CN117885103B - Flexible mechanical arm control method and system based on reduced-order expanded state observer - Google Patents
Flexible mechanical arm control method and system based on reduced-order expanded state observer Download PDFInfo
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Abstract
The invention discloses a flexible mechanical arm control method and system based on a reduced-order expanded state observer, and relates to the technical field of multi-joint flexible mechanical arm motion control. The invention solves the problem of differential explosion in the traditional back-step control method by improving the dynamic surface controller, effectively reduces the filtering error in the traditional dynamic surface control method by introducing a finite time filter, and improves the track tracking precision of the multi-joint flexible mechanical arm system; the system redundancy and phase lag problems of the traditional extended state observer are solved, the reduced-order extended state observer is designed, the calculation load of the system is reduced, the observation precision of the observer is improved, the designed multi-joint flexible mechanical arm control system can be suitable for disturbance suppression situations of various different types, and the system has good anti-interference performance and track tracking precision.
Description
Technical Field
The invention relates to the technical field of motion control of multi-joint flexible mechanical arms, in particular to a flexible mechanical arm control method and system based on a reduced-order extended state observer.
Background
With the development of industry, robots are increasingly important in the automation field and widely used to replace workers for performing a large number of repeated single productive activities. However, the safety of the robot is still not guaranteed, and in the field of industrial production, a fence is often required to separate the working area of the robot from a human. In recent years, in the field of industry and scientific research, a flexible multi-joint mechanical arm driven by a flexible actuator has been widely focused. The flexible actuator is a driving device which connects the elastic element in series among the driving source, the reduction gearbox and the output end load, and due to the buffer function of the elastic element, the rigid coupling between the output end load and the motor is cut off, so that the robot has passive flexibility, and the requirements of flexible action, self-adjusting contact and the like of the robot can be met. In addition, the multi-joint mechanical arm driven by the compliant actuator has the advantages of high load, impact resistance, low power consumption and the like. However, the introduction of elastic elements in the compliant actuator also makes the control of the mechanical arm more complex, namely, the introduction of the first elastic element increases the system order, becomes a fourth-order high-order system, the vibration of the elastic element decreases the system stability, and the control becomes difficult. The second and multi-joint flexible mechanical arm can be affected by load moment change, uncertain mechanical friction damping, nonlinearity of the elastic element and unknown external disturbance when in operation, so that the robustness performance of the flexible mechanical arm control system is poor, and the output performance is difficult to meet the set requirement.
At present, the backstepping control method is widely applied to the multi-joint flexible mechanical arm driven by the compliant actuator due to the systematic design process and wide applicability. However, considering that the actual multi-joint flexible mechanical arm is taken as a nonlinear system, the dynamic coupling is strong, and the conventional backstepping control method is difficult to meet the requirements of tracking precision and robustness in various complex environments; in addition, the design process of the traditional backstepping control method has the problem of differential explosion, the measurement noise of the sensor is further amplified in the multiple differential operations of the backstepping control law, serious interference is generated on the final control law, and the control effect of the compliant actuator is reduced.
Therefore, how to overcome the defect caused by the "differential explosion" problem in the conventional back-step control method and the system redundancy and phase lag defect of the conventional extended state observer are the problems to be solved in the prior art.
Disclosure of Invention
Aiming at the defects existing in the prior art, the invention aims to provide a flexible mechanical arm control method and a flexible mechanical arm control system based on a reduced-order extended state observer, solves the problem of differential explosion in the traditional backstepping control method by improving a dynamic surface controller, effectively reduces the filtering error in the traditional dynamic surface control method by introducing a finite time filter, and improves the track tracking precision of a multi-joint flexible mechanical arm system; the system redundancy and phase lag problems of the traditional extended state observer are solved, the reduced-order extended state observer is designed, the calculation load of the system is reduced, the observation precision of the observer is improved, the designed multi-joint flexible mechanical arm control system can be suitable for disturbance suppression situations of various different types, and the system has good anti-interference performance and track tracking precision.
In order to achieve the above object, the present invention is realized by the following technical scheme:
the first aspect of the invention provides a flexible mechanical arm control method based on a reduced-order expanded state observer, which comprises the following steps:
Acquiring dynamic parameters of the multi-joint flexible mechanical arm, and establishing a dynamic equation of the multi-joint flexible mechanical arm driven by the flexible actuator according to the working principle of the flexible actuator;
establishing a system state equation based on a dynamics equation according to a pre-selected state variable, and reconstructing the dynamics equation of the multi-joint flexible mechanical arm into an integral serial system state equation;
designing an unknown state vector of an observation system of a full-order expanded state observer and lumped time-varying disturbance according to an integral series system state equation;
reducing the full-order extended state observer into a reduced-order extended state observer through variable substitution;
Designing a virtual control law and a dynamic surface actual control law according to the Lyapunov stability theory and a backstepping design method, thereby forming a dynamic surface controller;
and designing a composite disturbance rejection controller according to the actual output signal of the dynamic surface controller and the observation value of the lumped time-varying disturbance of the system by combining the reduced-order expanded state observer, and controlling the multi-joint flexible mechanical arm by utilizing the output moment of the composite disturbance rejection controller.
Further, the specific steps of establishing a system state equation based on the dynamics equation according to a pre-selected state variable, and reconstructing the dynamics equation of the multi-joint flexible mechanical arm into an integral serial system state equation are as follows:
the method comprises the steps of selecting a joint angle, an angular speed, a motor rotation angle and a motor rotation angular speed as state variables of a system, and establishing a state space equation of the multi-joint flexible mechanical arm by considering system parameter uncertainty of the multi-joint flexible mechanical arm and unknown disturbance of a motor and a load end;
And re-selecting joint angles, angular velocities, angular accelerations and jerks as new state variables, expanding the lumped time-varying disturbance into the new state variables, and reconstructing the state equation of the multi-joint flexible mechanical arm into a state space equation in an integral serial form.
Further, the specific steps of designing the virtual control law and the dynamic surface actual control law according to the Lyapunov stability theory and the backstepping design method are as follows:
According to the Lyapunov stability theory and a backstepping design method, combining a system tracking error to construct a Lyapunov function;
And introducing a limited time filter output signal into the Lyapunov function, and deriving the corresponding Lyapunov function to obtain a virtual control law and an actual dynamic surface control law.
Further, the virtual control law signal is filtered by a finite time filter, and a system tracking error is defined according to the system state of the reduced-order extended state observer and the output signal of the finite time filter.
Further, in combination with the estimated value of the reduced-order extended state observer on the lumped disturbance and the actual control law of the dynamic surface, the composite anti-disturbance controller is designed as follows:
，
wherein, Is the output torque of the composite disturbance rejection controller. /(I)Gain conversion matrix for control law,/>，/>Is the joint acceleration, and the tracking error vector of the joint jerk,/>To control the gain coefficient,/>Filtered derivative of the third virtual control law for a finite time filter,/>Observations of the system lumped disturbance for the reduced state observer.
The second aspect of the present invention provides a flexible mechanical arm control system based on a reduced-order extended state observer, comprising:
The parameter acquisition module is configured to acquire dynamic parameters of the multi-joint flexible mechanical arm, and establishes a dynamic equation of the multi-joint flexible mechanical arm driven by the flexible actuator according to the working principle of the flexible actuator;
The equation reconstruction module is configured to establish a system state equation according to a pre-selected state variable based on the dynamics equation, and reconstruct the dynamics equation of the multi-joint flexible mechanical arm into an integral serial system state equation;
A state observer design module configured to design a full-order extended state observer to observe a system unknown state vector and a lumped time-varying disturbance according to an integral tandem system state equation; reducing the full-order extended state observer into a reduced-order extended state observer through variable substitution;
the control law design module is configured to design a virtual control law and a dynamic surface actual control law according to the Lyapunov stability theory and a backstepping design method so as to form a dynamic surface controller;
The composite controller design module is configured to design a composite disturbance rejection controller according to the actual output signal of the dynamic surface controller and the observation value of the lumped time-varying disturbance of the system by combining the reduced-order expanded state observer, and control the multi-joint flexible mechanical arm by utilizing the output moment of the composite disturbance rejection controller.
Further, the equation reconstruction module further comprises a system state equation establishment module and a state equation reconstruction module, wherein the system state equation establishment module is configured to select a joint angle, an angular speed, a motor rotation angle and a motor rotation angular speed as state variables of the system, and consider the uncertainty of system parameters of the multi-joint flexible mechanical arm and unknown disturbance of a motor and a load end to establish a state space equation of the multi-joint flexible mechanical arm;
The state equation reconstruction module is configured to re-select joint angles, angular velocities, angular accelerations and jerks as new state variables, expand lumped time-varying disturbances into the new state variables, and reconstruct the state equation of the multi-joint flexible mechanical arm into a state space equation in an integral series form.
Further, the control law design module is further configured to:
According to the Lyapunov stability theory and a backstepping design method, combining a system tracking error to construct a Lyapunov function;
And introducing a limited time filter output signal into the Lyapunov function, and deriving the corresponding Lyapunov function to obtain a virtual control law and a dynamic surface actual control law.
Further, the virtual control law signal is filtered by a finite time filter, and a system tracking error is defined according to the system state of the reduced-order extended state observer and the output signal of the finite time filter.
Further, in combination with the estimated value of the reduced-order extended state observer on the lumped disturbance and the actual output signal of the dynamic surface controller, the composite anti-disturbance controller is designed as follows:
，
wherein, Is the output torque of the composite disturbance rejection controller. /(I)Gain conversion matrix for control law,/>，/>Is the joint acceleration, and the tracking error vector of the joint jerk,/>To control the gain coefficient,/>Filtered derivative of the third virtual control law for a finite time filter,/>Observations of the system lumped disturbance for the reduced state observer.
The one or more of the above technical solutions have the following beneficial effects:
The invention discloses a flexible mechanical arm control method and a flexible mechanical arm control system based on a reduced-order extended state observer, which belong to an improved dynamic surface control method compared with a traditional back-step control method, solve the problem of differential explosion in the traditional back-step control method, reduce the filtering error in the traditional dynamic surface control method through a finite time filter, and improve the tracking precision of a multi-joint flexible mechanical arm system.
The method disclosed by the invention uses the reduced-order expanded state observer to estimate the disturbance of the system, and improves the anti-interference capability of the system in a disturbance compensation mode. Meanwhile, compared with the traditional extended state observer, the system has the advantages that the calculation load of the system is reduced, and the observation precision of the unknown state of the system and the lumped time-varying disturbance is improved.
Additional aspects of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the invention.
FIG. 1 is a schematic diagram of a compliant actuator driven multi-joint flexible mechanical arm in accordance with a first embodiment of the present invention;
FIG. 2 is a schematic diagram of a control method of a multi-joint flexible mechanical arm according to a first embodiment of the present invention;
FIG. 3 is a schematic diagram of the joint angle (a) and tracking error (b) under the ADRC control method according to the first embodiment of the present invention;
FIG. 4 is a schematic diagram of the joint angle (a) and tracking error (b) in the improved dynamic surface control method without disturbance compensation according to the first embodiment of the present invention;
FIG. 5 is a schematic diagram of a joint angle (a) and a tracking error (b) under a control method of a multi-joint flexible mechanical arm according to a first embodiment of the present invention;
FIG. 6 is a schematic diagram of a joint angle (a) and a tracking error (b) subjected to motor end and load end disturbance in a multi-joint flexible mechanical arm control method according to a first embodiment of the present invention;
Fig. 7 is a schematic diagram of observation (a) of joint speed and observation (b) of systematic lumped time-varying disturbance by a reduced-order expansion state observer subjected to motor-end and load-end disturbance in a multi-joint flexible mechanical arm control method according to an embodiment of the present invention;
fig. 8 is a schematic diagram of filtering output of a finite time filter subjected to motor-side and load-side disturbance on a first virtual control law (a), a second virtual control law (b) and a third virtual control law (c) in the multi-joint flexible mechanical arm control method according to the first embodiment of the present invention;
FIG. 9 is a schematic diagram of a joint angle (a) and a tracking error (b) considering parameter uncertainty in a multi-joint flexible mechanical arm control method according to a first embodiment of the present invention;
FIG. 10 is a schematic diagram of an observation of joint velocity (a) and an observation of systematic lumped time-varying disturbance (b) by a reduced-order extended state observer taking parameter uncertainty into consideration in a multi-joint flexible mechanical arm control method according to an embodiment of the present invention;
Fig. 11 is a schematic diagram of filtering output of a finite time filter to a first virtual control law (a), a second virtual control law (b) and a third virtual control law (c) under consideration of parameter uncertainty in the multi-joint flexible mechanical arm control method according to the first embodiment of the present invention.
Detailed Description
It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the invention. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of exemplary embodiments according to the present invention. As used herein, the singular is also intended to include the plural unless the context clearly indicates otherwise, and furthermore, it is to be understood that the terms "comprises" and/or "comprising" when used in this specification are taken to specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof;
Embodiment one:
In order to solve the problems of differential explosion of the existing back-step control method of the multi-joint flexible mechanical arm, the reduction of the control performance of a system and the system redundancy caused by an extended state observer due to the filtering error of a dynamic surface control method, the first embodiment of the invention provides a flexible mechanical arm control method based on a reduced-order extended state observer, as shown in fig. 1, the multi-joint flexible mechanical arm based on a flexible driver, and the actual joint angle position is obtained through a sensor by considering the motor side disturbance, the vibration of an elastic element, the unknown external disturbance and the uncertainty of a controller of an end load, and the moment input to the system is calculated by the controller according to the actual joint angle position so as to obtain the expected control effect. The control method can realize the rapid tracking control of the given track; the problem of differential explosion in the traditional back-step control is solved by a dynamic surface controller; the influence of filtering errors existing in the dynamic surface control method on the accuracy of the control system is reduced by designing a finite time filter; the disturbance detector is introduced into the controller to carry out disturbance compensation so as to improve the anti-disturbance capacity of the multi-joint flexible mechanical arm system driven by the flexible actuator; by performing reduced-order processing on the extended state observer, unnecessary redundancy and phase lag of the system are reduced, the observation precision of the observer on the multi-joint flexible mechanical arm system is improved, and more accurate motion control is realized. As shown in fig. 2, for the multi-joint flexible mechanical arm system, the joint position signal in the current sampling period and the composite disturbance rejection control law signal in the last sampling period are input into the reduced-order extended state observer to generate a system state estimation and a lumped disturbance estimation signal. And inputting the virtual control law generated by the dynamic surface controller of the previous sampling period into a finite time filter to generate a virtual control law filtering output signal and a virtual control law filtering derivative signal, wherein the virtual control law filtering signal is compared with the observed state of the reduced-order extended state observer to generate a system tracking error signal. The position error signal and the tracking error signal generated by comparing the joint position with the expected position are input into the robust dynamic surface controller to generate a virtual control law signal of the next sampling period. And inputting the lumped disturbance estimated signal generated by the reduced-order expanded state observer into a robust dynamic surface controller to generate a composite disturbance rejection control law signal of the next sampling period, and inputting the composite disturbance rejection control law signal into the multi-joint flexible mechanical arm system so as to realize corresponding control.
A flexible mechanical arm control method based on a reduced-order expanded state observer comprises the following steps:
s1: and acquiring dynamic parameters of the multi-joint flexible mechanical arm, and establishing a dynamic equation of the multi-joint flexible mechanical arm driven by the flexible actuator according to the working principle of the flexible actuator.
S2: and establishing a system state equation based on the dynamics equation according to the pre-selected state variable, and reconstructing the dynamics equation of the multi-joint flexible mechanical arm into an integral serial system state equation.
S3: and designing an unknown state vector of the observation system of the full-order expanded state observer and lumped time-varying disturbance according to an integral series system state equation.
S4: the full-order extended state observer is reduced to a reduced-order extended state observer through variable substitution.
S5: and designing a virtual control law and a dynamic surface actual control law according to the Lyapunov stability theory and a backstepping design method, thereby forming the dynamic surface controller.
S6: and designing a composite disturbance rejection controller according to the actual output signal of the dynamic surface controller and the observation value of the lumped time-varying disturbance of the system by combining the reduced-order expanded state observer, and finally generating an actual control moment by utilizing the composite disturbance rejection controller to control the multi-joint flexible mechanical arm.
In the S1, based on the working principle of the flexible actuator, dynamics of a motor end and a load end are comprehensively considered, and a dynamics equation of the multi-joint flexible mechanical arm driven by the flexible actuator is established, as shown in a formula (1).
（1）。
Wherein,Is the joint angular position vector,/>For the motor output shaft angular position vector,/>And/>Moment of inertia matrix at motor end and load end respectively,/>Is a damping coefficient matrix of the motor end,/>Is a centripetal-Coriolis force matrix,/>Is the gravity torque vector of the load end,Is a joint rigidity coefficient matrix,/>For the output torque of the controller,/>And/>The external disturbance vectors of the load end and the motor end of the system are respectively received.
In this embodiment, the parameter symbol is marked with'"And"/>"Means the first and second derivatives of this parameter, respectively. /(I)And/>Respectively express/>Dimensional real vector space sum/>The space of the real matrix is maintained.
S2, establishing a system state equation based on a dynamics equation according to a pre-selected state variable, and reconstructing the dynamics equation of the multi-joint flexible mechanical arm into an integral serial system state equation, wherein the method comprises the following specific steps of:
S2.1: and selecting a joint angle, an angular speed, a motor rotation angle and a motor rotation angular speed as state variables of a system, and establishing a state space equation of the multi-joint flexible mechanical arm by considering the uncertainty of system parameters of the multi-joint flexible mechanical arm and unknown disturbance of a motor and a load end.
Specifically, the joint angle, the angular velocity, the motor rotation angle and the motor rotation angular velocity are selected as the state variables of the system, namelyThe kinetic equation of the articulated flexible mechanical arm can be initially reconstructed into equation (2).
（2）。
Wherein the method comprises the steps ofTo/>And their respective derivatives as a function of the argument. /(I)And/>Respectively represent the/>, of the joint angleThird and fourth derivatives of (3).
From the above formula, it can be obtained:
（3）。
wherein, Gain conversion matrix for control law,/>For the nominal value of the control law gain conversion matrix,/>Respectively, joint rigidity coefficient matrix/>Motor end moment of inertia matrix/>And load side moment of inertia matrix/>Is a nominal value of (2). /(I)The system comprises a system parameter uncertainty and lumped time-varying disturbance of external disturbance of a motor end and a load end.
S2.2: re-selecting joint anglesAngular velocity/>Angular acceleration/>And jerk/>As a new state variable, i.e./>And expanding the lumped time varying disturbance to a new state variableThe state space equation of the multi-joint flexible mechanical arm can be reconstructed into a state space equation (4) in an integral series type form.
（4）。
Wherein,Is the first derivative of the lumped time varying disturbance. Based on the rewritten state space equation (4), the control method provided by the embodiment only needs to measure the joint angle of the load end, so that the number of sensors is reduced, and the hardware cost is reduced.
S3, according to the integral serial system state equation of the rewritten multi-joint flexible mechanical arm, combining the measured joint angle signalsOutput moment of controller/>The full-order extended state observer is designed as shown in formula (5).
（5）。
Wherein,State variables of the extended state observer are expressed as relative joint angles/>, respectivelyAngular velocity/>Angular acceleration/>And jerk/>Lumped time varying disturbance/>Is used for the estimation of (a),Is a diagonal matrix of coefficients corresponding to the observer.
S4, reconstructing the full-order extended state observer into a reduced-order extended state observer comprises the following specific steps:
S4.1 due to the joint angle Can be directly measured by a sensor, thus the system state/>No state observer is required for estimation. To reduce unnecessary system redundancy in the reduced-order extended state observer and to reduce the phase lag problem, the extended state observer in the reduced-order form is designed as formula (6).
（6）。
Wherein the method comprises the steps ofIs a diagonal matrix of coefficients corresponding to the observer.
S4.2 the above-mentioned state observer (6) however uses in the calculation processDue to/>Is calculated by pairThe differential operation results in a new phase lag being introduced to the state observer and the noise of the sensor being amplified, thus in order to avoid/>Use of (C) to perform variable substitution/> And brings it into equation (6), cancel/>A reduced-order extended state observer (7) can be obtained.
（7）。
Wherein,For the reduction of state variables of the extended state observer (7), and the extended state observer (5) are respectively used for/>Corresponding to each other. /(I)Is a coefficient diagonal matrix corresponding to the reduced-order extended state observer (7).
In S5, the embodiment uses the filter to implement the filtering of the virtual control law according to the dynamic surface theory to solve the differential explosion problem in the traditional back-step control theory, and introduces the finite time filter to replace the first order filter in the traditional dynamic surface controller to reduce the filtering error of the filter and improve the tracking precision of the system.
Specifically, according to Lyapunov (Lyapunov) stability theory and a backstepping design method, an improved dynamic surface controller is designed, the differential explosion problem caused by continuous derivation of an original backstepping control algorithm is solved through the dynamic surface controller, and a first-order filter of the traditional dynamic surface controller is replaced by a finite-time filter, so that filtering errors caused by the filter are reduced.
In a specific embodiment, the specific steps of designing the virtual control law and the dynamic surface actual control law according to the Lyapunov stability theory and the backstepping design method are as follows:
s5.1: and constructing the Lyapunov function by combining a system tracking error according to the Lyapunov stability theory and a backstepping design method. The virtual control law signal is filtered through a finite time filter, and a system tracking error is defined according to the system state of the reduced-order extended state observer and the output signal of the finite time filter.
S5.1.1: in combination with desired joint angular position signalsJoint angular position signal/>The reduced state observer observes signals/>, for each stage state of the systemOutput signal of finite time filter/>Calculating a system tracking error:
（8）。
wherein, Pair virtual control law/>, a finite time filterIs provided.
S5.1.2: according to Lyapunov stability and a backstepping control theory, a system tracking error is combinedFinite time Filter output Signal/>By constructing Lyapunov function/>、、/>And/>。
S5.2: and introducing a limited time filter output signal into the Lyapunov function, and deriving the corresponding Lyapunov function to obtain a virtual control law and a dynamic surface actual control law.
S5.2.1: the finite time filter expression is shown in equation (9).
（9）。
Wherein,For finite time instruction filter/>State variable of/>For finite time instruction filter/>State variable of/> Is an intermediate variable of the system and has no practical meaning. /(I)Is a finite time instruction filter/>Input variables of (2) which are virtual control signals in a virtual control law given in a later step,/>For finite time instruction filter/>Coefficient of/>Is a standard sign function. In addition, finite time instruction filter/>State variable/>Is recorded as/>For inputting variablesFiltered signal,/>Is recorded as/>Can be approximated as input variable/>Time derivative of/>。
S5.2.2: the corresponding Lyapunov function is derived, and the dynamic surface controller can be obtained as follows:
（10）。
wherein, The virtual control law signals in the dynamic surface controller designed according to the backstepping control theory are respectively defined as a first virtual control law, a second virtual control law and a third virtual control law,/>To improve the actual control law signal in a dynamic surface controller,/>To improve dynamic surface controller coefficients.
And S6, according to the actual output signal of the improved dynamic surface controller, adding the observed disturbance of the reduced state observer to compensate the parameter uncertainty suffered by the multi-joint flexible mechanical arm and the disturbance from the motor end and the load end. The composite immunity controller is designed as shown in formula (11).
（11）。
Wherein,Is the output torque of the composite disturbance rejection controller. /(I)Gain conversion matrix for control law,/>，/>Is the joint acceleration, and the tracking error vector of the joint jerk. /(I)To control the gain coefficient,/>Filtered derivative of the third virtual control law for a finite time filter,/>The observed signal for the system lumped disturbance is a reduced order state observer.
The performance of the control method disclosed by the invention is verified by simulation on the two-joint flexible mechanical arm driven by the flexible actuator. First, in the absence of external disturbance, i.e.Under the condition of the invention, the superiority of the control algorithm of the invention is proved by comparing and verifying the dynamic surface control method with the traditional ADRC control method and the control method of the invention.
Fig. 3, fig. 4 and fig. 5 show graphs of results of simulation comparison of joint angle position output and tracking error of a two-joint flexible mechanical arm driven by a compliant actuator under the condition that interference is zero in a conventional control method and a control method disclosed by the invention, specifically, fig. 3 shows a schematic diagram of joint angle and tracking error under an ADRC control method, wherein (a) in fig. 3 represents a schematic diagram of joint angle under the ADRC control method, and (b) in fig. 3 represents a schematic diagram of tracking error under the ADRC control method; fig. 4 shows a schematic diagram of joint angle and tracking error under a dynamic surface control method without disturbance compensation, where (a) in fig. 4 shows a schematic diagram of joint angle under a dynamic surface control method without disturbance compensation, and (b) in fig. 4 shows a schematic diagram of tracking error under a dynamic surface control method without disturbance compensation; fig. 5 shows a schematic diagram of joint angles and tracking errors under the control method of the multi-joint flexible mechanical arm, wherein (a) in fig. 5 shows a schematic diagram of joint angles under the control method of the multi-joint flexible mechanical arm, and (b) in fig. 5 shows a schematic diagram of tracking errors under the control method of the multi-joint flexible mechanical arm; compared with the traditional control method, the control method disclosed by the invention shows the highest tracking precision, and proves the superiority of the control method disclosed by the invention.
And then, respectively applying disturbance at the motor end and the load end of the two-joint flexible mechanical arm so as to verify the inhibition capability of the control algorithm to external disturbance: applying a load-side disturbance for a period of 10 seconds to 20 seconds as，/>Applying a motor end disturbance of/>, over a period of 20 seconds to 30 seconds，/>. Wherein/>Representing joints/>Load side perturbation,/>Representing joints/>Is provided.
Fig. 6 shows a graph of joint angle output and tracking error results of two-joint flexible mechanical arms subjected to motor end and load end disturbance, wherein (a) in fig. 6 shows a schematic diagram of joint angles subjected to motor end and load end disturbance under the control method of the multi-joint flexible mechanical arm, and (b) in fig. 6 shows a schematic diagram of tracking error subjected to motor end and load end disturbance under the control method of the multi-joint flexible mechanical arm. As can be seen from the figure, compared with the case that the external disturbance is zero, the external disturbance at the motor end and the load end only causes slight reduction of tracking precision of the control method of the invention, and the superior disturbance inhibition performance of the control method of the invention on the external disturbance is proved.
Fig. 7 shows the observed values of the reduced-order extended state observer for the joint angular velocity and the lumped time-varying disturbance of the two-joint flexible mechanical arm in this case, wherein (a) in fig. 7 shows a schematic view of the reduced-order extended state observer subjected to the motor-end and the load-end disturbance of the multi-joint flexible mechanical arm control method for observing the joint velocity, and (b) in fig. 7 shows a schematic view of the reduced-order extended state observer subjected to the motor-end and the load-end disturbance of the multi-joint flexible mechanical arm control method for observing the lumped time-varying disturbance of the system. From the figure, the reduced-order expanded state observer disclosed by the invention can realize accurate observation of the joint angular velocity, and can also realize estimation of external disturbance under the condition that the flexible mechanical arm system suffers from the external disturbance.
Fig. 8 shows the filtering results of the finite time filter disclosed in the present invention for the first virtual control law, the second virtual control law and the third virtual control law, wherein (a) in fig. 8 represents the filtering output result of the first virtual control law, (b) in fig. 8 represents the filtering output result of the second virtual control law, and (c) in fig. 8 represents the filtering output result of the third virtual control law. It can be seen from the figure that the finite time filter disclosed by the invention can realize effective filtering of high-frequency noise in the virtual control law.
Finally, the rigidity coefficient of the flexible mechanical arm at two joints is assumedLoad mass/>And centroid/>The corresponding parameters were changed by 20% during 15 seconds and 30 seconds to verify the robustness of the control algorithm of the present invention to parameter uncertainty.
Fig. 9 shows a graph of joint angle output and tracking error results of two articulated flexible mechanical arms under the condition of uncertain parameters, wherein (a) in fig. 9 shows a schematic diagram of joint angles taking into consideration uncertainty of parameters in a control method of the articulated flexible mechanical arms, and (b) in fig. 9 shows a schematic diagram of tracking errors taking into consideration uncertainty of parameters in a control method of the articulated flexible mechanical arms. Compared with the condition that the parameters are not changed, the parameter change of the two-joint flexible mechanical arm has little influence on the tracking precision under the control method of the invention, and the superior resistance performance of the control method disclosed by the invention to the uncertainty of the system parameters is proved.
Fig. 10 shows a schematic diagram of the observation values of the reduced-order extended state observer for the joint angular velocity and the lumped time-varying disturbance of the two-joint flexible mechanical arm in this case, where (a) in fig. 10 represents the schematic diagram of the reduced-order extended state observer for the joint velocity taking into consideration the parameter uncertainty in the multi-joint flexible mechanical arm control method, and (b) in fig. 10 represents the schematic diagram of the reduced-order extended state observer for the system lumped time-varying disturbance taking into consideration the parameter uncertainty in the multi-joint flexible mechanical arm control method. Under the condition that system parameters are changed, the obvious change of the estimation value of the reduced-order extended state observer to the lumped time-varying disturbance can be obviously seen, meanwhile, the accurate estimation of the angular velocity of the system joint is maintained, and the effectiveness of the reduced-order extended state observer disclosed by the invention is verified.
Fig. 11 shows the filtering results of the finite time filter disclosed in the present invention for the first virtual control law, the second virtual control law and the third virtual control law, wherein (a) in fig. 11 represents the filtering output result of the first virtual control law, (b) in fig. 11 represents the filtering output result of the second virtual control law, and (c) in fig. 11 represents the filtering output result of the third virtual control law.
Based on simulation results of the two-joint flexible mechanical arm under different disturbance scenes and comparison analysis with other control methods, the control method disclosed by the invention can realize high-precision motion control of the multi-joint flexible mechanical arm driven by the flexible actuator and is suitable for various different types of disturbance inhibition conditions.
Embodiment two:
The second embodiment of the invention provides a flexible mechanical arm control system based on a reduced-order expanded state observer, which comprises the following components:
The parameter acquisition module is configured to acquire dynamic parameters of the multi-joint flexible mechanical arm, and establishes a dynamic equation of the multi-joint flexible mechanical arm driven by the compliant actuator according to the working principle of the compliant actuator.
The equation reconstruction module is configured to establish a system state equation according to a pre-selected state variable based on the dynamics equation and reconstruct the multi-joint flexible mechanical arm dynamics equation into an integral serial system state equation.
The equation reconstruction module further comprises a system state equation establishment module and a state equation reconstruction module, wherein the system state equation establishment module is configured to select a joint angle, an angular speed, a motor rotation angle and a motor rotation angular speed as state variables of the system, and consider the uncertainty of system parameters of the multi-joint flexible mechanical arm and unknown disturbance of a motor and a load end to establish a state space equation of the multi-joint flexible mechanical arm;
The state equation reconstruction module is configured to re-select joint angles, angular velocities, angular accelerations and jerks as new state variables, expand lumped time-varying disturbances into the new state variables, and reconstruct the state equation of the multi-joint flexible mechanical arm into a state space equation in an integral series form.
A state observer design module configured to design a full-order extended state observer to observe a system unknown state vector and a lumped time-varying disturbance according to an integral tandem system state equation; the full-order extended state observer is reduced to a reduced-order extended state observer through variable substitution.
The reduced-order extended state observer is configured to: and obtaining a total time-varying disturbance estimated signal of a motor end of the compliant actuator and a high-order derivative observed signal of the joint position according to the actual angle signal of the joint and the output signal of the composite disturbance rejection dynamic surface controller.
And the control law design module is configured to design a virtual control law and a dynamic surface actual control law according to the Lyapunov stability theory and a backstepping design method so as to form the dynamic surface controller.
The control law design module is further configured to:
And constructing the Lyapunov function by combining a system tracking error according to the Lyapunov stability theory and a backstepping design method. The virtual control law signal is filtered through a finite time filter, and a system tracking error is defined according to the system state of the reduced-order extended state observer and the output signal of the finite time filter. The finite time instruction filter is configured to: and obtaining a filtering signal of the virtual control signal and a first derivative signal of the filtering signal according to the virtual control signal of the dynamic surface controller.
And introducing a limited time filter output signal into the Lyapunov function, and deriving the corresponding Lyapunov function to obtain a virtual control law and a dynamic surface actual control law.
The composite controller design module is configured to design a composite disturbance rejection controller according to the actual output signal of the dynamic surface controller and the observation value of the lumped time-varying disturbance of the system by combining the reduced-order expanded state observer, and control the multi-joint flexible mechanical arm by utilizing the output moment of the composite disturbance rejection controller.
In this embodiment, the composite antijamming dynamic surface controller is configured to: and according to the expected reference angle signal of the joint, the actual angle signal of the joint, the lumped time-varying disturbance estimation signal of the motor end of the actuator, the filtering signal of the virtual control law and the first derivative signal of the filtering signal, combining the backstepping control method and Lyapunvo stability theory, obtaining the virtual control signal of the virtual control law and the actual output control moment signal.
The composite anti-interference controller is designed as follows:
（11）。
wherein, Is the output torque of the composite disturbance rejection controller. /(I)Gain conversion matrix for control law,/>，/>Is the joint acceleration, and the tracking error vector of the joint jerk. /(I)To control the gain coefficient,/>Filtered derivative of the third virtual control law for a finite time filter,/>The observed signal for the system lumped disturbance is a reduced order state observer. /(I)
The steps involved in the second embodiment correspond to those of the first embodiment of the method, and the detailed description of the second embodiment can be found in the related description section of the first embodiment.
It will be appreciated by those skilled in the art that the modules or steps of the invention described above may be implemented by general-purpose computer means, alternatively they may be implemented by program code executable by computing means, whereby they may be stored in storage means for execution by computing means, or they may be made into individual integrated circuit modules separately, or a plurality of modules or steps in them may be made into a single integrated circuit module. The present invention is not limited to any specific combination of hardware and software.
While the foregoing description of the embodiments of the present invention has been presented in conjunction with the drawings, it should be understood that it is not intended to limit the scope of the invention, but rather, it is intended to cover all modifications or variations within the scope of the invention as defined by the claims of the present invention.
Claims (6)
1. The flexible mechanical arm control method based on the reduced-order expanded state observer is characterized by comprising the following steps of:
Acquiring dynamic parameters of the multi-joint flexible mechanical arm, and establishing a dynamic equation of the multi-joint flexible mechanical arm driven by the flexible actuator according to the working principle of the flexible actuator;
establishing a system state equation based on a dynamics equation according to a pre-selected state variable, and reconstructing the dynamics equation of the multi-joint flexible mechanical arm into an integral serial system state equation;
designing an unknown state vector of an observation system of a full-order expanded state observer and lumped time-varying disturbance according to an integral series system state equation;
reducing the full-order extended state observer into a reduced-order extended state observer through variable substitution;
Designing a virtual control law and a dynamic surface actual control law according to the Lyapunov stability theory and a backstepping design method, thereby forming a dynamic surface controller;
Designing a composite disturbance rejection controller according to an actual output signal of the dynamic surface controller and an observation value of the lumped time-varying disturbance of the system by combining a reduced-order expansion state observer, and controlling the multi-joint flexible mechanical arm by utilizing an output moment of the composite disturbance rejection controller;
The method comprises the specific steps of establishing a system state equation based on a dynamics equation according to a pre-selected state variable, and reconstructing the dynamics equation of the multi-joint flexible mechanical arm into an integral serial system state equation, wherein the specific steps are as follows:
Selecting a joint angle, an angular velocity, a motor rotation angle and a motor rotation angular velocity as state variables of a system, and reconstructing a dynamic equation of the multi-joint flexible mechanical arm into a state space equation by considering the uncertainty of system parameters of the multi-joint flexible mechanical arm and unknown disturbance of a motor end and a load end;
re-selecting joint angles, angular velocities, angular accelerations and jerks as new state variables, expanding lumped time-varying disturbance into new state variables, and reconstructing a state equation of the multi-joint flexible mechanical arm into an integral serial system state space equation;
the specific steps for designing the virtual control law and the dynamic surface actual control law according to the Lyapunov stability theory and the backstepping design method are as follows:
According to the Lyapunov stability theory and a backstepping design method, combining a system tracking error to construct a Lyapunov function;
Introducing a limited time filter output signal into the Lyapunov function, and obtaining a virtual control law and a dynamic plane actual control law by deriving the corresponding Lyapunov function;
Filtering the virtual control law signal through a finite time filter, and defining a system tracking error according to the system state of the reduced-order extended state observer and the output signal of the finite time filter;
the composite anti-interference controller is designed as follows: ，
wherein, Output moment of composite disturbance rejection controller,/>Gain conversion matrix for control law,/>，/>Is the joint acceleration, and the tracking error vector of the joint jerk,/>To control the gain coefficient,/>Filtered derivative of the third virtual control law for a finite time filter,/>The observed signal for the system lumped disturbance is a reduced order state observer.
2. A flexible robotic arm control system based on the reduced order extended state observer of the method of claim 1, comprising:
The parameter acquisition module is configured to acquire dynamic parameters of the multi-joint flexible mechanical arm, and establishes a dynamic equation of the multi-joint flexible mechanical arm driven by the flexible actuator according to the working principle of the flexible actuator;
The equation reconstruction module is configured to establish a system state equation according to a pre-selected state variable based on the dynamics equation, and reconstruct the dynamics equation of the multi-joint flexible mechanical arm into an integral serial system state equation;
A state observer design module configured to design a full-order extended state observer to observe a system unknown state vector and a lumped time-varying disturbance according to an integral tandem system state equation; reducing the full-order extended state observer into a reduced-order extended state observer through variable substitution;
the control law design module is configured to design a virtual control law and a dynamic surface actual control law according to the Lyapunov stability theory and a backstepping design method so as to form a dynamic surface controller;
The composite controller design module is configured to design a composite disturbance rejection controller according to the actual output signal of the dynamic surface controller and the observation value of the lumped time-varying disturbance of the system by combining the reduced-order expanded state observer, and control the multi-joint flexible mechanical arm by utilizing the output moment of the composite disturbance rejection controller.
3. The flexible mechanical arm control system based on the reduced-order extended state observer according to claim 2, wherein the equation reconstruction module further comprises a system state equation establishment module and a state equation reconstruction module, wherein the system state equation establishment module is configured to select a joint angle, an angular velocity, a motor rotation angle and a motor rotation angular velocity as state variables of the system, and to establish a state space equation of the multi-joint flexible mechanical arm in consideration of system parameter uncertainty of the multi-joint flexible mechanical arm and unknown disturbance of the motor and the load end;
The state equation reconstruction module is configured to re-select joint angles, angular velocities, angular accelerations and jerks as new state variables, expand lumped time-varying disturbances into the new state variables, and reconstruct the state equation of the multi-joint flexible mechanical arm into a state space equation in an integral series form.
4. The reduced order extended state observer based flexible robotic arm control system according to claim 2, wherein the control law design module is further configured to:
According to the Lyapunov stability theory and a backstepping design method, combining a system tracking error to construct a Lyapunov function;
And introducing a limited time filter output signal into the Lyapunov function, and deriving the corresponding Lyapunov function to obtain a virtual control law and a dynamic surface actual control law.
5. The reduced-order extended state observer based flexible mechanical arm control system according to claim 4, wherein the virtual control law signal is filtered by a finite time filter, and a system tracking error is defined according to a system state of the reduced-order extended state observer and an output signal of the finite time filter.
6. The reduced order extended state observer based flexible robotic arm control system according to claim 2, wherein the composite immunity controller is designed to:
，
wherein, Output moment of composite disturbance rejection controller,/>Gain conversion matrix for control law,/>，/>Is the joint acceleration, and the tracking error vector of the joint jerk,/>To control the gain coefficient,/>Filtered derivative of the third virtual control law for a finite time filter,/>The observed signal for the system lumped disturbance is a reduced order state observer.
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