A stability determination method for a variable impedance model of a robot system using a variable augmented matrix

Abstract

The present application belongs to the technical field of robot control, and particularly relates to a stability judgment method of a robot system variable impedance model using a variable augmented matrix, which enables the robot to verify its stability offline when the stiffness changes. Impedance control is a general force control scheme suitable for various manipulator tasks, one of which is human-robot collaboration: the manipulator performs force-position hybrid control on the object according to the impedance control law, while the operator applies control force to the object or the robot to realize human-robot collaboration and complete work such as assembly, carrying, etc. According to different manipulator tasks, variable stiffness parameters can be used to reflect flexibility and compliance. Not only can the dynamic relationship between external force and robot motion be controlled, but also these dynamic behaviors can be flexibly changed in a continuous manner during the task. Impedance control can be implemented in joint coordinates, Cartesian coordinates or other task-specific coordinates. Impedance targets can be achieved in different ways.

Description

Technical Field

[0001] This invention belongs to the field of robot control technology, and in particular relates to a method for determining the stability of a robot system with a variable impedance model using a variable augmented matrix. Background Technology

[0002] Collaborative robots are a type of robot specifically designed to work alongside humans. Compared to traditional industrial robots, collaborative robots are more flexible and safer, enabling them to work directly alongside humans in the same workspace without requiring special safety measures.

[0003] Impedance control is a robot control strategy designed to make a robot's response to external forces or torques resemble the impedance response in an electrical circuit. This control strategy allows robots to flexibly adapt to changes in the environment, and is particularly suitable for collaborative robots interacting with humans or unknown environments. One application of impedance control is human-robot collaboration: a robotic arm performs force-position hybrid control on an object according to an impedance control law, while an operator applies control forces to the object or robot to achieve human-robot collaboration, completing tasks such as assembly and handling. Compared to constant impedance control, variable impedance control provides the flexibility to continuously adjust dynamics according to task requirements and environmental conditions. This adaptability enhances the robot's capabilities, enabling it to undertake more complex tasks in unstructured environments.

[0004] Based on the research and understanding of these background technologies, this application proposes a stability judgment method for a robot system with a variable impedance model using a variable augmented matrix to solve the stability problem in the variable stiffness impedance control of a robotic arm, and proves its feasibility. Summary of the Invention

[0005] The purpose of this invention is to provide a method for determining the stability of a robot system with a variable impedance model using a variable augmented matrix.

[0006] To solve the above-mentioned technical problems, the specific technical solution of the present invention is as follows:

[0007] In some embodiments of this application, a method for determining the stability of a robot system with a variable impedance model using a variable augmented matrix is ​​provided, comprising the following steps:

[0008] Step 1) Obtain the state parameter information of the robot arm through the joint sensors;

[0009] Step 2) Construct the desired variable impedance-trajectory model;

[0010] Step 3) In the impedance controller, user-defined impedance parameters are used, and the stability of the variable impedance is analyzed;

[0011] Step 4) In variable impedance control, an energy function is constructed by introducing a variable augmented matrix into the estimation of the derivative of the Lyapunov function and by weighting the velocity error and position error, thus establishing a stability condition independent of the state related to variable stiffness.

[0012] Step 5) Based on step 4), set the damping parameters and mass parameters;

[0013] Step 6) Calculate the posture error using the posture information of the robotic arm obtained by the sensor and the desired posture information of the robotic arm. Calculate the control torque required for each task based on the error and the hierarchical variable impedance control model, and input it into the robotic arm to achieve control of the robotic arm.

[0014] In some embodiments of this application, the desired variable impedance-trajectory model is constructed in step 2) as follows:

[0015] The dynamic equations of a robotic arm with n degrees of freedom are as follows:

[0016]

[0017] Where M(q)∈R n×n Let the inertia matrix be the inertia matrix in the joint space of the robotic arm. It is the Coriolis force matrix of the robotic arm, G(q)∈R n×1 It is the gravity term of the robotic arm, J(q)∈R m×n Let be the Jacobian matrix of the robotic arm. Let m be the dimension of the end effector's task space, τ∈R. n×1 The input torque to the joint is τ. e The external force acting on the joint;

[0018] In the joint space, the external torque τ of the mass-damped spring system e and position error Represented as:

[0019]

[0020] Where H∈R n×n , D∈R n×n K(t)∈R n×n Let H, D, and K(t) be the desired inertia, damping, and variable stiffness matrices. H, D, and K(t) are positive definite and symmetric.

[0021] Let τ c To control the torque, as shown in equation (3)

[0022]

[0023] Define the new control input μ as:

[0024]

[0025] Substitute equations (3) and (4) into (1) and introduce symbols The closed-loop dynamics in equation (2) are obtained.

[0026] In some embodiments of this application, step 3) analyzes the stability of the variable impedance using the Lyapunov function:

[0027]

[0028] Will Let V1 be abbreviated as V1. Differentiate V1 along the trajectory of equation (2) by τ. e =0, H and D are both constant matrices;

[0029]

[0030] The symmetry of the stiffness matrix was used to eliminate and Mixed terms, but containing The positive terms still exist.

[0031]

[0032] In some embodiments of this application, step 4) constructs an energy function by weighting the velocity error and position error, establishing a state-independent stability condition related to variable stiffness, and introducing a time-varying augmented free weight matrix to reduce the conservatism of the energy function. The Lyapunov function is constructed as follows:

[0033]

[0034] Where β(t) and γ(t) are symmetric, positive semi-definite, and continuously differentiable matrices, and V2 is differentiated along the trajectory of equation (2), where τ e =0, and H is a constant matrix;

[0035]

[0036] To eliminate simultaneous inclusion and The term, defining β(t) as

[0037] β(t)=K(t)+αD+γ(t)K(t)-α 2 H.#(10);

[0038] Let D-αH+minK(t) / α≥0#(11), such that β(t) for all t>0

[0039] It is positive semidefinite, and β(t) and Substituting into equation (9), we can obtain

[0040]

[0041] In some embodiments of this application, the stability condition under dynamic decoupling is defined as Theorem 1, where H and D are constant, symmetric, and positive definite matrices; and K(t) is a symmetric, positive definite, and continuously differentiable variable stiffness matrix. If there exist α>0, γ(t)≥0, and... Make Then it exists It is semi-negative definite. If it is semi-negative definite, then the system in equation (2) is globally uniformly stable when τe = 0.

[0042] In some embodiments of this application, in the first condition of Theorem 1, the degrees of freedom provided by the free weighting matrix help reduce the conservatism of the stability judgment method, and γ(t) is chosen to ensure that Theorem 1 is maintained.

[0043] The stability inequality derived from Theorem 1 gives equation (13). Equations (13) and (11) are simplified to...

[0044] D-αH+min{Dγ(0),minK(t) / α}≥0.#(14)

[0045] Express the second condition of Theorem 1 as an inequality and move the terms containing the time-varying free weight matrix to the left side;

[0046]

[0047] Integrating over the t-domain yields...

[0048]

[0049] let have

[0050] Compared to existing technologies, the advantages of this invention lie in the fact that variable impedance control can not only control the dynamic relationship between external forces and robot motion, but also flexibly change these dynamic behaviors continuously during a task. Impedance control can be implemented in joint coordinates, Cartesian coordinates, or other task-specific coordinates. Impedance targets can be achieved in various ways. Attached Figure Description

[0051] Various other advantages and benefits will become apparent to those skilled in the art upon reading the following detailed description of preferred embodiments. The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Furthermore, the same reference numerals denote the same parts throughout the drawings. In the drawings:

[0052] Figure 1 This is a schematic diagram of the offline verification block for variable impedance stability provided in an embodiment of the present invention;

[0053] Figure 2 This is a schematic diagram of a variable impedance control experiment provided in an embodiment of the present invention;

[0054] Figure 3 A schematic diagram of the change in energy as a function of stiffness curve adjustment for the robotic arm error provided in this embodiment of the invention.

[0055] Figure 4 This is a schematic diagram of the change in energy as a function of stiffness curve adjustment for the robotic arm error provided in an embodiment of the present invention.

[0056] Figure 5 A schematic diagram showing the changes in stiffness, error, and energy of the robotic arm variable impedance controller provided in an embodiment of the present invention under external disturbances.

[0057] Figure 6 The diagram below illustrates the changes in stiffness, error, and energy of the robotic arm variable impedance controller provided in this embodiment of the invention under external disturbances. Detailed Implementation

[0058] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0059] To better understand the purpose, structure, and function of this invention, the invention will be described in further detail below with reference to the accompanying drawings.

[0060] See appendix Figure 1-6 As shown, according to some embodiments of this application, the following steps are included:

[0061] Step 1) Obtain the state parameter information of the robot arm through the joint sensors;

[0062] Step 2) Construct the desired variable impedance-trajectory model;

[0063] Step 3) In the impedance controller, user-defined impedance parameters are used, and the stability of the variable impedance is analyzed;

[0064] Step 4) In variable impedance control, an energy function is constructed by introducing a variable augmented matrix into the estimation of the derivative of the Lyapunov function and by weighting the velocity error and position error, thus establishing a stability condition independent of the state related to variable stiffness.

[0065] Step 5) Based on step 4), set the damping parameters and mass parameters;

[0066] Step 6) Calculate the posture error using the posture information of the robotic arm obtained by the sensor and the desired posture information of the robotic arm. Calculate the control torque required for each task based on the error and the hierarchical variable impedance control model, and input it into the robotic arm to achieve control of the robotic arm.

[0067] Variable impedance control not only controls the dynamic relationship between external forces and robot motion, but also allows for the continuous and flexible alteration of these dynamic behaviors during a task. Impedance control can be implemented in joint coordinates, Cartesian coordinates, or other task-specific coordinates. Impedance targets can be achieved in various ways, such as actively in a torque-controlled reversible drive manipulator or passively by placing an elastic element between the actuator and the joint. By maintaining a constant gain, impedance control makes the closed-loop system passive, thus becoming passive (and stable) in interaction with a passive environment. However, if arbitrary variations in the impedance parameters are allowed, the passive characteristic is lost. Therefore, an analytical or control method is needed to ensure the stable execution of variable impedance tasks. This application starts with the desired variable impedance and aims to verify whether it can be executed without sacrificing stability.

[0068] The method includes: a stability verification method for a robot variable impedance system that does not depend on state variables, which can verify the system stability after stiffness change offline according to the expected change stiffness curve. A variable augmented matrix is ​​introduced into the judgment method, which reduces the conservatism of the stability judgment method and increases the degree of freedom of stability judgment.

[0069] The desired variable impedance-trajectory model is constructed as follows:

[0070] The dynamic equations of a robotic arm with n degrees of freedom are as follows:

[0071]

[0072] Where M(q)∈R n×n Let the inertia matrix be the inertia matrix in the joint space of the robotic arm. It is the Coriolis force matrix of the robotic arm, G(q)∈R n×1 It is the gravity term of the robotic arm, J(q)∈R m×n Let be the Jacobian matrix of the robotic arm. m is the dimension of the end effector's task space, typically 6. τ∈R n×1The input torque to the joint is τ. e The external force acting on the joint.

[0073] The goal of a variable impedance controller is to maintain the external torque τ e and position error The dynamic relationship between them. In the joint space, the external torque τ of the mass-damped spring system. e and position error Represented as:

[0074]

[0075] Where H∈R n×n , D∈R n×n K(t)∈R n×n Let H, D, and K(t) be the desired inertia, damping, and variable stiffness matrices. H, D, and K(t) are positive definite and symmetric.

[0076] Let τ c To control the torque, as shown in (3)

[0077]

[0078] Define the new control input μ as (4):

[0079]

[0080] Substitute (3) and (4) into (1) and introduce symbols The closed-loop dynamics in (2) are obtained.

[0081] In an impedance controller, user-defined impedance parameters, such as the inertia matrix H, damping matrix D, and variable stiffness matrix K(t), determine the robot's resistance to external torque τ. e The system behaves asymptotically for any symmetric positive definite impedance parameter matrix H, D, and K if the chosen impedance parameters are constant. In this work, this application focuses on variable impedance control. Specifically, this application assumes that H and D remain constant, while K(t) is treated as a time-varying function. Furthermore, to achieve impedance control, the external force τ... e It is obtained through a joint torque sensor.

[0082] To analyze the stability of the variable impedance in (2), consider the following candidate Lyapunov functions:

[0083]

[0084] This application will Let it be abbreviated as V1. Differentiate V1 along the trajectory of (2) by τ e =0, H and D are both constant matrices.

[0085]

[0086] The symmetry of the stiffness matrix was used to eliminate and Mixed terms, but containing The positive terms still exist.

[0087]

[0088] In (7), for negative semi-definite It is negative semi-definite. Therefore, the conclusion of origin stability can only be drawn when all eigenvalues ​​of the stiffness matrix are constant or decreasing. Assume... Increasing the stiffness eigenvalue can inject potential energy into the system, and it is clear that this will lead to unstable behavior.

[0089] As shown in (7), the classical Lyapunov energy function cannot determine stability when dealing with varying stiffness. As observed in (7), a significant approach to ensuring system stability is to design a novel controller that suppresses the energy injected into the system due to stiffness variations. However, this approach has several drawbacks, the most important of which is...

[0090] The permissible range of variation of the variable impedance curve is determined by the robot's state, which cannot be predicted in advance;

[0091] 2) It cannot be guaranteed that the controller can perform the desired change in stiffness.

[0092] Experimental evidence from variable stiffness control suggests that reasonable variable stiffness parameters generally do not lead to instability. This has prompted researchers to explore Lyapunov candidate functions that are less conservative than (5). In variable impedance control, an energy function is constructed by weighting the velocity error and position error, establishing a state-independent stability condition related to variable stiffness. A time-varying augmented free weight matrix is ​​introduced to reduce the conservatism of the energy function. The Lyapunov function is constructed as follows:

[0093]

[0094] Where β(t) and γ(t) are symmetric, positive semi-definite, and continuously differentiable matrices. Differentiating V2 along the trajectory of (2), where τ e =0, and H is a constant matrix.

[0095]

[0096] To eliminate simultaneous inclusion and The term, defining β(t) as

[0097] β(t)=K(t)+αD+γ(t)K(t)-α 2 H.#(10)

[0098] make

[0099] D-αH+minK(t) / α≥0#(11)

[0100] This makes β(t) positive semidefinite for all t>0.

[0101] Let β(t) and Substituting into (9), we can obtain

[0102]

[0103] Theorem 1: (Stability Condition under Dynamic Decoupling) Let H and D be constant, symmetric, and positive definite matrices. Let K(t) be a symmetric, positive definite, and continuously differentiable variable stiffness matrix. Then, if there exist α>0, γ(t)≥0 and... Make

[0104] exist

[0105] It is semi-negative definite.

[0106] It is semi-negative definite.

[0107] Then, in equation (2), the system is globally uniformly stable when τe = 0.

[0108] Note 1: In the first condition of Theorem 1, the degrees of freedom provided by the free weighting matrix help to reduce the conservatism of the stability judgment method.

[0109] Choose an appropriate γ(t) to ensure that Theorem 1 is maintained.

[0110]

[0111] Subsequently, the stability inequality derived from Theorem 1 yields (13), and (13) and (11) are simplified to

[0112] D-αH+min{Dγ(0),minK(t) / α}≥0.#(14)

[0113] Express the second condition of Theorem 1 as an inequality and move the terms containing the time-varying free weight matrix to the left side.

[0114]

[0115] Integrating over the t-domain yields...

[0116]

[0117] let have

[0118]

[0119] Note 2: (17) involves a free weighting matrix, which relaxes the constraints of existing stability assessment methods. All simplified conditional methods (14) and (17) include γ(0). Choosing a larger γ(0) will undoubtedly increase the degree of freedom restriction of the stability assessment criterion.

[0120] In fact, in impedance control, stiffness parameters have the most significant impact on task performance. Therefore, prioritizing stiffness design is usually a prudent approach. Subsequently, damping parameters can be selected to ensure stable execution of the desired variable stiffness, typically by using critical damping parameters. Examining the constraints in Theorem 1 shows that the least conservative constraint is determined by the parameter α; therefore, choosing the largest eigenvalue of D-αH+min{Dγ(0),minK(t) / α} to remain negative throughout execution achieves the least conservative constraint. Therefore, to obtain the least conservative constraint, α should be chosen as...

[0121]

[0122] in and δ Let represent the maximum and minimum eigenvalues, respectively. With this choice, the first condition of Theorem 1 is essentially satisfied for all t. Therefore, only the second condition of Theorem 1 remains to be verified. A simple verification procedure for the stable impedance curve is as follows:

[0123] Choose ideal H, D and K(t) matrices and determine the value of α according to equation (18).

[0124] Verify that for all t>0,

[0125] If the verification fails, improve the stiffness parameters and re-verify.

[0126] Under the second condition of Theorem 1, the larger the initial value of γ(t), the wider the range of selectable variable stiffness. In this application, we select an initial value of the time-varying free weight matrix γ(t) that exceeds minK(t) / αD. Therefore, (13) is transformed into D-αH+minK(t) / α.

[0127] A simulation experiment was conducted on the Franka robotic arm using the Gazebo platform. The experiment focused on the fourth joint of the Franka robotic arm.

[0128] The system is selected with a mass parameter of 10 kg, a damping parameter of 1 Ns / m, and a stiffness curve of K(t) = 12 + 10sin(t). Figure 3 The graph shows the variation curves of variable stiffness, joint error, and energy function V². The error depicted in the figure is divergent, and the V² energy curve shows upward fluctuations, clearly indicating unstable behavior. The stability criterion proposed in this application cannot determine the stability of this variable impedance system. The system can be transformed from an unstable state to a stable state simply by modifying the impedance parameters. The damping parameter is adjusted to 3 Ns / m, and the variable stiffness is adjusted to 18 + 10sin(t)e. -0.02t As shown in Figure 4, this figure illustrates the evolution of the adjusted variable stiffness, joint error, and energy function V². The joint error and energy curves in the figure converge to zero, indicating that the system is stable. The stability criterion provided in this application confirms the stability of the adjusted variable stiffness curve. When evaluated using general stability criteria, the adjusted impedance parameter does not indicate stability because the variable impedance curve violates the second condition in the criterion. It has been found that conventional stability methods are more conservative, while the new stability method proposed in this application is more flexible and offers greater freedom.

[0129] Figure 5 In an experiment where the variable stiffness was set to K(t) = 15 + 10sin(0.1πt), stiffness, joint error, and energy data were extracted over 75 seconds. The system encountered four external disturbances, as shown in the shaded area. A standard impedance controller was used to ensure that the actual stiffness matched the expected setting and to guarantee system stability. The energy decrease after removing the external disturbances confirmed this result. (b) had the same experimental parameters as (a), but the system used a tank-based controller. When the energy in the tank was insufficient, the system could not maintain the variable stiffness and had to use a constant stiffness value.

[0130] In the description of this application, it should be understood that the terms "center", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this application.

[0131] The terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this application, unless otherwise stated, "a plurality of" means two or more.

[0132] In the description of this application, it should be noted that, unless otherwise expressly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection between two components. Those skilled in the art can understand the specific meaning of the above terms in this application based on the specific circumstances.

[0133] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0134] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for determining the stability of a robot system with a variable impedance model using a variable augmented matrix, characterized in that, Includes the following steps: Step 1) Obtain the state parameter information of the robot arm through the joint sensors; Step 2) Construct the desired variable impedance-trajectory model, where, has The dynamic equations of a robotic arm with one degree of freedom are as follows: ； in, Let the inertia matrix be the inertia matrix in the joint space of the robotic arm. It is the Coriolis force matrix of the robotic arm. It is the gravity term of the robotic arm. For the Jacobian matrix of the robotic arm, For the dimension of the task space of the end effector, Input torque to the joint, where The external force acting on the joint; In the joint space, the external torque of the mass-damped spring system and position error Represented as: ； in, , , For the desired inertia, damping, and variable stiffness matrices, It is positive definite and symmetrical; make To control the torque, as shown in equation (3) ； Define new control input for: ； Substitute equations (3) and (4) into (1) and introduce symbols The closed-loop dynamics in equation (2) are obtained. This represents the actual joint angle position of the robot. The desired joint angle position; Step 3) In the impedance controller, user-defined impedance parameters are used, and the stability of the variable impedance is analyzed; Step 4) In variable impedance control, an energy function is constructed by introducing a variable augmented matrix into the estimation of the derivative of the Lyapunov function and by weighting the velocity error and position error, thus establishing a stability condition that is independent of the state and is related to the variable stiffness. An energy function is constructed by weighting the velocity and position errors, establishing a state-independent stability condition related to variable stiffness, and introducing a time-varying augmented free weight matrix to reduce the conservatism of the energy function. The Lyapunov function is constructed as follows: ； in and It is a symmetric, positive semi-definite, continuously differentiable matrix, along the trajectory of equation (2) Perform differentiation, where ,and It is a constant matrix; ； To eliminate simultaneous inclusion and The item will Defined as ； make , making For all They are all semi-positive definite. and Substituting into equation (9), we can obtain ； Step 5) Based on Step 4), set the damping parameters and mass parameters; Step 6) Calculate the posture error using the posture information of the robotic arm obtained by the sensor and the desired posture information of the robotic arm. Calculate the control torque required for each task based on the error and the hierarchical variable impedance control model, and input it into the robotic arm to achieve control of the robotic arm. In the stability condition under dynamic decoupling, defined as Theorem 1, if there exists... and , making ,in, As an intermediate variable, For any time range, then there exists It is semi-negative definite. If it is semi-negative definite, then in equation (2), the system in The time is globally consistent and stable; In the first condition of Theorem 1, the degrees of freedom provided by the free weighting matrix help reduce the conservatism of the stability judgment method. To ensure that Theorem 1 is preserved ; The stability inequality derived from Theorem 1 yields equation (13). Equations (13) and (11) are simplified to... ; Express the second condition of Theorem 1 as an inequality and move the terms containing the time-varying free weight matrix to the left side; ； exist Integrating the domain with respect to it yields ； let have .

2. The stability judgment method for a robot system variable impedance model using a variable augmented matrix according to claim 1, characterized in that, In step 3), the stability of the variable impedance is analyzed using the Lyapunov function: ； Will Let V1 be abbreviated as V1. Differentiate V1 along the trajectory of equation (2). H and D are both constant matrices; ； The symmetry of the stiffness matrix was used to eliminate and Mixed terms, but containing The positive terms still exist; 。

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