CN114047773B - Underwater ore collection robot backstepping sliding mode self-adaptive attitude control method based on extended state observer - Google Patents
Underwater ore collection robot backstepping sliding mode self-adaptive attitude control method based on extended state observer Download PDFInfo
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Abstract
The invention belongs to the field of robots, and discloses a back-stepping sliding mode self-adaptive attitude control method of an underwater ore collecting robot based on an extended state observer, which solves the problem of external interference of the underwater ore collecting robot in actual attitude movement and improves the accurate attitude control effect under the condition that the dynamic model of the underwater ore collecting robot is uncertain. Combining with a dynamics model, designing an extended state observer to estimate the composite disturbance and realizing feedforward compensation; the robustness of the system is enhanced by utilizing a back-step design method and combining a sliding mode control method, and meanwhile, an integration link is introduced, so that the steady-state error of the system is reduced and the rapidity of the system is enhanced; designing an adaptive switching gain to reduce buffeting in the system; the control stability of the control system is verified through Lyapunov stability criteria. The invention overcomes the external interference and the unmodeled uncertainty of the dynamic model, realizes the accurate and rapid tracking of the gesture, and ensures that the control system has stronger stability and robustness.
Description
Technical Field
The invention relates to an attitude control method of an underwater ore collecting robot, in particular to a back-stepping sliding mode self-adaptive attitude control method of the underwater ore collecting robot based on an extended state observer.
Background
Mineral resources are an important material foundation for the development of human society, and as the development of marine mineral resources by human beings is more and more advanced, marine development equipment and technology have become hot spots for competitive research in various countries. The underwater ore collecting robot is used as an important component unit of a deep sea mining system, can be used for underwater operations such as underwater mineral sampling, port dredging, marine pipeline maintenance, deep sea accident investigation and the like after being provided with special tools besides being used for ore collecting operation, and has relatively controllable high-strength and multi-form operation and observation performance. The space attitude control of the underwater ore collecting robot is an important component of the bottom layer motion control, and along with the expansion of the application range of the underwater ore collecting robot, higher requirements are put on the aspects of the accuracy, the reliability and the response rapidness and the intellectualization of the attitude control of the underwater ore collecting robot. The complex water flow environment enables the motion of the underwater ore collecting robot to have the characteristics of strong nonlinearity, coupling, time variability and the like, so that the research on the motion gesture control method of the underwater ore collecting robot enables the control system to have strong robustness so as to overcome the external interference and the unmodeled uncertainty of the dynamic model, has strong self-adjusting capacity and reliable working capacity, and has very important practical significance.
The current motion control method for the underwater mine collecting robot mainly relates to sliding mode variable structure control, fuzzy control, self-adaptive control, backstepping control, active disturbance rejection control, PID control and the like. In engineering practice, in order to achieve a better control effect, a control strategy combining two or more control methods is often adopted, such as fuzzy sliding mode variable structure control. The sliding mode control has the advantages that the uncertainty of the system can be overcome, the sliding mode is irrelevant to the external interference and parameter change of the system, and therefore the robustness of the sliding mode variable structure control system is stronger than that of a common continuous system. However, when the sliding mode controller is used for controlling the underwater ore collecting robot, the buffeting problem occurs because a larger switching gain is required to overcome larger system disturbance. The extended state observer (Extended State Observer, ESO) can estimate the state and aggregate interference of the system by using little model information, and the adopted high gain error feedback ensures that the dynamic state of the observer is far higher than that of the system, so that the rapid convergence of the observation error and enough estimation precision can be ensured, the total internal and external disturbance of the system can be estimated in real time, and the control quantity can be compensated. The back-step control divides the nonlinear system into a plurality of subsystems, then constructs corresponding Lyapunov functions and virtual control amounts for each decomposed subsystem in sequence, and finally recursively completes the construction of the Lyapunov functions of the whole research target system according to the sequence, thereby guaranteeing the stability and dynamic performance of the whole system, but the back-step method has higher requirements on an accurate mathematical model, so the size of modeling errors can influence the tracking effect of the system.
Disclosure of Invention
The invention aims at: the invention provides a back-stepping sliding mode self-adaptive attitude control method of an underwater ore collecting robot based on an extended state observer, which aims to realize real-time estimation of total disturbance inside and outside a system and compensation of control quantity, and simultaneously solve the buffeting problem when sliding mode control is adopted, so that the underwater ore collecting robot can track the expected track of an upper attitude angle, realize stable control of the attitude of the underwater ore collecting robot and ensure that a control system has stronger robustness.
The invention solves the technical problems by adopting the technical scheme that:
an underwater ore collecting robot backstepping sliding mode self-adaptive attitude control method based on an extended state observer is characterized by comprising the following steps of: the control method comprises the following steps:
step 1, considering a six-degree-of-freedom dynamics model of an underwater ore collecting robot, and establishing a trim, trim and heading attitude motion equation by decoupling, simplifying the model and considering the influence of a coupling item, a model uncertainty item of the system and external interference on the system;
step 2, establishing a state space equation by utilizing the information in the step 1, and designing an extended state observer of the underwater robot;
step 3, determining that a control target is convergence of tracking error, constructing a Lyapunov function based on a back-stepping method, designing a virtual control variable, simultaneously introducing an integration link to reduce steady-state error possibly brought by the system, and enhancing the rapidity of the system;
step 4, estimating composite disturbance by using the extended state observer in the step 2, designing a sliding mode surface by using the information in the step 3, completing the design of control rate by constructing a Lyapunov function, and designing self-adaptive switching gain to reduce buffeting phenomenon in the system;
and step 5, verifying that the control system can quickly converge and reach stability by constructing a Lyapunov function containing a backstepping design error, a sliding mode function and a self-adaptive estimation error.
The invention also includes the following features:
1. the step 1 specifically comprises the following steps:
consider an underwater ore collection robot six-degree-of-freedom dynamics model:
where M is a generalized inertial matrix,is the position and attitude angle of the underwater ore collecting robot under the geodetic coordinate system, v= [ u v w p q r ]] T For the velocity vector of the underwater ore collecting robot under the motion coordinate system, < >>Model uncertainty term and external disturbance representing system, C (v) is a Kelvin matrix, D9v is a fluid damping matrix, g (eta) is a restoring matrix composed of gravity and buoyancy, J (eta) 2 ) Is a coordinate transformation matrix, τ c ＝[τ X τ Y τ Z τ K τ M τ N ] T Is a control force and control moment vector;
decoupling and simplifying the formula (1), comprehensively considering the influence of coupling relation, model parameter uncertainty and external interference on a system, and establishing a trim, trim and heading attitude motion equation:
wherein the method comprises the steps ofIs a transverse inclination angle, theta is a longitudinal inclination angle, phi is a heading angle, I x 、I y 、I z The moment of inertia of the X axis, the Y axis and the Z axis of the underwater ore collecting robot is>Is inertial waterCoefficient of dynamic (x) B ,y B ,z B ) Is the coordinates of the floating center in the motion coordinate system, K p 、M q 、N r 、K p|p| 、M q|q| 、N r|r| The first-order water damping coefficient and the second-order water damping coefficient of the transverse, longitudinal and fore motions of the underwater ore collecting robot are shown, B is the buoyancy born by the underwater ore collecting robot, and d' K 、d′ M 、d′ N Model uncertainty terms and external disturbances of the yaw, pitch and heading movements, respectively.
2. The step 2 specifically comprises the following steps:
defining a state variable:
x＝[x 1 x 2 ] T (3)
wherein the method comprises the steps ofx 2 ＝[p q r] T Establishing a state space equation according to the formula (2) and the formula (3):
wherein u= [ τ ] K τ M τ N ] T For controllable input of the system, y E R 3×1 For the controllable output of the system, A is the linear term coefficient matrix of the system,b is a positive constant matrix, +.>d F ＝[d K d M d N ] T Composite disturbances (including coupled terms, model uncertainty terms of the system, and external disturbances) representing the motion components, whose derivatives are present and bounded;
definition z= [ z 1 z 2 ] T ＝[x 1 x 2 ] T The mathematical model of the expanded formula (4) is:
wherein z is 3 Is d F Expanding into a new state quantity, wherein h (t) is a bounded uncertainty function;
designing a second order extended state observer according to the system described by equation (5):
wherein the method comprises the steps ofIs the state observation vector, beta 1 、β 2 、β 3 The following conditions are satisfied by adopting a bandwidth-based configuration method for the gain parameters and the normal number diagonal matrix:
β＝[β 1 β 2 β 3 ] T ＝[3w 0 I 3 3w 0 2 I 3 w 0 3 I 3 ] T (7)
in formula (7) I 3 Is a unitary matrix, w 0 Is bandwidth and w 0 > 0, when beta 1 、β 2 、β 3 When properly set, there areThrough the extended state observer, the composite disturbance d of each motion component of the underwater ore collecting robot can be estimated F And then compensates in real time in the control law.
3. The step 3 specifically comprises the following steps:
defining a system attitude angle tracking error as follows:
e 1 ＝x 1 -x d (8)
wherein the method comprises the steps ofRepresenting attitude angle tracking error, +.>Representing the desired attitude angle, wherein->x θd 、x ψd Respectively representing an expected transverse inclination angle, an expected longitudinal inclination angle and an expected heading angle;
deriving formula (8):
introducing a first order tracking error integral to reduce steady state error:
introducing a virtual control quantity alpha, and defining an error variable:
e 2 ＝x 2 -α (11)
wherein the method comprises the steps of
Construction of Lyapunov function V 1 ：
Deriving formula (12):
designing a virtual control quantity alpha:
wherein the method comprises the steps ofAre normal number diagonal matrix;
the compounds are obtained by the formulas (9) to (14):
from the formulas (14) and (15):
4. the step 4 specifically comprises the following steps:
introducing a sliding mode control item, and defining a sliding mode surface function:
s＝ke 1 +e 2 (17)
wherein the method comprises the steps ofIs a normal number diagonal matrix;
is obtained by the following formulas (16) and (17):
since k+c > 0, if s=0, e 1 ＝0，e 2 =0 andtherefore, the system needs to be designed in the next step;
construction of Lyapunov function V 2 ：
Deriving formula (19):
to ensure thatDesigning a control law u:
wherein the method comprises the steps ofFor the observer to observe the estimated value of the complex interference, +.>In order to observe errors and to be bounded, for example a transverse channel,get->An upper bound of delta, then->And->For normal number diagonal matrix->
In order to reduce buffeting problem in the sliding mode control process, the adaptive index approach rate is designed:
wherein the method comprises the steps ofFor the estimate of delta under the adaptive approach law,/->Then the estimation error of delta ∈>Lambda is the normal number diagonal matrix,>assuming that the observed error varies slowly +.>
The control law u is obtained from the formulas (21) and (22):
5. the step 5 specifically comprises the following steps:
ensuring the stability of the system under the self-adaptive law, and constructing a Lyapunov function:
deriving formula (24):
matrix in (25)The method comprises the following steps:
obtained by the formula (26):
wherein the method comprises the steps ofSelect->Such that:
thereby ensuring E θ The positive rules are:
the same applies to formulas (26) to (29):
is obtained by the following formulas (25), (29) and (30)Meets the consistent gradually-tending characteristics;
according to LaSalle invariance theory, takeWhen (I)>e θ ≡0，e ψ When t.ident.0, s.ident.0, t.fwdarw.infinity, e 1 →0，e 2 0, s.fwdarw.0, x 1 →x d ，/>The attitude closed-loop control system converges, and stable tracking of an attitude angle and an attitude angle rate can be realized.
The beneficial effects of the invention are as follows: the back-stepping method is combined with sliding mode control, so that the robustness of the system is enhanced, meanwhile, an integration link is introduced, steady-state errors possibly brought by the system are reduced, and the rapidity of the system is ensured; reducing buffeting in the system through an adaptive switching law; and designing an extended state observer of the underwater ore collecting robot to observe and estimate the composite disturbance and realize feedforward compensation. The underwater ore collecting robot can realize stable control of the gesture under the conditions of time-varying interference and uncertain system models, and compared with the traditional sliding mode control, the controller has high tracking precision and better robustness.
Drawings
Fig. 1 is a schematic diagram of a control flow of the underwater mine collection robot of the present invention.
Fig. 2 is a diagram of an attitude angle tracking effect of the underwater mine collection robot of the present invention.
Fig. 3 is a diagram showing the effect of tracking the angular velocity of the underwater mine collection robot.
Fig. 4 is a schematic diagram of an observation value of the composite interference by the extended state observer of the underwater mine collecting robot.
Fig. 5 is a graph showing a variation of a control moment u of the underwater mining robot according to the present invention.
FIG. 6 is a graph showing the variation of the s function of the sliding mode surface of the underwater mining robot.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1, the method for controlling the back-stepping sliding mode self-adaptive posture of the underwater ore collecting robot based on the extended state observer is realized by the following steps:
step 1, considering a six-degree-of-freedom dynamics model of an underwater ore collecting robot, and establishing a trim, trim and heading attitude motion equation by decoupling, simplifying the model and considering the influence of a coupling item, a model uncertainty item of the system and external interference on the system;
step 2, establishing a state space equation by utilizing the information in the step 1, and designing an extended state observer of the underwater ore collecting robot;
step 3, determining that a control target is convergence of tracking error, constructing a Lyapunov function based on a back-stepping method, designing a virtual control variable, simultaneously introducing an integration link to reduce steady-state error possibly brought by the system, and enhancing the rapidity of the system;
step 4, estimating composite disturbance by using the extended state observer in the step 2, designing a sliding mode surface by using the information in the step 3, completing the design of control rate by constructing a Lyapunov function, and designing self-adaptive switching gain to reduce buffeting phenomenon in the system;
and step 5, verifying that the control system can quickly converge and reach stability by constructing a Lyapunov function containing a backstepping design error, a sliding mode function and a self-adaptive estimation error.
Specifically, the method for controlling the back-stepping sliding mode self-adaptive posture of the underwater ore collecting robot based on the extended state observer is realized through the following steps:
step 1, considering a six-degree-of-freedom dynamics model of the underwater ore collecting robot:
where M is a generalized inertial matrix,is the position and attitude angle of the underwater ore collecting robot under the geodetic coordinate system, v= [ u v w p q r ]] T For the velocity vector of the underwater ore collecting robot under the motion coordinate system, < >>Model uncertainty term and external disturbance of the representation system, C (v) is a Kelvin matrix, D (v) is a fluid damping matrix, g (eta) is a restoring force matrix composed of gravity and buoyancy, J (eta) 2 ) Is thatCoordinate transformation matrix, τ c ＝[τ X τ Y τ Z τ K τ M τ N ] T Is a control force and control moment vector;
decoupling and simplifying the formula (1), comprehensively considering the influence of coupling relation, model parameter uncertainty and external interference on a system, and establishing a trim, trim and heading attitude motion equation:
wherein the method comprises the steps ofIs a transverse inclination angle, theta is a longitudinal inclination angle, phi is a heading angle, I x 、I y 、I z The moment of inertia of the X axis, the Y axis and the Z axis of the underwater ore collecting robot is>Is the coefficient of inertial hydrodynamic force, (x) B ,y B ,z B ) Is the coordinates of the floating center in the motion coordinate system, K p 、M q 、N r 、K p|p| 、M q|q| 、N r|r| The first-order water damping coefficient and the second-order water damping coefficient of the transverse, longitudinal and fore motions of the underwater ore collecting robot are shown, B is the buoyancy born by the underwater ore collecting robot, and d' K 、d′ M 、d′ N Model uncertainty terms and external disturbances of the yaw, pitch and heading movements, respectively.
Step 2, firstly defining state variables:
x＝[x 1 x 2 ] T (3)
wherein the method comprises the steps ofx 2 ＝[p q r] T Establishing a state space equation according to the formula (2) and the formula (3):
wherein u= [ τ ] K τ M τ N ] T For controllable input of the system, y E R 3×1 For the controllable output of the system, A is the linear term coefficient matrix of the system,b is a positive constant matrix, +.>d F ＝[d K d M d N ] T Composite disturbances (including coupled terms, model uncertainty terms of the system, and external disturbances) representing the motion components, whose derivatives are present and bounded;
definition z= [ z 1 z 2 ] T ＝[x 1 x 2 ] T The mathematical model of the expanded formula (4) is:
wherein z is 3 Is d F Expanding into a new state quantity, wherein h (t) is a bounded uncertainty function;
designing a second order extended state observer according to the system described by equation (5):
wherein the method comprises the steps ofIs the state observation vector, beta 1 、β 2 、β 3 The following conditions are satisfied by adopting a bandwidth-based configuration method for the gain parameters and the normal number diagonal matrix:
β＝[β 1 β 2 β 3 ] T ＝[3w 0 I 3 3w 0 2 I 3 w 0 3 I 3 ] T (7)
in formula (7) I 3 Is a unitary matrix, w 0 Is bandwidth and w 0 > 0, when beta 1 、β 2 、β 3 When properly set, there areThrough the extended state observer, the composite disturbance d of each motion component of the underwater ore collecting robot can be estimated F And then compensates in real time in the control law.
Step 3, firstly defining a system attitude angle tracking error as follows:
e 1 ＝x 1 -x d (8)
wherein the method comprises the steps ofRepresenting attitude angle tracking error, +.>Representing the desired attitude angle, wherein->x θd 、x ψd Respectively representing an expected transverse inclination angle, an expected longitudinal inclination angle and an expected heading angle;
deriving formula (8):
introducing a first order tracking error integral to reduce steady state error:
introducing a virtual control quantity alpha, and defining an error variable:
e 2 ＝x 2 -α (11)
wherein the method comprises the steps of
Construction of Lyapunov function V 1 ：
Deriving formula (12):
designing a virtual control quantity alpha:
wherein the method comprises the steps ofAre normal number diagonal matrix;
the compounds are obtained by the formulas (9) to (14):
from the formulas (14) and (15):
step 4, introducing a sliding mode control item, and defining a sliding mode surface function:
s＝ke 1 +e 2 (17)
wherein the method comprises the steps ofIs a normal number diagonal matrix;
is obtained by the following formulas (16) and (17):
since k+c > 0, if s=0, e 1 ＝0，e 2 =0 andtherefore, the system needs to be designed in the next step;
construction of Lyapunov function V 2 ：
Deriving formula (19):
to ensure thatDesigning a control law u:
wherein the method comprises the steps ofFor the observer to observe the estimated value of the complex interference, +.>In order to observe errors and to be bounded, for example a transverse channel,get->An upper bound of delta, then->And->For normal number diagonal matrix->
In order to reduce buffeting problem in the sliding mode control process, the adaptive index approach rate is designed:
wherein the method comprises the steps ofFor the estimate of delta under the adaptive approach law,/->Then the estimation error of delta ∈>Lambda is the normal number diagonal matrix,>assuming that the observed error varies slowly +.>
The control law u is obtained from the formulas (21) and (22):
step 5, ensuring system stability under the self-adaptive law, and constructing a Lyapunov function:
deriving formula (24):
matrix in (25)The method comprises the following steps:
obtained by the formula (26):
wherein the method comprises the steps ofSelect->Such that:
thereby ensuring E θ The positive rules are:
the same applies to formulas (26) to (29):
is obtained by the following formulas (25), (29) and (30)Meets the consistent gradually-tending characteristics;
according to LaSalle invariance theory, takeWhen (I)>e θ ≡0，e ψ When t.ident.0, s.ident.0, t.fwdarw.infinity, e 1 →0，e 2 0, s.fwdarw.0, x 1 →x d ，/>The attitude closed-loop control system converges, and stable tracking of an attitude angle and an attitude angle rate can be realized. />
In order to verify that the designed back-stepping sliding mode self-adaptive attitude controller of the underwater ore collecting robot based on the extended state observer can realize effective control of the attitude of the underwater ore collecting robot under the conditions of model uncertainty and external interference, the following simulation tasks are designed:
the parameters were given as follows:
setting an initial attitude angle x of an underwater ore collecting robot in static state 0 ＝[0.5 0.5 0.5] T The attitude angle tracking instruction is: x is x d ＝[sint cost sint] T The composite disturbance moment is as follows:
d η ＝[-2×sin(0.5t) 1.5×sin(0.5t) 1.5×cos(0.5t)] T 。
robot model parameters: the moment of inertia matrix is:the inertial hydrodynamic coefficients are: />Water damping coefficient: k (K) p ＝-12.76,M q ＝2.38,N r ＝3.40,K p|p| ＝38.69,M q|q| ＝37.59,N r|r| ＝27.71。
System control parameters: w (w) 0 ＝100，k＝diag{10,10,10}，c＝diag{18,15,25}，n＝diag{8,8,8}，ε＝diag{10,18,12}，λ＝diag{0.5,0.5,0.5}。
The simulation results obtained on the MATLAB platform by using the control method are shown in figures 2-6.
Fig. 2 is a diagram of an attitude angle tracking effect of the underwater mine collection robot of the present invention. The transverse inclination angle convergence time is 0.5s, the longitudinal inclination angle convergence time is 0.4s, the heading angle convergence time is 0.4s, and then each component enters a steady-state follow-up stage, as shown in fig. 2, and the system can track the expected value.
Fig. 3 is a diagram showing the effect of tracking the angular velocity of the underwater mine collection robot. The transverse inclination angle speed convergence time is 0.6s, the longitudinal inclination angle convergence time is 0.5s, the heading angle speed convergence time is 0.6s, and then all components enter a steady-state follow-up stage, as shown in fig. 3, and the system can track the expected value.
Fig. 4 is a schematic diagram of an observation value of the composite interference by the extended state observer of the underwater mine collecting robot. The designed extended observer realizes the observation and estimation of the transverse disturbance moment, the longitudinal disturbance moment and the heading disturbance moment in 1.1s, 1.0s and 1.1s respectively, and the observation errors are all within +/-0.2 N.m after entering a stable stage, as shown in figure 4, the extended state observer can rapidly and accurately estimate the composite disturbance and compensate the system, and has good compensation effect.
Fig. 5 is a graph showing a variation of a control moment u of the underwater mining robot according to the present invention. The output quantity of the controller is larger in the initial adjustment stage, the maximum moment reaches 27 N.m, the maximum moment is stable, the output of the controller reaches the stable stage within the range of +/-15 N.m, and the control change can be within a reasonable range by modifying corresponding control options, limiting or smoothing filtering in the actual application.
FIG. 6 is a graph showing the variation of the s function of the sliding mode surface of the underwater mining robot. The convergence time of the trim s function is 1.5s, the convergence time of the trim s function is 1.2s, and the convergence time of the heading s function is 1.2s, as shown in fig. 6, the sliding mode surface convergence condition of the system is good.
From the analysis, the underwater ore collecting robot has good dynamic characteristics, ESO can rapidly and accurately estimate composite disturbance, the sliding mode surface convergence condition is good, the attitude angle and the angular rate can rapidly track an upper expected track, and the output of the controller is stable.
In summary, the method for controlling the self-adaptive gesture of the back-stepping sliding mode of the underwater ore collecting robot based on the extended state observer can realize accurate and rapid tracking of the gesture under the condition that the dynamic model of the underwater ore collecting robot is uncertain and is subject to external interference, and enables the control system to have stronger stability and robustness.
In addition to the above embodiments, the present invention may be embodied in other forms without departing from the spirit or essential characteristics thereof, and the scope of the present invention is defined in the appended claims.
Claims (5)
1. An underwater ore collecting robot backstepping sliding mode self-adaptive attitude control method based on an extended state observer is characterized by comprising the following steps of: the control method comprises the following steps:
step 1, considering a six-degree-of-freedom dynamics model of an underwater ore collecting robot, and establishing a trim, trim and heading attitude motion equation by decoupling, simplifying the model and considering the influence of a coupling item, a model uncertainty item of the system and external interference on the system;
step 2, establishing a state space equation by utilizing the information in the step 1, and designing an extended state observer of the underwater robot;
step 3, determining that a control target is convergence of tracking error, constructing a Lyapunov function based on a back-stepping method, designing a virtual control variable, simultaneously introducing an integration link to reduce steady-state error possibly brought by the system, and enhancing the rapidity of the system;
step 4, estimating composite disturbance by using the extended state observer in the step 2, designing a sliding mode surface by using the information in the step 3, completing the design of control rate by constructing a Lyapunov function, and designing self-adaptive switching gain to reduce buffeting phenomenon in the system;
step 5, verifying that the control method can be quickly converged and stable by constructing a Lyapunov function containing a backstepping design error, a sliding mode function and a self-adaptive estimation error;
the step 1 specifically comprises the following steps:
consider an underwater ore collection robot six-degree-of-freedom dynamics model:
where M is a generalized inertial matrix,is the position and attitude angle of the underwater ore collecting robot under the geodetic coordinate system, v= [ u v w p q r ]] T For the velocity vector of the underwater ore collecting robot under the motion coordinate system, < >>Model uncertainty term and external disturbance of the representation system, C (v) is a Kelvin matrix, D (v) is a fluid damping matrix, g (eta) is a restoring force matrix composed of gravity and buoyancy, J (eta) 2 ) Is a coordinate transformation matrix, τ c ＝[τ X τ Y τ Z τ K τ M τ N ] T Is a control force and control moment vector;
decoupling and simplifying the formula (1), comprehensively considering the influence of coupling relation, model parameter uncertainty and external interference on a system, and establishing a trim, trim and heading attitude motion equation:
wherein the method comprises the steps ofIs a transverse inclination angle, theta is a longitudinal inclination angle, phi is a heading angle, I x 、I y 、I z The moment of inertia of the X axis, the Y axis and the Z axis of the underwater ore collecting robot is>Is the coefficient of inertial hydrodynamic force, (x) B ,y B ,z B ) Is the coordinates of the floating center in the motion coordinate system, K p 、M q 、N r 、K p|p| 、M q|q| 、N r|r| The first-order water damping coefficient and the second-order water damping coefficient of the transverse, longitudinal and fore motions of the underwater ore collecting robot are shown, B is the buoyancy born by the underwater ore collecting robot, and d' K 、d′ M 、d′ N Model uncertainty terms and external disturbances of the yaw, pitch and heading movements, respectively.
2. The method for controlling the back-stepping sliding mode self-adaptive posture of the underwater ore collection robot based on the extended state observer, which is characterized by comprising the following steps of: the step 2 specifically comprises the following steps:
defining a state variable:
x＝[x 1 x 2 ] T (3)
wherein the method comprises the steps ofx 2 ＝[p q r] T Establishing a state space equation according to the formula (2) and the formula (3):
wherein u= [ τ ] K τ M τ N ] T For controllable input of the system, y E R 3×1 For the controllable output of the system, A is the linear term coefficient matrix of the system,b isPositive constant matrix, < >>d F ＝[d K d M d N ] T A composite disturbance representing each motion component, the derivative of which exists and is bounded;
definition z= [ z 1 z 2 ] T ＝[x 1 x 2 ] T The mathematical model of the expanded formula (4) is:
wherein z is 3 Is d F Expanding into a new state quantity, wherein h (t) is a bounded uncertainty function;
designing a second order extended state observer according to the system described by equation (5):
wherein the method comprises the steps ofIs the state observation vector, beta 1 、β 2 、β 3 The following conditions are satisfied by adopting a bandwidth-based configuration method for the gain parameters and the normal number diagonal matrix:
β＝[β 1 β 2 β 3 ] T ＝[3w 0 I 3 3w 0 2 I 3 w 0 3 I 3 ] T (7)
in formula (7) I 3 Is a unitary matrix, w 0 Is bandwidth and w 0 > 0, when beta 1 、β 2 、β 3 When properly set, there areBy the above extended state observer, the underwater can be estimatedComposite disturbance d of each motion component of ore collecting robot F And then compensates in real time in the control law.
3. The method for controlling the back-stepping sliding mode self-adaptive posture of the underwater ore collection robot based on the extended state observer, which is characterized by comprising the following steps of: the step 3 specifically comprises the following steps:
defining a system attitude angle tracking error as follows:
e 1 ＝x 1 -x d (8)
wherein the method comprises the steps ofRepresenting attitude angle tracking error, +.>Representing the desired attitude angle, wherein->x θd 、x ψd Respectively representing an expected transverse inclination angle, an expected longitudinal inclination angle and an expected heading angle;
deriving formula (8):
introducing a first order tracking error integral to reduce steady state error:
introducing a virtual control quantity alpha, and defining an error variable:
e 2 ＝x 2 -α (11)
wherein the method comprises the steps of
Construction of Lyapunov function V 1 ：
Deriving formula (12):
designing a virtual control quantity alpha:
wherein the method comprises the steps ofAre normal number diagonal matrix;
the compounds are obtained by the formulas (9) to (14):
from the formulas (14) and (15):
4. the method for controlling the back-stepping sliding mode self-adaptive posture of the underwater ore collection robot based on the extended state observer, according to claim 3, is characterized in that: the step 4 specifically comprises the following steps:
introducing a sliding mode control item, and defining a sliding mode surface function:
s＝ke 1 +e 2 (17)
wherein the method comprises the steps ofIs a normal number diagonal matrix;
is obtained by the following formulas (16) and (17):
since k+c > 0, if s=0, e 1 ＝0，e 2 =0 andtherefore, the system needs to be designed in the next step;
construction of Lyapunov function V 2 ：
Deriving formula (19):
to ensure thatDesigning a control law u:
wherein the method comprises the steps ofFor the observer to observe the estimated value of the complex interference, +.>In order to observe errors and to be bounded, for example a transverse channel,get->An upper bound of delta, then->And->For normal number diagonal matrix->
In order to reduce buffeting problem in the sliding mode control process, the adaptive index approach rate is designed:
wherein the method comprises the steps ofFor the estimate of delta under the adaptive approach law,/->Then the estimation error of delta ∈>Lambda is the normal number diagonal matrix,>assuming that the observed error varies slowly +.>
The control law u is obtained from the formulas (21) and (22):
5. the underwater ore collection robot backstepping sliding mode self-adaptive attitude control method based on the extended state observer, which is characterized by comprising the following steps of: the step 5 specifically comprises the following steps:
ensuring the stability of the system under the self-adaptive law, and constructing a Lyapunov function:
deriving formula (24):
matrix in (25)The method comprises the following steps:
obtained by the formula (26):
wherein the method comprises the steps ofSelect->Such that:
thereby ensuring E θ The positive rules are:
the same applies to formulas (26) to (29):
is obtained by the following formulas (25), (29) and (30)Meets the consistent gradually-tending characteristics;
according to LaSalle invariance theory, takeWhen (I)>e θ ≡0，e ψ When t.ident.0, s.ident.0, t.fwdarw.infinity, e 1 →0，e 2 0, s.fwdarw.0, x 1 →x d ，/>The attitude closed-loop control system converges, and stable tracking of an attitude angle and an attitude angle rate can be realized.
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| CN115327925B (en) * | 2022-09-09 | 2023-12-15 | 北京理工大学 | Combined body posture finite time control method based on extended state observer |
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