An adaptive preset performance control method based on disturbance observer

Abstract

The embodiment of the present disclosure relates to a kind of self-adapting preset performance control method based on disturbance observer, comprising: the dynamic model of mechanical arm system is established;Preset performance function is constructed, preset performance function includes lower bound performance function and upper bound performance function, the angular displacement error of mechanical arm is constrained using lower bound performance function and upper bound performance function in the same direction;Under the constraint of lower bound performance function and upper bound performance function, barrier function is introduced to constrain the angular displacement error of each joint of mechanical arm, and the control error of each joint of mechanical arm is determined based on barrier function;The external disturbance of mechanical arm system is estimated using nonlinear disturbance observer, and disturbance estimation result is obtained;Controller is designed based on control error and disturbance estimation result, the external disturbance of mechanical arm system is compensated in real time using controller, and the action of each joint of mechanical arm is controlled.The present disclosure can well control angular displacement error overshoot, improve mechanical arm system stability.

Description

TECHNICAL FIELD

[0001] The present disclosure relates to the technical field of mechanical arm system, and particularly relates to a self-adaptive preset performance control method based on a disturbance observer. BACKGROUND

[0002] In actual industrial production, in order to meet the high standard operation requirements, the automatic control of the mechanical arm is very important. However, due to the inevitable limitation of different external disturbance conditions in the operation process of the mechanical arm, it is necessary to set performance constraints for the mechanical arm system in advance to ensure that the mechanical arm can meet the operation requirements in actual operation.

[0003] In the prior art, most of the preset performance control of the mechanical arm is achieved by using a decay exponential type preset performance function to impose performance constraints on the mechanical arm system. Although the decay exponential type preset performance function can guarantee the steady-state performance of the output error of the mechanical arm system in most cases, it cannot well control the overshoot of the angular displacement error, resulting in poor transient performance of the mechanical arm system. In addition, most of the external disturbances of the system are processed by using a simple scaling processing method, so that the disturbance error cannot be accurately compensated, which is not conducive to the robustness and stability of the mechanical arm system.

[0004] Therefore, it is necessary to provide a new technical scheme to improve one or more problems in the above-mentioned scheme.

[0005] It should be noted that the information disclosed in the above background section is only used to strengthen the understanding of the background of the present disclosure, and therefore can include information that does not constitute prior art known to those of ordinary skill in the art SUMMARY

[0006] The purpose of the embodiments of the present disclosure is to provide a self-adaptive preset performance control method based on a disturbance observer, which can well control the overshoot of the angular displacement error and improve the transient performance of the mechanical arm system.

[0007] According to the embodiments of the present disclosure, a self-adaptive preset performance control method based on a disturbance observer is provided, which comprises:

[0008] establishing a dynamic model of the mechanical arm system;

[0009] constructing a preset performance function, the preset performance function comprising a lower bound performance function and an upper bound performance function, and using the lower bound performance function and the upper bound performance function in the same direction to constrain the angular displacement error of the mechanical arm;

[0010] under the constraint of the lower bound performance function and the upper bound performance function, introducing a barrier function to constrain the angular displacement error of each joint of the mechanical arm, and determining the control error of each joint of the mechanical arm based on the barrier function.

[0011] A nonlinear disturbance observer is designed to estimate external disturbances of the manipulator system and obtain a disturbance estimation result;

[0012] A controller is designed based on the control error and the disturbance estimation result to compensate for external disturbances of the manipulator system in real time and control the movement of each joint of the manipulator.

[0013] In an example embodiment of the present disclosure, the dynamic model of the manipulator system is represented as:

[0014]

[0015] wherein M(q) represents an inertia matrix of the manipulator, C(q, q) represents a Coriolis matrix of the manipulator, G(q) represents a gravity term of the manipulator, d represents a preset external disturbance, F(q, q) represents a friction term, τ represents a control input torque of the manipulator system, q represents an angular displacement of the manipulator, q n represents an angular displacement of an nth joint of the manipulator, represents an angular velocity of the manipulator, represents an angular acceleration.

[0016] In an example embodiment of the present disclosure, the lower bound performance function and the upper bound performance function are represented as:

[0017]

[0018] wherein ρ l (t) represents a lower bound performance function, ρ l,0 represents an initial performance lower bound, ρ l,0 > 0, ρ l,∞ represents a steady-state lower bound performance boundary, ρ l,∞ > 0, λ l represents a first performance decay coefficient, ρ u (t) represents an upper bound performance function, ρ u,0 represents an initial performance upper bound, ρ u,0 > 0, ρ u,∞ represents a steady-state upper bound performance boundary, ρ u,∞ > 0, λ u represents a second performance decay coefficient, t represents a time, δ represents an initial error threshold, e(0) represents an angular displacement error of the manipulator at t = 0, and sgn(·) represents a sign function.

[0019] In an example embodiment of the present disclosure, the step of constraining the angular displacement error of the robot arm using the lower bound performance function and the upper bound performance function in the same direction comprises:

[0020] limiting the initial performance lower bound, the steady-state lower bound performance bound, the initial performance upper bound and the steady-state upper bound performance bound by formula (3) according to an initial error threshold, so that the lower bound performance function and the upper bound performance function are in the same direction;

[0021] The formula (3) is expressed as:

[0022]

[0023] Wherein, |δ| represents the absolute value of the initial error threshold, |ρ l,0 | represents the absolute value of the initial performance lower bound, |ρ l,∞ | represents the absolute value of the steady-state lower bound performance bound, |ρ u,0 | represents the absolute value of the initial performance upper bound, |ρ u,∞ | represents the absolute value of the steady-state upper bound performance bound.

[0024] In an example embodiment of the present disclosure, the step of constraining the angular displacement error of the robot arm using the lower bound performance function and the upper bound performance function in the same direction comprises:

[0025]

[0026] Wherein, e(t) represents the angular displacement error of the robot arm system at time t, ρ l (t) represents the lower bound performance function, ρ u (t) represents the upper bound performance function.

[0027] In an example embodiment of the present disclosure, the step of introducing a barrier function to constrain the angular displacement error of each joint of the robot arm system comprises:

[0028] Design a barrier function based on barrier function theory, and the barrier function is expressed as:

[0029]

[0030] Wherein, z i represents the barrier function variable, e i represents the angular displacement error of the i-th joint of the robot arm system, ρ l,i represents the lower bound performance function corresponding to the angular displacement error of the i-th joint, ρ u.i represents the upper bound performance function corresponding to the angular displacement error of the i-th joint.

[0031] Under the constraints of the lower bound performance function and the upper bound performance function, the angular displacement error of the ith joint satisfies the following formula (6) derived from the barrier function and formula (4):

[0032]

[0033] wherein e i (t) represents the angular displacement error of the ith joint at time t, e i (0) represents the angular displacement error of the ith joint at time t = 0, p l,i (t) represents the lower bound performance function corresponding to the angular displacement error of the ith joint at time t, p u,i (t) represents the upper bound performance function corresponding to the angular displacement error of the ith joint at time t.

[0034] In an example embodiment of the present disclosure, in the step of determining the control error of each joint of the robot arm based on the barrier function, the control error is represented as:

[0035]

[0036] wherein s i represents the control error corresponding to the ith joint, z i represents the barrier function variable corresponding to the ith joint, represents the derivative of z i , Λ i is a constant, and Λ i > 0.

[0037] In an example embodiment of the present disclosure, the nonlinear disturbance observer is represented as:

[0038]

[0039] wherein w represents the nonlinear disturbance observer state, represents the derivative of w, L(q) represents a preset gain matrix, τ represents the control torque vector of the robot arm, V represents the Coriolis force matrix of the robot arm, G represents the gravity term of the robot arm, F v represents the friction force term, q represents the angular displacement of the robot arm, q n represents the angular displacement of the nth joint of the robot arm, represents the angular velocity of the robot arm, represents a positive definite matrix, M represents the inertia matrix of the robot arm, represents the angular acceleration, represents the disturbance estimation result.

[0040] In an example embodiment of the present disclosure, in the step of designing a controller based on the disturbance estimation result, the controller is represented as:

[0041]

[0042] where τ represents a control input torque, represents a disturbance estimation result, represents a parameter estimation, represents a parameter adaptation rate, φ represents a smooth and continuous positive definite function, and c represents a set parameter, c > 0, represents the minimum eigenvalue of a matrix represents the transpose of χ, χ represents a vector, s represents a control error of the robot arm, s = [s1, s2, …, sn] ∈ Rn, n n s n represents the nth element of s, γ represents a set parameter, γ > 0, and σ represents a set parameter, σ > 0.

[0043] In an example embodiment of the present disclosure, the disturbance observer-based adaptive preset performance control method further comprises:

[0044] constructing a quadratic Lyapunov candidate function:

[0045]

[0046] where V represents the quadratic Lyapunov candidate function, s T represents the transpose of s, m represents a positive boundary constant, represents a parameter estimation error, θ represents a preset parameter, represents a positive boundary constant, λ min (L(q)) represents the minimum eigenvalue of a preset gain matrix L(q), represents a disturbance estimation error, d represents a preset external disturbance, represents the transpose of

[0047] deriving the derivative of the quadratic Lyapunov candidate function, to obtain:

[0048]

[0049] where represents the derivative of the quadratic Lyapunov candidate function, M represents the inertia matrix of the robot arm, represents the derivative of χ, ​​​derivative of the angular displacement error, derivative of T, T represents a variable, and A represents an arbitrary positive constant, representing a preset expected angular acceleration;

[0050] The stability of the nonlinear disturbance observer is verified by using the derivative of the quadratic form Lyapunov candidate function.

[0051] The technical solutions provided by the present disclosure can include the following beneficial effects:

[0052] In the embodiments of the present disclosure, on the one hand, the lower bound performance function and the upper bound performance function in the same direction are used to constrain the angular displacement error of the robot arm, and under the constraint of the preset performance function, a barrier function is further introduced to constrain the angular displacement error of each joint of the robot arm, which is conducive to better controlling the angular displacement error overshoot and improving the transient performance of the robot arm system. On the other hand, the control error of each joint of the robot arm is determined based on the barrier function, the external disturbance of the robot arm system is accurately estimated by using a nonlinear disturbance observer, the disturbance estimation result is obtained, and a controller is designed based on the control error of the robot arm and the disturbance estimation result, so as to realize real-time compensation for the external disturbance of the robot arm system and control the action of each joint of the robot arm, thereby improving the robustness and stability of the robot arm system.

[0053] It should be understood that the above general description and the following detailed description are only exemplary and explanatory, and cannot limit the present disclosure. BRIEF DESCRIPTION OF DRAWINGS

[0054] The accompanying drawings incorporated in the specification and constituting a part of the specification illustrate embodiments consistent with the present disclosure and serve together with the specification to explain the principles of the present disclosure. Obviously, the drawings in the following description are only some embodiments of the present disclosure, and other drawings can be obtained by those skilled in the art without creative labor on the basis of these drawings.

[0055] Figure 1 A step flowchart of the adaptive preset performance control method based on the disturbance observer in the exemplary embodiments of the present disclosure is shown;

[0056] Figure 2 A simulation schematic diagram showing the error overshoot phenomenon of the preset performance function adopted in the prior art in the exemplary embodiments of the present disclosure is shown;

[0057] Figure 3 A simulation result schematic diagram showing the constraint of the angular displacement error by using the preset performance function designed in the present application in the exemplary embodiments of the present disclosure is shown Figure 1 ;

[0058] Figure 4This illustration shows simulation results of constraining angular displacement error using a preset performance function designed in this application in an exemplary embodiment of the present disclosure. Figure 2 ;

[0059] Figure 5 This illustration shows simulation results of estimating external disturbances in a robotic arm system using the nonlinear disturbance observer designed in this application, as shown in an exemplary embodiment of this disclosure. Figure 1 ;

[0060] Figure 6 This illustration shows simulation results of estimating external disturbances in a robotic arm system using the nonlinear disturbance observer designed in this application, as shown in an exemplary embodiment of this disclosure. Figure 2 . Detailed Implementation

[0061] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, they are provided so that this disclosure will be more comprehensive and complete, and will fully convey the concept of the exemplary embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.

[0062] Furthermore, the accompanying drawings are merely illustrative of this disclosure and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and therefore repeated descriptions of them will be omitted. Some block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities may be implemented in software, in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.

[0063] This example implementation first provides an adaptive preset performance control method based on a disturbance observer, referencing... Figure 1 As shown, the method may include the following steps:

[0064] Step S101: Establish the dynamic model of the robotic arm system;

[0065] Step S102: Construct a preset performance function, which includes a lower bound performance function and an upper bound performance function. Use the lower bound performance function and the upper bound performance function, which are located in the same direction, to constrain the angular displacement error of the robotic arm.

[0066] Step S103: introducing a barrier function to constrain the angular displacement error of each joint of the robot arm under the constraint of the lower bound performance function and the upper bound performance function, and determining the control error of each joint of the robot arm based on the barrier function;

[0067] Step S104: designing a nonlinear disturbance observer to estimate the external disturbance of the robot arm system and obtain a disturbance estimation result;

[0068] Step S105: designing a controller based on the control error of the robot arm and the disturbance estimation result, using the controller to compensate for the external disturbance of the robot arm system in real time, and controlling the movement of each joint of the robot arm.

[0069] In an embodiment of the present disclosure, on the one hand, the angular displacement error of the robot arm is constrained by the lower bound performance function and the upper bound performance function in the same direction, and the barrier function is further introduced to constrain the angular displacement error of each joint of the robot arm under the constraint of the preset performance function, which is beneficial to better control the angular displacement error overshoot and improve the transient performance of the robot arm system. On the other hand, the control error of each joint of the robot arm is also determined based on the barrier function, the external disturbance of the robot arm system is accurately estimated by using the nonlinear disturbance observer, the disturbance estimation result is obtained, and the controller is designed based on the control error of the robot arm and the disturbance estimation result. The external disturbance of the robot arm system is compensated in real time, the movement of each joint of the robot arm is controlled, and thus the robustness and stability of the robot arm system are improved.

[0070] It should be noted that in the robot arm system dynamic model established in the present embodiment, the robot arm includes multiple joints, and by constraining the angular displacement error of the entire robot arm and the angular displacement error of each joint of the robot arm, the output error of the robot arm system can be more accurately regulated and controlled, and the transient performance of the robot arm system can be improved.

[0071] In the following, each step of the above method in the present example embodiment will be described in more detail.

[0072] Specifically, in step S101, the total kinetic energy and total potential energy of the robot arm system can be analyzed according to the Lagrange formula method, and the control torque of each joint of the robot arm can be solved by using the law of rotation combined with the mechanical model of the pneumatic artificial muscle, so as to obtain the dynamics model of the robot arm system, wherein the robot arm system dynamics model is represented as:

[0073]

[0074] wherein M(q) represents the inertia matrix of the robot arm, C(q) represents the Coriolis force matrix of the robot arm, G(q) represents the gravity term of the robot arm, and d represents a preset external disturbance. represents a friction term, τ represents a control input torque of the robot arm system, q represents an angular displacement of the robot arm, q n represents an angular displacement of the nth joint of the robot arm, represents an angular velocity of the robot arm, represents an angular acceleration.

[0075] It should be noted that establishing the dynamic model of the robot arm system is a routine technical means in the art, and will not be described here.

[0076] In the present embodiment, the following assumption 1 is given: the preset expected angular displacement q d = where q dn represents the expected angular displacement of the nth joint of the robot arm. Then, by subtracting the angular displacement q of the robot arm in the system from the preset expected angular displacement q d , the corresponding angular displacement error can be obtained, including the angular displacement error of the entire robot arm and the angular displacement error of each joint of the robot arm.

[0077] It should be noted that, for the dynamic model of the robot arm system established above, the angular displacement error of the robot arm can be controlled within a specified performance range to ensure that the robot arm moves accurately along the preset motion trajectory.

[0078] It should be explained that, in order to illustrate the difference between the preset performance function proposed in the present application and the preset performance function used in the prior art, Figure 2 a simulation diagram showing the error overshoot phenomenon of the preset performance function used in the prior art is given in the prior art. The performance change direction of this decaying exponential type of preset performance function is opposite, and the upper and lower boundaries of its performance are located in opposite directions, which can also be understood as the upper and lower boundaries of the performance being located on both sides of the x-axis, thereby causing the angular displacement error to possibly overshoot in the same direction or different directions, resulting in poor transient performance of the system.

[0079] In step S102, the lower boundary performance function and the upper boundary performance function of the preset performance function constructed by the present embodiment are represented as:

[0080]

[0081] where ρ l (t) represents the lower boundary performance function, ρ l,0 represents the initial performance lower boundary, ρ l,0 > 0, ρ l,∞ represents the steady-state lower boundary performance, ρ l,∞ > 0, λ l represents the first performance decay coefficient, ρ u(t) represents an upper bound performance function, p u,0 represents an initial performance upper bound, p u,0 > 0, p u,∞ represents a steady-state upper bound performance boundary, p u,∞ > 0, l u represents a second performance decay coefficient, t represents a time, d represents an initial error threshold, e(0) represents an angular displacement error of the robot arm at t = 0, sgn(·) represents a sign function,

[0082] In an example, the step of constraining the angular displacement error of the robot arm in step S102 using the lower bound performance function and the upper bound performance function in the same direction further comprises the following steps:

[0083] The initial performance lower bound, the steady-state lower bound performance boundary, the initial performance upper bound, and the steady-state upper bound performance boundary are limited according to the initial error threshold, so that the lower bound performance function and the upper bound performance function are in the same direction.

[0084] Specifically, in the step of limiting the initial performance lower bound, the steady-state lower bound performance boundary, the initial performance upper bound, and the steady-state upper bound performance boundary according to the initial error threshold, the limiting is performed by formula (3):

[0085]

[0086] Where |d| represents the absolute value of the initial error threshold, |p l,0 | represents the absolute value of the initial performance lower bound, |p l,∞ | represents the absolute value of the steady-state lower bound performance boundary, |p u,0 | represents the absolute value of the initial performance upper bound, |p u,∞ | represents the absolute value of the steady-state upper bound performance boundary.

[0087] The parameters |p l,0 |, |p l,∞ |, |p u,0 |, |p u,∞ | are limited by introducing the absolute value |d| of the initial error threshold, so that the preset lower bound p l (t) and the preset upper bound p u (t) are always in the same direction at any time t. Here, being in the same direction can be understood as being in the direction of approaching positive infinity along the x-axis, to ensure that after constraining the angular displacement error at any time, the overshoot phenomenon of the angular displacement error can be prevented, thereby improving the transient performance of the robot arm system.

[0088] Specifically, in the step of constraining the angular displacement error of the robot arm by the lower bound performance function and the upper bound performance function in the same direction, the constraint is performed by formula (4):

[0089]

[0090] wherein e(t) represents the angular displacement error of the robot arm system at time t, p l (t) represents the lower bound performance function, p u (t) represents the upper bound performance function.

[0091] By designing the preset performance functions p l (t) and p u (t) represented in formula (2) to formula (4) to constrain the angular displacement error, it is beneficial to better control the angular displacement error overshoot and improve the transient performance of the robot arm system.

[0092] Referring to Figure 3 and Figure 4 , the simulation result schematic diagram of the angular displacement error constrained by the preset performance function designed in the embodiment is shown, it can be seen that, Figure 3 when e(0) is greater than or equal to 0, Figure 4 when e(0) is less than 0, the angular displacement error can be prevented from overshooting, and the transient performance of the robot arm system is improved.

[0093] In the embodiment, in order to ensure that the angular displacement error falls in the set , a barrier function is further introduced, and the angular displacement error of each joint of the robot arm system is constrained by the barrier function to ensure that the angular displacement error falls in the set . The e i (0) represents the angular displacement error of the i th joint at time t = 0, e i (t) represents the angular displacement error of the i th joint at time t, and e i is the angular displacement error of the i th joint at any time.

[0094] For example, the step of introducing the barrier function to constrain the angular displacement error of each joint of the robot arm system in step S103 includes:

[0095] The barrier function is designed based on the barrier function theory, and the barrier function is represented as:

[0096]

[0097] wherein z i represents the barrier function variable, ei denotes the angular displacement error of the i-th joint of the robot arm system, ρ l,i denotes the lower bound performance function corresponding to the angular displacement error of the i-th joint, ρ u.i denotes the upper bound performance function corresponding to the angular displacement error of the i-th joint;

[0098] Under the constraints of the lower bound performance function and the upper bound performance function, the angular displacement error of the i-th joint satisfies the following formula (6) derived from the barrier function and formula (4):

[0099]

[0100] wherein e i (t) denotes the angular displacement error of the i-th joint at time t, e i (0) denotes the angular displacement error of the i-th joint at time t=0, ρ l,i (t) denotes the lower bound performance function corresponding to the angular displacement error of the i-th joint at time t, ρ u,i (t) denotes the upper bound performance function corresponding to the angular displacement error of the i-th joint at time t.

[0101] It should be noted that, under the constraints of the preset performance function, in order to make the angular displacement error of the i-th joint satisfy formula (6) derived from the barrier function and formula (4), the following premise conditions need to be met first:

[0102] The first premise condition is that e i (0) needs to satisfy the following formula (12):

[0103]

[0104] wherein e i (0) denotes the angular displacement error of the i-th joint at time t=0, ρ l,i (0) denotes the lower bound performance function corresponding to the angular displacement error of the i-th joint at time t=0, ρ u,i (0) denotes the upper bound performance function corresponding to the angular displacement error of the i-th joint at time t=0.

[0105] The second premise condition is to ensure that the barrier function variable z i is bounded.

[0106] In this way, it can be ensured that the angular displacement error e i (t) of the i-th joint at any time will never reach the boundary ρ l,i (t) and ρ u,i (t), thereby realizing corresponding performance constraints on the angular displacement error of each joint of the robot arm and achieving more stable dynamic control.

[0107] It should be noted that in order to realize the controller more accurately regulate the movement of the robot arm, the control error of the robot arm is further introduced in this embodiment, and in the step of determining the control error of each joint of the robot arm based on the barrier function, the control error is represented as:

[0108]

[0109] Wherein, s i represents the control error corresponding to the i th joint, z i represents the barrier function variable corresponding to the i th joint, represents the derivative of z i , Λ i is a constant, and Λ i > 0

[0110] In this embodiment, in order to compensate for the influence of external disturbance d on the robot arm system, a nonlinear disturbance observer is further designed.

[0111] In step S104, the nonlinear disturbance observer designed in this embodiment is represented as:

[0112]

[0113] Wherein, w represents the nonlinear disturbance observer state, represents the derivative of w, L(q) represents the preset gain matrix, τ represents the control torque vector of the robot arm, V represents the Coriolis force matrix of the robot arm, G represents the gravity term of the robot arm, F v represents the friction term, q represents the angular displacement of the robot arm, q n represents the angular displacement of the n th joint of the robot arm, represents the angular velocity of the robot arm, represents a positive definite matrix, M represents the inertia matrix of the robot arm, represents the angular acceleration, represents the disturbance estimation result.

[0114] According to the above designed nonlinear disturbance observer, the following assumption 2 is given: there is a positive constant such that

[0115] And introduce the following theorem 1: for any external disturbance d satisfying assumption 2, if the preset gain matrix of the nonlinear disturbance observer is positive definite (L(q) > 0), then the nonlinear disturbance observer can exponentially estimate the external disturbance d of the robot arm system, and get accurate disturbance estimation result

[0116] The following gives the proof process that the nonlinear disturbance observer designed in this embodiment can accurately estimate the disturbance estimation result :

[0117] First, according to the preset external disturbance d and the disturbance estimation result observed by the nonlinear disturbance observer, the disturbance estimation error is obtained.

[0118] Then, the disturbance estimation error is differentiated, and the following equation is obtained:

[0119]

[0120] wherein denotes the derivative of the disturbance estimation error, denotes the derivative of the disturbance estimation result, denotes a positive definite matrix, denotes the derivative of the nonlinear disturbance observer state;

[0121] Then, the Lyapunov candidate function is considered, and the Lyapunov candidate function V d is differentiated, and the following equation is obtained:

[0122]

[0123] wherein λ min (L(q)) denotes the minimum eigenvalue of the preset gain matrix L(q), denotes the derivative of the Lyapunov candidate function, denotes the transpose of .

[0124] Since the preset gain matrix L(q) > 0, λ min (L(q)) > 0, and thus the following equation is obtained: That is, as the time t increases, the disturbance estimation error tends to 0, which indicates that the nonlinear disturbance observer designed in this embodiment can accurately estimate the external disturbance of the manipulator system.

[0125] Referring to Figure 5 and Figure 6 , the simulation result schematic diagram of the estimation of the external disturbance of the manipulator system by the nonlinear disturbance observer designed in this embodiment is shown. From Figure 5As can be seen, the actual disturbance curve and the disturbance estimation result are shown. The actual disturbance curve represents the preset external disturbance. When the preset external disturbance is a constant external disturbance, i.e., d = 1.2, the disturbance estimation result is obtained by using the nonlinear disturbance observer. At time t = 1s, it is already very close to the preset external disturbance d. From Figure 6 As can be seen from this, when the preset external disturbance is a time-varying external disturbance, that is... At that time, the disturbance estimation results obtained by using the nonlinear disturbance observer are... The external disturbance d is close to the preset value.

[0126] In this embodiment, based on the accurate estimation of external disturbances by the aforementioned nonlinear disturbance observer, in order to compensate for the impact of constant / time-varying external disturbances on the performance of the robotic arm system and ensure that the angular displacement error remains within the performance range specified by the aforementioned preset performance function, a controller is further designed, and the controller parameters need to be updated adaptively over time.

[0127] In step S104, the controller designed in this embodiment is represented as follows:

[0128]

[0129] Where τ represents the control input torque, This indicates the result of the disturbance estimation. Indicates parameter estimation, Let represent the adaptive rate of the parameters, φ represent a smooth and continuous positive definite function, and c represent the set parameters, where c > 0. Representation matrix The smallest eigenvalue, Let χ denote the transpose of χ, where χ represents a vector, and s represents the column vector of the control error of the robotic arm, s = [s1, s2, ..., s]. n ]∈R n s n Let represent the nth element of s, γ represent the set parameter (γ>0), and σ represent the set parameter (σ>0). In the above controller, since ∥s∥ 2 φ≥0 and Heng was established.

[0130] Based on the real-time estimated disturbance estimation results, the control error of the robotic arm, and the adaptively updated parameter estimation, the controller can update the controller parameters in real time to control the robotic arm system and adjust the movements of each joint of the robotic arm in real time.

[0131] In one embodiment, the disturbance observer based adaptive preset performance control method proposed in the present embodiment further comprises a stability analysis process, which can comprise the following steps:

[0132] constructing a quadratic Lyapunov candidate function:

[0133]

[0134] wherein V represents the quadratic Lyapunov candidate function, represents the transpose of s, m represents a positive boundary constant, represents a parameter estimation error, represents a preset parameter, represents a positive boundary constant, λ min (L(q)) represents the minimum eigenvalue of a preset gain matrix L(q), represents a disturbance estimation error, d represents a preset external disturbance, represents the transpose of

[0135] deriving the derivative of the quadratic Lyapunov candidate function, to obtain:

[0136]

[0137] wherein, represents the derivative of the quadratic Lyapunov candidate function, M is M(q), and represents the inertia matrix of the robot arm, represents the derivative of χ, represents the derivative of the angular displacement error, represents the derivative of T, T represents a variable, and Λ represents an arbitrary positive constant, represents a preset expected angular acceleration;

[0138] the stability of the nonlinear disturbance observer is verified by using the derivative of the quadratic Lyapunov candidate function.

[0139] It should be noted that before constructing the quadratic Lyapunov candidate function, the derivative of the barrier function variable z i can be represented as:

[0140]

[0141] wherein, χ i represents the i-th element of the vector X, t​i denotes a variable, e i is the angular displacement error of the ith joint at any time, denotes the derivative of e i , denotes the derivative of p l,i , denotes the derivative of p u,i ;

[0142] Further, since the control error is substituted into the expression of the control error s i of the robot arm, and the control error s i is derived, the derivative of the control error is:

[0143]

[0144] wherein, denotes the derivative of the control error s i , denotes the derivative of x i , denotes the derivative of t i , denotes the derivative of ;

[0145] The above expression (16) can be rewritten in a compact form as shown in the following expression (17):

[0146]

[0147] wherein, Λ = diag{Λ1, Λ2, …, Λ n} ∈ R n×n , x = diag{χ1, χ2, …, χ n} ∈ R n×n , diag{Λ1, Λ2, …, Λ n} denotes a set consisting of diagonal elements in {Λ1, Λ2, …, Λ n}, diag{χ1, χ2, …, χ n} denotes a set consisting of diagonal elements in {χ1, χ2, …, χ n}, e = [e1, e2, …, e n ] ∈ R n , T = [t1, t2, …, t n ] ∈ R n , denotes the derivative of the column vector s of the control error.

[0148] For example, the step of verifying the stability of the nonlinear disturbance observer by using the derivative of a quadratic Lyapunov candidate function comprises:

[0149] Based on the mechanical arm system dynamic model, according to Young inequality, there is:

[0150]

[0151] where ||s|| represents the norm of s, ||χ|| represents the norm of χ, represents the norm of, ||Ξ|| represents the norm of Ξ, represents a positive boundary constant, represents a positive constant;

[0152] Substituting formula (18) into formula (11), there is:

[0153]

[0154] where, represents the derivative of a quadratic Lyapunov candidate function, φ represents a smooth continuous positive definite function, and θ represents a preset parameter, represents a positive boundary constant, represents a positive constant;

[0155] Substituting the control input torque τ of the controller into in formula (19), there is:

[0156]

[0157] where c represents a set parameter;

[0158] Substituting formula (20) into formula (19), there is:

[0159]

[0160] where, represents the derivative of a quadratic Lyapunov candidate function, and γ represents a set parameter;

[0161] Substituting the parameter adaptive rate of the controller into formula (21), there is:

[0162]

[0163] where, represents the derivative of a quadratic Lyapunov candidate function number;

[0164] Based on formula (22) The following formula (23) is satisfied. Substituting formula (23) into formula (22) yields formula (24), which verifies the stability of the nonlinear disturbance observer.

[0165] Formulas (23) and (24) are expressed as follows:

[0166]

[0167] Where, θ 2 Represents the square of θ;

[0168]

[0169] Where κ=min{2c m ,σ},

[0170] From formula (24), we can know that This demonstrates that the nonlinear perturbation observer is convergent, proving its stability.

[0171] Furthermore, it should be noted that the controller designed in this embodiment can be used to control the angular displacement error of the robotic arm system to meet the preset performance of the preset performance function. According to formula (24), the quadratic Lyapunov candidate function V is bounded, therefore the control error s of the robotic arm is... i and parameter estimation error Both are bounded, because Therefore, parameter estimation It is also bounded; because Therefore, the barrier function variable z i The derivative of the barrier function variable Both are bounded; because Therefore, the angular displacement error e of the i-th joint at any given time i Bounded and satisfied This ensures that the controller designed in this embodiment can control the angular displacement error of the robotic arm system to meet the preset performance specified by the preset performance function.

[0172] Furthermore, it should be noted that this embodiment also provides an explanation of the boundedness of all closed-loop signals, since ρ l,i ρ u,i , e i Both are bounded, and due to the above formula (15) Therefore, element χ i and variable t i Both are bounded; furthermore, due to the limitations imposed by formula (14) and the barrier function variable is differentiable is bounded, so is also bounded, so and further, the control input torque τ and the parameter adaptation rate are bounded, so it can be shown that all closed loop signals are bounded.

[0173] It is also noted that in the foregoing description, various features are described as being implemented with certain components in certain embodiments. These features are, however, separable from the components that implement them and can be implemented with other components in other embodiments. For example, the barrier function variable is differentiable

[0174] Assumption 3: There exists a positive constant such that

[0175] Property 1: The matrix is skew-symmetric.

[0176] Property 2: There exist some positive bounding constants m , such that

[0177] It is noted that while the various steps in the methods of the present disclosure are described in a particular order in the figures, this is not required or implied, and the desired results can be achieved in a different order, or all of the steps shown can not be performed. Additionally or alternatively, certain steps can be combined into a single step, multiple steps can be performed in a single step, and / or a single step can be divided into multiple steps. It is also readily appreciated that the steps can be performed synchronously or asynchronously, for example, in multiple modules / processes / threads.

[0178] Other embodiments of the present disclosure will be apparent to those skilled in the art from consideration of the specification and practice of the present disclosure. This application is intended to cover any variations, uses, or adaptations of the present disclosure following, in general, the principles of the present disclosure and including such departures from the present disclosure that come within known or customary practice in the art to which the present disclosure pertains. The specification and examples are to be regarded as exemplary only, and the true scope and spirit of the present disclosure are indicated by the appended claims.

Claims

1. A disturbance observer based adaptive preset performance control method, characterized in that, The method comprises the following steps: a mechanical arm system dynamic model is established; a preset performance function is constructed, the preset performance function comprising a lower bound performance function and an upper bound performance function, and the angle displacement error of the mechanical arm is constrained by the lower bound performance function and the upper bound performance function in the same direction; a barrier function is introduced to constrain the angle displacement error of each joint of the mechanical arm under the constraint of the lower bound performance function and the upper bound performance function, and the control error of each joint of the mechanical arm is determined based on the barrier function; a nonlinear disturbance observer is designed to estimate the external disturbance of the mechanical arm system, and a disturbance estimation result is obtained; a controller is designed based on the control error and the disturbance estimation result, the external disturbance of the mechanical arm system is compensated in real time by using the controller, and the movement of each joint of the mechanical arm is controlled; the lower bound performance function and the upper bound performance function are expressed as: （2） wherein, represents a lower bound performance function, represents an initial lower bound performance, , represents a steady-state lower bound performance boundary, , represents a first performance decay coefficient, represents an upper bound performance function, represents an initial upper bound performance, , represents a steady-state upper bound performance boundary, , represents a second performance decay coefficient, represents a time instant, represents an initial error threshold, represents an angular displacement error of the robotic arm at the time instant , represents a sign function, ; the barrier function is introduced to constrain the angle displacement error of each joint of the mechanical arm, comprising: a barrier function is designed based on barrier function theory, and the barrier function is expressed as: （5） in, Represents the barrier function variable. The first part of the robotic arm system Angular displacement error of each joint Indicates the first The lower bound performance function corresponding to the angular displacement error of each joint. Indicates the first The upper bound performance function corresponding to the angular displacement error of each joint.

2. The disturbance observer based adaptive preset performance control method according to claim 1, wherein, the mechanical arm system dynamic model is expressed as: （1） in, The inertia matrix of the robotic arm is represented. The Coriolis force matrix of the robotic arm is represented. This represents the gravity term of the robotic arm. This indicates a pre-defined external disturbance. Represents the friction term. This represents the control input torque of the robotic arm system. This represents the angular displacement of the robotic arm. , The first part of the robotic arm Angular displacement of each joint This represents the angular velocity of the robotic arm. It represents angular acceleration.

3. The disturbance observer based adaptive preset performance control method according to claim 2, wherein, the lower bound performance function and the upper bound performance function in the same direction are used to constrain the angle displacement error of the mechanical arm, comprising: the initial error threshold is used to limit the initial performance lower bound, the steady-state lower bound performance boundary, the initial performance upper bound and the steady-state upper bound performance boundary by formula (3), so that the lower bound performance function and the upper bound performance function are in the same direction; the formula (3) is expressed as: （3） wherein, denotes an absolute value of an initial error threshold, denotes an absolute value of an initial performance lower bound, denotes an absolute value of a steady state lower performance bound, denotes an absolute value of an initial performance upper bound, denotes an absolute value of a steady state upper performance bound.

4. The disturbance observer based adaptive preset performance control method according to claim 3, wherein, in the step of using the lower bound performance function and the upper bound performance function in the same direction to constrain the angle displacement error of the mechanical arm, the constraint is performed by formula (4): （4） wherein represents angular displacement error of the robot system at the time instant, represents a lower bound performance function, represents an upper bound performance function.

5. The disturbance observer based adaptive preset performance control method according to claim 4, wherein, Under the constraint of the lower bound performance function and the upper bound performance function, the first i The angular displacement error of the first joint satisfies the following formula (6): （6） in, express The first moment Angular displacement error of each joint Indicates in The first moment of time Angular displacement error of each joint express The first moment The lower bound performance function corresponding to the angular displacement error of each joint. express The first moment The upper bound performance function corresponding to the angular displacement error of each joint.

6. The disturbance observer based adaptive preset performance control method according to claim 5, wherein, in the step of determining the control error of each joint of the mechanical arm based on the barrier function, the control error is expressed as: （7） wherein, denotes the control error of the jth joint, denotes the barrier function variable of the jth joint, denotes the derivative of is a constant, and .​​​ 7. The disturbance observer based adaptive preset performance control method according to claim 2, wherein, the nonlinear disturbance observer is expressed as: （8） wherein denotes a nonlinear disturbance observer state, denotes a derivative of , denotes a preset gain matrix, denotes a control torque vector of the robot arm, denotes a Coriolis force matrix of the robot arm, denotes a gravity term of the robot arm, denotes a friction term, denotes an angular displacement of the robot arm, denotes an angular displacement of the nth joint of the robot arm, denotes an angular velocity of the robot arm, denotes a positive definite matrix, , denotes an inertia matrix of the robot arm, denotes an angular acceleration, denotes a disturbance estimation result.

8. The disturbance observer based adaptive preset performance control method according to claim 7, wherein, in the step of designing the controller based on the control error of the mechanical arm and the disturbance estimation result, the controller is expressed as: （9） wherein denotes a control input torque, denotes a disturbance estimate, denotes a parameter estimate, denotes a parameter adaptation rate, denotes a smooth, continuous, positive definite function, denotes a set parameter, , denotes the smallest eigenvalue of the matrix , denotes , denotes the transpose of , denotes a control error of the robot arm, , is the i-th element of , denotes a control error of the i-th joint of the robot arm, denotes a set parameter, , denotes a set parameter, .

9. The disturbance observer based adaptive preset performance control method according to claim 8, wherein, the adaptive preset performance control method based on the disturbance observer further comprises: a quadratic Lyapunov candidate function is constructed: （10） wherein denotes a quadratic Lyapunov candidate function, denotes the transpose of denotes a positive boundary constant, denotes a parameter estimation error, , denotes a preset parameter, denotes a positive boundary constant, denotes a preset gain matrix the minimum eigenvalue of denotes a disturbance estimation error, , denotes a preset external disturbance, denotes the transpose of the derivative of the quadratic Lyapunov candidate function is calculated to obtain: （11） wherein, denotes the derivative of a quadratic Lyapunov candidate function, denotes the inertia matrix of the robot arm, , denotes the derivative of denotes the derivative of the angular displacement error, denotes the derivative of denotes a variable, denotes an arbitrary positive constant, denotes a preset desired angular acceleration;​​ the stability of the nonlinear disturbance observer is verified by using the derivative of the quadratic Lyapunov candidate function.

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