A method for dynamic modeling and motion input shaping design of joint flexible robotic arm

Abstract

This application relates to a method for dynamic modeling and motion input shaping design of a joint-flexible robotic arm. This method establishes a dynamic model of a joint-flexible robotic arm, designs an input shaping function by solving the robotic arm's dynamic characteristic parameters, and then combines the shaped robotic arm's handling stroke with the inverse kinematics relationship to design the robotic arm's joint stroke. The method develops a dynamic modeling method for a rigid-rod-flexible joint robotic arm that satisfies the assumption that the drive motor control error is negligible, reduces the scale of solving the robotic arm's motion equations, and proposes a vibration-suppressing input shaping motion design method based on the numerical solution of inverse kinematics. This method reduces the low-frequency vibration generated by the handling system composed of the robotic arm and the object being handled during handling operations, thereby facilitating the precise positioning of the robotic arm and shortening the posture stabilization time.

Description

Technical Field

[0001] The present application relates to the technical field of robot arm posture control, and in particular to a method for dynamic modeling and motion input shaping design of a joint-flexible robot arm. Background Art

[0002] Space manipulators are essential components for spacecraft on-orbit missions. Space structure handling technologies based on these arms play an irreplaceable role in supporting rendezvous and docking, space station construction, and satellite launches. Due to their large range of motion, light weight, and high payload capacity, these manipulators possess remarkable flexible motion characteristics. Furthermore, these manipulators are commonly used for on-orbit missions such as module capture and transfer, equipment installation, repair, and replacement, payload manipulation, astronaut-assisted transfer, and extravehicular status monitoring, all of which require extremely high precision.

[0003] With respect to the above-mentioned related technologies, the inventors believe that the following defects exist: during the handling process, the robotic arm and the object being handled together constitute the handling system. In the process of dynamic modeling of the robotic arm, the actual control accuracy of the robotic arm drive motor is relatively high. In order to reduce the scale of solving the dynamic model equations and save computing costs, it is necessary to introduce the assumption of ignoring the joint angular displacement control error. For a spatial robotic arm, the stiffness of its connecting rod itself is very large and can be regarded as a rigid body relative to the connecting rod hinge. Therefore, the elastic link of the robotic arm mainly comes from the flexible hinge. Due to the existence of the elastic link, when the system is excited to perform the handling operation, the object being handled and the robotic arm together form low-frequency vibrations, which is not conducive to the precise positioning of the robotic arm. Moreover, the low-frequency vibration has a long attenuation time, which affects the robotic arm with fast posture positioning requirements. Therefore, it is necessary to implement appropriate vibration suppression measures for the handling system to reduce the decrease in the positioning accuracy of the robotic arm caused by vibration and shorten the posture stabilization time. Summary of the Invention

[0004] In response to the technical problems in the above-mentioned background technology, this application proposes a method for dynamic modeling and motion input shaping design of a joint flexible robotic arm.

[0005] In a first aspect, the present application proposes a method for dynamic modeling and motion input shaping design of a joint flexible robotic arm, the method comprising the following steps:

[0006] S1: Establish a dynamic model of a flexible joint manipulator;

[0007] S2: Design input shaping function based on the dynamic characteristic parameters of the robot arm;

[0008] S3: Use input shaping function to design input shaping of the robot arm’s handling stroke;

[0009] S4: Calculate the joint stroke after reshaping based on the inverse kinematics numerical solution method.

[0010] By adopting the above technical solution, the established robot arm dynamics model is used to solve the inherent vibration characteristic parameters, the numerical solution of the inverse kinematics relationship of the robot arm's handling stroke is combined with the input shaping method, and the joint angular displacement stroke is designed. The low-frequency vibration generated by the handling system composed of the rigid rod-flexible joint robot arm and the handled object during the handling operation is reduced, thereby facilitating the precise positioning of the robot arm and shortening the posture stabilization time.

[0011] Preferably, in S1, the robotic arm is a branchless n-link robotic arm, and the links are connected by rotating joints driven by a driving mechanism; the robotic arm dynamics model is established based on three assumptions: the robotic arm links are regarded as rigid bodies, the articulated joints are flexible, and the joint angular displacement control error of the driving mechanism is ignored, that is, the robotic arm has n joints and degrees of freedom.

[0012] Preferably, the S1 specifically includes:

[0013] S11: For an n-link robotic arm, its motion is described using n-dimensional generalized coordinates θ, where represents the angular displacement vector of the hinge joint of the robotic arm, and the kinetic energy of the rigid arm is:

[0014]

[0015] Where M(θ) is the inertia matrix of the rigid link of the manipulator with respect to the current configuration θ.

[0016] S12: Calculate the total potential energy of the system, i.e. the elastic potential energy of the flexible joint

[0017] S13: Let the Lagrangian function be Will and Substituting into the Lagrange equation, we can obtain the driving angular displacement θ of the driving mechanism in the θ coordinate representation. m Dynamic equations of the manipulator in motion.

[0018] Preferably, in S13, the dynamic equation of the robotic arm is:

[0019]

[0020] in, is the force / torque vector including the Coriolis force and centripetal force.

[0021] Preferably, the S2 specifically includes:

[0022] According to θ=θ0 and the dynamic parameters of the connecting rod, the inertia matrix M(θ0) of the manipulator in the initial configuration (ie, t=0) is obtained, combined with the manipulator joint stiffness matrix K s , by solving the generalized eigenvalue problem of the system:

[0023] K s φ=ω 2 Mφ

[0024] Where φ is the eigenvector and ω is the corresponding natural frequency. Calculate the natural frequency under this configuration, and then set the damping ratio of each order to 0.01 to obtain the corresponding shaper formula.

[0025] Preferably, the S3 specifically includes:

[0026] S31: According to the starting point p0 and end point p of the end of the robot arm n , the velocity function v(t) is selected as a typical uniform acceleration, uniform speed and uniform deceleration trapezoidal speed change process;

[0027] S32: Reshape the function v(t) to obtain the reshaped velocity function

[0028] S33: Use numerical quadrature method to calculate the velocity Get the Cartesian coordinates of the end of the robotic arm after shaping:

[0029]

[0030] in, is the unit vector from the starting point to the end point of the robot arm; the upper limit of the loop variable i after shaping is n(1+2mΔT / T), at which time the movement speed of the robot arm end is zero;

[0031] Since the shaped velocity function satisfies the condition

[0032] v(t)=0, when t<0 or t>T

[0033] That is, the function v(t) is equal to zero outside the interval [0,T], and the shaper constant also satisfies the corresponding shaper formula, so we can get

[0034]

[0035] The shaped velocity function is then integrated, and the shaped stroke is used to make the end of the robotic arm reach the end point before the shaping.

[0036] Preferably, in S4, the inverse kinematics numerical solution method is based on the Jacobian matrix, specifically the Jacobian transpose and Jacobian pseudo-inverse forms.

[0037] Preferably, in S4, the inverse kinematics numerical solution method specifically includes:

[0038] Based on the Jacobi pseudo-inverse form, given x d As the desired configuration of the manipulator, the forward kinematic function:

[0039] x=f(q)={f j (q)},j=1,2,…,6

[0040] Expanded to about configuration q d Taylor series of , and retaining the first-order small quantity, rearrange it to get:

[0041] q=J -1 (q d )[f(q)-f(q d )]+q d

[0042] J(q d ) is the Jacobian matrix. When the Jacobian matrix J is not a square matrix or is irreversible, the inverse matrix J in the formula -1 Using its pseudo-inverse Instead,

[0043] Then estimate the initial variable q0, according to the iterative algorithm:

[0044]

[0045] Form an estimation sequence q0,q1,q2,…; after the iteration converges, the desired joint coordinate q can be obtained d .

[0046] This application establishes the dynamic equations of an n-link robotic arm based on the three assumptions that the robotic arm links are regarded as rigid bodies, the flexibility of the articulated joints is considered, and the control error of the angular displacement of the joints of the driving mechanism is ignored. The joint angular displacement stroke is designed by applying the numerical solution of the inverse kinematics relationship, and then combined with the input shaping method, a rigid rod-flexible joint robotic arm dynamic modeling method is proposed that meets the assumption that the control error of the driving motor can be ignored, reducing the scale of solving the robotic arm motion equations, and proposing a vibration suppression input shaping motion design method based on the numerical solution of inverse kinematics, thereby reducing the low-frequency vibration generated by the handling system composed of the rigid rod-flexible joint robotic arm and the handled object during the handling operation, thereby facilitating the precise positioning of the robotic arm and shortening the posture stabilization time. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] The accompanying drawings are included to provide a further understanding of the embodiments and are incorporated into and constitute a part of this specification. The accompanying drawings illustrate the embodiments and, together with the description, serve to explain the principles of the present application. Other embodiments and many of the expected advantages of the embodiments will be readily apparent as they become better understood by reference to the following detailed description. The elements of the drawings are not necessarily to scale with respect to each other. Like reference numerals designate corresponding similar parts.

[0048] Figure 1 This is a flow chart of a method for dynamic modeling and motion input shaping design of a joint flexible robotic arm in this application.

[0049] Figure 2 It is a schematic diagram of an n-link flexible joint series robot arm model in one embodiment of the present application.

[0050] Figure 3 This is a schematic diagram of a 7-DOF robotic arm in one embodiment of the present application.

[0051] Figure 4 It is a schematic diagram of a 7-DOF robotic arm member coordinate system in one embodiment of the present application.

[0052] Figure 5 It is a schematic diagram of the forward / inverse kinematic relationship in one embodiment of the present application.

[0053] Figure 6 This is a schematic diagram of the straight line trajectory of the starting and ending points of the end of the robotic arm in S104 in one embodiment of the present application.

[0054] Figure 7 Schematic diagram of the trapezoidal velocity function in S104 and its trajectory changing with time in one embodiment of the present application.

[0055] Figure 8 This is a schematic diagram of the angular displacement function of each joint of the robotic arm changing with time in S104 in one embodiment of the present application.

[0056] Figure 9 It is a schematic diagram of the terminal velocity function after shaping in S104 in one embodiment of the present application.

[0057] Figure 10 This is a schematic diagram of the angular displacement function of each joint of the robotic arm changing with time after the plastic surgery in S104 in one embodiment of the present application.

[0058] Figure 11 It is a schematic diagram of a curve showing the change of the front and rear joint tracking errors with respect to time in one embodiment of the present application. DETAILED DESCRIPTION

[0059] The present application will be further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely for the purpose of explaining the relevant invention and are not intended to limit the invention. It should also be noted that, for ease of description, only portions relevant to the relevant invention are shown in the accompanying drawings.

[0060] It should be noted that, in the absence of conflict, the embodiments and features of the embodiments in this application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0061] Figure 1 A flowchart of a method for dynamic modeling and motion input shaping design of a joint flexible robotic arm of the present application is shown. In conjunction with the reference figure, the method specifically includes the following steps:

[0062] S1: Establish a dynamic model of a flexible joint manipulator;

[0063] S2: Design input shaping function based on the dynamic characteristic parameters of the robot arm;

[0064] S3: Use input shaping function to design input shaping of the robot arm’s handling stroke;

[0065] S4: Calculate the joint stroke after reshaping based on the inverse kinematics numerical solution method.

[0066] Among them, in S1, the robotic arm is a branchless n-link robotic arm, and the links are connected by rotating joints driven by the drive mechanism; the robotic arm dynamics model is based on three assumptions: the robotic arm links are regarded as rigid bodies, the articulated joints are flexible, and the angular displacement control error of the drive mechanism joints is ignored, that is, the robotic arm has n joints and degrees of freedom.

[0067] Among them, S1 specifically includes:

[0068] S11: For an n-link robotic arm, its motion is described using n-dimensional generalized coordinates θ, where represents the angular displacement vector of the robotic arm hinge, and the kinetic energy of the rigid arm is:

[0069]

[0070] Where M(θ) is the inertia matrix of the rigid link of the manipulator with respect to the current configuration θ.

[0071] S12: Calculate the total potential energy of the system, i.e. the elastic potential energy of the flexible joint

[0072] S13: Let the Lagrangian function be Will and Substituting into the Lagrange equation, we can obtain the driving angular displacement θ of the driving mechanism in the θ coordinate representation. m Dynamic equations of the manipulator in motion.

[0073] Wherein, in S13, the dynamic equation of the robotic arm is:

[0074]

[0075] is the force / torque vector including the Coriolis force and centripetal force.

[0076] Among them, S2 specifically includes:

[0077] According to θ=θ0 and the dynamic parameters of the connecting rod, the inertia matrix M(θ0) of the manipulator in the initial configuration (ie, t=0) is obtained, combined with the manipulator joint stiffness matrix K s , by solving the generalized eigenvalue problem of the system:

[0078] K s φ=ω 2 Mφ

[0079] Where φ is the eigenvector and ω is the corresponding natural frequency. Calculate the natural frequency under this configuration, and then set the damping ratio of each order to 0.01 to obtain the corresponding shaper formula.

[0080] S3 specifically includes:

[0081] S31: According to the starting point p0 and end point p of the end of the robot arm n , the velocity function v(t) is selected as a typical uniform acceleration, uniform speed and uniform deceleration trapezoidal speed change process;

[0082] S32: Reshape the function v(t) to obtain the reshaped velocity function

[0083] S33: Use numerical quadrature method to calculate the velocity Get the Cartesian coordinates of the end of the robotic arm after shaping:

[0084]

[0085] in, is the unit vector from the starting point to the end point of the robot arm; the upper limit of the loop variable i after shaping is n(1+2mΔT / T), at which time the movement speed of the robot arm end is zero;

[0086] Since the shaped velocity function satisfies the condition

[0087] v(t)=0, when t<0 or t>T

[0088] That is, the function v(t) is equal to zero outside the interval [0,T], and the shaper constant also satisfies the corresponding shaper formula, so we can get

[0089]

[0090] The shaped velocity function is then integrated, and the shaped stroke is used to make the end of the robotic arm reach the end point before the shaping.

[0091] Among them, in S4, the numerical solution method of the inverse kinematics relationship is based on the Jacobian matrix, specifically the Jacobian transpose and Jacobian pseudo-inverse forms.

[0092] Specifically, the numerical solution method of S4 inverse kinematics relationship includes:

[0093] Based on the Jacobi pseudo-inverse form, given x d As the desired configuration of the manipulator, the forward kinematic function:

[0094] x=f(q)={f j (q)},j=1,2,…,6

[0095] Expanded to about configuration q d Taylor series of , and retaining the first-order small quantity, rearrange it to get:

[0096] q=J -1 (q d )[f(q)-f(q d )]+q d

[0097] J(q d ) is the Jacobian matrix. When the Jacobian matrix J is not a square matrix or is irreversible, the inverse matrix J in the formula -1 Using its pseudo-inverse Instead,

[0098] Then estimate the initial variable q0, according to the iterative algorithm:

[0099]

[0100] Form an estimation sequence q0,q1,q2,… After the iteration converges, the desired joint coordinate q can be obtained d .

[0101] In a specific embodiment, the following describes a method for dynamic modeling and motion input shaping design of a joint flexible robotic arm of the present application:

[0102] S101: Establish a dynamic model of a flexible joint manipulator;

[0103] Theoretical modeling of robotic arm dynamics considering joint flexibility: Consider a branchless n-link robotic arm, where the links are connected by motor-driven revolute joints, meaning the arm has n joints and degrees of freedom. The links and joints are numbered sequentially starting from the base. The joint between link i and link i-1 is denoted as joint i (i = 1, 2, ..., n), and its corresponding drive motor is denoted as Motor #i, as follows: Figure 2 shown.

[0104] The dynamic model of the robotic arm is established, satisfying the following assumptions:

[0105] (1) Compared with the hinge joints of the space robot arm connecting rod, the stiffness of the rod arm itself is very large and can be approximated as a rigid body.

[0106] (2) For a robotic arm with flexible joints, according to Spong theory, the flexible connection between the motor rotor and the connecting rod is simplified to a stiffness coefficient k s,i A linear torsion spring with no inertia, such as Figure 2 shown.

[0107] (3) The actual robot arm joint drive motor has high control accuracy. Ignoring the influence of the robot arm joint angular displacement control error, the scale of solving the motion equation is reduced. The angular displacement θ of the motor-driven joint is m This is the designed angular displacement.

[0108] For an n-link elastic hinge manipulator, the n-dimensional generalized coordinate θ is used to describe its motion, where represents the angular displacement vector of the robotic arm hinge, and the kinetic energy of the rigid arm is:

[0109]

[0110] in

[0111] M(θ)={m ij (θ)},i,j=1,2,…,n

[0112] is the inertia matrix of the rigid link of the manipulator with respect to the current configuration θ, and is always a positive definite matrix. The elements of the matrix M are:

[0113]

[0114] Where, J p is the pseudo-inertia matrix of rod p; T p is the transformation matrix from the rod p coordinate system to the base coordinate system.

[0115] Ignoring the influence of gravity, the total potential energy of the system comes from the elastic potential energy of the flexible joint:

[0116]

[0117] in, represents the stiffness matrix at the hinge, K s =diag{k s,i}.

[0118] The Lagrangian function is According to Lagrange equation:

[0119]

[0120] Will and Substituting into the Lagrange equation, we can obtain the driving angular displacement θ of the driving mechanism in the θ coordinate representation. m The dynamic equation of the manipulator in motion is:

[0121]

[0122] in, is the force / torque vector including the Coriolis force and centripetal force.

[0123] Take the 7-DOF robotic arm as an example. Figure 3 Based on the modified DH modeling method, the member coordinate system is established as follows Figure 4 As shown, the connecting rod length parameters are:

[0124] a0=0.28, d2=0.31, d3=0.31, a3=1.9, d4=0.27

[0125] a4=1.9, d5=0.27, d6=0.31, d7=0.65, d8=0.3

[0126] The unit is m. Each joint is embedded with its rotation direction, that is, the joint coordinate z i The linear torsion spring of the axis rotation has a stiffness of k s,i =2.0×10 5 Nm / rad. The relevant dynamic parameters of each rod are listed in Table 1.

[0127] Table 1 Center of mass position, mass and moment of inertia relative to the center of mass in the rod coordinate system

[0128] Link Center of mass (x,y,z) quality <![CDATA[Moment of inertia diag[I xx ,I yy ,I zz > 1 [0 -0.1 0] 50 diag[2 2 200] 2 [0 0.1 0] 50 diag[2 2 200] 3 [2 0 0] 100 diag[5 300 500] 4 [2 0 0.5] 100 diag[5 300 500] 5 [0 0 0.5] 50 diag[2 2 200] 6 [0 0 0.5] 50 diag[2 2 200] 7 [0 0.5 1] 100 diag[2 2 200]

[0129] Table 2 DH parameters of seven-DOF manipulator

[0130] Link <![CDATA[a i-1 ]]> <![CDATA[α i-1 ]]> <![CDATA[d i ]]> <![CDATA[θ i ]]> 1 0 0 0.28 <![CDATA[θ1]]> 2 0 90° 0.31 <![CDATA[θ2]]> 3 0 -90° 0.31 <![CDATA[θ3]]> 4 1.9 0 0.275 <![CDATA[θ4]]> 5 1.9 0 0.275 <![CDATA[θ5]]> 6 0 -90° 0.31 <![CDATA[θ6]]> 7 0 90° 0.65 <![CDATA[θ7]]>

[0131] S102: Designing an input shaping function based on the dynamic characteristic parameters of the robot arm;

[0132] During the transfer process, the manipulator's configuration changes continuously, and the inertia matrix M(θ) also changes accordingly. Taking the inertia matrix M(θ0) of the manipulator in its initial configuration (i.e., t=0) as a reference, assuming θ=θ0, and based on the connecting rod dynamic parameters listed in Table 1, we can obtain:

[0133]

[0134] Combined with the manipulator stiffness matrix K s , by solving the generalized eigenvalue problem of the system:

[0135] K s φ=ω 2 Mφ

[0136] Where φ is the eigenvector and ω is the corresponding natural frequency. Calculate the natural frequency for this configuration:

[0137] ω1=5.9772rad / s ω2=10.2607rad / s ω3=14.0052rad / s ω4=17.7754rad / s

[0138] ω5=19.0041rad / s ω6=24.9942rad / s ω7=39.8862rad / s

[0139] Then take the damping ratio of each order as 0.01, and obtain the corresponding shaper formula:

[0140]

[0141] Where m = 7, parameter C = 1 / (1 + a1 + a2 + ... + a 2m ). The input shaping parameters related to the above formula are as follows:

[0142] a1=0.0331, a2=2.1967, a3=0.2628, a4=2.7521, a5=0.4328, a6=3.1472,

[0143] a7=0.3414,a8=3.0487,a9=0.5384,a 10 =2.4602,a 11 =0.4423,a 12 =1.8253,

[0144] a 13 =0.1431,a 14 =0.7845,ΔT=0.092s

[0145] In specific applications, the input shaping object is the driving angular displacement function θ m(t), can be obtained by the inverse kinematics numerical solution method.

[0146] S103: Using the input shaping function to perform input shaping design on the handling stroke of the robot arm;

[0147] For a robotic arm with an n-DOF motion mechanism, the motion stroke input shaping process based on the inverse kinematics numerical solution is as follows:

[0148] (1) According to the starting point p0 and the end point p of the end of the robot arm n , the velocity function v(t) is selected as a typical uniform acceleration, uniform speed and uniform deceleration trapezoidal speed change process;

[0149] (2) Reshape the function v(t) to obtain the reshaped velocity function

[0150] (3) Using numerical quadrature method, the velocity Get the Cartesian coordinates of the end of the robotic arm after shaping:

[0151]

[0152] in, is the unit vector from the starting point to the end point of the robot arm; the upper limit of the loop variable i after shaping is n(1+2mΔT / T), at which time the movement speed of the robot arm end is zero;

[0153] Since the shaped velocity function satisfies the condition

[0154] v(t)=0, when t<0 or t>T

[0155] That is, the function v(t) is equal to zero outside the interval [0,T], and the shaper constant also satisfies the corresponding shaper formula, so we can get

[0156]

[0157] This shows that by integrating the shaped velocity function, the running distance obtained is the same as the result without shaping. The shaped stroke can ensure that the end of the robot arm reaches the same end point.

[0158] S104: Calculate the joint stroke after reshaping based on the inverse kinematics numerical solution method.

[0159] Knowing the position and posture of the end of the robotic arm (i.e., the posture matrix ) The process of solving the joint coordinate q is called the inverse kinematics problem, and vice versa is the forward kinematics problem. The relationship is as follows: Figure 5As shown. Unlike forward kinematics problems, inverse kinematics problems have the problem of uniqueness and existence of solutions. To solve inverse kinematics problems, closed solutions or numerical solutions can be used. The closed solution method usually requires the manipulator configuration to meet the Pieper criterion. The numerical method solves the given x by iterative solution. d Q d , which has better versatility. The solution process is mainly based on the Jacobian matrix, which includes Jacobian transpose, Jacobian pseudo-inverse and other forms.

[0160] Take the Jacobi pseudo-inverse form as an example. Given x d As the desired configuration of the manipulator, the forward kinematic function:

[0161] x=f(q)={f j (q)},j=1,2,…,6

[0162] Expanded to about configuration q d Taylor series of , and retaining the first-order small quantity, we can get:

[0163] f(q)=f(q d )+J(q d )(qq d )

[0164] where x d =f(q d );J(q d ) is the Jacobian matrix. Rearrange it to get:

[0165] q=J -1 (q d )[f(q)-f(q d )]+q d

[0166] When the Jacobian matrix J is not a square matrix or is not invertible, the inverse matrix J in the formula -1 Using its pseudo-inverse Instead,

[0167] Then estimate the initial variable q0, according to the iterative algorithm:

[0168]

[0169] Form an estimation sequence q0,q1,q2,… After the iteration converges, the desired joint coordinate q can be obtained d .

[0170] In order to compare the joint travel before and after the reshaping, the inverse kinematics joint coordinates of the robot arm without the reshaping are directly calculated:

[0171] For example Figure 3 The robotic arm shown in the figure has the cartesian coordinates of the starting point and the ending point of the actuator end as S(-2,0,0) and E(-2,3,0) respectively. The pose matrix corresponding to the robotic arm is:

[0172]

[0173] By solving the inverse kinematics function, the coordinates of the corresponding joint space are obtained:

[0174] θ0={0.6732, 1.0072, 1.4836, 2.0047, -0.3468, 1.0072, 2.4684}

[0175] θ e ={2.8296,-0.5992,0.0107,-1.0536,1.0430,0.5992,-2.8296}

[0176] The robot arm is in two positions T at the starting point and the end point s ,T e Between, design the path along the straight line in Cartesian space, such as Figure 6 As shown. Along the straight path, the velocity is in the form of a trapezoidal distribution. Given that the maximum velocity of the end of the robot is v m =0.2m / s, operation cycle T=20s, along The running distance in the direction is 3m. The trapezoidal velocity function and its trajectory changing with time are as follows Figure 7 As shown. According to the inverse kinematics relationship

[0177]

[0178] Solve and obtain the angular displacement function of each joint of the robot arm that has not been input into the plastic design as time changes, such as Figure 8 During the transfer process, the roll, pitch, and yaw angles of the end member of the robotic arm are always 0.

[0179] Re-pass

[0180]

[0181] Processing, re-calculation of inverse kinematics, to obtain the terminal velocity function after shaping, and the angular displacement function of each joint of the manipulator over time are respectively as follows: Figure 9 、 Figure 10 shown.

[0182] In a specific embodiment, the control effect analysis of the joint flexible manipulator dynamics modeling and motion input shaping design method of the present application is carried out: Figure 3The 7-DOF manipulator shown in the figure was numerically modeled and simulated using Matlab / Simulink. A corresponding Simulink simulation program was established based on the manipulator's dynamic model. The Simulink simulation time step was set to 0.01s, and the solver was set to automatic selection mode. The joint vibration response was calculated as the joint angular displacement tracking error:

[0183] e=θ-θ d

[0184] The vibration response of the manipulator with and without input shaping control was numerically simulated and analyzed to obtain the joint vibration response. Figure 11 The solid and dotted lines are shown in .

[0185] The peak values ​​of the errors (absolute values) when input shaping is applied and when no input shaping is applied to the i-th joint are respectively Define relative error disturbance The relative disturbances corresponding to each joint are shown in Table 3. As can be seen from Table 3, the relative disturbances of each joint are reduced to more than 50% of the former.

[0186] Table 3 Relative disturbance of joints after applying control

[0187] joint 1 2 3 4 5 6 7 Relative disturbance 0.4953 0.3486 0.4429 0.4158 0.3681 0.3784 0.3432

[0188] The above describes the specific embodiments of the present application, but the scope of protection of the present application is not limited thereto. Any changes or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in this application should be included in the scope of protection of the present application. Therefore, the scope of protection of the present application should be based on the scope of protection of the claims.

[0189] In the description of this application, it should be understood that the terms "upper", "lower", "inside", "outside", etc., indicating orientations or positional relationships, are based on the orientations or positional relationships shown in the accompanying drawings. They are only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the devices or elements referred to must have a specific orientation, be constructed and operated in a specific orientation. Therefore, they cannot be understood as limiting this application. The word "comprising" does not exclude the presence of elements or steps not listed in the claims. The word "one" or "an" preceding an element does not exclude the presence of multiple such elements. The simple fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used for improvement. Any reference signs in the claims should not be interpreted as limiting the scope.

Claims

1. A method for dynamic modeling and motion input shaping design of a joint flexible robotic arm, characterized by: The method comprises the following steps: S1: Establish a dynamic model of a flexible joint manipulator; S2: Design input shaping function based on the dynamic characteristic parameters of the robot arm; S3: Use input shaping function to design input shaping of the robot arm’s handling stroke; S4: Calculate the joint stroke after shaping based on the numerical solution method of inverse kinematics relationship. The numerical solution method of inverse kinematics relationship is based on the Jacobian matrix, specifically the Jacobian transpose and Jacobian pseudo-inverse forms, specifically including: Based on the Jacobi pseudo-inverse form, given x d As the desired configuration of the manipulator, the forward kinematic function: x=f(q)={f j (q)},j=1,2,…,6 Expanded to about configuration q d Taylor series of , and retaining the first-order small quantity, rearrange it to get: q=J -1 (q d )[f(q)-f(q d )]+q d J(q d ) is the Jacobian matrix. When the Jacobian matrix J is not a square matrix or is irreversible, the inverse matrix J in the formula -1 Using its pseudo-inverse Instead, Then estimate the initial variable q0, according to the iterative algorithm: Form an estimation sequence q0,q1,q2,…; after iterative convergence, obtain the desired joint coordinate q d .

2. The method for dynamic modeling and motion input shaping design of a joint flexible robotic arm according to claim 1, characterized in that: In S1, the robotic arm is a branchless n-link robotic arm, and the links are connected by rotating joints driven by a driving mechanism; the robotic arm dynamics model is established based on three assumptions: the robotic arm links are regarded as rigid bodies, the articulated joints are flexible, and the joint angular displacement control error of the driving mechanism is ignored, that is, the robotic arm has n joints and degrees of freedom.

3. The method for dynamic modeling and motion input shaping design of a joint flexible robotic arm according to claim 2, characterized in that: Said S1 specifically includes: S11: For an n-link robotic arm, its motion is described using n-dimensional generalized coordinates θ, where represents the angular displacement vector of the hinge joint of the robotic arm, and the kinetic energy of the rigid arm is: Where M(θ) is the inertia matrix of the rigid link of the manipulator with respect to the current configuration θ; S12: Calculate the total potential energy of the system, i.e. the elastic potential energy of the flexible joint S13: Let the Lagrangian function be Will and Substituting into the Lagrange equation, we can obtain the driving angular displacement θ of the driving mechanism in the θ coordinate representation. m Dynamic equations of the manipulator in motion.

4. The method for dynamic modeling and motion input shaping design of a joint flexible robotic arm according to claim 3, characterized in that: In S13, the dynamic equation of the robotic arm is: in, is the force / torque vector including the Coriolis force and centripetal force.

5. The method for dynamic modeling and motion input shaping design of a joint flexible robotic arm according to claim 1, characterized in that: The S2 specifically includes: According to θ=θ0 and the dynamic parameters of the connecting rod, the inertia matrix M(θ0) of the manipulator in the initial configuration (ie, t=0) is obtained, combined with the manipulator joint stiffness matrix K S , by solving the generalized eigenvalue problem of the system: in, is the eigenvector, ω is the corresponding natural frequency; calculate the natural frequency under this configuration, and then take the damping ratio of each order as 0.01 to obtain the corresponding shaper formula.

6. The method for dynamic modeling and motion input shaping design of a joint flexible robotic arm according to claim 1, characterized in that: The S3 specifically includes: S31: According to the starting point p0 and end point p of the end of the robot arm n , the velocity function v(t) is selected as a typical uniform acceleration, uniform speed and uniform deceleration trapezoidal speed change process; S32: Reshape the function v(t) to obtain the reshaped velocity function S33: Use numerical quadrature method to calculate the velocity Get the Cartesian coordinates of the end of the robotic arm after shaping: in, is the unit vector from the starting point to the end point of the robot arm; the upper limit of the loop variable i after shaping is n(1+2mΔT / T), at which time the movement speed of the robot arm end is zero; Since the shaped velocity function satisfies the condition v(t)=0, when t<0 or t>T That is, the function v(t) is equal to zero outside the interval [0,T], and the shaper constant also satisfies the corresponding shaper formula, so we can get The shaped velocity function is then integrated, and the shaped stroke is used to make the end of the robotic arm reach the end point before the shaping.

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