Kinematics Analysis and Singularity Avoidance of a Parallel Mechanism with Kinematic Redundancy

The kinematic redundancy is considered as a way to improve the performance of the parallel mechanism. In this paper, the kinematics performance of a three degree-of-freedom parallel mechanism with kinematic redundancy (3-DOF PM-KR) and the influence of redundant parts on the PM-KR are analyzed. Firstly, the kinematics model of the PM-KR is established. The inverse solutions, the Jacobian matrix, and the workspace of the PM-KR are solved. Secondly, the influence of redundancy on the PM-KR is analyzed. Since there exists kinematic redundancy, the PM-KR possesses fault-tolerant performance. By locking one actuating joint or two actuating joints simultaneously, the fault-tolerant workspace is obtained. When the position of the redundant part is changed, the workspace and singularity will be changed. The results show that kinematic redundancy can be used to avoid singularity. Finally, the simulations are performed to prove the theoretical analysis.


Introduction
The Parallel mechanism is defined as a closed-loop kinematic chain mechanism whose end-effector is linked to the fixed base by several independent kinematic chains [1]. Many scholars claim that it has some advantages of high stiffness, high accuracy, and small error accumulation. However, it also has some disadvantages, such as many singularities in the workspace and difficulty to eliminate self-motion. Many scholars tried to improve the performance of the parallel mechanism by adding redundant actuators or redundant structures.
Scientist Merlet [2] pointed out that redundancy is a way to improve parallel mechanism's performance. For parallel mechanisms with redundant actuators, the number of actuators is greater than the mobility of the parallel mechanism. Many researchers proved that redundant actuators could be used to solve the forward kinematics [3], avoid singularities [4,5], and optimize the distribution of force/torque [6,7,8]. However, the redundant actuators need strict movement coordination, which increases the control difficulty.
In order to overcome the shortcomings and realize the selfcoordination between inputs and outputs, some scholars proposed to design parallel mechanisms with redundant kinematic structure. For parallel mechanism with kinematic redundancy, the mobility of the parallel mechanism is equal to the number of actuators but greater than the degree of freedom of the moving platform. Except the earlier studies [9,10,11], many researchers play great attentions on parallel mechanism with kinematic redundancy in recent years.
The Gosselin group conducted a lot of relevant analysis for the kinematically redundant parallel mechanism. Wen and Gosselin [12,13] proposed a (6+3) DOF kinematically redundant hybrid parallel robot and did a lot of kinematic analysis on it, and proved all the singularities of this mechanism are avoidable, indicating that this mechanism has a large workspace. They introduced a new method for detecting mechanical interferences between two links that are not directly connected, proposed for evaluating the workspace of the kinematically redundant parallel mechanism. Landuré and Gosselin [14] proposed a spherical parallel robot and proved that its workspace is huge. The analysis of the mechanism's singularity shows that the kinematically redundant parallel mechanism's workspace has design advantages. Mats and Gosselin [15] provides an analysis of a class of kinematically redundant parallel manipulators. They explored the singularity of this kind of parallel mechanism with kinematic redundancy by using the screw theory.
Many researchers explored the type synthesis of the parallel mechanism with kinematic redundancy. Qu [16,17] discussed the difference between a parallel manipulator's mobility and the relative degree of freedom. Based on the proposed relative degree-of-freedom criterion, they performed the type synthesis of the parallel manipulator with open-loop limbs or closed-loop limbs. Besides, they synthesized a kind of kinematically redundant parallel manipulator based on the modified G-K formula and RDOF criterion. Li [18] summarized several types of parallel mechanisms with 2R1T redundant parallel mechanisms and 2T1R redundant parallel mechanisms, and the results showed that all kinematic redundant mechanisms have good performance. Wang [19] proposed a hybrid strategy based on the combination of linear decoupling geometric analysis method and highorder convergence iteration method to solve the positive solution of parallel mechanism. Wang [20] analyzed the topology and kinematics of the 6-DOF parallel mechanism with kinematic redundancy, and then solved the direct solution of the mechanism.
Studies had shown that the kinematics redundant parallel mechanism has dramatically improved its kinematic performance in its mechanism singularity and workspace [21,22,23]. Li [24] proposed a new 3-PRPR parallel mechanism with kinematic redundancy, and analyzed the workspace of the mechanism under different trace, and proved that the workspace of the parallel mechanism with kinematic redundancy was significantly increased. Zhao [25] proposed a 3-RPS/3-SPS parallel mechanism with redundant limbs and defined the evaluation index of the workspace. The results showed that the workspace and kinematics performance of the mechanism was improved. Jin [26] designed a type of 2RIT kinematics redundant parallel mechanism, analyzed the inverse kinematics and workspace of this mechanism, and proved that it had a larger two-dimensional rotation angle of the workspace. By studying the singularity and workspace of the parallel mechanism with kinematic redundancy, changing the mechanism's structure can be used to optimize the geometry of the mechanism.
For the redundant kinematics parallel mechanism, the influence of redundant kinematics components on the mechanism is explored by solving the statics and dynamics. Qu [28] analyzed the statics of a planar parallel mechanism with kinematic redundancy, and its actuated torque was significantly lower than that of the 3-RRR parallel mechanism. This showed that the introduction of the kinematic redundant components could change the force transmission of the mechanism and improve the statics performance. Rahmat Abadi [29] took a 3-RPRR planar kinematics redundant mechanism as the research object and used the principle of virtual work to obtain the inverse dynamics of the mechanism. The comparison with the nonredundant mechanism showed that the mechanism could avoid the singular position by adding redundant components. Schreiber [30] studied the dynamic trace planning for a 3-DOF space parallel robot and established its dynamic model.
The above analysis proved that adding kinematic redundant components to the parallel mechanism has a positive effect on the optimization of mechanical structure and the improvement of mechanical performance. This paper takes a 3-DOF parallel mechanism with kinematic redundancy (PM-KR) as the research object. By building its structure model, the singular position and workspace of the mechanism can be solved. Through analyzing the kinematic redundant component's effect on the singularity and workspace，we know that the parallel mechanism with kinematic redundancy has different kinematic performance compared to the general parallel mechanism. The kinematics simulation of the three singular positions of the mechanism is carried out, exploring the impact of changes in the structural parameters of redundant components on the performance of avoiding singular positions, which can prove that this kinematic redundant structure design of the mechanism has obvious advantages.

Structure description
The studied parallel mechanism with kinematic redundancy is shown in Fig. 1. This parallel mechanism consists of a fixed base, an equilateral triangular moving platform, three kinematic limbs with the same structure and a lifting platform. Qu [17] proved that the moving platform has three independent movements, and the mobility of the whole mechanism is equal to 4. The 4 prismatic joints connected to the base are selected as the actuated joints, respectively actuated joints L1, L2, L3 and L4.
Lifting platform L 1 Limb 2 Figure 1 Three-dimensional structure model. One side of the limbs of the mechanism is connected to the base by a prismatic joint, and these three prismatic joints are all actuated joints. The other side is connected to the moving platform through a ball joint. The rotation joints of each limb are passive and are parallel to each other. Ai，Ci，Di and Ei are the center points of the rotating joints in each limb, Bi is the center point of the spherical joint in each limb, P4 is the center point of the lifting platform, the link CiBi and the link DiEi pass through the rotating joint Di Connect, Di is a point of the link CiBi. Establish the coordinate systems O-xyz and P-uvw with the x-axis and the u-axis perpendicular to the axes of revolute joints of the limb 1.

Kinematics inverse solution
The specific process to solve the inverse kinematics solution of the mechanism is shown in Fig. 2 The inverse kinematics solution process of the mechanism.
As shown in Fig. 3, it is the geometric structure of the mechanism. The base coordinate system {O} is O-xyz with the y-axis pointing out of the paper and x-axis aligned with the axis of prismatic joint Pi. The moving platform coordinate system {P} is P-uvw with w-axis perpendicular to the u-axis and the v-axis.
Base Moving platform Figure 3 The geometric structure of the mechanism.
The position of the center point P of the moving platform can be defined as shown in the following Eq. (1).
Besides, the rotation matrix O RP is used to define the position of the moving platform relative to the base coordinate system {O}, using the expression of pitch-roll-yaw, as shown in the following Eqs.
In the above Eqs. (2)-(3), α, β and γ are respectively expressed as Euler angles of the moving platform about the x, y,and z axes. Therefore, according to the Eq. (4), the midpoint's representation on the moving platform in {O} can be obtained.
The positions of the three spherical joints in the moving platform coordinate system {B} are expressed as.
[ ] According to the Eq. (5), the transformation formula of the three spherical joints can be obtained, as shown in the following Eq. (6).
  (6) Through calculation, the basic coordinate vector representation of Bi (i=1,2,3) can be obtained, as shown in the following Eqs. (7)- (9 According to the position coordinates of Ai and Bi, the distance between the joint Ai and the joint Bi can be calculated, as shown in the following Eq. (11).
, the dot product of the two vectors is equal to 0, and the expressions of px，py and γ can be obtained, as shown in the following Eqs.
According to the above method, the coordinate representation of the remaining joints on the limb can be obtained sequentially.
The inverse solution of the mechanism is that the parameters α, β and zp of the moving platform are known, and the actuated lengths L1, L2 and L3 of the input prismatic joints are calculated. According to the vector closed-loop principle, a vector closed-loop equation is established in a limb of this mechanism, as shown in Fig. 4 below, the four vector combinations OA1, A1C1, C1B1, B1O are selected as the closed-loop.
Moving platform Base Figure 4 Diagram of closed-loop 1.
According to Fig. 4, the closed-loop equation 1 of limb 1 can be obtained, as shown in the following Eq. (16).

OP P A A C C B OB
According to the above Eq. (16), and replaced with the structural parameters of the mechanism, as shown in the following Eq.
Because the mechanism is a parallel mechanism with kinematic redundancy, its redundant structure is a lifting platform. The two closed-loop equations of the limb are solved simultaneously, as shown in Fig. 5 and 6 below.
Moving platform Base Figure 5 Diagram of closed-loop 2.
According to Fig. 5, the closed-loop equation 2 of limb 1 can be obtained as follows, as shown in the following Eq. (18).  (18) According to the above Eq. (18), and replaced with the mechanism's structural parameters, as shown in the following Eq. (19).

OP P A A C C D D E OE
Moving platform Base Figure 6 Diagram of closed-loop 3.
As shown in Fig. 6. The closed-loop equation 3 of limb 1 can be obtained, as shown in the following Eq. (20). 1 1 11 11 ED DB EB +=    (20) According to the above Eq. (20), and replaced with the structural parameters of the mechanism, as shown in the following Eq. (21). 4 13 3 Expand the above Eqs. (16)- (21) to get six equations, as shown in the following Eq. (22).
According to the coordinate expressions of B1 and E1, take D1 as the center of the circle, B1D1, and E1D1 as the radius. The equations of the two circles are listed respectively, and the coordinates of D1 can be solved, as shown in the following Eqs. (23)- (24).
According to the coordinates of B1 and D1 and the tangent formula of the triangle, the size of θ13 can be solved, as shown in the following Eq. (25).
According to the above Eq. (27), the expression of θ11 can be obtained, as shown in the following Eq. (28).
In the same way, the expressions for actuated lengths L2 and L3 can be obtained by the closed-loop equations of limb 2 and limb 3, as shown in the following Eqs. (30)-(31).

Singularity analysis
The positive Jacobian matrix Jq is the relationship between the twist χ  of the motion platform expressed by the six-dimensional vector and the rate of changing Li of the actuated length, as shown in the following Eq. (32).
The Jacobian matrix of the parallel mechanism with kinematic redundancy is the relationship between the angular velocity ω of the moving platform and the vector L1, L2, L3 of the actuated joint change rate, as shown in the Eq. (33).
The relationship between the inverse Jacobian matrix Jx, the threedimensional twist vector χ  of the moving platform, and the angular velocity ω is shown in the following Eq. (34).
In the Eq. (34), χ  represents the moving platform's twist, A vp is the speed of the moving platform, and ω is the angular velocity of the moving platform.
Calculate the partial derivative of the time t in the actuated parameters L1, L2, L3 and the moving platform motion parameters α, β, zp. After simplification, the Eqs. (38)-(40) can be obtained.   By analyzing the Jacobian matrix and singular image of the mechanism, the singular position of the mechanism can be obtained [31,32,33]. When θi1 =π/2 and θi2=0°of a limb, the mechanism will have a singular position①; When θi1 =θi2=π/2 of the three limbs, the mechanism will have a singular position②; When θi1 =0° of a limb, the mechanism produces a singular position③. The redundant components of the mechanism called lifting platform, its position will affect the singularity of the mechanism, which will be analyzed in the following content.
Analyze the workspace of the mechanism by adopting the workspace search method. Set the length of the actuating joint L4 as a fixed value and obtain the point set where the workspace satisfies the inverse solution of the mechanism. Finally, we can get the workspace diagram of the mechanism.
Analyzing limb 1, according to the expressions (25)-(28) of θ11, θ12, θ13 obtained by the inverse solution, substituting the above three equations into L1 expression (29), we can get L1's expression only have the moving platform kinematic parameters include α, β, and zp. Similarly, we can get L1, and L2's expression only has the moving platform kinematic parameters include α, β, and zp.
Set the value range of L1, L2, and L3 to [50,200], the range of zp to [50,350], and the value range of α and β to [-π/2, π/2]. L4 is used as a structural parameter, and the workspace of the mechanism can be calculated when L4=90mm, as shown in Fig. 9. Use the method of fitting a surface with a set of points and fitting the workspace point set of the mechanism, which can get the surface image of the workspace of the mechanism [34], as shown in Fig. 10 . Analyze the workspace in the above Fig. 10, and solve the workspace diagram of the mechanism when zp=230 /250 /270 /290mm.The moving platform parameters α and β are variables, and zp is a fixed value. As shown in Fig. 11  By analyzing the workspace images of the mechanism, we can get the changes of the moving platform kinematics parameters α and β when the moving platform parameter zp is a fixed value. As shown in Table 1 below. The above Table 1 shows that when the moving platform is in working condition, the maximum angle α and β of its rotation around the x and y axes can reach 66°. It shows that the moving platform of this mechanism has the characteristics of a large turning angle compared with the general parallel mechanism, which can meet the needs of specific working conditions.

The influence of redundant components on mechanism's kinematics
Adding a redundant part to a general parallel mechanism, the change of its structural parameters will affect the mechanism's kinematic characteristics, such as fault tolerance, workspace, and singularity [35]. This section conducts research on the impact of redundant components, analyzing the influence of the change of redundant part's parameters on the mechanism's kinematics.

Analysis of mechanism fault tolerance
Fault tolerance refers to the change in the mechanism's kinematic performance when a part of the mechanism fails. The better the faulttolerant mechanism is, the smaller the difference in the mechanism's kinematics when the components fail. While for the mechanism with poor fault-tolerant performance, the components' failure will significantly impact the overall kinematics of the mechanism.
There are many indexes for measuring the fault tolerance performance of a mechanism, such as changes in the workspace, changes in actuated torque, and each joint's integrity. In measuring the fault-tolerant performance of the 3-DOF PM-KR, the change of the workspace is selected as the performance index. When a specific actuated joint of the mechanism fails, the influence on the mechanism's workspace will be very obvious. By observing the failure of one actuated joint of the mechanism and the failure of two actuated joints simultaneously, the change of the workspace of the mechanism is used to test the fault tolerance performance of the mechanism.
As shown in Fig. 12(a)~(f) below. It is the image of the workspace of the mechanism when the machine is operating normally, the actuated joint P1 fails, the actuated joint P1 and P2 all fails.  In order to analyze the fault-tolerant performance of the mechanism more deeply, the point search method is adopted, and the distance between the points is 3mm. First, the number of points in the workspace when the mechanism is normal is used as the base. Then the effective proportion of the workspace is analyzed when one actuated joint fails and two actuated joints fail simultaneously. As shown in Table 2, it can be found that when the kinematic parameters of the moving platform zp=210/230/250/310mm, the workspace when the actuated joint P1 fails is about 60% of the normal workspace. When the actuated joint P1 and P2 fail simultaneously, the workspace is about equal to 30% of the normal workspace, indicating that the mechanism has better performance in this moving platform height.
When the moving platform's kinematic parameters are zp=270/290mm, the workspace when the actuated joint P1 fails is about 30% of the normal workspace. When the actuated joint P1 and P2 fail simultaneously, the workspace is about equal to 10% of the normal workspace. Compared with the fault tolerance of the mechanism obtained by the previous parameters, the mechanism's fault tolerance performance decreases.
The position where the actuated joint fails also has an impact on the fault tolerance of the mechanism. Measure the influence of the position where the actuated joint fails on the workspace. When the actuated joint P1 fails, P1 and P2 fail simultaneously. As shown in Table 3, it is the influence of the drive joint failure's position on the workspace. Through the analysis of the above table 3, we can find that when the actuated joint fails at [50mm, 100mm], the workspace when the actuated joint P1 fails is 10.91% of the normal workspace, the workspace when the actuated joint P1 and P2 fails simultaneously is 1.9% of the normal workspace. Because this position of the actuated joint is very close to the center of the whole mechanism, indicating that the actuated joint has a fault at this position, the fault tolerance of the mechanism is very poor. When the fault position of the actuated joint is [100mm, 150mm], the fault tolerance of the mechanism is improved, the workspace when the actuated joint P1 fails is 29.68% of the normal workspace, the workspace when the actuated joint P1 and P2 fail simultaneously is 11.86% of the normal workspace. When the actuated joint's fault position is [150mm, 200mm], the fault tolerance of the mechanism is excellent, and the position of the actuated joint is the farthest from the center of the mechanism.
In summary, the 3-DOF PM-KR has a better fault tolerance. By analyzing the different height values zp of the moving platform, the fault tolerance of the mechanism is improved, the workspace when the actuated joint P1 fails is a maximum 60.78% of the normal workspace, the workspace when the actuated joint P1 and P2 fail simultaneously is maximum 32.35% of the normal workspace. At the same time, the position where the actuated joint fails also affects the fault tolerance of the mechanism. When the actuated joint fails is farther away from the center, the better the fault tolerance of the mechanism; on the contrary, the worse the fault tolerance of the mechanism.

The impact of redundant components on the workspace
Adding redundant components to the parallel mechanism will have a specific impact on the entire mechanism's workspace [36]. In the 3-DOF PM-KR, the lifting platform is connected with the link CiBi through three redundant link DiEi (i=1, 2, 3) in rotating joints. The lifting platform's height change will cause the position and posture of each link in the limb to change, which will affect the workspace of the mechanism.
Set the kinematic parameter zp=250mm of the moving platform as a fixed value, and research the influence of the change of the height of the lifting platform L4 on the kinematic parameters α and β of the moving platform. As shown in Fig. 13 below, it is the changing image of the workspace of the mechanism when L4=60, 65, 70, 75mm. By observing the workspace' image, we can conclude: With the increase of the value of L4, the workspace of the mechanism also becomes larger, indicating that the increase in the structural parameters of redundant components will increase the workspace of the mechanism in this posture.
As shown in Fig. 14 below, it is an image of the workspace when L4=80, 85, 90, 95mm. In this posture, when the value of L4 increases from 80mm to 90mm, it can be observed that the workspace continues to increase; but when the value of L4 continues to increase, because the lifting platform is located in the center of the entire mechanism, it's position has little influence on the posture of the limb, so the change of the workspace of the mechanism is not apparent. Continue to increase the value of L4, and take the changing image of the workspace when L4=100, 105, 110, 115mm, as shown in Fig. 15 below. By observing the posture change of the workspace, we can explain that as the value of L4 increases in this range, the mechanism's workspace is continuously reduced. Since the kinematic parameter zp of the moving platform is set to a fixed value, in this posture, the lifting platform's height is constantly rising close to the moving platform. At this time, the mechanism's posture inhibits the movement of the limb, which makes the workspace of the mechanism continuously reduced. Analyzing the mechanism's workspace shows that the change of the kinematic redundant part's position significantly impacts the workspace. When the lifting platform is at the starting position, as its height value increases, the mechanism's workspace also becomes larger. Because each mechanism's workspace is fixed, when the position of the lifting platform reaches a specific height value, the change of the workspace of the mechanism is not obvious at this time. As the lifting platform's height value continues to increase, its position slowly approaches the moving platform of the mechanism, and the redundant part has a significant inhibitory effect on the movement of the limb, which causes the working space to occur significantly reduced.

The effect of redundant components on singularity
The structural parameter's change of the redundant part will impact the mechanism's kinematics, the most significant of which is to change the mechanism to avoid the singularity. Kinematics analysis obtains the Jacobian matrix of the mechanism and then solves the mechanism's singularity. Finally, we can obtain the singular position of the motion posture of the mechanism. Research on the kinematics parameter zp value of the moving platform at different height values by changing the values of the redundant part L4, the change of the singular images of the mechanism, and analyze the influence of the redundant components on the singularity avoidance of the mechanism.
As shown in Fig. 16 below, it is a singular image when the moving platform's kinematics parameter is zp=150mm. When the value of L4 is increased from 60mm to 100mm, the singular position's image is significantly reduced. Indicating that at this position, as the lifting platform's height value increased, the point set of the singular position of the mechanism continued to shrink. When the value of actuated joint L4 becomes larger, observe the change of the singular position's range of the mechanism. As shown in Fig. 17 below, it is a singular image when the kinematics parameter zp of the moving platform of the mechanism is 170mm, the value range of L4 is [60mm,100mm]. Compared with the singular images, when the height of the moving platform zp is equal to 150mm, the point set of the mechanism's singular position at this height is increased. However, with the increase of the lifting platform's height, the point set of the mechanism's singular position has decreased significantly.  Research shows that the change of the kinematic parameters of redundant components has an obvious influence on the mechanism's singular performance. When the moving platform is at a certain height, the lifting platform's height change will reduce the singular position of the mechanism. It also shows that in this 3-DOF parallel mechanism, the addition of redundant components will positively affect the kinematics of the mechanism. When the mechanism is in a singular position, the mechanism's posture can be changed by changing the height of the lifting platform，which can achieve the purpose of avoiding singularity for the mechanism.

Simulations of mechanism avoiding singular position under three actions
In the previous section, the study found that the 3-DOF PM-KR has the characteristics of avoiding singularity. When the mechanism occurs singular positions, it can avoid the singular positions by changing the lifting platform's height. In this section, the Adams software is used to simulate the three singular positions of the mechanism. By changing the position of the lifting platform, the singular position of the mechanism can be avoided, which can verify the kinematic characteristics of the mechanism.
The mechanism's initial position is shown in Fig. 20. The height of the moving platform zp=170mm, the deflection angle α=β=0°, the height of the lifting platform L4=60mm, and the three actuated parameters on the base L1=L2=L3=135mm. Set link AiCi as link 1, link DiEi as link 2, and link BiCi as link 3.
Lifting platform Figure 20 The initial position of the mechanism simulation. As shown in Fig. 21(a), it is the first singular position of the mechanism, and the actuated joint's positions are L1=215mm, L2=L3=100mm, L4=60mm. At this time, the angle θ11 between the mechanism link 1 and the base is approximately equal to 90°, the link 1 of the limb 1 is approximately perpendicular to the base, and the link 3 is approximately parallel to the base, and the moving platform occurs a large angle. To avoid this singular position, it can increase the height of the lifting platform. When L4 is equal to 100mm, the mechanism change's position and posture change, and the mechanism avoids the singular position, as shown in Fig. 21(b). In the simulation, the first stage is the process of singular positions of the simulation mechanism, the simulation time is 0~5s; the second stage is the process of avoiding the singular positions of the mechanism, the simulation time is 5~8s. In the simulation using Adams software, the rotation angle of the moving platform, revolving joint and spherical joint are measured to judge whether the mechanism successfully completes the action of avoiding singular positions.
As shown in Fig. 22, it is the numerical change curve of the angle θ12 between link 1 and link 3 in the simulation of the mechanism. At the 5th second, indicate that link 1 and link 3 are approximately perpendicular at this time. When the position of the lifting platform increases, the angle between the two links changes, which indicate that the result of avoiding singular positions is obvious.

Figure 22
The change curve of angle θ12 when the mechanism avoids singular position 1. Fig. 23 and Fig. 24 show the change curve of the angle of rotating joint C1 and spherical joint B1. At the 5th second, the deflection angle of the rotating joint C1 is approximately 83.17°, which indicates that the joint has happened a very large rotation, and the position of the mechanism is gradually approaching the singular position. After the 5th second, as the lifting platform's height increased, the deflection angle of the joint gradually decreased. At the 8th second, the angle was 50.33°, and the angle of the spherical joint B1 also changed, which indicates that the mechanism avoided singularity and returned to its normal position and posture.

Figure 23
The change curve of the angle of the rotating joint C1 when the mechanism avoids singular position 1.

Figure 24
Angle change curve of spherical pair B1 when the mechanism avoids singular position 1.
When the angle θi1 and θi2 of the mechanism are approximately equal to 90°, it is the mechanism's second singular position. The actuated joint's positions are L1=L2=L3=100mm and L4=100mm, the link 1 and link 3 of the three limbs are approximately perpendicular to the base, as shown in Fig. 25(a) below. By decreasing the height of the lifting platform, the mechanism can avoid singularity. When L4=60mm, the posture of the limb changes and the height of the moving platform decreases simultaneously, and the mechanism returns to its normal posture, as shown in Fig. 25(b) below. As shown in Fig. 26 below, it is the change curve of the angle of rotating joint A1 and rotating joint C1 when the mechanism avoids singular position 2. In the 5th second, the rotating joint A1 and rotating joint C1 angles reached their maximum values of 68.34° and 126.45°. After the 5th second, the lifting platform's height is reduced, and the angles of the two rotating joints are reduced to 21.28° and 58.27°, which indicates that the mechanism has completed the avoidance of singular positions.

Figure 26
The change curve of the angle of the rotating joint A1 and C1 when the mechanism avoids singular position 2.
When the angle θi1 approximately equal to 0° is the third singular position of the mechanism, the actuated joint's positions are L1=L2=L3=143mm and L4=60mm, and the link 1 of the three limbs is approximately parallel to the base, as follows, as shown in Fig. 27(a) below. By increasing the height of the lifting platform, the mechanism can avoid singularity. When L4=100mm, the moving platform's height is increased, and the overall posture of the mechanism changes.  Fig. 28 and Fig. 29 below show the change curve of the angle of rotating joint A1 and rotating joint D1 when the mechanism avoids singular position 3. At the 5th second, the angle of the rotating joint A1 is 4.18°, close to 0°, which indicates that link 1 is approximately parallel to the base at this position. If the actuated joint continues to move, the mechanism will be singular. After the 5th second, increasing the height of the lifting platform, the angle of the rotating joint A1 has increased significantly to 43.21°, and the angle of the rotating joint D1 has also changed significantly, which can indicate that the height of the lifting platform makes the mechanism successfully avoid the singularity.

Figure 28
The change curve of the angle of the rotating joint A1 when the mechanism avoids singular position 3.

Figure 29
The change curve of the angle of the rotating joint D1 when the mechanism avoids singular position 3.

Discussion
It is proved that the singularities can be avoided by changing the lifting platform's height when the mechanism is near or already in the singular positions. It also shows that the redundant kinematics part significantly improves the mechanism's kinematics performance and proves that the 3-DOF PM-KR has the advantages of structural design.
However, there are some problems in the simulation research. For example, the simulation of the mechanism is too ideal and doesn't fully consider the influence of the errors of the mechanism's materials and joints on the simulation results; When the mechanism reaches different singular positions, the force and deformation between joints and links will affect the overall mechanism. The parameters changing of the mechanism will also affect the effect of avoiding singular positions. In the follow-up research, we should solve these known problems as much as possible.

Conclusions
In this paper, the kinematics performance of a 3-DOF PM-KR and the influence of redundant part on the mechanism are analyzed. The main work and conclusions are drawn as follows: (1) The kinematics of the PM-KR is solved. Since there are infinite number of solutions for the inverse kinematics of PM-KR, the actuated parameter of kinematic redundancy should be set as a constant firstly. Different actuated parameter of kinematic redundancy lead to different solutions.
(2) The kinematic redundancy affects the workspace and singularity of the PM-KR. The workspace and singularity can be adjusted by changing the parameters of redundant part. When the parameter of the redundant part is increased, the singularity of the mechanism can be significantly reduced, which achieves the purpose of avoiding singularities of 3-DOF PM-KR.
(3) Since the existed kinematic redundancy, the PM-KR possesses the fault-tolerant performance. When the position of the failed actuated joint is far away from the center, the fault-tolerant performance of the PM-KR is better.