A singularity analysis method for redundant space manipulators

Abstract

The present invention provides a singularity analysis method for a redundant spatial manipulator, comprising: step 1: obtaining the motion spinor coordinates of the manipulator joints and establishing a coordinate system corresponding to the manipulator joints; step 2: using a correlated spinor suppression method to calculate the manipulator singularity conditions based on the manipulator joint motion spinor coordinates and the coordinate system corresponding to the manipulator joints, and obtaining the manipulator singular configuration based on the manipulator singularity conditions; and step 3: performing operability analysis and calculation on the manipulator singular configuration to complete the singularity analysis of the manipulator. The present invention uses spinor theory to establish the motion spinor coordinates of the manipulator joints, uses a correlated spinor suppression method to solve the corresponding basis matrix using the joint motion spinor coordinates, and gradually analyzes the conditions under which the basis matrix is not rank-full to obtain the singularity conditions of the manipulator. This simplifies the complex computational process of singularity analysis and provides a theoretical basis for the kinematic analysis and controller design of the manipulator.

Description

Technical Field

[0001] The present invention belongs to the technical field of singularity analysis of robotic arms, and in particular relates to a singularity analysis method for redundant spatial robotic arms. Background Art

[0002] With the advancement of aerospace technology, the scope of on-orbit services provided by space robots is expanding. Currently, most space robots are seven-degree-of-freedom (SSRMS) manipulator arms, with the same configuration and joint arrangement as the International Space Station's Remote Manipulator System (ISS). This type of manipulator is used in Canadarm 2, ERA, and the Tianhe core module and Wentian laboratory modules of the Chinese space station. They are playing an increasingly important role in missions such as extravehicular activities (EVA), space station maintenance and repair, and payload installation.

[0003] The SSRMS configuration manipulator greatly improves the flexibility of motion control due to the introduction of a redundant degree of freedom and structural symmetry. That is, for specific space operation tasks, different manipulator configurations can be selected according to different requirements such as obstacle avoidance, joint limit avoidance, and minimum joint torque.

[0004] The Jacobian matrix is a mapping between the end velocity of the robot arm and the joint velocity. When the Jacobian matrix becomes singular, that is, when the rank is reduced, the end effector loses at least one degree of freedom, the flexibility of the robot arm is reduced, and the joint velocity will suddenly change or become infinite; when the robot arm is in a singular configuration or near it, a small displacement of the end also requires a large joint angular velocity, which may exceed the driving capacity of the joint, causing the end trajectory to deviate from the expected trajectory, and even causing damage to the robot arm and the surrounding environment.

[0005] The above analysis demonstrates that singularity analysis of a robotic arm is crucial for understanding its kinematics and provides a theoretical foundation for its structural and controller design. However, the Jacobian matrix of a redundant robotic arm is non-square, making it impossible to directly determine its singularity conditions by simply solving for the matrix determinant to be equal to zero. Therefore, a simpler and more practical method is needed to analyze the singularity characteristics of a robotic arm. Summary of the Invention

[0006] The purpose of the present invention is to provide a singularity analysis method for redundant space manipulators, which solves the technical problem in the prior art that the Jacobian matrix of the redundant space manipulator is not a square matrix and the singularity condition of the manipulator cannot be obtained directly by solving the determinant of the Jacobian matrix to be equal to zero.

[0007] To achieve the above objectives, the present invention provides a singularity analysis method for a redundant spatial manipulator, comprising:

[0008] Step 1: Obtain the screw coordinates of the robot arm joint motion and establish the coordinate system of the corresponding robot arm joint;

[0009] Step 2: Using the correlation screw suppression method, the singular conditions of the manipulator are calculated based on the screw coordinates of the manipulator joint motion and the coordinate system of the corresponding manipulator joint, and the singular configuration of the manipulator is obtained based on the singular conditions of the manipulator;

[0010] Step 3: Perform operability analysis and calculation on the singular configuration of the robotic arm to complete the singularity analysis of the robotic arm.

[0011] Optionally, the step of obtaining the manipulator joint motion screw coordinates and the coordinate system of the corresponding manipulator joint in step 1 includes:

[0012] Step 1.1: Calculate the rotational coordinates of the manipulator joint motion based on the rotational parameters of the manipulator joint;

[0013] Step 1.2: The joint rotation parameters of the manipulator include the point on each joint's spiral axis, each joint's spiral axis, the unit vector of each joint's spiral axis, and the linear velocity of the point coinciding with the origin relative to each joint's spiral axis;

[0014] Step 1.3: Construct the coordinate system of the corresponding manipulator joint based on the screw coordinates of the manipulator joint motion;

[0015] The base coordinate system of the coordinate system corresponding to the robot arm joint is {x B y B z B}, the end coordinate system is {x E y E z E}.

[0016] Optionally, the calculation steps of the robot arm singularity condition in step 2 include:

[0017] Step 2.1: Solve the Jacobian matrix of the target coordinate system of the manipulator and simplify the Jacobian matrix of the target coordinate system to obtain the simplified Jacobian matrix;

[0018] Step 2.2: Solve the basis matrix based on the rotational motion of the robot joint;

[0019] Step 2.3: Solve the rank deficiency condition of the basis matrix based on the basis matrix;

[0020] Step 2.4: Based on the insufficient rank condition of the basis matrix and the simplified Jacobian matrix, solve the singular conditions of the manipulator;

[0021] Step 2.5: Obtain the singular configuration of the manipulator based on the singular conditions of the manipulator.

[0022] Optionally, the calculation steps of the Jacobian matrix of the target coordinate system in step 2.1 include:

[0023] Step 2.1.1: Obtain the rotation angle θ of each manipulator joint based on the manipulator joint motion screw coordinates i The posterior joint motion screw coordinates;

[0024] Step 2.1.2: Rotate the robot arm joints by θ i The angled joint motion spinor coordinates construct the Jacobian matrix of the target coordinate system;

[0025] The expression of the Jacobian matrix of the target coordinate system is:

[0026] J E =[J E1 ,…,J Ei ],(i=1,2,…,n) (1)

[0027]

[0028] In formulas (1) and (2), J E is the Jacobian matrix of the target coordinate system, J Ei The angle θ of the i-th joint of the robot arm i The joint rotation coordinate after En is the spinor coordinate of the nth joint of the robot arm, S i is the screw axis of the i-th joint of the robot arm, S n is the screw axis of the nth joint of the robotic arm, for The adjoint matrix of S is the screw axis of the robot arm joint n Rotation angle -θ n The matrix index after .

[0029] Optionally, the simplified Jacobian matrix of the target coordinate system is:

[0030]

[0031]

[0032]

[0033]

[0034]

[0035]

[0036]

[0037]

[0038]

[0039] In formulas (3) to (11), 6 T E is the pose matrix of the end coordinate system relative to the coordinate system {x6y6z6}, θ i ,θ j ,θ k ,θ f is the rotation angle of the screw axis of the robot joint, is a matrix j J i The transposed matrix of t is the length of the robot arm.

[0040] Optionally, the steps of solving the basis matrix in step 2.2 include:

[0041] Step 2.2.1: Select the joint rotation of the preset joint in the robot arm as the base rotation;

[0042] Step 2.2.2: When the manipulator is in a singular configuration, calculate the basis submatrix based on the basis spinors;

[0043] The expression of the basis matrix is:

[0044] k [J] i =[ k J1 … k J n-1 ] (12)

[0045] In formula (12), k [J] i is a matrix 6 Remove the i-th column J from [J] i The 6*6 basis matrix, k is the coordinate system of joint k {x k y k z k} is the reference coordinate system of the Jacobian matrix;

[0046] The determinant of the basis matrix is:

[0047] | k [J] i |=0 (13)

[0048] In formula (13), | k [J] i | is a matrix k [J] i The determinant of .

[0049] Optionally, the steps of solving the rank deficiency condition of the basis matrix in step 2.3 include:

[0050] Step 2.3.1: Analyze the basis matrix and obtain three prior conditions for the basis matrix to be rank-deficient.

[0051] Step 2.3.2: The three prior conditions for the basis matrix to be rank-deficient include: the first matrix rank-deficient prior condition, the second matrix rank-deficient prior condition, and the third matrix rank-deficient prior condition.

[0052] Step 2.3.3: Substitute the rank dissatisfaction prior condition of the first matrix into the basis matrix to obtain the first suppression matrix, and obtain the first rank dissatisfaction condition based on the rank dissatisfaction prior condition of the first matrix and the first suppression matrix;

[0053] Step 2.3.4: Substitute the rank dissatisfaction prior condition of the second matrix into the basis matrix to obtain the second suppression matrix. Based on the rank dissatisfaction prior condition of the second matrix and the second suppression matrix, the second rank dissatisfaction condition and the third rank dissatisfaction condition are obtained.

[0054] Step 2.3.5: Simplify the basis matrix and extract the sub-element matrix of the basis matrix. Based on the sub-element matrix of the basis matrix, obtain the third suppression matrix. Based on the third matrix rank dissatisfaction prior condition and the third suppression matrix, obtain the fourth rank dissatisfaction condition and the fifth rank dissatisfaction condition.

[0055] Optionally, the process of obtaining the singularity condition and singularity configuration of the manipulator includes:

[0056] Combining the first rank-deficient condition, the second rank-deficient condition, the third rank-deficient condition, the fourth rank-deficient condition, and the fifth rank-deficient condition to obtain a first singular condition, a second singular condition, a third singular condition, a fourth singular condition, and a fifth singular condition;

[0057] Perform singular feature analysis on the first singular condition, the second singular condition, the third singular condition, the fourth singular condition and the fifth singular condition to obtain the singular configuration of the robotic arm corresponding to the singular conditions.

[0058] Optionally, the maneuverability analysis of the manipulator's singular configuration includes:

[0059] Calculate the maneuverability of the manipulator's singular configuration corresponding to the singular condition, select the manipulator's singular configuration with a maneuverability greater than a preset threshold, and complete the singularity analysis of the manipulator;

[0060] The calculation formula for the maneuverability of the manipulator's singular configuration is:

[0061]

[0062] In formula (14), M is the maneuverability of the singular configuration of the manipulator, J is the Jacobian matrix of the manipulator, and J T is the transposed matrix of matrix J, and det(·) is the determinant of the matrix.

[0063] The technical effects of the present invention are:

[0064] 1. The present invention adopts the screw theory to establish the motion screw coordinates of the robot arm joint, adopts the related screw suppression method, solves the corresponding basis matrix through the joint motion screw coordinates, and gradually analyzes the conditions under which the basis matrix is not full of rank to obtain the singular conditions of the robot arm.

[0065] 2. The present invention can simplify the complex computational process of singularity analysis of a robotic arm and can be applied to different types of robotic arm systems, with strong applicability and flexibility. The singular condition analysis obtained by this method can be used to eliminate singular configurations that appear in the position-level inverse kinematics solution results, thereby ensuring the safety of the robotic arm's motion. It helps to optimize path planning, avoid sudden changes in joint velocity caused by the robotic arm entering a singular point, and ensure the stability and accuracy of the robotic arm's motion. It also provides a theoretical basis for the kinematic analysis and controller design of the robotic arm. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0067] The accompanying drawings, which constitute part of this application, are intended to provide a further understanding of this application. The exemplary embodiments and descriptions of this application are intended to explain this application and do not constitute an improper limitation on this application. In the accompanying drawings:

[0068] Figure 1 A flowchart of a singularity analysis method for a redundant spatial manipulator provided by an embodiment of the present invention;

[0069] Figure 2 A schematic diagram of the rotary coordinates of the joint motion of the robot arm in the zero-position configuration according to an embodiment of the present invention;

[0070] Figure 3 A coordinate system diagram of each link of the robotic arm provided by an embodiment of the present invention;

[0071] Figure 4 A singular configuration diagram of the robotic arm corresponding to the first singular condition provided in an embodiment of the present invention;

[0072] Figure 5A singular configuration diagram of the robotic arm corresponding to the second singular condition provided in an embodiment of the present invention;

[0073] Figure 6 A diagram of a singular configuration of the robotic arm corresponding to the third singular condition provided in an embodiment of the present invention;

[0074] Figure 7 A diagram of a singular configuration of the robotic arm corresponding to the fourth singular condition provided in an embodiment of the present invention;

[0075] Figure 8 This is a diagram of a singular configuration of the robotic arm corresponding to the fifth singular condition provided in an embodiment of the present invention. Example

[0076] In order to make the technical problems, technical solutions and beneficial effects to be solved by the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0077] pass Figure 1-8 This embodiment will be described.

[0078] Step 1: If Figure 1 As shown, {x B y B z B}、{x E y E z E} are the base coordinate system and the end coordinate system of the robot arm respectively, {x6y6z6} is the connected coordinate system attached to joint 6, and its origin is located at the intersection of the screw axes S5 and S6; a t {t=0,1,…,8} is the length of the robot arm connecting rod. The specific values are shown in Table 1. i {i=1,…,7} is a point on the screw axis of each joint, S i =(w i ,v i ){i=1,…,7} is the screw axis of each joint, w i {i=1,…,7} is the direction unit vector of each joint screw axis, v i =-w i ×q i ,{i=1,…,7}is the coordinate system of the end {x E y E z E The point where the origin coincides with the spiral axis S of each joint i Linear velocity, q i 、w i 、v i The expression is shown in Table 2, q i 、wi 、v i 、S i The end coordinate system {x E y E z E} is the reference coordinate system.

[0079] Table 1 Parameters of the connecting rod of the robotic arm

[0080]

[0081] Table 2 Robotic arm joint rotation parameters

[0082]

[0083]

[0084] Step 2: Solve the end coordinate system {x E y E z E} is the Jacobian matrix of the reference coordinate system, because the Jacobian matrix can reflect the mapping relationship between the end velocity of the robot arm and the joint velocity. E y E z E} is the Jacobian matrix J of the n-DOF manipulator in the reference coordinate system E The solution formula is:

[0085] J E =[J E1 ,…,J Ei ],(i=1,2,…,n) (1)

[0086]

[0087] In formulas (1) and (2), J E is the Jacobian matrix of the target coordinate system, J Ei The angle θ of the i-th joint of the robot arm i The joint rotation coordinate after En is the spinor coordinate of the nth joint of the robot arm, S i is the screw axis of the i-th joint of the robot arm, S n is the screw axis of the nth joint of the robotic arm, for The adjoint matrix of S is the screw axis of the robot arm joint n Rotation angle -θ n The matrix index after .

[0088] Step 3: Changing the reference coordinate system does not affect the kinematic singularity of the manipulator. Simplify the Jacobian matrix expression. To facilitate subsequent singularity analysis, it is necessary to simplify J E Expression of , reselect the reference coordinate system of the Jacobian matrix;

[0089] In the present invention, the coordinate system {x6y6z6} is reselected as the reference coordinate system, and the Jacobian matrix 6 The expression of [J] is:

[0090]

[0091] 6 J i The solution result is:

[0092]

[0093]

[0094]

[0095]

[0096]

[0097]

[0098]

[0099]

[0100] In formulas (3) to (11), 6 T E is the pose matrix of the end coordinate system relative to the coordinate system {x6y6z6}, θ i ,θ j ,θ k ,θ f is the rotation angle of the screw axis of the robot joint, is a matrix j J i The transposed matrix of t is the length of the robot arm.

[0101] Step 4: Select the spinor of joint 1 as the basis spinor and solve the basis matrix. Select the joint spinors of joints 2-7 as the basis spinors. The expression of the corresponding basis matrix is:

[0102] k [J] i =[ k J1 … k J n-1 ] (12)

[0103] In formula (12), k [J] i is a matrix 6 Remove the i-th column J from [J] i The 6*6 basis matrix, k is the coordinate system of joint k {x k y k z k} is the reference coordinate system of the Jacobian matrix;

[0104] When the robotic arm is in a singular configuration, 6 [J] is not rank-consistent, and the determinant of the basis matrix is calculated as:

[0105] | 6 [J] i |=0 (13)

[0106] In formula (13), | 6 [J] i | is a matrix 6 [J] i The determinant of 6 [J] i is the basis matrix.

[0107] Further calculation of formula (13) yields:

[0108] a3a5s4s6(a3s3+a5s 34 +a7s 345 )=0 (14)

[0109] Analyzing formula (14), we can get the prior condition that the basis matrix is not full of rank. t Because the length of the robotic arm is always greater than 0, when the above equation is established, there are three prior conditions for the basis matrix to be rank-deficient, namely:

[0110] The first type of sub-matrix rank-deficient prior condition: s4 = 0;

[0111] The second type of sub-matrix does not meet the rank priori condition: s6 = 0;

[0112] The third type of sub-matrix is not full of rank prior condition: a3s3+a5s 34 +a7s 345 =0.

[0113] Step 5: Solve the condition that the basis matrix is not full of rank;

[0114] Substitute s4=0 into the basis matrix 6 [J] 1 The following formula is obtained:

[0115]

[0116] From formula (14), we can see that 6 J3 is 6 J4 and 6 The linear combination of J5 is expressed as follows: 6 J3=[(a5+a3c4) / a5]· 6 J4-(a3c4 / a5) 6 J5. Therefore, 6 J3 is regarded as the relevant spinor, 6 [J] 3 is the suppression matrix.

[0117] Calculate the spinor matrix 6 [J] Rank deficiency is equivalent to calculating the suppression matrix 6 [J] 3 Not full rank, that is, under the premise of s4=0, | 6 [J] 3 |=0, and after simplification, we get the following formula:

[0118]

[0119] Case 2: s6 = 0

[0120] Substitute s6=0 into the basis matrix 6 [J] 1 In the formula, we get:

[0121]

[0122] 6 J7 is 6 J3, 6 J4, 6 The linear combination of J5, that is 6 J7=[a7s5c6 / (a3s4)]· 6 J3-[a7c6s 45 / (a5s4)]· 6 J4+[a7c6s 45 / (a5s4)-a7s5c6 / (a3s4)-c6]· 6 J5, and 6 J3, 6 J4, 6 J5 and 6 The rank of the matrix composed of J7 does not exceed 3, so we can 6 J7 is regarded as a related spinor, 6 [J] 7 is the suppression matrix.

[0123] Calculate the spinor matrix6 [J] Rank deficiency is equivalent to calculating the suppression matrix 6 [J] 7 Not full rank, that is, under the premise of s6=0, | 6 [J] 7 |=0, and after simplification, we get the following formula:

[0124]

[0125] There are two sets of solutions:

[0126]

[0127] The third case: a3s3+a5s 34 +a7s 345 =0

[0128] To facilitate analysis, we first calculate the basis matrix 6 [J] 1 Simplify and extract the elements of the 1st, 2nd, 3rd, 5th and 6th rows of the matrix to form a submatrix [ 6 J3 6 J4 6 J5 6 J6 6 J7] 1,2,3,5,6 , and solving the determinant to obtain:

[0129] |[ 6 J3 6 J4 6 J5 6 J6 6 J7] 1,2,3,5,6 |=a3a5s4s6 (19)

[0130] Because a t The arm length of the robot arm is always greater than 0. s4=0 and s6=0 have been discussed in the first two cases. In formula (19), s4 and s6 are not 0, so a3a5s4s6≠0, that is, the submatrix [ 6 J3 6 J4 6 J5 6 J6 6 J7] 1,2,3,5,6 Full rank.

[0131] According to the above conditions, 6 J3, 6 J4, 6 J5, 6 J6 and 6 J7 is linearly independent, so 6 J2 can be expressed as 6 J3,6 J4, 6 J5, 6 J6 and 6 The linear combination of J7, the corresponding inhibition matrix 6 [J] 2 As shown in the following formula:

[0132] 6 [J] 2 =[ 6 J1 6 J3 6 J4 6 J5 6 J6 6 J7] (20)

[0133] Simplifying its determinant yields the equation:

[0134] -a3a5s2s4s6(a1+a3c3+a5c 34 +a7c 345 )=0 (21)

[0135] The above equation has two sets of solutions, namely:

[0136]

[0137] Step 6: Solve and analyze the singular conditions of the robot arm based on the rank deficiency condition;

[0138] Analyze the singular conditions of the robot arm based on the five sets of solution results obtained in step 5, and analyze the singular characteristics.

[0139] The first singular condition:

[0140]

[0141] According to formula (23), the angle of joint 2 is 0° or ±180°, indicating that the axis of joint 1 of the robot arm is parallel to the axis of joint 3; the angle of joint 6 is 0° or ±180°, indicating that the axis of joint 5 of the robot arm is parallel to the axis of joint 7. At the same time, the axes of joints 3, 4, and 5 of the robot arm are parallel to each other in structural design.

[0142] Therefore, the singular configuration characteristics of the manipulator in this case are: the axes of joints 1, 3, 4, 5, and 7 are parallel. The singular configuration of the manipulator corresponding to this condition is as follows: Figure 4 As shown, the joint angles are (-15°, 180°, -30°, 60°, -30°, 180°, 60°) respectively.

[0143] The second singular condition:

[0144]

[0145] According to formula (24), the angle of joint 6 is 0° or ±180°, indicating that the axis of joint 5 is parallel to the axis of joint 7. Observing the second equation, we can find that it describes the constraint relationship of joints 3, 4, and 5 of the manipulator. Figure 3 The link coordinate systems shown represent the states of each joint before rotation, where the origin of the coordinate system {x2y2z2} is located at the intersection of the z1 axis and the x3 axis, and the origin of the coordinate system {x6y6z6} is located at the intersection of the x5 axis and the z7 axis.

[0146] according to Figure 3 Derive the transformation relationship between the coordinate system {x1y1z1} (rotated by joints 1 and 2 in sequence) and {x6y6z6} (rotated by joints 1, 2, 3, 4, and 5 in sequence) after each joint rotates a certain angle 6 T1 is shown in the following formula:

[0147]

[0148] according to 6 From the expression T1, we can see that the origin of joint 1 of the manipulator (i.e. the origin of the coordinate system {x1y1z1}) is located on the plane formed by the axis S5 of joint 5 and the axis S6 of joint 6. The singular configuration of the manipulator corresponding to this condition is as follows: Figure 5 As shown, the joint angles are (0°, -45°, -57°, 80°, -38°, 0°, -40°) respectively.

[0149] The third singular condition:

[0150]

[0151] According to formula (26), the angle of joint 2 is 0° or ±180°, indicating that the axis of joint 1 is parallel to the axis of joint 3. Observing the second equation, we can find that it also describes the constraint relationship of joints 3, 4, and 5. Figure 2 , derive the transformation relationship between the coordinate system {x2y2z2} (rotated by joints 1 and 2 in sequence) and {x7y7z7} (rotated by joints 1, 2, 3, 4, 5, 6 in sequence) after each joint rotates a certain angle 2 T7 is as follows:

[0152]

[0153] according to 2 From the expression T7, we can see that the origin of joint 7 (the origin of the coordinate system {x7y7z7}) is located on the plane formed by the axis S2 of joint 2 and the axis S3 of joint 3.

[0154] The singular configuration of the robotic arm corresponding to this condition is as follows Figure 6 As shown, the joint angles are (-74.14°, 0°, -68.34°, -158.59°, -26.64°, -189.83°, -9.82°).

[0155] The fourth singular condition:

[0156]

[0157] Calculated according to formula (28), formula (28) describes the constraint relationship between joints 3, 4, and 5. Figure 3 , derive the transformation relationship between the coordinate system {x2y2z2} (rotated by joints 1 and 2 in sequence) and {x7y7z7} (rotated by joints 1, 2, 3, 4, 5, 6, 7 in sequence) after each joint rotates a certain angle 2 T7 is as follows:

[0158]

[0159] According to the expression 2 From T7, we can see that the origin of joint 7 (the origin of the coordinate system {x7y7z7}) is located on the plane formed by the axis S2 of joint 2 and the axis S3 of joint 3, and the distance from the axis of joint 3 is a1. It can also be understood that the origin of joint 7 is located on the intersection of the following two planes: the plane formed by the axis of joint 2 and the axis of joint 3, and the plane formed by the axis of x B -y B flat.

[0160] The singular configuration of the robotic arm corresponding to this condition is as follows Figure 7 As shown, the joint angles are (-45°, 30°, 80°, 157.78°, 79.52°, 75°, 90°).

[0161] The fifth strange condition:

[0162]

[0163] According to formula (30), the angle of joint 4 is 0° or ±180°, indicating that the axes of joints 3, 4 and 5 are parallel. The second row of equations shows that there are angle constraints between joints 2, 3, 5 and 6. The present invention gives the singular configuration of the manipulator corresponding to this singular condition as follows: Figure 8 As shown, the joint angles are (95.89°, 17.47°, 173.33°, 0°, 159.64°, 52.31°, -79.71°).

[0164] Step 7: Verify the singular conditions of the robotic arm through operability;

[0165] In order to determine whether the singular configurations of the manipulator obtained through the singular conditions are valid, the manipulatable degree M of the manipulator corresponding to each singular configuration in step 6 is calculated using the following formula:

[0166]

[0167] In formula (31), M is the maneuverability of the singular configuration of the manipulator, J is the Jacobian matrix of the manipulator, and J T is the transposed matrix of matrix J, and det(·) is the determinant of the matrix.

[0168] The calculation results of each singular configuration angle and operability are shown in Table 3:

[0169] Table 3 Joint angles and maneuverability of the robot arm with singular configuration

[0170]

[0171] It can be seen from the calculation results of the maneuverability in Table 3 that the configurations in Table 3 obtained by the singularity analysis method of the robotic arm proposed in the present invention are all singular configurations.

[0172] The above description is merely a preferred embodiment of the present application, but the scope of protection of the present application is not limited thereto. Any changes or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in the present application should be included in the scope of protection of the present application. Therefore, the scope of protection of the present application should be based on the scope of protection of the claims.

Claims

1. A singularity analysis method for redundant spatial manipulators, characterized in that: The singularity analysis method for redundant spatial manipulators comprises the following steps: Step 1: Obtain the screw coordinates of the robot arm joint motion and establish the coordinate system of the corresponding robot arm joint; Step 2: Using the correlation screw suppression method, the singular conditions of the manipulator are calculated based on the screw coordinates of the manipulator joint motion and the coordinate system of the corresponding manipulator joint, and the singular configuration of the manipulator is obtained based on the singular conditions of the manipulator; The calculation steps for the singularity condition of the manipulator in step 2 include: Step 2.1: Solve the Jacobian matrix of the target coordinate system of the manipulator and simplify the Jacobian matrix of the target coordinate system to obtain the simplified Jacobian matrix; The calculation steps of the Jacobian matrix of the target coordinate system in step 2.1 include: Step 2.1.1: Obtain the rotation angle of each robotic arm joint based on the rotary coordinates of the robotic arm joint motion The posterior joint motion screw coordinates; Step 2.1.2: Rotate the joints of each robotic arm The angled joint motion spinor coordinates construct the Jacobian matrix of the target coordinate system; The expression of the Jacobian matrix of the target coordinate system is: （1） （2） In formulas (1) and (2), is the Jacobian matrix of the target coordinate system, For the robotic arm i Joint rotation angle The joint rotation coordinates after For the robotic arm n The rotation coordinates of the joints, For the robotic arm i Joint screw shaft, For the robotic arm n Joint screw shaft, for The adjoint matrix of For the screw axis of the robot arm joint rotation angle The matrix index after ; The Jacobian matrix of the target coordinate system after simplification in step 2.1 for: （3） （4） （5） （6） （7） （8） （9） （10） （11） In formulas (3) to (11), The end coordinate system is relative to the coordinate system The pose matrix of is the rotation angle of the screw axis of the robot joint, is a matrix The transposed matrix of is the length of the robotic arm; Step 2.2: Solve the basis matrix based on the rotational quantity of the joint motion of the manipulator; Step 2.3: solving the rank deficiency condition of the basis matrix based on the basis matrix; Step 2.4: Solve the singular conditions of the manipulator based on the insufficient rank condition of the basis matrix and the simplified Jacobian matrix; Step 2.5: Obtaining a singular configuration of the robotic arm based on the singular condition of the robotic arm; Step 3: Perform operability analysis and calculation on the singular configuration of the robotic arm to complete the singularity analysis of the robotic arm.

2. The singularity analysis method for redundant spatial manipulators according to claim 1, characterized in that: The steps for obtaining the manipulator joint motion screw coordinates and the corresponding manipulator joint coordinate system in step 1 include: Step 1.1: Calculate the rotational coordinates of the manipulator joint motion based on the rotational parameters of the manipulator joint; Step 1.2: The joint rotation parameters of the robotic arm include the point on each joint screw axis, the unit vector of each joint screw axis, and the linear velocity of the point coinciding with the origin relative to each joint screw axis; Step 1.3: Construct the coordinate system of the corresponding manipulator joint based on the screw coordinates of the manipulator joint motion; The base coordinate system of the coordinate system corresponding to the robot arm joint is , the end coordinate system is .

3. The singularity analysis method for redundant spatial manipulators according to claim 1, characterized in that: The method for solving the basis matrix in step 2.2 includes: Step 2.2.1: Select the joint rotation of the preset joint in the robot arm as the base rotation; Step 2.2.2: When the manipulator is in a singular configuration, calculate the basis submatrix according to the basis spinor; The expression of the basis matrix is: （12） In formula (12), is a matrix Remove the i List The 6*6 basis matrix, k For joints k Coordinate system is the reference coordinate system of the Jacobian matrix; The determinant of the basis matrix is: （13） In formula (13), is a matrix The determinant of .

4. The singularity analysis method for redundant spatial manipulators according to claim 1, characterized in that: The steps for solving the rank deficiency condition of the basis matrix in step 2.3 include: Step 2.3.1: Analyze the basis matrix to obtain three prior conditions for the basis matrix to be rank-deficient; Step 2.3.2: The three prior conditions for the basis matrix to be rank-deficient include: a first matrix rank-deficient prior condition, a second matrix rank-deficient prior condition, and a third matrix rank-deficient prior condition; Step 2.3.3: Substitute the first matrix rank dissatisfaction prior condition into the basis matrix to obtain a first suppression matrix, and obtain a first rank dissatisfaction condition based on the first matrix rank dissatisfaction prior condition and the first suppression matrix; Step 2.3.4: Substitute the second matrix rank dissatisfaction prior condition into the basis matrix to obtain a second suppression matrix, and obtain a second rank dissatisfaction condition and a third rank dissatisfaction condition based on the second matrix rank dissatisfaction prior condition and the second suppression matrix; Step 2.3.5: Simplify the basis matrix and extract the sub-element matrix of the basis matrix, obtain the third suppression matrix based on the sub-element matrix of the basis matrix, and obtain the fourth rank dissatisfaction condition and the fifth rank dissatisfaction condition based on the third matrix rank dissatisfaction prior condition and the third suppression matrix.

5. The singularity analysis method for redundant spatial manipulators according to claim 1, characterized in that: The process of obtaining the singular conditions and singular configurations of the manipulator includes: Combining the first rank-deficient condition, the second rank-deficient condition, the third rank-deficient condition, the fourth rank-deficient condition, and the fifth rank-deficient condition to obtain a first singular condition, a second singular condition, a third singular condition, a fourth singular condition, and a fifth singular condition; Perform singular feature analysis on the first singular condition, the second singular condition, the third singular condition, the fourth singular condition and the fifth singular condition to obtain a singular configuration of the robotic arm corresponding to the singular condition.

6. The singularity analysis method for redundant spatial manipulators according to claim 1, characterized in that: The operability analysis of the manipulator's singular configuration includes: Calculate the maneuverability of the manipulator's singular configuration corresponding to the singular condition, select the manipulator's singular configuration with a maneuverability greater than a preset threshold, and complete the singularity analysis of the manipulator; The calculation formula for the maneuverability of the manipulator's singular configuration is: （14） In formula (14), The maneuverability of the robot arm's unique configuration, is the Jacobian matrix of the robot arm, is a matrix The transposed matrix of is the determinant of the matrix.

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