Constraint following control method for dual-arm collaborative robot based on differential homeomorphism mapping

Abstract

The application discloses a constraint following control method of a dual-arm collaborative robot based on a differential homeomorphism mapping, and relates to the field of robot control; the nominal dynamics model of the robot is established, and a geometric constraint equation is defined according to a collaborative task; a constraint tracking variable and a complementary coordinate are constructed, a global differential homeomorphism mapping is established, and the system dynamics model is decoupled into a constraint tracking subsystem and a complementary subsystem; a robust adaptive controller fusing a sliding mode control and an adaptive law is designed, system uncertainty is estimated and compensated on line; the application effectively simplifies the constraint processing complexity through the differential homeomorphism mapping, significantly improves the constraint tracking precision and robustness of the system under model uncertainty and external disturbance, and realizes high-precision and high-stability collaborative task execution of the dual-arm robot in a dynamic environment.

Description

Technical Field

[0001] This invention belongs to the field of robot control technology, and more specifically, relates to a constraint following control method for a dual-arm collaborative robot based on differential homeomorphism mapping. Background Technology

[0002] In modern manufacturing, logistics, and medical rehabilitation, dual-arm collaborative robots are playing an increasingly important role due to their human-like operating mode and high flexibility. They can mimic human arms to collaboratively complete complex tasks such as precision assembly and handling large objects, which places extremely high demands on the robots' control precision and coordination capabilities.

[0003] Traditional control methods typically employ strategies such as impedance control, task-based priority control, or model predictive control. These techniques demonstrate good performance for deterministic systems in structured, known environments. They primarily handle constraints through force / potential hybrid control or optimization algorithms, heavily rely on accurate system dynamics models, and often assume that uncertainties in the operating environment are negligible or known.

[0004] However, existing technologies face significant challenges in practical applications. First, the inherent parameter uncertainties, unmodeled dynamics, and unpredictable external disturbances of robot systems can severely degrade the performance of controllers designed based on nominal models, leading to increased constraint following errors or even task failure. Second, many methods lack effective online adaptive mechanisms and are not robust enough when dealing with these uncertainties. Furthermore, directly integrating complex geometric constraints into the control law design in joint space often results in complex system equations and heavy computational burdens, making it difficult to achieve real-time and efficient control, thus limiting the widespread application of dual-arm robots in dynamic and unstructured environments. Summary of the Invention

[0005] (a) Technical problems to be solved

[0006] To address the problems in related technologies, this invention provides a constrained following control method for a dual-arm collaborative robot based on differential homeomorphism mapping, thereby overcoming the aforementioned technical problems in existing related technologies.

[0007] (II) Technical Solution

[0008] To solve the above-mentioned technical problems, the present invention is achieved through the following technical solution:

[0009] In a first aspect, the present invention provides a constraint-following control method for a dual-arm cooperative robot based on differential homeomorphism mapping, comprising the following steps:

[0010] S1. Establish the nominal system dynamics model of the dual-arm collaborative robot;

[0011] S2. Define geometric constraint equations in the task space according to the dual-arm collaborative task; construct a global differential homeomorphism and a complete system dynamics model based on the geometric constraint equations.

[0012] The global differential homeomorphism mapping is obtained by defining constraint tracking variables and constructing complementary coordinates; the complete system dynamics model is obtained by introducing the Jacobian matrix of the geometric constraint equations and uncertainty data into the nominal system dynamics model;

[0013] S3. Based on global differential homeomorphism mapping, the state space of the complete system dynamics model is transformed into the state space of the target system, and the mapped target system dynamics model is obtained.

[0014] S4. Based on the mapped target system dynamics model, define the tracking error and sliding surface, design the adaptive law, and synthesize the control law to obtain a robust adaptive controller.

[0015] The robust adaptive controller estimates the uncertainty of the mapped target system dynamics model online in real time and generates real-time control torque;

[0016] S5. Apply the real-time control torque to the dual-arm collaborative robot to drive the movement of each joint of the dual-arm collaborative robot;

[0017] Preferably, step S1 includes the following steps:

[0018] S11. Collect the angular displacement data and linear displacement data of all joints of the dual-arm robot to obtain the set of angular displacement of the joints of the dual-arm robot and the set of linear displacement of the joints of the dual-arm robot.

[0019] By performing vector combination operations on all angular displacement data in the set of angular displacements of the joints of the dual-arm robot and all linear displacement data in the set of linear displacements of the joints of the dual-arm robot, a generalized coordinate vector is obtained.

[0020] S12. Collect the mass data of each link of the dual-arm robot, the position data of the center of mass of the link in the link coordinate system, and the inertia tensor data of the link through robot design drawings, CAD models and parameter identification experiments.

[0021] Based on the generalized coordinate vector, the mass data of each link of the dual-arm robot, the position data of the center of mass of the link in the link coordinate system, and the inertia tensor data of the link, a positive definite inertia matrix, Coriolis force and centripetal force matrix and gravity vector are constructed.

[0022] S13. Based on the positive definite inertia matrix, Coriolis force and centripetal force matrix and gravity vector, a nominal system dynamic model of the dual-arm robot is constructed by utilizing Lagrange's mechanics formula and introducing a friction model; the nominal dynamic equation describes the dynamics of the dual-arm robot under ideal and uncertainty-free conditions.

[0023] Preferably, step S2 includes the following steps:

[0024] S21. Based on the specific task of dual-arm collaboration, the geometric relationship between the end effectors of the specific task of dual-arm collaboration is transformed into mathematical equations to obtain the initial geometric constraint equations.

[0025] The initial geometric constraint equations are transformed into algebraic vector equations. Based on the dimensions of rotation of the revolute joints and translation of the translational joints, the values ​​of the constraint dimensions in the algebraic vector equations are determined, and the geometric constraint equations are obtained.

[0026] S22. Calculate the Jacobian matrix of the geometric constraint equations and collect uncertainty data; introduce the Jacobian matrix of the geometric constraints and the uncertainty data into the nominal system dynamics model to obtain the complete system dynamics model;

[0027] S23. Based on geometric constraint equations, global differential homeomorphism is obtained by defining constraint tracking variables and constructing complementary coordinates.

[0028] Preferably, step S22 includes the following steps:

[0029] S221. By taking the total differential equation formula for the geometric constraint equation with respect to time, the Jacobian matrix of the constraint is obtained.

[0030] S222. Collect parameter uncertainty data, unmodeled dynamic data, external disturbance data, and friction model error data to obtain uncertainty term data;

[0031] S223. Input the Jacobian matrix of the constraint into the Lagrange multiplier method. When the geometric constraint equation is equal to zero, the constraint force is obtained.

[0032] By incorporating the aforementioned uncertainty data and constraints into the nominal system dynamics model, a complete system dynamics model is obtained;

[0033] Preferably, step S23 includes the following steps:

[0034] S231. Based on the geometric constraint equations and the complete system dynamics model, define constraint position error variables and constraint velocity error variables, and fuse the constraint position error variables and constraint velocity error variables to obtain constraint tracking variables.

[0035] S232. Based on the number of joints and constraint dimensions of the complete system dynamics model, and the dimensions of the complementary variables, obtain the complementary variables;

[0036] Set a nonlinear mapping function equal to the complementary variables to obtain the complementary coordinate transformation function; specifically: calculate the null space projection matrix of the constrained Jacobian matrix; let the first derivative of the complementary variables with respect to time be equal to the product of the transpose of the null space projection matrix and the first derivative of the generalized coordinate vector with respect to time, so that the gradient of the complementary coordinate transformation function is orthogonal to the space spanned by the row vectors of the constrained Jacobian matrix on the constrained manifold.

[0037] S233. Construct a new state vector based on complementary variables and constraint tracking variables; define an initial global differential homeomorphism based on the new state vector;

[0038] The invertibility of the initial global differential homeomorphism is verified based on the complementary coordinate transformation function; if the verification is successful, the initial global differential homeomorphism is taken as the global differential homeomorphism; otherwise, return to S232 to correct the complementary coordinate transformation function until the verification is successful and the global differential homeomorphism is obtained.

[0039] Preferably, step S231 includes the following steps:

[0040] S2311. Set a variable equal to the geometric constraint equation to obtain the constraint position error variable; the constraint position error variable is used to quantify the degree of deviation of the generalized coordinate vector from the ideal constraint when it is zero.

[0041] S2312. Calculate the first derivative of the constraint position error variable with respect to time to obtain the constraint velocity error variable;

[0042] S2313. The constraint position error variable and the constraint velocity error variable are fused to obtain the constraint tracking variable;

[0043] Preferably, step S3 includes the following steps:

[0044] S31. Using the chain rule, differentiate the constraint velocity error variable in the new state vector to obtain the first derivative variable of the constraint velocity error variable.

[0045] S32. Combine the complete system dynamics model equations with the constraint velocity error variables and their first derivatives to obtain the combined equations.

[0046] S33. Using the transformation relationship in the inverse mapping, all variables in the equations of the complete system dynamics model are represented by the variables in the new state vector, thus obtaining the variables after the transformation relationship;

[0047] S34. Substitute the transformed variables into the simultaneous equations and simplify to obtain the mapped target system dynamics model described by the new state vector; the target system dynamics model includes a constraint tracking sub-model and a complementary sub-model.

[0048] Preferably, step S4 includes the following steps:

[0049] S41. Define the control objective of the constrained tracking sub-model in the target system dynamics model as the value of the constrained tracking sub-model asymptotically converges to zero, and obtain the defined tracking error; define a sliding surface based on the control objective of the constrained tracking sub-model to accelerate the convergence of the constrained tracking sub-model and enhance its robustness, and obtain the equation of the sliding surface;

[0050] S42. Set an upper bound for the uncertainty term data in the target system dynamics model; based on the sliding mode surface, design the equation of the adaptive law to evaluate the upper bound online, and obtain the adaptive law;

[0051] S43. Based on Lyapunov stability theory, design the control torque vector τ to be designed to obtain the designed control law; the control law includes feedforward model compensation term, constraint force compensation term, PD feedback term and robust adaptive term.

[0052] The defined tracking error, together with the sliding surface, adaptive law, and control law, constitutes a robust adaptive controller.

[0053] Preferably, step S5 includes the following steps:

[0054] S51. By using encoders on the robot joints, the position and velocity of each joint are read in real time to obtain a real-time set of joint positions and a real-time set of joint velocities; the joint velocities are obtained through a tachogenerator.

[0055] S52. Substitute the joint positions and joint velocities from the real-time joint position set and the real-time joint velocity set into the differential homeomorphism mapping in S3 to calculate the target system state at the current moment.

[0056] S53. Estimate the real-time upper bound of the error signal and adaptive law in the target system state at the current moment, input the estimated value of the real-time upper bound into the control law equation in S4, and obtain the control torque applied to each joint to obtain the real-time control torque set.

[0057] S54. The real-time control torque of the real-time control torque concentration is transmitted to the corresponding joint actuators of the dual-arm robot to drive the robotic arm to move until the collaborative task is completed.

[0058] Secondly, the present invention also provides a constraint following control system for a dual-arm collaborative robot based on differential homeomorphism mapping, used to implement the aforementioned constraint following control method for a dual-arm collaborative robot based on differential homeomorphism mapping, the system comprising:

[0059] System modeling module: Constructs a nominal system dynamic model of the dual-arm robot, and establishes joint space dynamic equations including inertia, Coriolis force, gravity and friction using the Lagrange method;

[0060] Constraints and Reconstruction Module: Define geometric constraints based on collaborative tasks, introduce their Jacobian matrix, uncertainties and external disturbances, construct a complete system dynamics model, and establish a differential homeomorphism mapping between constraint tracking variables and complementary coordinates;

[0061] Model decoupling module: Using differential homeomorphism mapping, the complete system dynamics model of the dual-arm robot is transformed to a new state space, decoupling it into a target system dynamics model that includes a constraint tracking sub-model and a complementary sub-model;

[0062] Controller design module: Based on the target system dynamics model, a sliding surface and adaptive law are designed, the uncertainty boundary is estimated online, and a control law including model compensation, feedback and robust terms is synthesized to obtain a robust adaptive controller;

[0063] Control execution module: Reads joint status in real time, calculates system error and uncertainty estimation through differential homeomorphism mapping, generates joint torque by robust adaptive controller and drives robot to complete collaborative task.

[0064] (III) Beneficial Effects

[0065] The present invention has the following beneficial effects:

[0066] This invention constructs a global differential homeomorphism to efficiently decouple complex geometric constraints from joint space dynamics models, significantly improving the design and analysis efficiency of control systems. This method transforms the original nonlinear constraint system into two independent subsystems: constraint tracking and internal dynamics. This allows the controller to be designed directly for simplified error dynamics, effectively overcoming the problems of complex constraint processing calculations and poor real-time performance in traditional methods.

[0067] The robust adaptive controller designed in this invention integrates sliding mode control and parameter adaptation mechanisms, enabling online real-time estimation and compensation of system uncertainties. This design ensures that the dual-arm robot can still accurately and stably follow task constraints even in the presence of model errors and unknown disturbances, significantly enhancing the system's adaptability and robustness in unstructured environments.

[0068] This invention achieves integrated control of the entire process from task constraint definition to joint torque generation, combining theoretical rigor with engineering practicality. This control method not only ensures high-precision execution of dual-arm collaborative tasks, but also has good versatility due to its modular design, and can be widely applied to various robot collaborative operation scenarios that require meeting complex geometric constraints.

[0069] Of course, any product implementing this invention does not necessarily need to achieve all of the advantages described above at the same time. Attached Figure Description

[0070] To more clearly illustrate the technical solutions of the embodiments of the invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the invention. For those skilled in the art, the drawings can be obtained from these drawings without creative effort.

[0071] Figure 1 This is a flowchart illustrating the constraint following control method for a dual-arm collaborative robot based on differential homeomorphism mapping according to the present invention.

[0072] Figure 2 This is a schematic diagram of the constraint following control system of the dual-arm collaborative robot based on differential homeomorphism mapping according to the present invention. Detailed Implementation

[0073] The technical solutions of the embodiments of the invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the invention, and not all embodiments. Based on the embodiments of the invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the invention.

[0074] Taking the control of a traditional dual-arm collaborative robot as an example, when collaborative handling tasks need to be completed, impedance control methods based on precise model calculations are usually adopted. For example, when two robotic arms need to jointly handle a rigid object, the control system will obtain the precise mass and inertia parameters of the robotic arms in advance through a CAD model, and establish a dynamic model based on this. In an ideal laboratory environment, the system detects the interaction force between the end effectors through force sensors and adjusts the collaborative movement of the two arms using a preset impedance model.

[0075] However, when the actual load mass deviates from the nominal value by 10%, or when encountering unknown external disturbances (such as conveyor belt vibration), this control method relying on a precise model reveals significant flaws. Due to the lack of effective online parameter identification and disturbance compensation mechanisms, the control system cannot quickly adapt to these changes, resulting in continuous relative pose errors between end effectors. Experimental data shows that when a 15% deviation in load parameters and an external disturbance with an amplitude of ±5N coexist, the constraint tracking position error of the traditional method can reach ±15mm, the speed error can reach ±20mm / s, and the internal force fluctuation of the end effector can reach as high as 30%. This error will be converted into unnecessary internal forces, which will affect the stability of the handling and may damage the grasped object. Especially under high-speed motion conditions, the uncompensated Coriolis force effect will further amplify the tracking error, ultimately leading to the failure of the collaborative task.

[0076] As can be seen from the above embodiments of the prior art, the main problems of the prior art are: incorporation of parameter uncertainty and external interference, insufficient robustness and rigid constraint handling.

[0077] To resolve the above issues, please refer to [link / reference]. Figure 1 This invention discloses a constrained following control method for a dual-arm cooperative robot based on differential homeomorphism mapping, comprising the following steps:

[0078] S1. Establish the nominal system dynamics model of the dual-arm collaborative robot;

[0079] S2. Define geometric constraint equations in the task space according to the dual-arm collaborative task; construct a global differential homeomorphism and a complete system dynamics model based on the geometric constraint equations.

[0080] The global differential homeomorphism mapping is obtained by defining constraint tracking variables and constructing complementary coordinates; the complete system dynamics model is obtained by introducing the Jacobian matrix of the geometric constraint equations and uncertainty data into the nominal system dynamics model;

[0081] S3. Based on global differential homeomorphism mapping, the state space of the complete system dynamics model is transformed into the state space of the target system, and the mapped target system dynamics model is obtained.

[0082] S4. Based on the mapped target system dynamics model, define the tracking error and sliding surface, design the adaptive law, and synthesize the control law to obtain a robust adaptive controller.

[0083] The robust adaptive controller estimates the uncertainty of the mapped target system dynamics model online in real time and generates real-time control torque;

[0084] S5. Apply the real-time control torque to the dual-arm collaborative robot to drive the movement of each joint of the dual-arm collaborative robot;

[0085] The embodiments of the present invention construct a nominal dynamic model of a dual-arm robot and establish a global differential homeomorphism based on task geometric constraints, decoupling the complex constraint system into two independent subsystems: constraint tracking and internal dynamics. On this basis, a robust controller integrating sliding mode control and adaptive law is designed to estimate and compensate for system uncertainties online. This achieves a control effect that maintains high-precision constraint tracking even under model uncertainties and external disturbances, while realizing the ability of the dual-arm robot to stably and adaptively complete complex collaborative tasks in dynamic environments.

[0086] In one embodiment, step S1 includes the following steps:

[0087] S11. Collect the angular displacement data and linear displacement data of all joints of the dual-arm robot to obtain the set of angular displacements and the set of linear displacements of the joints of the dual-arm robot; the angular displacements are for rotary joints, and the linear displacements are for translational joints.

[0088] By performing vector combination operations on all angular displacement data in the set of angular displacements of the joints of the dual-arm robot and all linear displacement data in the set of linear displacements of the joints of the dual-arm robot, a generalized coordinate vector q∈R can be obtained. n ;

[0089] The q∈R n Where R represents the position value of each joint, which is a real number; n is the total joint degrees of freedom of the dual-arm robot. For example, if two 7-DOF robotic arms cooperate, then n=14. Furthermore, q=[left arm joint 1, left arm joint 2,..., left arm joint 7, right arm joint 1, right arm joint 2,..., right arm joint 7]. If both arms are at the zero point position, then q is a vector containing 14 zeros: q=[0,0,0,0,0,0,0,0,0,0,0,0,0,0];

[0090] S12. Collect the mass data of each link of the dual-arm robot, the position data of the center of mass of the link in the link coordinate system, and the inertia tensor data of the link through robot design drawings, CAD models and parameter identification experiments.

[0091] A positive definite inertia matrix M(q)∈R is constructed based on the generalized coordinate vector, the mass data of each link of the dual-arm robot, the position data of the center of mass of the link in the link coordinate system, and the inertia tensor data of the link. (n×n) The Coriolis force and centripetal force matrix C(q,q1)∈R (n×n) and the gravity vector G(q)∈R nWhere q1 represents the first derivative of q with respect to time, i.e. the velocity of all joints. If q is displacement, then q1 is velocity; if q is rotation angle, then q1 is angular velocity.

[0092] S13. Based on the positive definite inertia matrix M(q), the Coriolis force and centripetal force matrix C(q,q1), and the gravity vector G(q), a nominal system dynamic model of the dual-arm robot is constructed by utilizing Lagrange's mechanics formulas and introducing a friction model. The nominal dynamic equations describe the dynamics of the dual-arm robot under ideal, uncertainty-free conditions. The formulas for the nominal system dynamic model equations are as follows:

[0093] M(q)*q2+C(q,q1)*q1+G(q)+F(q1)=τ;

[0094] Where q2 represents the second derivative of q with respect to time, i.e., acceleration; if q is displacement, then q2 is velocity; if q is rotation angle, then q2 is angular velocity; τ∈R n , represents the nominal control torque vector to be designed; F(q1) represents the friction force vector; the friction force model includes viscous friction and Coulomb friction, F(q1) = F v *q1+F c *sgn(q1), where F v F c These represent the viscous friction coefficient and the Coulomb friction coefficient, respectively.

[0095] In specific implementation, step S12 is as follows: the positive definite inertia matrix M(q) is determined by comprehensively considering the mass and moment of inertia of all links and based on the robot's current posture (i.e., joint position q) to determine the required torque to generate a specific acceleration for the robot. This matrix quantifies the inertia of the system. The Coriolis force and centripetal force matrix C(q,q1) is obtained by capturing the Coriolis effect (similar to the effect of the Earth's rotation on airflow) and centripetal effect generated when the robot's joints move at high speed. The gravity vector G(q) is obtained by calculating the supporting torque that each joint needs to continuously provide when balancing its own weight, based on the mass and center of mass position of each link and the direction of gravitational acceleration.

[0096] The above embodiments integrate robot joint displacement, link mass, center of mass position, and inertial parameters to construct a nominal dynamic model that accurately characterizes the system's inertia, Coriolis force, gravity, and friction effects. This lays a precise model foundation for subsequent control design, ensuring that the controller design is highly compatible with the dynamic characteristics of the physical system, and effectively improving the rationality and reliability of the entire control architecture.

[0097] Step S2 above includes the following steps:

[0098] S21. Based on the specific task of the two-arm collaboration (such as jointly and stably grasping an object), the geometric relationship that the end effectors of the specific task of the two-arm collaboration must satisfy is transformed into mathematical equations to obtain the initial geometric constraint equations.

[0099] The initial geometric constraint equations are transformed into algebraic vector equations. Based on the dimensions of rotation of the revolute joint and translation of the translational joint, the value of the constraint dimension m in the algebraic vector equations is determined, and the geometric constraint equations are obtained.

[0100] S22. Calculate the Jacobian matrix of the geometric constraint equations and collect uncertainty data; introduce the Jacobian matrix of the geometric constraints and the uncertainty data into the nominal system dynamics model to obtain the complete system dynamics model;

[0101] S23. Based on geometric constraint equations, global differential homeomorphism is obtained by defining constraint tracking variables and constructing complementary coordinates.

[0102] In specific implementation, step S21 above is as follows: For rigid gripping, the constraint can be defined as maintaining a constant relative pose between the two end effectors; let and Let these represent the homogeneous transformation matrices of the left and right arm end effectors, respectively; at this point, the rigid gripping constraint equation is: T d It is the pre-set desired relative pose;

[0103] The rigid gripping constraint equation is equivalently transformed into the rigid gripping algebraic vector equation Φ(q)=0, where ; where Φ(q) represents the vector function of the generalized coordinate vector;

[0104] At this point, the constraint dimension is determined based on the number of rotational and translational joints. Once determined, the algebraic vector equation Φ(q) that determines the dimension is used as the geometric constraint equation. For example, if the relative poses of two end effectors are completely fixed in three-dimensional space, m=6 (3 translations and 3 rotations).

[0105] The above embodiments transform the collaborative task into geometric constraint equations, construct a complete system model by calculating the Jacobian matrix and introducing uncertainties, and establish a differential homeomorphism mapping; thus, the complex constraint system is decoupled into an easily tractable form, laying the foundation for subsequent controller design;

[0106] Step S22 above includes the following steps:

[0107] S221. By differentiating the geometric constraint equation Φ(q) with respect to time t using the total differential equation formula, the Jacobian matrix J(q) of the constraint is obtained; the total differential equation formula is as follows.

[0108] ;

[0109] Where d(Φ(q)) / dt represents the total differential of the geometric constraint equation Φ(q) with respect to time t, which is used to measure the rate at which the constraint is violated; if d(Φ(q)) / dt=0, it means that the robot is moving in a way that perfectly maintains the constraint; if d(Φ(q)) / dt≠0, it means that the robot is deviating from the required geometric constraint. Let dq / dt represent the partial derivative of the vector function Φ with respect to the generalized coordinate vector q; dq / dt represents the first derivative of q with respect to time t; J(q) describes how the small motion dq in the joint space causes the small motion dΦ in the task space (constraint space), establishing a mapping relationship from joint velocity to task velocity. When the constraints are satisfied, J(q)q1 = 0; at this time, ;

[0110] S222. Collect parameter uncertainty data, unmodeled dynamic data, external disturbance data, and friction model error data to obtain uncertainty term data τ. d ∈R n The parameter uncertainty data includes the deviation between the actual mass and inertia and the nominal value; the unmodeled dynamic data includes the flexibility of the linkage and the backlash of the transmission mechanism; the external disturbance data includes the external forces / torques acting on the robot; and the friction model error data includes the deviation between the actual friction force and F(q1).

[0111] S223. Input the Jacobian matrix J(q) of the constraint into the Lagrange multiplier method. When the geometric constraint equation Φ(q) = 0, the constraint force J is obtained. T (q)λ;

[0112] The uncertainty term data τ d and constraint force J T (q)λ is introduced into the nominal system dynamics model to obtain the complete system dynamics model; the formula of the complete system dynamics model is as follows.

[0113] M(q)*q2+C(q,q1)*q1+G(q)+F(q1)+τ d =τ+J T (q)λ;

[0114] The above embodiments construct a complete system dynamics model that includes constraints and uncertainties by calculating the Jacobian matrix of the constraint equations and taking into account practical factors such as parameter uncertainty, unmodeled dynamics, and external disturbances. This establishes a more accurate system description and lays a key foundation for the subsequent design of a robust controller that can actively compensate for disturbances.

[0115] Step S23 above includes the following steps:

[0116] S231. Based on the geometric constraint equations and the complete system dynamics model, define constraint position error variable z1 and constraint velocity error variable z2. Merge constraint position error variable z1 and constraint velocity error variable z2 to obtain constraint tracking variable h.

[0117] S232. To ensure that the transformation from the original state space (q,q1) to the new state space is global, smooth and reversible (i.e., differential homeomorphism), a set of variables complementary to the constraint tracking variables must be introduced.

[0118] Based on the number of joints n (degrees of freedom) and constraint dimension m of the complete system dynamics model, the dimension k of the complementary variable η is calculated; the dimension k of the complementary variable η = number of joints n - constraint dimension m; η represents the degree of freedom that the system still possesses under the premise that all constraints Φ(q) = 0; for example, when carrying an object with both arms, η can describe the overall translational and rotational motion of the object in space.

[0119] A nonlinear mapping function is set to be equal to the complementary variable η, resulting in the complementary coordinate transformation function T(q). This complementary coordinate transformation function T(q) ensures that the mapping from the generalized coordinate vector q to (z1, η) is differentially homeomorphic. In practical applications, T(q) is usually directly selected based on the physical meaning of the system. For example, when carrying an object with both arms, the most natural choice is to let η equal the pose (position and orientation) of the object in space, because the pose of the object naturally describes the internal motion of the system under the constraint of two-handed grasping. Specifically:

[0120] Calculate the null projection matrix N(q) of the constrained Jacobian matrix J(q); let This makes the gradient of the complementary coordinate transformation function T(q) The space spanned by the row vectors of the constrained manifold and the constrained Jacobian matrix J(q) is orthogonal, i.e., the complementary coordinate transformation function η=T(q) is a function that satisfies... The function; where η1 represents the first derivative of η with respect to time, q1 represents the transpose of the null projection matrix and q1 represents the first derivative of the generalized coordinate vector q with respect to time.

[0121] S233. Construct a new state vector based on the complementary variable η and the constraint tracking variable h; the new state vector The initial global differential homeomorphism Ψ is defined based on the new state vector; the initial global differential homeomorphism Ψ is the (q,z1) mapping (X);

[0122] The invertibility of the initial global differential homeomorphism is verified based on the complementary coordinate transformation function T(q); if the verification is successful, the initial global differential homeomorphism is taken as the global differential homeomorphism; otherwise, return to S232 to correct the complementary coordinate transformation function T(q) until the verification is successful and the global differential homeomorphism is obtained.

[0123] In specific implementation, the verification in step S233 above is as follows: constructing the inverse mapping Ψ of the initial global differential homeomorphism Ψ. -1 ; Calculate the inverse mapping Ψ of the initial global differential homeomorphism Ψ. -1 Is it possible to uniquely and smoothly recover the original state (q, q1) from the new state vector X, i.e., is (q, q1) equal to...? If they are equal, then the initial global differential homeomorphism is used as the global differential homeomorphism; otherwise, return to S232 to correct the complementary coordinate transformation function T(q) until (q,q1) equals ;

[0124] The above embodiments establish a global differential homeomorphism mapping from the original state space to the new state space by defining constraint tracking variables and constructing complementary coordinates; decoupling the complex constraint system into an independent constraint tracking subsystem and an internal dynamic subsystem not only ensures strict mathematical invertibility, but also greatly simplifies the design complexity of the subsequent controller.

[0125] Step S231 above includes the following steps:

[0126] S2311. Set a variable equal to the geometric constraint equation Φ(q) to obtain the constraint position error variable z1; the constraint position error variable z1 is used to quantify the degree of deviation of the generalized coordinate vector q from the ideal constraint Φ(q) = 0;

[0127] S2312. Calculate the first derivative of the constraint position error variable z1 with respect to time to obtain the constraint velocity error variable z2; the formula is as follows.

[0128] ;

[0129] Where z2 represents the constraint velocity error variable, and z2 represents the rate at which the constraint is violated; to make the complete system dynamics model stable on the constraint manifold, it is necessary to ensure that both z1 and z2 converge to zero at the same time.

[0130] S2313. The constraint position error variable and constraint velocity error variable are fused to obtain the constraint tracking variable h=[z1 T z2 T ] T The constraint tracking variables fully describe the dynamic error of the complete system dynamics model relative to the desired constraints.

[0131] The h=[z1 T z2 T ] T In the equation, h fully describes the dynamic error state of the system near the constrained manifold, capturing not only the deviation by z1 but also the trend of deviation z2; [ ] T Indicates vector transpose and stacking;

[0132] Taking driving a car while keeping it in the center line of the lane as an example:

[0133] z1 represents the lateral distance between the vehicle's current position and the lane centerline, and z2 represents the vehicle's current heading (or lateral velocity); even if the car is on the centerline (z1=0), if the car's heading is off-center (z1≠0), the car will deviate from the centerline in the next second; h=[z1 T z2 T ] T This is a complete "lane departure state"; at this point, it is necessary to observe both position and direction (i.e., the h vector) simultaneously, and then decide how to steer the wheel to bring the entire h state to zero.

[0134] The above embodiments construct a complete constraint tracking variable by defining constraint position error and constraint velocity error, comprehensively describing the dynamic error state of the system relative to the desired constraint; they fully characterize the degree and trend of the system deviating from the constraint, providing a key error state description foundation for subsequent realization of accurate constraint following control;

[0135] Step S3 above includes the following steps:

[0136] S31. Using the chain rule, differentiate the constraint velocity error variable z2 in the new state vector X to obtain the first derivative variable z3 of the constraint velocity error variable; the first derivative variable z3 of the constraint velocity error variable is used to express the acceleration relationship of the new state vector; the chain rule differentiation formula is as follows.

[0137] Where J(q)1 represents the time derivative of the Jacobian matrix J(q), and the specific calculation method is as follows: Similarly, by differentiating the first derivative η1 of the complementary variable η, we obtain the second derivative η2 of the complementary variable η.

[0138] S32. The complete system dynamic model equation M(q)*q2+C(q,q1)*q1+G(q)+F(q1)+τ d =τ+J T (q)λ and constraint velocity error variable The first derivative of the constraint velocity error variable z2, z3 = d(J(q)q1) / dt = J(q)q2+J(q)1q1, is combined with the equation to obtain the combined equation.

[0139] S33, Using the inverse mapping Ψ -1 The transformation relationship in the equations of the complete system dynamics model represents all variables (q, q1, q2) in the equations of the new state vector as variables (h, η, z1, η1), thus obtaining the variables after the transformation relationship.

[0140] S34. Substitute the transformed variables into the simultaneous equations to obtain the mapped target system dynamic model described by the new state vector X; the target system dynamic model includes a constrained tracking sub-model and a complementary sub-model; the equations of the constrained tracking sub-model and the complementary sub-model are as follows.

[0141] h1=Fh(h,η,η1)+Bh(h,η)*u+Δh;

[0142] eta2=Fn(h,n,n1)+Bn(h,n)*u+Δn;

[0143] Where h1 represents the derivative of h with respect to time, u represents the transformed control input, Δ represents the transformed uncertainty, and Δh and Δη are the manifestations of the original uncertainty τd in the constrained tracking sub-model and the complementary sub-model, respectively, after the transformation by the differential homeomorphism mapping; Fh(h,η,η1) represents the drift dynamics of the constrained tracking sub-model, which is a nonlinear vector function used to describe the dynamic behavior of the constrained tracking sub-model itself when the control input u=0; Bh(h,η)*u represents the input gain matrix of the constrained tracking sub-model, used to describe how the control input u affects the rate of change h1 of the constraint error state h; Fη(h,η,η1) represents the acceleration of the complementary sub-model (internal dynamics) itself when the control input u=0; Bη(h,η)*u represents the input gain matrix of the complementary sub-model, used to describe how the control input u affects the acceleration η2 of the internal motion;

[0144] The constraint tracking sub-model describes the dynamics of the system deviating from the constraints, and the goal of control is to stabilize this sub-system (i.e., force h to approach 0 infinitely); the complementary sub-model describes the internal motion of the system on the constraint manifold, i.e., the actual task execution dynamics (such as the trajectory of a transported object).

[0145] The above embodiments transform the complete system dynamics model into a target system consisting of a constraint tracking subsystem and a complementary subsystem through differential homeomorphism mapping; this achieves complete decoupling of the complex constraint system, enabling the controller to be designed independently for constraint error stabilization and internal motion control, greatly simplifying the control strategy.

[0146] Step S4 above includes the following steps:

[0147] S41. Define the control objective of the constrained tracking sub-model in the target system dynamics model as the asymptotic convergence of the value of the constrained tracking sub-model to zero, thus obtaining the defined tracking error. Based on the control objective of the constrained tracking sub-model, define a sliding surface s to accelerate the convergence of the constrained tracking sub-model and enhance its robustness. The equation of the sliding surface s is s = z1 + Λz2, where Λ is a positive definite diagonal matrix. When s = 0, ... This ensures that the z1 exponent converges to zero, and thus... It also converges to zero;

[0148] S42. Set the upper bound of the uncertainty term data in the target system dynamic model to r, ||τd||≤r; design an adaptive law r based on the sliding surface s. 1 Equation 1 is used to evaluate the upper bound r online, yielding the adaptive law; the equation for the adaptive law is r. 1 1 = Γ||s||, where r 1 r is an estimate of the upper bound. 1 1 is r 1 The first derivative with respect to time, Γ represents the adaptive gain coefficient, which is a value greater than zero, and ||s|| is the Euclidean norm of the sliding mode surface vector s;

[0149] S43. Based on Lyapunov stability theory, design the control torque vector τ to be designed, and obtain the designed control law τ1; the control law includes feedforward model compensation terms, constraint force compensation terms, PD feedback terms, and robust adaptive terms; the equation of the control law τ1 is as follows.

[0150] τ1=M(q) 1 u+C(q,q1) 1 q1+G(q) 1 +F(q1) 1 -J T (q)λ 1 -Kds-r 1 1*sgn(s);

[0151] in) 1 This represents the estimated value of the parameter; M(q) 1 M(q) represents the estimated value of the inertia matrix, u is the virtual control variable in the target system dynamics, and M(q) is the virtual control variable in the target system dynamics. 1 u represents the inertial force compensation term; C(q,q1) 1 q1 represents the Coriolis force and centripetal force compensation term, C(q,q1). 1 Estimates of the Coriolis force and centripetal force; G(q) 1 This represents the estimated value of the gravity vector, i.e., the gravity compensation term; F(q1)1 This represents the estimated value of the friction force model, i.e., the friction force compensation term; J T (q)λ 1 Indicates constraint force feedforward compensation, λ 1 Φ(q) = 0 represents the constraint force (Lagrange multiplier) estimated to maintain the constraint Φ(q) = 0; Kds represents the error feedback stabilization term, which is used by adjusting the magnitude of Kd and applying it to s through proportional-derivative control to reduce control error; r 1 1*sgn(s) represents the uncertainty compensation term, meaning that regardless of the uncertainty, the sign function sgn(s) will always generate a maximum control force opposite to the direction of the error s, reducing the uncertainty error. 1 1. Used to adaptively adjust the magnitude of this force;

[0152] The defined tracking error, together with the sliding surface, adaptive law, and control law, constitutes a robust adaptive controller.

[0153] The above embodiments construct a composite controller that includes model compensation, constraint force feedforward, PD feedback and robust terms by designing sliding surfaces and adaptive laws. This design is based on Lyapunov stability theory, which ensures the system's ability to estimate and compensate for uncertainties online, and ultimately achieves rapid convergence of constraint errors and high-precision stable tracking of the system under disturbances.

[0154] Step S5 above includes the following steps:

[0155] S51. The position q of each joint is read in real time through the encoder on the robot joint. m and the velocity q of each joint m1， Obtain the real-time joint position set and the real-time joint velocity set; the joint velocity q m1 Obtained through a tachogenerator;

[0156] S52, Combine the joint positions q from the real-time joint position set and the real-time joint velocity set. m and joint velocity q m1 Substituting into the differential homeomorphism mapping Ψ of S3, the current state of the target system is calculated;

[0157] S53. Estimate the real-time upper bound of the error signal and adaptive law in the target system state at the current moment, input the estimated value of the real-time upper bound into the control law equation in S4, and obtain the control torque applied to each joint to obtain the real-time control torque set.

[0158] S54. Transmit the real-time control torque of the real-time control torque concentration to the corresponding joint actuators of the dual-arm robot to drive the robotic arm to move until the collaborative task is completed.

[0159] The embodiment described above collects joint state data in real time, converts it into the target system state through differential homeomorphism mapping, and generates joint control torque online based on adaptive control law. This closed-loop control process realizes the complete transformation from theoretical control algorithm to physical execution, ensuring that the dual-arm robot can dynamically adjust the control torque according to the actual motion state, and ultimately achieve high-precision collaborative task execution.

[0160] The following is Embodiment 1 based on the present invention, as follows:

[0161] After adopting the constraint following control method based on differential homeomorphism mapping of the present invention, the control system exhibits significantly superior performance when facing the cooperative handling task in the prior art. When two 7-DOF robotic arms jointly carry a precision instrument with uncertain mass through a channel with vibration interference, the method of the present invention plays a key role.

[0162] The system transforms the task requirement of "keeping the relative pose of the end effector constant" into geometric constraint equations and monitors deviations in real time through defined constraint tracking variables. When external vibrations cause a slight deviation in the relative pose, the constraint position error variable immediately detects the deviation, while the constraint velocity error variable captures the trend of the deviation.

[0163] Through global differential homeomorphism mapping, the system decouples complex constraint dynamics into independent constraint tracking subsystems and internal dynamic subsystems. When the load parameters are unknown and vibration disturbances exist, the adaptive law converges rapidly within 0.1 seconds, estimating the upper bound of uncertainty online, and the robust adaptive controller then generates the corresponding compensation torque. The model feedforward compensation term in the control law overcomes the inherent dynamic effects of the system, while the adaptive robust term effectively suppresses the influence of vibration disturbances.

[0164] In actual operation tests, even if the load mass suddenly changes by 15%, the system can automatically adjust the control torque within 0.2 seconds, always keeping the relative pose error between the end effectors within ±0.5 mm and the speed tracking error within ±2 mm / s, which is a significant improvement over traditional methods. Under continuous external disturbances with an amplitude of ±8 N, the system's adaptive robustness term successfully suppressed 85% of the disturbance effects. This rapid adaptive capability ensures the stability and safety of precision instruments throughout the handling process, fully demonstrating the powerful advantages of this invention in dealing with uncertainties.

[0165] By transforming complex constraint processing into a simple error stabilization problem, this invention not only ensures control accuracy but also significantly reduces computational complexity. The control law calculation cycle is shortened from 200ms in the traditional method to 50ms, providing technical assurance for the reliable operation of dual-arm robots in dynamic environments.

[0166] For further details, please refer to Figure 2 A constraint following control system for a dual-arm collaborative robot based on differential homeomorphism mapping is used to implement the constraint following control method for the dual-arm collaborative robot based on differential homeomorphism mapping. The system includes an initial modeling module, a constraint modeling and system reconstruction module, a model transformation module, a controller design module, and a real-time control and execution module.

[0167] The initial modeling module is used to construct the nominal system dynamics model of the dual-arm collaborative robot. This module collects the displacement data of each joint of the robot and combines them into a generalized coordinate vector. Based on parameters such as the mass, center of mass position and inertia tensor of the robot links, it constructs the nominal dynamic equations describing the system inertia, Coriolis force, centripetal force, gravity and friction, providing a basic model for subsequent control design.

[0168] The constraint modeling and system reconstruction module is used to define the geometric constraint equations that must be satisfied between end effectors according to the collaborative task (such as jointly grasping an object) and transform them into mathematical form. The module further calculates the Jacobian matrix of the constraints and, taking into account practical factors such as parameter uncertainty and unmodeled dynamics, extends the nominal model into a complete system dynamics model that includes constraint forces and uncertainties. At the same time, by defining constraint tracking variables and constructing complementary coordinates, a global differential homeomorphism is established, laying the foundation for the transformation of the system state space.

[0169] The model transformation module is used to transform the complete system dynamics model based on joint space into a target system dynamics model in a new coordinate system with constraint tracking error and internal motion as state variables by utilizing the aforementioned global differential homeomorphism mapping. This target system is decoupled into two subsystems that describe constraint deviation dynamics and task execution dynamics respectively, which greatly simplifies the design of the controller.

[0170] The controller design module is used to design a robust adaptive controller for the transformed target system. The module first defines a sliding surface to ensure that the constraint error converges quickly, and designs an adaptive law to estimate the upper bound of the system uncertainty online. Then, based on Lyapunov stability theory, a control law is synthesized that includes model feedforward compensation, constraint force compensation, PD feedback and robust adaptive terms. This controller can actively resist uncertainty and accurately perform constraint following tasks.

[0171] The real-time control and execution module is used to read the robot's joint position and speed in real time through sensors, calculate the current target system state using differential homeomorphism mapping, input the state error and adaptive estimate into the control law, generate real-time joint control torque online, and finally apply these torques to the robot's joint actuators to drive the robotic arm to complete the predetermined collaborative task.

[0172] The aforementioned constraint-following control system for a dual-arm collaborative robot based on differential homeomorphism mapping constructs a complete constraint-following control solution for dual-arm robots. Starting from initial modeling, an accurate system model is established through constraint integration and reconstruction. Then, model transformation is used to decouple and simplify the complex system, thereby designing a robust adaptive controller. Finally, a real-time control module enables high-precision task execution. The efficient decoupling of the constraint system is achieved through differential homeomorphism mapping, transforming the complex constraint tracking problem into a simple error stabilization problem, significantly simplifying the controller design. The introduction of an adaptive sliding mode control mechanism enables the system to estimate and compensate for various uncertainties online, effectively improving control accuracy and robust stability under parameter fluctuations and external disturbances. The modular design of the entire process ensures a smooth transition from task constraints to joint torques, guaranteeing both theoretical rigor and engineering practicality, enabling the dual-arm robot to reliably complete high-precision collaborative tasks in dynamic and uncertain environments.

[0173] In the description of this specification, references to terms such as "an embodiment," "example," "specific example," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0174] The preferred embodiments of the invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to the specific implementations described. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention.

Claims

1. A constraint-following control method for a dual-arm cooperative robot based on differential homeomorphism mapping, characterized in that, Includes the following steps: S1. Establish the nominal system dynamics model of the dual-arm collaborative robot; S2. Define geometric constraint equations in the task space according to the dual-arm collaborative task; construct a global differential homeomorphism and a complete system dynamics model based on the geometric constraint equations. The global differential homeomorphism mapping is obtained by defining constraint tracking variables and constructing complementary coordinates; The complete system dynamics model is obtained by incorporating the Jacobian matrix of the geometric constraint equations and uncertainty data into the nominal system dynamics model; S3. Based on global differential homeomorphism mapping, the state space of the complete system dynamics model is transformed into the state space of the target system, and the mapped target system dynamics model is obtained. S4. Based on the mapped target system dynamics model, define the tracking error and sliding surface, design the adaptive law, and synthesize the control law to obtain a robust adaptive controller. The robust adaptive controller estimates the uncertainty of the mapped target system dynamics model online in real time and generates real-time control torque; S5. Apply the real-time control torque to the dual-arm collaborative robot to drive the movement of each joint of the dual-arm collaborative robot.

2. The constraint following control method for a dual-arm cooperative robot based on differential homeomorphism mapping according to claim 1, characterized in that, S1 includes the following steps: S11. Collect the angular displacement data and linear displacement data of all joints of the dual-arm robot to obtain the set of angular displacement of the joints of the dual-arm robot and the set of linear displacement of the joints of the dual-arm robot. By performing vector combination operations on all angular displacement data in the set of angular displacements of the joints of the dual-arm robot and all linear displacement data in the set of linear displacements of the joints of the dual-arm robot, a generalized coordinate vector is obtained. S12. Collect the mass data of each link of the dual-arm robot, the position data of the center of mass of the link in the link coordinate system, and the inertia tensor data of the link through robot design drawings, CAD models and parameter identification experiments. Based on the generalized coordinate vector, the mass data of each link of the dual-arm robot, the position data of the center of mass of the link in the link coordinate system, and the inertia tensor data of the link, a positive definite inertia matrix, Coriolis force and centripetal force matrix and gravity vector are constructed. S13. Based on the positive definite inertia matrix, Coriolis force and centripetal force matrix and gravity vector, a nominal system dynamic model of the dual-arm robot is constructed by utilizing Lagrange's mechanical formula and introducing a friction model.

3. The constraint following control method for a dual-arm cooperative robot based on differential homeomorphism mapping according to claim 1, characterized in that, S2 includes the following steps: S21. Based on the specific task of dual-arm collaboration, the geometric relationship between the end effectors of the specific task of dual-arm collaboration is transformed into mathematical equations to obtain the initial geometric constraint equations. The initial geometric constraint equations are transformed into algebraic vector equations. Based on the dimensions of rotation of the revolute joints and translation of the translational joints, the values ​​of the constraint dimensions in the algebraic vector equations are determined, and the geometric constraint equations are obtained. S22. Calculate the Jacobian matrix of the geometric constraint equations and collect uncertainty data; introduce the Jacobian matrix of the geometric constraints and the uncertainty data into the nominal system dynamics model to obtain the complete system dynamics model; S23. Based on geometric constraint equations, global differential homeomorphism is obtained by defining constraint tracking variables and constructing complementary coordinates.

4. The constraint following control method for a dual-arm cooperative robot based on differential homeomorphism mapping according to claim 3, characterized in that, S22 includes the following steps: S221. By taking the total differential equation formula for the geometric constraint equation with respect to time, the Jacobian matrix of the constraint is obtained. S222. Collect parameter uncertainty data, unmodeled dynamic data, external disturbance data, and friction model error data to obtain uncertainty term data; S223. Input the Jacobian matrix of the constraint into the Lagrange multiplier method. When the geometric constraint equation is equal to zero, the constraint force is obtained. By incorporating the uncertainty data and constraints into the nominal system dynamics model, a complete system dynamics model is obtained.

5. The constraint following control method for a dual-arm cooperative robot based on differential homeomorphism mapping according to claim 3, characterized in that, S23 includes the following steps: S231. Based on the geometric constraint equations and the complete system dynamics model, define constraint position error variables and constraint velocity error variables, and fuse the constraint position error variables and constraint velocity error variables to obtain constraint tracking variables. S232. Based on the number of joints and constraint dimensions of the complete system dynamics model, and the dimensions of the complementary variables, obtain the complementary variables; Set a nonlinear mapping function equal to the complementary variables to obtain the complementary coordinate transformation function; specifically: calculate the null space projection matrix of the constrained Jacobian matrix; let the first derivative of the complementary variables with respect to time be equal to the product of the transpose of the null space projection matrix and the first derivative of the generalized coordinate vector with respect to time, so that the gradient of the complementary coordinate transformation function is orthogonal to the space spanned by the row vectors of the constrained Jacobian matrix on the constrained manifold. S233. Construct a new state vector based on complementary variables and constraint tracking variables; define an initial global differential homeomorphism based on the new state vector; The invertibility of the initial global differential homeomorphism is verified based on the complementary coordinate transformation function; if the verification is successful, the initial global differential homeomorphism is taken as the global differential homeomorphism; otherwise, return to S232 to correct the complementary coordinate transformation function until the verification is successful and the global differential homeomorphism is obtained.

6. The constraint following control method for a dual-arm cooperative robot based on differential homeomorphism mapping according to claim 5, characterized in that, S231 includes the following steps: S2311. Set a variable equal to the geometric constraint equation to obtain the constraint position error variable; the constraint position error variable is used to quantify the degree of deviation of the generalized coordinate vector from the ideal constraint when it is zero. S2312. Calculate the first derivative of the constraint position error variable with respect to time to obtain the constraint velocity error variable; S2313. The constraint position error variable and the constraint velocity error variable are fused together to obtain the constraint tracking variable.

7. The constraint following control method for a dual-arm cooperative robot based on differential homeomorphism mapping according to claim 1, characterized in that, S3 includes the following steps: S31. Using the chain rule, differentiate the constraint velocity error variable in the new state vector to obtain the first derivative variable of the constraint velocity error variable. S32. Combine the complete system dynamics model equations with the constraint velocity error variables and their first derivatives to obtain the combined equations. S33. Using the transformation relationship in the inverse mapping, all variables in the equations of the complete system dynamics model are represented by the variables in the new state vector, thus obtaining the variables after the transformation relationship; S34. Substitute the variables after the transformation into the simultaneous equations and simplify to obtain the mapped target system dynamic model described under the new state vector; the target system dynamic model includes a constraint tracking sub-model and a complementary sub-model.

8. The constraint following control method for a dual-arm cooperative robot based on differential homeomorphism mapping according to claim 1, characterized in that, S4 includes the following steps: S41. Define the control objective of the constrained tracking sub-model in the target system dynamics model as the value of the constrained tracking sub-model asymptotically converges to zero, and obtain the defined tracking error; define a sliding surface based on the control objective of the constrained tracking sub-model to accelerate the convergence of the constrained tracking sub-model and enhance its robustness, and obtain the equation of the sliding surface; S42. Set an upper bound for the uncertainty term data in the target system dynamics model; based on the sliding mode surface, design the equation of the adaptive law to evaluate the upper bound online, and obtain the adaptive law; S43. Based on Lyapunov stability theory, the control torque vector to be designed is designed to obtain the designed control law; the control law includes feedforward model compensation term, constraint force compensation term, PD feedback term and robust adaptive term. The defined tracking error, together with the sliding surface, adaptive law, and control law, constitutes a robust adaptive controller.

9. The constraint following control method for a dual-arm cooperative robot based on differential homeomorphism mapping according to claim 1, characterized in that, S5 includes the following steps: S51. By using encoders on the robot joints, the position and velocity of each joint are read in real time to obtain a real-time set of joint positions and a real-time set of joint velocities; the joint velocities are obtained through a tachogenerator. S52. Substitute the joint positions and joint velocities from the real-time joint position set and the real-time joint velocity set into the differential homeomorphism mapping in S3 to calculate the target system state at the current moment. S53. Estimate the real-time upper bound of the error signal and adaptive law in the target system state at the current moment, input the estimated value of the real-time upper bound into the control law equation in S4, and obtain the control torque applied to each joint to obtain the real-time control torque set. S54. The real-time control torque of the real-time control torque concentration is transmitted to the corresponding joint actuators of the dual-arm robot to drive the robotic arm to move until the collaborative task is completed.

10. A constraint-following control system for a dual-arm cooperative robot based on differential homeomorphism mapping, characterized in that, The system implementing the constraint following control method for a dual-arm cooperative robot based on differential homeomorphism mapping as described in any one of claims 1-9, the system comprising: System modeling module: Constructs a nominal system dynamic model of the dual-arm robot, and establishes joint space dynamic equations including inertia, Coriolis force, gravity and friction using the Lagrange method; Constraints and Reconstruction Module: Define geometric constraints based on collaborative tasks, introduce their Jacobian matrix, uncertainties and external disturbances, construct a complete system dynamics model, and establish a differential homeomorphism mapping between constraint tracking variables and complementary coordinates; Model decoupling module: Using differential homeomorphism mapping, the complete system dynamics model of the dual-arm robot is transformed to a new state space, decoupling it into a target system dynamics model that includes a constraint tracking sub-model and a complementary sub-model; Controller design module: Based on the target system dynamics model, a sliding surface and adaptive law are designed, the uncertainty boundary is estimated online, and a control law including model compensation, feedback and robust terms is synthesized to obtain a robust adaptive controller; Control execution module: Reads joint status in real time, calculates system error and uncertainty estimation through differential homeomorphism mapping, generates joint torque by robust adaptive controller and drives robot to complete collaborative task.

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