Six-degree-of-freedom parallel platform based on iterative initial value estimation and motion positive solution method thereof

Abstract

The invention provides a six-degree-of-freedom parallel platform based on iterative initial value estimation and a forward motion solution method thereof. The six-degree-of-freedom parallel platform takes 6-UPU configuration as a research object; comprising a lower platform, an upper platform, six Hooke hinges fixed on the lower platform, six Hooke hinges fixed on the upper platform, and six electric push rods connected with the Hooke hinges of the lower platform and the Hooke hinges of the upper platform. The six-degree-of-freedom parallel platform motion positive solution method based on iteration initial value estimation comprises forward kinematics modeling of a six-degree-of-freedom parallel platform, an iteration initial value estimation method based on space geometry, an iteration initial value estimation method based on particle swarm optimization, an iteration initial value comparison method and a quasi-Newton method based on iteration initial value estimation.

Description

Technical Field

[0001] This application belongs to the field of aircraft manufacturing equipment technology, specifically relating to a six-degree-of-freedom parallel platform based on iterative initial value estimation and its forward kinematics method. Background Technology

[0002] The six-degree-of-freedom parallel platform (also known as the Stewart-Gough platform) is a space motion mechanism composed of six electrically connected actuators. Due to its high rigidity, high precision, high load capacity, and flexible multi-degree-of-freedom motion characteristics, it has become one of the core devices in the field of modern precision motion control. In the aerospace field, it is used in flight simulators, satellite antenna pointing mechanisms, and spacecraft docking platforms; in precision manufacturing, it is used in six-axis parallel machine tools, micro-nano manipulators, complex surface machining, and micron-level positioning; in the medical field, it is used in surgical robot positioning platforms and rehabilitation training equipment; and in experimental testing, it is used in earthquake simulation shaking tables and optical platform pose adjustment. However, the forward kinematics problem of the Stewart platform—that is, solving the platform's pose (position and attitude) based on the lengths of the six actuators—remains a key technical bottleneck restricting its performance improvement. The accuracy and real-time performance of the forward kinematics directly affect the platform's control accuracy, dynamic response capability, and adaptability to complex scenarios.

[0003] Current research on the forward kinematics of a Stewart six-DOF parallel platform mainly falls into three categories: numerical iterative methods, analytical methods, and intelligent optimization algorithms. Numerical iterative methods, such as Newton's method, approximate the forward solution by iteratively solving a system of nonlinear equations. They are versatile and applicable to complex configurations, but are highly sensitive to initial conditions, their convergence heavily depends on the initial values, and they lack real-time performance. Analytical methods, such as symbolic elimination and geometric construction methods, derive closed-form solutions through algebraic transformations or geometric analysis. They involve no iterative process, are computationally fast, but are highly complex, have limited applicability, and exhibit poor numerical stability. Intelligent optimization algorithms, such as particle swarm optimization and genetic algorithms, transform the forward kinematics problem into a global optimization problem searching for the optimal pose. They are insensitive to initial values ​​and can handle multiple solutions, but they involve large computational loads, are easily affected by parameter settings, and suffer from insufficient accuracy and reliability. Therefore, it is necessary to research an exact forward kinematics solution method. Summary of the Invention

[0004] Purpose of the invention: To address the above problems, this invention proposes a forward kinematics solution method for a Stewart six-DOF parallel platform based on iterative initial value estimation. The method estimates the initial values ​​for iteration using spatial geometry and intelligent optimization algorithms, and then selects the better solution for iterative solution using a quasi-Newton method. This improves the initial value sensitivity of numerical iteration methods and ensures the globality, applicability, and high accuracy of the proposed method.

[0005] In the first aspect, this application provides a six-degree-of-freedom parallel platform based on iterative initial value estimation, taking the 6-UPU configuration as the research object, including a lower platform, an upper platform, six Hooke hinges fixed on the lower platform, six Hooke hinges fixed on the upper platform, and six electric actuators connecting the Hooke hinges of the lower platform and the Hooke hinges of the upper platform. The lower platform Hooke's hinge includes a fixed lower platform Hooke's hinge joint, a movable lower platform Hooke's hinge joint, and a cross shaft. The two ends of the cross shaft are connected to the joints via bearings, allowing the two joints to rotate relative to each other in space. (Lower Hooke's hinge center) The center of the cross axis of the lower platform Hooke's hinge. The upper platform Hooke's hinge includes a fixed upper platform Hooke's hinge joint, a movable upper platform Hooke's hinge joint, and a cross shaft. The two ends of the cross shaft are connected to the joints via bearings, allowing the two joints to rotate relative to each other in space. The upper Hooke's hinge center... The center of the Hooke's hinge on the upper platform is the cross axis; the lower platform is fixedly connected to the lower platform coordinate system {B}, and the upper platform is fixedly connected to the upper platform coordinate system {P}.

[0006] Preferably, the lower Hooke's hinge center and the center of the upper Hooke hinge They are distributed on the circumcircle of the Hooke's hinge on the lower platform and the circumcircle of the Hooke's hinge on the upper platform. These are the circumradii of the Hooke's hinge on the lower platform and the circumradii of the Hooke's hinge on the upper platform, respectively. The lower Hooke's hinge center The central angle corresponding to the circumcircle of the Hooke's hinge on the lower platform. The center of the upper Hooke's hinge The central angle (in radians) corresponding to the circumcircle of the Hooke's hinge on the upper platform:

[0007]

[0008] Secondly, this application also provides a forward kinematics solution method for a six-DOF parallel platform based on iterative initial value estimation, which specifically includes the following steps: (1) Forward kinematic modeling of the Stewart six-DOF parallel platform; (2) An iterative initial value estimation method based on spatial geometry is used to obtain the iterative initial value. ; (3) The iterative initial value estimation method based on particle swarm optimization is used to obtain the iterative initial value. ; (4) Compare the initial values ​​of the iteration and The initial values ​​for the quasi-Newton method are obtained. ; (5) Based on the iterative solution of the quasi-Newton method with iterative initial value estimation, the forward kinematics solution of the Stewart six-DOF parallel platform is obtained. .

[0009] Preferably, step (1) specifically includes the following steps: (1) Calculate the center of the lower Hooke's hinge. Spatial coordinates in the lower platform coordinate system {B} : ; (2) Calculate the center of the upper Hooke's hinge. Spatial coordinates in the upper platform coordinate system {P} : ; (3) The pose of the upper platform relative to the lower platform , Representing the rigid body rotational motion in space using ZYX Euler angles, the homogeneous transformation matrix between the lower platform coordinate system {B} and the upper platform coordinate system {P} is solved. : ; ; ; (4) Based on the information already obtained , and The direction vector of the electric actuator was further calculated. : ; in, The length of the electric actuator. is the unit direction vector of the electric actuator.

[0010] (5) Based on the information already obtained Furthermore, the inverse kinematics model of the Stewart six-DOF parallel platform was obtained: ; (6) Based on the obtained inverse kinematics model, the forward kinematics model of the Stewart six-DOF parallel platform is further obtained: ; in, This is the theoretical (target) length of the electric actuator.

[0011] Preferably, step (2) specifically includes the following steps: (1) with Let the midpoint of the lower Hooke's hinge center be found. Spatial coordinates in the lower platform coordinate system {B} : ; in, The lower Hooke's hinge center and The center; The lower Hooke's hinge center and The center; The lower Hooke's hinge center and The center.

[0012] (2) Set intermediate parameters m and n: ; Where m is the hinge center of the upper Hooke's hinge. arrive The distance, due to the center of the upper Hooke's hinge. The distribution is symmetrical, therefore n is the center of the upper Hooke's hinge. arrive The distance, due to the center of the upper Hooke's hinge. The distribution is symmetrical, therefore .

[0013] (3) Construct a system of nonlinear equations and solve for the equations. Coordinates: ; in, The midpoint of the upper Hooke's hinge center. They are respectively The coordinate components of the Cartesian coordinate system; The center of the upper Hooke's hinge and The center; The center of the upper Hooke's hinge and The center; The center of the upper Hooke's hinge and The center; This is the tolerance threshold for the length of the electric actuator.

[0014] (4) Based on the information already obtained ,make for The unit direction vector is calculated. outer core : ; ; ; in, for circumcenter coordinate vector, for The coordinate components of the Cartesian coordinate system.

[0015] (5) Based on the information already obtained To obtain the initial value of the iteration : ; Preferably, step (3) specifically includes the following steps: (1) Set the number of particles, maximum number of iterations, dynamic inertia weight, individual learning factor, social learning factor and convergence threshold; (2) Construct the objective function: ; (3) Obtain the initial values ​​for iteration based on the particle swarm optimization solution. .

[0016] Preferably, step (4) specifically includes the following steps: (1) Based on the inverse kinematics model, calculate Corresponding electric actuator length : ; (2) Based on the inverse kinematics model, calculate Corresponding electric actuator length : ; (3) Based on the information already obtained and The initial values ​​for the quasi-Newton method are obtained. : ; in, This is a comparison function.

[0017] Preferably, step (5) specifically includes the following steps: (1) Constructing an iterative expression for the quasi-Newton method based on the inverse Broyden algorithm: ; in, , It is a Jacobian matrix; ; And there are .

[0018] (2) Set the maximum number of iterations and the error threshold; (3) Based on the obtained initial values ​​of the iteration Using the quasi-Newton method iterative expression, the forward kinematics solution of the Stewart six-DOF parallel platform is calculated iteratively. .

[0019] This invention has the following advantages and significant benefits: (1) This application reduces the initial value sensitivity of numerical iteration method in forward kinematics solution by using intelligent optimization algorithm to estimate the initial value of iteration; (2) This application adopts a hybrid strategy of “iterative initial value estimation + numerical iteration method solution”, which integrates the global search capability of intelligent optimization algorithm and the local efficiency of numerical iteration method, thus ensuring the globality, applicability and high accuracy of the proposed method. Attached Figure Description

[0020] Figure 1 This is a schematic diagram of the Stewart six-DOF parallel platform (6-UPU configuration) provided in the embodiments of this application; Figure 2 This is a schematic diagram of the lower platform Hooke's hinge structure provided in an embodiment of this application; Figure 3 This is a schematic diagram of the upper platform Hooke's hinge structure provided in an embodiment of this application; Figure 4 This is a simplified diagram of the Stewart six-DOF parallel platform mechanism provided in the embodiments of this application; Figure 5 This is a top view of the Stewart six-DOF parallel platform provided in the embodiments of this application; Figure 6 This is a schematic diagram of the hinge center of the moving and lower Hooke hinges after being merged, provided in an embodiment of this application. Figure 7 This is a schematic diagram of the center distance constraint of the merged upper Hooke's hinge provided in the embodiments of this application; Figure 8 This is a flowchart of the forward kinematics solution based on iterative initial value estimation provided in the embodiments of this application; Figure 9 This is the iterative convergence curve provided in the embodiments of this application; The components are: 1. Lower platform, 2. Upper platform, 3. Lower platform Hooke's hinge, 4. Upper platform Hooke's hinge, 5. Electric actuator, 6. Servo motor, 7. Lower platform Hooke's hinge fixed joint, 8. Lower platform Hooke's hinge movable joint, 9. Upper platform Hooke's hinge fixed joint, 10. Upper platform Hooke's hinge movable joint, 11. Cross shaft. Detailed Implementation

[0021] The specific implementation scheme of a six-degree-of-freedom parallel platform based on iterative initial value estimation provided in this application is as follows: Figure 1 As shown, the Stewart six-DOF parallel platform includes a lower platform 1, an upper platform 2, six lower platform Hooke hinges 3 fixed to the lower platform 1, six upper platform Hooke hinges 4 fixed to the upper platform 2, and six electric actuators 5 connecting the lower platform Hooke hinges 3 and the upper platform Hooke hinges 4. The electric actuators 5 are driven by servo motors 6. Figure 2 As shown, the lower platform Hooke's hinge 3 includes a fixed lower platform Hooke's hinge joint 7, a movable lower platform Hooke's hinge joint 8, and a cross shaft 11. Both ends of the cross shaft 11 are connected to the joints via bearings, allowing the two joints to rotate relative to each other in space. Figure 3 As shown, the upper platform Hooke hinge 4 includes an upper platform Hooke hinge fixed joint 9, an upper platform Hooke hinge movable joint 10, and a cross shaft 11. The two ends of the cross shaft 11 are connected to the joints through bearings, allowing the two joints to rotate relative to each other in space.

[0022] like Figure 4 He Ru Figure 5 As shown, the lower Hooke's hinge center The center of the cross shaft 11 of the lower platform Hooke's hinge 3, and the hinge center of the upper Hooke's hinge. The center of the cross axis 11 of the upper platform Hooke's hinge 4 is located at the lower platform 1, which is fixedly connected to the lower platform coordinate system {B}, and the upper platform 2 is fixedly connected to the upper platform coordinate system {P}.

[0023] The basic parameters of the Stewart six-DOF parallel platform are set as shown in Table 1: Table 1 Basic parameters of the Stewart six-DOF parallel platform

[0024] Set the target pose of platform 2 as follows: ; The position is in meters (m), and the attitude angle is in radians (rad).

[0025] Using the inverse kinematics model, the lengths of the six electric actuators 5 corresponding to this pose can be obtained as follows: ; The length of the electric actuator 5 is measured in meters (m).

[0026] In other embodiments of this application, a specific implementation scheme for a forward kinematics method for a six-DOF parallel platform based on iterative initial value estimation is provided, which specifically includes the following steps: (1) Forward kinematic modeling of a six-degree-of-freedom parallel platform; (2) An iterative initial value estimation method based on spatial geometry is used to obtain the iterative initial value. ; (3) The iterative initial value estimation method based on particle swarm optimization (PSO) is used to obtain the iterative initial value. ; (4) Compare the initial values ​​of the iteration and The initial values ​​for the quasi-Newton method are obtained. ; (5) Based on the iterative solution of the quasi-Newton method with iterative initial value estimation, the forward kinematics solution of the Stewart six-DOF parallel platform is obtained. .

[0027] Furthermore: The forward kinematics modeling method for a six-DOF parallel platform specifically includes the following steps: (1) Calculate the center of the lower Hooke's hinge. Spatial coordinates in the lower platform coordinate system {B} As shown in Table 2: Table 2 Lower Hooke's Hinge Center Spatial coordinates in the lower platform coordinate system {B}

[0028] (2) Calculate the center of the upper Hooke's hinge. Spatial coordinates in the upper platform coordinate system {P} As shown in Table 3: Table 3 Upper Hooke's Hinge Center Spatial coordinates in the upper platform coordinate system {P}

[0029] (3) The pose of the upper platform 2 relative to the lower platform 1 , Representing the rigid body rotational motion in space using ZYX Euler angles, the homogeneous transformation matrix between the lower platform coordinate system {B} and the upper platform coordinate system {P} is solved. : ; ; ; (4) Based on the information already obtained , and Further calculations were performed on the direction vector of electric actuator 5. : ; in, The length of the electric actuator is 5. The direction vector of the electric actuator is 5 units.

[0030] (5) Based on the information already obtained Furthermore, the inverse kinematics model of the Stewart six-DOF parallel platform was obtained: ; (6) Based on the obtained inverse kinematics model, the forward kinematics model of the Stewart six-DOF parallel platform is further obtained: ; in, The theoretical (target) length of the electric actuator is 5.

[0031] Further: The iterative initial value estimation method based on spatial geometry, wherein step (2) is characterized by specifically including the following steps: (1) The center of the moving and lower Hooke's hinge after the merger is as follows Figure 6 As shown, with Let the midpoint of the lower Hooke's hinge center be found. The spatial coordinates in the lower platform coordinate system {B} are shown in Table 4: Table 4 Spatial coordinates of the midpoint of the lower Hooke's hinge center in the lower platform coordinate system {B}

[0032] in, The lower Hooke's hinge center and The center; The lower Hooke's hinge center and The center; The lower Hooke's hinge center and The center.

[0033] (2) Set intermediate parameters m and n, such as Figure 7 As shown, we can obtain: ; Where m is the hinge center of the upper Hooke's hinge. arrive The distance, due to the center of the upper Hooke's hinge. The distribution is symmetrical, therefore n is the center of the upper Hooke's hinge. arrive The distance, due to the center of the upper Hooke's hinge. The distribution is symmetrical, therefore .

[0034] (3) Construct a system of nonlinear equations and solve for the equations. Coordinates: ; in, The midpoint of the upper Hooke's hinge center. They are respectively The coordinate components of the Cartesian coordinate system; The center of the upper Hooke's hinge and The center; The center of the upper Hooke's hinge and The center; The center of the upper Hooke's hinge and The center; This is the tolerance threshold for the length of the electric actuator.

[0035] Since the aforementioned nonlinear equations are all distance functions, they are typically continuous and differentiable, making them suitable for gradient-based optimization methods. The conjugate gradient method, by selecting a series of conjugate directions for iterative optimization, performs exceptionally well when the objective function is a quadratic function, making it an efficient iterative algorithm. Therefore, the conjugate gradient method is used to solve the aforementioned nonlinear equations, yielding... The coordinates are shown in Table 5: Table 5 Spatial coordinates of the midpoint of the upper Hooke's hinge center in the lower platform coordinate system {B}

[0036] (4) Based on the information already obtained ,make for The unit direction vector is calculated. outer core As shown in Table 6: Table 6 outer center Spatial coordinates in the lower platform coordinate system {B}

[0037] in, for circumcenter coordinate vector, for The coordinate components of the Cartesian coordinate system.

[0038] (5) Based on the information already obtained To obtain the initial value of the iteration : ; Further: The iterative initial value estimation method based on particle swarm optimization, wherein step (3) is characterized by specifically including the following steps: (1) Set the number of particles to 300, the maximum number of iterations to 200, the dynamic inertia weight to [0.4 0.9], the individual learning factor to 1.5, the social learning factor to 2.0, and the convergence threshold to... ; (2) Construct the objective function: ; (3) Obtain the initial values ​​for iteration based on the particle swarm optimization solution. : ; Further: The initial value of the iteration and The comparison method, step (4), is characterized by specifically including the following steps: (1) Based on the inverse kinematics model, calculate The corresponding electric actuator length 5 : ; (2) Based on the inverse kinematics model, calculate The corresponding electric actuator length 5 : ; (3) Based on the information already obtained and The initial values ​​for the quasi-Newton method are obtained. : ; ; ; in, This is a comparison function.

[0039] Going further: The iterative solution based on the quasi-Newton method using iterative initial value estimation, wherein step (5) is characterized by specifically including the following steps: (1) Constructing an iterative expression for the quasi-Newton method based on the inverse Broyden algorithm: ; in, , It is a Jacobian matrix; ; And there are .

[0040] (2) Set the maximum number of iterations to 1000 and the error threshold to 1000. ; (3) Based on the obtained initial values ​​of the iteration Using the quasi-Newton method iterative expression, the forward kinematics solution of the Stewart six-DOF parallel platform is calculated iteratively. : ; At the same time, the iterative convergence curve of the quasi-Newton method is obtained as follows: Figure 8 As shown.

[0041] The flowchart of the above method steps is as follows: Figure 9 As shown.

[0042] Market prospect assessment of the present invention: (1) Through algorithm optimization, it can be directly embedded into existing PLCs or motion controllers, reducing the development cost of the six-degree-of-freedom platform; (2) The stability of the forward kinematics solution is improved by using the iterative initial value estimation method.

Claims

1. A six-degree-of-freedom parallel platform based on iterative initial value estimation, characterized in that, The platform includes a lower platform, an upper platform, six Hooke hinges fixed to the lower platform, six Hooke hinges fixed to the upper platform, and six electric actuators connecting the Hooke hinges of the lower platform and the Hooke hinges of the upper platform. The lower platform Hooke's hinge includes a fixed lower platform Hooke's hinge joint, a movable lower platform Hooke's hinge joint, and a cross shaft. Both ends of the cross shaft are connected to the joints via bearings, allowing the two joints to rotate relative to each other in space. The lower Hooke's hinge center... The center of the cross axis of the lower platform Hooke's hinge. The upper platform Hooke's hinge includes a fixed upper platform Hooke's hinge joint, a movable upper platform Hooke's hinge joint, and a cross shaft. Both ends of the cross shaft are connected to the joint via bearings, allowing the two joints to rotate relative to each other in space. The upper Hooke's hinge center... The center of the cross axis of the Hooke's hinge on the upper platform is defined; the lower platform is fixedly connected to the lower platform coordinate system {B}, and the upper platform is fixedly connected to the upper platform coordinate system {P}.

2. The platform according to claim 1, characterized in that, Lower Hooke's hinge center and the center of the upper Hooke hinge They are distributed on the circumcircle of the Hooke's hinge on the lower platform and the circumcircle of the Hooke's hinge on the upper platform; in, These are the circumradii of the Hooke's hinge on the lower platform and the circumradii of the Hooke's hinge on the upper platform, respectively. The lower Hooke's hinge center The central angle corresponding to the circumcircle of the Hooke's hinge on the lower platform. The center of the upper Hooke's hinge The central angle corresponding to the circumcircle of the Hooke's hinge on the upper platform: 。 3. A forward kinematics method for a six-DOF parallel platform based on iterative initial value estimation, characterized in that, The method includes: (3-1) Forward kinematic modeling of a six-degree-of-freedom parallel platform; (3-2) An iterative initial value estimation method based on spatial geometry is used to obtain the initial value for iteration. ; (3-3) An iterative initial value estimation method based on particle swarm optimization is used to obtain the iterative initial value. ; (3-4) Compare initial values ​​during iteration and The initial values ​​for the quasi-Newton method are obtained. ; (3-5) Based on the iterative solution of the quasi-Newton method using iterative initial value estimation, the forward kinematics solution of the six-DOF parallel platform is obtained. .

4. The method according to claim 3, characterized in that, Step (3-1) specifically includes the following steps: (4-1) Calculate the center of the lower Hooke's hinge. Spatial coordinates in the lower platform coordinate system {B} : ； (4-2) Calculate the center of the upper Hooke's hinge. Spatial coordinates in the upper platform coordinate system {P} : ； (4-3) Position of the upper platform relative to the lower platform , Representing the rigid body rotational motion in space using ZYX Euler angles, the homogeneous transformation matrix between the lower platform coordinate system {B} and the upper platform coordinate system {P} is solved. : ； ； ； (4-4) Based on the information already obtained , and The direction vector of the electric actuator was further calculated. : ； in, The length of the electric actuator. The unit direction vector of the electric actuator; (4-5) Based on the information already obtained Furthermore, the inverse kinematics model of the six-degree-of-freedom parallel platform is obtained: ； (4-6) Based on the obtained inverse kinematics model, the forward kinematics model of the six-degree-of-freedom parallel platform is further obtained: ； in, The theoretical or target length of the electric actuator.

5. The method according to claim 4, characterized in that, Step (3-2) specifically includes the following steps: (5-1) with Let the midpoint of the lower Hooke's hinge center be found. Spatial coordinates in the lower platform coordinate system {B} : ； in, The lower Hooke's hinge center and The center; The lower Hooke's hinge center and The center; The lower Hooke's hinge center and The center; (5-2) Set intermediate parameters m and n: ； Where m is the hinge center of the upper Hooke's hinge. arrive The distance, due to the center of the upper Hooke's hinge. The distribution is symmetrical, therefore n is the center of the upper Hooke's hinge. arrive The distance, due to the center of the upper Hooke's hinge. The distribution is symmetrical, therefore ; (5-3) Construct a system of nonlinear equations and solve for... Coordinates: ； in, The midpoint of the upper Hooke's hinge center. They are respectively The coordinate components of the Cartesian coordinate system; The center of the upper Hooke's hinge and The center; The center of the upper Hooke's hinge and The center; The center of the upper Hooke's hinge and The center; This refers to the tolerance threshold for the length of the electric actuator. (5-4) Based on the information already obtained ,make for The unit direction vector is calculated. outer core : ； ； ； in, for circumcenter coordinate vector, for The coordinate components of the Cartesian coordinate system; (5-5) Based on the information already obtained To obtain the initial value of the iteration : 。 6. The method according to claim 5, characterized in that, Step (3-3) specifically includes the following steps: (6-1) Set the number of particles, maximum number of iterations, dynamic inertia weight, individual learning factor, social learning factor, and convergence threshold; (6-2) Construct the objective function: ； (6-3) Based on the particle swarm optimization solution, the initial values ​​for iteration are obtained. .

7. The method according to claim 6, characterized in that, Step (3-4) specifically includes the following steps: (7-1) Based on the inverse kinematics model, calculate... Corresponding electric actuator length : ； (7-2) Based on the inverse kinematics model, calculate... Corresponding electric actuator length : ； (7-3) Based on the information already obtained and The initial values ​​for the quasi-Newton method are obtained. : ； in, This is a comparison function.

8. The method according to claim 7, characterized in that, Step (3-5) specifically includes the following steps: (8-1) Construct the iterative expression for the quasi-Newton method based on the inverse Broyden algorithm: ； in, , It is a Jacobian matrix; ; And there are ; (8-2) Set the maximum number of iterations and the error threshold; (8-3) Based on the obtained initial values ​​for iteration Using the quasi-Newton method iterative expression, the forward kinematics solution of the six-DOF parallel platform is calculated iteratively. .

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