Kinematics modeling method and parameter optimization method of stepless amplitude-variable spatial four-bar mechanism

Abstract

The application relates to a kinematics modeling method and a parameter optimization method of a stepless amplitude-variable space four-bar mechanism, and solves the technical problem of how to continuously and high-precisely adjust the reciprocating swing amplitude of the output end of the space four-bar mechanism; firstly, a global reference coordinate system and a moving coordinate system are created for the stepless amplitude-variable space four-bar mechanism, and then an output swing angle constraint equation of an output swing arm is established. The application is widely applied to the space four-bar mechanism.

Description

Technical Field

[0001] This invention relates to the field of spatial linkage transmission mechanism technology, and more specifically, to a kinematic modeling method and parameter optimization method for a continuously variable amplitude spatial four-bar linkage. Background Technology

[0002] In modern mechanical transmission systems, converting the continuous rotational motion of the drive source into periodic reciprocating oscillation at the output end is a widely used motion conversion scheme. Although the common planar crank-rocker mechanism has a simple structure, the oscillation amplitude at its output end is uniquely determined by the inherent geometric dimensions of each link in the mechanism. Once the manufacturing process is completed, its output amplitude cannot be changed.

[0003] Spatial four-bar linkages (RSSR: revolute-spherical-spherical-revolute) offer significant advantages such as compact structure and strong adaptability to three-dimensional space due to the staggered arrangement of kinematic pairs, making them widely used in specialized machinery in textiles, knitting, and garment manufacturing. However, conventional spatial four-bar linkages also suffer from a fixed output amplitude. In many advanced industrial equipment applications, such as fine fluid propulsion, spatial actuators, material mixing, and multi-stage high-precision vibrating screens, it is often necessary to adjust the reciprocating amplitude of the output to meet the actual operating conditions, and even to dynamically, continuously, and with high precision without stopping the system. Therefore, improvements to conventional spatial four-bar linkages are needed to achieve continuous and high-precision adjustment of the output reciprocating amplitude. Summary of the Invention

[0004] This application aims to solve the technical problem of how to continuously and accurately adjust the reciprocating swing amplitude of the output end of a spatial four-bar linkage, and provides a kinematic modeling method and parameter optimization method for a continuously variable amplitude spatial four-bar linkage.

[0005] This application provides a kinematic modeling method for a continuously variable amplitude spatial four-bar linkage, including the following steps:

[0006] Step (1), define the coordinate system;

[0007] Define a global reference coordinate system The center of rotation of the turntable is the origin. , The shaft coincides with the axis of the drive motor output shaft. The axis is horizontal and perpendicular to axis, The axis is vertically upward;

[0008] Define a moving coordinate system Its origin and the origin coincide; The axis coincides with the rotation axis of the turntable; The axis is along the radial direction of the turntable and points towards the center of the first spatial spherical pair. ; The axis is determined by the right-hand rule;

[0009] Define a moving coordinate system Its origin The reference swing center of the output swing arm. The axis of the shaft coincides with the swing axis of the output swing arm. The shaft runs along the length of the output swing arm and points towards the center of the second spatial spherical pair. , The axis is determined by the right-hand rule;

[0010] This represents the length of the active crank, i.e., the distance from the center of rotation of the turntable to the center of the first spatial spherical pair. The distance between them; This indicates the distance from the reference swing center of the output swing arm to the center of the second spatial spherical pair. The distance between them; Indicates the length of the spatial link;

[0011] This indicates that the plane of rotation of the turntable itself is in Projection around the plane The rotation angle of the axis, express shaft and The angle between the axes, Indicates the rotation angle of the turntable. Indicates the output swing angle of the output swing arm. Indicates pressure angle; spatial transmission angle Pressure angle The complementary angle;

[0012] Step (2), establish constraint equations:

[0013]

[0014] In the formula, ,

[0015] ,

[0016]

[0017] This is the lateral offset parameter. For vertical offset parameters, This is the axial offset parameter.

[0018] Preferably, the continuously variable amplitude spatial four-bar linkage includes a drive motor, a ball cage universal joint assembly, a turntable, a first spatial spherical joint, a spatial connecting rod, a second spatial spherical joint, an output swing arm, a bracket, a support ring, a bearing, an amplitude adjustment connecting rod, and a linear module. The ball cage universal joint assembly includes a star-shaped sleeve, a ball cage cage, and multiple drive balls. The outer periphery of the star-shaped sleeve has multiple inner raceways, and the drive balls are located in the inner raceways of the star-shaped sleeve. The ball cage cage is fitted onto the star-shaped sleeve and has multiple windows, in which the drive balls are located. A receiving chamber is located at the center of the front of the turntable, and the receiving chamber has multiple outer raceways. The ball cage universal joint assembly is located in the receiving chamber, and the outer sides of the drive balls are embedded in the outer raceways. The output shaft of the drive motor is fixedly connected to the center of the star-shaped sleeve.

[0019] The first spatial spherical pair includes a shell base and a ball head pin. The shell base is fixedly connected to the edge of the turntable, and one end of the spatial connecting rod is fixedly connected to the ball head pin.

[0020] The second spatial spherical joint includes a shell base and a ball head pin. The other end of the spatial link is fixedly connected to the ball head pin of the second spatial spherical joint. The shell base of the second spatial spherical joint is fixedly connected to one end of the output swing arm. The other end of the output swing arm is hinged to the bracket.

[0021] The back of the turntable is rotatably connected to the support ring via a bearing; the linear module is equipped with a linear motion part, one end of the amplitude adjustment link is hinged to the support ring, and the other end of the amplitude adjustment link is hinged to the linear motion part.

[0022] This application also provides a parameter optimization method for a continuously variable amplitude spatial four-bar linkage, comprising the following steps:

[0023] For the continuously variable amplitude spatial four-bar linkage, the optimized variables are: ;

[0024] This represents the length of the active crank, i.e., the distance from the center of rotation of the turntable to the center of the first spatial spherical pair. The distance between them; This indicates the distance from the reference swing center of the output swing arm to the center of the second spatial spherical pair. The distance between them; Indicates the length of the spatial link; This is the lateral offset parameter. For vertical offset parameters, This is the axial offset parameter;

[0025] Based on the following kinematic modeling:

[0026]

[0027] In the formula, ,

[0028] ,

[0029]

[0030] This indicates that the plane of rotation of the turntable itself is in Projection around the plane The rotation angle of the axis, express shaft and The angle between the axes, Indicates the rotation angle of the turntable. Indicates the output swing angle of the output swing arm. Indicates pressure angle; spatial transmission angle Pressure angle The complementary angle;

[0031] Discriminant equation ,right , , , , , Optimization is required across the entire amplitude range. and the entire exercise cycle Internal satisfaction .

[0032] Preferably, parameter optimization is performed based on a genetic algorithm to adjust the lateral bias. Vertical offset The search range is fixed at 0, with only the axial offset retained. As an adjustable parameter;

[0033] Set the allowable transmission angle of the spatial linkage mechanism The minimum transmission angle must satisfy the following requirements throughout the entire amplitude range and the entire motion cycle: ;

[0034] Construct the dual-objective fitness function cost:

[0035]

[0036] In the formula, , ; , , As weight, ;

[0037] , To achieve the target maximum swing amplitude, To achieve the minimum swing amplitude, This represents the actual maximum swing amplitude. This represents the actual minimum swing amplitude.

[0038] The beneficial effects of this disclosure are that the reciprocating oscillation amplitude of the output end of the spatial four-bar linkage can be adjusted, and continuous stepless adjustment can be achieved, as well as high-precision adjustment. Furthermore, dynamic and high-precision adjustment can be achieved without stopping the machine. A wide range of amplitude adjustment is possible, and the adjustment process is highly efficient.

[0039] It can effectively reduce rotational inertia and suppress "center drift" during amplitude-changing operation.

[0040] The continuously variable amplitude spatial four-bar linkage has the characteristics of compact structure and small size of amplitude-changing mechanism.

[0041] Further features and aspects of this disclosure will be clearly described in the following detailed description with reference to the accompanying drawings. Attached Figure Description

[0042] Figure 1 It is an isometric drawing of a continuously variable amplitude spatial four-bar linkage;

[0043] Figure 2 This is the front view of a continuously variable amplitude spatial four-bar linkage;

[0044] Figure 3 This is a top view of a continuously variable amplitude spatial four-bar linkage;

[0045] Figure 4 It is an exploded view of the components consisting of a drive motor, a ball-cage universal joint assembly, a turntable, a support ring, and bearings;

[0046] Figure 5 This is a structural schematic diagram of a ball-cage universal joint assembly;

[0047] Figure 6 This is a schematic diagram of the turntable's structure;

[0048] Figure 7 This is a schematic diagram of the support ring structure;

[0049] Figure 8 This is a structural diagram of the linear module;

[0050] Figure 9 This is an isometric view of a continuously variable amplitude spatial four-bar linkage from another perspective;

[0051] Figure 10 It is a schematic diagram of the spatial angle between the rotation plane of the turntable itself and the axis of the output shaft of the drive motor;

[0052] Figure 11 This is a schematic diagram of the equivalent structure of a continuously variable amplitude spatial four-bar linkage;

[0053] Figure 12 This is a schematic diagram of the motion model of a continuously variable amplitude spatial four-bar linkage;

[0054] Figure 13 This is a schematic diagram of the swing axis of the output swing arm;

[0055] Figure 14 It is the spatial transmission angle Relationship with amplitude modulation angle α;

[0056] Figure 15 It is a graph showing the relationship between the instantaneous swing center of the output swing arm and the amplitude adjustment angle α;

[0057] Figure 16 This is a data graph showing the center drift range;

[0058] Figure 17 It is the law governing the change of output swing amplitude with amplitude modulation angle.

[0059] Explanation of symbols in the diagram:

[0060] 1. Drive motor; 2. Ball cage universal joint assembly; 2-1. Star sleeve; 2-1-1. Spline hole; 2-2. Transmission ball; 2-3. Ball cage cage; 3. Turntable; 3-1. Outer raceway; 3-2. Connecting boss; 4. First spatial spherical joint; 4-1. Shell base; 4-2. Ball pin; 5. Spatial connecting rod; 6. Second spatial spherical joint; 7. Output swing arm; 8. Follower; 9. Bracket. 10. Support ring, 10-1. Connecting ear plate, 11. Bearing, 11-1. Inner ring, 11-2. Outer ring, 12. Amplitude adjustment link, 12-1. First hinge end, 12-2. Second hinge end, 13. Linear module, 13-1. Linear motion part, 13-2. Motor, 13-3. Coupling, 13-4. Lead screw, 13-5. Slider, 13-6. Guide rail, 13-7. Base. Detailed Implementation

[0061] The application will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0062] The specific embodiments described below are merely preferred embodiments of this application, and the scope of protection of this application is not limited thereto. Those skilled in the art can make modifications or variations based on the principles, concepts, and spirit of this application, and the resulting technical solutions should all be covered within the scope of protection of this application.

[0063] like Figures 1-9As shown, the continuously variable amplitude spatial four-bar linkage disclosed in this invention includes a drive motor 1, a ball cage universal joint assembly 2, a turntable 3, a first spatial spherical joint 4, a spatial connecting rod 5, a second spatial spherical joint 6, an output swing arm 7, a driven member 8, a bracket 9, a support ring 10, a bearing 11, an amplitude adjustment connecting rod 12, and a linear module 13. The output shaft of the drive motor 1 is connected to the turntable 3 through the ball cage universal joint assembly 2. The ball cage universal joint assembly 2 includes a star-shaped sleeve 2-1, a ball cage cage 2-3, and six drive balls 2-2. The star-shaped sleeve 2-1 has six inner raceways evenly distributed circumferentially. A spline hole 2-1-1 is located at the center of the star-shaped sleeve 2-1, and the six drive balls 2-2 are respectively located in the six inner raceways of the star-shaped sleeve 2-1. The ball cage cage 2-3 is fitted onto the star-shaped sleeve 2-1 and has six windows, in which the drive balls 2-2 are located, each confined within one of the six windows. A receiving chamber is located at the center of the front of the turntable 3, and this chamber has six outer raceways 3-1. The ball cage universal joint assembly 2 is placed in this receiving chamber, with the outer sides of the drive balls 2-2 embedded in the outer raceways 3-1. This achieves the connection and engagement between the ball cage universal joint assembly 2 and the turntable 3. The output shaft of the drive motor 1 is provided with an external spline. The output shaft of the drive motor 1 is inserted into the spline hole 2-1-1 of the star sleeve 2-1. The external spline of the output shaft is connected to the spline hole 2-1-1, thereby realizing the center fixed connection between the output shaft of the drive motor 1 and the star sleeve 2-1.

[0064] One end of the spatial link 5 is connected to the edge of the turntable 3 through the first spatial spherical joint 4. The first spatial spherical joint 4 includes a shell base 4-1 and a ball head pin 4-2 that are connected together. The shell base 4-1 is fixedly connected to the edge of the turntable 3, and one end of the spatial link 5 is fixedly connected to the ball head pin 4-2.

[0065] The other end of the spatial link 5 is connected to one end of the output swing arm 7 through the second spatial spherical joint 6. The second spatial spherical joint 6 includes a housing base and a ball head pin. The other end of the spatial link 5 is connected to the ball head pin. The housing base is fixedly connected to one end of the output swing arm 7.

[0066] The other end of the output swing arm 7 is hinged to the bracket 9. The driven member 8 is fixedly connected to the output swing arm 7.

[0067] The back of the turntable 3 is rotatably connected to the support ring 10 via a bearing 11. The bearing 11 includes an inner ring 11-1 and an outer ring 11-2. The bearing 11 is installed in the support ring 10, and the outer ring 11-2 is fixed on the inner side wall of the support ring 10. The back of the turntable 3 is provided with a connecting boss 3-2. The inner ring 11-1 is fitted onto the connecting boss 3-2, and the connecting boss 3-2 is fixedly connected to the inner ring 11-1.

[0068] The linear module 13 is provided with a linear motion section 13-1 as an output section. One end of the amplitude adjustment link 12 is hinged to the support ring 10, and the other end of the amplitude adjustment link 12 is hinged to the linear motion section 13-1. Specifically, the amplitude adjustment link 12 is provided with a first hinge end 12-1 and a second hinge end 12-2, and the support ring 10 is provided with a connecting ear plate 10-1; a specific structure of the linear module 13 includes a linear motion part 13-1, a motor 13-2, a coupling 13-3, a lead screw 13-4, a nut, a slider 13-5, a guide rail 13-6, and a base 13-7. The motor 13-2 is mounted on the base 13-7. One end of the lead screw 13-4 is connected to the output shaft of the motor 13-2 through the coupling 13-3, and the other end of the lead screw 13-4 is rotatably connected to the base 13-7 through a bearing. The guide rail 13-6 is mounted on the base 13-7. The slider 13-5 is connected to the guide rail 13-6. The nut is connected and engaged with the lead screw 13-4. The linear motion part 13-1 is fixedly connected to the nut, and the bottom of the linear motion part 13-1 is fixedly connected to the slider 13-5. When motor 13-2 operates, it causes lead screw 13-4 to rotate, thereby causing linear motion part 13-1 to move in a straight line. The first hinge end 12-1 of amplitude regulating link 12 is hinged to connecting ear plate 10-1, and the second hinge end 12-2 of amplitude regulating link 12 is hinged to linear motion part 13-1.

[0069] It should be noted that for the structure of the ball cage universal joint assembly 2, six drive balls 2-2 is a preferred configuration. Those skilled in the art will understand that the number of drive balls 2-2 is not limited to six; generally, there are multiple drive balls 2-2. With multiple drive balls 2-2, correspondingly, multiple inner raceways are provided around the star-shaped sleeve 2-1, multiple windows are provided in the ball cage cage 2-3, and multiple outer raceways 3-1 are provided at the center of the turntable 3.

[0070] As can be seen, in the above-mentioned continuously variable amplitude spatial four-bar linkage, the drive motor 1 serves as the power source, providing the initial power for the continuous operation of the mechanism, and the output shaft of the drive motor 1 serves as the input rotation shaft. Regarding the ball cage universal joint assembly 2 and the turntable 3, the star-shaped sleeve 2-1 serves as the input end, and the turntable 3 serves as the output end. The geometric center of the star-shaped sleeve 2-1 and the geometric center of the turntable 3 always coincide absolutely at a fixed point in space (i.e., the center point of the ball cage). The outer sides of the six transmission balls 2-2 are embedded with six outer raceways 3-1, forming a rolling high pair. The transmission balls 2-2 are constrained by the bidirectional raceways. When the turntable 3 undergoes angular deflection under the action of the linear module 13, the transmission balls 2-2 perform pure rolling sliding along the raceways. At the same time, the axial limit of the raceways locks the relative axial displacement between the turntable 3 and the input rotation shaft. The shell base 4-1 of the first spatial spherical pair 4 and the turntable 3 belong to the same rigid entity. The spatial Euclidean distance between the rotation axis of the turntable 3 and the center of the first spatial spherical pair 4 is absolutely locked by the machining dimensions and is a physical constant.

[0071] The amplitude of the output swing arm 7 is infinitely adjustable by controlling the spatial deflection angle of the rotation axis of the turntable 3 through the amplitude adjustment linkage 12 and the linear module 13. The motor in the linear module 13 can typically be a servo motor or a stepper motor to provide precise position control power, ensuring accurate linear displacement of the linear motion unit 13-1.

[0072] The working process of the above continuously variable amplitude spatial four-bar linkage is described below:

[0073] The drive motor 1, the linear module 13, and the bracket 9 are fixed. When the system is working normally, the output shaft of the drive motor 1 rotates, and the torque is transmitted to the star sleeve 2-1 through the spline, and then transmitted at the same speed to the turntable 3, which is the outer star wheel, through the six transmission balls 2-2. Supported by the support ring 10 and the bearing 11, the turntable 3 rotates continuously at high speed around its rotation axis, driving the first spatial spherical pair 4 to perform spatial circular motion. Through the traction of the spatial connecting rod 5, the output swing arm 7 is driven to reciprocate around the bracket 9, thereby driving the driven member 8 to output a periodic oscillation with a fixed amplitude.

[0074] If the output amplitude needs to be adjusted without stopping the mechanism, the linear module 13 is activated, and the linear motion unit 13-1 moves linearly, pushing or pulling the support ring 10 through the amplitude adjustment linkage 12. Due to the torque, the support ring 10 applies a deflection thrust to the turntable 3 through the bearing 11, forcing the turntable 3 to deflect precisely around the center point of the ball cage, thereby changing the spatial angle between the rotation axis of the turntable 3 and the axis of the output shaft of the drive motor 1, that is, changing the angle between the rotation plane of the turntable 3 and the axis of the output shaft of the drive motor 1 (i.e., the amplitude adjustment angle α, such as...). Figure 10 (As shown).

[0075] The change in amplitude adjustment angle α directly alters the motion projection trajectory of spatial link 5, thereby changing the swing amplitude of output swing arm 7 and driven member 8 in real time and steplessly. During this amplitude adjustment process, the geometric distortion caused by the angular deflection between the main input rotation shaft (i.e., the output shaft of drive motor 1) and turntable 3 is completely decoupled and absorbed by the pure rolling of the balls inside the ball cage universal joint within the spatial raceway; simultaneously, the relative rotational interference between the high-speed rotating turntable 3 and the stationary amplitude adjustment link 12 is completely isolated by the stationary support ring 10 and bearing 11, thus ensuring smooth and uninterrupted operation of the mechanism under different deflection angles and high-frequency heavy-load conditions. The boundary conditions of the entire amplitude adjustment process are clear, and the kinematic input and output have a deterministic and unique mapping relationship.

[0076] As can be seen, the reciprocating swing amplitude of the output swing arm 7 and the driven member 8 can be dynamically and precisely steplessly adjusted over a wide range without stopping the machine. The ingenious use of a ball-cage universal joint structure makes the overall structure of the mechanism compact, effectively reducing the rotational inertia of the system. More importantly, this compact spatial fixed-point transmission architecture provides extremely high geometric constraint accuracy. Combined with the kinematic dimension design of the mechanism, it can effectively suppress "center drift" at the output end during the dynamic adjustment of the swing amplitude. ("Center drift" refers to the reciprocating swing center at the output end deviating from the equilibrium zero position, resulting in extreme asymmetry in the reciprocating swing angle.)

[0077] Follower 8 can be used to connect external functional modules to achieve stepless amplitude output, and can be configured into any functional structure according to the specific needs of external application scenarios.

[0078] It should be noted that, regarding the main input rotating shaft connected to the ball-cage universal joint assembly 2, when the drive motor 1 is not used as the power source, other power sources can also be used to connect to the main input rotating shaft and drive the main input rotating shaft to rotate.

[0079] The design of a spatial four-bar linkage is more complex than that of a planar crank-rocker mechanism. A kinematic model is established for the aforementioned continuously variable amplitude spatial four-bar linkage.

[0080] The first step is to define the coordinate system.

[0081] refer to Figure 12 Define a global reference coordinate system The center of rotation of turntable 3 is the origin. , The shaft coincides with the axis of the output shaft of drive motor 1; The axis is horizontal and perpendicular to Axis, and with The shafts together form the output swing plane. flat); The axes point vertically upwards, forming a right-handed Cartesian coordinate system.

[0082] Define a moving coordinate system that rotates synchronously with turntable 3. Its origin and the origin coincide; The axis coincides with the rotation axis of turntable 3; The shaft is along the radial direction of the turntable 3 and points towards the center of the first spatial spherical joint 4. ; The axis is determined by the right-hand rule.

[0083] Define a moving coordinate system that swings synchronously with turntable 3. Its origin This serves as the reference swing center for output swing arm 7; The axis of the shaft coincides with the swing axis of the output swing arm 7 (reference). Figure 13 The dashed lines in the middle represent axis); The shaft runs along the length of the output swing arm 7 and points towards the center of the second spatial spherical joint 6. ; The axis is determined by the right-hand rule.

[0084] Turntable 3 and the first spatial spherical joint 4 form an active crank. This indicates the length of the active crank, i.e., the distance from the center of rotation of turntable 3 to the center of the first spatial spherical joint 4. The distance between them. This indicates the distance from the rocker arm length, i.e., the reference swing center of the output rocker arm 7, to the center of the second spatial spherical joint 6. The distance between them. This indicates the length of spatial link 5.

[0085] h, p, q: Three-dimensional frame offset parameters, determining the fixed rotation center of output swing arm 7. The global spatial location.

[0086] like Figure 11 As shown, This indicates the tilt angle of the active crank rotation plane, that is, the amplitude angle between the rotation plane of turntable 3 itself and the axis of the output shaft of drive motor 1, i.e., the angle between the rotation plane of turntable 3 and the axis of the output shaft of drive motor 1. Projection around the plane The rotation angle of the axis is used as an independent amplitude adjustment variable.

[0087] Joystick mounting angle express shaft and The angle between axes.

[0088] This indicates the crank input angle, which is the angle through which the output shaft of drive motor 1 rotates, and also the rotation angle of turntable 3.

[0089] This indicates the output swing angle of output swing arm 7.

[0090] The pressure angle is the angle between the direction of the driving force exerted by the spatial link 5 on the output swing arm 7 at point B and the direction of the instantaneous absolute velocity at point B. This angle is acute.

[0091] Spatial transmission angle, reflecting the force transmission performance of a mechanism. Pressure angle The complementary angle, that is .

[0092] The second step, to establish the kinematic relationships of the mechanism, is to transform the motion vectors in each local coordinate system to the global reference coordinate system using rotation transformation matrices and displacement vectors. In this system, a unified description of the pose of each component is achieved. The motion of all components can be uniformly described through the transformation from "local moving coordinates to global fixed coordinates". The general transformation formula is:

[0093]

[0094] in, Let be the coordinate vector of the point in the moving coordinate system. Here is the total rotation transformation matrix. It is a translation vector.

[0095] Point A is in the coordinate system The coordinates in the image are fixed values:

[0096]

[0097] The global coordinates of point A are obtained by superimposing two rotation transformations: around Rotational transformation of the shaft: crank around The axis rotates through the input angle The corresponding rotation transformation matrix is:

[0098]

[0099] Attitude transformation: Transform the tilt plane containing the crank to the global coordinate system. The corresponding rotation transformation matrix is:

[0100]

[0101] The total rotation transformation matrix is Since the crank rotation center coincides with the global origin, the translation vector Therefore, the global coordinates of point A are:

[0102]

[0103] Substituting the values ​​into the matrix, we can obtain the global coordinates of point A:

[0104]

[0105] Similarly, point B in the coordinate system The coordinates in the image are fixed values:

[0106]

[0107] The global coordinates of point B are obtained by superimposing rotation and translation transformations, around which... Rotational transformation of the axis: rocker arm around Turn the output swing angle The corresponding rotation transformation matrix is:

[0108]

[0109] Attitude transformation: Transform the swing plane where the output swing arm 7 is located to the global coordinate system. The corresponding rotation transformation matrix is:

[0110]

[0111] Translation transformation: Output the reference swing center of swing arm 7 The translation vector relative to the global origin is determined by the rack offset parameters:

[0112]

[0113] The total rotation transformation matrix is Therefore, the global coordinates of point B are:

[0114]

[0115] Substituting the values ​​into the matrix, we can obtain the global coordinates of point B:

[0116]

[0117] After completing the coordinate transformation of points A and B, the two points are associated using the fixed length constraint of the connecting rod, and the spatial closed-loop vector constraint equation of the mechanism is established.

[0118] The connecting rod is a rigid component with a length of As a fixed constant, points A and B are the two ends of the link, and their spatial distance is always equal to the length of the link, thus satisfying the closed-loop vector constraint:

[0119]

[0120] Substituting the global coordinates of points A and B and expanding them, the equation can be rearranged into its standard form:

[0121]

[0122] Introducing half-width substitution The above equation can be transformed into a quadratic equation in one variable, and then solved. about and The exact parsed expression:

[0123]

[0124] In the formula,

[0125]

[0126]

[0127]

[0128] By pushing or pulling the amplitude adjustment link 12, the turntable 3 is precisely deflected around the center point of the ball cage, thereby changing the amplitude adjustment angle. To make turntable 3 change the amplitude angle It can still rotate 360° at a uniform speed under operating conditions, introducing a discriminant... Determine the continuity of motion of the mechanism ( (The timing mechanism can move normally), requiring the current amplitude adjustment angle. ,equation It must satisfy the condition that there is a solution in the range of 0° to 360°.

[0129] The above modeling process clarifies the input angle. Amplitude adjustment angle With output swing angle The mapping relationship can accurately solve the mechanism output response under arbitrary amplitude adjustment angle and arbitrary input phase, which is the basis for subsequent parameter optimization.

[0130] Next, we will perform parameter optimization based on a genetic algorithm.

[0131] To ensure that the amplitude output not only meets the requirement of continuous amplitude adjustment, but also that the swing center of the output swing arm 7 is strictly maintained at the design reference position throughout the entire amplitude adjustment process, that is, the mechanism output must have displacement symmetry. To avoid... During the change, the swing center of the output swing arm 7 undergoes an asymmetrical shift (center drift). Firstly... The analytical solution provides a general quantitative mathematical definition of center drift:

[0132] For any given The output swing arm 7 completes one motion cycle ( The output swing angle within ) is Let its maximum value be . The minimum value is Then the The instantaneous oscillation center below is:

[0133]

[0134] The corresponding single-frame swing amplitude is:

[0135]

[0136] Define the center drift range of the full amplitude range as:

[0137]

[0138] In the formula, This represents the effective range of the mechanism. If... If the mechanism's swing center remains fixed within the full amplitude range, there will be no drift; if Then the center of oscillation follows Changes cause center drift.

[0139] Extreme position of motion and The input component rotates, causing The instantaneous position of the extreme point. Differentiating the above analytical solution, we obtain the extreme point corresponding to... The value is Highly nonlinear function, i.e. .therefore, , And thus the center of the swing It must be The function, i.e. .

[0140] Therefore, a genetic algorithm is used for optimization:

[0141] Step S1: Design variables.

[0142] The optimization variables are:

[0143]

[0144] , , For three-dimensional bias parameters, It is a lateral offset. For vertical offset, The axial offset determines the global spatial position of the reference swing center of the output swing arm 7.

[0145] Lateral offset Vertical offset The search range is fixed at 0, with only the axial offset retained. As an adjustable parameter.

[0146]

[0147] Step S2, constraints.

[0148] All design variables are required to take values ​​strictly within the above search boundaries, which is automatically guaranteed by the boundary handling mechanism of the genetic algorithm.

[0149] Based on kinematic modeling Discriminant equation in It requires full amplitude range and the entire exercise cycle Internal satisfaction If there exists any set of parameters such that... If the mechanism cannot form a closed-loop motion, it is determined to be an invalid solution, and the set of parameters is directly eliminated.

[0150] Step S3, transmission performance constraints.

[0151] To ensure the force transmission performance of the mechanism and avoid dead spots and severe wear, spatial transmission angle constraints are added.

[0152] Spatial transmission angle The magnitude of the force transmission capacity reflects the quality of the force transmission performance.

[0153] Spatial transmission angle Calculated via the mixed product of space vectors:

[0154]

[0155] In the formula, Let B be the position vector of the reference swing center of the output swing arm 7; This is the unit direction vector of the link; This is to output the unit direction vector of swing arm 7.

[0156] Set the allowable transmission angle of the spatial linkage mechanism The minimum transmission angle must satisfy the following requirements throughout the entire amplitude range and the entire motion cycle: If the conditions are not met, the solution is considered invalid.

[0157] Step S4: Install the center tolerance constraint.

[0158] A tolerance mechanism is introduced to allow the algorithm to autonomously find the optimal swing center for output symmetry under full amplitude variation conditions within the engineering-permitted installation range. Considering the installation constraints in practical applications, the swing center constraint conditions are set as follows:

[0159]

[0160] In the formula, This represents the number of sampling points for the amplitude modulation angle, corresponding to the full amplitude range. Discrete sampling. This is the theoretical reference angle (taken as 0°). For engineering tolerance (take 45°).

[0161] Step S5, Objective Function.

[0162] A weighted summation method is used to construct a dual-objective fitness function cost, and the smaller the value of cost, the better the overall performance of the mechanism.

[0163] With equation Assign the highest weight to the core optimization metrics. This ensures that the algorithm converges to the solution that minimizes the center drift.

[0164]

[0165] Fine-grained amplitude tracking: Set the target maximum amplitude Minimum swing amplitude Swing tracking is a secondary performance indicator, and weights are assigned accordingly. and Balance the priority of optimization objectives to avoid weakening core optimization objectives:

[0166]

[0167] in, This represents the actual maximum swing amplitude. This represents the actual minimum swing amplitude.

[0168] In summary, the overall fitness function is:

[0169]

[0170] Step S6: After multiple rounds of parameter cross-validation and iterative iteration, the optimized parameters that balance global search and local convergence performance are finally determined. The specific settings are shown in Table 1.

[0171] Table 1. Optimization parameters of the genetic algorithm Population size 500 Expanding the initial sample space increases the probability of generating feasible solutions under nonlinear constraints, thus addressing constraint-sensitive problems. Maximum number of generations 1500 The algorithm is guaranteed to iterate sufficiently in the complex solution space to achieve stable convergence. Cross ratio 0.8 Preserving superior genetic traits to achieve efficient population recombination and evolution. Select function Tournament Selection To prevent dominant individuals from prematurely monopolizing the population, maintain genetic diversity, and mitigate convergence oscillations. Mutation function Adaptive feasible mutation Ensure that the mutated individuals strictly satisfy the boundary constraints, guarantee the validity of the solution, and adapt to the narrow feasible solution space. Elite retention count 25 To prevent the loss of optimal solutions, 5% of the best individuals in the population are retained. Mixing function fmincon Gradient-based local refinement significantly improves the accuracy of the optimal solution. Multiple starting point loop count 20 times By repeatedly changing random seeds to initialize the population, premature stagnation and local optima can be avoided.

[0172] Step S7: Optimize the results.

[0173] The optimized mechanism parameters are shown in Table 2.

[0174] Table 2 Optimization results of rod length and offset (unit: mm) crank m 30~60 37.69 37.5 swing arm n 40~50 40.00 40.0 Linkage l 80~200 171.91 172.0 bias q 80~200 179.86 180.0 bias h 0 0.00 0.0 bias p 0 0.00 0.0

[0175] After verification, the rounded parameters still meet all hard constraint requirements. Verification is as follows:

[0176] Boundary constraints: All optimization variables take values ​​within a preset initial range; Kinematic continuity constraints: The optimized mechanism maintains its rotational integrity throughout the entire amplitude range and motion cycle. The motion is consistently stable, without any singularities or jamming; transmission performance constraints: extract the minimum transmission angle within the entire motion cycle and the entire amplitude range, such as... Figure 14As shown, the minimum transmission angle of the optimized mechanism across the entire amplitude range is 41.12°, which satisfies the requirements of a spatial linkage mechanism. Permissible transmission angle requirements. Mounting center tolerance constraints: such as... Figure 15 As shown, the optimized mechanism's swing center across the entire amplitude range is 23°, meeting the set tolerance requirement of -45° to 45°. Regarding core performance... Figure 16 A quantitative comparison of the center drift characteristics of the mechanism before and after optimization is given. After optimization, the center drift range of the mechanism in the full amplitude range is reduced to 1.318°. Compared with the unoptimized conventional α-RSSR mechanism (center drift range of 4.319° in the full amplitude range), the center drift suppression effect is improved by 69.48% in terms of swing amplitude control. Figure 17 The variation of output swing with amplitude modulation angle is shown. The results indicate that the amplitude modulation angle... When dynamically adjusted within the range of 55° to 103°, the output swing of the mechanism can be continuously and steplessly adjusted within the range of 0.39° to 42.24°, fully covering the target working range.

Claims

1. A kinematic modeling method for a continuously variable amplitude spatial four-bar linkage, characterized in that, Includes the following steps: Step (1), define the coordinate system; Define global reference coordinate system , the rotation center of the turntable as the origin , The shaft coincides with the axis of the output shaft of the driving motor, The shaft is horizontally perpendicular to The shaft, The shaft is vertically upward; Define a moving coordinate system Its origin and the origin coincide; The axis coincides with the rotation axis of the turntable; The axis is along the radial direction of the turntable and points towards the center of the first spatial spherical pair. ; The axis is determined by the right-hand rule; Define a moving coordinate system Its origin The reference swing center of the output swing arm. The axis of the shaft coincides with the swing axis of the output swing arm. The shaft runs along the length of the output swing arm and points towards the center of the second spatial spherical pair. , The axis is determined by the right-hand rule; This represents the length of the active crank, i.e., the distance from the center of rotation of the turntable to the center of the first spatial spherical pair. The distance between them; This indicates the distance from the reference swing center of the output swing arm to the center of the second spatial spherical pair. The distance between them; Indicates the length of the spatial link; This indicates that the plane of rotation of the turntable itself is in Projection around the plane The angle of rotation of the axis, express shaft and The angle between the axes, Indicates the rotation angle of the turntable. This indicates the output swing angle of the output swing arm. Indicates pressure angle; spatial transmission angle Pressure angle The complementary angle; Step (2), establish constraint equations: In the formulae, , ， is a lateral offset parameter, is a vertical offset parameter, is an axial offset parameter.

2. The method of kinematic modeling of a continuously variable amplitude spatial four-bar mechanism according to claim 1, wherein, The continuously variable amplitude spatial four-bar linkage includes a drive motor, a ball-cage universal joint assembly, a turntable, a first spatial spherical joint, a spatial connecting rod, a second spatial spherical joint, an output swing arm, a bracket, a support ring, a bearing, an amplitude adjustment connecting rod, and a linear module. The ball-cage universal joint assembly includes a star-shaped sleeve, a ball-cage cage, and multiple drive balls. The outer periphery of the star-shaped sleeve has multiple inner raceways, and the drive balls are located in the inner raceways of the star-shaped sleeve. The ball-cage cage is fitted onto the star-shaped sleeve and has multiple windows, in which the drive balls are located. A receiving chamber is located at the center of the front of the turntable, and the receiving chamber has multiple outer raceways. The ball-cage universal joint assembly is located in the receiving chamber, and the outer sides of the drive balls are embedded in the outer raceways. The output shaft of the drive motor is fixedly connected to the center of the star-shaped sleeve. The first spatial spherical joint includes a shell base and a ball head pin. The shell base is fixedly connected to the edge of the turntable, and one end of the spatial connecting rod is fixedly connected to the ball head pin. The second spatial spherical joint includes a shell base and a ball head pin. The other end of the spatial connecting rod is fixedly connected to the ball head pin of the second spatial spherical joint. The shell base of the second spatial spherical joint is fixedly connected to one end of the output swing arm. The other end of the output swing arm is hinged to the bracket. The back of the turntable is rotatably connected to the support ring via a bearing; the linear module is provided with a linear motion part, one end of the amplitude adjustment link is hinged to the support ring, and the other end of the amplitude adjustment link is hinged to the linear motion part.

3. A parameter optimization method of a continuously variable amplitude spatial four-bar mechanism, characterized in that, Includes the following steps: For the stepless variable amplitude spatial four-bar mechanism, the optimization variables are: ; This represents the length of the active crank, i.e., the distance from the center of rotation of the turntable to the center of the first spatial spherical pair. The distance between them; This indicates the distance from the reference swing center of the output swing arm to the center of the second spatial spherical pair. The distance between them; Indicates the length of the spatial link; This is the lateral offset parameter. For vertical offset parameters, This is the axial offset parameter; Based on the following kinematic modeling: In the formulae, , ， This indicates that the plane of rotation of the turntable itself is in Projection around the plane The angle of rotation of the axis, express shaft and The angle between the axes, Indicates the rotation angle of the turntable. This indicates the output swing angle of the output swing arm. Indicates pressure angle; spatial transmission angle Pressure angle The complementary angle; Discriminant equation ,right , , , , , Optimization is required across the entire amplitude range. and the entire exercise cycle Internal satisfaction .

4. The method of parameter optimization of a continuously variable amplitude spatial four-bar mechanism according to claim 3, characterized in that, Parameter optimization based on genetic algorithm, lateral bias Vertical offset The search range is fixed at 0, with only the axial offset retained. As an adjustable parameter; Setting allowable transmission angle of spatial link mechanism , requiring minimum transmission angle in full amplitude range and full motion cycle to meet ; Construct the dual-objective fitness function cost: In the formula, , ; , , is a weight, ; , Target Max Swing, Target Min Swing, Actual Max Swing, Actual Min Swing.

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