A connecting rod finger structure and a method for optimizing structural parameters thereof

Abstract

The application provides a kind of connecting rod finger structure and its structural parameter optimization method, in particular to the field of robot technology.The connecting rod finger includes finger base, proximal phalanx, middle phalanx, distal phalanx, three linear actuators and connecting rod transmission assembly;The first linear actuator drives the proximal joint to be coupled with the distal joint to bend, and the second and third linear actuators are synchronously or differentially actuated through the mirror image setting of the drive branch chain to drive the metacarpophalangeal joint to bend or side swing.The method establishes a planar two-degree-of-freedom model, a mechanical transmission model and a spatial side swing model based on mirror symmetry, constructs a target function and constraint conditions containing motion range, fingertip position, output force and structural size, and optimizes the rod length and fixed angle using an improved differential evolution algorithm with adaptive crossover probability, nonlinear decreasing scaling factor and elite guided mutation to obtain a connecting rod finger parameter that balances motion flexibility, force transmission performance and compact structure.

Description

Technical Field

[0001] This application relates to the field of robotics, and more specifically, to a linkage finger structure and a method for optimizing its structural parameters. Background Technology

[0002] With the development of robotics technology, dexterous hands are widely used in industrial operations, special operations, medical assistance, and bionic robots. As the actuator of a dexterous hand that directly contacts external objects, the fingers' degrees of freedom of movement, joint coordination, gripping force, and structural compactness directly affect the dexterous hand's operational capabilities. To make robotic fingers resemble the movement patterns of the human hand, existing technologies typically incorporate metacarpophalangeal joints, proximal interphalangeal joints, and distal interphalangeal joints, employing gears, tendons, or linkage mechanisms to transmit motion between the drive components and each joint.

[0003] Early fully actuated dexterous hands typically used separate actuators for each joint. While this enabled relatively flexible joint control, the large number of actuators and transmission components led to a complex hand structure, increased size and weight, and higher control difficulty and manufacturing costs. Therefore, current technologies are increasingly employing underactuated or coupled actuation methods, using a single actuator to drive multiple joints in tandem. This reduces the number of actuators while still enabling finger flexion and object grasping.

[0004] For example, Chinese utility model patent CN206445813U discloses an underactuated humanoid dexterous hand device. It employs a modular finger structure, using different driving mechanisms to drive the base joints of the fingers, as well as the intermediate and distal joints coupled by linkage mechanisms. This allows the fingers to form an envelope according to the shape of the object being grasped, improving upon the problems of traditional underactuated fingers, such as difficulty in adjusting the coupled joints and high requirements for the initial position of the object. This solution has a positive effect on reducing the number of driving components, improving grasping adaptability, and simplifying overall control.

[0005] However, as the degrees of freedom and number of links in the linkage fingers increase, structural parameters such as link length, fixed angles between links, drive component installation position, and drive stroke are interrelated and jointly affect the range of motion of each joint, fingertip trajectory, fingertip output force, and overall finger size. Changing one parameter often causes changes in multiple kinematic and structural properties simultaneously. Especially for spatial linkage fingers that can both bend and laterally swing, their kinematic relationships exhibit strong nonlinearity and coupling, making it difficult to determine a reasonable combination of parameters directly based on a single dimension or simple experience.

[0006] The parameter design of existing multi-link fingers typically relies heavily on design experience, item-by-item adjustments, and repeated simulations, requiring iterative kinematic calculations and structural verifications for different parameter combinations. When there are many parameters to be optimized, the number of possible parameter combinations increases rapidly, leading not only to a large design and computational workload but also to problems such as low search efficiency or unsatisfactory overall performance of the obtained parameter combinations. It is difficult to balance joint range of motion, fingertip output capability, and structural compactness within a reasonable computation time. Therefore, improving the optimization efficiency of spatial multi-link finger structural and kinematic parameters, and obtaining parameter combinations with better overall performance, remains a technical problem that needs to be solved in this field. Summary of the Invention

[0007] The purpose of this application is to address the shortcomings of the prior art by providing a linkage finger structure and a method for optimizing its structural parameters.

[0008] To achieve the above objectives, the technical solution adopted in this application is as follows: One aspect of this application provides a linkage finger structure, including a finger base, a proximal phalanx, a middle phalanx, a distal phalanx, a first linear drive member, a second linear drive member, a third linear drive member, and a linkage transmission assembly; The middle phalanx hinges to the proximal phalanx to form the proximal interphalangeal joint, and the distal phalanx hinges to the middle phalanx to form the distal interphalangeal joint; The output ends of the first linear drive, the second linear drive, and the third linear drive are respectively slidably connected to the base of the finger along the length direction of the base of the finger; The linkage transmission assembly includes a first drive chain, a second drive chain, a third drive chain, and a distal finger coupling chain. The first drive chain is connected between the output end of the first linear drive member and the middle finger joint, and is used to convert the linear displacement output by the first linear drive member into the rotation of the middle finger joint relative to the proximal finger joint around the proximal finger joint. One end of the distal phalanx is hinged to the proximal phalanx, and the other end is hinged to the distal phalanx. The proximal phalanx, middle phalanx, distal phalanx, and distal phalanx form a closed-loop linkage mechanism to drive the distal phalanx to rotate around the distal phalanx when the middle phalanx rotates around the proximal phalanx. The second drive chain is connected between the output end of the second linear drive member and the proximal phalanx, and the third drive chain is connected between the output end of the third linear drive member and the proximal phalanx. The proximal phalanx is movably connected to the base of the finger through the second drive chain and the third drive chain, and forms a metacarpophalangeal joint between the proximal phalanx and the base of the finger. The second linear drive, the third linear drive, the second drive branch, and the third drive branch are mirror-symmetrical about the mid-surface of the finger base in the width direction. When the output ends of the second and third linear drive move synchronously, the proximal phalanx bends around the metacarpophalangeal joint. When the output ends of the second and third linear drive move differentially, the proximal phalanx swings laterally around the metacarpophalangeal joint.

[0009] Furthermore, the proximal phalanx includes a first proximal four-link and a second proximal four-link, which are mirror images of each other in the width direction about the base of the finger. The middle phalanx includes a middle three-link. The first drive chain includes a first push rod link, a proximal three-link, and a proximal staggered link. One end of the first push rod link is hinged to the output end of the first linear drive member, and the other end is hinged to the proximal three-link. The proximal three-link is also hinged to the proximal phalanx and the proximal staggered link, respectively. The end of the proximal staggered link away from the proximal three-link is hinged to the middle three-link, so that the proximal three-link, the proximal phalanx, the middle three-link, and the proximal staggered link form a proximal closed-loop linkage mechanism.

[0010] Furthermore, the distal phalanx includes a distal fingertip, and the first and second proximal four-linkage links are respectively hinged to the middle three-linkage link, which is hinged to the distal fingertip. The distal finger coupling branch includes a first and second middle-end staggered link arranged in a mirror image of the mid-surface in the width direction about the base of the finger. One end of the first middle-end staggered link is hinged to the first proximal four-linkage link, and the other end is hinged to the distal fingertip. One end of the second middle-end staggered link is hinged to the second proximal four-linkage link, and the other end is hinged to the distal fingertip. This forms a first distal closed-loop linkage mechanism with the first proximal four-linkage link, the middle three-linkage link, the distal fingertip, and the first middle staggered link. The second proximal four-linkage link, the middle three-linkage link, the distal fingertip, and the second middle staggered link form a second distal closed-loop linkage mechanism.

[0011] Furthermore, the second drive chain includes a second push rod link, and the third drive chain includes a third push rod link. One end of the second push rod link is hinged to the output end of the second linear drive member, and the other end is hinged to the first proximal fourth auxiliary rod. One end of the third push rod link is hinged to the output end of the third linear drive member, and the other end is hinged to the second proximal fourth auxiliary rod.

[0012] Furthermore, the proximal three-link, the middle three-link, and the distal fingertip each include three spaced-apart hinges and two fixed link segments. The two fixed link segments connect to two adjacent hinges and are fixed to each other, forming a fixed angle between the two fixed link segments. The first proximal four-link and the second proximal four-link each include four spaced-apart hinges and three fixed link segments. The three fixed link segments connect to two adjacent hinges and are fixed to each other in sequence, forming a fixed angle between the two adjacent fixed link segments.

[0013] Another aspect of this application provides a method for optimizing the structural parameters of a link finger based on improved differential evolution, comprising: A spatial three-degree-of-freedom model of the finger with the linkage to be optimized is established. The finger with the linkage to be optimized includes metacarpophalangeal joints, proximal finger joints, distal finger joints, a first linear drive, a second linear drive, a third linear drive, and a linkage transmission assembly. The linkage transmission assembly includes a first drive branch, a second drive branch, a third drive branch, and a distal finger coupling branch. The first linear drive drives the proximal and distal finger joints to bend in a coupled manner through the first drive branch and the distal finger coupling branch. The second and third linear drive acts on the metacarpophalangeal joints through mirror-symmetric second and third drive branches, respectively. When the driving stroke of the second and third linear drive changes synchronously, it drives the metacarpophalangeal joint to bend; when it changes differentially, it drives the metacarpophalangeal joint to swing laterally. Based on the mirror symmetry relationship between the second and third driving branches, one of the second and third driving branches is selected, and a planar two-degree-of-freedom model is established with the first driving branch and the distal finger coupling branch. Based on the planar two-degree-of-freedom model, a positive kinematic mapping relationship is established between the driving stroke of the first linear drive and the linear drive corresponding to the selected driving branch and the bending angle of the metacarpophalangeal joint, the angle of the proximal finger joint, the angle of the distal finger joint, and the fingertip position. Based on the positive kinematic mapping relationship, a mechanical transmission model between the driving force of the first linear drive and the linear drive corresponding to the selected drive branch and the fingertip output force is established. Based on the spatial three-degree-of-freedom model, a lateral swing motion model between the driving stroke of the second and third linear drive and the lateral swing angle of the metacarpophalangeal joint is established. The lengths of independent links, the lengths of each fixed link segment, and the fixed angles between adjacent fixed link segments within the same rigid component in the linkage transmission assembly are used as optimization variables. A comprehensive objective function and constraints are constructed based on the positive kinematic mapping relationship, the mechanical transmission model, and the lateral swing motion model. An improved differential evolution algorithm, including an adaptive crossover probability mechanism, a nonlinear decreasing scaling factor mechanism, and an elite-guided mutation mechanism, is adopted to iteratively optimize the optimization variables within the parameter range limited by the constraints. The independent link length, fixed link segment length, and fixed angle corresponding to the candidate individual with the smallest comprehensive objective function value are determined as the optimization structural parameters of the link finger to be optimized.

[0014] Furthermore, establishing a two-degree-of-freedom planar model includes: The output end of the first linear drive and the output end of the linear drive corresponding to the selected drive branch are respectively used as sliders in the crank-slider mechanism to construct a mutually coupled palm-finger driven crank-slider mechanism, a proximal-distal finger driven crank-slider mechanism, a first planar interlaced four-bar mechanism and a second planar interlaced four-bar mechanism. The planar vector analytical method is used to establish the vector closed-loop equations of the palm-finger driven crank-slider mechanism, the proximal-finger driven crank-slider mechanism, the first planar interlocked four-bar mechanism, and the second planar interlocked four-bar mechanism, respectively. Based on the coupling relationship between the four mechanisms, the forward solution is performed to obtain the cascade mapping relationship between the driving stroke of the first linear drive component and the linear drive component corresponding to the selected drive branch and the angles of each link, the bending angle of the palm-finger joint, the angle of the proximal finger joint, the angle of the distal finger joint, and the Cartesian coordinates of the fingertip.

[0015] Furthermore, establishing a force transmission model includes: The contact between the fingertip and the target object is equivalent to point contact. Under the quasi-static condition of ignoring the gravity and inertial forces of each component of the link finger to be optimized, the driving stroke of the first linear drive component and the linear drive component corresponding to the selected drive branch are combined into a driving stroke vector. The partial derivative of the fingertip Cartesian coordinate vector with respect to the driving stroke vector is obtained to construct the Jacobian matrix of the fingertip Cartesian coordinate vector with respect to the driving stroke vector. Based on the principle of virtual work, the Jacobian transpose relationship between the driving force vectors of the two linear actuators and the fingertip output force vector is established, thereby obtaining the mapping relationship between the driving force of the two linear actuators and the fingertip output force.

[0016] Furthermore, establishing the lateral swing motion model includes: Establish a global coordinate system and a local coordinate system fixed to the proximal phalanx. Transform the link vectors of the second and third drive branches in the local coordinate system to the global coordinate system using a rotation matrix. Based on the spatial vector closed-loop relationship between the second and third driving branches, the residual vector is constructed with the driving stroke of the second and third linear driving components as known inputs and the bending angle and lateral swing angle of the metacarpophalangeal joint as variables to be determined. Within the preset feasible range of the metacarpophalangeal joint flexion angle and the metacarpophalangeal joint lateral displacement angle, a grid search is performed, and the sampling point with the minimum sum of squares of the residual vector is determined as the coarse solution. Using the coarse solution as the initial value of Newton's iteration, the central difference method is used to calculate the numerical Jacobian matrix of the residual vector relative to the bending angle and the lateral swing angle of the metacarpophalangeal joint, and the bending angle and the lateral swing angle of the metacarpophalangeal joint are iteratively updated until the norm of the residual vector is less than the preset residual threshold or the norm of the angle correction is less than the preset correction threshold. The flexion angle and lateral swing angle of the metacarpophalangeal joint obtained after iterative convergence are used to establish a correspondence with the driving stroke of the second and third linear actuators.

[0017] Furthermore, the improved differential evolution algorithm is used to iteratively optimize the variables, including: Initialize the candidate population and generate the corresponding crossover probability for each candidate individual; In each iteration, a scaling factor that decreases according to a nonlinear law is determined based on the current iteration number. The fitness of each candidate individual in the current candidate population is sorted according to the comprehensive objective function value. Elite individuals are selected from the candidate individuals whose fitness ranking is within a preset range. At least two different random individuals are selected. Mutant individuals are generated based on the individual to be mutated, the elite individuals, and at least two random individuals. The mutation direction includes the guiding component from the individual to be mutated to the elite individual and the difference component between the random individuals. Based on the crossover probability corresponding to the individual to be mutated, crossover operation is performed on the mutated individual and the individual to be mutated to generate test individuals. The comprehensive objective function values ​​of the test individuals and the individuals to be mutated are compared, and the individuals with smaller comprehensive objective function values ​​are retained in the next generation of candidate population. When the test individuals are retained, the crossover probability corresponding to the individual to be mutated is assigned to the test individuals. When the individual to be mutated is retained, the crossover probability corresponding to the individual to be mutated is regenerated. Repeat the steps of determining the scaling factor, generating mutant individuals, crossover, and selection for the next generation of candidate populations until the preset iteration termination condition is met.

[0018] The beneficial effects of this application include: This application provides a linkage finger structure and its structural parameter optimization method. A mechanical coupling relationship is established between the proximal and distal finger joints through a first linear drive component, a first drive branch, and a distal finger coupling branch. This allows the linear displacement output by the first linear drive component to be sequentially converted into rotation of the middle phalanx around the proximal joint and rotation of the distal phalanx around the distal joint, thus achieving coordinated bending of the proximal and distal finger joints with a single drive input. This structure reduces the need for independent drive mechanisms at the proximal and distal joints, which helps reduce the structural complexity and moment of inertia of the middle and distal parts of the finger. It also allows the distal phalanx to gradually retract towards the target object along with the middle phalanx, forming a more natural envelope grasping posture. Simultaneously, the proximal, middle, and distal phalanxes, along with their corresponding interlaced links, form multiple closed-loop linkage mechanisms. Determined geometric constraints can limit the motion trajectory of each phalanx, establishing a stable correspondence between the linear drive stroke and the joint angle, which improves the controllability and repeatability of the motion.

[0019] The second and third linear actuators, along with their corresponding second and third drive chains, are arranged in a mirror-symmetric configuration. The movement of the metacarpophalangeal joints is controlled by the coordinated changes in the drive inputs from both sides. When the drive strokes of the second and third linear actuators change synchronously, the drive chains on both sides apply corresponding actions to the proximal phalanx, causing it to primarily bend around the metacarpophalangeal joint. When the drive strokes of the two actuators change differentially, the transmission states on both sides differ, causing the proximal phalanx to laterally swing around the metacarpophalangeal joint. Thus, the same set of symmetrical drive structures can achieve both metacarpophalangeal joint bending and lateral swinging, and can coordinate with the coupled bending of the proximal and distal phalanxes, giving the link fingers three independently controllable degrees of freedom. This enhances the fingers' adaptability to objects of different positions, shapes, and grasping directions. The mirror-symmetric structure on both sides also forms corresponding load-bearing and force transmission paths, which helps reduce unexpected deflection or torsion caused by unilateral drive, improving the stability of the link fingers during movement and force application.

[0020] This application utilizes the mirror symmetry relationship between the second and third driving branches to extract a single-sided palmar-finger driving structure from a three-degree-of-freedom spatial linkage finger, and establishes a planar two-degree-of-freedom model together with the first driving branch and the distal finger coupling branch. Through the cascaded solution of the palmar-finger driven crank-slider mechanism, the proximal-distal finger driven crank-slider mechanism, the first planar interlaced four-bar mechanism, and the second planar interlaced four-bar mechanism, a positive kinematic mapping relationship can be established between the linear driving stroke and the angles of each link, the bending angle of the palmar-finger joint, the angle of the proximal finger joint, the angle of the distal finger joint, and the fingertip position. This dimensionality reduction method preserves the main transmission characteristics of the main bending motion, reduces the number of unknowns, rotational transformations, and closed-loop equations involved in directly solving the complete spatial model, and uses a positive kinematic approach to sequentially obtain the joint posture and fingertip position from the driving input, thereby reducing the complexity of the model solution and facilitating rapid and continuous evaluation of a large number of candidate solutions during structural parameter optimization.

[0021] Based on the forward kinematics model, this application establishes the mechanical transmission relationship between the driving force of the linear drive component and the output force of the fingertip through the Jacobian matrix of the fingertip position with respect to the drive stroke. This allows the evaluation of structural parameters to go beyond the joint range of motion and fingertip trajectory, reflecting the influence of different structural parameters on the fingertip output force and force transmission performance. Simultaneously, for the metacarpophalangeal joint lateral swing motion that cannot be reflected by the planar model, a lateral swing motion model is established based on the spatial vector closed-loop relationship of the second and third drive branches. A reliable coarse solution is obtained using grid search, and then the coarse solution is used as the initial value for Newton iteration. The numerical Jacobian matrix is ​​calculated using the central difference method. This solution method reduces the sensitivity of Newton iteration to initial values, reduces the possibility of convergence to incorrectly assembled branches or iteration divergence, and balances the solution stability and computational accuracy of the lateral swing model.

[0022] This application further uses the length of independent connecting rods, the length of each fixed connecting rod segment, and the fixing angle between adjacent fixed connecting rod segments within the same rigid component as optimization variables. It also comprehensively considers driving stroke, joint range of motion, fingertip output force, metacarpophalangeal joint lateral swing capability, finger structural dimensions, installation space, and mechanism interference to construct a comprehensive objective function and constraints. By simultaneously invoking the forward kinematics model, force transmission model, and lateral swing motion model during the candidate scheme evaluation process, parameter combinations that cannot close, break during movement, exceed the joint range, or cause structural interference can be eliminated before prototype manufacturing. This ensures that the optimization results balance motion performance, force transmission performance, and engineering feasibility.

[0023] To address the issues of decreased population diversity, insufficient convergence speed, and sensitivity to algorithm parameters inherent in traditional differential evolution algorithms (DEE) in high-dimensional nonlinear parameter spaces, this application employs an improved DEE algorithm combining adaptive crossover probability, a nonlinear decreasing scaling factor, and elite-guided mutation. The adaptive crossover probability determines the inheritance or update of the crossover probability based on whether candidate individuals successfully produce better offspring, ensuring that different search regions form crossover strengths commensurate with their optimization effects. The nonlinear decreasing scaling factor maintains a large search range in the early stages of iteration and gradually shifts towards a finer search near the optimal solution in the later stages. Elite-guided mutation allows candidate individuals to maintain population diversity while moving closer to the currently optimal parameter region using random differencing. These mechanisms work together to balance global exploration capability and local convergence accuracy, reducing the risk of prematurely getting trapped in local optima during the optimization process.

[0024] In summary, this application combines a multi-closed-loop linkage finger structure with palmar-finger bending, palmar-finger lateral swing, and proximal-distal finger coupled bending capabilities with planar reduced-dimensional kinematic modeling, spatial lateral swing solution, Jacobian mechanics analysis, and an improved differential evolution optimization method. While maintaining the compactness of the linkage finger structure and the clarity of the transmission relationship, it achieves the coordinated design of motion range, fingertip trajectory, lateral swing capability, output force, and structural dimensions. This improves the efficiency and rationality of linkage finger structure parameter design, reduces the workload of repeatedly adjusting parameters manually and prototyping, and ensures that the resulting linkage finger possesses good motion flexibility, grasping envelope capability, force transmission stability, and engineering feasibility. Attached Figure Description

[0025] Figure 1 A schematic diagram of a spatial three-degree-of-freedom linkage finger structure is provided for this application; Figure 2 A schematic diagram of an equivalent model of a planar two-degree-of-freedom link finger provided in this application; Figure 3 This application provides schematic diagrams of a three-link and a four-link structure; Figure 4A schematic diagram of the spatial vector relationship of a metacarpophalangeal joint lateral swing motion model provided in this application.

[0026] Reference numerals: 1-First linear drive; 2-Second linear drive; 3-Third linear drive; 4-First push rod link; 5-Second push rod link; 6-Third push rod link; 7-Proximal three-link; 8-First proximal four-link; 9-Second proximal four-link; 10-Proximal interlaced link; 11-Middle three-link; 12-First middle interlaced link; 13-Second middle interlaced link; 14-Fingertip; 15-Output end of the first linear drive; 16-Output end of the palm and finger drive linear drive; 17-The... 18-Planar equivalent component of push rod connecting rod; 19-Planar equivalent component of palm-finger driven push rod connecting rod; 10-Planar equivalent component of proximal three-link rod; 1901-First fixed link segment of proximal three-link rod; 1902-Second fixed link segment of proximal three-link rod; 20-Planar equivalent component of proximal four-link rod; 2001-First fixed link segment of proximal four-link rod; 2002-Second fixed link segment of proximal four-link rod; 2003-Third fixed link segment of proximal four-link rod; 21-Planar equivalent component of proximal staggered connecting rod; 22-Middle three-link rod connecting rod connecting rod connecting rod; Planar equivalent component of the secondary link; 2201 - First fixed link segment of the middle three secondary links; 2202 - Second fixed link segment of the middle three secondary links; 23 - Planar equivalent component of the middle staggered link; 24 - Planar equivalent component of the distal fingertip; 2401 - First fixed link segment of the distal fingertip; 2402 - Second fixed link segment of the distal fingertip; 25 - Metacarpophalangeal joint; 26 - Proximal finger joint; 27 - Distal finger joint; 28 - Global coordinate system XYZ; 29 - Vector pointing from the origin of the global coordinate system to the origin of the local coordinate system; 30 - Local coordinate system The coordinate system is defined as follows: UVW; 31 - Vector pointing from the origin of the global coordinate system to the starting point of the second linear drive's stroke; 32 - Vector pointing from the origin of the global coordinate system to the starting point of the third linear drive's stroke; 33 - Stroke vector of the second linear drive; 34 - Stroke vector of the third linear drive; 35 - Vector of the second push rod connecting rod; 36 - Vector of the third push rod connecting rod; 37 - Vector pointing from the origin of the local coordinate system to the hinge point between the second push rod connecting rod and the proximal phalanx; 38 - Vector pointing from the origin of the local coordinate system to the hinge point between the third push rod connecting rod and the proximal phalanx. Detailed Implementation

[0027] The following, in conjunction with the accompanying drawings, further explains the structural configuration, planar and spatial kinematic models, mechanical transmission model, optimization objectives, and improved differential evolution process of the linkage finger. This embodiment is used to explain the connection relationships, kinematic relationships, and computational logic between various technical features and should not be construed as a limitation on the scope of protection. Regarding the differences in naming of the same technical feature in the spatial structure and the planar equivalent model, the following uses the names of the solid component and the planar equivalent component, respectively, and explains the correspondence between the two.

[0028] In this embodiment, the terms "first," "second," and "third" are primarily used to distinguish components with the same or similar functions and do not indicate any sequential order or importance between the components. A hinged connection refers to a connection method in which two components can rotate relative to each other around a corresponding hinge axis, which can be achieved using pins, shafts, bushings, or bearings. A sliding connection refers to a connection method in which the output end is guided by the base of a finger and moves back and forth along a predetermined linear direction. Each hinge position should have necessary axial clearance according to the component thickness and assembly method, and the component's movement along the hinge axis can be restricted by shims, shoulders, or limiting components.

[0029] like Figure 1 As shown, this embodiment provides a linkage finger structure that can be applied to robotic dexterous hands, bionic manipulators, or other end-effectors requiring envelope grasping, fingertip gripping, and lateral adjustment movements. The linkage finger structure includes a finger base, proximal phalanx, middle phalanx, distal phalanx, a first linear drive 1, a second linear drive 2, a third linear drive 3, and a linkage transmission assembly. The finger base serves as the load-bearing foundation for the entire finger structure, housing the three linear drives and providing linear motion guidance for their output ends. The proximal, middle, and distal phalanxes are arranged sequentially from the finger root towards the fingertip, forming a multi-phalanx structure capable of simulating the spatial movement of human hand fingers.

[0030] The length direction of the finger base is the longitudinal extension direction along the finger's extension direction, and the width direction of the finger base is the direction that intersects the length direction and is located between the two transverse sides of the finger. The mid-surface of the width direction of the finger base extends along the length direction and divides the finger base into two sides along the width direction. The first linear drive 1, the second linear drive 2, and the third linear drive 3 can all output linear displacement and driving force that vary along the length direction of the finger base, and their output ends can be push rods, sliders, or other output components that can move along a linear guide path.

[0031] The middle phalanx and proximal phalanx hinge to form a proximal phalanx joint 26, and the distal phalanx and middle phalanx hinge to form a distal phalanx joint 27. The hinge axes of the proximal phalanx 26 and distal phalanx 27 are set according to the bending direction of the finger body, allowing the middle and distal phalanxes to rotate towards the palmar side of the finger and gradually form a curled posture. A metacarpophalangeal joint 25 is formed between the proximal phalanx and the base of the finger. The metacarpophalangeal joint 25 can realize the bending and lateral movement of the proximal phalanx relative to the base of the finger. Therefore, the metacarpophalangeal joint 25 is not limited to a single-degree-of-freedom revolute joint with a single fixed axis, but forms spatial posture changes through the coordinated movement of two mirror-driven branches.

[0032] The output ends of the first linear actuator 1, the second linear actuator 2, and the third linear actuator 3 are slidably connected to the base of the finger along its length. The linear movements of the three output ends are independent of each other, so as to control the proximal-distal finger coupling flexion and the flexion and lateral movement of the metacarpophalangeal joint 25, respectively. In one specific installation method, the second linear actuator 2 and the third linear actuator 3 are located on both sides in front of the first linear actuator 1, and are mirror images of each other about the width direction of the base of the finger. The second linear actuator 2 and the third linear actuator 3 have the same maximum output stroke, but their actual output positions at any instant can be the same or have a difference.

[0033] The linkage assembly includes a first drive branch, a second drive branch, a third drive branch, and a distal finger coupling branch. The first drive branch is connected between the output end of the first linear drive 1 and the middle finger joint, converting the linear displacement of the first linear drive 1 into rotation of the middle finger joint relative to the proximal finger joint around the proximal finger joint 26. One end of the distal finger coupling branch is hinged to the proximal finger joint, and the other end is hinged to the distal finger joint, forming a closed-loop linkage mechanism with the proximal finger joint, the middle finger joint, the distal finger joint, and the distal finger coupling branch. When the middle finger joint rotates around the proximal finger joint 26, the geometric constraints of the closed-loop linkage mechanism cause the distal finger joint to rotate accordingly around the distal finger joint 27, thereby achieving coupled bending of the proximal finger joint 26 and the distal finger joint 27 using one input from the first linear drive 1.

[0034] The coupling between the proximal interphalangeal joint 26 and the distal interphalangeal joint 27 does not require the two joints to have the same rotation angle. Rather, it refers to the kinematic correspondence between the postures of the two joints, which is determined by the link length, hinge position, and fixation angle of the closed-loop linkage mechanism. By configuring the geometric parameters of each component, the distal phalanx can be gradually brought towards the palm during the flexion of the middle phalanx, allowing the fingertip to approach the target object along a suitable envelope grasping trajectory.

[0035] The second drive chain is connected between the output end of the second linear drive member 2 and the proximal phalanx, and the third drive chain is connected between the output end of the third linear drive member 3 and the proximal phalanx. The proximal phalanx is movably connected to the base of the finger via the second and third drive chains. The second linear drive member 2, the third linear drive member 3, the second drive chain, and the third drive chain are mirror-symmetrical about the mid-plane of the finger base in the width direction.

[0036] When the output ends of the second linear drive unit 2 and the third linear drive unit 3 move synchronously, the two drive chains apply basically symmetrical displacement constraints and driving forces to the proximal phalanx, causing the proximal phalanx to rotate mainly around the metacarpophalangeal joint 25 along the bending direction of the finger body. Synchronous movement can mean that the two output ends move the same displacement in the same direction, or it can mean that when there are manufacturing errors in the geometric relationship between the two mirrored chains, the outputs output mutually matching displacements according to the control calibration results. When the output ends of the second linear drive unit 2 and the third linear drive unit 3 move differentially, the motion states of the two drive chains are different, resulting in different displacement constraints on both sides of the proximal phalanx, which in turn causes it to swing around the metacarpophalangeal joint 25 to one side in the width direction. Changing the magnitude of the stroke difference between the two output ends can change the lateral swing amplitude, and exchanging the relative magnitudes of their strokes can change the lateral swing direction.

[0037] The synchronous and differential components of the second linear actuator 2 and the third linear actuator 3 can coexist, causing the proximal phalanx to simultaneously produce metacarpophalangeal flexion and metacarpophalangeal lateral movement. The first linear actuator 1 can also operate independently during metacarpophalangeal flexion or lateral movement, causing the coupled flexion of the proximal and distal phalanges 26 and the spatial movement of the metacarpophalangeal joint 25 to be superimposed. Thus, the linking finger possesses three degrees of freedom: proximal-distal coupled flexion, metacarpophalangeal flexion, and metacarpophalangeal lateral movement.

[0038] The first linear drive component 1, the second linear drive component 2, and the third linear drive component 3 are centrally mounted at the base of the finger, eliminating the need for large rotary drive mechanisms to be directly mounted on each phalanx. This helps reduce the mass and inertia of the fingertip. The closed-loop linkage mechanism can also form clear motion constraints and multi-path force transmission relationships within a smaller structural space, thus balancing structural compactness, motion flexibility, and gripping load-bearing capacity.

[0039] Furthermore, the proximal phalanx includes a first proximal quadrupole 8 and a second proximal quadrupole 9, mirror-imagely arranged in the mid-plane direction of the finger base width. The first proximal quadrupole 8 and the second proximal quadrupole 9 are located on both sides of the finger body, together forming the bilateral load-bearing skeleton of the proximal phalanx. The mid-phalanx includes a mid-terminal triplet 11, which is located between the proximal and distal phalanges and undertakes the motion transmission and grasping load transmission between the proximal interphalangeal joint 26 and the distal interphalangeal joint 27.

[0040] The first drive chain includes a first push rod link 4, a proximal three-link link 7, and a proximal interleaved link 10. One end of the first push rod link 4 is hinged to the output end of the first linear drive member 1, and the other end is hinged to the proximal three-link link 7. When the output end of the first linear drive member 1 moves along the length of the finger base, the first push rod link 4 oscillates relative to the hinged positions at both ends while translating, and transmits the thrust or pull force of the first linear drive member 1 to the proximal three-link link 7.

[0041] The proximal three-link 7 is hinged to the proximal phalanx and the proximal staggered link 10, respectively. The end of the proximal staggered link 10 away from the proximal three-link 7 is hinged to the middle three-link 11. The proximal three-link 7, proximal phalanx, middle three-link 11, and proximal staggered link 10 form a proximal closed-loop linkage mechanism. When the first linear drive member 1 is activated, the proximal three-link 7 changes its posture relative to the proximal phalanx and drives the middle three-link 11 through the proximal staggered link 10, causing the middle three-link 11 to rotate relative to the proximal phalanx according to the trajectory determined by the closed-loop mechanism.

[0042] In one feasible assembly configuration, the proximal third link 7 is located between the first proximal fourth link 8 and the second proximal fourth link 9, and is connected to the two proximal fourth links at corresponding hinged positions. The two hinged positions can be located on the same rotation axis, or they can be achieved separately through coaxial connecting shafts, ensuring that the proximal third link 7 maintains a consistent rotational reference relative to the first and second proximal fourth links 8 and 9. Whether a single through-shaft or two coaxial short shafts are used can be determined based on the component thickness, assembly space, and maintenance requirements.

[0043] The proximal closed-loop linkage mechanism establishes a definite relationship between the driving stroke of the first linear drive 1 and the rotational posture of the middle three linkages 11. Compared with the method of directly pushing the middle phalanx with a single push rod, the proximal closed-loop linkage mechanism can limit the movement trajectory of the middle phalanx through multiple hinge positions and allows the first linear drive 1 to be concentrated at the base of the finger.

[0044] Based on the proximal closed-loop linkage mechanism, the distal phalanx includes a distal fingertip 14, which is hinged to the middle three-link 11, forming a distal finger joint 27. The distal fingertip 14 serves both as the load-bearing body of the distal phalanx and as the fingertip area that contacts the target object. The fingertip contact area can be fitted with an anti-slip layer, a flexible pad, or a contact surface with a predetermined curvature, depending on the object being grasped, but these contact layers do not change the basic hinged and coupled transmission relationship between the distal fingertip 14 and the middle three-link 11.

[0045] The first proximal fourth link 8 and the second proximal fourth link 9 are respectively hinged to the middle third link 11. The distal finger coupling branch includes a first middle interlaced link 12 and a second middle interlaced link 13, which are mirror images of each other in the width direction about the base of the finger. One end of the first middle interlaced link 12 is hinged to the first proximal fourth link 8, and the other end is hinged to the distal fingertip 14; one end of the second middle interlaced link 13 is hinged to the second proximal fourth link 9, and the other end is hinged to the distal fingertip 14.

[0046] The first proximal four-link 8, the middle three-link 11, the distal fingertip 14, and the first middle staggered link 12 form a first distal closed-loop linkage mechanism; the second proximal four-link 9, the middle three-link 11, the distal fingertip 14, and the second middle staggered link 13 form a second distal closed-loop linkage mechanism. The two distal closed-loop linkage mechanisms share the middle three-link 11 and the distal fingertip 14, and are located on both sides of the finger body, respectively.

[0047] When the middle three-link 11 rotates around the proximal finger joint 26, the relative position between the middle three-link 11 and the distal fingertip 14 changes. Since the first middle interlocking link 12 and the second middle interlocking link 13 restrict the position of the distal fingertip 14 relative to the first proximal four-link 8 and the second proximal four-link 9 on both sides, the distal fingertip 14 is jointly constrained by the two distal closed-loop linkage mechanisms and rotates around the distal finger joint 27, thereby forming a coupled motion between the proximal finger joint 26 and the distal finger joint 27.

[0048] The double-sided mirror closed-loop structure can form corresponding motion constraints and force transmission paths on both sides of the fingertip 14. When the fingertip 14 is subjected to the reaction force of the target object, the load can be transmitted to the first proximal fourth link 8 and the second proximal fourth link 9 through the middle three-link 11, the first middle staggered link 12 and the second middle staggered link 13, which helps to reduce the unexpected torsion of the fingertip 14 caused by unilateral force transmission. In order to avoid the components on both sides rubbing against each other during movement, spacers or shims can be set along the hinge axis, and necessary axial misalignment can be maintained between the first middle staggered link 12, the second middle staggered link 13 and the middle three-link 11 and the fingertip 14.

[0049] Meanwhile, the second drive chain includes a second push rod link 5, and the third drive chain includes a third push rod link 6. One end of the second push rod link 5 is hinged to the output end of the second linear drive member 2, and the other end is hinged to the first proximal fourth auxiliary rod 8; one end of the third push rod link 6 is hinged to the output end of the third linear drive member 3, and the other end is hinged to the second proximal fourth auxiliary rod 9. Both ends of the second push rod link 5 and the third push rod link 6 are hinged to accommodate directional changes between the translation of the output end of the linear drive member and the spatial rotation of the proximal fourth auxiliary rod.

[0050] In the illustrated mirror embodiment, the second push rod link 5 and the third push rod link 6 are of the same length, and the hinge positions of the two push rod links on the proximal phalanx and the mounting positions of the second linear drive member 2 and the third linear drive member 3 are mirror images of each other. During synchronous drive, the second push rod link 5 and the third push rod link 6 apply corresponding driving actions to the first proximal fourth link 8 and the second proximal fourth link 9, respectively; during differential drive, the two push rod links form different spatial postures and cause the proximal phalanx to sway laterally.

[0051] The second linear drive unit 2 and the third linear drive unit 3 can use linear actuators, lead screws, slides, or other linear actuators of the same specifications to achieve the same maximum output stroke and similar driving force characteristics. In actual control, the positions of the two output ends can be detected in a closed loop using position sensors, and deviations caused by manufacturing errors or transmission backlashes can be corrected in synchronous mode.

[0052] like Figure 3 As shown, the proximal three-link 7, the middle three-link 11, and the distal fingertip 14 each include three spaced-apart hinges and two fixed link segments. The two fixed link segments connect to two adjacent hinges and are fixed to each other, forming a fixed angle between them. Here, "three-link" refers to the number of hinges on the rigid member and does not represent three independent members capable of rotating relative to each other. The proximal three-link 7, the middle three-link 11, and the distal fingertip 14 can rotate as a whole relative to other members hinged to them, but there is no relative rotation between the two internal fixed link segments.

[0053] Both the first proximal four-link 8 and the second proximal four-link 9 include four spaced-apart hinges and three fixed link segments. The three fixed link segments connect two adjacent hinges and are fixed in sequence, forming a fixed angle between adjacent fixed link segments. The four links are also named according to the four hinges provided on a rigid member, and the three fixed link segments together define the relative positions between the four hinges.

[0054] Each connecting rod segment can be formed using a one-piece machining method, or it can be connected by welding, screwing and then positioning and locking, gluing, or other methods that can maintain a fixed relative posture during normal movement and load-bearing. For situations where structural parameters need to be adjusted or a verification prototype needs to be made, a detachable connection with an angle positioning structure can be used first to determine the optimal rod length and fixing angle before machining it into a one-piece component.

[0055] The fixed angle differs from the metacarpophalangeal joint angle, proximal interphalangeal joint angle, and distal interphalangeal joint angle. The fixed angle is a geometric parameter that remains unchanged between adjacent fixed links within a rigid component after manufacturing or assembly, while the joint angle changes with finger movement. The length of each fixed link and the fixed angle together determine the transmission ratio, joint range of motion, fingertip trajectory, and force transmission characteristics of the closed-loop mechanism.

[0056] The three hinged parts of the proximal three-link 7 are used to connect the first push rod link 4, the proximal knuckle, and the proximal staggered link 10; the three hinged parts of the middle three-link 11 are used to connect the proximal knuckle, the proximal staggered link 10, and the distal fingertip 14; the three hinged parts of the distal fingertip 14 are used to connect the middle three-link 11, the first middle staggered link 12, and the second middle staggered link 13. The four hinged parts of the first proximal four-link 8 are used to connect the proximal three-link 7, the second push rod link 5, the middle three-link 11, and the first middle staggered link 12; the corresponding hinged parts of the second proximal four-link 9 are used to connect the proximal three-link 7, the third push rod link 6, the middle three-link 11, and the second middle staggered link 13, respectively.

[0057] To improve feasibility, sufficient solid material and bearing installation space should be reserved around each hinge. The cross-sectional dimensions of the fixed link section should be checked for strength and stiffness based on the maximum output force of the linear drive and the target gripping force. Rounded or chamfered corners can be provided on the edges of components to reduce the risk of collisions and stress concentrations during the closed-loop mechanism's movement. After the prototype is assembled, the stroke of the three linear drives can be traversed at low speed to check for interference, jamming, or assembly branch jumps in each component, before conducting a gripping test under rated driving force conditions.

[0058] This embodiment establishes a structural parameter optimization method based on the aforementioned link-finger structure. This method first utilizes mirror symmetry to equate the main bending portion of the spatial three-degree-of-freedom model to a planar two-degree-of-freedom model, establishing the positive kinematic relationship between the driving stroke and the angles of each link, joint angle, and fingertip position; then, it establishes a fingertip output force model and a metacarpophalangeal joint lateral swing model; subsequently, it constructs a comprehensive objective function and constraints, and uses an improved differential evolution algorithm to search for structural parameters.

[0059] Step 1: Establish a two-degree-of-freedom planar model and its forward kinematic mapping relationship The spatial model of the finger linkage to be optimized has three degrees of freedom: proximal-distal joint coupled bending, metacarpophalangeal joint bending, and metacarpophalangeal joint lateral swing. The second and third driving branches are mirror-symmetric about the mid-plane of the finger base width direction. When the second linear drive 2 and the third linear drive 3 move synchronously, the two driving branches have the same motion law in the main bending plane. Therefore, either side is extracted from the second and third driving branches and used together with the first driving branch and the distal finger coupled branch to establish a planar two-degree-of-freedom model. The planar model retains the metacarpophalangeal joint bending degree of freedom and the proximal-distal finger coupled bending degree of freedom, while the lateral swing degree of freedom is solved separately in the spatial model in step 2.

[0060] like Figure 1 and Figure 2 As shown, Figure 2This is a planar equivalent model established by selecting one side from the second and third drive branches. The output end of the first linear drive unit 1 is... Figure 2 The first linear drive unit's output end 15 is represented by the first push rod connecting rod 4. Figure 2 The first push rod connecting rod is represented by a planar equivalent component 17, and the near-end three auxiliary rods 7 are in Figure 2 The planar equivalent member 19 of the three proximal links is represented by the proximal staggered link 10. Figure 2 The planar equivalent member 21 of the near-end staggered link is represented by the three intermediate links 11. Figure 2 The planar equivalent component 22 of the three intermediate rods is represented by the end fingertip 14. Figure 2 The planar equivalent component 24 at the fingertip is used to represent the palm-finger drive. The output end 16 of the palm-finger drive linear actuator, the planar equivalent component 18 of the palm-finger drive push rod link, the planar equivalent component 20 of the proximal four-link link, and the planar equivalent component 23 of the mid-end staggered link correspond to the palm-finger drive structure on the selected side. In this embodiment, a planar two-degree-of-freedom model is established on the side where the second drive branch is located. Therefore, the output end 16 of the palm-finger drive linear actuator corresponds to the output end of the second linear actuator 2, the planar equivalent component 18 of the palm-finger drive push rod link corresponds to the second push rod link 5, the planar equivalent component 20 of the proximal four-link link corresponds to the first proximal four-link link 8, and the planar equivalent component 23 of the mid-end staggered link corresponds to the first mid-end staggered link 12. When a planar two-degree-of-freedom model is established on the side where the third drive branch is located, the above-mentioned planar equivalent components correspond to the output end of the third linear actuator 3, the third push rod link 6, the second proximal four-link link 9, and the second mid-end staggered link 13, respectively.

[0061] exist Figure 2 In the planar two-degree-of-freedom model shown, the planar equivalent component 17 of the first push rod, the first fixed link segment 1901 of the proximal three-link, the planar equivalent component 18 of the palm-finger drive push rod, the first fixed link segment 2001 of the proximal four-link, the second fixed link segment 1902 of the proximal three-link, the planar equivalent component 21 of the proximal staggered link, the first fixed link segment 2201 of the middle three-link, the second fixed link segment 2002 of the proximal four-link, the second fixed link segment 2202 of the middle three-link, the first fixed link segment 2401 of the distal fingertip, the planar equivalent component 23 of the middle staggered link, the third fixed link segment 2003 of the proximal four-link, and the second fixed link segment 2402 of the distal fingertip are denoted as follows: The planar equivalent component 17 of the first push rod, the first fixed link segment 1901 of the proximal three-link, the planar equivalent component 18 of the palm-finger drive push rod, the first fixed link segment 2001 of the proximal four-link, the second fixed link segment 1902 of the proximal three-link, the planar equivalent component 21 of the proximal staggered link, the first fixed link segment 2201 of the middle three-link, the second fixed link segment 2002 of the proximal four-link, the second fixed link segment 2202 of the middle three-link, the first fixed link segment 2401 of the fingertip, the planar equivalent component 23 of the middle staggered link, the third fixed link segment 2003 of the proximal four-link, and the second fixed link segment 2402 of the fingertip, with respect to the horizontal axis of the planar coordinate system, are denoted as follows: All directional angles adopt the same positive direction convention, for example, the counterclockwise direction is taken as the positive direction, and this convention is maintained in all closed-loop equations of the planar kinematic model.

[0062] The kinematic equations of the palm-finger driven crank-slider mechanism are:

[0063] In the formula, , and These represent the eccentricity of the output end 16 of the palm-finger drive linear actuator, the ordinate of the hinge point of the first fixed link segment 2001 of the proximal four-link at the palm-finger joint 25, and the drive stroke of the output end 16 of the palm-finger drive linear actuator, respectively. The eccentricity describes the lateral offset between the guide trajectory of the output end 16 of the palm-finger drive linear actuator and the reference hinge position of the palm-finger joint 25. Based on this set of equations, given the structural parameters and the stroke of the output end 16 of the palm-finger drive linear actuator, the orientation angles of the planar equivalent member 18 of the palm-finger drive push rod and the first fixed link segment 2001 of the proximal four-link can be solved, and the in-plane bending posture of the palm-finger joint 25 can be obtained.

[0064] When solving the crank-slider mechanism, multiple mathematical solutions may be obtained. In practice, the solution that is consistent with the initial assembly posture and meets the joint angle range is adopted. When calculating continuously along the drive stroke, the angle of the sampling point of the previous stroke can be used as the branch selection reference for the current sampling point, thereby avoiding the solution result from jumping to another assembly branch.

[0065] The kinematic equations of the near-far finger driven crank-slider mechanism and the first-plane interleaved four-bar mechanism are:

[0066] In the formula, , and The corresponding parameter concept is consistent with the output end 16 model of the aforementioned palm and finger driven linear drive component. This represents the lateral position parameter of the output end of the first linear drive unit, representing a stroke of 15. Because... and The orientation of the metacarpophalangeal joint 25, defined by the output end 16 of the metacarpophalangeal linear actuator, is related to the motion boundary of the output end 15 of the first linear actuator and the motion state of the output end 16 of the metacarpophalangeal linear actuator. This relationship reflects the influence of the bending of the metacarpophalangeal joint 25 on the driving geometry boundary of the proximal interphalangeal joint 26.

[0067] This indicates the fixing angle between the first fixed link segment 1901 and the second fixed link segment 1902 of the near-end three-link. This represents the fixing angle between the first fixed link segment 2001 and the second fixed link segment 2002 of the proximal four-link. The linear displacement of the output end 15 of the first linear drive member is transmitted to the planar equivalent member 21 of the proximal interlaced link via the planar equivalent member 17 of the first push rod and the planar equivalent member 19 of the proximal three-link, and is converted into the rotation of the proximal finger joint 26 by the proximal closed-loop mechanism. Each fixing angle is a fixed parameter within the rigid member and does not change during movement.

[0068] The kinematic equations of the second-plane staggered four-bar linkage are:

[0069] In the formula, This indicates the fixing angle between the first fixed link segment 2201 and the second fixed link segment 2202 of the middle three-link structure. The fixed angle between the second fixed link segment 2002 and the third fixed link segment 2003 of the proximal four-bar linkage is indicated. The second planar interleaved four-bar linkage further transmits the motion at the proximal finger joint 26 to the distal closed-loop mechanism to solve for the orientation of the first fixed link segment 2401 of the distal fingertip and the planar equivalent component 23 of the middle interleaved link, and to obtain the rotational state of the distal finger joint 27.

[0070] To facilitate the establishment of the vector closed-loop equation, the line connecting the non-common endpoints of the first fixed link segment 2401 and the second fixed link segment 2402 at the end fingertip is defined as an auxiliary vector, and its length is denoted as . , Not independent structural parameters, but rather determined by... , and fixed angle The uniquely determined angle of the line connecting the non-common endpoints of the first fixed link segment 2401 and the second fixed link segment 2402 at the end of the fingertip, relative to the horizontal axis of the planar coordinate system, is denoted as . :

[0071] The kinematic equation of the fingertip is:

[0072] In the formula, This indicates the binding angle between the first binding segment 2401 and the second binding segment 2402 of the fingertip. This set of relationships is used to describe the directional relationship between the two binding segments inside the planar equivalent member 24 of the fingertip and their combined position.

[0073] By combining the step-by-step solutions of each closed-loop mechanism, a cascaded nonlinear mapping relationship can be established between the driving stroke of the output end 15 of the first linear drive and the output end 16 of the palm and finger drive linear drive, the angles of the thirteen intermediate links, and the Cartesian position of the fingertips. Its comprehensive analytical expression is as follows:

[0074] In the formula, Indicates fingertips Coordinates about input and function expressions and Indicates fingertips Coordinates about input and The function expression is as follows. In actual calculations, the relevant angles of the metacarpophalangeal joints are first determined based on the output stroke of the linear actuator. Then, the geometric boundary of the closed loop corresponding to the output of the first linear actuator is updated. Next, the first and second planar interleaved four-bar linkages are solved, and finally, the fingertip position is calculated. Each set of candidate structural parameters should be solved continuously within the predetermined drive stroke range. If a real solution satisfying the link length closed loop, assembly branch, and angle constraints does not exist at a certain stroke position, the candidate structure is considered infeasible.

[0075] Step 2: Establish the Jacobian mechanical transmission model and the lateral swing motion model When establishing the mechanical transmission model, the contact between the fingertip and the target object is simplified to point contact, and quasi-static analysis is adopted. Quasi-static analysis mainly examines the static equilibrium relationship after the driving force is transmitted to the fingertip through the linkage mechanism, neglecting the dynamic effects caused by the gravity, inertial forces, and high-speed motion of each component. For the actual prototype, joint friction and transmission efficiency can be compensated for using correction coefficients calibrated through experiments.

[0076] Based on the mapping between the fingertip Cartesian coordinates obtained in step 1 and the two driving inputs, the partial derivative of the end position with respect to the driving stroke is calculated to establish the Jacobian matrix, and the mapping from driving force to fingertip output force is realized accordingly. The mechanical transmission relationship is as follows:

[0077] In the formula, The force vector at the fingertip For Jacobian matrices, and This represents the two equivalent driving forces in a two-degree-of-freedom planar model. When using this relationship, attitudes where the Jacobian matrix is ​​non-invertible or nearly singular should be avoided. For spatial structures where the second linear actuator 2 and the third linear actuator 3 operate synchronously, the second equivalent driving force in the planar model should reflect the combined contribution of the two mirror palm-finger driving branches.

[0078] When calculating the specific fingertip force, the input can be set according to the rated thrust of the selected linear actuator or the actual thrust measured in the experiment, and the fingertip force can be calculated under the predetermined gripping posture. The input forces of the three linear actuators can be given separately according to the actuator specifications, and they do not need to be exactly the same; when the second linear actuator 2 and the third linear actuator 3 grip simultaneously, the same or calibrated and matched input forces are usually used. A fingertip force of 30 N corresponds to a specific driving force and a specific gripping posture. During implementation, the corresponding stroke and joint posture should be recorded for verification.

[0079] like Figure 4 As shown, the lateral swing motion needs to be calculated in a three-dimensional coordinate system. A global coordinate system XYZ 28 is established at the base of the finger, and a local coordinate system UVW 30 is established on the proximal phalanx, moving with the proximal phalanx. The origin of the local coordinate system UVW 30 can be set at a fixed reference point on the proximal phalanx to facilitate the description of the connection position of the two metacarpophalangeal drive branches, and should remain consistent throughout the calculations; its specific dimensions and coordinates can be determined based on the actual structural model or three-dimensional design drawing.

[0080] The second linear drive 2 and the third linear drive 3 are located to the right front and left front of the first linear drive 1, respectively, and both have the same maximum total drive stroke. The vector pointing from the origin of the global coordinate system XYZ 28 to the origin of the local coordinate system UVW 30 is vector 29; the vectors pointing from the origin of the global coordinate system XYZ 28 to the starting points of the strokes of the second linear drive 2 and the third linear drive 3 are vectors 31 and 32, respectively; the stroke vectors of the second linear drive 2 and the third linear drive 3 are vectors 33 and 34, respectively; the vectors of the second push rod connecting rod 5 and the third push rod connecting rod 6 are vectors 35 and 36, respectively; the vectors pointing from the origin of the local coordinate system UVW 30 to the hinge points on the second push rod connecting rod 5 and the third push rod connecting rod 6 are vectors 37 and 38, respectively. The connection position vectors in the local coordinate system are transformed to the global coordinate system using a rotation matrix, and then combined with the actual rod lengths of the two push rods to establish a spatial closed-loop relationship. The vector relationships of the spatial lateral swing model are as follows:

[0081] In the formula, each vector corresponds to the positional relationship between the origin of the global coordinate system and the origin of the local coordinate system, the origin of the global coordinate system and the starting point of the stroke of the two linear actuators, the stroke of the two linear actuators, and the origin of the local coordinate system and the hinge point on the two push rods, respectively. Rotation matrix This is used to reflect the coordinate transformation under the combined effects of bending in the plane of the metacarpophalangeal joint and lateral sway in space. In the mirrored embodiment, the two push rods are of the same length, and the two stroke start points and the upper hinge position are mirrored about the mid-plane in the finger width direction.

[0082] Based on the above spatial closed-loop relationship, two push rod strokes and bending angles in the metacarpophalangeal joint plane can be established. and spatial lateral sway angle Function mapping between:

[0083] The spatial closed-loop relationship involves rotation matrices and nonlinear rod length constraints. Directly obtaining explicit analytical solutions easily leads to high-order equations and multiple solution selection problems. Therefore, a mesh search is first performed within the physically feasible domain of the palmar finger bending angle and lateral sway angle. The sum of squared residuals at each sampling point is calculated, and the minimum sum is selected as the coarse solution. This coarse solution is then used as the initial value for Newton iteration. The residual Jacobian in Newton iteration is calculated using the central difference method until the state correction or residual accuracy meets a preset threshold.

[0084] The grid spacing, central difference step size, and convergence threshold can be set according to the model scale and computational accuracy. For example, the grid search can first use angular intervals on the order of degrees to cover the entire feasible region, and then refine it near the coarse solution; the central difference can use a small step size that is sufficient to reflect local changes without causing significant rounding errors; the residual threshold and state correction threshold can be set uniformly according to the rod length unit and angular accuracy. If convergence is not achieved after reaching the maximum number of iterations, the corresponding drive stroke combination can be judged as having no valid solution.

[0085] When the strokes of the second linear drive 2 and the third linear drive 3 are the same, the calculated lateral sway angle should be close to zero; when there is a difference in their strokes, the proximal phalanx will sway; after exchanging the relative magnitudes of their strokes, the lateral sway direction should be opposite. This symmetry rule can be used to verify whether the spatial lateral sway model and program implementation are correct.

[0086] Step 3: Construct the comprehensive objective function and constraints The optimization objective comprehensively considers the linear drive component stroke, mechanism mounting spacing, fingertip force, lateral swing angle, total finger length, motion angles of the three joints, and local structural compactness. The corresponding comprehensive objective function is as follows:

[0087] In the formula, to These are used to adjust the weight of each evaluation item in the overall optimization. and The driving stroke of the output end 15 of the first linear drive and the output end 16 of the palm and finger drive linear drive are defined. and Used to characterize the dimensions of relevant local structures Used to reflect the spacing between the guide positions of related linear drive components. The vector represents the fingertip force, and the angle term is used to evaluate the range of motion of the metacarpophalangeal, proximal, and distal interphalangeal joints from the initial state to the final state. According to the following constraint inequality, the total length of the finger... It needs to be as small as possible while still being greater than 120mm.

[0088] In practice, the weights can be set according to application requirements and the numerical range of each indicator, so that indicators of different units have comparable influence in the comprehensive objective function. For the fingertip force, lateral swing angle, and joint range of motion that need to be maximized, the optimal structure should correspond to a more favorable objective function evaluation result; for the drive stroke and local dimensions, the optimal structure should keep them small while meeting the requirements of range of motion and gripping force.

[0089] To ensure that the candidate structure meets the actual assembly and motion requirements, the corresponding constraint inequality is:

[0090] The optimization process specifies the position of the fingers as follows: Figure 2 As shown in the straightened state, the difference between the ordinate of the fingertip and the ordinate of the hinge point of the first fixed link segment 2001 of the proximal four-bar linkage in the first frame is the total length of the finger. .

[0091] Four different constant values ​​are used to limit the tolerance for reaching the target value at each angle. =1, It is 1.5. It is 3. The value is 1. These four values ​​are only set during the optimization process of this application. If higher precision is required, these four values ​​can be further reduced.

[0092] The above constraints are used to limit the relevant link direction angles, linear drive component mounting spacing, and total finger length. Initial and final attitudes, as well as local structural dimensions. In addition to the constraints in the formula, the candidate structure should also meet the following requirements: each closed-loop mechanism can be assembled and move continuously; all link lengths and fixed angles are within the preset range; there is no obvious crossing or collision of each component within the complete stroke; and the stroke of the linear drive does not exceed the rated range.

[0093] Interference checks can begin by examining the intersection of line segments or component outlines in a planar model, followed by consideration of component thickness, hinge axes, gaskets, and actual assembly clearances in a complete 3D model. Candidates that do not meet hard constraints can be directly deemed infeasible; for candidates approaching the lower limit of safe clearances or near singular postures, unfavorable results can be set in the fitness evaluation to eliminate them in subsequent selections.

[0094] The optimized variables and their initialization ranges are as follows:

[0095] This inequality simultaneously defines the initialization ranges for installation position parameters, drive stroke, thirteen rod lengths, and five fixing angles. If an implementation method that optimizes only structural parameters is used, these parameters can be pre-fixed. , , , and The search only considers thirteen rod lengths and five fixed angles; if structural and installation parameters are optimized together, the candidate individuals will contain all of the above parameters.

[0096] Step 4: Iterative optimization using an improved differential evolution algorithm. The differential evolution algorithm includes population initialization, mutation, crossover, and greedy selection. Each candidate individual corresponds to a set of parameters to be evaluated. After the candidate individuals are generated, boundary and basic assembly feasibility checks are performed first, and then the kinematic model, mechanical model, and lateral sway model are called to calculate the comprehensive objective function value.

[0097] One specific algorithm initialization parameter is:

[0098] in, This represents the total number of evolutions. For population size, The initial scaling factor. This represents the crossover probability randomly selected from the interval [0,1]. Each candidate individual has a corresponding crossover probability, so that inheritance or updating can be performed based on the actual evolutionary effect of each individual.

[0099] During population initialization, parameters are randomly generated within the aforementioned boundary range. Individuals exceeding the geometric fitability conditions or lacking a valid kinematic solution can be regenerated until an initial population satisfying the basic structural conditions is obtained. This reduces the proportion of infeasible parameters in subsequent high-cost model calculations.

[0100] The nonlinear scaling factor and elite-guided mutation are related as follows:

[0101] In the formula, and This is an intermediate variable that changes with the number of iterations. This represents the total number of iterations. This represents the current iteration number. The scaling parameter used for the current mutation; It is a variant. For individuals that require mutation, and For two random individuals, This represents elite individuals with superior fitness. As iterations progress, the scaling factor changes according to a predefined pattern, enabling the algorithm to maintain strong global search capabilities in the early stages and enhance fine-grained search around optimal parameters in the later stages.

[0102] The direction of mutation includes both the guiding component from the individual to be mutated towards the elite individual and the difference component between the two random individuals. The elite individual can be selected from several individuals with the highest overall objective function values ​​in the current population, rather than always using a single optimal individual, to balance evolutionary directionality and population diversity. The two random individuals should be distinct from each other and differentiated from the individual to be mutated.

[0103] After generating mutants, parameter crossover is performed between the mutants and the original individuals based on the crossover probability corresponding to the individuals to be mutated, forming experimental individuals. During crossover, it should be ensured that the experimental individuals inherit at least one parameter from the mutants. If the experimental individuals exceed the parameter boundaries, they can be corrected by boundary truncation, reflection, or re-random assignment, maintaining consistency in boundary handling rules within the same optimization process.

[0104] After completing constraint checks and calculating the comprehensive objective function, the comprehensive objective function values ​​of the experimental individuals and the original individuals are compared. Individuals with smaller comprehensive objective function values ​​are retained for the next generation. When an experimental individual is superior to the original individual, the crossover probability that generated that experimental individual is reassigned to the experimental individuals entering the next generation; when an experimental individual is not superior to the original individual, the original individual is retained and its crossover probability is randomly generated again. Mutation, crossover, and selection are repeated until the preset number of iterations is reached or the convergence termination condition is met.

[0105] Step 5: Determine and verify the optimized structural parameters.

[0106] After iteration, from the final population and the historical best individuals saved during the iteration, the individual that satisfies all constraints and has the smallest comprehensive objective function value is selected. The thirteen rod lengths and five fixed angles of this individual are used as the optimized structural parameters of the link fingers; in the collaborative optimization method, , , , and This also serves as the installation position and drive stroke parameters that match the structure.

[0107] The parameter range and optimization results of a specific optimization example are shown in the table above. , , , , and to The unit is millimeters. to The unit is degrees.

[0108] After obtaining the optimization results, all parameters should be resubmitted into the complete three-degree-of-freedom model for independent verification. First, continuously solve the planar model within the complete stroke range of the first linear drive 1 and the second linear drive 2, checking whether the angles of each link and the attitudes of the three joints are continuous, and whether the closed-loop mechanism always has a valid solution that conforms to the initial assembly branch. Second, when the second linear drive 2 and the third linear drive 3 move synchronously, check whether the lateral swing angle is close to zero; when moving differentially, check whether the lateral swing direction and amplitude are consistent with the stroke difference.

[0109] Subsequently, the fingertip output force is calculated under the given input forces of the three linear actuators and the preset gripping posture, and the Jacobian matrix is ​​checked to see if it is close to singular. The specific input forces of the three linear actuators can be set according to the rated thrust, control current and transmission efficiency of the actual selected actuator; the verification report should record the drive stroke, joint posture, input force and fingertip force direction at the same time to reproduce the 30 N fingertip force result.

[0110] The entire range of motion should also be checked for collisions between the three linear actuators and their output ends, the three push rods, the two near-end four-links, the near-end three-links, the near-end staggered links, the mid-end three-links, the two mid-end staggered links, and the end fingertips. Interference checks should consider the actual thickness of the components, the ends of the hinge shafts, fasteners, and manufacturing tolerances, with safety clearances provided. For minor interferences that can be resolved by axial misalignment or local chamfering of the components, assembly details can be adjusted without altering the core closed-loop topology.

[0111] Under the specific driving force and structural load conditions, the optimized results enable the metacarpophalangeal joints to achieve approximately 90° of flexion and, within the limits of structural strength, a lateral swing exceeding 30°; the proximal interphalangeal joints achieve approximately 90° of flexion, the distal interphalangeal joints achieve approximately 80° of flexion, and the fingertip force reaches approximately 30 N. These results, corresponding to specific parameters, input force, and grasping posture, illustrate that this method can simultaneously improve range of motion, lateral swing capability, fingertip output force, and structural compactness.

[0112] Using the above structure and method, the complex motion of the three-degree-of-freedom spatial link finger is divided into planar body bending solution and spatial lateral swing solution, reducing the computational burden of repeatedly solving spatial rotation relationships during the optimization process; the improved differential evolution algorithm uses nonlinear scaling, adaptive crossover probability and elite-guided mutation to search in a large parameter space, and eliminates candidate parameters that are not engineering feasible through kinematic, mechanical, interference and size constraints.

[0113] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

Claims

1. A linkage finger structure, characterized in that, It includes the base of the finger, the proximal phalanx, the middle phalanx, the distal phalanx, the first linear drive, the second linear drive, the third linear drive, and the linkage transmission assembly; The middle phalanx hinges to the proximal phalanx to form the proximal interphalangeal joint, and the distal phalanx hinges to the middle phalanx to form the distal interphalangeal joint; The output ends of the first linear drive, the second linear drive, and the third linear drive are respectively slidably connected to the base of the finger along the length direction of the base of the finger; The linkage transmission assembly includes a first drive chain, a second drive chain, a third drive chain, and a distal finger coupling chain. The first drive chain is connected between the output end of the first linear drive member and the middle finger joint, and is used to convert the linear displacement output by the first linear drive member into the rotation of the middle finger joint relative to the proximal finger joint around the proximal finger joint. One end of the distal phalanx is hinged to the proximal phalanx, and the other end is hinged to the distal phalanx. The proximal phalanx, middle phalanx, distal phalanx, and distal phalanx form a closed-loop linkage mechanism to drive the distal phalanx to rotate around the distal phalanx when the middle phalanx rotates around the proximal phalanx. The second drive chain is connected between the output end of the second linear drive member and the proximal phalanx, and the third drive chain is connected between the output end of the third linear drive member and the proximal phalanx. The proximal phalanx is movably connected to the base of the finger through the second drive chain and the third drive chain, and forms a metacarpophalangeal joint between the proximal phalanx and the base of the finger. The second linear drive, the third linear drive, the second drive branch, and the third drive branch are mirror-symmetrical about the mid-surface of the finger base in the width direction. When the output ends of the second and third linear drive move synchronously, the proximal phalanx bends around the metacarpophalangeal joint. When the output ends of the second and third linear drive move differentially, the proximal phalanx swings laterally around the metacarpophalangeal joint.

2. The linkage finger structure according to claim 1, characterized in that, The proximal phalanx includes a first proximal four-link and a second proximal four-link, which are mirror images of each other in the width direction about the base of the finger. The middle phalanx includes a middle three-link. The first drive chain includes a first push rod link, a proximal three-link, and a proximal staggered link. One end of the first push rod link is hinged to the output end of the first linear drive member, and the other end is hinged to the proximal three-link. The proximal three-link is also hinged to the proximal phalanx and the proximal staggered link, respectively. The end of the proximal staggered link away from the proximal three-link is hinged to the middle three-link, so that the proximal three-link, proximal phalanx, middle three-link, and proximal staggered link form a proximal closed-loop linkage mechanism.

3. The linkage finger structure according to claim 2, characterized in that, The distal phalanx includes the distal fingertip. The first proximal four-link and the second proximal four-link are respectively hinged to the middle three-link, and the middle three-link is hinged to the distal fingertip. The distal finger coupling branch includes a first middle-end staggered link and a second middle-end staggered link arranged in a mirror image of the mid-surface in the width direction about the base of the finger. One end of the first middle-end staggered link is hinged to the first proximal four-link and the other end is hinged to the distal fingertip. One end of the second middle-end staggered link is hinged to the second proximal four-link and the other end is hinged to the distal fingertip. The first proximal four-link, the middle three-link, the distal fingertip and the first middle-end staggered link form a first distal closed-loop linkage mechanism, and the second proximal four-link, the middle three-link, the distal fingertip and the second middle-end staggered link form a second distal closed-loop linkage mechanism.

4. The linkage finger structure according to claim 2 or 3, characterized in that, The second drive chain includes a second push rod link, and the third drive chain includes a third push rod link. One end of the second push rod link is hinged to the output end of the second linear drive member, and the other end is hinged to the first proximal fourth auxiliary rod. One end of the third push rod link is hinged to the output end of the third linear drive member, and the other end is hinged to the second proximal fourth auxiliary rod.

5. The linkage finger structure according to claim 3, characterized in that, The proximal three-link, the middle three-link, and the distal fingertip each include three spaced-apart hinges and two fixed link segments. The two fixed link segments connect to two adjacent hinges and are fixed to each other, forming a fixed angle between the two fixed link segments. The first proximal four-link and the second proximal four-link each include four spaced-apart hinges and three fixed link segments. The three fixed link segments connect to two adjacent hinges and are fixed to each other in sequence, forming a fixed angle between the two adjacent fixed link segments.

6. A method for optimizing the structural parameters of a link finger based on improved differential evolution, characterized in that, include: A spatial three-degree-of-freedom model of the finger with the linkage to be optimized is established. The finger with the linkage to be optimized includes metacarpophalangeal joints, proximal finger joints, distal finger joints, a first linear drive, a second linear drive, a third linear drive, and a linkage transmission assembly. The linkage transmission assembly includes a first drive branch, a second drive branch, a third drive branch, and a distal finger coupling branch. The first linear drive drives the proximal and distal finger joints to bend in a coupled manner through the first drive branch and the distal finger coupling branch. The second and third linear drive acts on the metacarpophalangeal joints through mirror-symmetric second and third drive branches, respectively. When the driving stroke of the second and third linear drive changes synchronously, it drives the metacarpophalangeal joint to bend; when it changes differentially, it drives the metacarpophalangeal joint to swing laterally. Based on the mirror symmetry relationship between the second and third driving branches, one of the second and third driving branches is selected, and a planar two-degree-of-freedom model is established with the first driving branch and the distal finger coupling branch. Based on the planar two-degree-of-freedom model, a positive kinematic mapping relationship is established between the driving stroke of the first linear drive and the linear drive corresponding to the selected driving branch and the bending angle of the metacarpophalangeal joint, the angle of the proximal finger joint, the angle of the distal finger joint, and the fingertip position. Based on the positive kinematic mapping relationship, a mechanical transmission model between the driving force of the first linear drive and the linear drive corresponding to the selected drive branch and the fingertip output force is established. Based on the spatial three-degree-of-freedom model, a lateral swing motion model between the driving stroke of the second and third linear drive and the lateral swing angle of the metacarpophalangeal joint is established. The lengths of independent links, the lengths of each fixed link segment, and the fixed angles between adjacent fixed link segments within the same rigid component in the linkage transmission assembly are used as optimization variables. A comprehensive objective function and constraints are constructed based on the positive kinematic mapping relationship, the mechanical transmission model, and the lateral swing motion model. An improved differential evolution algorithm, including an adaptive crossover probability mechanism, a nonlinear decreasing scaling factor mechanism, and an elite-guided mutation mechanism, is adopted to iteratively optimize the optimization variables within the parameter range limited by the constraints. The independent link length, fixed link segment length, and fixed angle corresponding to the candidate individual with the smallest comprehensive objective function value are determined as the optimization structural parameters of the link finger to be optimized.

7. The method according to claim 6, characterized in that, Establishing a two-degree-of-freedom planar model includes: The output end of the first linear drive and the output end of the linear drive corresponding to the selected drive branch are respectively used as sliders in the crank-slider mechanism to construct a mutually coupled palm-finger driven crank-slider mechanism, a proximal-distal finger driven crank-slider mechanism, a first planar interlaced four-bar mechanism and a second planar interlaced four-bar mechanism. The planar vector analytical method is used to establish the vector closed-loop equations of the palm-finger driven crank-slider mechanism, the proximal-finger driven crank-slider mechanism, the first planar interlocked four-bar mechanism, and the second planar interlocked four-bar mechanism, respectively. Based on the coupling relationship between the four mechanisms, the forward solution is performed to obtain the cascade mapping relationship between the driving stroke of the first linear drive component and the linear drive component corresponding to the selected drive branch and the angles of each link, the bending angle of the palm-finger joint, the angle of the proximal finger joint, the angle of the distal finger joint, and the Cartesian coordinates of the fingertip.

8. The method according to claim 7, characterized in that, Establishing a mechanical transmission model includes: The contact between the fingertip and the target object is equivalent to point contact. Under the quasi-static condition of ignoring the gravity and inertial forces of each component of the link finger to be optimized, the driving stroke of the first linear drive component and the linear drive component corresponding to the selected drive branch are combined into a driving stroke vector. The partial derivative of the fingertip Cartesian coordinate vector with respect to the driving stroke vector is obtained to construct the Jacobian matrix of the fingertip Cartesian coordinate vector with respect to the driving stroke vector. Based on the principle of virtual work, the Jacobian transpose relationship between the driving force vectors of the two linear actuators and the fingertip output force vector is established, thereby obtaining the mapping relationship between the driving force of the two linear actuators and the fingertip output force.

9. The method according to claim 6, characterized in that, Establishing a lateral swing motion model includes: Establish a global coordinate system and a local coordinate system fixed to the proximal phalanx. Transform the link vectors of the second and third drive branches in the local coordinate system to the global coordinate system using a rotation matrix. Based on the spatial vector closed-loop relationship between the second and third driving branches, the residual vector is constructed with the driving stroke of the second and third linear driving components as known inputs and the bending angle and lateral swing angle of the metacarpophalangeal joint as variables to be determined. Within the preset feasible range of the metacarpophalangeal joint flexion angle and the metacarpophalangeal joint lateral displacement angle, a grid search is performed, and the sampling point with the minimum sum of squares of the residual vector is determined as the coarse solution. Using the coarse solution as the initial value of Newton's iteration, the central difference method is used to calculate the numerical Jacobian matrix of the residual vector relative to the bending angle and the lateral swing angle of the metacarpophalangeal joint, and the bending angle and the lateral swing angle of the metacarpophalangeal joint are iteratively updated until the norm of the residual vector is less than the preset residual threshold or the norm of the angle correction is less than the preset correction threshold. The flexion angle and lateral swing angle of the metacarpophalangeal joint obtained after iterative convergence are used to establish a correspondence with the driving stroke of the second and third linear actuators.

10. The method according to any one of claims 6 to 9, characterized in that, The improved differential evolution algorithm is used to iteratively optimize the variables, including: Initialize the candidate population and generate the corresponding crossover probability for each candidate individual; In each iteration, a scaling factor that decreases according to a nonlinear law is determined based on the current iteration number. The fitness of each candidate individual in the current candidate population is sorted according to the comprehensive objective function value. Elite individuals are selected from the candidate individuals whose fitness ranking is within a preset range. At least two different random individuals are selected. Mutant individuals are generated based on the individual to be mutated, the elite individuals, and at least two random individuals. The mutation direction includes the guiding component from the individual to be mutated to the elite individual and the difference component between the random individuals. Based on the crossover probability corresponding to the individual to be mutated, crossover operation is performed on the mutated individual and the individual to be mutated to generate test individuals. The comprehensive objective function values ​​of the test individuals and the individuals to be mutated are compared, and the individuals with smaller comprehensive objective function values ​​are retained in the next generation of candidate population. When the test individuals are retained, the crossover probability corresponding to the individual to be mutated is assigned to the test individuals. When the individual to be mutated is retained, the crossover probability corresponding to the individual to be mutated is regenerated. Repeat the steps of determining the scaling factor, generating mutant individuals, crossover, and selection for the next generation of candidate populations until the preset iteration termination condition is met.

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