Analysis of Singular Configuration of Robotic Manipulators

: Robotic manipulators inevitably encounter singular configurations in the process of movement, which seriously affects their performance. Therefore, the identification of singular configurations is extremely important. However, serial manipulators that do not meet the Pieper criterion cannot obtain singular configurations through analytical methods. A joint angle parameterization method, used to obtain singular configurations, is here creatively proposed. First, an analytical method based on the Jacobian determinant and the proposed method were utilized to obtain their respective singular configurations of the Stanford manipulator. The singular configurations obtained through the two methods were consistent, which suggests that the proposed method can obtain singular configurations correctly. Then, the proposed method was applied to a seven-degree-of-freedom (7-DOF) serial manipulator and a planar 5R parallel manipulator. Finally, the correctness of the singular configurations of the 7-DOF serial manipulator was verified through the shape of the end-effector velocity ellipsoid, the value of the determinant, the value of the condition number, and the value of the manipulability measure. The correctness of singular configurations of the planar 5R parallel manipulator was verified through the value of the determinant, the value of the condition number, and the value of the manipulability measure.


Introduction
It is well-known that the kinematics of robotic manipulators can be expressed on the velocity level, and the relationship between the joint velocities and end-effector (EE) velocities is described by a Jacobian matrix. Robotic manipulators that do not meet the Pieper criterion [1] only obtain a numerical solution to the inverse kinematics, and the Jacobian iteration algorithm is a commonly algorithm for numerical solution. However, when robotic manipulators approach a singular configuration, the Jacobian matrix becomes numerically unrealizable, and this is experienced in the form of high joint velocities, which are not conducive to motion control. Therefore, the identification of singular configurations is a key issue to avoid singularity. Singularity in robotic manipulators includes boundary singularity and internal singularity [2]. Boundary singularity appears at the working space boundary, and internal singularity is caused by the coincidence of two or more joint axes. Robotic manipulators are divided into serial manipulators and parallel manipulators from the perspective of mechanism [3]. Müller [4] roughly classified the singularities of serial manipulators and parallel manipulators based on active and passive joints and proposed an algorithm to determine the singularities of the mechanism. Most researchers have performed singularity analysis on serial manipulators or parallel manipulators. The following concerns parallel manipulators. Li [5] introduced a cell-division method for of a Stanford manipulator are obtained through an analytical method in Section 3.1. In Section 3.2, a joint angle parameterization method is proposed to identify singular configurations and applied to the Stanford manipulator, a 7-DOF serial manipulator, and a planar 5R parallel manipulator. In Section 4, the correctness of the singular configurations of the 7-DOF serial manipulator obtained by the proposed algorithm is verified through the shape of the EE velocity ellipsoid, the value of the determinant, the value of the condition number, and the value of the manipulability measure. The correctness of the singular configurations of the planar 5R parallel manipulator obtained by the proposed algorithm is verified through the value of the determinant, the value of the condition number, and the value of the manipulability measure. Finally, the conclusions and direction for future work are presented in Section 5.

Related Work
The forward kinematics on the velocity level can be described as follows: where .
x ∈ R m is the velocity vector of the EE, . q ∈ R n represents the velocity vector of the joint, and J ∈ R m×n is a Jacobian matrix.
The inverse kinematics on the velocity level for non-redundant manipulators is: .
x (2) and for redundant manipulators it is Equations (2) and (3) hold when J −1 or J + exists, and Equation (3) provides a least-norm solution.
Singularity occurs when the determinant of the Jacobian matrix is zero: Singular value decomposition [25,26] can explain the effect of singularity more clearly.

A Novel Singularity Identification Method
First, singular configurations of a Stanford manipulator are obtained through an analytical method based on the Jacobian determinant. Then, a singular configuration identification method based on joint angle parameterization is proposed and applied to the Stanford manipulator. Singular configurations obtained through the proposed method are compared with the results through the analytical method, which verifies the correctness of the proposed method. Furthermore, the proposed method is applied to a 7-DOF serial manipulator and a planar 5R parallel manipulator.

Determining Singular Configurations of a Stanford Manipulator through an Analytical Method
The Stanford manipulator [27] is a classic industrial manipulator that meets the Pieper criterion. Singular configurations of this manipulator can be obtained through an analytical method. The coordinate system and standard DH parameters are shown in Figure 1 and Table 1.

A Singular Configuration Identification Method Based on Joint Angle Parameterization
This section and Section 4 were developed using the MATLAB R2015a tool, and an Intel Core™ i5-2450M CPU @ 2.50 GHz and 2 GB RAM control platform was used to run it.
In view of the fact that singular configurations of robotic manipulators appear when a joint angle is 0 , π 2 ± , π ± , or more, the joint angles are here considered to be 0 , π 2 ± , π ± . A joint angle parameterization method to determine singular configurations based on these special angles is proposed. The method steps are as follows: (1) A group of joint positions to be applied in the subsequent steps are arbitrarily chosen to satisfy (2) First, all joint positions are set to 0 , π 2 ± , π ± , respectively, and substituted into Equation (4) or Equation (5). If the determinant is not zero, it means that this group of joint positions will not produce singularity, and these joint positions can be ig-

A Singular Configuration Identification Method Based on Joint Angle Parameterization
This section and Section 4 were developed using the MATLAB R2015a tool, and an Intel Core™ i5-2450M CPU @ 2.50 GHz and 2 GB RAM control platform was used to run it.
In view of the fact that singular configurations of robotic manipulators appear when a joint angle is 0, ± π/2, ±π, or more, the joint angles are here considered to be 0, ± π/2, ±π. A joint angle parameterization method to determine singular configurations based on these special angles is proposed. The method steps are as follows: (1) A group of joint positions to be applied in the subsequent steps are arbitrarily chosen to satisfy det(J) = 0; (2) First, all joint positions are set to 0, ± π/2, ±π, respectively, and substituted into Equation (4) or Equation (5). If the determinant is not zero, it means that this group of joint positions will not produce singularity, and these joint positions can be ignored in the subsequent steps. Then, on the basis of the set of joint positions in step 1, a joint position is selected and set to 0, ± π/2, ±π. From the remaining joints, a joint is selected and varied within its range, and the other joint positions remain unchanged. Finally, the distribution of the minimum singular value with the change in a joint position is obtained. For example, for a 6-DOF manipulator, q 1 = 0 , q 3 = q 4 = q 5 = q 6 = π/3, and q 2 ∈ [−π, π] are set. Finally, the distribution of the minimum singular value with the change in q 2 is obtained. In the same way, the distributions of the minimum singular values with the changes of q 3 , q 4 , q 5 , and q 6 are also obtained; (3) On the basis of the set of joint positions in step 1, two joint positions are selected one by one and set to 0, ± π/2, ±π. From the remaining joints, a joint is selected and varied within its range, and the other joint positions remain unchanged. Finally, the distribution of the minimum singular value with the change in a joint position is obtained. For example, for a 6-DOF manipulator, q 1 = q 2 = 0 , q 4 = q 5 = q 6 = π/3, and q 3 ∈ [−π, π] are set. Finally, the distribution of the minimum singular value with the change in q 3 is obtained. In the same way, the distributions of the minimum singular values with the changes in q 4 , q 5 , and q 6 are also obtained; (4) The rest may be deduced by analogy: the distributions of the minimum singular values with the changes in all combined joint positions are obtained. When the minimum singular value is zero, singular configurations occur.
It can be seen that the proposed method does not need to calculate det(J) = 0 to obtain singular configurations. In other words, the proposed method eliminates the more complex mathematical derivation. Especially for redundant manipulators that do not meet the Pieper criterion, it is very difficult to obtain singular configurations through determinant transformation. Fortunately, the minimum singular value distribution curve obtained through the proposed method can clearly show some singular configurations.

Singular Analysis of the Stanford Manipulator Based on the Proposed Method
Analyzing the proposed method steps, in step (1), by setting q 1 = q 2 = q 4 = q 5 = q 6 = π/3 and d 3 = 0.3, and by using Equation (7), σ min = 0.1447; this set of joint positions is applied to the subsequent steps in this section.
In step (2), by setting q 1 = q 2 = q 4 = q 5 = q 6 = π/2, d 3 = 0.3, and by using Equation (7), σ min = 0.209; this shows that singularity does not occur. Furthermore, by setting q 1 = q 2 = q 4 = q 5 = q 6 = 0, ±π, d 3 = 0.3, and by using Equation (7), σ min = 0; this shows that singular configurations occur (one or more joints are 0, ±π). In addition, it can be seen in Figure 2 that the minimum singular value does not change with the change in q 1 and q 6 . Therefore, q 1 and q 6 can be ignored when using the proposed method to obtain singular configurations. meet the Pieper criterion, it is very difficult to obtain singular configurations through determinant transformation. Fortunately, the minimum singular value distribution curve obtained through the proposed method can clearly show some singular configurations. ; this set of joint positions is applied to the subsequent steps in this section.
In addition, it can be seen in Figure 2 that the minimum singular value does not change with the change in 1 q and 6 q . Therefore, 1 q and 6 q can be ignored when using the proposed method to obtain singular configurations. On the basis of the set of joint positions in step 1, a joint position is selected and set to 0 , π ± rad. From the remaining joints, a joint is selected and varied within its range, and the other joint positions remain unchanged. The distributions of the minimum singular values with the changes in 2 q , 4 q , and 5 q can then be obtained, as shown in Figure   3. The distribution of the minimum singular value with the change in 3 d can also be obtained, as shown in Figure 4. On the basis of the set of joint positions in step 1, a joint position is selected and set to 0, ±π rad. From the remaining joints, a joint is selected and varied within its range, and the other joint positions remain unchanged. The distributions of the minimum singular values with the changes in q 2 , q 4 , and q 5 can then be obtained, as shown in Figure 3. The distribution of the minimum singular value with the change in d 3 can also be obtained, as shown in Figure 4. (b) setting 4 = 0 q , π ± rad, the distributions of the minimum singular values with the changes in 2 q and 5 q ; (c) setting ± rad, the distributions of the minimum singular values with the changes in 2 q and 4 q .
As only 2 q , 3 d , 4 q , and 5 q need to be analyzed, there is no need to continue with the subsequent steps of the proposed method. Figures 3 and 4 show that, when 2 0 q = , π ± ; 3 0 d = ; and 5 0 q = , π ± , singular configurations occur. The results are the same as the

Singular Analysis of a 7-DOF Serial Manipulator Based on the Proposed Method
This section involves a 7-DOF serial manipulator developed by us for laparoscopic surgery that does not meet the Pieper criterion. The coordinate system is established as shown in Figure 5 and the modified DH parameters are shown in Table 2.  As only q 2 , d 3 , q 4 , and q 5 need to be analyzed, there is no need to continue with the subsequent steps of the proposed method. Figures 3 and 4 show that, when q 2 = 0, ±π; d 3 = 0 ; and q 5 = 0 , ±π, singular configurations occur. The results are the same as the singular configurations obtained through the analytical method in Section 3.1. This shows that the proposed method is correct.

Singular Analysis of a 7-DOF Serial Manipulator Based on the Proposed Method
This section involves a 7-DOF serial manipulator developed by us for laparoscopic surgery that does not meet the Pieper criterion. The coordinate system is established as shown in Figure 5 and the modified DH parameters are shown in Table 2.
According to the proposed method steps, in step (1), by setting q 2 = q 3 = q 4 = q 5 = q 6 = q 7 = π/3 and d 1 = 0.3, and by using Equation (8), σ min = 0.000472; this set of joint positions is applied to the subsequent steps in this section.
In step (2), by setting q 2 = q 3 = q 4 = q 5 = q 6 = q 7 = π/2 and d 1 = 0.3, and by using Equation (8), σ min = 0.000442; this shows that singularity does not occur when all joint positions are π/2. Furthermore, by setting q 2 = q 3 = q 4 = q 5 = q 6 = q 7 = 0, ±π and d 1 = 0.3, and by using Equation (8), σ min = 0; this shows that singular configurations occur (one or more joints are 0, ±π). In addition, it is found that the minimum singular  Figure 6. Also, the order of magnitude of the minimum singular value varying with q 6 is 10 −6 . Therefore, d 1 , q 6 , and q 7 can be ignored when using the proposed method to obtain singular configurations. According to the proposed method steps, in step (1), by setting , and by using Equation (8), min 0 σ = ; this shows that singular configurations occur (one or more joints are 0 , π ± ). In addition, it is found that the minimum singular value does not change with the change of 1 d and 7 q through Figure 6. Also, the order of magnitude of the minimum singular value varying with 6 q is 10 −6 . Therefore, 1 d , 6 q , and 7 q can be ignored when using the proposed method to obtain singular configurations.
On the basis of the set of joint positions in step 1, a joint position is selected and set to 0, ±π rad. From the remaining joints, a joint is selected and varied within its range, and the other joint positions remain unchanged. The distributions of the minimum singular values with the changes in q 2 , q 3 , q 4 , and q 5 can then be obtained, as shown in Figure 7.
In step (3), on the basis of the set of joint positions in step 1, two joint positions are selected one by one and set to 0, ±π rad. From the remaining joints, a joint is selected and varied within its range, and the other joint angles remain unchanged. The distributions of the minimum singular values with the changes in q 2 , q 3 , q 4 , and q 5 can then be obtained, as shown in Figure 8. On the basis of the set of joint positions in step 1, a joint position is selected and set to 0 , π ± rad. From the remaining joints, a joint is selected and varied within its range, and the other joint positions remain unchanged. The distributions of the minimum singular values with the changes in 2 q , 3 q , 4 q , and 5 q can then be obtained, as shown in Figure 7. In step (3), on the basis of the set of joint positions in step 1, two joint positions are selected one by one and set to 0 , π ± rad. From the remaining joints, a joint is selected and varied within its range, and the other joint angles remain unchanged. The distributions of the minimum singular values with the changes in 2 q , 3 q , 4 q , and 5 q can then be obtained, as shown in Figure 8.
In step (4), on the basis of the set of joint positions in step 1, three joint positions are selected one by one and set to 0 , π ± rad. From the remaining joints, a joint is selected and varied within its range, and the other joint angles remain unchanged. The distributions of the minimum singular values with the changes in 4 q and 5 q can then be obtained, as shown in Figure 9.  On the basis of the set of joint positions in step 1, a joint position is selected and set to 0 , π ± rad. From the remaining joints, a joint is selected and varied within its range, and the other joint positions remain unchanged. The distributions of the minimum singular values with the changes in 2 q , 3 q , 4 q , and 5 q can then be obtained, as shown in Figure 7. In step (3), on the basis of the set of joint positions in step 1, two joint positions are selected one by one and set to 0 , π ± rad. From the remaining joints, a joint is selected and varied within its range, and the other joint angles remain unchanged. The distributions of the minimum singular values with the changes in 2 q , 3 q , 4 q , and 5 q can then be obtained, as shown in Figure 8.
In step (4), on the basis of the set of joint positions in step 1, three joint positions are selected one by one and set to 0 , π ± rad. From the remaining joints, a joint is selected and varied within its range, and the other joint angles remain unchanged. The distributions of the minimum singular values with the changes in 4 q and 5 q can then be obtained, as shown in Figure 9.   , π ± rad, the distributions of the minimum singular values with the changes in 3 q , 4 q , and 5 q ; (b) setting 3 = 0 q , π ± rad, the distributions of the minimum singular values with the changes in 2 q , 4 q , and 5 q ; (c) setting 4 = 0 q , π ± rad, the distributions of the minimum singular values with the changes in 2 q , 3 q , and 5 q ; (d) setting 5 = 0 q , π ± rad, the distributions of the minimum singular values with the changes in 2 q , 3 q , and 4 q .  (b) setting q 2 = q 4 = 0, ±π rad, the distributions of the minimum singular values with the changes in q 3 and q 5 ; (c) setting q 2 = q 5 = 0, ±π rad, the distributions of the minimum singular values with the changes in q 3 and q 4 ; (d) setting q 3 = q 4 = 0, ±π rad, the distributions of the minimum singular values with the changes in q 2 and q 5 ; (e) setting q 3 = q 5 = 0, ±π rad, the distributions of the minimum singular values with the changes in q 2 and q 4 ; (f) setting q 4 = q 5 = 0, ±π rad, the distributions of the minimum singular values with the changes in q 2 and q 3 .
In step (4), on the basis of the set of joint positions in step 1, three joint positions are selected one by one and set to 0, ±π rad. From the remaining joints, a joint is selected and varied within its range, and the other joint angles remain unchanged. The distributions of the minimum singular values with the changes in q 4 and q 5 can then be obtained, as shown in Figure 9.

Singular Analysis of a Planar 5R Parallel Robot Based on the Proposed Method
To verify that the proposed method is also suitable for parallel manipulators, a planar 5R parallel manipulator [28] was taken as an example for analysis. The mechanism diagram is shown in Figure 10  Under the premise that the output pose is X and the drive is q , the robot's input-output relationship is: Figure 9. (a) Setting q 2 = q 3 = q 4 = 0, ±π rad, the distribution of the minimum singular value with the change in q 5 ; (b) setting q 2 = q 3 = q 5 = 0, ±π rad, the distribution of the minimum singular value with the change in q 4 . Figures 7-9 show that when q 2 = q 3 = q 4 = 0, ±π; q 2 = q 3 = 0 , ±π, and q 5 = ±π/2; q 4 = q 5 = 0 , ±π, singular configurations occur.

Singular Analysis of a Planar 5R Parallel Robot Based on the Proposed Method
To verify that the proposed method is also suitable for parallel manipulators, a planar 5R parallel manipulator [28] was taken as an example for analysis. The mechanism diagram is shown in Figure 10. A 1 and A 2 represent the drive pair, and the other end of the drive link is represented by B 1 and B 2 . The common intersection point of the two branch chains is represented by P(x, y) as the output point. The origin of the coordinate system is at the center of A 1 A 2 , x is along the direction of A 1 A 2 , and the y-axis is perpendicular to A 1 A 2 . OA 1 = OA 2 and PB 1 = PB 2 . Under the premise that the output pose is X and the drive is q, the robot's input-output relationship is: Electronics 2021, 10, 2189 13 of 20 The relationship on the velocity level is as follows [29]: is the configuration Jacobian matrix and x r r q y r q x r r q y r q is the mechanism Jacobian matrix.  We can analyze the configuration Jacobian matrix according to the proposed method steps. In step (1), by setting 1 π 3 = q , 2 = π 3 q , and by using Equation (8) ; this set of joint positions is applied to the subsequent steps in this section.
In step (2), on the basis of the set of joint positions in step 1, a joint position is selected and set to π / 3 rad. From the remaining joints, a joint is selected and varied within its range, and the other joint positions remain unchanged. The distributions of the minimum singular values with the changes in 1 q and 2 q can then be obtained, as shown in Figure 11a.
We can also analyze the mechanism Jacobian matrix according to the proposed The relationship on the velocity level is as follows [29]: where J x = y cos q 1 − (x + r 3 ) sinq 1 0 0 y cos q 2 + (−x + r 3 ) sinq 2 * r 1 is the configuration Jacobian matrix and J q = x + r 3 − r 1 cos q 1 y − r 1 sin q 1 x − r 3 − r 1 cos q 2 y − r 1 sin q 2 is the mechanism Jacobian matrix. e = r 1 (sin q 1 −sin q 2 ) 2r 3 +r 1 cos q 2 −r 1 cos q 1 , f = r 1 r 3 (cos q 1 +cos q 2 ) 2r 3 +r 1 cos q 2 −r 1 cos q 1 We can analyze the configuration Jacobian matrix according to the proposed method steps. In step (1), by setting q 1 = π/3,q 2 = π/3, and by using Equation (8), σ min = 0.4714; this set of joint positions is applied to the subsequent steps in this section. In step (2), on the basis of the set of joint positions in step 1, a joint position is selected and set to π/3 rad. From the remaining joints, a joint is selected and varied within its range, and the other joint positions remain unchanged. The distributions of the minimum singular values with the changes in q 1 and q 2 can then be obtained, as shown in Figure 11a.

Singularity Configurations of the 7-DOF Serial Manipulator Verified through the EE Velocity Ellipsoid
The flexibility of robotic manipulators is the key aspect of research in kinematics, and the manipulability measure is an evaluable index. Yoshikawa [30] defined the manipulability measure as follows: Corke [31] further proposed the EE velocity ellipsoid on the basis of the manipulability measure, as shown in Figure 12. This ellipsoid describes the flexibility of robotic manipulators' motion at the geometric level more vividly, and it defines the joint velocities of robotic manipulators as a unit sphere; i.e., where n represents the number of joints.
2 q  is mapped to the ellipsoid of the task space through the Jacobian matrix J ; i.e., The direction of each axis of the ellipsoid is consistent with the eigenvector of  We can also analyze the mechanism Jacobian matrix according to the proposed method steps. In step (1), by setting q 1 = π/3,q 2 = π/3, and by using Equation (8), σ min = 0.8485; this set of joint positions is applied to the subsequent steps in this section. In step (2), on the basis of the set of joint positions in step 1, a joint position is selected and set to π/3 rad. From the remaining joints, a joint is selected and varied within its range, and the other joint positions remain unchanged. The distributions of the minimum singular values with the changes in q 1 and q 2 can then be obtained, as shown in Figure 11b. It can be seen that there is no mechanism singularity and when q 1 = π/3 rad q 2 = 1.40924 rad; q 1 = π/3 rad, q 2 = 2.70535 rad; q 1 = π/3 rad, q 2 = 1.96856 rad; and q 1 = 0.86476 rad q 2 = π/3 rad, internal singularities appear.

Singularity Configurations of the 7-DOF Serial Manipulator Verified through the EE Velocity Ellipsoid
The flexibility of robotic manipulators is the key aspect of research in kinematics, and the manipulability measure is an evaluable index. Yoshikawa [30] defined the manipulability measure as follows: Corke [31] further proposed the EE velocity ellipsoid on the basis of the manipulability measure, as shown in Figure 12. This ellipsoid describes the flexibility of robotic manipulators' motion at the geometric level more vividly, and it defines the joint velocities of robotic manipulators as a unit sphere; i.e., where n represents the number of joints.
. q 2 is mapped to the ellipsoid of the task space through the Jacobian matrix J; i.e., .
x  The direction of each axis of the ellipsoid is consistent with the eigenvector of (JJ T ) −1 .
The length of each axis is equal to the reciprocal of the square root of its eigenvalue, and it is also equal to the singular value of J. When robotic manipulators approach a singular configuration, ω(q) = 0 and the elliptical plate has almost zero thickness [31]. The translational velocity ellipsoid and rotational velocity ellipsoid of the EE corresponding to the singular configurations of the 7-DOF serial manipulator obtained in Section 3.2.2 are shown in Figures 13-15. To illustrate the singular configurations, the 7-DOF serial manipulator is returned to a nominal configuration and the corresponding translational velocity and rotational velocity ellipsoids are computed, as shown in the Figure 16. Figure 16 clearly indicates that the EE translational velocity and rotational velocity ellipsoids are both standard ellipsoids. However, Figures 13b, 14b and 15b illustrate that the translational velocity ellipse evolves into an elliptical plate with a thickness of zero, indicating that the manipulability measure of the 7-DOF serial manipulator is zero, i.e., the 7-DOF serial manipulator approaches singular configurations. It is thus verified that the singular configurations of the 7-DOF serial manipulator obtained by the proposed method are correct.

Singularity Configurations of the 7-DOF Serial Manipulator and the Planar 5R Parallel Manipulator Verified through an Analytical Method
In Equation (6) It is used to define whether a Jacobian matrix is "good" or "ill-conditioned". When ( ) = 1 q K , the flexibility of the manipulator's movement is optimal. When ( ) q K is infinite, the manipulator is in a singular configuration.
For the 7-DOF serial manipulator, the values for det( ( )) J q , ( ) q ω , and ( ) q K corresponding to the singular configurations obtained through the proposed method can be calculated using Equations (5), (14), and (17), as shown in Table 3. It can be seen that the corresponding values for det( ( )) J q and ( ) q ω for the first to the third rows are zero, and

Singularity Configurations of the 7-DOF Serial Manipulator and the Planar 5R Parallel Manipulator Verified through an Analytical Method
In Equation (6), σ 1 and σ r are the maximum and minimum singular values, respectively. The condition number is defined as follows [32]: It is used to define whether a Jacobian matrix is "good" or "ill-conditioned". When K(q) = 1 , the flexibility of the manipulator's movement is optimal. When K(q) is infinite, the manipulator is in a singular configuration.
For the 7-DOF serial manipulator, the values for det(J(q)), ω(q), and K(q) corresponding to the singular configurations obtained through the proposed method can be calculated using Equations (5), (14), and (17), as shown in Table 3. It can be seen that the corresponding values for det(J(q)) and ω(q) for the first to the third rows are zero, and the corresponding values for K(q) tend to infinity. This further indicates that the singular configurations obtained by the proposed method are correct. In addition, det(J(q)) and ω(q) for the fourth to the thirteenth row are near zero, and the corresponding values for K(q) are very large. The larger the K(q), the closer the configuration is to a singular configuration. This indicates that the flexibility of the manipulator is very poor, so configurations with singular values of zero or approximately zero should be avoided in the actual motion of robotic manipulators. Table 3. det(J(q)), ω(q), and K(q) with singularity and approximate singularity.

Joint Position
det(J(q)) ω(q) K(q) Finally, the proposed method is compared with the analytical method in terms of three aspects: the complexity of the determinant transformation, whether they are able to solve det(J(q)) = 0, and whether they are able to obtain singular configurations, as shown in Tables 5 and 6. The results show that the proposed method can obtain some singular configurations of serial manipulators and parallel manipulators and that it is able to eliminate complex determinant transformation and obtain the solution of det(J(q)) = 0.

Conductions and Future Work
For serial manipulators that do not meet the Pieper criterion, it is difficult to obtain singular configurations through the analytical method. A joint angle parameterization method to be used to obtain singular configurations for robotic manipulators was here proposed. First, an analytical method was used to analyze singular configurations of the Stanford manipulator. Then, the singular configurations of the Stanford manipulator were obtained through the proposed method and compared with the results obtained with the analytical method. The correctness of the proposed method was verified. Next, the proposed method was applied to a 7-DOF serial manipulator and a planar 5R parallel manipulator. Finally, the translational velocity ellipsoid of the EE under singular configurations of the 7-DOF serial manipulator obtained through the proposed method was found to be a plane, and the values for det(J(q)) = 0, ω(q) = 0, and K(q) → ∞ corresponding to singular configurations were calculated. The correctness of the proposed method was verified from these two aspects. For the planar 5R parallel manipulator, by calculating the values for det(J(q)) = 0, ω(q) = 0, and K(q) → ∞ corresponding to singular configurations, the correctness of the proposed method was verified. This showed that the proposed method can be applied to both serial manipulators and parallel manipulators and that it can eliminate complex determinant transformation and obtain the solution of det(J(q)) = 0.
The proposed method can only obtain singular configurations of robotic manipulators at a specific angle, but cannot obtain singular configurations of multiple angles satisfying a certain equation. For example, one singular configuration of PUMA 560 is d 4 sin(q 2 + q 3 ) + a 2 cos(q 2 ) + a 3 cos(q 2 + q 3 ) = 0, and one singular configuration of ABB IRB 1400 is a 3 sin(q 2 + q 3 ) − d 4 cos(q 2 + q 3 ) + a 2 sin(q 2 ) − a 1 = 0. As there are countless combination angles satisfying these two equations, the proposed method fails. Although the proposed method cannot be guaranteed to find all singular configurations, in reality, singular configurations obtained with the proposed method can be set in the initial parameters of manipulators to avoid the corresponding configurations, which is similar to avoiding predetermined fixed obstacles in the working environment. Otherwise, these singular configurations can only be solved by the singularity avoidance algorithm, which reduces the pose accuracy of the EE and makes the calculation more complex. In addition, in the fourth to thirteenth rows of Table 3 (as a few examples; in fact, there are many similar situations), the determinants and the manipulability measures are near zero, and the corresponding condition numbers are very large. In these cases, the velocities of some joints are also very high, which can seriously affect the motion performance. We plan to solve the problem of joint velocities caused by these two situations using a damped least square algorithm in the future. On this basis, we will continue to work on inverse kinematics analysis and trajectory planning.

Conflicts of Interest:
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.