A mechanical arm nonlinear impedance control method based on gaussian damping and exponential stiffness

Abstract

The application belongs to the technical field of robot control, and particularly relates to a mechanical arm nonlinear impedance control method based on Gaussian damping and exponential stiffness, inspired by the nonlinear mechanical characteristics of biological soft tissues, and specifically comprises: constructing a dynamics model of the mechanical arm in the Cartesian space; designing a nonlinear damping function based on a Gaussian function, so that the nonlinear damping function increases at a small error to suppress oscillation and decreases at a large error to improve response speed; designing a nonlinear stiffness function based on an exponential function and a saturation function, so that the nonlinear stiffness function nonlinearly increases with the increase of the error to provide a correction force, and the saturation function is used to prevent the stiffness from increasing infinitely; and substituting the designed nonlinear impedance parameters into an impedance control law to realize closed-loop control. The application has smaller trajectory deviation, faster recovery capability and better tracking accuracy under dynamic interference, and is suitable for human-machine interaction and precision operation tasks of a collaborative mechanical arm.

Description

Technical Field

[0001] This invention belongs to the field of robot control technology, specifically relating to a nonlinear impedance control method for a robotic arm based on Gaussian damping and exponential stiffness. Background Technology

[0002] With the widespread application of collaborative robots in intelligent manufacturing, medical rehabilitation, and service fields, the physical interactions between robots and humans and unknown environments are becoming increasingly frequent and complex. Since Hogan proposed the impedance control framework, impedance control and admittance control have become fundamental methods for achieving compliant human-robot interaction. Impedance control, by establishing a dynamic mapping relationship between force and motion, enables robots to flexibly adjust their motion to respond to external forces, thereby ensuring the safety and compliance of the interaction process.

[0003] However, in practice, most impedance models are linear or variable-coefficient linear. While these linear models can achieve basic compliance, they often exhibit slow response, large tracking errors, and insufficient adaptability when faced with the nonlinear and time-varying dynamics of real-world human-computer interaction. Traditional linear impedance control, due to its use of fixed impedance parameters (inertia parameters, damping parameters, and stiffness parameters), shows significant limitations in dynamic interactive tasks: it is difficult to simultaneously achieve high tracking accuracy and low impedance compliance during the free motion phase; it is prone to generating large impact forces or oscillations during the contact transition phase; and it lacks the adaptive capability to dynamically adjust according to task errors in time-varying interactive environments.

[0004] Current research and applications of nonlinear impedance control still face significant challenges. While most studies focus on the nonlinear dynamics of the controlled object (such as a robotic system), the impedance controllers designed to ensure system stability often employ linear structures. For example, the auxiliary regulating controller proposed by Morbi et al., although called a nonlinear impedance controller, primarily exhibits nonlinearity in its threshold-switching-based auxiliary regulating strategy, while the inner-loop position control still uses a linear controller to ensure stability. Similarly, the passive impedance control scheme proposed by Kim et al. for nonlinear robot dynamics embeds a linear PI control law into a passive interconnect framework to handle system nonlinearity, but the impedance tracker itself remains linear. Bascetta et al. established a nonlinear human arm impedance model for manual control tasks, incorporating posture stiffness, muscle co-contraction, and neural response delay; however, to achieve stable compliant interaction, their proposed solution is still a standard second-order linear impedance filter. These studies indicate that even in scenarios where the controlled object is highly nonlinear, impedance controllers themselves rarely employ nonlinear designs.

[0005] It is worth noting that various biological tissues and materials in nature possess nonlinear mechanical properties, and their constitutive relationships can provide biomimetic inspiration for the design of nonlinear impedance controllers. For example, the mechanical behavior of biological soft tissues (such as arterial walls) exhibits typical nonlinear hardening characteristics: within the normal physiological range (small strain), they have low stiffness to maintain normal compliance; when subjected to abnormal loads (large strain), their stiffness increases sharply, forming a J-shaped stress-strain curve, thereby protecting the tissue from excessive deformation damage. This adaptive hardening mechanism of "becoming stiffer under stress" provides a biomechanical basis for designing nonlinear damping stiffness functions that can maintain compliance under small errors and quickly correct deviations under large errors.

[0006] Therefore, there is an urgent need for a method that can draw on the nonlinear mechanical properties of biological soft tissues and incorporate state-dependent adaptive impedance characteristics into controller design, in order to overcome the problems of fixed parameters and insufficient adaptive capability of traditional linear impedance control, while providing strict stability theoretical guarantees. Summary of the Invention

[0007] The purpose of this invention is to provide a nonlinear impedance control method for robotic arms based on Gaussian damping and exponential stiffness. By introducing state-dependent nonlinear damping and nonlinear stiffness functions, the robotic arm can intelligently adjust its dynamic characteristics according to real-time position and velocity errors. When the error is small, the damping is enhanced to ensure compliance and stability, and when the error is large, the stiffness is increased sharply to prioritize task accuracy and convergence speed, thereby achieving comprehensive performance optimization in dynamic and uncertain environments.

[0008] The specific technical solution adopted by this invention is as follows: A nonlinear impedance control method for a robotic arm based on Gaussian damping and exponential stiffness includes the following steps: Step S1: Establish a dynamic model of the multi-degree-of-freedom robotic arm and map the joint space dynamics to the operating space to obtain the end effector dynamic model of the robotic arm; Step S2: Define the position tracking error of the robotic arm's end effector and its derivative A nonlinear impedance control target is established based on the error; Step S3: Based on position tracking error An exponential-saturation nonlinear stiffness function is constructed to achieve adaptive adjustment of stiffness that increases nonlinearly with increasing error and eventually saturates. Step S4: Based on position error and speed error A Gaussian nonlinear damping function is constructed to achieve nonlinear adjustment of damping, which increases when the error is small and decreases when the error is large. Step S5: Construct a nonlinear impedance model based on the exponential-saturated stiffness function and the Gaussian damping function, and design the impedance control law from Cartesian space to joint space; Step S6: Construct a Lyapunov function to perform stability analysis on the closed-loop system, proving that the system is globally asymptotically stable at the equilibrium point when there is no external force; when there is an external force, the system is passive between the input force and the output velocity, thus ensuring the closed-loop stability when interacting with any passive environment.

[0009] Furthermore, in step S1, for an n-DOF serial robotic arm (a redundant robotic arm with n>6), a joint space dynamics model is first established, and then the Moore-Penrose pseudo-inverse of the Jacobian matrix is ​​used. Mapping the joint space dynamics model to Cartesian space yields an end-effector dynamics model suitable for redundant robotic arms: (1) in, The inertial matrix in Cartesian space; Let the Coriolis force be the vector of centrifugal force in Cartesian space; Let Cartesian gravity vector be the vector; vector and These represent the external force and the control force in Cartesian space, respectively. The end-effector dynamics model is used to construct the subsequent nonlinear impedance control law.

[0010] Furthermore, the estimation error modeling method in step S2 specifically includes: Define the position tracking error of the robotic arm end effector. and its derivative ,in This represents the actual pose of the end effector. To track the desired trajectory; Based on the tracking error, the objective equation for nonlinear impedance control is established as follows: (2) in, Let be the desired inertia matrix, and be a positive definite diagonal constant matrix; as well as These are the end-position tracking error and its derivative, respectively. The nonlinear damping matrix to be designed; The nonlinear stiffness matrix to be designed; The external force applied to the end effector of the robotic arm; The objective equation defines the desired dynamic relationship of the robotic arm end effector under external force, providing a unified framework for the subsequent design of nonlinear stiffness and damping functions.

[0011] Furthermore, the exponential-saturation nonlinear stiffness function constructed in step S3 is specifically as follows: (3) in, The constant positive definite stiffness matrix, For the nonlinear stiffness increment based on the exponential function and the hyperbolic tangent saturation function, its first... The diagonal elements are defined as follows: (4) In the formula, For the first Position error components in the dimension; This is the stiffness gain coefficient, which determines the upper limit of stiffness increment saturation. and These are shape parameters that collectively control the rate at which stiffness increases with error; It is the saturation coefficient, used to adjust the smoothness of stiffness as it approaches the upper limit of saturation; It is the hyperbolic tangent function; The nonlinear stiffness function in position error When smaller, The stiffness increases slowly with the error, and the system remains compliant; when the error... When the value is large, the stiffness exponent term increases rapidly, through The function causes the stiffness increment to approach a saturation value. This provides strong correction while avoiding instability caused by excessive stiffness.

[0012] Furthermore, the Gaussian nonlinear damping function constructed in step S4 is specifically as follows: (5) in, The damping matrix is ​​a constant positive definite matrix. For the nonlinear damping increment based on the Gaussian function, its first... The diagonal elements are defined as follows: (6) In the formula, and The first Position error components and velocity error components in dimensionality; The damping gain coefficient determines the maximum value of the damping increment; and These are the position error width parameter and the velocity error width parameter, respectively, used to adjust the sensitive range of damping as the error changes; The nonlinear damping function is in position error and speed error As the Gaussian function term approaches zero, the system damping at the base value... The maximum value of the superimposed layer This enhances energy dissipation and suppresses oscillations and overshoot; as the error increases, the Gaussian function term decays exponentially, and the system damping approaches the base value. This is to avoid slow response caused by overdamping.

[0013] Furthermore, the impedance control law from Cartesian space to joint space designed in step S5 is specifically as follows: The desired terminal acceleration is calculated by solving the nonlinear impedance objective equation. : (7) Combining the Cartesian space end dynamics model Through the pseudo-inverse of the Jacobian matrix And its derivative, calculate the expected acceleration in joint space. : ; The joint space impedance control torque is finally obtained. for: (8) in, , , These are the joint space inertia matrix, the Coriolis force and centrifugal force vectors, and the gravity vector, respectively. This is a term for compensation of joint friction.

[0014] Furthermore, the closed-loop system stability analysis in step S6 specifically includes: In the absence of external force ( Under the given conditions, the Lyapunov function is selected as follows: (9) in as well as These are the end-position tracking error and its derivative, respectively. For positive definite expected inertia matrix, and These are the damping matrix and stiffness matrix, respectively; potential energy function. satisfy ,and ,in It is any null space vector orthogonal to the main task; calculate Time derivative along the closed-loop system trajectory: (10) Depend on The positive definiteness can be known ,and If and only if According to the LaSalle invariant set principle, the system asymptotically converges to the maximal invariant set. ,Depend on The positive definiteness is derived This proves that the system is at equilibrium when there are no external forces. The overall situation is asymptotically stable; Under external force Under the given conditions, select a positive definite storage function. Calculate its derivative: (11) Define system input as The output is ,but The condition is met, and according to the definition of passivity, the system is passive between the input force and the output velocity. By the passivity theorem, when the system interacts with any passive environment, the closed-loop system remains stable, thus ensuring the safety and robustness of the human-computer interaction process.

[0015] A robotic arm control system includes: The sensor module is used to collect position and force signals at the end of the robotic arm; The control module is used to execute control methods; The drive module is used to drive the robotic arm to move according to the torque commands output by the control module.

[0016] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method as described in any one of claims 1 to 7.

[0017] An electronic device includes a memory, a processor, and a computer program stored in the memory, wherein the processor executes the program to implement a nonlinear impedance control method for a robotic arm based on Gaussian damping and exponential stiffness.

[0018] The technical effects achieved by this invention are as follows: This invention presents a nonlinear impedance control method for a robotic arm based on Gaussian damping and exponential stiffness. By introducing state-dependent Gaussian nonlinear damping and exponential-saturation nonlinear stiffness, the impedance parameters can be adaptively adjusted according to real-time pose and velocity errors, overcoming the limitations of traditional linear impedance control methods that have fixed parameters and limited performance in complex dynamic tasks. The nonlinear stiffness provides strong correction under large errors to ensure tracking accuracy, while the nonlinear damping enhances energy dissipation to suppress oscillations under small errors. The synergistic effect of these two methods enables the robotic arm to have smaller trajectory deviations and faster recovery capabilities under external disturbances. At the same time, rigorous Lyapunov stability proofs and passive analysis theoretically guarantee the global asymptotic stability and interaction safety of the system during free motion and interaction with passive environments. Furthermore, this method is not dependent on a specific robotic arm configuration and is applicable to various series redundant robotic arms, exhibiting good engineering versatility and promotional value. Attached Figure Description

[0019] Figure 1 This is a schematic diagram comparing the trajectory tracking and error of the method of this invention and linear impedance control in a simulation environment; Figure 2 This is a schematic diagram illustrating the performance improvement of the method of the present invention compared to linear impedance control; Figure 3 This is a schematic diagram of the experimental platform and the seven-DOF Franka Emika Panda robotic arm of the present invention.

[0020] Figure 4 This is a schematic diagram comparing the actual end-point running trajectory of the method of the present invention and linear impedance control in the yz plane; Figure 5 This is a schematic diagram comparing the tracking trajectory and error of the method of the present invention and linear impedance control under external interference; Figure 6 This is a schematic diagram comparing the changes in external force, damping parameters, and stiffness parameters applied under external disturbances with those of the method of this invention and linear impedance control over time. Detailed Implementation

[0021] To make the objectives and advantages of this invention clearer, the invention will be specifically described below with reference to embodiments. It should be understood that the following text is merely used to describe one or more specific embodiments of the invention and does not strictly limit the scope of protection specifically claimed by the invention.

[0022] like Figures 1-6As shown, this embodiment uses the 7-DOF Franka Emika Panda collaborative robotic arm as the controlled object, and implements nonlinear impedance control (GE-IC) based on Gaussian damping and exponential stiffness in Cartesian space. The control algorithm is implemented on a remote PC running the Ubuntu operating system, and is integrated with the robotic arm's underlying controller through the Robot Operating System (ROS), enabling real-time communication and control at a frequency of 1kHz.

[0023] A nonlinear impedance control method for a robotic arm based on Gaussian damping and exponential stiffness includes the following steps: Step S1: Robotic Arm Dynamics Modeling and Cartesian Space Mapping Consider an n-DOF serial robotic arm, whose joint space dynamics equations are: (12) in, The joint angle vector; For joint velocity and acceleration; The inertia matrix of the symmetric positive definite robotic arm; The vectors of the Coriolis force and the centrifugal force; It is the gravitational moment vector; For the joint driving torque, External force applied to the end; It is a Jacobian matrix.

[0024] The relationship between the velocity of the end effector in Cartesian space and the joint velocity is as follows: (13) Differentiating equation (13) yields the Cartesian acceleration: (14) The Franka Emika Panda robotic arm used in this embodiment is a 7-DOF serial robotic arm, where the number of degrees of freedom n=7 is greater than the motion dimension m=6 of the end effector in Cartesian space. The Jacobian matrix describes the mapping relationship between joint space velocity and Cartesian space velocity. It is not a square matrix, but its regular inverse is... No. For such redundant robotic arms, the Moore-Penrose pseudoinverse of the Jacobian matrix must be used to derive their spatial dynamics transformation: (15) This pseudo-inverse satisfies When considering only the primary task of the end-effector motion, the joint space velocity can be expressed as: Substituting joint accelerations into the joint space dynamics equations, a Cartesian space dynamics model suitable for redundant robotic arms can be derived: (16) in, (17) (18) (19) In the formula, Let be the Cartesian space inertia matrix of the redundant robotic arm; This is the equivalent control force in Cartesian space.

[0025] Step S2: Error Definition and Nonlinear Impedance Control Objective Establishment Define the position tracking error of the robotic arm's end effector as: (20) Its derivative is the velocity tracking error: (twenty one) in, This represents the actual pose of the end effector. To track the desired trajectory.

[0026] Based on the above definition of error, the objective equation for nonlinear impedance control is established as follows: (2) in, The desired inertia matrix is ​​designed as a positive definite diagonal constant matrix. The nonlinear damping matrix to be designed; The nonlinear stiffness matrix to be designed; The external force applied to the end of the robotic arm.

[0027] Step S3: Design of exponential-saturated nonlinear stiffness function Inspired by the biomimetic principle that stiffness increases nonlinearly with deformation in the stress-strain relationship of blood vessels, this embodiment designs the following nonlinear stiffness function with saturation characteristics to achieve precise control while maintaining compliance in the small error region, accelerate convergence in the large error region, and prevent the control force from saturating or becoming unstable due to infinite stiffness growth at extremely large errors: (3) in, The constant positive definite stiffness matrix; For nonlinear stiffness increments, its first... The diagonal elements are defined as follows: (4) In the formula, This is the stiffness gain coefficient, which determines the upper limit of stiffness increment saturation. The parameter is the exponential growth rate. For power-order shape parameters, and Together they control the rate at which stiffness increases with error; It is the saturation coefficient, used to adjust the smoothness of stiffness as it approaches the upper limit of saturation; As a hyperbolic tangent function, and as a smooth saturation function, it can limit the stiffness increment to a certain value. Within the range.

[0028] The physical meaning of this nonlinear stiffness function is: when the position error When smaller, The stiffness increases slowly with the error, and the system remains compliant; when the error... When the value is large, the exponent term increases rapidly, through The function causes the stiffness increment to approach a saturation value. This provides greater resilience to accelerate convergence while avoiding excessive stiffness that could lead to instability.

[0029] In this embodiment, parameters are set for the three dimensions of translational motion: basic stiffness K0 = 100I K 0=100 I N / m, stiffness gain Shape parameters , saturation coefficient .

[0030] Step S4: Design and Implementation of Nonlinear Impedance Control Law The desired terminal acceleration is calculated by solving the nonlinear impedance objective equation (1.23). : (7) Combining the dynamics model at the end of the Cartesian space, and through the pseudo-inverse of the Jacobian matrix... And its derivative, calculate the expected acceleration in joint space. : (twenty two) The joint space impedance control torque is finally obtained. for: (8) in, , , These are the joint space inertia matrix, the Coriolis force and centrifugal force vectors, and the gravity vector, respectively. This is a term for compensation of joint friction.

[0031] Step S6: Stability and passivity analysis of the closed-loop system The stability proof provides a rigorous mathematical foundation for the proposed nonlinear impedance control algorithm. In the absence of external forces ( Under the given conditions, the Lyapunov function is chosen as the total energy of the system: (9) Where the potential energy function satisfy Defined as .

[0032] calculate Time derivative along the closed-loop system trajectory: (10) because Symmetric positive definite, therefore , and only if According to the LaSalle invariant set principle, the system asymptotically converges to the maximal invariant set. The positive definiteness is derived This proves that the system is globally asymptotically stable at the equilibrium point when there are no external forces.

[0033] Under external force conditions Next, select the stored function. Calculate its derivative: Define system input as The output is ,but This is true. According to the definition of passivity, the system is passive between the input force and the output velocity. The passivity theorem states that the system can maintain closed-loop stability when interacting with any passive environment.

[0034] Step S7: Simulation Experiment Verification A two-degree-of-freedom robotic arm simulation model was built in the MATLAB / Simulink environment. The dynamic parameters are shown in Table 1.

[0035] To verify the effectiveness and advantages of the proposed GE-IC method, a comparative experiment was conducted with linear impedance control (L-IC) under identical design parameters. The controller parameters were set as follows: L-IC control group:

[0036] GE-IC method: ; The desired trajectory is a circular path: ; The experiment was divided into three phases: steady-state tracking phase (3.0-5.0 s), disturbance application phase (5.0-5.5 s, with applied step force). ), recovery phase (5.5-7.5s).

[0037] Simulation results show that GE-IC significantly outperforms L-IC in all performance indicators: GE-IC exhibits higher trajectory tracking accuracy during steady-state tracking; during disturbance application, GE-IC's maximum transient deviation and steady-state offset are both smaller than L-IC's; and during recovery, GE-IC demonstrates faster recovery speed and smaller recovery overshoot. (Appendix) Figure 1 To be continued Figure 2 The results of the above comparison are presented intuitively.

[0038] Step S8: Physical Experiment Verification To further verify the effectiveness of the method of this invention in a real physical system, a physical experiment was conducted on the 7-DOF Franka EmikaPanda robotic arm platform. The experimental platform is attached. Figure 3 As shown.

[0039] The desired trajectory focuses on translational motion control at the end point, setting the position in the x-direction to remain constant, while tracking sinusoidal trajectories in the y and z directions respectively: ; During the operation of the robotic arm, the operator applies step-like force pulses in the y and z directions to simulate sudden environmental contact or human intervention in actual tasks.

[0040] Controller parameters are set as follows: L-IC uses: ; GE-IC adopts Basic damping Damping gain Position error width parameter Speed ​​error width parameter Basic stiffness stiffness gain Shape parameters , saturation coefficient .

[0041] The experimental results are attached. Figure 4 As shown. (Attached) Figure 4The comparison between the actual and desired motion trajectories of the robotic arm's end effector in the yz plane under L-IC and GE-IC control is presented. In the absence of external force, GE-IC significantly outperforms L-IC in tracking accuracy. When external force is applied, the trajectory under L-IC control deviates noticeably and recovers slowly with oscillations after the disturbance is removed, while the trajectory under GE-IC control deviates less and converges back to the desired trajectory faster and more smoothly after the disturbance is removed.

[0042] Appendix Figure 5 The comparison of trajectories and errors in the y and z directions is shown. During the application of external force, the peak error of GE-IC is lower than that of L-IC, and it converges rapidly after the disturbance is removed. (Appendix) Figure 6 The experiment recorded the changes in applied external forces, damping, and stiffness parameters. During the stable tracking phase without interference, the nonlinear damping term remained at a high level to suppress oscillations. When a sudden increase in external force led to an increase in error, the nonlinear stiffness rapidly increased to generate a corrective force, while the damping increased accordingly to consume excess energy and suppress overshoot. This adaptive adjustment mechanism enables GE-IC to simultaneously possess the robustness of rapid error correction and the smoothness of oscillation suppression.

[0043] In summary, the nonlinear impedance control method based on Gaussian damping and exponential stiffness proposed in this invention effectively overcomes the problems of fixed parameters and insufficient adaptive capability in traditional linear impedance control through a state-adaptive stiffness and damping mechanism. It achieves comprehensive optimization of trajectory accuracy, disturbance rejection robustness, and interaction compliance in dynamic interactive tasks. This invention is applicable to various types of serial collaborative robotic arms and has good engineering application prospects and promotional value.

[0044] The above description is merely a preferred embodiment of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention. Structures, devices, and operating methods not specifically described or explained in this invention are implemented according to conventional methods in the art unless otherwise specified or limited.

Claims

1. A nonlinear impedance control method for a robotic arm based on Gaussian damping and exponential stiffness, characterized in that: Includes the following steps: Step S1: Establish a dynamic model of the multi-degree-of-freedom robotic arm and map the joint space dynamics to the operating space to obtain the end effector dynamic model of the robotic arm; Step S2: Define the position tracking error of the robotic arm's end effector and its derivative A nonlinear impedance control target is established based on the error; Step S3: Based on position tracking error An exponential-saturation nonlinear stiffness function is constructed to achieve adaptive adjustment of stiffness that increases nonlinearly with increasing error and eventually saturates. Step S4: Based on position error and speed error A Gaussian nonlinear damping function is constructed to achieve nonlinear adjustment of damping, which increases when the error is small and decreases when the error is large. Step S5: Construct a nonlinear impedance model based on the exponential-saturated stiffness function and the Gaussian damping function, and design the impedance control law from Cartesian space to joint space; Step S6: Construct a Lyapunov function to perform stability analysis on the closed-loop system, proving that the system is globally asymptotically stable at the equilibrium point when there is no external force; when there is an external force, the system is passive between the input force and the output velocity, thus ensuring the closed-loop stability when interacting with any passive environment.

2. The nonlinear impedance control method for a robotic arm based on Gaussian damping and exponential stiffness according to claim 1, characterized in that: In step S1, for an n-DOF serial robotic arm, a joint space dynamics model is first established, and then the Moore-Penrose pseudo-inverse of the Jacobian matrix is ​​used. Mapping the joint space dynamics model to Cartesian space yields an end-effector dynamics model suitable for redundant robotic arms: (1) in, The inertial matrix in Cartesian space; Let the Coriolis force be the vector of centrifugal force in Cartesian space; Let Cartesian gravity vector be the vector; vector and These represent the external force and the control force in Cartesian space, respectively.

3. The nonlinear impedance control method for a robotic arm based on Gaussian damping and exponential stiffness according to claim 2, characterized in that: The estimation error modeling method in step S2 includes: Define the position tracking error of the robotic arm end effector. and its derivative ,in This represents the actual pose of the end effector. To track the desired trajectory; Based on the tracking error, the objective equation for nonlinear impedance control is established as follows: （2） in, Let be the desired inertia matrix, and be a positive definite diagonal constant matrix; The nonlinear damping matrix to be designed; The nonlinear stiffness matrix to be designed; The external force applied to the end of the robotic arm.

4. The nonlinear impedance control method for a robotic arm based on Gaussian damping and exponential stiffness according to claim 3, characterized in that: The exponential-saturation nonlinear stiffness function constructed in step S3 is specifically as follows: （3） in, The constant positive definite stiffness matrix, For the nonlinear stiffness increment based on the exponential function and the hyperbolic tangent saturation function, its first... The diagonal elements are defined as follows: （4） In the formula, For the first Position error components in the dimension; This is the stiffness gain coefficient, which determines the upper limit of stiffness increment saturation. and These are shape parameters that collectively control the rate at which stiffness increases with error; It is the saturation coefficient, used to adjust the smoothness of stiffness as it approaches the upper limit of saturation; It is the hyperbolic tangent function.

5. The nonlinear impedance control method for a robotic arm based on Gaussian damping and exponential stiffness according to claim 4, characterized in that: The Gaussian nonlinear damping function constructed in step S4 is specifically as follows: （5） in, The damping matrix is ​​a constant positive definite matrix. For the nonlinear damping increment based on the Gaussian function, its first... The diagonal elements are defined as follows: （6） In the formula, and The first Position error components and velocity error components in dimensionality; The damping gain coefficient determines the maximum value of the damping increment; and These are the position error width parameter and the velocity error width parameter, respectively, used to adjust the sensitivity range of damping as the error changes.

6. The nonlinear impedance control method for a robotic arm based on Gaussian damping and exponential stiffness according to claim 5, characterized in that: The impedance control law from Cartesian space to joint space designed in step S5 is as follows: The desired terminal acceleration is calculated by solving the nonlinear impedance objective equation. : （7） Combining the dynamics model at the end of the Cartesian space, and through the pseudo-inverse of the Jacobian matrix... And its derivative, calculate the expected acceleration in joint space. : ； The joint space impedance control torque is finally obtained. for: （8） in, , , These are the joint space inertia matrix, the Coriolis force and centrifugal force vectors, and the gravity vector, respectively. This is a term for compensation of joint friction.

7. The nonlinear impedance control method for a robotic arm based on Gaussian damping and exponential stiffness according to claim 6, characterized in that: The closed-loop system stability analysis in step S6 specifically includes: Under the condition of no external force, the Lyapunov function is selected as follows: （9） Where the potential energy function satisfy ,and ; calculate Time derivative along the closed-loop system trajectory: （10） Depend on The positive definiteness can be known ,and If and only if According to the LaSalle invariant set principle, the system asymptotically converges to the maximal invariant set. ,Depend on The positive definiteness is derived This proves that the system is at equilibrium when there are no external forces. The overall situation is asymptotically stable; Under external force Under the given conditions, select a positive definite storage function. Calculate its derivative: （11） Define system input as The output is ,but This holds true. According to the definition of passivity, the system is passive between the input force and the output velocity. By the passivity theorem, the closed-loop system remains stable when the system interacts with any passive environment.

8. A robotic arm control system, characterized in that: include: The sensor module is used to collect position and force signals at the end of the robotic arm; A control module is configured to execute the control method according to any one of claims 1 to 7; The drive module is used to drive the robotic arm to move according to the torque commands output by the control module.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by the processor, it implements the control method according to any one of claims 1 to 7.

10. An electronic device, characterized in that: It includes a memory, a processor, and a computer program stored in the memory, wherein the processor executes the program to implement the control method according to any one of claims 1 to 7.

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