Continuous body robot model-free shape control method, system and device and medium

Abstract

The invention discloses a continuum robot model-free shape control method, system and device and a medium, and the method comprises the steps: applying a zero neural network model to a shape feature parameter error function, obtaining an update formula of an actuator according to a first design formula, and calculating the variable quantity of the actuator according to an estimated value of a Jacobian matrix; applying a zero neural network model to the parameter speed error function, obtaining an updating formula of a Jacobian matrix according to a second design formula, and calculating an estimated variable quantity of the Jacobian matrix according to an acceleration value of an actual shape characteristic parameter fed back by the sensor and the estimated acceleration of the actuator; and performing learning according to the variable quantity of the actuator and the estimated variable quantity of the Jacobian matrix to obtain actuator control input of the continuum robot at the next moment and an estimated value of the Jacobian matrix, and performing shape control on the continuum robot at the next moment. The model-free shape control efficiency and accuracy of the continuum robot are improved, and the model-free shape control method can be applied to the technical field of robot control.

Description

Technical Field

[0001] The present invention relates to the field of robot control technology, and in particular to a method, system, device and medium for model-free shape control of a continuum robot. Background Art

[0002] Robots have broad application prospects in various fields. Traditional robots have very limited degrees of freedom, poor flexibility, safety and adaptability, and perform poorly in some unstructured environments or constrained environments where the surrounding environment is not fixed and changes, and it is difficult to achieve ideal control effects. In contrast, continuum robots can interact with the environment more safely due to their interactive safety and high softness and agility, reducing the risk of harm to themselves and the environment, and are widely used in the above-mentioned unstructured or constrained complex environments. Therefore, continuum robot control is a hot topic in the field of robotics recently.

[0003] However, in situations where collision avoidance or full arm path planning is required (such as minimally invasive surgery on the human body), the robot's shape needs to be globally controlled. The shape control of a continuum robot generally allows the arm to move from an initial shape to a specified shape, thereby avoiding collisions with the environment.

[0004] Continuum robots are similar to human arms, and their arms are usually made of materials with a Young's modulus close to or lower than that of biological skin. There are also many ways to drive continuum robots, including rope-driven robots, pneumatic robots, concentric tube robots, etc. The mathematical model of the robot can be divided into a kinematic model and a dynamic model. The kinematic model describes the mapping relationship between the angles of each joint of the arm and the position of a module on the arm (such as the end of the arm) in the task space, while the dynamic model describes the mapping relationship between the force, torque and joint angle applied by each joint of the arm. The control of the arm can usually be divided into inverse kinematic control and inverse dynamics control. Inverse kinematics control refers to the calculation of the required joint angles based on the desired posture of the arm, while inverse dynamics control refers to the calculation of the required joint torque based on the desired joint angles.

[0005] The control method of the continuum robot in the prior art has the following disadvantages:

[0006] 1) It is highly dependent on the mathematical model of the robot. In fact, existing technical solutions are basically model-based control solutions. For robots with different drives and different modeling, the relationship between the actuator and shape feature parameters is different. Researchers cannot know the mathematical models of all continuum robots in advance, so the application scenarios of this type of method are relatively limited.

[0007] 2) For model-free control methods, the control needs to be pre-trained by the network, which requires a large amount of data sets. And similarly, since this type of method needs to be pre-trained, it essentially requires pre-learning the mathematical model of the robot. If the robot is replaced for shape control, pre-training needs to be performed again, which takes a lot of time. The creation of data sets is also very troublesome and requires a lot of resources. Therefore, the versatility of this type of method is still poor and cannot be applied to more robots. Therefore, it is necessary to study a shape control method with a wider range of application scenarios, less resource consumption, and higher efficiency.

[0008] 3) The current model-free control method does not achieve ideal control results due to the lack of a model, and the relative error of the controlled shape exceeds 1%. Although a 1% relative error is not considered large in shape control tasks, it may bring unnecessary risks in some tasks with high precision requirements. Therefore, a more accurate model-free shape control method needs to be studied. Summary of the invention

[0009] The purpose of the present invention is to solve one of the technical problems existing in the prior art to at least a certain extent.

[0010] To this end, an object of an embodiment of the present invention is to provide a model-free shape control method for a continuum robot, which improves the efficiency and accuracy of the model-free shape control of the continuum robot.

[0011] Another object of an embodiment of the present invention is to provide a model-free shape control system for a continuum robot.

[0012] In order to achieve the above technical objectives, the technical solutions adopted by the embodiments of the present invention include:

[0013] In a first aspect, an embodiment of the present invention provides a model-free shape control method for a continuum robot, comprising the following steps:

[0014] Initialize the actuators of the continuum robot arm, control system related parameters, the initial value of the Jacobian matrix of the robot system itself, and control task related parameters;

[0015] A shape feature parameter error function is defined for the continuum robot system, a zeroing neural network model is applied to the shape feature parameter error function, an update formula of the actuator is obtained according to the first design formula, and a change amount of the actuator is calculated according to an estimated value of the Jacobian matrix;

[0016] A parameter velocity error function is defined for the continuum robot system, a zeroing neural network model is applied to the parameter velocity error function, an update formula of the Jacobian matrix is ​​obtained according to the second design formula, and an estimated change of the Jacobian matrix is ​​calculated according to the acceleration value of the actual shape characteristic parameter fed back by the sensor and the estimated actuator acceleration;

[0017] According to the change of the actuator and the estimated change of the Jacobian matrix, the actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment are learned, and the shape of the continuum robot at the next moment is controlled according to the actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment.

[0018] Furthermore, in one embodiment of the present invention, the control system related parameters include convergence parameters of the zeroing neural network model, the control task related parameters include control task duration and shape characteristic parameters corresponding to the expected shape, the shape characteristic parameter error function is a function of the error between the expected shape characteristic parameters and the actual shape characteristic parameters fed back by the sensor with respect to time, and the parameter speed error function is a function of the error between the derivative value of the actual shape characteristic parameters fed back by the sensor and the product of the estimated value of the Jacobian matrix at the previous moment and the change in the actuator with respect to time.

[0019] Furthermore, in one embodiment of the present invention, the shape feature parameter error function is:

[0020] e(t)=x d (t)-x a (t)

[0021] Among them, e(t) represents the shape feature parameter error at time t, x d (t) represents the expected shape characteristic parameters at time t, x a (t) represents the actual shape characteristic parameters fed back by the sensor at time t;

[0022] The first design formula of the zeroing neural network model is:

[0023]

[0024] in, represents the derivative value of the shape feature parameter error function at time t, γ 1 >0 is the convergence parameter of the shape feature parameter error function, and Φ(·) represents the activation function of the error.

[0025] Furthermore, in one embodiment of the present invention, the updating formula of the actuator is:

[0026]

[0027] in, represents the change of the actuator, represents the estimate of the Jacobian matrix, express The Moore-Penrose generalized inverse of x d represents the expected shape characteristic parameter, x a Represents the actual shape characteristic parameters of the robot.

[0028] Furthermore, in one embodiment of the present invention, the parameter speed error function is:

[0029]

[0030] Among them, ∈(t) represents the parameter speed error at time t, Represents the derivative value of the actual shape characteristic parameter fed back by the sensor at time t, represents the estimated value of the Jacobian matrix at time t, Represents the change of the actuator at time t;

[0031] The second design formula of the zeroing neural network model is:

[0032]

[0033] in, represents the derivative value of the parameter velocity error function at time t, γ 2 >0 is the convergence parameter of the parameter speed error function, and Φ(·) represents the activation function of the error.

[0034] Furthermore, in one embodiment of the present invention, the updating formula of the Jacobian matrix is:

[0035]

[0036] in, represents the estimated change in the Jacobian matrix, Represents the derivative value of the actual shape characteristic parameter fed back by the sensor, represents the estimate of the Jacobian matrix, represents the change of the actuator, The acceleration value representing the actual shape characteristic parameter fed back by the sensor, represents the estimated actuator acceleration, express The Moore-Penrose generalized inverse of .

[0037] Further, in one embodiment of the present invention, the actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment are calculated by the following formula:

[0038]

[0039] Among them, q(t n+1 ) indicates t n+1 The actuator control input of the continuum robot at the moment, q(t n ) indicates t n The actuator control input of the moment continuum robot, Indicates t n The change of the actuator at the moment, Indicates t n+1 The estimated value of the Jacobian matrix at time instant, Indicates t n The estimated value of the Jacobian matrix at time instant, Indicates t n The estimated change of the Jacobian matrix at time t, Δt represents the preset time step.

[0040] In a second aspect, an embodiment of the present invention provides a continuum robot model-free shape control system, comprising:

[0041] Parameter initialization module, used to initialize the actuator of the continuum robot arm, control system related parameters, the initial value of the Jacobian matrix of the robot system itself, and control task related parameters;

[0042] an actuator change calculation module, used to define a shape feature parameter error function for a continuum robot system, apply a zeroing neural network model to the shape feature parameter error function, obtain an actuator update formula according to a first design formula, and then calculate the actuator change amount according to an estimated value of a Jacobian matrix;

[0043] A Jacobian matrix change calculation module is used to define a parameter velocity error function for a continuum robot system, apply a zeroing neural network model to the parameter velocity error function, obtain an update formula for the Jacobian matrix according to a second design formula, and then calculate an estimated change of the Jacobian matrix according to the acceleration value of the actual shape feature parameter fed back by the sensor and the estimated actuator acceleration;

[0044] The shape control module is used to learn the actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment based on the change of the actuator and the estimated change of the Jacobian matrix, and to perform shape control on the continuum robot at the next moment based on the actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment.

[0045] In a third aspect, an embodiment of the present invention provides a model-free shape control device for a continuum robot, comprising:

[0046] at least one processor;

[0047] at least one memory for storing at least one program;

[0048] When the at least one program is executed by the at least one processor, the at least one processor implements the above-mentioned model-free shape control method for a continuum robot.

[0049] In a fourth aspect, an embodiment of the present invention further provides a computer-readable storage medium, which stores a program executable by a processor, and the program executable by the processor is used to execute the above-mentioned continuum robot model-free shape control method when executed by the processor.

[0050] The advantages and beneficial effects of the present invention will be partly given in the following description, partly become apparent from the following description, or be understood through the practice of the present invention:

[0051] The embodiment of the present invention can control the shape of the arm when the model of the continuum robot system is unknown, thereby solving the challenges brought by model uncertainty or unknown models, avoiding the complicated process of modeling the robot arm, and the control scheme has good portability for continuum robots of different types and structures; the embodiment of the present invention only needs to use the parameters and their derivative data fed back by the sensor to calculate the shape control of the continuum robot in real time, avoiding a large number of pre-training processes in other model-free methods, and greatly reducing the resources consumed to complete the task; the embodiment of the present invention can realize the precise shape control of the continuum robot, and the control effect is comparable to or even exceeds the model-based shape control method, thereby expanding the application scenarios of the continuum robot. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] In order to more clearly illustrate the technical solution in the embodiments of the present invention, the following introduction is made to the drawings required for use in the embodiments of the present invention. It should be understood that the drawings introduced below are only for the convenience of clearly describing some embodiments of the technical solution of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative work.

[0053] Figure 1 A flowchart of the steps of a model-free shape control method for a continuum robot provided by an embodiment of the present invention;

[0054] Figure 2 It is a schematic diagram of a piecewise constant curvature model in the prior art;

[0055] Figure 3 It is a schematic diagram of shape parameters under the piecewise constant curvature model in the prior art;

[0056] Figure 4A schematic diagram of the principle of model-free shape control of a continuum robot based on an annihilation neural network provided by an embodiment of the present invention;

[0057] Figure 5 A schematic diagram of a flow chart of model-free shape control of a continuum robot based on an annihilation neural network provided by an embodiment of the present invention;

[0058] Figure 6 A schematic diagram of the shape and posture of a continuum robot provided by an embodiment of the present invention for model-free shape control;

[0059] Figure 7 A shape error variation diagram of the model-free shape control of a continuum robot provided by an embodiment of the present invention;

[0060] Figure 8 A graph showing the error and evaluation index variation of shape feature parameters in a shape control task of a rope-driven robot provided in an embodiment of the present invention;

[0061] Fig. 9 A diagram showing changes in actuators in a shape control task of a rope-driven robot provided by an embodiment of the present invention;

[0062] Fig.10 A structural block diagram of a model-free shape control system for a continuum robot provided by an embodiment of the present invention;

[0063] Fig.11 A structural block diagram of a model-free shape control device for a continuum robot provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0064] The embodiments of the present invention are described in detail below, and examples of the embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and are not to be construed as limitations of the present invention. For the step numbers in the following embodiments, they are only provided for the convenience of explanation, and the order between the steps is not limited in any way, and the execution order of each step in the embodiment can be adaptively adjusted according to the understanding of those skilled in the art.

[0065] In the description of the present invention, the meaning of "a plurality" is two or more than two. If there is a description of "a first" or "a second", it is only used for the purpose of distinguishing technical features, and cannot be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features or implicitly indicating the order of the indicated technical features. In addition, unless otherwise defined, all technical and scientific terms used in this document have the same meaning as those commonly understood by those skilled in the art.

[0066] Since different robots are driven in different ways, their mathematical models are likely to be different. Therefore, in order to unify the control description of the robot, "actuator" is used to represent the control input quantity consisting of its kinematic parameters. In robots with different drives, the actuator may include parameters such as the length of the robot's rope, the deviation angle of the rope, and the rotation angle of the control motor. Generally speaking, the kinematic model of a robot can be expressed by the following formula:

[0067] f(q(t))=x(t) (1)

[0068] in represents the value of the control actuator of the arm at time t, (·) T Represents the transpose operator. represents the position of the arm in space. Specifically, in the robot's trajectory tracking task, x(t) represents the position of the end effector, and in the robot's shape control task, x(t) represents the robot's shape posture. is a nonlinear mapping function that describes the kinematics of the robot arm. The kinematic model of the robot can usually also be described at the velocity level. By taking the derivative of both sides of formula (1) with respect to time t, we can obtain:

[0069]

[0070] in represents the partial derivative of the function f(·) with respect to the actuator q(t), i.e., the Jacobian matrix of the robot arm, and the column vector Represents the change speed of the actuator, column vector It represents the speed of change of the arm posture, which is expressed as the speed of change of the robot's shape posture in the shape control task. The shape control problem of the robot is to obtain the corresponding required actuator value q(t) according to the specified expected shape posture, that is:

[0071]

[0072] The obtained actuator value q(t) is then used to control the robot arm so that it moves in the target shape and posture x d (t) Transformation, i.e., the actual shape and posture of the arm x a (t) tends to the target shape and pose x d (t). The function that represents the robot shape and posture to the actuator is the inverse function of f(·), that is, the inverse kinematic mapping. Since the kinematic mapping function f(·) is nonlinear in the continuum robot, it is difficult to directly obtain it through mathematical derivation. At the same time, due to the different kinematic models of different robots, their inverse mapping The shapes of the robots are also different, so it is difficult to use a unified control method to control the shape of the robot.

[0073] The piecewise constant curvature model is a commonly used model in continuum robot modeling. This model divides the flexible arm of the continuum robot into multiple segments, each of which is composed of a series of tangent arcs with constant curvature. Under this assumption, the curvature of each arc is constant in space but variable in time. The advantage of this model is that a complex continuum robot with nearly infinite degrees of freedom can be regarded as multiple simple arcs, and the robot's position and posture can be determined by knowing the geometric characteristics of each arc. The schematic diagram of the piecewise constant curvature model is shown in the figure. Figure 2 As shown, S represents the endpoint of the arm segment, O represents the center of the arm segment, k represents the curvature of the arm segment, and l represents the length of the arm segment.

[0074] Under the assumption of the piecewise constant curvature model, some shape characteristic parameters of the arc corresponding to the arm can be used to represent the shape of the arm. Common shape characteristic parameters include curvature k, rotation angle θ, deflection angle The arm length l, etc. The deflection angle is the angle between the plane passing through each arc segment and the xz coordinate plane of the starting point of the segment. The rotation angle θ represents the center angle of the arc segment. Figure 3 is a schematic diagram of these shape parameter characteristics, {S i}-{x i ,y i ,z i} represents the local coordinate system of the arm segment. When the selected shape feature parameters can uniquely determine the arm shape, the shape control problem of the continuum robot can be transformed into the control of the arm shape feature parameters. At this time, x in formula (1) to formula (3) represents the value of the corresponding shape feature parameter.

[0075] The existing continuum robot shape control methods are as follows:

[0076] First, based on the piecewise constant curvature assumption, the method selects the shape characteristic parameters as curvature k, torque τ, deflection angle The shape control problem of the continuum robot arm is converted into the control of the arm's curvature, torque, and deflection angle. In the task space, the curvature, torque, and deflection angle can uniquely determine the shape of the arm. This method uses a rope-driven robot as the control object, and the arm is driven by four ropes. The relationship between each driving rope and the shape parameters can be obtained based on the robot's parameters:

[0077]

[0078] Where l is the length of the center line of the arm under the initial condition, d is the parameter of the robot itself, β 1 =0,β2 =π / 2,β 3 =π,β 4 =3π / 2. By taking the derivative of formula (4) with respect to the three shape characteristic parameters, we can obtain the velocity-level relationship:

[0079]

[0080] in is the Jacobian matrix of the robot shape control, and x still represents the shape feature parameters. Since shape control requires the arm shape to be controlled to the expected posture x d , so formula (5) can be rewritten as:

[0081]

[0082] where x d -x represents the expected change in the shape parameter, and Indicates the change value of the driving rope in the rope-driven robot, K p Represents the proportional gain, which can control the convergence speed.

[0083] Formula (6) can calculate the expected change value of the actuator at the next moment through the error between the robot shape feature parameters and the expected shape parameters, thus completing the shape control of the continuum robot.

[0084] From formula (4) and formula (5), we can know that this method requires knowing the mathematical model of the continuum robot in advance. In fact, most of the current shape control methods also require knowing the mathematical model of the robot as a prerequisite. For example, the PID method, the method based on artificial potential field, and the method based on adaptive visual servoing are all methods with good shape control effects.

[0085] In addition to these methods, there are a few continuum robot shape control methods that can complete shape control tasks without knowing the robot model in advance. Since the robot model is unknown, these methods are essentially to predict the mathematical model of the robot and then complete the shape control task. For example, the mathematical model of the continuum robot is predicted by a neural network, and the neural network is used to learn [q i ,x d ,x]→q i+1 The mapping relationship is then used to complete the shape control of the robot.

[0086] From the above content, it can be seen that the control method of the continuum robot in the prior art has the following disadvantages:

[0087] 1) It is highly dependent on the mathematical model of the robot. In fact, existing technical solutions are basically model-based control solutions. For robots with different drives and different modeling, the relationship between the actuator and shape feature parameters is different. Researchers cannot know the mathematical models of all continuum robots in advance, so the application scenarios of this type of method are relatively limited.

[0088] 2) For model-free control methods, the control needs to be pre-trained by the network, which requires a large amount of data sets. And similarly, since this type of method needs to be pre-trained, it essentially requires pre-learning the mathematical model of the robot. If the robot is replaced for shape control, pre-training needs to be performed again, which takes a lot of time. The creation of data sets is also very troublesome and requires a lot of resources. Therefore, the versatility of this type of method is still poor and cannot be applied to more robots. Therefore, it is necessary to study a shape control method with a wider range of application scenarios, less resource consumption, and higher efficiency.

[0089] 3) The current model-free control method does not achieve ideal control results due to the lack of a model, and the relative error of the controlled shape exceeds 1%. Although a 1% relative error is not considered large in shape control tasks, it may bring unnecessary risks in some tasks with high precision requirements. Therefore, a more accurate model-free shape control method needs to be studied.

[0090] In view of the shortcomings of the prior art, the present invention has the following purposes:

[0091] Firstly, the present invention is to design a scheme to make the shape control of a continuum robot more universal, which is not limited by the robot model. To this end, the present invention designs a model-free continuum robot shape control method, which can complete the control of the continuum robot without knowing the robot mathematical model.

[0092] Secondly, the second purpose of the present invention is to solve the problem that the current model-free shape control method requires pre-training and consumes a lot of resources. To this end, the control method designed by the present invention only needs to use the data of the robot controlled by the sensor to complete the real-time shape control of the continuum robot, without pre-training, which greatly saves the resources consumed to complete the shape control task.

[0093] Finally, the third purpose of the present invention is to solve the problem that the current shape control method has unsatisfactory control effect. To this end, the present invention defines the shape error based on the zeroing neural network design formula, estimates the Jacobian matrix of the continuum robot according to the shape data sent back by the sensor, and allows the system to autonomously learn the mathematical model of the robot, thereby obtaining a shape control method with better control effect, which is better than the model-based control method.

[0094] Reference Figure 1 The embodiment of the present invention provides a model-free shape control method for a continuum robot, which specifically includes the following steps:

[0095] S101, initializing the actuator of the continuum robot arm, control system related parameters, the initial value of the Jacobian matrix of the robot system itself, and control task related parameters;

[0096] S102, defining a shape feature parameter error function for the continuum robot system, applying a zeroing neural network model to the shape feature parameter error function, deriving an update formula for the actuator according to the first design formula, and then calculating a change amount of the actuator according to an estimated value of the Jacobian matrix;

[0097] S103, defining a parameter velocity error function for the continuum robot system, applying a zeroing neural network model to the parameter velocity error function, obtaining an update formula for the Jacobian matrix according to the second design formula, and then calculating an estimated change in the Jacobian matrix according to the acceleration value of the actual shape feature parameter fed back by the sensor and the estimated actuator acceleration;

[0098] S104. The actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment are learned based on the change of the actuator and the estimated change of the Jacobian matrix, and the shape control of the continuum robot at the next moment is performed based on the actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment.

[0099] Further as an optional implementation, the control system related parameters include the convergence parameters of the zeroing neural network model, the control task related parameters include the control task duration, the shape feature parameters corresponding to the expected shape, the shape feature parameter error function is a function of the error between the expected shape feature parameters and the actual shape feature parameters fed back by the sensor with respect to time, and the parameter speed error function is a function of the error between the derivative value of the actual shape feature parameters fed back by the sensor and the product of the estimated value of the Jacobian matrix at the previous moment and the change of the actuator with respect to time.

[0100] like Figure 4 The figure shows a schematic diagram of the principle of model-free shape control of a continuum robot based on a zeroing neural network provided by an embodiment of the present invention. The input of the control system is the expected shape feature parameter defined by the user, and the output q of the control system can be used to control the shape posture of the continuum robot at the next moment. ZNN represents zeroing neural network.

[0101] like Figure 5 The figure is a schematic flow chart of the model-free shape control of a continuum robot based on an annihilation neural network provided by an embodiment of the present invention, which is as follows:

[0102] A. Initialization t 0 = 0, the actuator q(t 0 ), control system related parameters (mainly the convergence parameter γ of the zeroing neural network 1 ,γ 2 ), the initial value of the Jacobian matrix of the robot system itself Control task related parameters (including control task duration T d , shape feature parameter x corresponding to the expected shape d )

[0103] B. Define the nth moment (t n ) shape characteristic parameter error e(t n ), the error function is the error between the expected shape feature parameters and the actual shape feature parameters of the robot. The zeroing neural network model is applied to the parameter error, and the actuator update formula is obtained according to the design formula. Then, t=t is calculated based on the estimated Jacobian matrix value. n The change of the actuator at the moment

[0104] C. Define the velocity error ∈(t n ), the error function is the derivative value of the shape characteristic parameter fed back by the sensor The error between the Jacobian matrix estimated at the last moment and the derivative value of the actuator is formed. The annihilation neural network model is applied to the velocity error, and the update formula of the Jacobian matrix is ​​obtained according to the design formula, and then the acceleration value of the shape characteristic parameter fed back by the sensor is calculated. and the estimated actuator acceleration Calculation shows that t = t n The estimated change in the Jacobian matrix at time

[0105] D. According to step B and step C, t = t n The change of the actuator at the moment and the estimated change in the Jacobian matrix Learn to get the next moment t = t n+1 The actuator control input q(t n+1 ) and the value of the Jacobian matrix

[0106] After completing step D, repeat steps B to D until the entire shape control task is completed.

[0107] In the simulation experiment, a three-segment continuum robot based on the piecewise constant curvature hypothesis (i.e., a continuum arm composed of three segments of constant curvature curves, with smooth joints and the same tangents) is taken as an example. Here, each segment of the robot arm is controlled by three different cables, so the three-segment continuum robot is controlled by a total of nine cables. First initialize t 0 = 0, the actuator q(t 0 ) = [0.20, 0.21, 0.19, 0.20, 0.21, 0.19, 0.20, 0.21, 0.19] T m, where the actuator represents the length of the rope that controls the corresponding segment. The convergence parameter γ of the zeroing neural network in the initial control system 1 =1,γ 2 =5. The above parameters will be defined in the further technical solution. In order to initialize the Jacobian matrix of the arm, it is necessary to give the angle of the j-th actuator (j=1,2,…,9) a sufficiently small increment Δq j , thereby changing the state of the control rope and measuring the characteristic parameter change vector Δx of the arm shape, so the (i, j)th element of the Jacobian matrix of the arm is initialized as follows:

[0108]

[0109] Where Δx i represents the i-th element (i=1,2,…,9) of the shape feature parameter change Δx. Therefore, according to all elements of the shape feature parameter change Δx and the change Δq of the j-th actuator j , we can determine all the elements in the jth column of the initial value of the Jacobian matrix. Similarly, repeating the above process for all actuators of the arm (j = 1, 2, ..., 9) can determine the Jacobian matrix J(t 0 ) The values ​​of all elements. After determining the initial value of the Jacobian matrix, initialize the characteristic parameters corresponding to the expected shape in, Respectively represent the curvature, deflection angle and length of the i-th arm, and their units are m -1 , rad, m. Define the total duration T of the control task d =20 seconds.

[0110] Based on the above technical solution, the continuum robot system is controlled to complete the shape control task. The mathematical operations in the above process can be completed by the MATLAB software in the host computer, and then the input control signal obtained by the solution is transmitted to the continuum robot. Here, the root mean squared error (RMSE) of the shape feature parameters and the Cartesian distance error (Car_err) of the shape are used to evaluate the effect of the continuum robot shape control. The definition of RMSE is as follows:

[0111]

[0112] where x d (t) and x a (t) represent the shape feature parameters corresponding to the expected shape posture and the current shape posture of the robot. The Cartesian coordinate error is defined as follows:

[0113]

[0114] The arm is divided into m segments according to the principle of equal spacing, p d (t,i) and p a (t,i) represents the 3D position of the i-th endpoint of the expected shape and the current shape respectively. The Cartesian coordinate error indicates the degree of fit between the expected and current poses in space.

[0115] Figure 6 and Figure 7 The shape posture result diagram and error change diagram of the continuum robot using the annihilation neural network to complete the above shape control task are shown respectively. d = After 20 seconds of control, the mean square error RMSE of the shape feature parameters is 1.12×10 -9 , the Cartesian coordinate error Car_err is 7.18×10 -11 m. It can be seen that the arm posture has already coincided with the expected shape posture when the task time is halfway through, which is difficult to distinguish with the naked eye. In addition, the mean square error of the shape feature parameters and the Cartesian coordinate error of the shape converge rapidly.

[0116] In addition, in order to reflect the superiority of the control performance of the present invention, the shape control task with the same parameter setting is applied to the model-based shape control method with better control effect. Table 1 shows the shape control effects of different control algorithms. The computational cost refers to the computational time for the computer to complete the shape control task. It can be found that except for the PID method, the control error of the present invention is much smaller than that of other model-based control algorithms, and the control effect is much better than that of other model-based shape control algorithms. At the same time, the computational cost of the present invention is much smaller than the duration T of the shape control task. d , indicating that it can be applied in practice, and the relatively small computational cost also shows that the present invention consumes very little resources. Therefore, the design algorithm of the present invention is highly efficient and superior in controlling the shape of a continuum robot.

[0117] Table 1

[0118]

[0119] In some optional embodiments, for step B, the specific scheme is as follows:

[0120] First, at the nth moment, the shape characteristic parameter error of shape control is defined for the continuum robot system, that is, the error function between the expected shape and the characteristic parameters corresponding to the robot shape measured by the current sensor:

[0121] e(t)=x d (t)-x a (t) (10)

[0122] Among them, e(t) represents the shape feature parameter error at time t, x d (t) represents the expected shape characteristic parameters at time t, x a (t) represents the actual shape characteristic parameters fed back by the sensor at time t;

[0123] According to the design formula of the zeroing neural network:

[0124]

[0125] in, represents the derivative value of the shape feature parameter error function at time t, γ 1 >0 is the convergence parameter of the shape feature parameter error function, Φ(·) represents the activation function of the error, and the activation function is set as a linear activation function, that is, Φ(x) = x. For convenience, the variables in the subsequent formulas are omitted (t). Expanding formula (11), we get:

[0126]

[0127] Since the basic shape control problem is to control the arm shape to the desired shape, the characteristic parameter value of the desired control shape should be constant, so Substituting into formula (12) we get The expression is:

[0128]

[0129] in, represents the change of the actuator, represents the estimate of the Jacobian matrix, express The Moore-Penrose generalized inverse of x d represents the expected shape characteristic parameter, x a Represents the actual shape characteristic parameters of the robot. Since the dimensions of the actuator q and the characteristic parameters x may be inconsistent, the estimated Jacobian matrix It may not be a square, so here express The Moore-Penrose generalized inverse of . This gives Figure 4 Change of actuator The estimation formula of .

[0130] In some optional embodiments, for step C, the specific scheme is as follows:

[0131] After completing step B, define the velocity error of the robot's current shape feature parameters. The error function is the derivative value of the shape feature parameters fed back by the sensor. and the Jacobian matrix estimated at the previous moment Derivative value with actuator The error between them is:

[0132]

[0133] Among them, ∈(t) represents the parameter speed error at time t, Represents the derivative value of the actual shape characteristic parameter fed back by the sensor at time t, represents the estimated value of the Jacobian matrix at time t, Represents the change of the actuator at time t;

[0134] Similarly, according to the design formula of the zeroing neural network:

[0135]

[0136] in, represents the derivative value of the parameter velocity error function at time t, γ 2>0 is the convergence parameter of the parameter speed error function, Φ(·) represents the activation function of the error, the activation function is set as the linear activation function, and (t) is omitted in the formula. Expanding formula (15), we get:

[0137]

[0138] Thus, the change in the estimated value of the robot Jacobian matrix can be obtained The expression is:

[0139]

[0140] in, represents the estimated change in the Jacobian matrix, Represents the derivative value of the actual shape characteristic parameter fed back by the sensor, represents the estimate of the Jacobian matrix, represents the change of the actuator, The acceleration value representing the actual shape characteristic parameter fed back by the sensor, represents the estimated actuator acceleration, express The Moore-Penrose generalized inverse of Figure 4 Estimated change in the Jacobian matrix

[0141] In some optional embodiments, for step D, the specific scheme is as follows:

[0142] Observing the calculation formula (17), it can be found that the calculation of the estimate requires the derivative value of the shape feature parameter change and the acceleration of the actuator changes The derivative values ​​of the parameters can be obtained through sensor feedback, and the acceleration of the actuator changes It can be solved by the backward Euler method:

[0143]

[0144] Where Δt represents the time step of the calculation program execution, which is set to 0.01s in the simulation task and 0.1s in the experimental task. Finally, the change of the actuator is solved according to formula (13) and formula (17) respectively: and the Jacobian matrix to estimate the change The values ​​of the actuator and robot Jacobian matrices at the next moment can be calculated:

[0145]

[0146]

[0147] Among them, q(t n+1 ) indicates t n+1 The actuator control input of the continuum robot at the moment, q(t n ) indicates t n The actuator control input of the moment continuum robot, Indicates t n The change of the actuator at the moment, Indicates t n+1 The estimated value of the Jacobian matrix at time instant, Indicates t n The estimated value of the Jacobian matrix at time instant, Indicates t n The estimated change of the Jacobian matrix at time t, Δt represents the preset time step, and the calculated actuator values ​​and the predicted Jacobian matrix will be used for the continuum robot shape control at the next moment.

[0148] In the physical experiment, a one-segment rope-driven robot based on the piecewise constant curvature assumption is taken as the control object. The rope-driven robot is controlled by four motors, which are divided into two groups and are opposite to each other. The robot's actuator is the rotation angle of each group of driving motors, that is, First initialize t 0 = 0, the actuator q(t 0 )=[0,0] T rad, where the actuator represents the length of the rope that controls the corresponding segment. The convergence parameter γ of the zeroing neural network in the initial control system 1 =γ 2 = 0.1. The initialization of the Jacobian matrix of the arm is shown in the above formula (7). Since the length of the rope-driven robot cannot be changed, the curvature and deflection angle of the above shape parameters can determine the shape of the arm. Initialize the characteristic parameters corresponding to the expected shape in Respectively represent the curvature and deflection angle of the i-th arm, and their units are m -1 , rad. Define the total duration of the control task T d =10 seconds.

[0149] Based on the above technical solution, the rope-driven robot is controlled to complete the shape control task. The mathematical operations in the above process can be completed by the MATLAB software in the host computer, and then the input control signal obtained by the solution is transmitted to the robot. Similar to the simulation part, the mean square error (RMSE) of the shape feature parameters and the Cartesian coordinate error (Car_err) of the shape are used here to evaluate the effect of the continuum robot shape control.

[0150] Figure 8 and Fig. 9The error and evaluation index change diagrams of the shape feature parameters of the control task and the change diagram of the actuator during the task are shown respectively. φ represents the deflection angle error, e k represents the curvature error, q 1 and q 2 Represents the actuator value under different control tasks. d = After 10 seconds of control, the mean square error of the shape feature parameters is 8.74×10 -3 , the Cartesian coordinate error is 1.56×10 -4 m. It can be seen that at t = 6 seconds, the shape and posture have almost converged to the expected shape and posture. This illustrates the effectiveness of the control scheme of the present invention for the shape control of the physical robot.

[0151] The above describes the method steps and experimental results of the embodiment of the present invention. It can be recognized that the embodiment of the present invention can realize the control of the arm shape when the model of the continuum robot system is unknown, thereby solving the challenges brought by model uncertainty or unknown models, avoiding the complex process of modeling the robot arm, and the control scheme has good portability for continuum robots of different types and structures; the embodiment of the present invention only needs to use the parameters and their derivative data fed back by the sensor to calculate the shape control of the continuum robot in real time, avoiding a large number of pre-training processes in other model-free methods, and greatly reducing the resources consumed to complete the task; the embodiment of the present invention can realize the precise shape control of the continuum robot, and the control effect is comparable to or even exceeds the model-based shape control method, thereby expanding the application scenarios of the continuum robot.

[0152] Compared with the prior art, the embodiments of the present invention also have the following advantages:

[0153] 1) The present invention designs a solution to make the shape control of a continuum robot more versatile. This solution does not need to be restricted by the robot model, that is, a model-free continuum robot shape control method. This method can complete the control of the continuum robot without knowing the mathematical model of the robot.

[0154] 2) The present invention can solve the problem that the current model-free shape control method requires pre-training and consumes a lot of resources. To this end, the control method designed by the present invention only needs to use the data of the robot controlled by the sensor to complete the real-time shape control of the continuum robot, without the need for pre-training, which greatly saves the resources consumed to complete the shape control task.

[0155] 3) The present invention can solve the problem that the current shape control method has unsatisfactory control effect. To this end, the present invention defines the shape error based on the zeroing neural network design formula, estimates the Jacobian matrix of the continuum robot according to the shape data sent back by the sensor, and allows the system to autonomously learn the mathematical model of the robot, thereby obtaining a shape control method with better control effect, which is better than the model-based control method.

[0156] Reference Fig.10 , an embodiment of the present invention provides a continuum robot model-free shape control system, comprising:

[0157] Parameter initialization module, used to initialize the actuator of the continuum robot arm, control system related parameters, the initial value of the Jacobian matrix of the robot system itself, and control task related parameters;

[0158] An actuator change calculation module is used to define a shape feature parameter error function for a continuum robot system, apply a zeroing neural network model to the shape feature parameter error function, obtain an actuator update formula according to the first design formula, and then calculate the actuator change amount according to the estimated value of the Jacobian matrix;

[0159] A Jacobian matrix change calculation module is used to define a parameter velocity error function for a continuum robot system, apply a zeroing neural network model to the parameter velocity error function, obtain an update formula for the Jacobian matrix according to a second design formula, and then calculate an estimated change of the Jacobian matrix according to the acceleration value of the actual shape feature parameter fed back by the sensor and the estimated actuator acceleration;

[0160] The shape control module is used to learn the actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment based on the change of the actuator and the estimated change of the Jacobian matrix, and to perform shape control on the continuum robot at the next moment based on the actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment.

[0161] The contents of the above method embodiments are all applicable to the present system embodiments. The functions specifically implemented by the present system embodiments are the same as those of the above method embodiments, and the beneficial effects achieved are also the same as those achieved by the above method embodiments.

[0162] Reference Fig.11 , an embodiment of the present invention provides a model-free shape control device for a continuum robot, comprising:

[0163] at least one processor;

[0164] at least one memory for storing at least one program;

[0165] When the at least one program is executed by the at least one processor, the at least one processor implements the above-mentioned model-free shape control method for a continuum robot.

[0166] The contents of the above method embodiments are all applicable to the present device embodiments. The functions specifically implemented by the present device embodiments are the same as those of the above method embodiments, and the beneficial effects achieved are also the same as those achieved by the above method embodiments.

[0167] An embodiment of the present invention further provides a computer-readable storage medium, in which a program executable by a processor is stored. When the program executable by the processor is executed by the processor, it is used to execute the above-mentioned continuum robot model-free shape control method.

[0168] A computer-readable storage medium according to an embodiment of the present invention can execute a model-free shape control method for a continuum robot provided by an embodiment of the method of the present invention, can execute any combination of implementation steps of the method embodiment, and has the corresponding functions and beneficial effects of the method.

[0169] The embodiment of the present invention also discloses a computer program product or a computer program, wherein the computer program product or the computer program includes computer instructions, and the computer instructions are stored in a computer-readable storage medium. A processor of a computer device can read the computer instructions from the computer-readable storage medium, and the processor executes the computer instructions, so that the computer device executes Figure 1 The method shown.

[0170] In some selectable embodiments, the function / operation mentioned in the block diagram may not occur in the order mentioned in the operation diagram. For example, depending on the function / operation involved, the two boxes shown in succession can actually be executed substantially simultaneously or the above-mentioned boxes can sometimes be executed in reverse order. In addition, the embodiment presented and described in the flow chart of the present invention is provided by way of example, for the purpose of providing a more comprehensive understanding of technology. The disclosed method is not limited to the operation and logic flow presented herein. Selectable embodiments are expected, wherein the order of various operations is changed and the sub-operation of a part for which is described as a larger operation is performed independently.

[0171] In addition, although the present invention is described in the context of functional modules, it should be understood that, unless otherwise specified to the contrary, one or more of the above-mentioned functions and / or features can be integrated into a single physical device and / or software module, or one or more functions and / or features can be implemented in a separate physical device or software module. It is also understood that a detailed discussion of the actual implementation of each module is unnecessary for understanding the present invention. More specifically, in view of the properties, functions and internal relationships of the various functional modules in the device disclosed herein, the actual implementation of the module will be understood within the conventional skills of the engineer. Therefore, those skilled in the art can implement the present invention set forth in the claims without excessive experimentation using ordinary techniques. It is also understood that the specific concepts disclosed are merely illustrative and are not intended to limit the scope of the present invention, which is determined by the full scope of the appended claims and their equivalents.

[0172] If the above functions are implemented in the form of software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art or the part of the technical solution, can be embodied in the form of a software product. The computer software product is stored in a storage medium, including several instructions for a computer device (which can be a personal computer, a server, or a network device, etc.) to perform all or part of the steps of the above methods of each embodiment of the present invention. The aforementioned storage medium includes: U disk, mobile hard disk, read-only memory (ROM, Read-Only Memory), random access memory (RAM, Random Access Memory), disk or optical disk and other media that can store program codes.

[0173] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as an ordered list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by an instruction execution system, device or apparatus (such as a computer-based system, a system including a processor, or other system that can fetch instructions from an instruction execution system, device or apparatus and execute instructions), or in conjunction with such instruction execution systems, devices or apparatuses. For the purposes of this specification, "computer-readable medium" can be any device that can contain, store, communicate, propagate or transmit a program for use by an instruction execution system, device or apparatus, or in conjunction with such instruction execution systems, devices or apparatuses.

[0174] More specific examples of computer-readable media (a non-exhaustive list) include the following: an electrical connection with one or more wires (electronic device), a portable computer disk case (magnetic device), a random access memory (RAM), a read-only memory (ROM), an erasable and editable read-only memory (EPROM or flash memory), an optical fiber device, and a portable compact disk read-only memory (CDROM). In addition, the computer-readable medium may even be a paper or other suitable medium on which the above-mentioned program is printed, since the above-mentioned program can be obtained electronically, for example, by optically scanning the paper or other medium, followed by editing, deciphering or processing in other suitable ways as necessary, and then stored in a computer memory.

[0175] It should be understood that the various parts of the present invention can be implemented by hardware, software, firmware or a combination thereof. In the above-mentioned embodiments, a plurality of steps or methods can be implemented by software or firmware stored in a memory and executed by a suitable instruction execution system. For example, if implemented by hardware, as in another embodiment, it can be implemented by any one of the following technologies known in the art or their combination: a discrete logic circuit having a logic gate circuit for implementing a logic function for a data signal, a dedicated integrated circuit having a suitable combination of logic gate circuits, a programmable gate array (PGA), a field programmable gate array (FPGA), etc.

[0176] In the above description of this specification, the description with reference to the terms "one embodiment / example", "another embodiment / example" or "certain embodiments / examples" etc. means that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representation of the above terms does not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described may be combined in any one or more embodiments or examples in a suitable manner.

[0177] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the claims and their equivalents.

[0178] The above is a specific description of the preferred implementation of the present invention, but the present invention is not limited to the above embodiments. Those skilled in the art may make various equivalent modifications or substitutions without violating the spirit of the present invention. These equivalent modifications or substitutions are all included in the scope defined by the claims of this application.

Claims

1. A model-free shape control method for a continuum robot, characterized in that: The following steps are involved: Initialize the actuators of the continuum robot arm, control system related parameters, the initial value of the Jacobian matrix of the robot system itself, and control task related parameters; A shape feature parameter error function is defined for the continuum robot system, a zeroing neural network model is applied to the shape feature parameter error function, an update formula of the actuator is obtained according to the first design formula, and a change amount of the actuator is calculated according to an estimated value of the Jacobian matrix; A parameter velocity error function is defined for the continuum robot system, a zeroing neural network model is applied to the parameter velocity error function, an update formula of the Jacobian matrix is ​​obtained according to the second design formula, and an estimated change of the Jacobian matrix is ​​calculated according to the acceleration value of the actual shape characteristic parameter fed back by the sensor and the estimated actuator acceleration; According to the change of the actuator and the estimated change of the Jacobian matrix, the actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment are learned, and the shape of the continuum robot at the next moment is controlled according to the actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment.

2. A model-free shape control method for a continuum robot according to claim 1, characterized in that: The control system related parameters include the convergence parameters of the zeroing neural network model, the control task related parameters include the control task duration and the shape characteristic parameters corresponding to the expected shape, the shape characteristic parameter error function is a function of the error between the expected shape characteristic parameters and the actual shape characteristic parameters fed back by the sensor with respect to time, and the parameter speed error function is a function of the error between the derivative value of the actual shape characteristic parameters fed back by the sensor and the product of the estimated value of the Jacobian matrix at the previous moment and the change of the actuator with respect to time.

3. A model-free shape control method for a continuum robot according to claim 2, characterized in that: The shape feature parameter error function is: e(t)=x d (t)-x a (t) Among them, e(t) represents the shape feature parameter error at time t, x d (t) represents the expected shape characteristic parameters at time t, x a (t) represents the actual shape characteristic parameters fed back by the sensor at time t; The first design formula of the zeroing neural network model is: in, represents the derivative value of the shape feature parameter error function at time t, γ1>0 is the convergence parameter of the shape feature parameter error function, and Φ(·) represents the activation function of the error.

4. A continuum robot model-free shape control method according to claim 3, characterized in that: The update formula of the executor is: in, represents the change of the actuator, represents the estimate of the Jacobian matrix, express The Moore-Penrose generalized inverse of x d represents the expected shape characteristic parameter, x a Represents the actual shape characteristic parameters of the robot.

5. A continuum robot model-free shape control method according to claim 2, characterized in that: The parameter velocity error function is: Among them, ∈(t) represents the parameter speed error at time t, Represents the derivative value of the actual shape characteristic parameter fed back by the sensor at time t, represents the estimated value of the Jacobian matrix at time t, Represents the change of the actuator at time t; The second design formula of the zeroing neural network model is: in, represents the derivative value of the parameter velocity error function at time t, γ2>0 is the convergence parameter of the parameter velocity error function, and Φ(·) represents the activation function of the error.

6. A continuum robot model-free shape control method according to claim 5, characterized in that: The update formula of the Jacobian matrix is: in, represents the estimated change in the Jacobian matrix, Represents the derivative value of the actual shape characteristic parameter fed back by the sensor, represents the estimate of the Jacobian matrix, represents the change of the actuator, The acceleration value representing the actual shape characteristic parameter fed back by the sensor, represents the estimated actuator acceleration, express The Moore-Penrose generalized inverse of .

7. A model-free shape control method for a continuum robot according to any one of claims 1 to 6, characterized in that: The estimated values ​​of the actuator control input and Jacobian matrix of the continuum robot at the next moment are calculated as follows: Among them, q(t n+1 ) indicates t n+1 The actuator control input of the continuum robot at the moment, q(t n ) indicates t n The actuator control input of the moment continuum robot, Indicates t n The change of the actuator at the moment, Indicates t n+1 The estimated value of the Jacobian matrix at time instant, Indicates t n The estimated value of the Jacobian matrix at time instant, Indicates t n The estimated change of the Jacobian matrix at time t, Δt represents the preset time step.

8. A model-free shape control system for a continuum robot, characterized in that: include: Parameter initialization module, used to initialize the actuator of the continuum robot arm, control system related parameters, the initial value of the Jacobian matrix of the robot system itself, and control task related parameters; an actuator change calculation module, used to define a shape feature parameter error function for a continuum robot system, apply a zeroing neural network model to the shape feature parameter error function, obtain an actuator update formula according to a first design formula, and then calculate the actuator change amount according to an estimated value of a Jacobian matrix; A Jacobian matrix change calculation module is used to define a parameter velocity error function for a continuum robot system, apply a zeroing neural network model to the parameter velocity error function, obtain an update formula for the Jacobian matrix according to a second design formula, and then calculate an estimated change of the Jacobian matrix according to the acceleration value of the actual shape feature parameter fed back by the sensor and the estimated actuator acceleration; The shape control module is used to learn the actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment based on the change of the actuator and the estimated change of the Jacobian matrix, and to perform shape control on the continuum robot at the next moment based on the actuator control input and the estimated value of the Jacobian matrix of the continuum robot at the next moment.

9. A model-free shape control device for a continuum robot, characterized in that: include: at least one processor; at least one memory for storing at least one program; When the at least one program is executed by the at least one processor, the at least one processor implements the model-free shape control method for a continuum robot as described in any one of claims 1 to 7.

10. A computer-readable storage medium storing a program executable by a processor, characterized in that: The processor-executable program is used to execute a continuum robot model-free shape control method as claimed in any one of claims 1 to 7 when executed by the processor.

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