Kinematics and Transmission Performance Analyses of a 2 T 2 R Type 4-DOF Spatial Parallel Manipulator

A 2-RPU&2-SPS spatial parallel mechanism (SPM) is researched. Firstly, the number and property of degrees of freedom (DOF) of the SPM are analyzed by screw theory.There are two rotational and two translational movements (2R2T) of themechanism that can be achieved. Secondly, the position analyses are researched. For the inverse position analysis, the explicit expression can be obtained from the independent motion parameters of the given mechanism, and the forward position problem is solved by calculating a set of nonlinear equation systems. Then we obtained the workspace of the mechanism based on the analytic formulae of the inverse displacement. Finally, by establishing the Jacobian matrix of the mechanism, the singularity of the mechanism is obtained, and the kinematics transmission performance of the mechanism is studied by using the index of the output efficiency of the limb output of the mechanism. This work will provide the theoretical basis for prototype development and application of the mechanism.


Introduction
In recent years, with the deepening of the relevant theoretical research of the parallel mechanism and the continuous expansion of the application field, the lower-mobility mechanisms have become the research hotspot in the field of parallel mechanisms.In the comparison between 6-DOF parallel mechanism and lower-mobility mechanism, the latter has the advantages of simpler mechanical design, low manufacturing cost, larger workspace, high accuracy, high velocity, and high stiffness and the accumulated error of positional is little.In the fields of mechanical processing, medical devices, space technology, and sensors and other fields, the lower-mobility mechanism has received increasing attention from industry and academia in various countries.
The 4-DOF parallel mechanism is the most important kind of mechanism in the lower-mobility mechanism, which can be divided into 3T1R, 2T2R, and 3R1T according to the property of DOF.At present, the research depth and breadth of the 4-DOF SPM are far from reaching the research level of 3-DOF SPM.One of the principal reasons is the case that the configuration of the 4-DOF SPM is much smaller than the 3-DOF SPM.In view of this situation, domestic and foreign scholars have carried on the related theoretical research and application to the 4-DOF SPM.
In 1983, Hunt [1] proposed a classical mixed 3-RPS PM, and many literatures could be found about the 3-RPS PM related research work.The kinematic analyses concerning instantaneous motions analysis, dynamic modeling, synthesis, workspace analysis, position analysis, and the like can be found in the literature [2][3][4][5][6][7][8].The 2-DOF PMs of Diamond [9], the 3-DOF PMs of Trivariant [10], Tricept [11], and Delta [12] have been widely applied for industrial treatment.Zeng [13] proposed a series of 2-DOF rotational decoupled PMs based on the mechanism of four-bar linkage.Li [14] researched the type synthesis of the RPR equivalent PM and proposed a new architecture with encouraging potential in practice.Kong [15] firstly used the virtual chain method carried out the PMs type synthesis with a variety of operating modes.The type synthesis of 3-DOF cable driven PMs was also researched by Khakpour [16].Compared with the 2-DOF or 3-DOF counterparts, the 4-DOF PMs are seldom used in practice.Rolland [17] used parallelogram to investigate two 4-DOF parallel mechanisms to eliminate the rotation.Zhao [18] investigated a 4-URU parallel mechanism with the symmetrical topology, which has one rotational DOF and three translational DOFs along the normal of the base.Araujo-Gómez [19] proposed a two translational and two rotational (2T2R) four-degrees-of-freedom (DOF) parallel kinematic mechanism (PKM) designed as a knee rehabilitation and diagnosis mechatronics system.Q. zhao [20] presents the evolution process of pressure angles from planar parallel mechanisms to spatial parallel mechanisms.L. M. Zhang [21] deals with dynamic dimensional synthesis of the Delta robot using the pressure/transmission angle constraints.
In this paper, a 2T2R 4-DOF 2-RPU & 2-SPS SPM is investigated and it has a lot of advantages.Firstly, the mechanism moving platform can realize the output of two rotational and two mobile decoupling motions.Secondly, because of the symmetrical arrangement of four moving limbs, the mechanism has a large translational motion space and rotational motion space.Thirdly, the inverse and forward analysis of the mechanism is simple, which simplifies the series of technical problems such as trajectory planning, control, and correction, which are conducive to the application of the mechanism in the field of parallel machine tool equipment.In this paper, firstly, the number and property of degrees of freedom (DOF) of the SPM are analyzed by screw theory.There are two rotational and two translational movements (2R2T) of the mechanism that can be achieved.Secondly, the inverse and forward position analyses of the mechanism are researched.Then we obtained the workspace of the mechanism based on the analytic formulae of the inverse displacement.Finally, by establishing the Jacobian matrix of the mechanism, the singularity of the mechanism is obtained, and the kinematics transmission performance of the mechanism is studied by using the index of the output efficiency of the limb output of the mechanism.This work will provide the theoretical basis for prototype development and application of the mechanism.

Structure Characteristics of the 2-RPU&2-SPS SPM
As shown in Figure 1, the investigated 4-DOF 2-RPU&2-SPS SPM composed of a moving platform and a base platform and both platforms are attached by two same SPS limbs and two same RPU limbs.The spherical, revolute, prismatic, and universal joints can be abbreviated as S, R, P, and U, respectively, and the underlined p represents actuated joints.Right for the researched SPM, the two same RPU limbs connect the base platform to the moving platform with R, P, and U joints in sequence.Similarly, two identical SPS limbs connect the base platform to the moving platform by S, P, and S joints in sequence.
In order to establish the position analysis model of the mechanism as shown in Figure 2, place the reference frame  −  attached to the base platform with  located in the center of this quadrilateral, and the  axis is collinear with  2  4 and points from point  to point  2 .The  axis is collinear with  1  3 and points from point  to point  3 , and  the  axis is perpendicular to the base platform.The moving frame  − V is established on the moving platform, where point  is the center of the moving platform, the  axis is collinear with  2  4 and points from point A to point  2 , the V axis is collinear with  1  3 and points from point  to point  3 , and the  axis is perpendicular to the moving platform.Here,  2 and  4 are the rotational axes of the revolute joints,  1 ,  3 ,  1 , and  3 are the centers of the spherical joint, and  2 and  4 , respectively, represent the universal joints center.
Note that, for the RPU limb, the axis of the revolute joint connected to the base platform is parallel to the  axis and the axis of the universal joint connected to the moving platform is also parallel to the  axis.The other axis of the universal joint is parallel to the  axis; the two axes of the universal joints are perpendicular to each other.For the SPS limb, the line connecting the two spherical joints connected to the base platform coincides with the  axis and the connection is symmetrical about the  axis on the base platform.The line connecting the two spherical joints connected to the moving platform coincides with the V axis and the connection is symmetrical about the  axis on the moving platform.
Through the spatial arrangement of the abovementioned kinematic pair, the two RPU limbs can only move in the plane of  and the movement of the two SPS limbs can make the moving platform produce a rotation around the  axis from the perspective of geometric constraints.This structure determines that the motion trajectory of the moving platform is the motion of a plane and the two rotations of the V axis and the  axis, respectively, thus forming a 2T2R 4-DOF SPM.
In order to research the 2-RPU&2-SPS SPM conveniently, the moving coordinate system  − V is rotated relative to the reference coordinate system  −  by  −  − , the corresponding Euler angles of the rotation transformation of each coordinate system are , , and , and the composition of the rotation matrix is where sin and cos can be abbreviated to "" and "", respectively, and u, k, and w represents the measure of the unit vector of the three coordinate axes of the moving coordinate system in the reference coordinate system.

The DOF Analysis of the 2-RPU&2-SPS SPM
The number of DOF of the 4-DOF 2-RPU&2-SPS SPM can be obtained by the calculation of the criterion ℎ −  ü   =  ( −  − 1) where  denotes the task space order,  stands for the links number,  is the kinematic pairs number,   represents the degrees of freedom of joint , V denotes the number of redundant constraints, and  represents the local degree of freedom.
We can quickly find the number of DOF of the 2-RPU&2-SPS SPM by using (2).However, it should be pointed out that one shortcoming of the criterion ℎ −  ü  is that it can only obtain the number of DOFs of the mechanism rather than indicate the attributes of the DOF, whether they are translational or rotational DOFs.Otherwise, we can analyze the motion of a 2-RPU&2-SPS SPM via the analysis of the screw theory effectively; then we can easily obtain motions of translation and rotation in three-dimensional space.In Figure 3 through the analysis of the screw theory, the following can be drawn: where Considering the reciprocity between twist and wrench, the constraint system for the RPU limb can be described as Similarly, kinematics of screw and reverse screw of the SPS limbs can also be obtained; since the kinematics screw of SPS limb is composed of six linear independence screws without constraint reverse screw, the two SPS limbs do not have constraints on the mechanism.
It can be seen from ( 4) that a plane along the y axis and a rotation around the z axis are constrained so that the 2-RPU & 2-SPS SPM has two translational degrees of freedom (along the x and z axes direction) and two rotational degrees of freedom (rotation around the x and y axes).

Mechanism Analysis of the Inverse Position.
As shown in Figure 2, the closed-loop vector equation is constructed, and the position vector r = (  )  of point A can be represented in the reference coordinate system − which gives where a  , b  are the position vectors of   and   , respectively.w  is the unit vector of limb . a  = Ra 0 , R is the rotation matrix of the mechanism, and a 0 is the measure of a  in the coordinate system.From the structure of the mechanism, we can see where  1 ,  Right for the RPU limb, making the dot product with c  on both sides of (5) leads to where c 2 = c 4 = (0 1 0)  .In the same way, adapting to the SPS limb, making the dot product with c  on both sides of (5), yields where c 1 = Rc 10 , c 3 = Rc 30 and c 10 = c 30 = (1 0 0).As shown in Figure 2, based on the mechanism geometric constraint, the two RPU limbs can just only move in the plane , so we can get Through ( 7), (8), and (9), we can obtain The position vector r = (  )  for a given reference point  can be determined by (10) for the corresponding attitude angle and the corresponding moving platform relative to the rotation table  of the base platform, and the length of each limb can be obtained by 4.2.Mechanism Analysis of the Forward Position.The relative position of the moving coordinate system in the reference coordinate system is obtained from the given length of the mechanism and a certain mathematical operation.Then according to (11) the forward position analysis of the 2-RPU&2-SPS SPM can be obtained Then according to (11) the mechanism analysis of the forward position can be obtained as Since the positive solution constrained equations of the 4-DOF SPM are complex, this paper will directly call the mathematical calculation software to solve the numerical solution of the nonlinear Equations ( 12)-(15).  1 and Figure 4 by calculating two given different sets of inputs (  ).Adapting to the mechanism analysis of the forward position, for the purpose of testifying the correlation of the analyses of the inverse and forward position, the outputs value of the inverse position analysis are served as the inputs ( 1  2  3  4 ) of the forward position analysis, and the output results can be obtained by solving the nonlinear equation system as shown in Table 2 and Figure 5.As shown in Figures 4 and 5, we can see that the output of the positive and inverse solution of the mechanism is consistently, which proves the accuracy of the calculation.

The Overall Jacobian Matrix of the Mechanism. Equation
(5) at time differentiating can be written as where "×" represents the cross product of the vectors, q  denotes the velocity of the linear actuator , and w  stands for the angular velocity of link, and     , ṙ = ( ẋ ẏ ż )  , and  = (      ), respectively, stand for the spatial linear velocity and the spatial angular velocity of the moving platform.Through dot multiplying on both sides of ( 16) with w  , the passive variables w  are eliminated; this leads to When the SPM is not in a singular configuration, (17) can be expressed as where Equation ( 18) denotes the solution of the inverse velocity for the 2-RPU&2-SPS SPM.Equations ( 7) and ( 8) at time differentiating, respectively, lead to When the mechanism is no singularity, ( 20) and ( 21) can be expressed as where ] and 0 represents a zero matrix of 4 × 1.
Simultaneously to (18) and ( 22) we can get where q = [ q  0 ], J 0 = [ J  J  ] is the overall Jacobian matrix of the 2-RPU&2-SPS SPM.Equation ( 10) at time differentiating can be written as The constrained Jacobian matrix of the2-RPU&2-SPS SPM is In combination with the geometric constraints of the 2-RPU&2-SPS SPM, the input parameters of the mechanism are set as −40 ∘ ⩽  ⩽ 40 ∘ , −40 ⩽  ⩽ 40 ∘ and the overall Jacobian of the mechanism is shown in Figure 6.

Singularity Analysis.
Singular configuration is in the process of the parallel mechanism operation to achieve a state special configuration; under this kind of posture or nearby, mechanism can get out of control, mainly for mechanism freedom change; stiffness and transmission performance are reduced, which more serious when it causes the damage of organization structure, so the singularity is necessary.
(1) When |A| ̸ =0 and |D|=0, which means that the determinant value of the input coefficient matrix is 0, then, the mechanism kinematic limb is in a boundary singular positon.That is, the output vector corresponding to a nonzero input vector is 0, so no velocity vector is generated at the output.In this configuration, the output element will lose one or more degrees of freedom while resisting one or more forces or moments when no torque is applied to the input.
(2) When |A|=0 and |D| ̸ =0, that is, when the determinant value of the output coefficient matrix is 0, the corresponding mechanism reaches a singular position in the workspace, which is called actuated singularity of the mechanism.In other words, when all the actuated elements are locked, the output elements of the mechanism can still be partially moved.In this configuration, the output element will gain one or more degrees of freedom while being unable to resist one or more forces or moments when the input is locked.
(3) When |A|=0 and |D|=0, the input-output coefficient matrix determinant value is also 0. If the mechanism result parameter satisfies certain specific conditions and the positional relationship reached by the limb satisfies the kinematics equation configuration, the configuration of the mechanism is called hybrid singularity.This singularity is characterized by the ability of the limb to withstand limited movement when the actuated element is locked or the limited input does not produce any output.Equation ( 23) shows the relation of the singularity configuration of the mechanism: A is the output coefficient matrix.D is the input coefficient matrix.
According to (27), the singularity configuration of the mechanism may be obtained: (1) Because the length of the limb is not zero, there is no configuration singularity and hybrid singularity. ( there is an actuated singularity, then we obtain  = 77.63∘ or 0 ∘ .

Velocity and Acceleration
. Simultaneous (23) and ( 24) can obtain the spatial linear velocity and the spatial angular velocity of the moving platform.
Differentiating (30) and (31) with respect to time, respectively, can obtain the spatial linear acceleration and the spatial angular acceleration of the moving platform.

Workspace Analysis
The working space of the parallel mechanism is one of the indexes of the comprehensive performance of the mechanism.It directly reflects the working ability of the mechanism, and its analysis results provide a theoretical basis for the design and application of the 2-RPU&2-SPS SPM.In Section 3, on the position analysis, the working space of the SPM is obtained by utilizing the search algorithm combined with the structural characteristics of 2-RPU&2-SPS SPM, the inverse kinematic solution, and the limitation of the limb length.
6.1.Constraints on Workspace.The parallel mechanism moving platform and base platform are relate to the limb, which needs to use the spherical joint and revolute joint; then the conditions of the spherical joint and revolute joint angle range need to be set.
(1) Limit of Limb where   represents the length of limb ,  min denotes the minimum length of limb , and  max stands for the maximum length of limb .
(2) Limit of Rotation Angle where   represents the angle of revolute joint ,  min denotes the minimum angle of revolute joint , and  max stands for the maximum angle of revolute joint .
where   represents the angle of revolute joint ,  min denotes the minimum angle of revolute joint , and  max stands for the maximum angle of revolute joint .

The Analysis of the Transmission Performance
One of the most important factors in designing and analyzing of the 2-RPU&2-SPS SPM is the evaluation of the performance, and there are many indicators of performance evaluation, such as dexterity, transmission angle, and torque/force transmission performance.Traditional transmission pressure angle is usually used to assess the limb of a single loop and  should not be utilized for the evaluation of spatial parallel mechanism.However, in this paper, the 2-RPU&2-SPS SPM with two identical RPU limbs and two identical SPS libms and the two RPU limbs move in the same plane, due to the these two rotating joint axes being parallel to each other.Based on the special structure of the research mechanism, the pressure angle can be served to evaluate the transmission performance of the SPM torque/force.The analysis of the 2-RPU&2-SPS SPM transmission performance in this paper is based on the pressure angle of the limbs, and the definition of the pressure angle is shown in Figures 11 and 12.
Right for the RPU limb, the acute angle   shown in Figure 11 is angle between the force (w 2 (w 4 )) generated at the same point of the two RPU limbs and the velocity (Qw 4 (−Qw 2 )) of point .It indicates that when the actuated joint is locked, there is transmission capability of the force/motion from limb 2(4) to limb 4(2);  is the angle between the w 2 and w 4 .Right for the SPS limb, the acute angle   shown in Figure 12 is angle between the force (w 1 (w 3 )) generated at the same point of the two SPS limbs and the velocity (Nw 1 (−Nw 3 )) of point .It indicates that when the actuated joint is locked, there is transmission capability of the force/motion from limb 1(3) to limb 3 (1). is the angle between the w 1 and w 3 .
For evaluating the transmission performance of the mechanism, the definitions of the transmission performance indicators are When the transmission efficiency index   of the limb is closer to 1, it indicates that the transmission efficiency of the mechanism is higher.On the contrary it is relatively low, and the range of transmission efficiency index is 0 ∼ 1.Through the research of the 2-RPU&2-SPS SPM and the definition and analysis of the transmission performance, the  software is used to simulate the transmission performance of the mechanism.According to the structural characteristics of the 2-RPU & 2-SPS SPM, the scale parameters of the mechanism are selected as   As shown in Figures 13 and 14, the output transmission performance of each limb increases with the increase of  under the condition of the scale parameter and working parameter of a given mechanism, which shows that the output transmission performance of the mechanism increases with the increase of  value which is big and getting better.When the  value is a certain value, the output transmission performance of the RPU limb increases with the increase of the angle .When  = 10 ∘ , the limb output transmission performance is the worst and the output transmission performance of the limb is independent of the size of the angle .Similarly, for the SPS limb, when the value of  is certainly, the output transmission performance of the limb decreases with the increase of the angle .when  = 0 ∘ , the output transmission performance of the limb is the best, at the same time, the output transmission performance of the limb increases with the increase of . when  = 10 ∘ , the output transmission performance of the mechanism is the worst.

Conclusions
In this paper, the DOF of the 2-RPU&2-SPS SPM is analyzed by using the screw theory.At the same time, the position analysis of the 2-RPU&2-SPS SPM is obtained by utilizing the vector method, and the inverse kinematic solution of the 2-RPU&2-SPS SPM is acquired.The forward kinematic solution of the 2-RPU&2-SPS SPM is obtained by using the mathematical calculation software.Based on the screw theory, the entire Jacobian matrix of the 2-RPU&2-SPS SPM is established and the mechanism working space is studied.Finally, through the definition of output efficiency index of limb output, the kinematic transmission performance of the mechanism is evaluated by MATLAB software simulation, which provides some reference value for the optimization design and dynamic research of the follow-up mechanism.

Figure 3 :
Figure 3: Twist system of the RPU limb.

4. 3 .
Numerical Examples.According to the forward and inverse position analysis of Sections 4.1 and 4.2, the selections of the mechanism parameters are  1 =  3 = 150 mm,  2 =  4 = 75 mm,  1 =  3 = 550 mm,  2 =  4 = 350 mm.Right for the inverse position analysis, the calculated output results are shown in Table

Figure 6 :
Figure 6: The overall Jacobian diagram of the mechanism.

6. 2 .
Workspace Analysis.According to the structural characteristics of the 2-RPU & 2-SPS SPM, the scale parameters of the mechanism are selected as  1 =  3 = 150 mm,  2 =  4 = 75 mm,  1 =  3 = 550 mm, and  2 =  4 = 350 mm, and the constraints of the mechanism are −30 ∘ ≤   ≤ 30 ∘ , −45 ∘ ≤   ≤ 45 ∘ , and 700 ≤   ≤ 1000, and the working space is shown in Figure 7.Because the 2-RPU&2-SPS SPM can only move along the  and  axes in the  plane and the rotation of the  and  axes, the working space of the mechanism in the  plane is about the symmetric distribution of the  axis in Figure8, and the working space is roughly in the range of −410 ≤  ≤ 410 (mm) and 655 ≤  ≤ 860 (mm); compared to the scale parameters of the mechanism, the work space of the mechanism is larger.The workspaces of the mechanism in  plane and  plane are shown in Figure9and Figure10.

Figure 7 :Figure 8 :
Figure 7: The entire workspace of the mechanism.

Figure 9 :
Figure 9: The workspace of the mechanism in  − .

Figure 10 :
Figure 10: The workspace of the mechanism in y -z.

Table 1 :
Mechanism analysis of the inverse position.

Table 2 :
Mechanism analysis of the forward position.