Method for calculating Jacobian matrix of Stewart platform

Abstract

The invention discloses a Stewart platform jacobian matrix calculation method, which belongs to the technical field of robot kinematics, and comprises the following steps: respectively establishing an inverse kinematics model of UPS-Stewart and a position inverse kinematics model of UPRU-Stewart; according to the inverse kinematics model of the UPS-Stewart and the inverse kinematics model of the position of the UPRU-Stewart, the relative angular velocity of the branch chains of the UPRU-Stewart platform is established; and according to the relative angular velocity of the branch chains of the UPRU-Stewart platform, a Jacobian matrix of the URPU-Stewart platform is obtained. According to the method, the passive helix of the UPRU or UPU branch chain is compensated, and the effects that the calculation process is simple and visual, and the calculation result is accurate and reliable are achieved.

Description

Technical Field

[0001] The present invention discloses a method for calculating the Jacobian matrix of a Stewart platform, belonging to the technical field of robot kinematics. Background Art

[0002] Six-degree-of-freedom platforms, such as 6-UPS and 6-UPRU / UPU Gough-Stewart platforms, etc., have the advantages of high stiffness, strong load-bearing capacity, and high precision, and are widely used in industrial production lines, motion simulators, parallel machine tools, positioning devices, etc. The 6-UPRU / UPU Gough-Stewart platform has the advantages of large load-bearing capacity, convenient installation, and low cost because both the upper and lower connecting hinges are Hooke joints. Compared with the 6-UPS Gough-Stewart platform, it has been more widely used. However, the kinematics and dynamics of the UPRU (or UPU, both are expressed as UPRU in this disclosure) type are far from as perfect as those of the UPS type, and are also more complex than the UPS type in theory. In view of this, people approximate the UPU type as the UPS type for use in many scenarios. This approximation is feasible in some low-requirement occasions, but in the high-precision field, this approximation will cause non-negligible errors and often cannot meet the required precision requirements.

[0003] To obtain an accurate dynamic theory model of the UPRU type Stewart platform, an accurate kinematic model is indispensable. How to obtain an accurate kinematic model requires considering the passive rotation of the Hooke joint in the UPRU type platform, which will cause the passive screw phenomenon of the entire branch chain, resulting in different rotational angular velocities of the upper and lower branch chains and increasing the difficulty of solution. Summary of the Invention

[0004] In order to overcome the problem that the existing UPRU or UPU-Stewart platform does not have a complete method for calculating the Jacobian matrix, and the approximate use of the Jacobian matrix velocity inverse kinematics of the UPS-Stewart platform causes large motion errors, the present invention discloses and provides a method for calculating the angular velocity of the branch chain of the UPU type Stewart platform based on the passive rotation of the Hooke joint to solve the problems existing in the above-mentioned prior art, compensate for the passive screw of the UPRU or UPU branch chain, and achieve the effects of simple and intuitive calculation process and accurate and reliable calculation results.

[0005] The present invention discloses and provides a method for calculating the Jacobian matrix of a Stewart platform, including:

[0006] Step S10, respectively establish the inverse kinematic model of UPS-Stewart and the position inverse kinematic model of UPRU-Stewart;

[0007] Step S20: Establish the relative angular velocity of the UPRU-Stewart platform's linkages through the inverse kinematics model of the UPS-Stewart and the position inverse kinematics model of the UPRU-Stewart.

[0008] Step S30: Obtain the Jacobian matrix of the URPU-Stewart based on the relative angular velocity of the UPRU-Stewart platform's linkages.

[0009] Furthermore, the specific steps of Step S10 include:

[0010] Step S11: Establish the position inverse kinematics model of the UPS-Stewart.

[0011] Step S12: Based on the position inverse kinematics model of the UPS-Stewart, establish the velocity inverse kinematics model of the UPS-Stewart.

[0012] Step S13: Based on the velocity inverse kinematics model of the UPS-Stewart, obtain the Jacobian matrix of the UPS-Stewart's inverse kinematics.

[0013] Step S14: Based on the passive screw mechanism of the UPRU-type linkage, obtain the relationship between the passive screw angle and the passive additional length of the UPRU-type linkage.

[0014] Step S15: Based on the relationship between the passive screw angle and the passive additional length of the UPRU-type linkage, establish the position inverse kinematics model of the UPRU-Stewart.

[0015] Furthermore, the relationship between the passive screw angle and the passive additional length of the UPRU-type linkage is shown in the following formula (1):

[0016]

[0017] Where: Δθ i is the passive screw angle of the UPRU-type linkage, Δl i is the passive additional length of the UPRU-type linkage, s is the helix direction, s = 1 for right-handed, s = -1 for left-handed, and p is the lead.

[0018] Furthermore, the specific steps of Step S20 include:

[0019] Step S21: Based on the position inverse kinematics model of the UPRU-Stewart, establish the velocity inverse kinematics model of the UPRU-Stewart.

[0020] Step S22: Based on the velocity inverse kinematics model of the UPRU-Stewart and the position inverse kinematics model of the UPS-Stewart, the angular velocity of the lower link of the UPRU-Stewart platform and the angular velocity of the upper link of the UPRU-Stewart platform are obtained respectively;

[0021] Step S23: Based on the angular velocity of the lower link of the UPRU-Stewart platform and the angular velocity of the upper link of the UPRU-Stewart platform, the relative angular velocity of the links of the UPRU-Stewart platform is obtained.

[0022] Furthermore, the Jacobian matrix of the URPU-Stewart is shown in the following formula (2):

[0023] J u =(J s -J Δ J p ) (2)

[0024] Where: J u is the Jacobian matrix of the URPU-Stewart, J s is the Jacobian matrix of the inverse kinematics of the UPS-Stewart, J Δ is the manifestation of the influence of the "passive screw phenomenon" on the motion of the UPRU-Stewart platform. Let

[0025]

[0026] Then its expression is

[0027]

[0028] J p is the transformation matrix from the velocity of the moving platform to the generalized velocity and its expression is

[0029]

[0030] Principle and beneficial effects of the technical solution of the present disclosure:

[0031] (1) By analyzing the "passive screw" motion mechanism of the UPRU type link, the present invention obtains a calculation method for the influence of the joint velocity affected by the passive screw. Through its compensation, a complete and accurate velocity inverse kinematics calculation method of the UPRU-Stewart is obtained. Furthermore, through further mathematical derivation and arrangement, the calculation formula of the Jacobian matrix of the UPRU-Stewart is obtained;

[0032] The main difference between the UPRU-type Stewart platform and the UPS-type Stewart platform lies in the upper or lower hinge points. The upper or lower hinge points of the UPS-type are spherical hinges, while those of the UPRU-type are Hooke's joints. Since a spherical hinge has three degrees of freedom and a Hooke's joint has only two degrees of freedom, it directly causes different motion modes of the upper and lower chains of the driving electric cylinder, manifested as relative rotation in the axial direction of the chain. To calculate the magnitude of this relative rotation speed, kinematic analysis of the UPRU-type chain is necessary, and the angular velocities of the upper and lower chains need to be calculated separately. After calculating the relative angular velocity, through the analysis of the passive screw mechanism, compensation can be implemented, and the calculation expression of the Jacobian matrix of the accurate and complete UPRU-Stewart platform can be obtained;

[0033] (2) The relative rotation of the UPRU-Stewart platform chain is a relative rotation between the cylinder and the push rod caused by the mechanical structure, which cannot be measured by the servo motor encoder of the electric cylinder. Therefore, in practical applications, the kinematics of the UPS-Stewart is often used for the UPRU-type Stewart platform, and the Jacobian matrix used is also the Jacobian matrix of the UPS-Stewart. This will cause theoretical errors in the research processes such as the optimization of the structural parameters of the UPRU-type platform, the exploration of the working space and singularity, the accuracy and error analysis, and the identification of dynamic parameters, greatly reducing the performance of the UPRU-Stewart platform;

[0034] (3) Its correctness is verified through simulation, greatly enhancing the application potential of the UPRU-Stewart platform in terms of precision and accuracy, laying a solid theoretical foundation for the improvement and efficient application of the comprehensive performance of the UPRU-Stewart platform system, and providing a theoretical basis for the characteristic analysis, structure and control system design of the UPRU-Stewart platform. Description of the Drawings

[0035] Figure 1 It is a schematic diagram of the Stewart platform in the calculation method of the Jacobian matrix of a Stewart platform according to the present invention.

[0036] Figure 2 It is a schematic diagram of the UPRU-type chain and the UPS-type chain of the Stewart platform in the calculation method of the Jacobian matrix of a Stewart platform according to the present invention.

[0037] Figure 3 It is a curve graph of the expected motion speed and the simulated motion speed calculated by the method disclosed in the UPRU-Stewart platform in the calculation method of the Jacobian matrix of a Stewart platform according to the present invention;

[0038] Figure 4It is the curve graph of the expected motion speed of the UPRU-Stewart platform and the simulated motion speed calculated by using the method disclosed in a method for calculating the Jacobian matrix of a Stewart platform according to the present invention;

[0039] Figure 5 It is the curve graph of the expected motion speed of the UPRU-Stewart platform and the simulated motion speed calculated by using the method disclosed in a method for calculating the Jacobian matrix of a Stewart platform according to the present invention;

[0040] Figure 6 It is the curve graph of the expected motion speed of the UPRU-Stewart platform and the simulated motion speed calculated by using the method disclosed in a method for calculating the Jacobian matrix of a Stewart platform according to the present invention;

[0041] Figure 7 It is the curve graph of the expected motion speed of the UPRU-Stewart platform and the simulated motion speed calculated by using the method disclosed in a method for calculating the Jacobian matrix of a Stewart platform according to the present invention;

[0042] Figure 8 It is the curve graph of the expected motion speed of the UPRU-Stewart platform and the simulated motion speed calculated by using the method disclosed in a method for calculating the Jacobian matrix of a Stewart platform according to the present invention;

[0043] Figure 9 It is the error curve graph of the expected motion speed and the simulated motion speed of the UPRU-Stewart platform disclosed in the present disclosure;

[0044] Figure 10 It is the error curve graph of the expected motion speed and the simulated motion speed of the UPRU-Stewart platform disclosed in the present disclosure;

[0045] Figure 11 It is the error curve graph of the expected motion speed and the simulated motion speed of the UPRU-Stewart platform disclosed in the present disclosure;

[0046] Figure 12 It is the error curve graph of the expected motion speed and the simulated motion speed of the UPRU-Stewart platform disclosed in the present disclosure;

[0047] Figure 13 It is the error curve graph of the expected motion speed and the simulated motion speed of the UPRU-Stewart platform disclosed in the present disclosure;

[0048] Figure 14 It is the error curve graph of the expected motion speed and the simulated motion speed of the UPRU-Stewart platform disclosed in the present disclosure;

[0049] Figure 15 is the velocity inverse kinematics error map of the UPRU-Stewart platform of the present disclosure using the Jacobian matrix of the UPS-Stewart platform;

[0050] Figure 16 is the velocity inverse kinematics error map of the UPRU-Stewart platform of the present disclosure using the Jacobian matrix of the UPS-Stewart platform;

[0051] Figure 17 is the velocity inverse kinematics error map of the UPRU-Stewart platform of the present disclosure using the Jacobian matrix of the UPS-Stewart platform;

[0052] Figure 18 is the velocity inverse kinematics error map of the UPRU-Stewart platform of the present disclosure using the Jacobian matrix of the UPS-Stewart platform;

[0053] Figure 19 is the velocity inverse kinematics error map of the UPRU-Stewart platform of the present disclosure using the Jacobian matrix of the UPS-Stewart platform;

[0054] Figure 20 is the velocity inverse kinematics error map of the UPRU-Stewart platform of the present disclosure using the Jacobian matrix of the UPS-Stewart platform. Detailed implementation manners

[0055] The following further describes the present invention according to the attached Figures 1 - 20 description:

[0056] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are some but not all of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0057] In the description of the present invention, it should be noted that the orientation or positional relationship indicated by the terms "center", "upper", "lower", "left", "right", "vertical", "horizontal", "inner", "outer", etc. is based on the orientation or positional relationship shown in the accompanying drawings. It is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and thus should not be construed as a limitation of the present invention.

[0058] In the description of the present invention, it should be noted that unless otherwise clearly specified and defined, the terms "installation", "connection", and "coupling" should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be directly connected or indirectly connected through an intermediate medium, and it can be the communication inside two components. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0059] The first embodiment of the present invention provides a method for calculating the Jacobian matrix of a Stewart platform on the basis of the prior art. The specific steps are as follows:

[0060] Step S10: Establish the inverse kinematics model of UPS-Stewart and the position inverse kinematics model of UPRU-Stewart respectively. The specific steps include:

[0061] Step S11: Establish the position inverse kinematics model of UPS-Stewart. The specific content is as follows:

[0062] As Figure 1 shown, a i and b i on the branch chain i satisfy the following formula:

[0063] a i = x + W R P P a i (1)

[0064] where x is the position vector of O p under {W}, W R P is the rotation matrix of {P} relative to, P a i represents the description of the vector under. a i and l i satisfy

[0065] a i = l i n i + b i (2)

[0066] where n i is the direction vector of the branch chain i. The angular velocity and angular acceleration of {P} are represented by ω and α. The generalized position is represented by q. When the platform posture is represented by ZYX Euler angles (α, β, and γ), q = [x α β γ] T. The relationships between the rotation matrix and Euler angles, angular velocity, angular acceleration, and the time derivatives of ZYX Euler angles are respectively

[0067]

[0068] where c* and s* represent cos(*) and sin(*) respectively.

[0069] With formula (1), the vector L of branch i i can be written as

[0070] L i = a i - b i (6)

[0071] Then, the length of branch i can be simply calculated as:

[0072]

[0073] Stack the lengths of the six branches into a vector, thus obtaining the vector form expression of the inverse kinematics model of the UPS-Stewart platform position:

[0074]

[0075] Meanwhile, the calculation formula of n i is:

[0076] n i = L i / l i (9)

[0077] Step S12, based on the inverse kinematics model of the UPS-Stewart position, establish the inverse kinematics model of the UPS-Stewart velocity, and the specific content is as follows:

[0078] Taking the derivative of formula (1) with respect to time, we get:

[0079]

[0080] Then project it onto the direction of branch i to obtain the inverse kinematics model of the UPS-Stewart velocity:

[0081]

[0082] Step S13, based on the inverse kinematics model of the UPS-Stewart velocity, obtain the Jacobian matrix of the UPS-Stewart inverse kinematics, and the specific content is as follows:

[0083] To obtain the Jacobian matrix of the UPS-Stewart platform, the equations (11) of the six joints are written in matrix form:

[0084]

[0085] where q is the generalized position of {P} in {W}, and J s is the Jacobian matrix of the inverse kinematics of the UPS-Stewart. Meanwhile, in this disclosure, it is also referred to as the "Jacobian matrix of l". When the attitude of {P} is represented by ZYX Euler angles (α, β, and γ), J s can be calculated by the following formula:

[0086] J s = J sω J p (13)

[0087] where,

[0088]

[0089] where, I 3×3 and O 3×3 represent the 3×3 identity matrix and zero matrix.

[0090] Step S14, based on the passive screw mechanism of the UPRU-type link, obtain the relationship between the passive screw angle and the passive additional length of the UPRU-type link. The specific content is as follows:

[0091] The difference between the UPRU link type and the UPS link type lies in the different connecting hinges at one end. As shown in Figure 2 and Figure 3 , one section of the UPS link is a spherical hinge, while both ends of the UPRU link are Hooke's joints. Due to the passive rotation characteristic of the Hooke's joint, when the UPRU link moves from position 1 to position 2, relative rotation will occur between the upper and lower parts of the electric cylinder in the axial direction n i . This can be obtained from the analysis of the degrees of freedom of the link. Due to the screw pair transmission between the lead screw and the screw nut of the electric cylinder, the relative rotation between the upper and lower parts of the electric cylinder will generate an additional displacement in the n i direction. Since this phenomenon is passively generated and is under the action of the screw motion pair, we call this phenomenon the "passive screw of the UPRU link". Due to the existence of the spherical hinge, the UPS-type link does not have the passive screw phenomenon. The relative rotation angle is called the passive screw angle, as shown in Figure 2 where Δθ i . And θ i can be calculated by the following formula for Δl i

[0092] θ i= cos -1 (v i ·k i ) (16)

[0093] The existence of the passive helix phenomenon makes the length of the branch chain not completely controlled by the electric cylinder. The part of the rod length determined by the electric cylinder is called the "controlled length of the branch chain", denoted by , and is represented by a square dot in Figure 2 . Δl i is the passive additional length, and is represented by the length between the triangular dot and the square dot in Figure 2 . The UPS type branch chain does not produce passive helix, so there is no triangular dot. Then the passive helix angle Δθ i and Δl i satisfy

[0094]

[0095] where p is the lead. However, a problem worthy of note is the positive and negative of Δθ i and Δl i . The magnitude of Δθ i can be obtained by subtracting the θ i at position 2 from that at position 1. The positive and negative of Δθ i should satisfy the right-hand helix rule, that is, the four fingers rotate from the k i of displacement 1 to the k i of position 2. If the thumb points to the positive direction of n i , it is positive, otherwise it is negative. The positive and negative of are determined not only by the positive and negative of Δθ i , but also by considering the helix direction of the helical pair in the electric cylinder. Let s represent the helix direction, 1 for right-handed and -1 for left-handed. Then the relationship between Δθ i and Δl i is updated to

[0096]

[0097] where: Δθ i is the passive helix angle of the UPRU type branch chain, Δl i is the passive additional length of the UPRU type branch chain, s is the helix direction, s = 1 for right-handed and s = -1 for left-handed, and p is the lead.

[0098] When calculating the inverse kinematics of the velocity or acceleration of UPRU-Stewart, the relationship between their derivatives will be used, that is

[0099]

[0100] where and represent the upper and lower parts of the branch chain in ni The difference in the [direction].

[0101] Step S15: Based on the relationship between the passive helix angle and the passive additional length of the UPRU-type branch chain, establish the inverse position kinematic model of UPRU-Stewart, and the specific content is as follows:

[0102] Due to the existence of the passive helix phenomenon of the UPRU branch chain, the length of the branch chain is not completely determined by the electric cylinder. The part of the rod length determined by the electric cylinder is called the "control length of the branch chain". Then, the inverse position kinematics of UPRU-Stewart is to solve the "control length of the branch chain". After knowing the operation mechanism of the passive helix phenomenon, it is natural to think that a compensation method can be used to solve the "control length of the branch chain":

[0103]

[0104] Through Equation (18), the "control length of the branch chain" can be calculated.

[0105] Step S20: Establish the relative angular velocity of the branch chain of the UPRU-Stewart platform through the inverse kinematic model of the UPS-Stewart and the inverse position kinematic model of the UPRU-Stewart. The specific steps are as follows:

[0106] Step S21: Based on the inverse position kinematic model of UPRU-Stewart, establish the inverse velocity kinematic model of UPRU-Stewart, and the specific content is as follows:

[0107] The passive helix phenomenon will not only have an impact on the position, but also have an impact on the velocity and acceleration dimensions. The velocity kinematics also requires compensation. By taking the derivative of the compensation calculation expression of the inverse position kinematics, that is, Equation (20), with respect to time, the compensation calculation expression of the inverse velocity kinematics can be obtained, that is

[0108]

[0109] Among them, can be solved according to Equation (19), where the key lies in is the angular velocity of the upper part of the branch chain relative to the lower part in the n i direction, that is:

[0110]

[0111] Here, we use "1" to represent the upper part of the branch chain i, and "2" to represent the lower part of the branch chain i. The angular velocity of the upper part of the branch chain i is expressed as ω 1i and the lower part is ω 2i . and The superscript "a" of i indicates the projection in the n i direction.

[0112] Step S22: Based on the inverse velocity kinematic model of UPRU-Stewart and the inverse position kinematic model of UPS-Stewart, the angular velocity of the lower branch chain of the UPRU-Stewart platform and the angular velocity of the upper branch chain of the UPRU-Stewart platform are obtained respectively, and the specific content is as follows:

[0113] Since the motion of the lower part of the i-th chain of UPRU-Stewart is exactly the same as that of the whole chain of UPS-Stewart, then ω 2i can be obtained by kinematic calculation of the chain of UPS-Stewart.

[0114] First, the angular velocity of the lower branch chain can be expressed as

[0115] ω 2i = ω ui u i + ω vi v i (22)

[0116] At the same time, ω 2i can also be obtained by differentiating Equation (2), and we get:

[0117]

[0118] Substitute Equation (22) into Equation (23), and multiply both sides of the formula by v i and u i respectively, and the calculation formulas for ω ui and ω vi are obtained:

[0119]

[0120] In the formula, can be obtained by calculating Equation (3).

[0121] Then, can be directly obtained by projecting ω 2i in the n i direction. At the same time, considering that v i ·n i = 0:

[0122]

[0123] For the upper part of the UPRU chain, the angular velocity can also be written in a form similar to Equation (22), but except for j i and ki There is a corresponding ω in the ji direction, and ω ki component, d i should also have an ω di component in the direction. This is because the upper part of the branch chain is connected to the moving platform through a Hooke hinge, and there may be a component of the angular velocity of the platform in the d i direction. Therefore, ω 1i can be written in the following form:

[0124] ω 1i = ω ji j i + ω ki k i + ω di d i (26)

[0125] where ω di = ω·d i .

[0126] Observing Equation (26), it can be seen that ω 2i can also be replaced by ω 1i , because the angular velocities of the upper and lower branch chains only differ in the axial direction, and the components in the normal direction of the branch chain must be equal. Moreover, the cross product in Equation (23) is only related to the normal component. Therefore,

[0127]

[0128] Similarly, substituting Equation (26) into Equation (27), and then dot-multiplying both sides of the formula by k i and j i respectively, the calculation formulas for ω ji and ω ki are obtained:

[0129]

[0130] Then, can be directly obtained by projecting ω 1i onto the n i direction. Considering that k i ·n i = 0:

[0131]

[0132] Step S23: Based on the angular velocity of the lower branch chain of the UPRU-Stewart platform and the angular velocity of the upper branch chain of the UPRU-Stewart platform, obtain the relative angular velocity of the branch chain of the UPRU-Stewart platform. The specific content is as follows:

[0133] According to step S23, the axial angular velocity of the upper part of the UPRU-Stewart branch chain i relative to the lower branch chain can be calculated as follows:

[0134]

[0135] Step S30, obtaining the Jacobian matrix of the URPU-Stewart based on the relative angular velocity of the UPRU-Stewart platform branch chain, the specific content is as follows:

[0136] Let ω di = ω·d i and substitute ω in Equation (29) ji into Equation (30), and at the same time consider (n i ·j i ) 2 +(n i ·d i ) 2 = 1:

[0137]

[0138] Similarly, substitute ω ui into Equation (29):

[0139]

[0140] Then substitute the above two equations, Equation (31) and Equation (32), into Equation (30), considering j i = k i ×d i , u i = v i ×c i and applying the scalar triple product, it is simplified to:

[0141]

[0142] where i.e., the derivative of Equation (1), is arranged in the form of matrix multiplication:

[0143]

[0144] Then substitute it back into Equation (33) and arrange it in the form of matrix multiplication:

[0145]

[0146] where

[0147]

[0148] Finally, stack the 6 from i = 1 to 6 and write it as Arrange it in the form of matrix multiplication and separate it from it

[0149]

[0150] Among them,

[0151]

[0152] Thus It can be calculated by the following formula

[0153]

[0154] J u is the "Jacobian matrix of l u ", that is, the Jacobian matrix of UPRU-Stewart:

[0155] J u =(J s -J Δ J p )(41)

[0156] Among them: J u is the Jacobian matrix of URPU-Stewart, J s is the Jacobian matrix of the inverse kinematics of UPS-Stewart, J Δ is the manifestation of the influence of the "passive screw phenomenon" on the movement of the UPRU-Stewart platform, and the expression is formula (39), J p is the conversion matrix from the moving platform velocity to the generalized velocity , and its expression is formula (15).

[0157] To verify the correctness of the calculation method of the present disclosure, in the python environment, referring to the Stewart platform structure parameters in Table 1, the velocity kinematics of the UPRU-Stewart platform based on the Jacobian matrix was compiled and verified by Adams simulation.

[0158] Table 1 Stewart platform structure parameters

[0159] Parameter Data Unit <![CDATA[Radius r of the hinge point of the moving platform P > 0.25172 m <![CDATA[Fixed base hinge point radius r W > 0.5612 m <![CDATA[The included angle a between adjacent hinge points of the fixed base W > 20.95 Degree (°) <![CDATA[The included angle α between adjacent hinge points of the moving platform P > 10.04 Degree (°) <![CDATA[Height h of the moving platform P > 0.7328377909 m Moving platform position q <![CDATA[(0,0,h P ,0,0,0)]]> m <![CDATA[Fixed base Hooke joints u1 and u2]]> (1，0，0) \ <![CDATA[Fixed base Hooke joints u3 and u4]]> (-0.5，0.866，0) \ <![CDATA[Fixed base Hooke joint u5 and u6]]> (-0.5，-0.866，0) \ <![CDATA[Hook joints j1 and j6 of the moving platform]]> (0.5，-0.866，0) \ <![CDATA[Hook joints j2 and j3 of the moving platform]]> (0.5，0.866，0) \ <![CDATA[Hook joints j4 and j5 of the moving platform]]> (-1，0，0) \

[0160] The Stewart platform structure parameters in Table 1 include the hinge point radii of the moving platform and the fixed base, the included angle between adjacent hinge points, and the position and velocity of the moving platform, as well as the Hooke joint direction under the initial position, that is, u i and j i , and these parameters are independent of the configuration of the Stewart platform branch chain.

[0161] Now, let the moving platform of UPRU-Stewart perform sinusoidal motion with an amplitude of 0.1 m and a period of 2 s in all directions. Then, at each moment, the Jacobian matrix of the UPRU-Stewart platform is calculated using the method of the present disclosure, and the joint velocities are calculated using this Jacobian matrix. At the same time, a model is built in Adams, and the calculated joint velocities are used as the driving input for simulation to obtain the motion data of the moving platform, such as Figures 3 - 8 the "expected" and "simulation" curves in

[0162] From Figures 3 - 8 the curves in Figures 9 - 14 it can be seen that the expected motion of the UPRU-Stewart moving platform is almost exactly the same as the simulation result in Adams. Subtracting the numerical values of the two gives the error, as shown in -8 It can fully demonstrate the correctness and stability of the method of the present disclosure that the accuracy of the inverse kinematics of the velocity of URPU-Stewart reaches the order of magnitude of 10

[0163] Furthermore, to illustrate the advantage of the method of the present disclosure compared with the approximate calculation using the UPS-Stewart Jacobian matrix, the following simulation of this approximation behavior is carried out: The inverse kinematics of the velocity of the UPS-Stewart platform with the same structural parameters as in Table 1 is written using Python. The expected motion of the platform is the same as the "expected" motion curve in Figures 3 - 8 The joint velocities are calculated and used as the driving input for the Adams model of the UPRU-Stewart platform. Subtracting the curve of the platform motion data measured in the Adams simulation from the "expected" motion curve gives the error magnitude of the inverse kinematics of the velocity of UPRU-Stewart approximated by the UPS-Stewart Jacobian matrix, as shown in Figures 15 - 20 From Figures 15 - 20 it can be seen that using the UPS-Stewart platform Jacobian matrix to approximate the inverse kinematics of the velocity of the UPRU-Stewart platform will cause a motion error of approximately the order of magnitude of 10 -3 This will produce non-negligible theoretical errors in the research processes such as the structural parameter optimization, workspace and singularity exploration, accuracy and error analysis, and dynamic parameter identification of the UPRU type platform, thus highlighting the significance of the method of the present disclosure.

[0164] Although the embodiments of the present invention have been disclosed as above, they are not limited to the applications listed in the specification and the embodiments. It can be fully applied to various fields suitable for the present invention. For those skilled in the art, additional modifications can be easily achieved. Therefore, without departing from the general concept defined by the claims and the equivalent scope, the present invention is not limited to the specific details and the illustrated examples herein.

Claims

1. A method for calculating the Jacobian matrix of a Stewart platform, characterized in that: include: Step S10, establishing a UPS-Stewart inverse kinematics model and a UPRU-Stewart position inverse kinematics model respectively; Step S20, establishing the relative angular velocity of the UPRU-Stewart platform branch chain through the UPS-Stewart inverse kinematics model and the UPRU-Stewart position inverse kinematics model; Step S30, obtaining the Jacobian matrix of the URPU-Stewart according to the relative angular velocity of the UPRU-Stewart platform branch.

2. The method for calculating the Jacobian matrix of the Stewart platform according to claim 1, characterized in that: The step S10 specifically includes: Step S11, establishing the UPS-Stewart position inverse kinematics model; Step S12, establishing the UPS-Stewart velocity inverse kinematics model based on the UPS-Stewart position inverse kinematics model; Step S13, obtaining the Jacobian matrix of UPS-Stewart inverse kinematics based on the UPS-Stewart velocity inverse kinematics model; Step S14, obtaining the relationship between the passive helical angle and the passive additional length of the UPRU type branch chain based on the passive helical mechanism of the UPRU type branch chain; Step S15, establishing the UPRU-Stewart position inverse kinematics model based on the relationship between the passive helix angle and the passive additional length of the UPRU type branch chain.

3. The method for calculating the Jacobian matrix of the Stewart platform according to claim 2, characterized in that: The relationship between the passive helix angle and the passive additional length of the UPRU type branch chain is shown in the following formula (1): Where: Δθ i is the passive helix angle of the UPRU type branch, Δl i is the passive additional length of the UPRU type branch chain, s is the rotation direction, s=1 is right-handed, s=-1 is left-handed, and p is the lead.

4. The method for calculating the Jacobian matrix of the Stewart platform according to claim 3, characterized in that: The step S20 specifically includes: Step S21, establishing a UPRU-Stewart velocity inverse kinematics model based on the UPRU-Stewart position inverse kinematics model; Step S22, based on the UPRU-Stewart velocity inverse kinematics model and the UPS-Stewart position inverse kinematics model, respectively obtain the angular velocity of the branch chain under the UPRU-Stewart platform and the angular velocity of the branch chain on the UPRU-Stewart platform; Step S23, obtaining the relative angular velocity of the branch chain on the UPRU-Stewart platform based on the angular velocity of the branch chain under the UPRU-Stewart platform and the angular velocity of the branch chain on the UPRU-Stewart platform.

5. The method for calculating the Jacobian matrix of the Stewart platform according to claim 4, characterized in that: The Jacobian matrix of the URPU-Stewart is shown in the following formula (2): I u =(J s -J Δ I p ) (2) Among them: J u is the Jacobian matrix of URPU-Stewart, J s is the Jacobian matrix of UPS-Stewart inverse kinematics, J Δ The passive spiral phenomenon affects the motion of the UPRU-Stewart platform. Then its expression is J p The velocity of the moving platform is the generalized velocity The transformation matrix is ​​expressed as

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