A gravity compensation shape control method for a rope-driven variable stiffness flexible arm

Abstract

The application provides a gravity compensation shape control method of a rope-driven variable stiffness flexible arm, and belongs to the technical field of flexible robot control; the application takes the Cosserat rod theory as a core theoretical basis, and establishes an explicit plane partial differential equation (PDE) dynamics model containing a gravity term in combination with an extended Hamilton principle around the rope-driven variable stiffness flexible arm. Meanwhile, taking the expected shape of the flexible arm as a control target, a control method based on a static equilibrium inverse solution gravity compensation feedforward term and combined with state feedback is proposed, a distributed gravity compensation shape control law is designed to eliminate the steady-state residual of the system caused by gravity, and finally, accurate shape control of the flexible arm in the actual gravity environment is realized. The method provided by the application provides a new idea for modeling and controller design of the flexible arm considering the non-uniform characteristics of space parameters, and lays an important theoretical foundation for the practical engineering application of related flexible arm control schemes.

Description

Technical Field

[0001] This invention belongs to the field of flexible robot control technology, and particularly relates to a gravity-compensated shape control method for a rope-driven variable stiffness flexible arm. Background Technology

[0002] Flexible robotic arms, due to their excellent compliance, lightweight characteristics, and good human-robot interaction safety, have broad application prospects in fields such as medical surgery, pipeline inspection, and space operation. Among them, rope-driven flexible arms achieve bending deformation and shape control of the arm body by distributing the tension of the driving rope. They have advantages such as simple structure, remote drive, and light weight, and are one of the important directions of current flexible robot research.

[0003] In the dynamic modeling of flexible arms, existing research methods are mainly divided into two categories: geometric models and mechanical models. Among geometric models, the constant curvature method is often used for real-time kinematics solving and control design due to its simplicity. However, this method assumes uniform curvature in each arm segment, leading to significant errors when subjected to external forces (such as gravity), making it difficult to accurately describe the actual deformation state, and it is also sensitive to errors in the propagation of external forces. Regarding mechanical models, early studies often used Euler-Bernoulli beam theory and Timoshenko beam theory to model flexible arms. These methods are based on linear assumptions and are only suitable for small deformation scenarios. Their accuracy is insufficient when the flexible arm undergoes large deflection nonlinear deformation, making it difficult to meet practical engineering needs. In recent years, modeling methods based on Cosserat rod theory have gradually become the mainstream framework for continuous body dynamics modeling. This method can accurately describe the geometric nonlinear large deformation behavior of the arm in space, such as bending, torsion, shearing and stretching. It can also be combined with Hamilton's principle for modeling, and derive partial differential equations (PDEs) describing the dynamic characteristics of the arm. Therefore, it is considered one of the most representative continuous body dynamics modeling frameworks.

[0004] However, existing research still has the following major shortcomings in the application of the above modeling framework: First, existing research largely ignores the influence of gravity, leading to significant discrepancies between models and reality. In a large body of existing literature, the gravitational effect on flexible structures is typically ignored to simplify the modeling process. However, in practical engineering applications, especially with low-stiffness materials or flexible structures of considerable length, gravity significantly impacts their static equilibrium state. For example… Figure 3 As shown, under the same cable tension conditions, there is a significant difference between the simulation results that ignore gravity and the actual shape that takes gravity into account. This indicates that the dynamic model that ignores the gravity term is difficult to accurately reflect the real working conditions and cannot meet the requirements for precise shape control.

[0005] Second, existing modeling methods are typically based on the assumption of uniform stiffness, which makes it difficult to describe the spatial parameter variations in actual structures. During the manufacturing or design process of real flexible structures, their cross-sectional dimensions or material distribution often change along the length, resulting in a non-uniform distribution of bending stiffness. However, most existing models still employ the assumption of uniform stiffness, ignoring these spatial parameter variations, thus limiting model accuracy and consequently affecting subsequent control performance.

[0006] Third, the coupled effects of gravity and spatial parameter non-uniformity on the dynamic behavior of systems lack systematic research. The bending stiffness of flexible structures is closely related to their cross-sectional parameters, and in actual structures, these parameters vary along the length, resulting in spatially non-uniform dynamics. Furthermore, gravity, as a distributed load, further influences the equilibrium state and dynamic response of the system. However, existing studies often consider these factors separately, lacking a systematic analysis method that integrates gravity and spatial non-uniformity into the dynamic model, leading to insufficient model prediction accuracy under actual operating conditions.

[0007] Fourth, research on distributed control methods for flexible structures that simultaneously exhibit gravitational influence and spatial non-uniformity remains insufficient. Existing control methods are mostly based on the assumption of uniform stiffness. For systems with actual parameter variations, their dynamic characteristics are more complex, increasing the difficulty of controller design and stability analysis. The lack of systematic theoretical support and verification makes it difficult to achieve high-precision morphological control.

[0008] In summary, existing rope-driven flexible structures generally suffer from problems in dynamic modeling and control, such as neglecting the influence of gravity, failing to consider the non-uniformity of spatial parameters, and lacking system analysis and control methods for the coupling effect between gravity and spatial parameters. This leads to insufficient motion prediction accuracy and steady-state errors in shape control under actual working conditions. Therefore, there is an urgent need for a modeling and control method that can simultaneously consider the effects of gravity and the non-uniformity of spatial parameters to achieve high-precision shape control. Summary of the Invention

[0009] To address the common problem that existing flexible structure dynamics modeling often neglects the influence of gravity and fails to consider the non-uniformity of spatial parameters, resulting in insufficient motion prediction accuracy and steady-state errors in shape control under actual working conditions, this invention focuses on a rope-driven variable stiffness flexible arm. Based on Cosserat rod theory and combined with an extended Hamiltonian principle, it establishes a planar PDE dynamic model that explicitly includes a gravity term. Simultaneously, taking the desired shape of the flexible arm as the control objective, it proposes a shape control method based on static equilibrium inverse solution of gravity compensation feedforward terms combined with feedback control. A distributed gravity compensation shape control law is designed to eliminate static deviations in the system, ultimately achieving high-precision shape control under gravity conditions.

[0010] This invention provides a gravity-compensated shape control method for a rope-driven variable stiffness flexible arm, comprising the following processes: S1. Construct a flexible arm structure model with variable cross-section and variable stiffness. In the structure model, the cross-section radius changes linearly along the arc length coordinate, passively realizing the non-uniform distribution of bending stiffness along the axial direction, and using the tangent angle as a generalized coordinate to describe the bending shape of the flexible arm body in the plane. S2. Under the condition of ignoring shear effects and assuming that the flexible arm body is inextensible, based on the Cosserat rod theory and the extended Hamilton principle, combined with the variable cross-section and variable stiffness flexible arm structure model described in S1, a plane partial differential equation dynamic model containing gravity terms is derived and established. S3, set the time derivative term in the dynamic model described in S2 to zero to obtain the static equilibrium equation containing the gravity term. Substitute the desired shape into the static equilibrium equation to solve the rope tension required to maintain the desired shape and determine the gravity compensation feedforward term. S4, based on the dynamic model described in S2, defines the shape error with the desired shape of the flexible arm as the control target, and combines the gravity compensation feedforward term and feedback control term determined in S3 to construct a distributed gravity compensation shape control law to drive the flexible arm to converge to the desired shape, thereby realizing shape control under gravity environment.

[0011] Preferably, in step S1, the construction of a flexible arm structure model with variable cross-section and variable stiffness is specifically as follows: The cross-sectional parameters of the flexible arm are set to vary continuously along the arc length direction. The variation can be a linear function, a nonlinear function, or a piecewise function to adapt to different stiffness adjustment requirements. Set the cross-sectional radius Along arc length coordinates Linear change: ; in, The length of the flexible arm, For the fixed end cross-section radius, The radius of the free end section is... It is the arc length; The bending stiffness EI(s) is calculated based on the cross-sectional geometric parameters, making it axially non-uniformly distributed, thereby achieving passive variable stiffness characteristics without introducing an active adjustment mechanism.

[0012] Preferably, the bending stiffness The mathematical expression for a non-uniform distribution along the axial direction is: ; Where E is the elastic modulus of the material, and the bending stiffness is continuously and linearly adjusted along the axial direction by changing the fourth power of the cross-sectional radius.

[0013] Preferably, the position vector of any point on the flexible arm body... Through the tangent angle The integral yields: ; in, All are arc lengths, used within the integral. To avoid exceeding the upper limit Confusion, arc length Tangent angle at the point, For time, Let be an infinitesimal segment along the arc length direction, representing the arc length. Find the infinitesimal element of the integral.

[0014] Preferably, the specific process of S2 is as follows: S21, Establish an inertial coordinate system at the fixed end, with the tangent angle... Describe the bending shape of the arm in the plane; S22, Establish the system energy expression, including translational kinetic energy, rotational kinetic energy, elastic potential energy, and explicitly introduced gravitational potential energy; S23. In the calculation of gravitational potential energy, the expression of distributed load containing gravity terms is derived by changing the order of integration using Fubini's theorem. S24. Construct a nonconservative force virtual work term that includes distributed viscous damping, rope control torque and internal force constraints, and derive a set of partial differential equations describing the dynamic behavior of the system by combining extended Hamilton's principle.

[0015] Preferably, in step S23, the derivation of the distributed load expression containing the gravity term specifically includes: using Fubini's theorem to change the order of integration, and converting the arc length... The gravitational potential energy of a mass element can be expressed as: ; in, It is the acceleration due to gravity. The linear density of the flexible arm. It is gravitational potential energy. All coordinates are arc length coordinates, used in integrals. To avoid exceeding the upper limit Confusion, arc length Tangent angle at the point, For time, All are infinitesimal segments along the arc length direction, representing the arc length. Find the infinitesimal element of the integral, used in double integrals. As an inner-layer integration variable, using As an outer layer, to avoid confusion.

[0016] Preferably, in step S4, the feedback control term is used to correct the dynamic error of the system, and the feedback control term adopts velocity feedback control or proportional-derivative control; When velocity feedback control is used, the distributed gravity compensation shape control law The specific form is as follows: ; in, This is the gravity compensation feedforward term obtained by inverse solution of the static equilibrium relationship. The velocity feedback term is constructed based on the rate of change of error. This is a positive feedback gain.

[0017] Preferably, the variation of the cross-sectional radius along the length direction in S1 further includes a nonlinear function or a piecewise function, and the number of driving units of the flexible arm is a plurality of discretely distributed rope actuators.

[0018] Compared with the prior art, the present invention has the following beneficial effects: 1. Improved Model Accuracy under Gravity: This invention introduces gravity into the dynamic modeling process, enabling the established model to more accurately describe the static equilibrium state and dynamic response of flexible structures under gravity. Compared to models that ignore gravity, this invention significantly reduces model error and improves motion prediction accuracy under conditions of low stiffness and large aspect ratio. Simulation results show that, under the same rope tension, the end tangent angles of the gravity-ignoring model and the gravity-included model deviate significantly, proving that ignoring the gravity term leads to significant model distortion. 2. Improved modeling capability for non-uniform structures: Existing models are usually based on the assumption of uniform stiffness, which makes it difficult to describe actual structures with parameters varying along the axial direction. The model framework established in this invention is applicable to general flexible structures with cross-sectional parameters continuously varying along the axial direction. Compared with existing methods, it has a wider coverage and can more accurately reflect the mechanical behavior of actual structures. 3. Improved shape control accuracy and reduced steady-state error: This invention constructs a feedforward control term based on static equilibrium inverse kinematics and combines it with feedback control to achieve shape adjustment. This method can effectively compensate for static deviations caused by gravity, significantly reduce or eliminate system steady-state errors, and improve the shape control accuracy and stability of flexible structures under actual gravity environments. As the simulation experiment 2 comparison results show, the control scheme without gravity compensation in the simulated model has a non-zero steady-state residual of about 0.132 rad, which cannot reach the desired shape; the shape error of the gravity compensation scheme of this invention monotonically decays and converges to zero within 2 seconds, accurately reaching the desired shape, and the difference in shape control accuracy between the two is significant. Attached Figure Description

[0019] Figure 1 This is a flowchart of the overall process of the present invention.

[0020] Figure 2 This is a schematic diagram of a flexible arm.

[0021] Figure 3 The diagram shows a comparison of the morphology under gravity and gravity-free conditions with varying stiffness in Example 2, as well as a comparison of the bending angle distribution.

[0022] Figure 4 This is a dynamic diagram of the gravity-compensated flexible arm in Example 3.

[0023] Figure 5 This is a dynamic diagram of the gravity-compensated flexible arm in Example 3.

[0024] Figure 6 This is an asymptotically stable convergence curve of the deformation error in Example 3. Detailed Implementation

[0025] This invention provides a gravity-compensated shape control method for a rope-driven variable stiffness flexible arm, the overall process of which is as follows: Figure 1 As shown in the figure. The present invention will be further described below with reference to specific embodiments.

[0026] Example 1: The gravity compensation shape control method for a cable-driven variable stiffness flexible arm provided in this embodiment specifically includes the following steps: Step 1: Construct a flexible arm structure model with variable cross-section and variable stiffness: A schematic diagram of the flexible arm constructed in this invention is shown below. Figure 2 As shown, at the fixed end Establish an inertial coordinate system ,in , with tangent angle Describe the bending shape of the arm in the plane, where , ; in,{ Using a body coordinate system, Transform the inertial coordinate system and the body coordinate system.

[0027] Determine the structural model of the flexible arm with variable cross-section and variable stiffness, wherein the flexible arm adopts a variable cross-section design along the axial direction, and the cross-section radius D(s) is along the arc length coordinate. Linear change, specifically in the following form: ; in, The length of the flexible arm, For the fixed end cross-section radius, The radius of the free end section is... It is the arc length; Therefore, the moment of inertia per unit length and the second moment of the cross section are respectively: and ; in, The moment of inertia per unit length, Let be the second moment of the cross section.

[0028] Therefore, the specific form of the non-uniform distribution of bending stiffness along the axial direction is as follows: ; in, The elastic modulus of the material. For bending stiffness.

[0029] Finally, the position vector of any point on the arm is obtained by integrating the tangent angle: ; in, All are arc lengths, used within the integral. To avoid exceeding the upper limit Confusion, arc length Tangent angle at the point, For time, Let be an infinitesimal segment along the arc length direction, representing the arc length. Find the infinitesimal element of the integral.

[0030] This allows the variable cross-section geometry and the kinematic description of the arm body to be unified under the same framework.

[0031] Step 2: Based on Cosserat rod theory and Hamilton's principle, establish the dynamic equations of the plane PDE containing the gravity term: Based on the kinematic model established in Step 1, assuming the skeleton is inextensible and shear effects are negligible, the tangent directions satisfy: ; in, is the unit vector of the tangent direction.

[0032] Taking its derivative with respect to t, we obtain the velocity constraint relationship: ; in, This represents the angular velocity error.

[0033] Analyzing the flexible arm, the system consists of kinetic energy With elastic potential energy It consists of two parts.

[0034] in, Linear velocity, Angular velocity, For local curvature.

[0035] In arc length The height of the prime element is By changing the order of integration using Fubini's theorem, the gravitational potential energy is: ; in, It is the acceleration due to gravity. The linear density of the flexible arm. It is gravitational potential energy. For time, All are infinitesimal segments along the arc length direction, representing the arc length. Find the infinitesimal element of the integral, used in double integrals. As an inner-layer integration variable, using As an outer layer, to avoid confusion.

[0036] Then, the non-conservative force virtual work consists of three parts: First, the virtual work of distributed viscous damping, the distributed viscous damping torque density along the angular direction is... The corresponding virtual skill is: ; in, It is the viscous damping coefficient. This represents the virtual displacement of the tangent angle.

[0037] Secondly, the rope control torque is a virtual work, and the flexible arm is... The rope is in position Segment drive, the first The distance of the rope from the center line is The rope tension is , generating concentrated torque .forward The first torque acts within the domain, the second... A torque is applied as a natural boundary condition at the end. The virtual work done by the rope torque within the domain is: ; in, The Dirac delta function represents the moment concentrated at the point where the torque is concentrated. Location.

[0038] Thirdly, internal force is virtual, introducing inextensible constraining internal force. Then, internal energy ineffectiveness is: ; in, This is the imaginary displacement of the position vector. The gradient of the inextensible constraint internal force along the arc length. Let be the partial derivative of the imaginary displacement of the position vector with respect to the arc length.

[0039] In summary, the total amount of non-conservative and superficial work is as follows: ; Finally, the extended Hamiltonian principle is employed: ; Expand each item and make them separately and Since the coefficients are zero, the boundary terms are extracted using integration by parts, resulting in the following set of dynamic equations containing gravity PDEs, which can be used as a dynamic model of plane partial differential equations: ; in, For linear acceleration, Angular acceleration, For fixed-end position vectors, For fixed end tangent angle, The second moment of the free end section, For the local curvature of the free end, For the first Rope tension, The internal force constrained by the inextensible free end.

[0040] Step 3: Solve the static equilibrium equations to determine the gravity compensation feedforward terms: Setting all time derivative terms in the dynamic equations of step two to zero, we obtain the static equilibrium equations containing gravity terms: ; in, For the desired tangent angle, Let the desired tangent direction be a unit vector. For the desired local curvature, For the first The gravity-compensated feedforward term required for the rope to maintain the desired tangent angle. Let the desired tangent direction be a unit vector at the fixed end. For the desired local curvature at the fixed end, For the desired local curvature at the free end, For the first The gravity-compensated feedforward term required for the rope to maintain the desired tangent angle.

[0041] Given the desired tangent angle Substituting this into the static equilibrium equation, the inverse solution yields the gravity compensation feedforward term required to maintain the desired tangent angle. This allows the arm to achieve static equilibrium at the desired tangent angle under gravity, thus providing a feedforward compensation basis for the control law design in step four.

[0042] Step 4: Design the gravity-compensated shape control law and perform stability analysis: Based on the gravity-incorporated PDE dynamic equations established in step two, with the desired shape and To control the target, define the morphological error: ; ; in, For angular error, This represents the positional error.

[0043] Control Law Design: Design a distributed control law for the i-th rope, comprising gravity compensation feedforward and velocity feedback components. ; The gravity compensation feedforward term is obtained by inverse solution in step three: ; ; in, arc length The second moment of the cross section, arc length The second moment of the cross section, arc length The second moment of the cross section, arc length Desired local curvature arc length Desired local curvature arc length Desired local curvature arc length The radius of the cross section at that location, arc length The radius of the cross section at that location.

[0044] Used to counteract static bending moment deviations caused by gravity; the feedback term is constructed based on velocity feedback, and .

[0045] Stability analysis: Constructing the Lyapunov function: ; in, For linear velocity error, For angular velocity error, For curvature error, It is a very small positive number.

[0046] A series of calculations guarantee that the constructed function is positive definite. By differentiating the function with respect to time and substituting it into the control law, and using the boundary conditions, the system can eventually be made asymptotically stable, meaning that the arm shape converges to the desired shape under gravity.

[0047] Example 2: This embodiment utilizes a variable cross-section, variable stiffness flexible arm model for static deformation simulation to verify the necessity and effectiveness of the proposed explicit gravity-included dynamic model in accurately predicting equilibrium states. The simulation compares the states under variable stiffness conditions with and without gravity, as well as the distribution of bending angles. Figure 3 As shown.

[0048] The basic physical and cross-sectional parameters of the flexible arm described in this embodiment are specifically set as follows: arm length linear density Material elastic modulus Gravitational acceleration The fixed end cross-sectional radii and the free end cross-sectional radii are set as follows: D(0) = 0.05 m and D(L) = 0.03 m, respectively. The working positions of the two drive ropes are 0.3 m and 0.6 m, respectively.

[0049] Under the same desired shape and inverse rope tension, the traditional model ignoring the gravity term and the model explicitly including the gravity term established in this embodiment are compared. Simulation results show that, affected by gravity, the arm body under gravity-included conditions exhibits a significant additional deflection in the gravity direction, and the two shape curves show obvious deviations. This quantitatively verifies that under actual low-stiffness conditions, ignoring gravity leads to significant model distortion, and explicitly introducing the gravity term is an important prerequisite for achieving high-precision shape control.

[0050] Example 3: This embodiment utilizes a two-segment rope-driven variable stiffness flexible arm system for dynamic process simulation to verify the superiority of the proposed shape control law with gravity-compensated feedforward term in eliminating steady-state errors and achieving high-precision tracking. A comparison of dynamic processes without and with gravity compensation is provided. Figure 4 , Figure 5 As shown, the asymptotically stable convergence curve of the deformation error is as follows: Figure 6 As shown.

[0051] In this embodiment, the basic physical parameters of the flexible arm in the dynamic simulation are set as follows: arm length L = 0.6 m, linear density... The material's elastic modulus E = 20000 Pa, and its viscous damping coefficient... Gravitational acceleration .

[0052] The cross-sectional radius also varies linearly along the arc length, with the fixed end and end dimensions being D(0) = 0.10 m and D(L) = 0.08 m, respectively. The target shape is set as two opposing "S"-shaped curves. For the above control objective, a comparative analysis was conducted between the "gravity-free scheme (feedforward term without gravity integral)" and the "gravity-compensated scheme (control law designed in this invention)". The velocity feedback gain in the control law is uniformly set to... .

[0053] Simulation results show that the gravity-free compensation scheme, due to the lack of gravity integral in the feedforward force, effectively introduces a continuously acting gravity residual disturbance, causing the shape error to eventually converge to a non-zero steady-state residual of about 0.132 rad, which cannot accurately fit the target shape. In contrast, the control law proposed in this invention accurately cancels the bending moment deviations caused by gravity in each segment through the feedforward term based on the static equilibrium inverse solution. Its shape error norm decreases monotonically with time and converges to zero within about 2 seconds, perfectly verifying the conclusion of the Lyapunov stability analysis and realizing high-precision shape control under gravity environment.

[0054] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

[0055] While the specific embodiments of the present invention have been described above, they are not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of the present invention are still within the scope of protection of the present invention.

Claims

1. A gravity-compensated shape control method for a rope-driven variable stiffness flexible arm, characterized in that, The process includes the following: S1. Construct a flexible arm structure model with variable cross-section and variable stiffness. In the structure model, the cross-section radius changes linearly along the arc length coordinate, passively realizing the non-uniform distribution of bending stiffness along the axial direction, and using the tangent angle as a generalized coordinate to describe the bending shape of the flexible arm body in the plane. S2, neglecting shear effects and assuming the flexible arm is inextensible, based on Cosserat rod theory and the extended Hamilton's principle, and combined with the variable cross-section, variable stiffness flexible arm structure model described in S1, a dynamic model of planar partial differential equations containing gravity terms is derived and established; the specific process is as follows: S21, Establish an inertial coordinate system at the fixed end, with the tangent angle... Describe the bending shape of the arm in the plane; S22, Establish the system energy expression, including translational kinetic energy, rotational kinetic energy, elastic potential energy, and explicitly introduced gravitational potential energy; S23, In the calculation of gravitational potential energy, by changing the order of integration using Fubini's theorem, the expression for the distributed load containing the gravity term is derived. Specifically, this includes: changing the order of integration using Fubini's theorem, and converting the arc length... The gravitational potential energy of a mass element can be expressed as: ； in, It is the acceleration due to gravity. The linear density of the flexible arm. It is gravitational potential energy. arc length Tangent angle at the point, For time, All are infinitesimal segments along the arc length direction, representing the arc length. Find the infinitesimal element of the integral, used in double integrals. As an inner-layer integration variable, using As the outer layer; The length of the flexible arm, It is the arc length; S24, construct a nonconservative force virtual work term that includes distributed viscous damping, rope control torque and internal force constraints, and derive a set of partial differential equations describing the dynamic behavior of the system by combining extended Hamilton's principle; S3, set the time derivative term in the dynamic model described in S2 to zero to obtain the static equilibrium equation containing the gravity term. Substitute the desired shape into the static equilibrium equation to solve the rope tension required to maintain the desired shape and determine the gravity compensation feedforward term. S4, based on the dynamic model described in S2, defines the shape error with the desired shape of the flexible arm as the control target, and combines the gravity compensation feedforward term and feedback control term determined in S3 to construct a distributed gravity compensation shape control law to drive the flexible arm to converge to the desired shape, thereby realizing shape control under gravity environment.

2. The gravity compensation shape control method for a cable-driven variable stiffness flexible arm as described in claim 1, characterized in that: In S1, a flexible arm structure model with variable cross-section and variable stiffness is constructed, specifically as follows: The cross-sectional parameters of the flexible arm are set to vary continuously along the arc length direction. The variation can be a linear function, a nonlinear function, or a piecewise function to adapt to different stiffness adjustment requirements. Set the cross-sectional radius Along arc length coordinates Linear change: ； in, The length of the flexible arm, For the fixed end cross-section radius, The radius of the free end section is... It is the arc length; The bending stiffness EI(s) is calculated based on the cross-sectional geometric parameters, making it axially non-uniformly distributed, thereby achieving passive variable stiffness characteristics without introducing an active adjustment mechanism.

3. The gravity compensation shape control method for a rope-driven variable stiffness flexible arm as described in claim 2, characterized in that: The bending stiffness The mathematical expression for a non-uniform distribution along the axial direction is: ； Where E is the elastic modulus of the material, and the bending stiffness is continuously and linearly adjusted along the axial direction by changing the fourth power of the cross-sectional radius.

4. The gravity compensation shape control method for a cable-driven variable stiffness flexible arm as described in claim 2, characterized in that: The position vector of any point on the flexible arm body. Through the tangent angle The integral yields: ； in, All are arc lengths. arc length Tangent angle at the point, For time, Let be an infinitesimal segment along the arc length direction, representing the arc length. Find the infinitesimal element of the integral.

5. The gravity compensation shape control method for a rope-driven variable stiffness flexible arm as described in claim 1, characterized in that: In S4, the feedback control term is used to correct the dynamic error of the system, and the feedback control term adopts speed feedback control or proportional-derivative control. When velocity feedback control is used, the distributed gravity compensation shape control law The specific form is as follows: ； in, This is the gravity compensation feedforward term obtained by inverse solution of the static equilibrium relationship. The velocity feedback term is constructed based on the rate of change of error. This is a positive feedback gain.

6. The gravity compensation shape control method for a cable-driven variable stiffness flexible arm as described in claim 1, characterized in that: The variation form of the cross-sectional radius along the length direction described in S1 also includes a nonlinear function or a piecewise function, and the number of driving units of the flexible arm is multiple discretely distributed rope actuators.

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