Passive maneuvering digital control method for robot arm under input constraints

Abstract

The application provides a kind of passive maneuvering digital control method of mechanical arm under input constraint, comprising: constructing the dynamics model of mechanical arm with input constraint;Desired geometric path in workspace is converted into joint space by inverse kinematics method;Desired passive maneuvering controller is designed based on the dynamics model of mechanical arm;When control input does not satisfy constraint, the input that satisfies constraint and has minimum deviation is selected by quadratic programming to replace current control input;Continuous control that satisfies constraint is converted into digital control, and passive maneuvering digital control of mechanical arm is realized.The application guarantees stability by passivity;Optimization algorithm is used to process constraint, to ensure that the system operates stably within physical limit;Pulse width modulation technology is used to convert continuous control that satisfies constraint into digital control with low power consumption, anti-interference and easy implementation, to complete maneuvering task.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of mechanical arm control, and particularly relates to a passive maneuvering digital control method for a mechanical arm under input constraints. BACKGROUND

[0002] In many applications of mechanical arms, such as cutting, welding, etc., the main task is to control the movement of the mechanical arm end in the workspace along a continuously parameterized geometric path. However, satisfying the required dynamic behavior along the path is a second and secondary task. This problem is called the maneuvering problem, which is a problem between tracking and path following. However, most control techniques for mechanical arms are used to deal with tracking or path following problems.

[0003] In engineering practice, the control input of the mechanical arm is subject to various physical constraints. In the process of mathematical modeling, the constraints need to be converted into boundary conditions (such as inequality constraints) of the control algorithm. If the control input does not meet the constraints, the system performance will be greatly affected, and even the mechanical equipment will be damaged. Therefore, it is crucial to select an input that meets the constraint conditions and achieves the best expected performance.

[0004] Most control algorithms of existing mechanical arms are continuous or piecewise continuous, have high power consumption, are weak in noise resistance, and are not easy to physically implement. Therefore, it is necessary to provide a passive maneuvering digital control method, which guarantees stability through passivity, processes constraints by using an optimization algorithm, reduces power consumption through digital control, is resistant to noise, and realizes maneuvering control. SUMMARY

[0005] According to the above technical problems, a passive maneuvering digital control method for a mechanical arm under input constraints is provided. The present application is a passive maneuvering digital control method, which guarantees stability through passivity, processes constraints by using an optimization algorithm, converts the continuous control that meets the constraints into digital control with low power consumption, noise resistance, and easy implementation by using pulse width modulation technology, and realizes maneuvering control, that is, completes the expected geometric task and dynamic task.

[0006] To achieve the above purpose, the technical means adopted by the present application are as follows:

[0007] A passive maneuvering digital control method for a mechanical arm under input constraints comprises the following steps:

[0008] A mechanical arm dynamics model with input constraints is constructed;

[0009] An inverse kinematics method is used to convert the expected geometric path in the workspace into the joint space;

[0010] An expected passive maneuvering controller is designed based on the mechanical arm dynamics model;

[0011] When the control input does not satisfy the constraint, the input that has the minimum deviation and satisfies the constraint is selected by using quadratic programming to replace the current control input;

[0012] The continuous control satisfying the constraint is converted into digital control, so as to realize passive maneuvering digital control of the mechanical arm.

[0013] Further, the mechanical arm dynamics model with input constraints is constructed, and specifically includes:

[0014] The dynamics model of the mechanical arm driven by the DC motor and having n degrees of freedom is represented as:

[0015]

[0016] Wherein, q=[q1,…,q n ] T , q i represents the joint variable of the i th joint, i=1,…,n, I=[I1,…,I n ] T , I i represents the current of the DC motor of the i th joint, L=diag{L1,…,L n}, L i represents the armature inductance of the DC motor of the i th joint, R=diag{R1,…,R n}, R i represents the armature resistance of the DC motor of the i th joint, U m =diag{U m1 ,…,U mn}, U mi represents the torque constant of the DC motor of the i th joint, U e =diag{U e1 ,…,U en}, U ei represents the back electromotive force coefficient of the DC motor of the i th joint; is the generalized inertia matrix satisfying M(q)=M T (q), represents the set of n×n-dimensional real matrices, is the Coriolis centripetal matrix, represents the generalized gravity, represents the set of n-dimensional real vectors, represents the damping force, τ=U m I is the torque provided by the DC motor, u=[u1,…,u n ] T is the control input, u i represents the voltage of the DC motor of the i th joint. q represents the first derivative of q, represents the second derivative of q, denotes the first derivative of I.

[0017] The control input u satisfies the constraint condition due to the physical constraint of the DC motor:

[0018] |u i |≤U i ,i=1,2,…,n

[0019] wherein U i >0 denotes the rated voltage of the DC motor of the ith joint.

[0020] Further, the converting the desired geometric path in the work space into the joint space by the inverse kinematics method specifically comprises:

[0021] representing the continuous parameterized desired geometric path of the end effector of the robot arm as:

[0022] o d (x d (θ),y d (θ),z d (θ)) T

[0023] wherein (x d ,y d ,z d ) T is the coordinate of the end of the robot arm in the fixed coordinate system, is the path variable.

[0024] The path in the work space is converted into the joint space by the inverse kinematics method. The desired joint geometric path is: r (θ)=(q r1 (θ),q r2 (θ),…,q rn (θ)) T .

[0025] Further, the designing the desired passive maneuvering controller based on the dynamics model of the robot arm specifically comprises:

[0026] Under the condition of not considering the input constraint, for the bounded desired joint position path q r satisfying the 3-order continuous differentiability and the bounded feasible velocity distribution satisfying the 2-order continuous differentiability for the first variable and the 1-order continuous differentiability for the second variable, an error variable is introduced in the dynamics model of the robot arm:

[0027] z1=q-q r

[0028]

[0029] z3 = I - a

[0030] where, is q r The derivative of a with respect to q, c1 > 0 is a design parameter, is v s The partial derivative of a with respect to t, c2 > 0 is a design parameter.

[0031] The feedback passive controller is designed as:

[0032]

[0033] where, a q is the partial derivative of a with respect to q, is the partial derivative of a with respect to a t is the partial derivative of a with respect to t, c3 > 0 is a design parameter, is the derivative of a with respect to q, is the derivative of a with respect to q, is the derivative of a with respect to q, is v s the derivative of a with respect to q.

[0034] The error dynamics of the manipulator is calculated as:

[0035]

[0036] where, denotes the first derivative of z i , is the velocity assignment error, denotes the first derivative of q. The error system is a strictly passive system with w s as input and as output, and the storage function is

[0037] According to the passivity theorem, the velocity assignment is designed as:

[0038]

[0039] where, k > 0 and g > 0 are design parameters, denotes the first derivative of w s .

[0040] In summary, the passive dynamic maneuvering controller is:

[0041]

[0042] The closed-loop system is represented as a strictly passive error system and the storage function is a strictly passive system with a negative feedback connection, i.e.

[0043]

[0044] The Lyapunov function is calculated The derivative along the closed-loop system is:

[0045]

[0046] According to the stability theorem, under the condition of not considering the input constraint, the designed controller completes the maneuvering task, i.e. the output error and the speed distribution error satisfy

[0047]

[0048] Further, when the control input does not satisfy the constraint, a quadratic programming is used to select an input that has the minimum deviation and satisfies the constraint to replace the current control input, and the specific process comprises the following steps.

[0049] When the control input does not satisfy the constraint, a quadratic programming (QP) is given for the synthesis method of the bounded controller with the minimum deviation.

[0050]

[0051] Wherein, u i is the i-th component of u, u si is the i-th component of u s .

[0052] By verifying the Karush-Kuhn-Tucker condition, a closed-form solution of the QP problem is given

[0053] u * =sat(u s )

[0054] Wherein, sat(u s )=[sat(u s1 ),sat(u s2 ),…,sat(u sn )] T is a saturation function vector, which is described as

[0055]

[0056] Further, the continuous control satisfying the constraint is converted into digital control, and the specific process comprises the following steps.

[0057] By introducing a duty cycle function μ i (t) = (u i (t) + U i ) / 2U i , the components u s (i = 1, …, n) of the bounded continuous control u = sat(u i ) are converted from the interval [-U i , U i ] to the interval [0, 1].

[0058] By comparing circuit, the duty cycle function μ i is compared with the carrier u0(t) to obtain two-stage pulse signal:

[0059]

[0060] Wherein, u0(t) selects a triangle wave with a frequency of 1 / T and an amplitude of 1, and T is an adjustable parameter.

[0061] Allowing the motor to rotate freely in both directions, the digital control is designed as:

[0062] u di = σ i U i , σ i = 2μ Ti -1

[0063] And the digital control u d = (u d1 , …, u dn ) T is physically realized by the H-bridge circuit with gate trigger.

[0064] The digital control signal of the servo motor is equivalent to the bounded continuous control signal in time integration.

[0065] Compared with the prior art, the present application has the following advantages:

[0066] The input-constrained passive maneuvering digital control method for a mechanical arm provided by the present application constructs a mechanical arm dynamics model with input constraints; uses inverse kinematics method to convert the expected geometric path in the workspace into the joint space; designs an expected passive maneuvering controller based on the mechanical arm dynamics model; when the control input does not satisfy the constraints, uses quadratic programming to select the input that has the minimum deviation and satisfies the constraints to replace the current control input; converts the continuous control that satisfies the constraints into digital control, and realizes passive maneuvering digital control of the mechanical arm.

[0067] The present application guarantees the energy dissipation characteristics of the system through passive control design, and avoids the instability phenomenon caused by energy accumulation. The present application combines an optimization algorithm to process input constraints, and ensures stable operation of the system within physical limits. The pulse width modulation technology is used to convert the continuous control satisfying the constraints into digital control with low power consumption, anti-interference and easy implementation, and complete the maneuvering task.

[0068] Based on the above reasons, the present application can be widely popularized in the field of mechanical arm control. BRIEF DESCRIPTION OF DRAWINGS

[0069] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments or prior art description will be briefly introduced. Obviously, the drawings in the following description are some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor.

[0070] Figure 1 The flow chart of the passive maneuvering digital control method of the mechanical arm under input constraints in the present application.

[0071] Figure 2 The schematic diagram of the i-th joint of the mechanical arm driven by the direct current motor in the present application.

[0072] Figure 3 The schematic diagram of the expected passive maneuvering control closed-loop system in the present application.

[0073] Figure 4 The schematic diagram of the passive maneuvering digital control system of the mechanical arm under input constraints in the present application.

[0074] Figure 5 The schematic diagram of the planar elbow type mechanical arm driven by the direct current motor in the embodiment of the present application.

[0075] Figure 6 The schematic diagram of the geometric task of the mechanical arm in the embodiment of the present application.

[0076] Figure 7 The geometric task response simulation result in the embodiment of the present application.

[0077] Figure 8 The digital control response simulation result in the embodiment of the present application.

[0078] Figure 9 The output error and speed distribution response simulation result in the embodiment of the present application. DETAILED DESCRIPTION

[0079] It should be noted that the embodiments and features of the present application can be combined with each other, if there is no conflict. The present application will be described in detail below with reference to the drawings and embodiments.

[0080] In order to make the objects, technical solutions and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings of the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, but not all the embodiments. The description of the at least one exemplary embodiment below is actually only illustrative, but not as any limitation on the present application and its application or use. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative labor fall within the scope of the present application.

[0081] It should be noted that the terms used herein are only intended to describe specific embodiments, and are not intended to limit the exemplary embodiments according to the present application. As used herein, the singular form is intended to include the plural form, unless the context clearly indicates otherwise, and it should also be understood that when the terms "comprise" and / or "include" are used in the specification, there is a feature, step, operation, device, component and / or combination thereof.

[0082] Unless specifically stated otherwise, the relative arrangement of components and steps, numerical expressions, and numerical values set forth in the various embodiments described herein are not limiting. It should be understood that for the convenience of description, the sizes of the various parts shown in the drawings are not drawn according to the actual proportion relationship. The technology, methods and devices known to those of ordinary skill in the relevant art can not be discussed in detail, but should be considered as part of the authorized specification. In all examples shown and discussed herein, any specific value should be interpreted as merely exemplary, and not as a limitation. Therefore, other examples of exemplary embodiments can have different values. It should be noted that similar reference numbers and letters represent similar items in the following drawings, so once an item is defined in one drawing, it does not need to be further discussed in subsequent drawings.

[0083] In the description of the present application, it should be understood that the orientation words such as "front, back, up, down, left, right", "transverse, vertical, perpendicular, horizontal" and "top, bottom" and the like indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, and are only for the convenience of describing the present application and simplifying the description, and in the absence of the opposite description, these orientation words do not indicate and imply that the devices or elements referred to must have a particular orientation or be constructed and operated in a particular orientation, and therefore cannot be understood as a limitation on the scope of protection of the present application: the orientation words "inner, outer" refer to the inner and outer of the contour of each component itself.

[0084] For the convenience of description, spatial relative terms such as "over", "above", "upper surface", "upper" and the like can be used herein to describe the spatial positional relationship of one device or feature with other devices or features as shown in the drawings. It should be understood that the spatial relative terms are intended to include different orientations in use or operation in addition to the orientation of the devices described in the drawings. For example, if the devices in the drawings are inverted, the device described as "above" or "over" other devices or structures will be positioned "below" or "under" other devices or structures. Thus, the exemplary term "above" can include both "above" and "below" orientations. The device can also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatial relative descriptions used herein are interpreted accordingly.

[0085] In addition, it should be noted that the use of the words "first", "second" and the like to define parts is only for the convenience of distinguishing the corresponding parts, and the above words have no special meaning unless otherwise stated, and therefore cannot be understood as a limitation on the scope of protection of the present application.

[0086] As Figure 1 shown, the present application provides a passive maneuvering digital control method for a mechanical arm under input constraints, comprising:

[0087] constructing a mechanical arm dynamics model with input constraints;

[0088] In specific implementation, as a preferred embodiment of the present application, the construction of the mechanical arm dynamics model with input constraints specifically includes:

[0089] As Figure 2 shown, the dynamics model of the mechanical arm driven by the DC motor with n degrees of freedom is represented as:

[0090]

[0091] where q = [q1, …, qn]T is the joint angle vector, J(q) is the Jacobian matrix, τ is the joint torque vector, and B is the damping matrix. n ]T q i Let I represent the joint variables of the i-th joint, i = 1, ..., n, and I = [I1, ..., I2]. n ] T I i Let L represent the current of the DC motor at the i-th joint, where L = diag{L1, ..., L2} n}, L i Let R = diag{R1, ..., R2} be the armature inductance of the DC motor at the i-th joint. n}, R i U represents the armature resistance of the DC motor at the i-th joint. m =diag{U m1 ,…,U mn}, U mi U represents the torque constant of the DC motor at the i-th joint. e =diag{U e1 ,…,U en}, U ei This represents the back electromotive force coefficient of the DC motor at the i-th joint; Is it satisfying M(q)=M T The generalized inertia matrix of (q), Denotes the set of all n×n dimensional real matrices. For the Coriolis centripetal matrix, Represents generalized gravity. Let n represent the set of all n-dimensional real vectors. Representing the damping force, τ = U m I is the torque provided by the DC motor, u = [u1, ..., u] n ] T It is the control input, u i This represents the voltage of the DC motor at the i-th joint. Denotes the first derivative of q. Denotes the second derivative of q. Let I represent the first derivative of I.

[0092] Due to the physical constraints of the DC motor, the control input u satisfies the following constraints:

[0093] |u i |≤U i i = 1, 2, ..., n

[0094] Among them, U i >0 indicates the rated voltage of the DC motor in the i-th joint.

[0095] The desired geometric path in the workspace is transformed into the joint space using inverse kinematics methods.

[0096] In particular implementation, as the preferred embodiment of the present application, the desired geometric path in the workspace is converted into the joint space by using inverse kinematics method, specifically including:

[0097] In many applications of the robot arm, such as cutting or welding, the continuous parameterized desired geometric path of the robot arm end effector is represented as:

[0098] o d (θ)=[x d (θ),y d (θ),z d (θ)] T

[0099] Where (x d ,y d ,z d ) T is the coordinates of the robot arm end in the base coordinate system, is the path variable.

[0100] The path in the workspace is converted into the joint space by using inverse kinematics method. The desired joint geometric path is: r (θ)=(q r1 (θ),q r2 (θ),…,q rn (θ)) T .

[0101] For a bounded desired joint configuration path q r satisfying 3-order continuous differentiability and a bounded feasible velocity assignment satisfying 2-order continuous differentiability for the first variable and 1-order continuous differentiability for the second variable, the control objective is to design a desired passive maneuvering controller without considering input constraints, so that the system can complete the maneuvering task, i.e. to meet the geometric task and the dynamic task Design a digital controller to modify the desired controller in a minimal way without violating the input constraints, so that the system can complete the maneuvering task as much as possible.

[0102] Design the desired passive maneuvering controller based on the dynamics model of the robot arm.

[0103] In particular implementation, as the preferred embodiment of the present application, the desired passive maneuvering controller is designed based on the dynamics model of the robot arm, specifically including:

[0104] Without considering the input constraints, the controller based on passivity is designed to achieve the control objective, for Introduce error variable:

[0105] z1=q-q r

[0106]

[0107] where, is q r The derivative with respect to θ, c1 > 0 is a design parameter, and the design

[0108]

[0109] where, is v s The partial derivative with respect to t, c2 > 0 is a design parameter. For the whole manipulator system τ = U m I is only a state variable, not the actual control, to achieve the control objective, the expected value of the current I des is expressed as:

[0110]

[0111] Let z3 be the difference between I and the expected value I des :

[0112] z3 = I - a

[0113] The feedback passive controller is designed as:

[0114]

[0115] where, a q is the partial derivative of a with respect to q, is the partial derivative of a with respect to a t is the partial derivative of a with respect to t, c3 > 0 is a design parameter, is the derivative with respect to θ, is the partial derivative with respect to θ, is the derivative with respect to θ, is v s the partial derivative with respect to θ.

[0116] The manipulator error dynamics model is calculated as:

[0117]

[0118] where, denotes the first derivative of z i , is the velocity assignment error, denotes the first derivative of θ.

[0119] Consider the storage function:

[0120]

[0121] The derivative of the storage function V along the error system is:

[0122]

[0123] where, This shows that the error system is a strictly passive system with w s as input and as output,

[0124] According to the passivity theorem, the velocity assignment is designed as:

[0125]

[0126] where, k > 0 and γ > 0 are design parameters, denotes the first derivative of w s . The derivative of with respect to time t is:

[0127]

[0128] This shows that is a strictly passive system with as input and w s as output;

[0129] In summary, the passive dynamic maneuvering controller is:

[0130]

[0131] As shown in Fig. 2, the closed-loop system is represented as a negative feedback connection of a strictly passive error system and a strictly passive system with the storage function Figure 3 , i.e.,

[0132] The derivative of the Lyapunov function along the closed-loop system is:

[0133]

[0134] According to the stability theorem, it is known that the designed controller completes the maneuvering task, i.e., the output error and the velocity assignment error satisfy:

[0135]

[0136] ​​

[0137] When the control input does not satisfy the constraint, the input with the minimum deviation and satisfying the constraint is selected by quadratic programming to replace the current control input;

[0138] In particular implementation, as the preferred embodiment of the present application, when the control input does not satisfy the constraint, the input with the minimum deviation and satisfying the constraint is selected by quadratic programming to replace the current control input, specifically including:

[0139] When the control input does not satisfy the constraint, the synthesis method of the bounded controller with the minimum deviation is given by quadratic programming (QP) as follows:

[0140]

[0141] Wherein, u i is the i-th component of u, u si is the i-th component of u s .

[0142] The Lagrange function of the above-mentioned QP problem is as follows:

[0143]

[0144] Wherein, λ 1i > 0, λ 2i > 0, i = 1, 2, …, n are Lagrange multipliers, and the Karush-Kuhn-Tucker condition is as follows:

[0145]

[0146] λ 1i (u i -U i ) = 0

[0147] λ 2i (u i + U i ) = 0

[0148] Solving the KKT condition obtains the closed-form solution of the QP problem

[0149] u * = sat(u s )

[0150] Wherein, sat(u s ) = [sat(u s1 ), sat(u s2 ), …, sat(u sn )] T is a saturation function vector, which can be described as

[0151]

[0152] The continuous control satisfying the constraints is converted into digital control, and passive maneuvering digital control of the robot arm is realized.

[0153] In the implementation, as a preferred embodiment of the present application, the continuous control satisfying the constraints is converted into digital control, and specifically includes:

[0154] By introducing a duty cycle function μ i (t)=(u i (t)+U i ) / 2U i , the components u s (i=1,…,n) of the bounded continuous control u=sat(u i ) are converted from the interval [-U i ,U i ] to the interval [0,1].

[0155] By comparing the duty cycle function μi with the carrier u0(t) through a comparison circuit, two-level pulse signals are obtained:

[0156]

[0157] Wherein, u0(t) is a triangular wave with a frequency of 1 / T and an amplitude of 1, and T is an adjustable parameter.

[0158] The motor is allowed to rotate freely in a bidirectional manner, and the digital control is designed as:

[0159] u di =σ i U i ,σ i =2μ Ti -1

[0160] And the digital control u d =(u d1 ,…,u dn ) T is physically realized through an H-bridge circuit with a gate trigger.

[0161] The digital control signal of the servo motor is equivalent to the bounded continuous control signal in time integration.

[0162] In summary, the passive maneuvering digital control closed-loop system of the robot arm under input constraints is shown in Figure 4 .

[0163] Embodiment

[0164] This embodiment is as shown in Figure 5A planar elbow manipulator driven by DC motors is simulated. l1 and l2 represent the lengths of the two links, m1 and m2 represent the masses of the links, and g represents the acceleration of gravity. The dynamics model of the manipulator with 2 degrees of freedom is:

[0165]

[0166] where,

[0167] As Figure 6 shown, consider the manipulator to complete the geometric task of drawing a circle with the base position as the center A and r as the radius, that is, the continuous parameterized geometric path of the manipulator end effector C is:

[0168] x d (θ)=rcosθ,y d (θ)=rsinθ

[0169] where the path variable θ is the angle of the manipulator end effector deviating from the x-axis. The corresponding geometric path q r (θ) in the joint space is r1 (θ),q r2 (θ)) T The geometric method of inverse kinematics is obtained, and the specific method is:

[0170] Without loss of generality, let the elbow of the manipulator point downward, as Figure 6 shown, draw a perpendicular line from the elbow B to AC, and divide AC into a1 and a2, and b is the length of the perpendicular line, then their geometric relationship is

[0171]

[0172] According to Figure 6 , the desired joint path is:

[0173] q r1 (θ)=θ-atan2(b,a1)

[0174] q r2 (θ)=atan2(b,a1)+atan2(b,a2)

[0175] where, The atan2(y,x) function returns the azimuth angle from the origin to the point (x,y).

[0176] The desired velocity assignment is:

[0177]

[0178] The initial values of the system are q0=[0;0]rad, v0=[0;0]rad / s, I0=[0;0]A, θ0=0 rad, w s0 =0 rad / s.

[0179] The parameters of the system are m1=0.5 kg, m2=0.1 kg, l1=l2=0.6 m, g=9.8 m / s 2 , ρ1=ρ2=10 N / m, L1=0.01 H, L2=0.04 H, R1=R2=2 Ω, U m1 =32 N m / A, U m2 =30 N m / A, U e1 =0.01 V / (rad / s), U e2 =0.04 V / (rad / s), U1=U2=2.

[0180] The design parameters are c1=75, c2=60, c3=60, k=1, μ=0.1.

[0181] The simulation results are shown in Figures 7-9 From Figure 7 , it can be seen that the end effector of the manipulator gradually converges to the expected circular path during the movement, which indicates that the geometric task is successfully completed. From Figure 8 , it can be seen that the digital signals of the servo motor are approximately equivalent to the time integral of the continuous expected control signal. From Figure 9 , it can be seen that the output error and the speed distribution error are approximately converged to zero, and the maneuverability task is completed.

[0182] Finally, it should be noted that: the above examples are only used to illustrate the technical solutions of the present application, but not to limit it; although the present application has been described in detail with reference to the foregoing examples, those skilled in the art should understand that: it can still modify the technical solutions recorded in the foregoing examples, or make equivalent replacement for part or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present application.

Claims

1. A passive motion digital control method for a robotic arm under input constraints, characterized in that, include: Construct a dynamic model of the robotic arm with input constraints; Based on robotics and electromagnetism, the degrees of freedom for DC motor drive are established as follows: The dynamic model of the robotic arm: in, , Indicates the first Joint variables of each joint, , , Indicates the first The current of the DC motor in each joint, , Indicates the first The armature inductance of a DC motor with one joint. , Indicates the first Armature resistance of a DC motor with one joint. , Indicates the first The torque constant of a DC motor for each joint , Indicates the first The back electromotive force coefficient of the DC motor at each joint; It is to satisfy The generalized inertia matrix, Representing the whole A set of real matrices of dimension 1 For the Coriolis centripetal matrix, Represents generalized gravity. Representing the whole A set of real vectors of dimension 1 Indicates damping force. The torque is provided by a DC motor. It is a control input. Indicates the first The voltage of the DC motor in each joint express The first derivative, express The second derivative, express The first derivative; Control input Due to the physical constraints of the DC motor, the following constraint conditions are met: in, Indicates the first The rated voltage of the DC motor for each joint; The desired geometric path in the workspace is transformed into that in the joint space using inverse kinematics; the continuously parameterized desired geometric path of the robotic arm end effector is represented as: in, These are the coordinates of the robotic arm's end effector in a fixed coordinate system. It is a path variable; Using inverse kinematics, the path in the workspace is transformed into the joint space, and the desired joint geometry path is: ; Design a passive motion controller based on a robotic arm dynamics model; without considering input constraints, for a bounded desired joint configuration path that is continuously differentiable by order 3. And a bounded feasible velocity assignment that satisfies the condition that the first variable is continuously differentiable by order 2 and the second variable is continuously differentiable by order 1. Introduce error variables into the robotic arm dynamics model: in, , yes about The derivative, For design parameters, , yes about The partial derivatives, , For design parameters; The passive feedback controller is designed as follows: in, , yes about The partial derivatives, yes about The partial derivatives, yes about The partial derivatives, It is a design parameter. , yes about The derivative, yes about The partial derivatives, yes about The derivative, yes about The partial derivatives; The calculated error dynamic model of the robotic arm is as follows: in, It is a speed distribution error. express The first derivative; the error system is based on As input, with For a strictly passive system as output, the energy storage function is: ; According to the passivity theorem, the velocity distribution is designed as follows: in, and For design parameters, express The first derivative; In summary, the desired passive dynamic motion controller is: The closed-loop system is represented as a strictly passive error system and the energy storage function is: Strictly passive system The negative feedback connection, i.e.: Calculate the Lyapunov function The derivative along the closed-loop system is: According to the stability theorem, without considering input constraints, the designed controller completes the maneuvering task, meaning the output error and velocity distribution error satisfy the following: ； When the control input does not meet the constraints, use quadratic programming to select the input with the smallest deviation that meets the constraints to replace the current control input; By converting continuous control that satisfies constraints into digital control, passive digital control of the robotic arm's motion is achieved.

2. The passive motion digital control method for a robotic arm under input constraints according to claim 1, characterized in that, When the control input does not meet the constraints, the method of using quadratic programming to select the input with the smallest deviation that meets the constraints to replace the current control input specifically includes: When the control input does not meet the constraints, the quadratic programming QP method for synthesizing the bounded controller with the minimum deviation is: in, , yes The One portion, yes The One component; By verifying the Karush-Kuhn-Tucker conditions, a closed-form solution to the QP problem is given: in, It is a saturation function vector, described as .

3. The passive motion digital control method for a robotic arm under input constraints according to claim 2, characterized in that, The conversion of continuous control satisfying constraints into digital control specifically includes: By introducing the duty cycle function Bounded continuous control The amount From the interval Transform to range superior; By using a comparator circuit, the duty cycle function is... With carrier Two-stage pulse signals were obtained through comparison: in, Selected frequency is And a triangular wave with an amplitude of 1, It is an adjustable parameter; The motor is allowed to rotate freely in both directions, and the digital control design is as follows: Digital control is physically implemented using an H-bridge circuit with gate flip-flops. ; The digital control signal of a servo motor is equivalent to the bounded continuous control signal in terms of time integration.

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