help of commercial software. Combined with related theoretical and experimental research, a finite-element analysis model of mechanism stiffness was derived. A novel method for force analysis of the over-constrained parallel mechanisms was proposed by Xu et al. (2015) and the correctness of the proposed method for force analysis is effectively verified based on Adams simulation software. Zhang et al. (2019) verified the analytical results of kinematics and dynamics through the co-simulation solution of Simulink and Recurdyn. Cao and Ding (2018) proposed a general method for solving 3R2T parallel mechanisms with complex structures and multiple redundant constraints, and then verified the correctness of the established stiffness model through finite-element analysis.

The main contribution of this paper is as follows: a novel redundantly over constrained 2PRU-2SPR parallel mechanism is presented. What is more, the relationship between the pose of moving platform and external load is derived and the overall stiffness matrix of the mechanism is obtained. To have a deeper understanding of the internal structure of stiffness matrix, an eigenscrew decomposition method is utilized to illustrate the physical interpretation of stiffness matrix that can be expressed as linear superposition of simple springs based on the screw decomposition. Finally, the stiffness modeling and analysis methods are effective and correct after proving by finite-element software analysis, and the stiffness performances are compared with the traditional 2RPU-SPR parallel mechanism.

Relevant concepts associated with stiffness analysis of parallel mechanism.

In order to meet the processing requirements of large complex special-shaped workpiece, an over-constrained 2RPU-2SPR redundantly actuated parallel mechanism is applied to the machine tool equipment combined with the characteristics of parallel mechanism and the purpose of machining complex curved surface. The over-constrained parallel mechanism not only has the characteristics of parallel mechanism but also has the advantages of high stiffness, high precision, strong carrying capacity and avoiding singularity. Wu et al. (2013) revealed the relationship between the stiffness performance index of the parallel manipulator and its natural frequency through the study of a 3-DOF parallel manipulator. And the simulation results demonstrate that additional redundant legs can increase the natural frequency of the parallel mechanism and improve the stiffness performance. As shown in Fig. 1, the 2RPU-2SPR parallel mechanism is mainly composed of a fixed platform, a moving platform, an end effector, and two joints of actuated branches with identical structures and symmetrically arranged. The fixed platform is connected with the moving platform through the two joints of actuated branches. One joint of branches is RPU moving branch chain including a revolute joint, a prismatic joint and a universal joint. The other joints of branches are SPR moving branches including a spherical joint, a prismatic joint and a revolute joint.

As shown in Fig. 2, the coordinate system is established on the structure
diagram of the over-constrained parallel mechanism so as to facilitate the
later modeling and optimization calculation. With

Five-axis linkage processing equipment.

2RPU-2SPR parallel mechanism.

According to the screw theory, the constraint force and constraint moment on
each kinematic branch chain do no work on the center of the moving platform,
that is,

Equations (1) and (2) are written into matrix form and the relationship between
constraints is as follows:

It is known that the Kutzbach degree of freedom calculation formula is as
follows:

Since there is neither constraint couple with the same direction nor
coaxial constraint force in the constraint screw of the mechanism, the
mechanism has no common constraint, i.e.,

The degree of freedom of the parallel mechanism can be obtained by
substituting Eq. (5):

The parallel mechanism studied in this paper has three degrees of freedom
which can rotate around two axes and move along one axis, namely,

Set

Assuming that the moving platform

The inverse position solution of Eq. (8) can be expressed as follows:

Due to the introduction of the rotating joint

Assuming that the velocity vector of the moving platform reference point

Then, the axial speed

In matrix form, it can be expressed as follows:

Combining Eqs. (14) and (15), the generalized Jacobian matrix

According to the dual relationship between velocity mapping and force
mapping, the force mapping relationship between each branch of the mechanism
and the moving platform can be obtained from Eq. (16).

The five-coordinate hybrid machine tool has high requirements for the kinematic and static properties of the mechanism when performing high-speed milling tasks in the aerospace field, so the stiffness analysis and optimization design is the key content that must be considered in the mechanism design.

When constructing the stiffness analytical equation of 2RPU-2SPR over
constrained parallel mechanism, it does not lose generality. Firstly, the
moving and fixed platforms of the parallel mechanism are regarded as ideal
rigid bodies and it is assumed that the deformation of other parts belongs
to the category of linear elasticity and small deformation, ignoring the
small deformation of revolute, spherical and universal joint. The unit
actuating force screw and unit constraining force screw applied each branch at the center point A of the
moving platform are represented by

According to the basic knowledge of material mechanics, non-rigid parts will
produce various deformations under different types of forces, so it is
assumed that the tensile deformation of each branch under the driving screw

The specific relationship between force and deformation and stiffness
coefficient is as follows:

The above formula is expressed in matrix form as follows:

According to the above analysis, the elastic deformation vector generated by
the actuating branch and moving platform under the constraint force and
actuating force of each branch can be expressed as follows:

Substituting Eqs. (22) and (23) into Eq. (17)

Then, the stiffness matrix of the parallel mechanism is as follows:

Deformation with force/moment imposed.

If the parameters and pose of the mechanism are given, the stiffness matrix
of the mechanism will be determined and some stiffness indices can be
defined to evaluate the stiffness characteristics of the mechanism. It
should be noted that the parallel mechanism designed in this paper is mainly
applied to the machine tool performing milling operation. Therefore, the
linear stiffness and angular stiffness can be defined as follows:

Geometric parameters, configuration parameters and physical parameters.

According to the structural parameters, position parameters and physical
parameters given in Table 2, the stiffness matrix of the mechanism under
this typical posture can be expressed as follows:

In order to testify the correctness and effectiveness of the stiffness
model, finite-element analysis of the parallel mechanism is also conducted. The finite-element analysis of the parallel mechanism is carried out by
employing the Ansys Workbench software. Figure 3a–c show the deformation of the parallel manipulator under force along the direction of

The results obtained by the analysis calculation and finite-element analysis are listed in Table 3. Clearly, the results obtained by the finite-element analysis are very consistent with the results of analysis calculation. FEA is the abbreviation of finite-element analysis.

The results of calculation and finite-element analysis.

Under the premise of ignoring the deformation of the kinematic
joint, the calculation results of the analytical model and the finite-element analysis model are very similar. The error of

Assuming that the dynamic and static platforms are rigid bodies and ignoring
the deformation of the moving pair, the calculation results of the
analytical model and the finite-element model are very close. Therefore, the established stiffness analytical model is effective and it can be used for
static stiffness analysis. In addition, the workspace is defined
as

Stiffness distributions in the workspace when

Normally, the end deformation

Therefore, the decomposition of the stiffness matrix is transformed into the
decomposition of the

The eigenscrew decomposition is applied to the stiffness matrix

The physical interpretation of stiffness matrix

Equivalent spring constants.

Stiffness matrix spring distribution diagram.

It can be seen from Table 4 and Fig. 5 that the stiffness matrix can be equivalent to a simple superposition of six springs. Six springs can be divided into three groups. Two springs in each group meet at one point which has the same spring stiffness and the pitches are opposite numbers. Since the mechanism has two branches that are not the same from each other, the mechanism does not show a certain symmetry and the resulting spring distribution is not regular.

In order to evaluate the stiffness of some positions in the workspace, the
maximum eigenvalues of the stiffness matrix

Parallel mechanism stiffness surface.

It can be seen from Fig. 6 that the stiffness values of mechanism
decrease with the increase of

It is obvious from Fig. 7 that the kinematic performance of redundantly actuated 2RPU-2SPR parallel mechanism is better than traditional 2RPU-SPR parallel mechanism which has a bright engineering practical prospect. Therefore, the introduction of redundant branch chain can effectively improve the kinematic performance of 2RPU-SPR parallel mechanism which lays a theoretical foundation for later optimization design.

Taking the 2RPU-2SPR over-constrained parallel mechanism applied to complex surface machining as the research object, it starts from the mechanism characteristic description and the DOF characteristics. The position mapping relationship between the input member and the output member is established and the Jacobian matrix representing the velocity mapping relationship between the input joint and the output joint of the mechanism is obtained. It is worth noting that the stiffness performance of the parallel mechanism directly determines the motion accuracy and stability of the system when the mechanism performs the surface machining task. Based on the screw theory, the stiffness of the 2RPU-2SPR over-constrained parallel mechanism is deduced. Then the eigenscrew decomposition method can be adopted to decompose the stiffness matrix.

Comparison of maximum stiffness.

In this study, aiming at the large-scale and complex surface machining tasks in the aerospace field, a novel over-constrained redundantly actuated parallel mechanism is proposed. Furthermore, the correctness of the derivation of the stiffness matrix through finite-element simulation analysis is verified which can be used to study the static stiffness characteristics of the parallel mechanism. Simultaneously, the stiffness matrix is equivalent to three groups of springs with more definite physical meanings. Finally, compared with the stiffness of the 2RPU-SPR parallel mechanism, the passive over-constrained parallel mechanism has higher stiffness, indicating that the introduction of the redundantly actuated over-constrained branch chain SPR can improve the stiffness performance of the mechanism and can also effectively improve the stiffness distribution of the mechanism.

In our future research, the mechanism proposed in this paper has great
stiffness performance; it can be connected with

All the data used in this article can be made available upon reasonable request. Please contact the corresponding author (hqzhang@sdust.edu.cn).

HZ proposed the novel parallel robot and developed the theoretical analysis; JT and CY wrote the manuscript draft under the guidance of HZ; and GC, MZ and YY supervised the whole research work.

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 in published maps and institutional affiliations.

The authors would like to acknowledge the financial support of the Fundamental Research Funds for Shanghai Collaborative Innovation Center of Intelligent Manufacturing Robot Technology for Large Components under grant no. ZXP20211101 and Key Laboratory of Vehicle Advanced Manufacturing, Measuring and Control Technology (Beijing Jiaotong University), Ministry of Education, China.

This research has been supported by the Fundamental Research Funds for Shanghai Collaborative Innovation Center of Intelligent Manufacturing Robot Technology for Large Components (grant no. ZXP20211101) and Key Laboratory of Vehicle Advanced Manufacturing, Measuring and Control Technology (Beijing Jiaotong University), Ministry of Education, China.

This paper was edited by Wuxiang Zhang and reviewed by three anonymous referees.