A pieper configuration mechanical arm inverse solution screening method based on safe posture screening and energy function optimization

Abstract

The application discloses a Pieper configuration mechanical arm inverse solution screening method based on safe posture screening and energy function optimization, faces the safety requirement of a mechanical arm grabbing posture, combines a mechanical arm configuration, and proposes a kinematics inverse solution safety screening strategy, divides 8 groups of inverse solution postures into 3 categories, analyzes the 3 categories of postures according to common grabbing postures, and obtains the conditions of conforming to safe grabbing postures; aiming at the requirement of low energy consumption and short planning time of motion planning, an energy function is introduced to calculate the energy loss corresponding to the solution after safety posture screening, so as to select the kinematics inverse solution with the optimal energy function. The application effectively guarantees the safety of the grabbing posture and effectively reduces the energy loss of planning.

Description

Technical Field

[0001] This invention belongs to the field of kinematics planning technology for robotic arms, specifically relating to an inverse kinematics screening method for Pieper configuration robotic arms based on safe posture screening and optimal energy function. Background Technology

[0002] Describing any position in space requires six degrees of freedom: three positional degrees of freedom and three orientation degrees of freedom. Because of these six degrees of freedom, a six-axis Pieper-configuration robotic arm can reach any point in the workspace in any orientation through motion planning. However, the inverse kinematics solution for a six-degree-of-freedom robotic arm is not unique; at most eight solutions exist at non-singular points. Multiple solutions need to be selected to obtain the optimal inverse kinematics solution.

[0003] Solving inverse kinematics (IK) problems for robotic arms typically employs two methods: analytical and numerical solutions. Compared to numerical solutions, analytical solutions offer advantages such as higher accuracy and faster computation speed. However, due to the unique structure of the Pieper-configuration robotic arm (with three parallel or perpendicular axes), the IK solution for the same end-effector position is not unique. How to select the optimal solution from multiple IK solutions is a problem that robotic arm researchers have been continuously studying. Currently, researchers often employ a shortest path strategy for optimal solution selection, which only considers the planning time of the robotic arm without considering whether the end-effector posture conforms to safety regulations.

[0004] In robotic arm motion planning, unhealthy inverse kinematic solutions can lead to unsafe end-effector postures, increasing the likelihood of planning task failure and potentially injuring engineers. Therefore, selecting a solution that ensures both posture safety and minimal energy consumption, while guaranteeing both speed and safety in planning, is crucial. This significantly reduces the risks of robotic arm kinematic planning and minimizes energy loss to some extent, making an inverse kinematic solution selection strategy that considers both the shortest path and posture safety extremely important. Summary of the Invention

[0005] This invention provides a method for inverse kinematics screening of Pieper configuration robotic arms based on safe posture screening and optimal energy function. This method screens out solutions that can ensure both posture safety and minimum energy consumption, thus ensuring the speed and safety of planning.

[0006] To achieve the above objectives, the present invention adopts the following technical solution:

[0007] A method for inverse kinematics selection of a Pieper-configuration robotic arm based on safe posture selection and optimal energy function includes the following steps:

[0008] S1: Based on the two-dimensional model of the robotic arm, a DH (modify) link coordinate model of the robotic arm is established according to the forward kinematics and geometric parameters, using... Establish the inverse solution system of equations;

[0009] S2: To meet the requirements of end-effector posture specifications, a safe posture screening strategy is proposed. The end-effector postures of the 8 sets of inverse kinematics are divided into three categories. By analyzing the three categories of postures, sufficient conditions for meeting the safe end-effector posture are obtained. Furthermore, the joint angles corresponding to joint2 and joint3 meet the sufficient conditions for the safe end-effector posture, thus realizing the screening of the safe end-effector posture.

[0010] S3: To meet the shortest path requirement, an optimal energy function strategy is proposed. An energy function is constructed based on the joint angles corresponding to the initial and final poses. The solution obtained in S2, which meets the safe pose requirements, is then processed through the optimal energy function strategy to obtain the inverse kinematic solution that meets both the safe pose and energy optimization requirements.

[0011] Beneficial Effects: This invention provides an inverse kinematics selection method for Pieper-configuration robotic arms based on safe posture selection and optimal energy function. It improves the inverse kinematics solution algorithm for six-DOF Pieper-configuration robotic arms by addressing existing analytical solution-solving and optimal solution-selection strategies. This effectively simplifies the complexity of the equation system, reduces the difficulty of deriving the analytical kinematics solution, lowers the difficulty of solving the inverse kinematics, simplifies the symbolic expression of the analytical solution for easier programming implementation, and makes the inverse kinematics result expression more concise. This invention aims to provide a new solution for inverse kinematics solution solving and selection, unlike existing technologies based on... A system of inverse equations was established to solve the inverse kinematics of the robotic arm, and a method was proposed to use... A system of inverse kinematic equations was established, and a kinematic inverse kinematic equation safety screening strategy was proposed based on the robotic arm configuration. The eight sets of inverse kinematic postures were divided into three categories. Analysis of these three categories based on common grasping postures yielded the conditions for safe grasping postures. To address the requirements of low energy consumption and short planning time in motion planning, an energy function was introduced to calculate the energy loss corresponding to the solution selected after the safe posture screening, thereby selecting the kinematic inverse kinematic equation with the optimal energy function. Simulation experiments show that for a six-axis Pieper configuration robotic arm, this invention can effectively solve the kinematic inverse kinematic equation of the Pieper configuration robotic arm and select the kinematic inverse kinematic equation considering both safe posture and optimal energy function, effectively ensuring the safety of the grasping posture and effectively reducing the energy loss of planning. The inverse kinematic equation algorithm can still operate normally even when there are few solutions due to singular configurations, demonstrating a certain degree of robustness. This provides a complete strategy for solving kinematic inverse kinematic equations with optimal energy consumption and safe posture. Attached Figure Description

[0012] Figure 1 This is the DH kinematic model of the 6-DOF robotic arm in this embodiment of the invention;

[0013] Figure 2The posture classification of the robotic arm on the 2nd and 3rd axes in this embodiment of the invention;

[0014] Figure 3 The distribution of analytical solutions in this embodiment of the invention;

[0015] Figure 4 This is the pseudocode for the inverse algorithm based on safe attitude selection and optimal energy function in this embodiment of the invention;

[0016] Figure 5 This is the MATLAB GUI simulation interface in this embodiment of the invention;

[0017] Figure 6 This is the inverse solution result in the embodiment of the present invention;

[0018] Figure 7 This is a schematic diagram of the interpolation trajectory planning results in an embodiment of the present invention, where (a) is the trajectory planning result of the robotic arm, and (b) is the planning result of the angles of each joint of the robotic arm;

[0019] Figure 8 This is a schematic diagram of the inverse kinematics algorithm for the robotic arm in an embodiment of the present invention;

[0020] Figure 9 This is a schematic diagram of the safe pose selection algorithm and the energy function optimization algorithm in an embodiment of the present invention. Detailed Implementation

[0021] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments:

[0022] A method for inverse kinematics selection of a Pieper-configuration robotic arm based on safe posture selection and optimal energy function includes the following steps:

[0023] 1. Improved kinematic analytical solution

[0024] (1) Kinematic modeling of a six-degree-of-freedom robotic arm

[0025] Based on the two-dimensional model of the robotic arm, a model is established according to the forward kinematics and geometric parameters as follows: Figure 1 The DH (modify) link coordinate model of the robotic arm is shown in Table 1, where the DH parameter values ​​are as follows:

[0026] Table 1 DH (modify) Parameter Table

[0027]

[0028] The parameters in the DH parameter table are defined as follows:

[0029] (1)a i-1 : will (x i-1 y i-1 , zi-1 ) along x i-1 Directional translation a i-1 Distance, such that z i-1 axis and z i Axis coincidence;

[0030] (2)α i-1 : will (x i-1 y i-1 , z i-1 ) around x i-1 Rotate the axis in the right-hand helix direction α i-1 Angle, making z i-1 axis and z i The axes are parallel;

[0031] (3)d i : will (x i-1 y i-1 , z i-1 ) along z i Directional translation d i Distance, such that x i-1 axis and x i Axis coincidence;

[0032] (4)θ i : Joint i around z i The angle of rotation in the right-hand helix direction of the axis;

[0033] The homogeneous transformation matrix between the links of the robotic arm is:

[0034]

[0035] The forward kinematic equations of the robotic arm are:

[0036]

[0037] (2) Solving inverse kinematics

[0038] The robot's axes 2, 3, and 4 are parallel to each other. According to the sufficient condition for the existence of the inverse kinematics in Pieper's criterion, the robot arm satisfies the Pieper configuration. Therefore, the inverse kinematics of the robot arm has a closed-form solution. The derivation of the inverse solution is based on conventional... The analytical solution derivation is improved, and a concise inverse solution derivation process is derived:

[0039]

[0040] Where c1 = cos(θ1), c 23 =cos(θ2+θ3), c 234 =cOs(θ2+θ3+θ4), s1 = sin(θ1), s 23 =sin(θ2+θ3), s234 = sin(θ2+θ3+θ4), where n, o, and a are the column vectors corresponding to the rotation matrix, and p is the position vector;

[0041] Let the matrix elements of equation (3) and Equal to each other, we get:

[0042]

[0043] make Among them, A, B, r, As an intermediate variable used to solve for θ1; then:

[0044]

[0045] when θ1 has a solution:

[0046]

[0047] Where atan2(x, y) is the arctangent function that can automatically switch quadrants based on the signs of x and y. After finding θ1, θ5 can be calculated based on θ1:

[0048]

[0049] Let the matrix elements of equation (3) and Equal to each other (θ5≠0), we get:

[0050]

[0051] Let the matrix elements of equation (3) and Equal to each, we get:

[0052]

[0053] Let the matrix elements of equation (3) and Equal to each, we get:

[0054]

[0055] Let p x c1+p y s1+d5s 234 -d6c 234 s5 = B1, p z -d1-d5c 234 -d6s 234 s5 = B2, therefore:

[0056]

[0057] make Where B1, B2, and β are intermediate variables used to solve θ2, when We can obtain:

[0058]

[0059] Based on the obtained θ2, θ3 can be solved:

[0060]

[0061] Finally, based on the obtained θ2 and θ3, solve for θ4:

[0062] θ4=(θ2+θ3+θ4)-(θ2+θ3) (14)

[0063] At this point, the angles of the six joints have been solved, resulting in a total of 8 inverse solutions. The next section will analyze these 8 inverse solutions and select the optimal solution.

[0064] 2. Safe posture screening technology

[0065] Before trajectory planning, it is necessary to select the most suitable solution. The safety of joint pose, planning time, energy consumption, etc. must be considered after trajectory planning and during the process. Before introducing the algorithm for selecting the optimal solution, the properties of the eight sets of solutions need to be analyzed to provide a theoretical basis for the algorithm.

[0066] Trajectory planning based on inverse kinematics cannot only consider speed; safety must also be taken into account. Otherwise, it may lead to serious engineering safety incidents. The working state of planar grasping operations should also be considered. Figure 3 The first and eighth solutions are not suitable for grasping, even though their distance from the initial angle may be the smallest.

[0067] The main factors affecting the overall pose of the robotic arm are joint2 and joint3. The strategy for eliminating unsafe solutions is based on the commonly used planar grasping posture, and the posture obtained by the inverse solution is analyzed.

[0068] First, we categorize the values ​​of joint2 and joint3, disregarding the value of joint1. The categorization results for joint2 and joint3 are as follows:

[0069] (1) joint2≥0, joint3≥0

[0070] (2) joint2≥0, joint3≤0

[0071] (3) joint2≤0, joint3≥0

[0072] (4) joint2≤0, joint3≤0

[0073] Considering the object to be grasped is directly in front of the robotic arm, the value of joint1 can be either 0 or π, which can be broadly divided into the 0 half-zone and the π half-zone. The robotic arm's 0 pose is shown in Figure (1). The initial state of the robotic arm's kinematic model is as follows: (joint1 = 0, joint2 ≥ 0, joint3 ≥ 0) and (joint1 = π, joint2 ≤ 0, joint3 ≤ 0) have the same 2 and 3 axis poses; (joint1 = 0, joint2 ≤ 0, joint3 ≥ 0) and (joint1 = π, joint2 ≥ 0, joint3 ≤ 0) have the same 2 and 3 axis poses. However, when joint1 is in the 0 position, the 2 and 3 axis poses are not the same. When t1 = 0, it is impossible for joint2 ≤ 0 and joint3 ≤ 0. At this time, the robot arm pose is opposite to the workspace. Similarly, when joint1 = π, it is impossible for joint2 ≥ 0 and joint3 ≥ 0. In summary, there are three main poses of joint2 and joint3 facing the workspace, as shown in Figure (2). Pose 2 and pose 3 are safe poses, and pose 1 is an unsafe pose. The sufficient condition for the inverse pose to be a safe pose is: Ψ ≤ π, where Ψ is the angle at which link 2 is rotated counterclockwise to link 3 with link 3 as the center. The joint angle corresponding to the sufficient condition for the safe pose is the safe pose solution.

[0074] Furthermore, by modifying the sufficient condition for a safe attitude, the sufficient condition for satisfying the inverse attitude to be a safe attitude is:

[0075] (θ2+θ3≤0 and θ3≤0)or(θ2+θ3≥0 and θ3≥0)

[0076] Consider the end-effector pose as follows:

[0077] Location: [p] x p y p z = [121.5 -368.37 + 701.37] Unit: cm

[0078] Posture: [r x r y r z = [-π 0 0] Unit: rad

[0079] The four sets of solutions generated are shown in Figure (3) (there are 4 sets of singular configurations here).

[0080] The first set of solutions, joint1, lies in the π half-region, where joint2 ≥ 0 and joint3 ≤ 0; the second set of solutions, joint1, lies in the π half-region, where joint2 ≤ 0 and joint3 ≤ 0; the seventh set of solutions, joint1, lies in the 0 half-region, where joint2 ≥ 0 and joint3 ≥ 0; the eighth set of solutions, joint1, lies in the 0 half-region, where joint2 ≤ 0 and joint3 ≥ 0. According to the sufficient condition for joint angles in a safe posture, the solutions in the above four sets that satisfy the sufficient condition for a safe posture are the second set of solutions and the seventh set of solutions. After obtaining the solutions that meet the requirements for a safe posture, the inverse kinematic solution that satisfies both the safe posture and energy optimization can be obtained through the energy function optimal strategy in the next subsection.

[0081] 3. Shortest distance algorithm

[0082] (1) Minimum travel method

[0083] According to the minimum stroke method criterion, there is usually only one optimal solution. Therefore, an energy function is constructed with the joint angle distance as the unit. This function considers both the joint angle distance and the energy loss under unit angle motion of joints with different masses.

[0084]

[0085] Where m(n) is the mass of link n, theta safe For one set of safe solutions, theta start This is the initial joint angle;

[0086] The pseudocode for the inverse kinematics algorithm of the Pieper configuration robotic arm based on safe posture selection and optimal energy function is shown in Figure (4). Consider:

[0087] Initial position: θ = [0 0 0 0 0 0]

[0088] End position: [p x p y p z = [121.5 -368.37 + 701.37] Unit: cm

[0089] End attitude: [r x r y r z = [-π 0 0] Unit: rad

[0090] The results of the algorithm shown in Figure (4) are shown in Table 2. At this point, the solution that considers both attitude safety and energy optimization is the seventh solution. The inverse solution results are shown in Figure (6).

[0091] Table 2 shows the solution results of the inverse algorithm.

[0092]

[0093]

[0094] (2) Continuity of the optimal solution

[0095] Based on the continuity of the robotic arm's motion, it can be concluded that: when the solution sequence selected by the inverse solution of the trajectory planning calculation interpolation point is consistent with the solution sequence corresponding to the end point, without considering the passage through singular configurations and working boundaries, the robotic arm can continuously move along the predetermined path. This ensures that posture safety, energy loss is minimized, and joint speed does not change abruptly.

[0096] The proof is as follows:

[0097] The analytical solution is described as a mapping function from the end-effector pose to the joint angle. Where i is the sequence number corresponding to the optimal solution, and the differential property of the mapping function is:

[0098]

[0099] Where δt is the differential rotation vector about the base coordinate system, and d is the differential motion vector, therefore, when δt→0, we have dT→0. According to the continuity of elementary functions, when ΔT→0 approaches 0, f i (T+ΔT)-f i (T)→0; when the serial numbers are inconsistent. Discontinuities in joint angular velocities occur, and the optimal energy function is not satisfied, potentially leading to unsafe end-effector postures. The effect of combining kinematic interpolation programming with cubic trajectory planning is as follows: Figure 7 (a), where the initial angle is [0 0 0 00 0], and the final inverse solution angle is... The inverse solution angle corresponding to the interpolation point is The planning results for each joint angle are as follows: Figure 7 (b), where the interpolation point is at t=20s. It can be seen that the introduction of the interpolation point has little impact on the overall angle change trajectory, and the trajectory is very smooth. This is because the inverse solution sequence of the interpolation point and the sequence selected at the endpoint are both optimal solutions.

[0100] The above are merely preferred embodiments of the present invention and should not be construed as limiting the scope of protection of the present invention. Any modifications made to the technical solution based on the technical concept proposed in this invention shall fall within the scope of protection of this invention.

Claims

1. A method for inverse kinematics selection of a Pieper-configuration robotic arm based on safe posture selection and optimal energy function, characterized in that, Includes the following steps: S1: Based on the two-dimensional model of the robotic arm, a coordinate model of the DH link of the robotic arm is established according to the forward kinematics and geometric parameters, using... Establish a system of inverse equations and solve for the joint angles to generate multiple sets of inverse solutions; S2: To meet the requirements of end-effector attitude specification, a safe attitude screening strategy is proposed. The end-effector attitudes of the multiple inverse kinematics obtained in S1 are classified. Through attitude analysis, sufficient conditions for satisfying the inverse kinematics attitude as a safe attitude are derived, and further... and The corresponding joint angles meet the sufficient conditions for a safe end-effector posture, thus enabling the screening of safe end-effector postures. The inverse solution of the end-effector posture is divided into three categories, specifically: factors affecting the overall posture of the robotic arm are... and ;right and Classify the values, without considering The value of , and The classification results are as follows: （1） ， ， （2） ， ， （3） ， ， （4） ， ， Considering the object to be grasped is directly in front of the robotic arm, at this time... The value can be either 0 or 2. Broadly speaking, it is divided into the 0 half-zone and Half-region, the robot arm's 0 pose reference robot arm kinematic model initial state, where , , and , , The 2nd and 3rd axis attitudes are consistent; , , and , , The 2nd and 3rd axis attitudes are consistent, however when It is impossible for it to occur , At this point, the robotic arm's pose is opposite to that of the workspace, and similarly... It is impossible for it to occur , In summary , There are three possible attitudes facing the workspace; the sufficient condition for satisfying the inverse kinematics attitude to be a safe attitude is: ; S3: To address the shortest path requirement, an optimal energy function strategy is proposed. The energy function is constructed based on the joint angles corresponding to the initial and final poses. The constructed energy function is as follows: (15)， in for The mass of the connecting rod, For one set of safe solutions, This is the initial joint angle; By applying the solution obtained in S2 that meets the safe attitude requirements, we can obtain the inverse kinematic solution that meets both the safe attitude and energy optimality through the energy function optimal strategy.

2. The inverse kinematics screening method for Pieper-configuration robotic arms based on safe posture screening and optimal energy function as described in claim 1, characterized in that, S1 uses The inverse system of equations is as follows: (3)， in: , , .

3. The inverse kinematics screening method for Pieper-configuration robotic arms based on safe posture screening and optimal energy function as described in claim 2, characterized in that, In S1, the six joint angles are solved by solving the inverse equation system, resulting in eight sets of inverse solutions.

4. The inverse kinematics screening method for Pieper configuration robotic arms based on safe posture screening and optimal energy function as described in claim 3, characterized in that, The solution process is as follows: Let the matrix elements of equation (3) be... and Equal to each other, we get: (4)， make , , ,but: (5)， when , A solution exists: (6)， in, In order to be able to and The arctangent function whose sign automatically switches quadrants; find Afterwards, it can be based on Seeking : (7)， Let the matrix elements of equation (3) and They are equal respectively. ,have to: (8)， Let the matrix elements of equation (3) and Equal to each, we get: (9)， Let the matrix elements of equation (3) and Equal to each, we get: (10)， make , We can obtain: (11)， make , ,when We can obtain: (12)， According to the solution It can be solved : (13)， Finally, based on the obtained... , Solve : (14)， The angles of the six joints have now been determined.

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