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This paper presents a novel parallel manipulator with one translational and two rotational (1T2R) degrees of freedom that can be employed to form a five-degree-of-freedom hybrid kinematic machine tool for large heterogeneous complex structural component machining in aerospace field. Compared with serial or parallel machine, hybrid machine has the merits of high stiffness, high speed, large workspace, and complicated surface processing ability. To increase stiffness, three-degree-of-freedom redundantly actuated and overconstrained 2

Parallel kinematic manipulator tools were claimed to possess the inherent advantages such as high stiffness, high loading capability, high precision, low error accumulation, quick response speed, and high orientation capability. However, parallel manipulators suffer inherently from the unfavorable workspace. Therefore, it is of crucial significance to have a larger workspace so as to satisfy the working capability [

Hybrid kinematic machine is expected to integrate the respective merits of pure serial and parallel machine, which has bigger workspace and better dynamic performance, higher precision and higher rigidity, more complicated surface processing ability, and more flexible orientation capability and has been successfully employed as machine tools and robots for high speed milling, drilling, and welding in aerospace and automotive industry for free surface processing, as well as assembly operations of aluminum structural parts [

Hybrid machine tool underwent fast improvements and drew particular interests for numerous researchers, since they satisfy the increasing demanding task requirements of many various applications such as in machine tools, assembly lines, and high speed machining used in automotive, railway, and construction industries. For instance, the German DS Technologie launched five-axis machining center Ecospeed spindle for aircraft structure components with complex geometries, and the spindle head was mounted on the end effector of parallel manipulator, which can realize rotation about the x- and y-axis and translation along z-axis, and translation in X and Y direction can be realized by two very long tracks [

The remainder of the paper is organized as follows. The research background of hybrid parallel machine tools and traditional serial-parallel machines is presented firstly. In the subsequent section, the required degree of freedom for high speed milling machining of large heterogeneous complex freedom surface is briefly addressed, and a redundantly actuated and overconstrained 2

The purpose of this paper is to design a hybrid machine tool used in the aerospace field for a large heterogeneous free surface high speed milling; the workpiece magnitude is shown in Figures

Limit rotational angle about x-axis.

Limit rotational angle about y-axis.

According to the characteristics of the curve surface, five relative motion between the surface and the cutter are processed, the x- and y-axis motion are configured by the ball screw motion unit in series to satisfy the requirements of large workspace, and translation along

Absolute position accuracy: parallel manipulator configuration instead of traditional series configuration can reduce error accumulation.

Normal precision: the parallel manipulator has good orientation capability to ensure the normal contact between the normal of free surface and machine tool point, so as to keep its normal position precision.

Workspace: workspace is generated by adopting compound spherical joint to increase orientation workspace and serial X-Y long tracks to expand the position workspace of the hybrid machine tool.

High stiffness: hybrid machine tool will produce heavy cutting force in the high speed machining process, so as to maintain higher machining accuracy, and the machine tool should be able to bear the heavy force and resist the external force deformation. Thus, the parallel manipulator should have higher stiffness characteristics.

High quality: the high orientation capability of parallel manipulator is employed instead of manual milling, which is beneficial to ensure the machining quality.

According to the above processing requirements, the parallel manipulator tool requires high stiffness and good orientation capability for the high speed machining of the free surface. Considering development trend of the hybrid machine tool, a five-axis hybrid machine tool can be constructed by adopting 1T2R three-degree-of-freedom parallel manipulator with two long X-Y tracks, which is the best choice to realize the machining task requirements.

It is configuration innovation of the 1T2R three degrees of freedom that is the kernel of the hybrid machine tool. In order to complete the surface process with high efficiency and high precision, it is of crucial importance and significance for novel 1T2R lower-degree of freedom parallel manipulator with high stiffness, large workspace, and high orientation capability. There is an abundance of research on 1T2R mechanism. Kong and Li [

To increase the workspace of the parallel manipulator, enhance the kinematics performance of the machine tool, and improve the stiffness characteristic and dynamic characteristic, this paper adopts the redundantly actuated and overconstrained 2

A hybrid kinematic machine with five DOFs.

A novel redundantly actuated and overconstrained 2-

The schematic diagram of the 2-

To facilitate analysis, a fixed coordinate system_{1}_{2} be_{1}_{2},_{1}_{2}, and_{1}_{2} be_{1}_{2},_{1}_{2}, and_{1}_{2}_{3}_{1}_{2}_{3}_{1},_{2}, and_{3} are (0_{1},_{2}, and_{3} are (_{1}_{1}_{1}), (_{2}_{2}_{2}), and (_{3}_{3}_{3}), respectively. The layout angle of the fixed actuators is defined as _{i} joint and

Lower-mobility parallel manipulators whose independent degrees of freedom of the end effector are usually less than six can be implemented in many applications. In order to determine the motion pattern of the redundantly actuated and overconstrained 2

It is assumed that the axes of the fixed coordinate system and the moving coordinate system are parallel to each other in the initial pose. Firstly, the first _{1} point and parallel to

Similarly, the twist screw of the second

The wrench screw of (_{2} and parallel to the

The twist screw of the third _{3} and_{3} are direction cosines of the three prismatic joints in limb 3.

The wrench screw of (_{3} and parallel to

The constraint screw of all the limbs acting on the moving platform can be obtained as

It is worth noting that_{1}=_{2}=_{3}=

Similarly, the twist screw of the first _{i} denotes the twist of the P joint and_{i} and_{i} represent the direction cosine of the second revolute axis of the Hooke joint of

Then, the wrench screw of limb 1 is easily obtained as_{1} and parallel to

As previously mentioned, the twist screw of the second

The wrench screw of (_{2} and parallel to

The twist screw of the third _{3} is a constant, and_{3} and_{3} represent the direction cosine of the third joint axis.

Subsequently, the wrench screw of (

So far the constraint screw of the parallel manipulator can be expressed as

It is noteworthy that the direction vector_{1}_{2} coincides with the axes of the second joint axis of the Hooke joint in the two

By combining (_{3} and parallel to_{2} and parallel to the_{3} and parallel to the

Generally, the degree of freedom of parallel manipulator can be calculated by the modified

There was neither constraint couple in the same direction nor constraint force in collinearity among the constraint screw in the parallel manipulator; therefore, there is no common constraint, that is,

Thus, based on the revised

In summary, the redundantly actuated and overconstrained 2

The inverse kinematics solution is based on the determination of the structural parameters of the parallel manipulator, when the position and orientation of the moving platform are given, so as to solve the input displacement of the prismatic joints.

_{i} and_{i}, and the coordinate of each joint in the fixed coordinate system can be, respectively, expressed in matrix form as

Because of the arrangement of revolute joint in

Selecting parameters

The close-loop vector method is used to establish the equation of vector_{i}_{i} in the fixed coordinate system

Equation (

Position inverse solution of the 2_{i} in (_{1}=_{2}=_{3} is an extensible and compressible link.

To better describe the orientation capability of the parallel manipulator, Liu and Bonev [

Figures

The relation of

The relation of

The relation of

The orientation workspace of parallel manipulator tool is an important performance index, which is the set of all practically feasible orientation of the moving platform. By analyzing the workspace, we can deduce the problem whether the expected machining range can be realized. The main analysis methods contain analytical method and numerical method; in this paper, the limit boundary searching method of reachable workspace was adopted, and the mechanism is divided into several single limbs, and the boundary of single limb space is obtained by using the surface enveloping theory; finally, the whole workspace of the mechanism is obtained by using the surface intersection technique. The difficulty of the hybrid machine tool workspace is to solve the workspace of parallel manipulator, while the workspace analysis of parallel manipulator is mainly to investigate the reachable workspace of the end effector. For 1T2R parallel manipulator, the workspace of the manipulator mainly considers the rotational capability about the

In the actual machining process of parallel manipulator tool, the following limitations should be considered.

_{ i }

_{ 4 }

_{ 2i }

_{ 3i }

_{ 53 }

The orientation workspace of the manipulator tool is generated according to the structure parameters and constraint conditions of the parallel manipulator aforementioned. Based on the inverse position solution, the constraint conditions of each step are judged, respectively, and the detailed search algorithm is seen in the flowchart as shown as Figure

Flowchart of singularity-free workspace calculation.

Numerical examples are presented to verify the validity of the theory, and its main structure parameters of the redundant actuation parallel manipulator are as follows (unit:_{1}=_{2}=1045,

Because the redundantly actuated and overconstrained parallel manipulator has a parasitic motion in

Parasitic motion relation between

To illustrate the analysis process of the orientation workspace comprising reachable workspace, task workspace, and joint workspace, the novel parallel manipulator proposed has been taken as an example. By adopting a numerical searching method aforementioned, the reachable workspace and task workspace of the redundantly actuated and overconstrained parallel manipulator are generated as shown in Figure

Singularity-free reachable and task workspace.

By specifying the value of the_{4} (that is link length of_{3}), we can get a surface in three-dimensional joint workspace (_{1},_{2},_{3}) formed by the three prismatic joints. And the surface contains all the possible actuated parameters which affect the motion of the moving platform. Figure _{4}=1150

Joint workspace of the parallel manipulator.

The reachable workspace of the proposed parallel manipulator and the overconstrained 2PRU-PRS parallel manipulator, with the same degree of freedom, is illustrated in Figure

Singularity-free workspace comparison of two parallel manipulators.

In this paper, a novel serial-parallel hybrid kinematic machine tool with high stiffness, high orientation capability, and large workspace has been presented, which is a combination of 1T2R parallel manipulator and two long X-Y tracks and can be applied to high speed machining of a large heterogeneous complex freedom surface in the aerospace field.

The mobility of proposed redundantly actuated and overconstrained 2

The orientation workspace, the performance evaluation index, can serve as an alternative for the description of the orientation capability. Compared with overstrained 2PRU-PRS parallel manipulator, the corresponding results illustrate that the redundant actuation parallel manipulator can effectively increase the workspace and potentially improve its orientation capability, which lays a theoretical foundation for the stiffness analysis and optimal design of the parallel manipulator in the future work.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

The authors would like to acknowledge the financial support of the Fundamental Research Funds for the Central Universities under Grant no. 2018JBZ007, no. 2018YJS136, and no. 2017YJS158 and the National Natural Science Foundation of China (NSFC) under Grants 51675037 and 51505023.