A flexible-joint robot arm safety control method and system

Abstract

This invention relates to a safety control method and system for a flexible joint robotic arm, comprising: constructing a strict feedback system model and primary safety constraints based on the state vector of the robotic arm; designing the nominal tracking control input of the robotic arm based on the strict feedback system model and state vector while ignoring the primary safety constraints; constructing a composite control barrier function of the strict feedback system model in conjunction with the primary safety constraints; constructing a dimensionality-reduced regressor merging vector based on the composite control barrier function, deriving the global worst-case violation bound of the strict feedback system model, and constructing a robust safety lower bound for the time derivative of the composite control barrier function; constructing the nominal safety sandbox boundary of the strict feedback system model based on the composite control barrier function, the dimensionality-reduced regressor merging vector, and the robust safety lower bound for the time derivative of the composite control barrier function; and constructing the physical control law for controlling the robotic arm based on the nominal tracking control input and the nominal safety sandbox boundary. This invention enables safe control of the robotic arm.

Description

Technical Field

[0001] This invention relates to the field of robotic arm control technology, and in particular to a safety control method and system for a flexible joint robotic arm. Background Technology

[0002] Flexible joint robotic arms have been widely used in medical assistance, space exploration, and industrial collaboration due to their advantages such as lightweight structure, high safety in human-machine interaction, and effective absorption of impact energy. However, the introduction of flexible joints makes robotic arm systems inherently high-order systems with highly nonlinear and parametric coupling (typically exhibiting a fourth-order strict feedback form). In real-world physical environments, uncertainties in physical friction models, unknown variations in end-effector loads, and unavoidable external disturbances pose significant challenges to high-precision trajectory tracking and stringent safety constraint control in such systems.

[0003] Currently, to address the aforementioned problems, existing methods for controlling flexible joint robotic arms in this field are mainly classified into the following categories, but all of them have obvious drawbacks or shortcomings:

[0004] First, traditional tracking control methods based on nominal models.

[0005] Current practices: Traditional methods often employ control algorithms based on known and accurate dynamic models, such as the torque calculation method, classical feedback linearization, or backstepping under ideal conditions.

[0006] Disadvantages and limitations: This type of method is highly dependent on the accuracy of the system's nominal model. However, in actual operation, flexible joint robotic arms face complex unmodeled dynamics (such as nonlinear stiffness and hysteresis characteristics) and unknown external disturbances. When system parameters become uncertain, control algorithms based on ideal nominal models are prone to performance degradation, leading to decreased control accuracy and even system instability and safety constraint violations.

[0007] Second, the Standard Control Barrier Function (CBF-QP) safety control method.

[0008] Current Practices: In recent years, the combination of Control Barrier Functions (CBFs) and Quadratic Programming (QP) has been widely used to ensure system safety. The main idea is to construct barrier functions to transform safety into inequality constraints on the derivatives of the system state, and then modify the nominal control commands within the QP framework in a minimally invasive manner to ensure that the system state does not exceed the safety set.

[0009] Disadvantages and limitations: Relative order limitation: Standard CBF is mainly applicable to systems where the relative order of the control input to the safety constraint function is 1. However, the relative order of a flexible joint manipulator from the actual control input (motor torque) to the main constrained state (such as link angle) is usually 4. Directly applying standard CBF cannot solve this problem, and conventional high-order CBF extension methods often ignore the transient errors caused by the underlying cascade dynamics, which can easily lead to the breach of the actual physical safety boundary.

[0010] Poor robustness: The safety verification of traditional CBF relies heavily on accurate models. In the presence of severe parameter perturbations and external bounded disturbances, the evolution trend of the "safe set" calculated by the model deviates from the real physical system, causing the nominal CBF framework to fail and unable to provide absolute forward invariance guarantees.

[0011] Third, robust safety control methods based on high-dimensional adaptive or neural networks.

[0012] Current practices: In order to overcome uncertainty, some existing advanced processes attempt to introduce radial basis neural networks (RBFNNs) or adaptive control into the CBF framework to compensate for the calculation error of the security filter by estimating unknown parameters and disturbances online.

[0013] Disadvantages and limitations: Curse of dimensionality and computational latency: The high-order dynamics of flexible joint robotic arms cause unknown parameters to exhibit high-dimensional coupling characteristics. This is particularly problematic for high-dimensional parameter spaces (such as the overall unknown parameter vector of the system). Performing multi-channel joint adaptive estimation is not only prone to parameter drift, but also brings huge computational burden ("curse of dimensionality"), which makes the real-time deployment of the algorithm on actual hardware platforms extremely difficult.

[0014] Transient safety cannot be guaranteed: Most adaptive CBF methods only focus on the asymptotic convergence of parameter estimation, but ignore the risk of short-term safety constraint violations caused by negative nonlinear cross terms in the transient process at the beginning of parameter estimation (i.e. when the parameter guessing error is large).

[0015] In summary, existing technologies struggle to balance tracking performance, absolute safety (zero violations), and computational efficiency when dealing with the control of high-order flexible joint robotic arms with complex uncertainties. Summary of the Invention

[0016] Therefore, the technical problem to be solved by the present invention is to overcome the difficulty in the prior art in controlling flexible joint robotic arms to simultaneously achieve tracking performance, safety and computational efficiency.

[0017] To address the aforementioned technical problems, this invention provides a safety control method for a flexible joint robotic arm, comprising:

[0018] Step S1: Obtain the state vector of the flexible joint robot arm and based on the state vector A rigorous feedback system model reflecting the physical dynamics of the robotic arm is constructed, and at the same time, based on the state vector... Define the primary safety constraints for robotic arms ;

[0019] Step S2: Ignoring the primary security constraints Under the premise of the aforementioned strict feedback system model and its corresponding state vector Design the nominal tracking control input for the robotic arm. ;

[0020] Step S3: In conjunction with the primary safety constraints Construct the composite control barrier function of the strict feedback system model. ;

[0021] Step S4: Based on the composite control barrier function Constructing a merged vector of dimensionality-reduced regressors Derive the global worst-case violation bound of the rigorous feedback system model caused by external lumped disturbances. And construct the time derivative of the composite control barrier function. The robust safety lower bound;

[0022] Step S5: Based on the composite control barrier function Dimensionality reduction regression vector merging Time derivative of composite control barrier function Robust safety lower bound, and nominal safety sandbox boundary of the strictly feedback system model constructed by synthesis. ;

[0023] Step S6: Based on the nominal tracking control input Nominal security sandbox boundary Constructing the optimal physical control law Through the optimal physical control law The motor driver controls the robotic arm to achieve control of the robotic arm.

[0024] In one embodiment of the present invention, step S1 obtains the state vector of the flexible joint robotic arm. and based on the state vector Methods for constructing rigorous feedback system models that reflect the physical dynamics of robotic arms include:

[0025] The state vector of the flexible joint robotic arm is obtained as follows:

[0026] (1);

[0027] in, Let be the state vector of the robotic arm. These are the link angles of the robotic arm. Angular velocity of the connecting rod Motor angle Motor angular velocity Linkage angle Angle with motor The two are connected by joint stiffness Flexible links enable elastic coupling;

[0028] Based on state vector A rigorous feedback system model with a fourth-order rigorous feedback form, reflecting the physical dynamics of the robotic arm, is constructed and represented as follows:

[0029] (2);

[0030] in, for The first derivative; Let the nominal drift function of the connecting rod be _____. Given the known link damping, inertia, and joint stiffness; For virtual control gain; This is the motor's nominal drift function. Given the known motor damping and inertia; Physical control gain; The actual input physical control torque; Given an unknown constant parameter vector, and with given upper and lower bounds for the constant vector. and , making Valid; Regression Vector and Given a smooth function; and The first and second unknown bounded external disturbances are respectively applied to the linkage channel and the motor channel, and both have known upper and lower bounds. and , making For any time All are true. .

[0031] In one embodiment of the present invention, step S1 is based on the state vector. Defining the primary safety constraints for flexible articulated robotic arms The methods include:

[0032] The link angle in the strict feedback system model The constraint is limited to a safe region, wherein the safe region is defined as a continuously differentiable function. The 0-level set is used as the primary security constraint. :

[0033] (3);

[0034] in, As the primary safety constraint, For state vectors, Let be the set of real numbers. The set safety boundary function for the link angle, This is the preset maximum allowable rotation limit for the link angle.

[0035] In one embodiment of the invention, step S2 ignores the primary security constraint. Under the premise of the aforementioned strict feedback system model and its corresponding state vector Design the nominal tracking control input for the robotic arm. The methods include:

[0036] Ignoring the primary security constraints Under the premise of the state vector Define the sequential tracking error variable as:

[0037] (4);

[0038] (5);

[0039] (6);

[0040] (7);

[0041] in, These are the tracking error variables for the link angle, link angular velocity, motor angle, and motor angular velocity, respectively. They are state vectors The link angle, link angular velocity, motor angle, and motor angular velocity of the robotic arm. For the desired trajectory, These are the first, second, and third virtual control laws;

[0042] The definition of formula (2) includes an unknown constant parameter vector. item and For unknown continuous nonlinear dynamic functions and , represented as:

[0043] (8-1);

[0044] (8-2);

[0045] Using a pre-defined radial basis function to solve the unknown function and To approximate, it can be represented as:

[0046] (8-3);

[0047] (8-4);

[0048] in, This represents the optimal ideal weight vector for the neural network. These are the Gaussian function vectors corresponding to the input states. For bounded approximation error;

[0049] To estimate the optimal ideal weight vector in formulas (8-3) and (8-4) online. The design includes - Modified weight adaptive update law:

[0050] (8);

[0051] (9);

[0052] in, These are the neural network weight estimates. The first derivative, Yes The estimated value, For adaptive learning rate, Correction coefficients to prevent parameters from increasing indefinitely;

[0053] Combining formula (8) Calculate the first to third virtual control laws ,in,

[0054] Design the first virtual control law Used to stabilize the link angle tracking error in formula (4) :

[0055] (10-1);

[0056] Design the second virtual control law Used to stabilize the link angular velocity tracking error in formula (5) :

[0057] (10-2);

[0058] Design the third virtual control law Used to stabilize the motor angle tracking error in formula (6) :

[0059] (10-3);

[0060] Combining formula (9) and The nominal tracking control input for driving the motor is calculated. This is used to stabilize the motor angular velocity tracking error in formula (7). :

[0061] (10-4);

[0062] in, For the desired trajectory The first time derivative, The feedback gain is a preset first, second, third, and fourth constant. The first, second, and third virtual control laws are respectively. The full-time derivative.

[0063] In one embodiment of the invention, step S3 incorporates the primary safety constraint. Construct the composite control barrier function of the strict feedback system model. The methods include:

[0064] Link angles based on the aforementioned rigorous feedback system model Introducing the link angular velocity in a strict feedback system model Virtual security control laws that need to be tracked :

[0065] (11-1);

[0066] in, The preset safety convergence gain;

[0067] Based on primary safety constraints Link angle safety boundary function The virtual security control law Formula (11) must be satisfied to ensure the primary safety constraint. It has forward invariance:

[0068] (11);

[0069] in, For the default extension Class function, defined as , For the defense evolution rate coefficient; The set safety boundary function for the link angle, The preset maximum allowable rotation limit for the link angle. The link angle of the robotic arm;

[0070] Based on virtual security control law The first safe execution error in the design of a strict feedback system model is defined as follows: Based on this, a design was developed for the motor angle in a strict feedback system model. The second virtual security control law :

[0071] (12-1);

[0072] in, This is the nominal drift compensation item for the connecting rod. For the safety gradient coupling term, As the first design weight, This is the feedforward term of the time derivative of the first virtual control law. This is the first layer of error suppression term. The first attenuation rate coefficient;

[0073] Based on the second virtual security control law The second safe execution error in the strict feedback system model design is defined as follows: Based on this, a design was developed for the motor angular velocity in a strict feedback system model. The third virtual security control law :

[0074] (12-2);

[0075] in For cross-level cascaded compensation items, As the second design weight, This is the feedforward term of the time derivative of the second virtual control law. This is the second layer error suppression term. This is the second attenuation rate coefficient;

[0076] Based on the third virtual security control law The third safety execution error in the design of a strict feedback system model is defined as follows: ;

[0077] Through primary safety constraints Link angle safety boundary function Compared with the first to third safety execution errors To construct a composite control barrier function that includes the entire rigorous feedback system model. :

[0078] (12);

[0079] in, Design weights for the first, second, and third.

[0080] In one embodiment of the present invention, step S4 is based on the composite control barrier function. Constructing a merged vector of dimensionality-reduced regressors The methods include:

[0081] Calculate the composite control barrier function time derivative The time derivative is derived using the chain rule of multivariable calculus. Expanding, we get:

[0082] (13-1);

[0083] in, For composite control barrier function State The partial derivatives;

[0084] Substituting formula (2) into formula (13-1) yields:

[0085] (13-2);

[0086] Extract and merge all vectors containing unknown constant parameters from formula (13-2). The coefficient of the term, i.e. and Construct a merged vector of dimensionality-reduced regressors :

[0087] (13);

[0088] Therefore, formula (13-2) contains an unknown constant parameter vector. The item, namely and Equivalently merged into a total item containing parameters .

[0089] In one embodiment of the present invention, step S4 is based on the composite control barrier function. Derivation of the global worst-case violation bound of the rigorous feedback system model caused by external lumped disturbances. And construct the time derivative of the composite control barrier function. Methods for establishing robust safe lower bounds include:

[0090] Extracting the first and second unknown bounded external disturbances from formula (13-2) and Caused item and By calculating separately and The known perturbation boundary defined by formula (2) and Minimum value and As the worst-case physical lower bound, the globally most unfavorable violation bound of the rigorous feedback system model caused by external lumped disturbances is calculated. :

[0091] (15);

[0092] in, For finding the minimum value operator;

[0093] Select the known parameter boundary as defined in formula (2) Internal parameter reference values Define a unique parameter: the error distance scalar. for:

[0094] (14);

[0095] in, It is the Euclidean norm; and formula (14) satisfies , The known upper limit of distance, determined by physical limits;

[0096] For the total term containing parameters obtained from formula (13) Perform algebraic reconstruction using parameter reference values. Decompose it into known parameter terms With unknown parameter perturbation term sum:

[0097] (16-1);

[0098] To ensure the absolute safety of the strict feedback system model, a robust safety lower bound is extracted from formula (13-2): the perturbation term of the unknown parameter in formula (16-1) is removed. The worst-case attenuation lower limit for the safety boundary is set as follows: By combining formula (14), we can derive the corresponding formula (13-2). The robust safety lower bound expression is:

[0099] (16-2);

[0100] in, For composite control barrier function The Lie derivative obtained from the rigorous feedback system model is expressed as: ; For the composite control obstacle function with respect to the link angular velocity Partial derivatives of the state, For the composite control obstacle function with respect to the motor angular velocity Partial derivatives of the state; This represents the worst-case attenuation limit caused by perturbations of the actual parameters.

[0101] In one embodiment of the present invention, step S5 is based on the composite control barrier function. Dimensionality reduction regression vector merging Time derivative of composite control barrier function Robust safety lower bound, and nominal safety sandbox boundary of the strictly feedback system model constructed by synthesis. The methods include:

[0102] The dimensionality reduction regression vector based on formula (13) is merged. Online estimation of the parameter error distance scalar of formula (14) The estimated value :

[0103] (16);

[0104] in, for The first derivative, For parameter error distance scalar The online estimate, and All are adaptive leakage constants that are greater than zero. It is the Euclidean norm;

[0105] Constructing a continuous and smooth nonlinear robust compensation term :

[0106] (18);

[0107] in, It is the Euclidean norm. It is a design constant and satisfies ;

[0108] Based on composite control barrier function Dimensionality reduction regression vector merging Time derivative of composite control barrier function The robust safety lower bound, the parameter error distance scalar corresponding to formula (16) Adaptive update law, nonlinear robust compensation term Construct the nominal safe sandbox boundary of a rigorous feedback system model. :

[0109] (19).

[0110] In one embodiment of the present invention, step S6 is based on the nominal tracking control input. Nominal security sandbox boundary Constructing the optimal physical control law The methods include:

[0111] According to the nominal tracking control input Nominal security sandbox boundary Define the optimization objective and constraints:

[0112] (20-1);

[0113] (20);

[0114] in, The actual physical control input variable to be optimized represents the total torque command ultimately issued to the motor driver; For the composite control obstacle function with respect to the motor angular velocity Partial derivatives of the state, Physical control gain;

[0115] Based on the optimization objective of formula (20-1) and the constraints of formula (20), the optimal physical control law is constructed. :

[0116] (twenty one);

[0117] in, This is the operator for finding the maximum value.

[0118] To address the aforementioned technical problems, this invention provides a safety control system for a flexible joint robotic arm, comprising:

[0119] First building block: Used to obtain the state vector of the flexible joint robotic arm. and based on the state vector A rigorous feedback system model reflecting the physical dynamics of the robotic arm is constructed, and at the same time, based on the state vector... Define the primary safety constraints for robotic arms ;

[0120] Design module: used to ignore the primary security constraints Under the premise of the aforementioned strict feedback system model and its corresponding state vector Design the nominal tracking control input for the robotic arm. ;

[0121] Second building module: used to incorporate the primary security constraints Construct the composite control barrier function of the strict feedback system model. ;

[0122] The third construction module: used to determine the composite control barrier function Constructing a merged vector of dimensionality-reduced regressors Derive the global worst-case violation bound of the rigorous feedback system model caused by external lumped disturbances. And construct the time derivative of the composite control barrier function. The robust safety lower bound;

[0123] Fourth building module: used for the composite control barrier function Dimensionality reduction regression vector merging Time derivative of composite control barrier function Robust safety lower bound, and nominal safety sandbox boundary of the strictly feedback system model constructed by synthesis. ;

[0124] Control module: used to track control inputs according to the nominal value. Nominal security sandbox boundary Constructing the optimal physical control law Through the optimal physical control law The motor driver controls the robotic arm to achieve control of the robotic arm.

[0125] Compared with the prior art, the above-described technical solution of the present invention has the following advantages:

[0126] The flexible joint robotic arm safety control method described in this invention can effectively balance tracking performance, safety and computational efficiency when controlling the flexible robotic arm. This invention can achieve effective decoupling of performance and safety in a dual perturbation environment with unknown parameters and disturbances, and strictly guarantee the absolute safety of the physical system with low computational cost.

[0127] The composite control barrier function constructed in this invention To address the problem of traditional control obstacle functions failing due to excessively high relative order in flexible joint robotic arm systems, this paper proposes a method that uses the step-by-step reverse derivation of virtual control laws to transform the single primary constraint on the link angle into a composite safety constraint encompassing the entire system state, thus providing a feasible mathematical basis for the safety control of high-order nonlinear systems.

[0128] The global worst-case destruction boundary constructed in this invention For external unknown bounded disturbances, by evaluating the minimum value (worst case) caused by the disturbance to the system under known physical boundaries, the control system can build an absolutely safe physical fallback protection without needing to accurately obtain the real-time model of the external disturbance, which greatly enhances the robustness of the system in the face of complex external environments.

[0129] This invention extracts The robust safety lower bound, at the algebraic derivation level, successfully decouples the known nominal dynamics of the system from the unknown uncertainties (parameter perturbations and external disturbances). This provides a rigorous benchmark for the subsequent design of single-dimensional parameter adaptive laws and nonlinear robust compensation terms, and theoretically eliminates the possibility of the system going out of bounds under extreme perturbation conditions.

[0130] The nominal security sandbox boundary constructed by this invention This is the core key to achieving "computational efficiency" in this invention. The boundary condition highly reduces and integrates complex adaptive approximation errors, nonlinear robust compensation terms, and defensive gains, completely avoiding the dimensionality curse and computational explosion caused by high-dimensional uncertainty. While ensuring absolute security with a tight seal, it achieves extremely low online computation costs. Attached Figure Description

[0131] To make the content of this invention easier to understand, the invention will be further described in detail below with reference to specific embodiments and accompanying drawings.

[0132] Figure 1 This is a flowchart of the method of the present invention. Detailed Implementation

[0133] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0134] Example 1

[0135] Reference Figure 1 As shown, this invention relates to a safety control method for a flexible joint robotic arm, comprising:

[0136] Step S1: Obtain the state vector of the flexible joint robot arm and based on the state vector A rigorous feedback system model reflecting the physical dynamics of the robotic arm is constructed, and at the same time, based on the state vector... Define the primary safety constraints for robotic arms ;

[0137] Step S2: Ignoring the primary security constraints Under the premise of the aforementioned strict feedback system model and its corresponding state vector Design the nominal tracking control input for the robotic arm. ;

[0138] Step S3: In conjunction with the primary safety constraints Construct the composite control barrier function of the strict feedback system model. ;

[0139] Step S4: Based on the composite control barrier function Constructing a merged vector of dimensionality-reduced regressors Derive the global worst-case violation bound of the rigorous feedback system model caused by external lumped disturbances. And construct the time derivative of the composite control barrier function. The robust safety lower bound;

[0140] Step S5: Based on the composite control barrier function Dimensionality reduction regression vector merging Time derivative of composite control barrier function Robust safety lower bound, and nominal safety sandbox boundary of the strictly feedback system model constructed by synthesis. ;

[0141] Step S6: Based on the nominal tracking control input Nominal security sandbox boundary Constructing the optimal physical control law Through the optimal physical control law The motor driver controls the robotic arm to achieve control of the robotic arm.

[0142] The following is a detailed description of this embodiment:

[0143] Further, step S1 obtains the state vector of the flexible joint robotic arm. and based on the state vector Methods for constructing rigorous feedback system models that reflect the physical dynamics of robotic arms include:

[0144] The state vector of the flexible joint robotic arm is obtained as follows:

[0145] (1);

[0146] in, Let be the state vector of the robotic arm. These are the link angles of the robotic arm. Angular velocity of the connecting rod Motor angle Motor angular velocity In this embodiment, the connecting rod angle Angle with motor The two are connected by joint stiffness Flexible links achieve elastic coupling; torque is controlled physically. It acts directly on the motor end, changing the motor angle. The flexible link is deformed, and the resulting elastic restoring force is used to drive the linkage movement.

[0147] Based on the state vector of formula (1) A rigorous feedback system model with a fourth-order rigorous feedback form, reflecting the physical dynamics of the robotic arm, is constructed and represented as follows:

[0148] (2);

[0149] in, for The first derivative; Let the nominal drift function of the connecting rod be _____. Given the known link damping, inertia, and joint stiffness; For virtual control gain; This is the motor's nominal drift function. Given the known motor damping and inertia; Physical control gain; The actual input physical control torque; Given an unknown constant parameter vector, and with given upper and lower bounds for the constant vector. and , making Valid; Regression Vector and Given a smooth function; and The first and second unknown bounded external disturbances are respectively applied to the linkage channel and the motor channel, and both have known upper and lower bounds. and , making For any time All are true. .

[0150] Further, step S1 is based on the state vector Defining the primary safety constraints for flexible articulated robotic arms The methods include:

[0151] The link angle in the strict feedback system model The constraint is limited to a safe region, wherein the safe region is defined as a continuously differentiable function. The 0-level set is used as the primary security constraint. :

[0152] (3);

[0153] in, As the primary safety constraint, For state vectors, Let be the set of real numbers. The set safety boundary function for the link angle, This is the preset maximum allowable rotation limit for the link angle.

[0154] Furthermore, step S2 ignores the primary security constraint. Under the premise of the aforementioned strict feedback system model and its corresponding state vector Design the nominal tracking control input for the robotic arm. The methods include:

[0155] Ignoring the primary security constraints (i.e., under the premise of formula (3)), according to the state vector) Define the sequential tracking error variable as:

[0156] (4);

[0157] (5);

[0158] (6);

[0159] (7);

[0160] in, These are the tracking error variables for the link angle, link angular velocity, motor angle, and motor angular velocity, respectively. They are state vectors The link angle, link angular velocity, motor angle, and motor angular velocity of the robotic arm. For the desired trajectory, These are the first, second, and third virtual control laws.

[0161] The definition of formula (2) includes an unknown constant parameter vector. item and For unknown continuous nonlinear dynamic functions and , represented as:

[0162] (8-1);

[0163] (8-2);

[0164] Using pre-defined radial basis function neural networks (RBFNNs) to solve unknown functions and To approximate, it can be represented as:

[0165] (8-3);

[0166] (8-4);

[0167] in, This represents the optimal ideal weight vector for the neural network. These are the Gaussian function vectors corresponding to the input states. This is the bounded approximation error.

[0168] This embodiment aims to estimate the optimal ideal weight vector in formulas (8-3) and (8-4) online. The design includes - Modified weight adaptive update law:

[0169] (8);

[0170] (9);

[0171] in, These are the neural network weight estimates. The first derivative, Yes The estimated value, For adaptive learning rate, Correction coefficients are used to prevent parameters from increasing indefinitely.

[0172] Furthermore, combining formula (8) Calculate the first to third virtual control laws ,in,

[0173] Design the first virtual control law Used to stabilize the link angle tracking error in formula (4) :

[0174] (10-1);

[0175] Design the second virtual control law Used to stabilize the link angular velocity tracking error in formula (5) By introducing a neural network compensation term To offset the unknown nonlinear dynamics in formula (2) :

[0176] (10-2);

[0177] Design the third virtual control law Used to stabilize the motor angle tracking error in formula (6) :

[0178] (10-3);

[0179] Combining formula (9) and The nominal tracking control input for driving the motor is calculated. This is used to stabilize the motor angular velocity tracking error in formula (7). This is used to stabilize the motor angular velocity tracking error in formula (7). By introducing a neural network compensation term To offset the unknown nonlinear dynamics in formula (2) :

[0180] (10-4);

[0181] in, For the desired trajectory The first time derivative, The feedback gain is a preset first, second, third, and fourth constant. The first, second, and third virtual control laws are respectively. The full-time derivative.

[0182] Furthermore, in step S3, the aforementioned primary security constraints are incorporated. Construct the composite control barrier function of the strict feedback system model. The methods include:

[0183] Link angle based on the aforementioned strict feedback system model (i.e., formula (2)) Introducing a method for addressing the angular velocity of the connecting rod Virtual security control laws that need to be tracked :

[0184] (11-1);

[0185] in, This is the preset safety convergence gain.

[0186] The link angle safety boundary function defined by formula (3) The virtual security control law Formula (11) must be satisfied to ensure the primary safety constraint. It has forward invariance:

[0187] (11);

[0188] in, For the default extension Class function, defined as , The defense evolution rate coefficient; the virtual security control law in formula (11) This constitutes the angular velocity of the connecting rod. The desired safe trajectory is determined by constraining the angular velocity of the connecting rod. The desired trajectory is used to ensure the linkage angle. Absolute safety.

[0189] To ensure strict adherence to the safety constraints of formula (11), a multi-step control barrier function (CBF) backstepping method is adopted, targeting the virtual safety control law for each state of the strict feedback system model in formula (2). Suppress transient errors generated during execution, and define the first to third safe execution errors in sequence. The corresponding lower-level virtual security control law is designed as follows:

[0190] Based on virtual security control law The first safe execution error in the rigorous feedback system model design of formula (2) is defined as follows: Based on this, a design was developed for the motor angle. The second virtual security control law :

[0191] (12-1);

[0192] in, This is the nominal drift compensation term for the connecting rod, used to counteract the nonlinear dynamic disturbances of the connecting rod itself; For the safety gradient coupling term, As the first design weight; For the time derivative feedforward term of the first virtual control law; This is the first layer of error suppression term. This is the first attenuation rate coefficient.

[0193] Based on the second virtual security control law The second safe execution error in the rigorous feedback system model design of formula (2) is defined as follows: Based on this, a design was developed targeting the motor's angular velocity. The third virtual security control law :

[0194] (12-2);

[0195] in For cross-level cascaded compensation items, This is the second design weight, used to dynamically offset the residual safety error generated at the connecting rod end through the motor end; For the time derivative feedforward term of the second virtual control law; This is the second layer error suppression term. This is the second attenuation coefficient, used to ensure the tightness of motor angle tracking.

[0196] Based on the third virtual security control law The third safety execution error in the rigorous feedback system model design of formula (2) is defined as follows: ;

[0197] Finally, the safety boundary function of the link angle in formula (3) is used. Compared with the first to third safety execution errors We weight and combine the quadratic terms to construct a composite control barrier function that includes the entire strict feedback system model. :

[0198] (12);

[0199] in, Weights are designed for the first, second, and third errors to adjust the contribution ratio of each order of error to the total safety margin.

[0200] Further, step S4 is based on the composite control barrier function. Constructing a merged vector of dimensionality-reduced regressors The methods include:

[0201] Calculate the composite control barrier function of formula (12) time derivative The time derivative is derived using the chain rule of multivariable calculus. Expanded to the sum of the inner products of the partial derivatives of each state and the rate of change of the state:

[0202] (13-1);

[0203] in, For composite control barrier function State The partial derivatives of .

[0204] Substituting formula (2) into formula (13-1) yields:

[0205] (13-2);

[0206] Extract and merge all vectors containing unknown constant parameters from formula (13-2). The coefficient of the term, i.e. and Construct a merged vector of dimensionality-reduced regressors :

[0207] (13);

[0208] Therefore, formula (13-2) contains an unknown constant parameter vector. The item (i.e.) and The terms were equivalently merged into a total item containing parameters. .

[0209] Further, step S4 is based on the composite control barrier function. Derivation of the global worst-case violation bound of the rigorous feedback system model caused by external lumped disturbances. And construct the time derivative of the composite control barrier function. Methods for establishing robust safe lower bounds include:

[0210] Extracting the first and second unknown bounded external disturbances from formula (13-2) and Caused item and By calculating separately and The known perturbation boundary defined by formula (2) and Minimum value and As the worst-case physical lower bound, the globally most unfavorable violation bound of the rigorous feedback system model caused by external lumped disturbances is calculated. :

[0211] (15);

[0212] in, To find the minimum value operator, All of these are known external disturbance boundary parameters defined by formula (2).

[0213] Introducing a parameter distance scalarization strategy, we select a known parameter boundary located in formula (2). Internal parameter reference values Define a unique parameter: the error distance scalar. for:

[0214] (14);

[0215] in, For an unknown constant parameter vector, It is the Euclidean norm; and formula (14) satisfies , The known upper limit of distance is determined by physical limits.

[0216] Combined with the known upper limit of distance determined by physical limits The high-dimensional constant parameter vector in formula (2) The resulting complex, high-dimensional uncertainty is simplified into a single-dimensional distance value. .

[0217] For the total term containing parameters obtained from formula (13) Perform algebraic reconstruction using parameter reference values. Decompose it into known parameter terms With unknown parameter perturbation term sum:

[0218] (16-1);

[0219] To ensure the absolute safety of the strict feedback system model, a robust safety lower bound is extracted from formula (13-2): combining the mathematical properties of the vector inner product, the perturbation term of the unknown parameter in formula (16-1) is... The worst-case attenuation lower limit for the safety boundary is set as follows: By combining formula (14), we can derive the corresponding formula (13-2). The robust safety lower bound expression is:

[0220] (16-2);

[0221] in, For composite control barrier function The Lie derivative obtained from the rigorous feedback system model is expressed as: ; For the composite control obstacle function with respect to the link angular velocity Partial derivatives of the state, For the composite control obstacle function with respect to the motor angular velocity Partial derivatives of the state, For physical control gain, For physical control torque; This represents the most unfavorable destruction boundary in the overall system. For known parameter terms; This represents the worst-case attenuation limit caused by perturbations of the actual parameters.

[0222] Furthermore, step S5 is based on the composite control barrier function. Dimensionality reduction regression vector merging Time derivative of composite control barrier function Robust safety lower bound, and nominal safety sandbox boundary of the strictly feedback system model constructed by synthesis. The methods include:

[0223] The dimensionality reduction regression vector based on formula (13) is merged. Online estimation of the parameter error distance scalar of formula (14) The estimated value :

[0224] (16);

[0225] in, for The first derivative, For parameter error distance scalar The online estimate, and All are adaptive leakage constants that are greater than zero. It is the Euclidean norm.

[0226] Combined with the estimated value obtained by online updating based on formula (16) The composite control barrier function constructed with formula (12) Constructing a conservative virtual security boundary :

[0227] (17);

[0228] in, The true value of the parameter error distance Compared with the estimated value The parameter estimation error between them is defined as ; An error penalty term to reflect the uncertainty of parameter estimation;

[0229] Formula (17) Abbreviated as Through the Find the time derivative and substitute it into formula (16), then The original evolutionary expansion is expressed as:

[0230] (17-1);

[0231] Due to objective physical parameters As a constant, leading to Substitute these into formula (16-2) and the adaptive update law formula (16), and make the positive and negative signs... An offset occurs, at this time The following inequalities must be satisfied:

[0232] (17-2);

[0233] in, For Li Daoshu.

[0234] As can be seen from formula (17-2), the main residual destructive factor leading to the reduction of system security is the semi-negative definite nonlinear cross term left after the adaptive law is canceled out. To effectively suppress and compensate for the cross terms remaining in formula (17-2) Explicitly introduce a continuous and smooth nonlinear robust compensation term. :

[0235] (18);

[0236] in, For nonlinear robust compensation terms, It is the Euclidean norm. A design constant is used to prevent division by zero, and it satisfies... .

[0237] To ensure virtual security boundaries The evolution satisfies the composite control barrier function The forward invariance condition is introduced, and a mandatory lower bound for the target is imposed. Among them, defensive bonus Furthermore, due to the lower bound of this mandatory target The algebraic expansion will produce a target error term. The adaptive leakage constant is strictly configured as follows: To perform algebraic compensation.

[0238] target error term The derivative error term in formula (17-2) Move the terms to the same side and combine them to obtain the simplified terms with perfect squares. Combined with parameter error distance scalar physical limits The perfect square simplified term is derived from this. The lower bound of the mathematical limit in the worst case is: ; the lower bound of the mathematical limit in the worst case Substituting into formula (17-2), the physical control input is derived. The absolute defense inequality that must be satisfied:

[0239] (18-1);

[0240] All expressions in formula (18-1) related to physical control inputs Irrelevant items are moved to the same side and merged for extraction, combining them to form the nominal safe sandbox boundary of the strict feedback system model. :

[0241] (19);

[0242] in, For Li Daoshu, The most unfavorable destruction boundary in the overall situation. The known parameter feedforward term generated for algebraic reconstruction; The reference value for the parameter set in formula (14) Nonlinear robust compensation terms and their smoothing limits are introduced to eliminate adaptive residual cross terms; Corresponding to the defensive gain term including the limit error boundary; defensive gain .

[0243] The nominal safety sandbox boundary in this embodiment Used to define physical control inputs in subsequent steps. The absolute safety range that must be met.

[0244] Further, step S6 involves tracking the control input based on the nominal tracking input. Nominal security sandbox boundary Constructing the optimal physical control law The methods include:

[0245] According to the nominal tracking control input Nominal security sandbox boundary Define the optimization objective and constraints:

[0246] (20-1);

[0247] (20);

[0248] in, The actual physical control input variable to be optimized represents the total torque command ultimately issued to the motor driver;

[0249] Based on the optimization objective of formula (20-1) and the constraints of formula (20), and according to the KKT optimality condition, the optimal physical control law is derived. :

[0250] (twenty one);

[0251] in, This is the operator for finding the maximum value.

[0252] Example 2

[0253] This embodiment provides a safety control system for a flexible joint robotic arm, including:

[0254] First building block: Used to obtain the state vector of the flexible joint robotic arm. and based on the state vector A rigorous feedback system model reflecting the physical dynamics of the robotic arm is constructed, and at the same time, based on the state vector... Define the primary safety constraints for robotic arms ;

[0255] Design module: used to ignore the primary security constraints Under the premise of the aforementioned strict feedback system model and its corresponding state vector Design the nominal tracking control input for the robotic arm. ;

[0256] Second building module: used to incorporate the primary security constraints Construct the composite control barrier function of the strict feedback system model. ;

[0257] The third construction module: used to determine the composite control barrier function Constructing a merged vector of dimensionality-reduced regressors Derive the global worst-case violation bound of the rigorous feedback system model caused by external lumped disturbances. And construct the time derivative of the composite control barrier function. The robust safety lower bound;

[0258] Fourth building module: used for the composite control barrier function Dimensionality reduction regression vector merging Time derivative of composite control barrier function Robust safety lower bound, and nominal safety sandbox boundary of the strictly feedback system model constructed by synthesis. ;

[0259] Control module: used to track control inputs according to the nominal value. Nominal security sandbox boundary Constructing the optimal physical control law Through the optimal physical control law The motor driver controls the robotic arm to achieve control of the robotic arm.

[0260] Example 3

[0261] This embodiment provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of the flexible joint robotic arm safety control method described in Embodiment 1.

[0262] Example 4

[0263] This embodiment provides a computer-readable storage medium storing a computer program thereon. When the computer program is executed by a processor, it implements the steps of the flexible joint robotic arm safety control method described in Embodiment 1.

[0264] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code. The solutions in the embodiments of this application can be implemented in various computer languages, such as the object-oriented programming language Java and the interpreted scripting language JavaScript.

[0265] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0266] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0267] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0268] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A safety control method for a flexible joint robotic arm, characterized in that, include: Step S1: Obtain the state vector of the flexible joint robot arm and based on the state vector A rigorous feedback system model reflecting the physical dynamics of the robotic arm is constructed, and at the same time, based on the state vector... Define the primary safety constraints for robotic arms ; Step S2: Ignoring the primary security constraints Under the premise of the aforementioned strict feedback system model and its corresponding state vector Design the nominal tracking control input for the robotic arm. ; Step S3: In conjunction with the primary safety constraints Construct the composite control barrier function of the strict feedback system model. ; Step S4: Based on the composite control barrier function Constructing a merged vector of dimensionality-reduced regressors Derive the global worst-case violation bound of the rigorous feedback system model caused by external lumped disturbances. And construct the time derivative of the composite control barrier function. The robust safety lower bound; Step S5: Based on the composite control barrier function Dimensionality reduction regression vector merging Time derivative of composite control barrier function Robust safety lower bound, and nominal safety sandbox boundary of the strictly feedback system model constructed by synthesis. ; Step S6: Based on the nominal tracking control input Nominal security sandbox boundary Constructing the optimal physical control law Through the optimal physical control law The motor driver controls the robotic arm to achieve control of the robotic arm.

2. The safety control method for a flexible joint robotic arm according to claim 1, characterized in that: Step S1 obtains the state vector of the flexible joint robotic arm. and based on the state vector Methods for constructing rigorous feedback system models that reflect the physical dynamics of robotic arms include: The state vector of the flexible joint robotic arm is obtained as follows: （1）； in, Let be the state vector of the robotic arm. These are the link angles of the robotic arm. Angular velocity of the connecting rod Motor angle Motor angular velocity Linkage angle Angle with motor The two are connected by joint stiffness Flexible links enable elastic coupling; Based on state vector A rigorous feedback system model with a fourth-order rigorous feedback form, reflecting the physical dynamics of the robotic arm, is constructed and represented as follows: （2）； in, for The first derivative; Let the nominal drift function of the connecting rod be _____. Given the known link damping, inertia, and joint stiffness; For virtual control gain; This is the motor's nominal drift function. Given the known motor damping and inertia; Physical control gain; The actual physical control torque input; Given an unknown constant parameter vector, and with given upper and lower bounds for the constant vector. and , making Valid; Regression Vector and Given a smooth function; and The first and second unknown bounded external disturbances are respectively applied to the linkage channel and the motor channel, and both have known upper and lower bounds. and , making For any time All are true. .

3. The safety control method for a flexible joint robotic arm according to claim 1, characterized in that: Step S1 is based on the state vector Defining the primary safety constraints for flexible articulated robotic arms The methods include: The link angle in the strict feedback system model The constraint is limited to a safe region, wherein the safe region is defined as a continuously differentiable function. The 0-level set is used as the primary security constraint. : （3）； in, As the primary safety constraint, For state vectors, Let be the set of real numbers. The set safety boundary function for the link angle, This is the preset maximum allowable rotation limit for the link angle.

4. The safety control method for a flexible joint robotic arm according to claim 2, characterized in that: Step S2 ignores the primary security constraint. Under the premise of the aforementioned strict feedback system model and its corresponding state vector Design the nominal tracking control input for the robotic arm. The methods include: Ignoring the primary security constraints Under the premise of the state vector Define the sequential tracking error variable as: （4）； （5）； （6）； （7）； in, These are the tracking error variables for the link angle, link angular velocity, motor angle, and motor angular velocity, respectively. They are state vectors The link angle, link angular velocity, motor angle, and motor angular velocity of the robotic arm. For the desired trajectory, These are the first, second, and third virtual control laws; The definition of formula (2) includes an unknown constant parameter vector. item and For unknown continuous nonlinear dynamic functions and , represented as: （8-1）； （8-2）； Using a pre-defined radial basis function to solve the unknown function and To approximate, it can be represented as: （8-3）； （8-4）； in, This represents the optimal ideal weight vector for the neural network. These are the Gaussian function vectors corresponding to the input states. For bounded approximation error; To estimate the optimal ideal weight vector in formulas (8-3) and (8-4) online. The design includes - Modified weight adaptive update law: （8）； （9）； in, These are the neural network weight estimates. The first derivative, Yes The estimated value, For adaptive learning rate, Correction coefficients to prevent parameters from increasing indefinitely; Combining formula (8) Calculate the first to third virtual control laws ,in, Design the first virtual control law Used to stabilize the link angle tracking error in formula (4) : (10-1)； Design the second virtual control law Used to stabilize the link angular velocity tracking error in formula (5) : (10-2)； Design the third virtual control law Used to stabilize the motor angle tracking error in formula (6) : (10-3)； Combining formula (9) and The nominal tracking control input for driving the motor is calculated. This is used to stabilize the motor angular velocity tracking error in formula (7). : (10-4)； in, For the desired trajectory The first time derivative, The feedback gain is a preset first, second, third, and fourth constant. The first, second, and third virtual control laws are respectively. The full-time derivative.

5. The safety control method for a flexible joint robotic arm according to claim 1, characterized in that: In step S3, the primary safety constraint is incorporated. Construct the composite control barrier function of the strict feedback system model. The methods include: Link angles based on the aforementioned rigorous feedback system model Introducing the link angular velocity in a strict feedback system model Virtual security control laws that need to be tracked : （11-1）； in, The preset safety convergence gain; Based on primary safety constraints Link angle safety boundary function The virtual security control law Formula (11) must be satisfied to ensure the primary safety constraint. It has forward invariance: （11）； in, For the default extension Class function, defined as , For the defense evolution rate coefficient; The set safety boundary function for the link angle, The preset maximum allowable rotation limit for the link angle. The link angle of the robotic arm; Based on virtual security control law The first safe execution error in the design of a strict feedback system model is defined as follows: Based on this, a design was developed for the motor angle in a strict feedback system model. The second virtual security control law : （12-1）； in, This is the nominal drift compensation item for the connecting rod. For the safety gradient coupling term, As the first design weight, This is the feedforward term of the time derivative of the first virtual control law. This is the first layer of error suppression term. The first attenuation rate coefficient; Based on the second virtual security control law The second safe execution error in the strict feedback system model design is defined as follows: Based on this, a design was developed for the motor angular velocity in a strict feedback system model. The third virtual security control law : （12-2）； in For cross-level cascaded compensation items, As the second design weight, This is the feedforward term of the time derivative of the second virtual control law. This is the second layer error suppression term. This is the second attenuation rate coefficient; Based on the third virtual security control law The third safety execution error in the design of a strict feedback system model is defined as follows: ; Through primary safety constraints Link angle safety boundary function Compared with the first to third safety execution errors To construct a composite control barrier function that includes the entire rigorous feedback system model. : (12)； in, Design weights for the first, second, and third.

6. The safety control method for a flexible joint robotic arm according to claim 2, characterized in that: Step S4 is based on the composite control barrier function. Constructing a merged vector of dimensionality-reduced regressors The methods include: Calculate the composite control barrier function time derivative The time derivative is derived using the chain rule of multivariable calculus. Expanding, we get: （13-1）； in, For composite control barrier function State The partial derivatives; Substituting formula (2) into formula (13-1) yields: （13-2）； Extract and merge all vectors containing unknown constant parameters from formula (13-2). The coefficient of the term, i.e. and Construct a merged vector of dimensionality-reduced regressors : （13）； Therefore, formula (13-2) contains an unknown constant parameter vector. The item, namely and Equivalently merged into a total item containing parameters .

7. The safety control method for a flexible joint robotic arm according to claim 6, characterized in that: Step S4 is based on the composite control barrier function. Derivation of the global worst-case violation bound of the rigorous feedback system model caused by external lumped disturbances. And construct the time derivative of the composite control barrier function. Methods for establishing robust safe lower bounds include: Extracting the first and second unknown bounded external disturbances from formula (13-2) and Caused item and By calculating separately and The known perturbation boundary defined by formula (2) and Minimum value and As the worst-case physical lower bound, the globally most unfavorable violation bound of the rigorous feedback system model caused by external lumped disturbances is calculated. : （15）； in, For finding the minimum value operator; Select the known parameter boundary as defined in formula (2) Internal parameter reference values Define a unique parameter: the error distance scalar. for: (14)； in, It is the Euclidean norm; and formula (14) satisfies , The known upper limit of distance, determined by physical limits; For the total term containing parameters obtained from formula (13) Perform algebraic reconstruction using parameter reference values. Decompose it into known parameter terms With unknown parameter perturbation term sum: （16-1）； To ensure the absolute safety of the strict feedback system model, a robust safety lower bound is extracted from formula (13-2): the perturbation term of the unknown parameter in formula (16-1) is removed. The worst-case attenuation lower limit for the safety boundary is set as follows: By combining formula (14), we can derive the corresponding formula (13-2). The robust safety lower bound expression is: （16-2）； in, For composite control barrier function The Lie derivative obtained from the rigorous feedback system model is expressed as: ; For the composite control obstacle function with respect to the link angular velocity Partial derivatives of the state, For the composite control obstacle function with respect to the motor angular velocity Partial derivatives of the state; This represents the worst-case attenuation limit caused by perturbations of the actual parameters.

8. The safety control method for a flexible joint robotic arm according to claim 7, characterized in that: Step S5 is based on the composite control barrier function. Dimensionality reduction regression vector merging Time derivative of composite control barrier function Robust safety lower bound, and nominal safety sandbox boundary of the strictly feedback system model constructed by synthesis. The methods include: The dimensionality reduction regression vector based on formula (13) is merged. Online estimation of the parameter error distance scalar of formula (14) The estimated value : （16）； in, for The first derivative, For parameter error distance scalar The online estimate, and All are adaptive leakage constants that are greater than zero. It is the Euclidean norm; Constructing a continuous and smooth nonlinear robust compensation term : （18）； in, It is the Euclidean norm. It is a design constant and satisfies ; Based on composite control barrier function Dimensionality reduction regression vector merging Time derivative of composite control barrier function The robust safety lower bound, the parameter error distance scalar corresponding to formula (16) Adaptive update law, nonlinear robust compensation term Construct the nominal safe sandbox boundary of a rigorous feedback system model. : （19）。 9. The safety control method for a flexible joint robotic arm according to claim 1, characterized in that: Step S6 is based on the nominal tracking control input. Nominal security sandbox boundary Constructing the optimal physical control law The methods include: According to the nominal tracking control input Nominal security sandbox boundary Define the optimization objective and constraints: （20-1）； （20）； in, The actual physical control input variable to be optimized represents the total torque command ultimately issued to the motor driver; For the composite control obstacle function with respect to the motor angular velocity Partial derivatives of the state, Physical control gain; Based on the optimization objective of formula (20-1) and the constraints of formula (20), the optimal physical control law is constructed. : （21）； in, This is the operator for finding the maximum value.

10. A safety control system for a flexible joint robotic arm, characterized in that, include: First building block: Used to obtain the state vector of the flexible joint robotic arm. and based on the state vector A rigorous feedback system model reflecting the physical dynamics of the robotic arm is constructed, and at the same time, based on the state vector... Define the primary safety constraints for robotic arms ; Design module: used to ignore the primary security constraints Under the premise of the aforementioned strict feedback system model and its corresponding state vector Design the nominal tracking control input for the robotic arm. ; Second building module: used to incorporate the primary security constraints Construct the composite control barrier function of the strict feedback system model. ; The third construction module: used to determine the composite control barrier function Constructing a merged vector of dimensionality-reduced regressors Derive the global worst-case violation bound of the rigorous feedback system model caused by external lumped disturbances. And construct the time derivative of the composite control barrier function. The robust safety lower bound; Fourth building module: used for the composite control barrier function Dimensionality reduction regression vector merging Time derivative of composite control barrier function Robust safety lower bound, and nominal safety sandbox boundary of the strictly feedback system model constructed by synthesis. ; Control module: used to track control inputs according to the nominal value. Nominal security sandbox boundary Constructing the optimal physical control law Through the optimal physical control law The motor driver controls the robotic arm to achieve control of the robotic arm.

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