Modelling method and system for solving rope flexible deformation vibration problem of rope-driven robot arm

Abstract

This invention provides a modeling method and system for solving the problem of rope flexible deformation and vibration in a rope-driven robotic arm. The method includes: establishing a mapping relationship between the rope and joint angles; establishing the elastic potential energy of the rope deformation based on the changes in rope winding and unwinding driven by a motor-driven winch and the current rope mapping length at the joints; defining the kinetic and potential energy of the joint rotation and arm motion of the motor-driven rope-driven robotic arm, and constructing a Lagrangian function; deriving the dynamic equations of the rope-driven robotic arm system based on the second kind of Lagrangian equation, which relates the rope change control input to the mapping relationship between the robotic arm joint angles; optimizing the model, proving its stability, and verifying the effectiveness and reliability of the algorithm. This invention solves the technical problems of nonlinear characteristics such as flexible vibration, friction, rope deformation, and hysteresis in circuits, rope flexibility, and robotic arm systems, which lead to servo defects in robotic arm reliability, low accuracy in tracking predetermined desired trajectories, and high-frequency vibrations.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of robot motion and control technology, in particular to a rope-driven manipulator, a modeling method and system test for solving the problems of rope deformation and flexible vibration. BACKGROUND

[0002] At present, one of the key difficulties in constructing a large-torque robot is to design a joint module that meets the requirements. The joint module determines a series of important characteristics of the robot, such as the motion range, positioning accuracy, work stability, joint compliance, and load capacity. In order to achieve the design requirements of integration, modularity, and convenient replacement of the robot, people often adopt the design concept of modular rotary joints to integrate driving, transmission, sensing, control, and other technologies in the joint space, creating a highly integrated electromechanical component. The traditional robot structure is mainly composed of modular rotary joints and metal or carbon fiber composite pipe arms, and the joint interior integrates torque sensors, encoders, harmonic reducers, servo motors, motor control boards, and other components. The high integration leads to a crowded joint space, increases the joint volume and mass, and makes the design more complex. This kind of rigid manipulator composed of traditional joint modules cannot meet the requirements of long distance, light weight, large adjustable stiffness, and large extension ratio.

[0003] The central nervous system adjusts the mechanical properties of the limbs according to the task requirements by simultaneously activating the antagonistic muscles around the joints, including maintaining posture and limb movement. The biological skeletal muscle system is different from the traditional robotic system, and the natural human motion is more smooth, agile and stable than any existing robot. We can try to imitate the skeletal muscle system to break through the limitations of traditional rotary joints. 1) The tendon is elastic, so it can only pull but not push; 2) The joint movement degree of freedom is driven by multiple pairs of agonist and antagonist. NASA describes a new type of robotic arm-TALISMAN, which combines tendon-driven joints and a new type of hinge joint. The hinge allows the connected link to rotate 360 degrees and can combine auxiliary and passive tension to enhance it. A boom structure is set at the joint to obtain a longer torque arm between the tendon tension and the joint rotation axis, thereby obtaining the mechanical advantage of generating torque to drive the joint. However, the hinge joint relies on 4 rotating shafts, and the actual controllability of the rotating shafts is not ideal. The working condition analysis is complex, and it is necessary to improve the joint structure to meet the application requirements of high precision and high reliability. Based on the skeletal muscle system, the antagonistic arrangement of the tendon is used as the driving tendon, similar to muscle, which can only provide contraction tension, and the elastic tendon gives the joint good compliance characteristics. By controlling the winch driven by the motor, the agonist muscle is actively contracted, and the antagonist muscle on the other side is also contracted under the control of the winch, thereby controlling the smooth movement of the joint, while having a large range of stiffness adjustment capability. The front end of the muscle is generally differentiated into multiple heads, which are connected to different bone positions, while the end of the muscle is gathered together and connected to the position near the driven joint. This multi-head connection mechanism at the front end gives the joint more powerful movement ability. Similarly, we can also use multi-branch tendon drive, and each branch is connected to different positions on the arm or different arms, and different branch drive strategies are designed according to the driving task requirements. However, the attachment point at the end of the joint makes the torque arm between the muscle and the joint shaft very small, and the advantages of tendon drive are not fully utilized. A boom structure is set at the joint to increase the length of the torque arm between the tendon tension and the joint rotation axis, improve the torque output performance of the skeletal muscle system, and realize the large torque output of the joint. Due to the mechanical limitations of the skeleton, muscle and joint hinge, the movement range of the skeletal muscle system is very limited, and the hinge joint cleverly reduces the mechanical limitations at the joint, which can realize full 360° rotation. In view of this, a new type of joint structure is designed, which only needs 2 rotating shafts to realize full 360° rotation, and the model is simple, has good control accuracy, and is more easily applied.Based on the above design strategy, we completed the design of a planar single-degree-of-freedom boom with a full 360° rotation range, a torque arm amplification boom, a modular boom, and a motor and winch drive module, see below. Figure 1 The tendon-driven joint module structure shown.

[0004] Nonlinear characteristics of cable-driven mechanisms, such as flexible vibration, dead zone-hysteresis, and frictional deformation, are common in systems like circuits, gears, and robotic arms. As is well known, robotic arms are a typical type of nonlinear system, playing a crucial role in industrial manufacturing, deep-sea exploration, and aerospace. However, achieving rapid and precise tracking and control of cable-driven robotic arm systems with flexible cable deformation remains a pressing problem and a hot research topic.

[0005] The existing invention patent application document CN117245670A, entitled "A Modeling and Testing Method and System for a Rope-Driven Spatial Robotic Arm," describes a method that includes: assuming parameters for flexible links globally; calculating coordinate transformation equations in the state space of the topological robotic arm system through coordinate transformation, vector operations, and matrix processing to establish kinematic equations; integrating motors, reducers, winches, and other devices in the rope-driven system to drive rope tension changes, which are mapped to the motion of the joint axes of the robotic arm to control the target object at the end of the robotic arm; and deriving the dynamic equations of the flexible joint and flexible arm spatial robot system with uncontrolled carrier position and controlled attitude using the second type of Lagrange equations. However, the aforementioned prior art cannot fully utilize the mapping relationship of the rope, resulting in a high system error. Furthermore, this prior art struggles to simultaneously address nonlinear coupling problems and reduce vibrations caused by the flexible rope. The rope self-adjustment modeling method and the system computational performance of the model need improvement, and the rope length variation established by this existing modeling method has a lack of representativeness.

[0006] In summary, existing technologies suffer from technical problems such as reliability defects in robotic arms, low accuracy in tracking predetermined trajectories, and high-frequency vibrations due to nonlinear characteristics in circuits, rope flexibility, and robotic arm systems, including flexible vibrations, friction, rope deformation, and hysteresis. Summary of the Invention

[0007] The technical problem to be solved by this invention is: how to solve the technical problems in the prior art of nonlinear characteristics such as flexible vibration, friction, rope deformation and hysteresis in circuits, rope flexibility and robotic arm systems, which lead to servo defects in the reliability of robotic arms, low accuracy in tracking predetermined expected trajectories and high-frequency vibrations.

[0008] This invention solves the above-mentioned technical problems using the following technical solution: A modeling method for solving the problem of rope flexible deformation and vibration in a rope-driven robotic arm includes:

[0009] S1. Based on the joint module motion form, joint module state, arm, lifting device and joint rotation axis angle, establish the mapping relationship from rope to joint angle; construct a kinematic model to describe the changes in the robot system motion process, and obtain the coordinate transformation equation in the state space of the topological robot through coordinate transformation operation, vector operation and matrix calculation.

[0010] S2. Obtain and establish an elastic potential energy model of rope deformation based on the changes in rope winding and unwinding driven by the motor-driven winch and the current mapped length of the joint rope; define the kinetic and potential energy of the joint rotation and arm movement of the motor-driven rope transmission mechanical arm in order to construct the second type of Lagrange equation;

[0011] S3. Based on the second kind of Lagrange equation, the dynamic equation of the rope-driven robotic arm system is derived, which relates the control input of the rope change and the mapping relationship of the joint angle of the robotic arm.

[0012] S4. Optimize the dynamic model and elastic potential energy model, and use the Lyapunov stability principle to prove the stability of the dynamic model and elastic potential energy model. Then, use a pre-built physical prototype for verification.

[0013] This invention addresses the design of a control input term for the change in rope-driven mechanical arm system with flexible deformation. The reliability and stability of the model creation were verified based on a physical prototype. A feasibility analysis was conducted on a novel tendon-driven mechanical arm mechanism, and multi-rope antagonistic drive was introduced to complete kinematic modeling. This invention also addresses the problems of rope deformation and flexible vibration caused by force, proposing a dynamic model for the change in rope winding and unwinding based on elastic potential energy and the Lagrangian function of kinetic energy conservation.

[0014] In a more specific technical solution, S1 includes:

[0015] S11. Express the distance between the end of the lifting device and the joint pivot using the following logic:

[0016]

[0017] In the formula, d i (i = 1, 2, 3…n) represents the distance from the pivot of each joint module to the center of the lifting device; h i (i = 1, 2, 3…n) is the distance from the end of the lifting device to the center of the lifting device; S ij (i = 1, 2, 3…n; j = 1, 2, 3, 4) represents the distance from the end of the lifting device to the center of the rotating shaft;

[0018] S12. Using the following logic, express the rope length from the contact end of the lifting rope to the contact point of the winch rope:

[0019]

[0020] In the formula, L i (i = 1, 2, 3…n, n+1) represents the length of each arm in the cable-driven robotic arm system, l ij (i = 1, 2, 3…n; j = 1, 2, 3, 4) represents the rope length of each part in the rope-driven robotic arm system; q = [q1 q2…q n ] T : is the column vector of relative rotation angles of each arm of the tethered spatial manipulator; q i (i = 1, 2, 3…n) represents the boom L i+1 Compared to L i Angle of rotation;

[0021] θ is the angle between the line connecting the end of the lifting device to the shaft and the central axis of the lifting device; r is the radius of the winch.

[0022] S13. Using the following logic, differentiate over time t to obtain the differential relationship between the change in rope length and the rotation angle q of the joint axis:

[0023] dl ij =J ij dq i (3)

[0024] In the formula,

[0025] In a more specific technical solution, S2 includes:

[0026] S21. Treating the linkage rope as an equivalent geometric constraint spring with high stiffness, the following logic is used to express the change in the rope winding and unwinding of the drum-driven robotic arm when the motor rotor rotates by θ:

[0027] Δl θ =K r θ;

[0028] S22. Using the following logic, express the change in length from the lifting device to the motor drum corresponding to the joint shaft rotation angle q:

[0029] Δl=l-l0

[0030] S23. Define K as the stiffness coefficient matrix of each rope, and use the following logic to establish the elastic potential energy model of the flexible rope:

[0031] S24. Using coefficient logic, establish the second type of Lagrange equation:

[0032] L = EU (5).

[0033] This invention addresses the reliability servo problem of a novel tendon-driven robotic arm mechanism caused by uncertain disturbances such as multi-rope antagonistic drive, rope elastic deformation, and rope flexible vibration. It establishes a dynamic model of the tendon-driven robotic arm system using the second kind of Lagrange theorem, conservation of kinetic energy, and the elastic potential energy of rope changes, providing a foundation for the design of a high-performance servo controller.

[0034] In a more specific technical solution, in S21, the change in the amount of rope release and take-up is equivalent to a geometric constraint, let:

[0035] Δl θ =l θ -l0

[0036] In the formula, l θ θ is a variable.

[0037] In a more specific technical solution, S3 includes:

[0038] S31. Using the second type of Language method and the system momentum conservation relationship, the dynamic equations of the space manipulator driven by the motor and the flexible rope are derived according to the following logic:

[0039]

[0040] In the formula, q = [q1 q2 q3] T Let M(q) ∈ R be the column vector of the actual joint rotation angles of each robotic arm. n×n , These are the positive definite symmetric inertia matrix and column vector containing Coriolis force and centrifugal force at the link end of the robotic arm, respectively, Δl θ J is the control input for each rope variation matrix. kq ∈R 4n×n This is the mapping matrix from the stiffness coefficients of each rope to the rotational joints;

[0041] S32. Based on the dynamic equations of the space manipulator, the dynamic equations of the cable-driven space manipulator system are derived:

[0042]

[0043] In the formula, Δl θ This represents the deformation of the motor rope.

[0044] In a more specific technical solution, in S4, the state space of the cable-driven robotic arm is expressed using the following logic:

[0045]

[0046] In a more specific technical solution, in S4, the stability proof operations include: open-loop control based on the inverse dynamics model and error-based feedback control.

[0047] In a more specific technical solution, in S4, the motion of the inverse dynamics joint model in the dynamic model and the actuator is used to obtain the generalized force of each joint. The joint error is then controlled by feedback through the mapping relationship between the motor and the rope. The actual control output of the rope-driven robotic arm system is defined using the following logic:

[0048] q = [q1 q2…q n ] T The corresponding ideal state control output is q. d ;

[0049] Express the expected perspective using the following logic: q d =[q 1d q 2d …q nd ] T ;

[0050] Define the system tracking error using the following logic:

[0051]

[0052] Based on the feedback control term of the system tracking error and combined with the controlled object's rope-driven robotic arm model, stability can be determined using a Lyapunov function.

[0053] In a more specific technical solution, in S4, the input format of the controller is set using the following logic:

[0054]

[0055] In the formula, e, K represents the joint angle tracking error, joint angular velocity tracking error, and joint angular acceleration tracking error, respectively, as shown in equation (10). d K p It is a homogeneous quadratic differential function under specific parameters, where the error e converges exponentially to zero, and the system reaches dynamic stability.

[0056] For rope-driven robotic arm systems, not only must uncertain interference factors such as rope deformation and flexible vibration be considered, but the proposed novel mechanism must also consider issues such as system nonlinearity, friction, and parameter perturbation. In order to ensure that the rope-driven robotic arm system with antagonistic interference can achieve precise control, this invention designs PID control for the input control term of rope length variation to gradually converge the tracking error to zero. The reliability and stability of the method are verified through simulation and physical prototype.

[0057] This invention utilizes an angle encoder to collect the rotation angle of the joint shaft, compares it with the predetermined desired trajectory signal to form an error feedback closed loop, and designs a PID control scheme for the rope take-up and release, enabling the tendon-driven robotic arm prototype to track the predetermined desired trajectory with high precision, and greatly eliminating the high-frequency vibration problem caused by rope flexibility and rope deformation.

[0058] In a more specific technical solution, the modeling system for solving the problem of rope flexible deformation and vibration in a rope-driven robotic arm includes:

[0059] The kinematic model construction module is used to establish the mapping relationship between ropes and joint angles based on the joint module motion form, joint module state, arm, lifting device and joint rotation axis angle; construct the kinematic model to describe the changes in the robot system motion process, and obtain the coordinate transformation equation in the state space of the topological robot through coordinate transformation operations, vector operations and matrix calculations.

[0060] The Lagrange equation construction module is used to obtain and establish an elastic potential energy model of rope deformation based on the changes in the rope winding and unwinding driven by the motor-driven winch and the current mapped length of the rope at the joint; it defines the kinetic and potential energy of the joint rotation and arm movement of the motor-driven rope-driven robotic arm to construct the second type of Lagrange equations. The Lagrange equation construction module is connected to the kinematic model construction module.

[0061] The dynamic equation construction module is used to derive the dynamic equation of the rope-driven robotic arm system based on the second kind of Lagrange equation, which is the mapping relationship between the control input of the rope change and the joint angle of the robotic arm. The dynamic equation construction module is connected to the Lagrange equation construction module.

[0062] The model verification module is used to optimize the dynamic model and the elastic potential energy model. It uses the Lyapunov stability principle to prove the stability of the dynamic model and the elastic potential energy model, and performs verification operations using a pre-set physical prototype. The model verification module is connected to the dynamic equation construction module and the kinematic model construction module.

[0063] The present invention has the following advantages over the prior art:

[0064] This invention addresses the design of a control input term for the change in rope-driven mechanical arm system with flexible deformation. The reliability and stability of the model creation were verified based on a physical prototype. A feasibility analysis was conducted on a novel tendon-driven mechanical arm mechanism, and multi-rope antagonistic drive was introduced to complete kinematic modeling. This invention also addresses the problems of rope deformation and flexible vibration caused by force, proposing a dynamic model for the change in rope winding and unwinding based on elastic potential energy and the Lagrangian function of kinetic energy conservation.

[0065] This invention addresses the reliability servo problem of a novel tendon-driven robotic arm mechanism caused by uncertain disturbances such as multi-rope antagonistic drive, rope elastic deformation, and rope flexible vibration. It establishes a dynamic model of the tendon-driven robotic arm system using the second kind of Lagrange theorem, conservation of kinetic energy, and the elastic potential energy of rope changes, providing a foundation for the design of a high-performance servo controller.

[0066] For rope-driven robotic arm systems, not only must uncertain interference factors such as rope deformation and flexible vibration be considered, but the proposed novel mechanism must also consider issues such as system nonlinearity, friction, and parameter perturbation. In order to ensure that the rope-driven robotic arm system with antagonistic interference can achieve precise control, this invention designs PID control for the input control term of rope length variation to gradually converge the tracking error to zero. The reliability and stability of the method are verified through simulation and physical prototype.

[0067] This invention utilizes an angle encoder to collect the rotation angle of the joint shaft, compares it with the predetermined desired trajectory signal to form an error feedback closed loop, and designs a PID control scheme for the rope take-up and release, enabling the tendon-driven robotic arm prototype to track the predetermined desired trajectory with high precision, and greatly eliminating the high-frequency vibration problem caused by rope flexibility and rope deformation.

[0068] Compared with the existing patent with publication number CN117245670A, this invention can effectively utilize the mapping relationship of rope changes to greatly reduce system errors and improve accuracy. It solves the nonlinear coupling problem while greatly reducing the vibration caused by flexible ropes. The relatively better rope self-adjustment modeling method and model reduce the computational performance of the system. It has a good scope of practical application in prototypes and accurately establishes a representative modeling method for rope length changes, providing technical support and means for the research and application of rope mechanism systems.

[0069] This invention solves the technical problems in the prior art, such as nonlinear characteristics in circuits, ropes, and robotic arm systems, including flexible vibration, friction, rope deformation, and hysteresis, which lead to servo defects in the reliability of robotic arms, low accuracy in tracking predetermined desired trajectories, and high-frequency vibrations. Attached Figure Description

[0070] Figure 1 This is a schematic diagram of the tendon-driven joint module configuration in the background technology of this invention;

[0071] Figure 2 This is a schematic diagram of the basic steps of the modeling method for solving the problem of flexible deformation and vibration of ropes in the rope-driven robotic arm according to Embodiment 1 of the present invention.

[0072] Figure 3 This is a schematic diagram of the cable-driven robotic arm system configuration according to Embodiment 1 of the present invention;

[0073] Figure 4This is a schematic diagram of the transmission structure of the novel rope-driven space robotic arm according to Embodiment 1 of the present invention;

[0074] Figure 5 This is a schematic diagram of the PID control principle in Embodiment 2 of the present invention;

[0075] Figure 6 This is a tracking curve of the first joint angle in Embodiment 2 of the present invention;

[0076] Figure 7 This is a tracking curve of the second joint angle in Embodiment 2 of the present invention;

[0077] Figure 8 This is a tracking error curve diagram of Embodiment 2 of the present invention;

[0078] Figure 9 This is a physical prototype drawing of Embodiment 2 of the present invention;

[0079] Figure 10 This is a schematic diagram of joint angle tracking in Embodiment 2 of the present invention;

[0080] Figure 11 This is a schematic diagram of the joint angle tracking error in Embodiment 2 of the present invention;

[0081] Figure 12 This is a schematic diagram of the rope forces of the rope-driven robotic arm in Embodiment 2 of the present invention;

[0082] Figure 13 This is a schematic diagram of joint angle tracking in Embodiment 2 of the present invention;

[0083] Figure 14 This is a schematic diagram of the joint angle tracking error in Embodiment 2 of the present invention;

[0084] Figure 15 This is a schematic diagram of the rope forces of the rope-driven robotic arm in Embodiment 2 of the present invention. Detailed Implementation

[0085] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below in conjunction with the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0086] Example 1

[0087] like Figure 2 and Figure 3 As shown, the modeling method for solving the problem of flexible deformation and vibration of ropes in a rope-driven robotic arm provided by this invention includes the following basic steps:

[0088] S1. Based on the motion pattern and state of the joint module, establish the mapping relationship between the rope and the joint angle according to the boom, lifting device and joint rotation axis angle;

[0089] like Figure 3 As shown, in this embodiment, for the novel rope-driven spatial robotic arm system, the two ends of the joint module are driven by a motor-driven winch that transmits power to the rope, forming the arm L. i —lifting gear d i —arm L i-1 The joint module configuration of the rope-driven space robotic arm system, where O is the initial point of the robotic arm lever, and L... i1 and L i2 L i3 and L i4 These components consist of two motors on the same rope and a single boom, working in tandem to drive the rope. A constant-force spring maintains the rope tension, and a rope-retracting mechanism is used. The rope drives the boom and the lifting device's shaft to rotate, enabling the end effector of the robotic arm to grasp the object. See the attached diagram for its structure. Figure 3 Considering that the cable-driven spatial manipulator system performs planar motion, a simplified structural model needs to be established and its dynamics analyzed to achieve high-precision control. The kinematic model describes the changes in the robot system's motion process, using coordinate transformations, vector operations, and matrix calculations to determine the coordinate transformation equations in the topological robot's state space. In the cable-driven system, integrated motors, worm gears, planetary reducers, winches, and other devices drive the changes and tension of the cable, which are then mapped to the motion of the manipulator's joint axes, enabling the capture of the target object at the manipulator's end effector.

[0090] L i (i = 1, 2, 3…n, n+1): represents the length of each arm in the rope-driven robotic arm system;

[0091] m i (i = 1, 2, 3…n, n+1): represents the mass of each arm in the rope-driven robotic arm system;

[0092] I i (i = 1, 2, 3…n, n+1): represents the moment of inertia of each robotic arm in the rope-driven robotic arm system;

[0093] m si (i = 1, 2, 3…n): represents the mass of the lifting device in each joint module of the rope-driven robotic arm system;

[0094] I si (i = 1, 2, 3…n): the moment of inertia of the lifting device and gears in each joint module;

[0095] d i(i = 1, 2, 3…n): the distance from the pivot of each joint module to the center of the lifting device;

[0096] h i (i = 1, 2, 3…n): the distance from the end of the lifting device to the center of the lifting device;

[0097] l ij (i = 1, 2, 3…n; j = 1, 2, 3, 4): represents the rope length of each part in the rope-driven robotic arm system;

[0098] f ij (i = 1, 2, 3…n; j = 1, 2, 3, 4): represents the rope tension in each part of the rope-driven robotic arm system;

[0099] S ij (i = 1, 2, 3…n; j = 1, 2, 3, 4): is the distance from the end of the lifting device to the center of the rotating shaft;

[0100] The angle between the line connecting the end of the spreader to the pivot and the central axis of the spreader;

[0101] q i (i = 1, 2, 3…n): represents the boom L i+1 Compared to L i Angle of rotation;

[0102] q = [q1 q2…q n ] T : is the column vector of relative rotation angles of each arm of the rope-driven space robot;

[0103] r: is the radius of the winch.

[0104] In this embodiment, the kinematic model describes the changes in motion of the rope-driven spatial manipulator system. It uses coordinate transformation, vector operations, and matrix calculations to determine the coordinate transformation equations in the topological robot's state space. In the rope-driven system, integrated motors, worm gears, planetary reducers, winches, and other devices drive rope changes and tension, which are then mapped to the motion of the manipulator's joint axes, enabling the capture of the target object at the manipulator's end effector. The distance between the end of the lifting device and the joint axis is:

[0105]

[0106] In this embodiment, the rope length from the contact end of the lifting rope to the contact point of the winch rope is expressed as follows:

[0107]

[0108] In this embodiment, the derivative of equation (1.2) with respect to time t yields the differential relationship between the change in rope length and angle q as follows:

[0109] dl ij =J ij dq i (14)

[0110] in,

[0111] S2. Due to the deformation of the rope due to its flexibility, the elastic potential energy of the rope deformation is established based on the changes in the rope winding and unwinding driven by the motor-driven winch and the current mapped length of the rope at the joint. The kinetic and potential energy of the joint rotation and arm movement of the motor-driven rope-transmitted robotic arm are defined, and the Lagrangian function is constructed.

[0112] In this embodiment, the rope-driven spatial manipulator system is studied mainly by a transmission mode of motor drum-rope-arm. The linkage rope used is a flexible rope, which is equivalent to a geometrically constrained spring with high stiffness. The change in the rope winding and unwinding of the manipulator driven by the drum when the motor rotor rotates θ is Δl. θ =K r θ, which is equivalent to a geometric constraint, let Δl θ =l θ -l0, where l θ For variables related only to θ, the change in length from the lifting device to the motor drum corresponding to the joint axis rotation q is Δl = l - l0. Define K as the stiffness coefficient matrix of each rope, then the elastic potential energy in the flexible rope is:

[0113]

[0114] In this embodiment, when moving in a space without gravitational potential energy, the kinetic energy E of the system is the sum of the kinetic energies of each boom and lifting device, and the potential energy of the system is the sum of the elastic potential energies of each rope. The second type of Language function can be represented as follows:

[0115] L = EU (16)

[0116] S3. Based on the second kind of Lagrange equation, derive the dynamic equation of the rope-driven manipulator system that relates the control input of the rope change to the mapping relationship of the manipulator joint angle;

[0117] In this embodiment, assuming that the tendon-driven components of the spatial manipulator move on a plane and ignoring the weight of the rope, the manipulator system is a multi-body system driven by a linkage rope without external force. Then, using the second type of Language method and the system momentum conservation relationship, the dynamic equation of the space manipulator driven by the motor and the flexible rope is derived as follows:

[0118]

[0119] Where q = [q1 q2 q3] TLet M(q) ∈ R be the column vector of the actual joint rotation angles of each robotic arm. n×n , These are the positive definite symmetric inertia matrix and column vector containing Coriolis force and centrifugal force at the link end of the robotic arm, respectively, Δl θ J is the control input for each rope variation matrix. kq ∈R 4n×n This is the mapping matrix from the stiffness coefficients of each rope to the rotational joints.

[0120] In this embodiment, the dynamic equations of the rope-driven space manipulator system can be derived from equations (17) and (18) as follows:

[0121]

[0122] Where, Δl θ This represents the deformation of the motor rope.

[0123] S4. Considering the influence of rope-driven flexible vibration, optimize the system model, prove the stability using the Lyapunov stability principle, and verify the effectiveness and reliability of the method on a physical prototype.

[0124] In this embodiment, considering a flexible rope-driven multi-input multi-output height nonlinear manipulator system, the state-space expression of the rope-driven manipulator can be expressed as follows, according to equation (19):

[0125]

[0126] In the design and stability verification of the control method in this embodiment, the methods used to address the robotic arm control problem include, but are not limited to: open-loop control based on inverse dynamics models and error-based feedback control. Since open-loop control systems cannot control the generated errors, this study utilizes the dynamics model and the actuators to derive the generalized forces of each joint from the motion of the inverse dynamics joint model. Feedback control of joint errors is achieved through the mapping relationship between the motor and the cable. The actual control output of the cable-driven robotic arm system is defined as: q = [q1 q2…q ... n ] T The corresponding ideal state control output is q. d The expected angle is expressed as: q d =[q 1d q 2d …q nd ] T The tracking error of the system is defined as:

[0127]

[0128] In this embodiment, based on the feedback control term of the error and the controlled object rope-driven manipulator model, stability can be determined through the Lyapunov function. The core of the PID control method is to transform the dynamic model of the rope-driven manipulator system (controller + controlled object) into a necessarily stable closed-loop system as shown in equation (20) by setting the controller. That is, the form of the controller input can be expressed as follows:

[0129]

[0130] Among them, e, These represent the joint angle tracking error, joint angular velocity tracking error, and joint angular acceleration tracking error, respectively, as shown in equation (21), K d K p It is a homogeneous quadratic differential function under specific parameters, where the error e converges exponentially to zero, and the system reaches dynamic stability.

[0131] Example 2

[0132] like Figure 5 As shown, in this embodiment, in order to verify the reliability of the method designed by the present invention and the high control performance of the control design, the novel architecture system is verified by adopting the form of a rope-driven robotic arm dual-joint module. The design method is verified based on the PID control scheme, with a sampling frequency of L = 0.001s and a sampling period of 50s.

[0133] Table 1 Characteristic parameters of each component of the dual-joint rope-driven robotic arm

[0134]

[0135] In this embodiment, to verify the effectiveness of the control scheme, the control scheme proposed in this invention is simulated. The control parameters based on the PID controller are designed as follows, with parameter K. p =diag(50,50),K d =diag(10,10), K i =diag(0.05,0.05), where the expected trajectory of each joint axis joint angle is as follows:

[0136]

[0137] In this embodiment, the novel rope-driven robotic arm system is designed as shown in equation (22), and its simulation results are as follows: Figures 5 to 7 As shown, Figure 5 This is a simulation curve of the joint angle 1 of the robotic arm system. Figure 6 The figure shows the simulation curve of joint angle 2 of the robotic arm system. Figure 7 The simulation curves for tracking the joint angle errors of the robotic arm system are shown.

[0138] like Figure 8 As shown in the figure, in this embodiment, the test process is based on the output error between the angle encoder and the desired trajectory. The PID control design for the rope length change is as shown in equation (22). A dynamic trajectory of ±30° of single joint motion angle is established, and the position signal of the four motor drive is imported. The desired trajectory of the single joint is designed as follows:

[0139]

[0140] In this embodiment, the trajectory planning period is defined as 0.02s, the total planning time is 100s, and the first 5s are the rope pretensioning time. The tracking performance of the robotic arm prototype is tested by adjusting the PID parameters. The average error is 0.0107°, and the maximum error is 3.22° at the beginning of joint rotation. The tracking results of the test are shown in [reference needed]. Figures 9 to 11 .

[0141] In this embodiment, the high-frequency oscillation problem existing in the above-mentioned rope-driven robotic arm prototype test was addressed by adjusting parameters and adding filtering to design the desired trajectory. The test results are as follows: Figures 12 to 14 As shown.

[0142] In this embodiment, when the initial sampling point has a large error, the rope tensioning is first set for 5 seconds. The average error of the joint angle trajectory tracking is 0.1737°, and the maximum error is 3.8697° during the tensioning stage of the initial movement of the joint rotation. The scheme and modeling method studied in this invention have high reliability and efficiency in the high-precision control of the rope-driven robotic arm system.

[0143] In summary, this invention addresses the design of a control input term for the change in rope-driven mechanical arm system with flexible deformation. The reliability and stability of the model were verified based on a physical prototype. A feasibility analysis was conducted on a novel tendon-driven mechanical arm mechanism, and multi-rope antagonistic drive was introduced to complete kinematic modeling. This invention also addresses the problems of rope deformation and flexible vibration caused by force, proposing a dynamic model for the change in rope winding and unwinding based on elastic potential energy and the Lagrangian function of kinetic energy conservation.

[0144] This invention addresses the reliability servo problem of a novel tendon-driven robotic arm mechanism caused by uncertain disturbances such as multi-rope antagonistic drive, rope elastic deformation, and rope flexible vibration. It establishes a dynamic model of the tendon-driven robotic arm system using the second kind of Lagrange theorem, conservation of kinetic energy, and the elastic potential energy of rope changes, providing a foundation for the design of a high-performance servo controller.

[0145] For rope-driven robotic arm systems, not only must uncertain interference factors such as rope deformation and flexible vibration be considered, but the proposed novel mechanism must also consider issues such as system nonlinearity, friction, and parameter perturbation. In order to ensure that the rope-driven robotic arm system with antagonistic interference can achieve precise control, this invention designs PID control for the input control term of rope length variation to gradually converge the tracking error to zero. The reliability and stability of the method are verified through simulation and physical prototype.

[0146] This invention utilizes an angle encoder to collect the rotation angle of the joint shaft, compares it with the predetermined desired trajectory signal to form an error feedback closed loop, and designs a PID control scheme for the rope take-up and release, enabling the tendon-driven robotic arm prototype to track the predetermined desired trajectory with high precision, and greatly eliminating the high-frequency vibration problem caused by rope flexibility and rope deformation.

[0147] This invention solves the technical problems in the prior art, such as nonlinear characteristics in circuits, ropes, and robotic arm systems, including flexible vibration, friction, rope deformation, and hysteresis, which lead to servo defects in the reliability of robotic arms, low accuracy in tracking predetermined desired trajectories, and high-frequency vibrations.

[0148] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A modeling method for solving the problem of flexible deformation and vibration of ropes in a rope-driven robotic arm, characterized in that, The method includes: S1. Based on the joint module motion form, joint module state, arm, lifting device and joint rotation axis angle, establish the mapping relationship from rope to joint angle; construct a kinematic model to describe the changes in the robot system motion process, and obtain the coordinate transformation equation in the state space of the topological robot through coordinate transformation operation, vector operation and matrix calculation. S2. Obtain and establish an elastic potential energy model of rope deformation based on the changes in rope winding and unwinding driven by the motor-driven winch and the current mapped length of the joint rope; define the kinetic and potential energy of the joint rotation and arm movement of the motor-driven rope transmission mechanical arm in order to construct the second type of Lagrange equation; S2 includes: S21. Treat the linkage rope as an equivalent geometric constraint spring with high stiffness, and use the following logic to express the rotation of the motor rotor. θ At that time, the change in the amount of rope winding and unwinding driven by the drum robotic arm: ； S22. Express the rotation angle of the joint axis using the following logic. q The change in length from the lifting device to the motor drum: S23, Definition K Given the stiffness coefficient matrix of each rope, the elastic potential energy model of the flexible rope is established using the following logic: (4) S24. Using coefficient logic, establish the second type of Lagrange equation: (5) S3. Based on the second kind of Lagrange equation, the dynamic equation of the rope-driven robotic arm system is derived, which relates the control input of the rope change and the mapping relationship of the joint angle of the robotic arm. S3 includes: S31. Using the second type of Language method and the system momentum conservation relationship, the dynamic equations of the space manipulator driven by the motor and the flexible rope are derived according to the following logic: (6) (7) In the formula, This is a column vector of the actual joint rotation angles of each robotic arm. , These are the positive definite symmetric inertia matrix of the robotic arm's link end and the column vector containing the Coriolis force and centrifugal force, respectively. These are the control inputs for the variation matrices of each rope. This is the mapping matrix from the stiffness coefficients of each rope to the rotational joints; S32. Based on the dynamic equations of the space manipulator, the dynamic equations of the cable-driven space manipulator system are derived: (8) In the formula, This refers to the deformation of the motor rope. S4. Optimize the dynamic model and elastic potential energy model, and use the Lyapunov stability principle to prove the stability of the dynamic model and elastic potential energy model. Then, use a pre-built physical prototype for verification.

2. The modeling method for solving the problem of flexible deformation and vibration of ropes in a rope-driven robotic arm according to claim 1, characterized in that, S1 includes: S11. Express the distance between the end of the lifting device and the joint pivot using the following logic: (1) In the formula, This refers to the distance from the pivot point of each joint module to the center of the lifting device. The distance from the end of the spreader to the center of the spreader; This is the distance from the end of the lifting device to the center of the pivot. S12. Using the following logic, express the rope length from the contact end of the lifting rope to the contact point of the winch rope: (2) In the formula, For each arm link in the rope-driven robotic arm system, The length of the rope in each part of the rope-driven robotic arm system; : is the column vector of relative rotation angles of each arm of the rope-driven space robot; For boom Compared to Angle of rotation; The angle between the line connecting the end of the spreader to the pivot and the central axis of the spreader; The radius of the winch; S13. Using the following logic, differentiate over time t to obtain the change in rope length and the rotation angle of the joint axis. q Differential relation: (3) In the formula, , , , .

3. The modeling method for solving the problem of flexible deformation and vibration of ropes in a rope-driven robotic arm according to claim 1, characterized in that, In step S21, the change in the rope length is equivalent to a geometric constraint, let: In the formula, To and θ Relevant variables.

4. The modeling method for solving the problem of flexible deformation and vibration of ropes in a rope-driven robotic arm according to claim 1, characterized in that, In S4, the state space of the cable-driven robotic arm is expressed using the following logic: (9)。 5. The modeling method for solving the problem of flexible deformation and vibration of ropes in a rope-driven robotic arm according to claim 1, characterized in that, In S4, the stability proof operation includes: open-loop control based on the inverse dynamics model and error-based feedback control.

6. The modeling method for solving the problem of flexible deformation and vibration of ropes in a rope-driven robotic arm according to claim 1, characterized in that, In step S4, the motion of the inverse dynamics joint model in the dynamic model and the driver is used to obtain the generalized force of each joint. The joint error is then controlled by feedback through the mapping relationship between the motor and the rope. The actual control output of the rope-driven robotic arm system is defined using the following logic: The corresponding ideal state control output is ; Express your expectations using the following logic: ; Define the system tracking error using the following logic: (10) Based on the feedback control term of the system tracking error and combined with the controlled object rope-driven manipulator model, stability can be determined by Lyapunov function.

7. The modeling method for solving the problem of flexible deformation and vibration of ropes in a rope-driven robotic arm according to claim 6, characterized in that, The input format of the controller is set using the following logic: (11) In the formula, The joint angle tracking error, joint angular velocity tracking error, and joint angular acceleration tracking error are respectively represented by equation (10). The homogeneous second-order differential function takes specific parameters. and At that time, error The system converges exponentially to zero, and the system error converges to zero, at which point the system reaches dynamic equilibrium.

8. A modeling system for solving the problem of flexible deformation and vibration of ropes using a rope-driven robotic arm, used to execute the modeling method for solving the problem of flexible deformation and vibration of ropes using a rope-driven robotic arm according to any one of claims 1 to 7, characterized in that, The system includes: The kinematic model construction module is used to establish the mapping relationship between ropes and joint angles based on the joint module motion form, joint module state, arm, lifting device and joint rotation axis angle; construct the kinematic model to describe the changes in the robot system motion process, and obtain the coordinate transformation equation in the state space of the topological robot through coordinate transformation operations, vector operations and matrix calculations. The Lagrange equation construction module is used to obtain and establish an elastic potential energy model of rope deformation based on the changes in the rope winding and unwinding driven by the motor-driven winch and the current mapped length of the joint rope; it defines the kinetic and potential energy of the joint rotation and arm movement of the motor-driven rope-transmitted robotic arm to construct a second type of Lagrange equation. The Lagrange equation construction module is connected to the kinematic model construction module. The dynamic equation construction module is used to derive the dynamic equation of the rope-driven robotic arm system based on the second type of Lagrange equation, which is the mapping relationship between the control input of the rope change and the joint angle of the robotic arm. The dynamic equation construction module is connected to the Lagrange equation construction module. The model verification module is used to optimize the dynamic model and the elastic potential energy model. It uses the Lyapunov stability principle to prove the stability of the dynamic model and the elastic potential energy model, and performs verification operations using a pre-set physical prototype. The model verification module is connected to the dynamic equation construction module and the kinematic model construction module.

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