METHOD AND SYSTEM FOR TRAJECTORY TRACKING CONTROL OF VEHICLE-MANIPULATOR COUPLING SYSTEM WITH FINITE TIME PRESCRIBED PERFORMANCE

Abstract

The disclosure provides a method and system for trajectory tracking control of a vehicle-manipulator coupling system with finite time prescribed performance. Specifically, the trajectory tracking error of the vehicle-manipulator coupling system is obtained. A finite time performance function is designed to constrain the trajectory tracking error. In a case the constraint conditions corresponding to the finite time performance function are satisfied, the trajectory tracking error is converted to obtain a transformed error. The sliding mode surface is designed based on the transformed error to control the transformed error to converge in a finite time, and the external disturbance of the vehicle-manipulator coupling system is observed based on non-linear disturbance observer. The control input of the vehicle-manipulator coupling system is designed based on the sliding mode surface and the non-linear disturbance observer output. This ensures that the vehicle-manipulator coupling system can be controlled to operate precisely along the desired trajectory.

Claims

1. A method for trajectory tracking control of a vehicle-manipulator coupling system with finite time prescribed performance, the method is applied to the vehicle-manipulator coupling system to control a motion trajectory of the vehicle-manipulator coupling system, the vehicle-manipulator coupling system comprises an underwater vehicle and a robotic arm, and the method comprises following steps: obtaining, by a processor, a present motion state and a desired trajectory of the vehicle-manipulator coupling system, so as to calculate a difference between the present motion state and the desired trajectory to obtain a trajectory tracking error; designing, by the processor, a finite time performance function to constrain the trajectory tracking error so that the vehicle-manipulator coupling system reaches a steady state in response to the trajectory tracking error converging to a preset convergence boundary; and a gradient of the finite time performance function is not zero in response to an operation time of the vehicle-manipulator coupling system exceeding a preset convergence time, so as to avoid generating a singularity in a calculation of the state of the vehicle-manipulator coupling system and to ensure that a controller of the vehicle-manipulator coupling system does not diverge; converting, by the processor, the trajectory tracking error to obtain a corresponding transformed error in a case that constraint conditions corresponding to the finite time performance function are satisfied; designing, by the processor, a sliding mode surface of the vehicle-manipulator coupling system based on the transformed error to control the transformed error to converge in a finite time, and observing an external disturbance of the vehicle-manipulator coupling system based on a non-linear disturbance observer and the sliding mode surface; designing, by the processor, a control input of the vehicle-manipulator coupling system based on the sliding mode surface and the external disturbance observer, so that the vehicle-manipulator coupling system is controlled to operate according to the desired trajectory; designing the control input of the vehicle-manipulator coupling system based on the sliding mode surface and the external disturbance observer specifically is: ${}_m=A_m^{-1}\left[-B_m-{\stackrel{}{F}}_{dm}-{}^{-1}\left(H_m+k_{}{.Math.s_m.Math.}^{\frac{1}{2}}\mathrm{sign}\left(s_m\right)+{}_0^t{k}_{}\mathrm{sign}\left(s_m\left(\right)\right)d\right)\right],$ wherein T.sub.m represents a control input, k.sub. and k.sub. represent controller parameters to be designed, sign(.Math.) represents a signum function, represents the time, S.sub.m() represents a value of S.sub.m at the time , S.sub.m represents the sliding mode surface, .sub.0.sup.tk.sub. sign(s.sub.m())d represents an integral of k.sub. sign(s.sub.m) at a time interval [0,t]; A.sub.m and B.sub.m represent a matrix related to a dynamics model of the vehicle-manipulator coupling system; {circumflex over (F)}.sub.dm represents the observation of the unknown external disturbance; $=\frac{\left({}_u+{}_l\right){}_m}{2\left({}_l{}_m+{}_e\right)\left({}_u{}_m-{}_e\right)},$ .sub.l and .sub.u represent performance boundary coefficients, .sub.e represents the trajectory tracking error, .sub.m is an abbreviation of .sub.m(t), and .sub.m(t) represents a trajectory tracking error boundary; and $H_m=\left({}_m+\stackrel{}{}\right)\left({\stackrel{}{}}_e-\frac{{}_e{\stackrel{}{}}_m}{{}_m}\right)-.Math.\frac{{\stackrel{}{}}_e{\stackrel{}{}}_m{}_m+{}_e{\stackrel{.Math.}{}}_m{}_m-{}_e{\stackrel{}{}}_m^2}{{}_m^2}-{\stackrel{.Math.}{}}_d,$ {dot over ()} represents the first order derivative of , {umlaut over ()}.sub.m is an abbreviation of {umlaut over ()}.sub.m(t) being the second order derivative of .sub.m(t), {umlaut over ()}.sub.d represents the second order derivative of the desired trajectory represents .sub.d, {dot over ()}.sub.d the first order derivative of the trajectory tracking error .sub.e, .sub.m represents a diagonal sliding mode surface coefficient matrix, and {dot over ()}.sub.m is an abbreviation of {dot over ()}.sub.m(t) being the first order derivative of .sub.m(t), wherein the improved finite time performance function is: ${}_m\left(t\right)=\{\begin{array}{c}{\left({}_0^a-t\right)}^{1/}+{}_c,0t<T_m\\ \left({}_c-{}_{}\right)e^{-k\left(t-T_m\right)}+{}_{},t{T}_m\end{array},$ wherein .sub.0 represents a preset initial error boundary, and .sub.0>0; .sub.c represents a trajectory tracking error preset convergence boundary, and 0<.sub.c<<.sub.0; .sub. represents a trajectory tracking error asymptotic convergence boundary, and 0<.sub.>.sub.c; , , k represent prescribed performance parameters, configured to adjust a convergence rate and a convergence time of the finite time performance function; e represents a natural constant; t represents a time process; and T.sub.m represents a preset convergence time, and T.sub.m=.sub.0.sup./(), wherein .sub.e represent the trajectory tracking error, .sub.e=.sub.d, represents the present motion state, and .sub.d represents the desired trajectory; the finite time performance function constraining the trajectory tracking error specifically is: $-{}_l{}_m\left(t\right)<{}_e<{}_u{}_m\left(t\right),$ wherein error transformation is performed on the trajectory tracking error .sub.e in order to satisfy constraining the finite time performance function, the transformed error is represented by .sub.; ${}_{}=\frac{1}{2}\ln \frac{+{}_l}{{}_u-},$ wherein =.sub.e/.sub.m, and the first order derivative {dot over ()}.sub. of the transformed error .sub. is: ${\stackrel{.}{}}_{}=\left({\stackrel{.}{}}_e-\frac{{}_e{\stackrel{.}{}}_m}{{}_m}\right),$ wherein a motion model of the vehicle-manipulator coupling system is represented by: $\{\begin{array}{c}{\stackrel{.}{x}}_1=x_2\\ {\stackrel{.}{x}}_2=A_m{}_m+B_m+F_{\mathrm{dm}}\end{array},$ wherein x.sub.1=, x.sub.2 represents a speed state vector, F.sub.dm represents the unknown external disturbance received by the vehicle-manipulator coupling system; designing the sliding mode surface S.sub.m based on the transformed error .sub.; $s_m={}_m{}_{}+{\stackrel{.}{}}_{},$ wherein observing the external disturbance of the vehicle-manipulator coupling system based on the non-linear disturbance observer and the sliding mode surface specifically is: $\{\begin{array}{c}{\hat{F}}_{\mathrm{dm}}={}_{\mathrm{dm}}+K_{\mathrm{dm}}{s}_m\\ {\stackrel{.}{}}_{\mathrm{dm}}=-L_{\mathrm{dm}}{}_{\mathrm{dm}}{L}_{\mathrm{dm}}\left(K_{\mathrm{dm}}{s}_m+{}^{-1}{H}_m+A_m{}_m+B_m\right)\end{array},$ wherein .sub.dm represents an auxiliary intermediate variable of the non-linear disturbance observer, L.sub.dm and K.sub.dm represent gain coefficients of the non-linear disturbance observer, and K.sub.dm satisfies K.sub.dm=L.sub.dm.sup.1, wherein a problem of the trajectory tracking control under an external unknown disturbance is solved, and control precision, robustness, and transient-state performance of the vehicle-manipulator coupling system is ensured.

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7. A system for trajectory tracking control of a vehicle-manipulator coupling system with finite time prescribed performance, the control system is applied to control a motion trajectory of the vehicle-manipulator coupling system, the vehicle-manipulator coupling system comprises an underwater vehicle and a robotic arm, and the control system comprises a processor, and the processor is configured to: obtain a present motion state and an desired trajectory of the vehicle-manipulator coupling system, so as to calculate a difference between the present motion state and the desired trajectory to obtain a trajectory tracking error; design a finite time performance function to constrain the trajectory tracking error so that the vehicle-manipulator coupling system reaches a steady state in response to the trajectory tracking error converging to a preset convergence boundary; and a gradient of the finite time performance function is not zero in response to an operation time of the vehicle-manipulator coupling system exceeding a preset convergence time, so as to avoid generating a singularity in a calculation of the state of the vehicle-manipulator coupling system and to ensure that a controller of the vehicle-manipulator coupling system does not diverge; convert the trajectory tracking error to obtain a corresponding transformed error in a case that constraint conditions corresponding to the finite time performance function are satisfied; design a sliding mode surface of the vehicle-manipulator coupling system based on the transformed error to control the transformed error to converge in a finite time, and observe an external disturbance of the vehicle-manipulator coupling system based on a non-linear disturbance observer and the sliding mode surface; and design a control input of the vehicle-manipulator coupling system based on the sliding mode surface and the external disturbance observed, so that the vehicle-manipulator coupling system is controlled to operate according to the desired trajectory; designing the control input of the vehicle-manipulator coupling system based on the sliding mode surface and the external disturbance observed specifically is: ${}_m=A_m^{-1}\left[-B_m-F_{\mathrm{dm}}-{}^{-1}\left(H_m+k_{}{.Math.s_m.Math.}^{\frac{1}{2}}\mathrm{sign}\left(s_m\right)+{}_0^t{k}_{}\mathrm{sign}\left(s_m\left(\right)\right)d\right)\right],$ wherein .sub.m represents a control input, k.sub. and k.sub. represent controller parameters to be designed, sign(.Math.) represents a signum function, represents a time, S.sub.m() represents a value of S.sub.m at the time , S.sub.m represents the sliding mode surface, .sub.0.sup.tk.sub. sign(s.sub.m())d represents an integral of k.sub. sign(s.sub.m) at a time interval [0,t]; A.sub.m and B.sub.m represent a matrix related to a dynamics model of the vehicle-manipulator coupling system; {circumflex over (F)}.sub.dm represents the observation of the unknown external disturbance; $=\frac{\left({}_u+{}_l\right){}_m}{2\left({}_l{}_m+{}_e\right)\left({}_u{}_m-{}_e\right)},$ .sub.l and .sub.u represent performance boundary coefficients, .sub.e represents the trajectory tracking error, .sub.m is an abbreviation of .sub.m(t), and .sub.m(t) represents a trajectory tracking error boundary; and $H_m=\left({}_m+\stackrel{.}{}\right)\left({\stackrel{.}{}}_e-\frac{{}_e{\stackrel{.}{}}_m}{{}_m}\right)-.Math.\frac{{\stackrel{.}{}}_e{\stackrel{.}{}}_m{}_m+{}_e{\stackrel{.Math.}{}}_m{}_m-{}_e{\stackrel{.}{}}_m^2}{{}_m^2}-{\stackrel{.Math.}{}}_d,\stackrel{.}{}$ represents the first order derivative of , {umlaut over ()}.sub.m is an abbreviation of {umlaut over ()}.sub.m(t) being the second order derivative of .sub.m(t), {umlaut over ()}.sub.d represents the second order derivative of the desired trajectory .sub.d, {dot over ()}.sub.e represents the first order derivative of the trajectory tracking error .sub.e, .sub.m represents a diagonal sliding mode surface coefficient matrix, and {dot over ()}.sub.m is an abbreviation of {dot over ()}.sub.m(t) being the first order derivative of .sub.m(t), wherein the improved finite time performance function designed by the processor is: ${}_m\left(t\right)=\{\begin{array}{c}{\left({}_0^a-t\right)}^{1/}+{}_c,0t<T_m\\ \left({}_c-{}_{}\right)e^{-k\left(t-T_m\right)}+{}_{},t{T}_m\end{array},$ wherein .sub.0 represents a preset initial error boundary, and .sub.0>0; .sub.c represents a trajectory tracking error preset convergence boundary, and 0<.sub.c<<.sub.0; .sub. represents a trajectory tracking error asymptotic convergence boundary, and 0<.sub.<.sub.c; , , k represent prescribed performance parameters, configured to adjust a convergence rate and a convergence time of the finite time performance function; e represents a natural constant; t represents a time process; and T.sub.m represents a preset convergence time, and T.sub.m=.sub.0.sup./(), wherein the trajectory tracking error obtained by the processor is represented by .sub.e, .sub.e=.sub.d, represents the present motion state, and .sub.d represents the desired trajectory; the finite time performance function designed by the processor constraining the trajectory tracking error specifically is: $-{}_l{}_m\left(t\right)<{}_e<{}_u{}_m\left(t\right),$ the processor performs error transformation on the trajectory tracking error .sub.e, the transformed error resulted is represented by ${}_{}:{}_{}=\frac{1}{2}\ln \frac{+{}_l}{{}_u-},$ wherein =.sub.e/.sub.m, and the first order derivative {dot over ()}.sub. of the transformed error .sub. is: ${\stackrel{.}{}}_{}=\left({\stackrel{.}{}}_e-\frac{{}_e{\stackrel{.}{}}_m}{{}_m}\right),$ wherein a motion model of the vehicle-manipulator coupling system is represented by: $\{\begin{array}{c}{\stackrel{.}{x}}_1=x_2\\ {\stackrel{.}{x}}_2=A_m{}_m+B_m+F_{\mathrm{dm}}\end{array},$ wherein x.sub.1=, x.sub.2 represents a speed state vector, F.sub.dm represents the unknown external disturbance received by the vehicle-manipulator coupling system; designing the sliding mode surface S.sub.m based on the transformed error .sub.; s.sub.m=.sub.m.sub.+{dot over ()}.sub., wherein observing the external disturbance of the vehicle-manipulator coupling system based on the non-linear disturbance observer and the sliding mode surface specifically is: $\{\begin{array}{c}{\hat{F}}_{\mathrm{dm}}={}_{\mathrm{dm}}+K_{\mathrm{dm}}{s}_m\\ {\stackrel{.}{}}_{\mathrm{dm}}=-L_{\mathrm{dm}}{}_{\mathrm{dm}}{L}_{\mathrm{dm}}\left(K_{\mathrm{dm}}{s}_m+{}^{-1}{H}_m+A_m{}_m+B_m\right)\end{array},$ wherein .sub.dm represents an auxiliary intermediate variable of the non-linear disturbance observer, L.sub.dm and K.sub.dm represent gain coefficients of the non-linear disturbance observer, and K.sub.dm satisfies K.sub.dm=L.sub.dm.sup.1, wherein a problem of the trajectory tracking control under an external unknown disturbance is solved, and control precision, robustness, and transient-state performance of the vehicle-manipulator coupling system is ensured.

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Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0042] FIG. 1 is a flow chart of a method for trajectory tracking control of a vehicle-manipulator coupling system with finite time prescribed performance according to an embodiment of the disclosure.

[0043] FIG. 2 is a schematic diagram of an underwater vehicle-manipulator system according to an embodiment of the disclosure.

[0044] FIG. 3 is a block diagram of the finite time prescribed performance control according to an embodiment of the disclosure.

[0045] FIG. 4 is a curve diagram of a trajectory tracking performance of the underwater robotic arm system according to an embodiment of the disclosure.

[0046] FIG. 5 is a curve diagram of a trajectory tracking error of the underwater robotic arm system according to an embodiment of the disclosure.

[0047] FIG. 6 is a curve diagram of a control input of the underwater robotic arm system according to an embodiment of the disclosure.

[0048] FIG. 7 is a curve diagram of an observation performance of a non-linear disturbance observer according to an embodiment of the disclosure.

[0049] FIG. 8 is a curve diagram of an observation error of the non-linear disturbance observer according to an embodiment of the disclosure.

[0050] FIG. 9 is a curve diagram of a dynamic coupling disturbance received by the underwater vehicle according to an embodiment of the disclosure.

[0051] FIG. 10 is a curve diagram of errors in the position tracking from dynamic positioning of the underwater vehicle according to an embodiment of the disclosure.

[0052] FIG. 11 is a curve diagram of errors in the attitude tracking from dynamic positioning of the underwater vehicle according to an embodiment of the disclosure.

[0053] FIG. 12 is a diagram of an architecture of a system for controlling the finite time prescribed performance of trajectory tracking of the vehicle-manipulator coupling system according to an embodiment of the disclosure.

DESCRIPTION OF THE EMBODIMENTS

[0054] In order to make the purpose, technical solutions, and advantages of the disclosure more comprehensible, the disclosure will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described here are merely used to explain the disclosure and the embodiments are not used to limit the disclosure.

[0055] The purpose of this disclosure is to, under the uncertainty, complex coupling disturbance, and unknown external disturbance of the model of the vehicle-manipulator coupling system, a control method is designed to realize the trajectory tracking control of the underwater vehicle-manipulator system, the trajectory tracking error should satisfy the preset transient-state and steady-state performance constraints, and the convergence time may be preset. In addition, the control method should have the ability to handle unknown external disturbances and weaken control buffeting to ensure the robustness and control precision of the vehicle-manipulator coupling system.

[0056] FIG. 1 is a flow chart of the trajectory tracking control of a vehicle-manipulator coupling system with finite time prescribed performance according to an embodiment of the disclosure. As shown in FIG. 1, the following steps are included.

[0057] S101. A present motion state and an desired trajectory of a vehicle-manipulator coupling system are obtained, so as to calculate a difference between the present motion state and the desired trajectory to obtain a trajectory tracking error.

[0058] A structure of an underwater vehicle-manipulator coupling system is shown in FIG. 2.

[0059] S102. A finite time performance function is designed to constrain the trajectory tracking error so that when the trajectory tracking error converges to a preset convergence boundary, the vehicle-manipulator coupling system reaches a steady state; and when an operation time of the vehicle-manipulator coupling system exceeds a preset convergence time, a gradient of the finite time performance function is not zero, so as to avoid generating a singularity in a calculation of the state of the vehicle-manipulator coupling system and to ensure that a controller of the vehicle-manipulator coupling system does not diverge.

[0060] S103. In a case that constraint conditions corresponding to the finite time performance function are satisfied, the trajectory tracking error is converted to obtain a corresponding transformed error.

[0061] S104. A sliding mode surface of the vehicle-manipulator coupling system is designed based on the transformed error to control the transformed error to converge in a finite time, and an external disturbance of the vehicle-manipulator coupling system is observed based on a non-linear disturbance observer and the sliding mode surface.

[0062] S105. A control input of the vehicle-manipulator coupling system is designed based on the sliding mode surface and the external disturbance observed, so that the vehicle-manipulator coupling system is controlled to operate according to the desired trajectory.

[0063] Specifically, a method for controlling the finite time prescribed performance of a dynamic trajectory tracking task directed to the underwater vehicle-manipulator system according to the disclosure, as shown in FIG. 3, includes the following: [0064] (1) An improved finite time performance function is proposed to constrain a trajectory tracking error of the system; [0065] (2) An error transformation method directed to the performance function is designed; and (3) A super-twisting sliding mode control algorithm based on the improved finite time prescribed performance of the non-linear disturbance observer is designed.

[0066] The improved finite time performance function proposed is as follows:

[00013]${}_m\left(t\right)=\{\begin{array}{c}{\left({}_0^{}-t\right)}^{1/}+{}_c,0t<T_m\\ \left({}_c-{}_{}\right)e^{-k\left(t-T_m\right)}+{}_{},t{T}_m\end{array}$

[0067] In the formula, .sub.m(t) represents a trajectory tracking error boundary; .sub.0 represents a preset initial error boundary; .sub.c represents a trajectory tracking error preset convergence boundary; .sub. represents the trajectory tracking error asymptotic convergence boundary; , , represent prescribed performance parameters, used to adjust a convergence rate and a convergence time of the performance function; e represents a natural constant; t represents a time process; and T.sub.m represents a convergence time of the performance function.

[0068] The improved finite time performance function has characteristics as follows:

[0069] (1) The convergence time T.sub.m of the finite time performance function is calculated as: T.sub.m=.sub.0.sup./(); (2) An actual initial error boundary of the finite time performance function is: .sub.m(0)=.sub.0+.sub.c; (3) When the trajectory tracking error converges to .sub.c, the system should reach a steady state. Therefore, when designing parameters, .sub.c should be designed to be small enough, usually 0.01, to satisfy the system convergence condition; (4) When the system time tT.sub.m is used, in order to ensure that the gradient of the performance function designed is not zero, the design range of .sub. is 0<.sub.<.sub.c.

[0070] The formula of the performance function constraining the system trajectory tracking error is:

[0071] .sub.el<.sub.e<.sub.eu

[0072] In the formula, .sub.e represents the system trajectory tracking error, and .sub.el and .sub.eu represent a lower limit and an upper limit of the trajectory tracking error prescribed performance respectively and are defined by the following formula:

[00014]${}_{el}=-{}_l{}_m\left(t\right),{}_{eu}={}_u{}_m\left(t\right)$

[0073] In the formula, .sub.l and .sub.u represent performance boundary coefficients, and values thereof are in a range of 0<.sub.l1 and in a range of 0<.sub.u1 respectively. The system trajectory tracking error is defined as:

[00015]${}_e=-{}_d$

[0074] In the formula, .sub.e represents the trajectory tracking error, represents a system motion state, and .sub.d represents a system desired trajectory.

[0075] An error transformation method is represented by the following formula:

[00016]$S\left({}_{}\right)=\frac{{}_u{e}^{{}_{}}-{}_l{e}^{-{}_{}}}{e^{{}_{}}+e^{-{}_{}}}$

[0076] In the formula, .sub. represents an transformed system error, e represents the natural constant, S represents the error transformation method, and there are:

[00017]$S\left({}_{}\right)=\frac{{}_e}{{}_m}$

[0077] Therefore, below is obtained:

[00018]${}_{}=S{\left(\frac{{}_e}{{}_m}\right)}^{-1}=\frac{1}{2}\ln \frac{+{}_l}{{}_u-}$

[0078] In the formula, =.sub.e/.sub.m.

[0079] For the super-twisting sliding mode control algorithm based on the improved finite time prescribed performance of the non-linear disturbance observer proposed, the design process is as follows.

[0080] Firstly, the system model expression is given:

[00019]$\{\begin{array}{c}{\stackrel{.}{x}}_1=x_2\\ {\stackrel{.}{x}}_2=A_m{}_m+B_m+F_{\mathrm{dm}}\end{array}$

[0081] In the formula, x.sub.1= represents a system position state vector, x.sub.2 represents a system speed state vector, A.sub.m and B.sub.m represent a matrix related to a dynamics model of the system, F.sub.dm represents the external disturbance unknown received by the system, and .sub.m represents a control input of the system.

[0082] Secondly, the sliding mode surface s.sub.m is designed based on the transformed error .sub.:

[00020]$s_m={}_m{}_{}+{\stackrel{}{}}_{}$

[0083] In the formula, .sub.m represents a diagonal sliding mode surface coefficient matrix.

[0084] Further, since F.sub.dm is the unknown external disturbance, the non-linear disturbance observer is designed as follows to perform real-time estimation on F.sub.dm:

[00021]$\{\begin{array}{c}{\hat{F}}_{\mathrm{dm}}={}_{\mathrm{dm}}+K_{\mathrm{dm}}{s}_m\\ {\stackrel{.}{}}_{\mathrm{dm}}=-L_{\mathrm{dm}}{}_{\mathrm{dm}}-L_{\mathrm{dm}}\left(K_{\mathrm{dm}}{s}_m+{}^{-1}{H}_m+A_m{}_m+B_m\right)\end{array}$

[0085] In the formula, {circumflex over (F)}.sub.dm represents the observation of the unknown external disturbance, .sub.dm represents an auxiliary intermediate variable of the non-linear disturbance observer, L.sub.dm represents a gain coefficient of the observer to be designed, and K.sub.dm satisfies K.sub.dm=L.sub.dm.sup.1.

[0086] Finally, the control algorithm is designed according to the following:

[00022]${}_m=A_m^{-1}\left[-B_m-{\stackrel{}{F}}_{\mathrm{dm}}-{}^{-1}\left(H_m+k_{}{.Math.s_m.Math.}^{\frac{1}{2}}\mathrm{sign}\left(s_m\right)+{}_0^t{k}_{}\mathrm{sign}\left(s_m\left(\right)\right)d\right)\right]$

[0087] In the formula, k.sub. and k.sub. represent controller parameters to be designed.

[0088] It should be noted that the design idea of the control algorithm is as follows. The super-twisting sliding mode control approaching law is adopted so that the transformed error converges within a finite time and that the system tracking error ne satisfies the prescribed performance convergence boundary, and the system chattering effect can be weakened. The non-linear disturbance observer is introduced to improve system robustness under system uncertainty and external disturbance.

[0089] The disclosure uses the following embodiment to verify the above technical solution.

[0090] Basic parameters of the underwater vehicle-manipulator system according to an embodiment are shown in Table 1.

TABLE-US-00001 TABLE 1 Basic parameters of the underwater vehicle-manipulator system Parameter name Parameter description Parameter value M.sub.v Underwater vehicle mass 226 kg L.sub.v Underwater vehicle length 1.2 m D.sub.v Underwater vehicle 0.6 m equivalent diameter G.sub.v Underwater vehicle gravity 2217N B.sub.v Underwater vehicle buoyancy 2217N I.sub.v, xx Underwater vehicle moment 8 kg .Math. m.sup.2 of inertia-X I.sub.v, yy Underwater vehicle moment 20 kg .Math. m.sup.2 of inertia-Y I.sub.v, zz Underwater vehicle moment 20 kg .Math. m.sup.2 of inertia-Z M.sub.1 Robotic arm link 1 mass 3.39 kg M.sub.2 Robotic arm link 2 mass 3.39 kg L.sub.1 Robotic arm link 1 length 0.5 m L.sub.2 Robotic arm link 2 length 0.5 m D.sub.1 Robotic arm link 1 diameter 0.16 m D.sub.2 Robotic arm link 2 diameter 0.16 m C.sub.m Inertia coefficient 1.0 C.sub.d Drag coefficient 1.1 Water density 1000 kg .Math. m.sup.3 g Acceleration of gravity 9.8 m .Math. s.sup.2

[0091] This embodiment uses simulation to illustrate the effectiveness and advancement of the method proposed by this disclosure. In the simulation, the underwater vehicle completes a dynamic positioning control task, and the robotic arm mounted thereon completes complex sinusoidal motion to reflect the effectiveness of the control method proposed by this disclosure. Next, simulation parameter settings are introduced. For the dynamic positioning task of the underwater vehicle, an initial pose is

[00023]${}_{v,0}={\left[\begin{array}{cccccc}1m& 0m& 0.5m& \frac{0.1}{2}\mathrm{rad}& \frac{0.1}{2}\mathrm{rad}& \frac{0.3}{2}\mathrm{rad}\end{array}\right]}^T,$

and an expected pose is .sub.v,d=[3 m 3 m 4 m 0 rad 0 rad 0 rad].sup.T. For the robotic arm 10 trajectory tracking task, an initial joint angle is q.sub.0=[0 rad 0 rad].sup.T, and the expected joint angle changes with time t as

[00024]$q_d={\left[-0.5-\frac{}{4}\begin{array}{cc}\sin \left(t\right)\mathrm{rad}& -0.5-\frac{}{4}\sin \left(t\right)\mathrm{rad}\end{array}\right]}^T.$

In the control parameter aspect, for the robotic arm system, parameters are designed follows: =0.4, =0.8, k=0.2, .sub.0=[2 2].sup.T, .sub.c=[0.03 0.03].sup.T, .sub.=[0.02 0.02].sup.T, .sub.u=1, .sub.1=0.7, .sub.m=diag(0.05,0.05), k.sub.=diag(25,25), k.sub.=diag(20,20). For the underwater vehicle, control parameters are designed follows: =0.2, =0.8, k=0.3, .sub.0=[6 6 6 1 1 1].sup.T, .sub.c=[0.15 0.15 0.15 0.1 0.1 0.1].sup.T, .sub.=[0.1 0.1 0.1 0.05 0.05 0.05].sup.T, .sub.u=1, .sub.1=1, .sub.m=diag(0.1,0.1,0.1,0.1,0.1,0.1), k.sub.=diag(500,500,500,400,400, 400), k.sub.=diag(300,300,300,300,300,300), in which diag represents the diagonal matrix. The external disturbance received by the system is set to d.sub.1=3+8 sin(0.6t)+5 cos(0.3t), and d.sub.2=4+7 sin(0.7t)+4 cos(0.2t). A parameter of the non-linear disturbance observer in the robotic arm system is designed as L.sub.dm=diag(20,18), and a parameter of the observer of the underwater vehicle is designed as L.sub.dm=diag(100,100,100,100,100,100).

[0092] The simulation results are shown in FIG. 4 to FIG. 11. FIG. 4 shows trajectory tracking results of the robotic arm with two joints, and FIG. 5 shows trajectory tracking errors of the two joints. The simulation results show that the convergence time of the method proposed by this disclosure is less than 4 s, the convergence is fast, and the convergence time is controllable. In addition, the trajectory tracking steady-state error of the method according to the disclosure is less than 0.01 rad. The Calculation shows that for a joint 1 and a joint 2, average steady-state tracking errors of the method according to the disclosure are 6.3010.sup.4 rad and 2.3310.sup.3 rad respectively, and the control precision is high.

[0093] FIG. 6 shows control inputs of the joint 1 and the joint 2. It may be concluded from FIG. 6 that when the method according to the disclosure is adopted, the chattering effect is significantly weakened. FIG. 7 and FIG. 8 show the observation performance of the non-linear disturbance observer, the observer may stably estimate external disturbance, and the observation values may converge in a short time. Specifically, the average steady-state estimation errors of disturbance 1 and disturbance 2 are 4.1810.sup.2 Nm and 1.2910.sup.2 Nm respectively. Finally, the dynamic positioning control performance of the underwater vehicle is briefly demonstrated. When working together with the robotic arm, the underwater vehicle is affected by the dynamic coupling effect. Since the robotic arm of 2 degrees of freedom merely operates in an xoz plane of an underwater vehicle boat body coordinate system, the dynamic coupling effect merely affects three degrees of freedom, x, z, and of the underwater vehicle, as shown in FIG. 9. FIG. 10 and FIG. 11 respectively show dynamic positioning position errors and attitude errors of the underwater vehicle under the action of external disturbance and coupling disturbance, in which steady-state position errors on three degrees of freedom, x, y, z, are respectively 1.3910.sup.4 m, 3.0510.sup.3 m, 4.3010.sup.4 m, and steady-state attitude errors on three degrees of freedom , , are respectively 9.2110.sup.3 rad, 1.4210.sup.2 rad, 2.0510.sup.4 rad, the control precision is high, and the convergence time is fast, approximately 4 s. The above analysis shows that the method proposed by the disclosure has good transient-state and steady-state control performances.

[0094] In summary, the disclosure discloses the method for the finite time prescribed performance control of the dynamic trajectory tracking task directed to the underwater vehicle-manipulator system. Under the influence of model uncertainty, dynamic coupling effects, and external disturbances, the trajectory tracking control of the underwater vehicle-manipulator systems faces big challenges. In order to ensure the transient-state and steady-state performances of the system, the improved finite time performance function is designed to ensure that the preset tracking precision is achieved in a specified convergence time. The performance function proposed can avoid generating a singularity in the system, and the robustness of the system is improved. In order to reduce the influence of unknown external disturbances on the system, the non-linear disturbance observer is adopted to process unknown disturbances. Finally, the super-twisting sliding mode control framework based on the improved finite time prescribed performance of the non-linear disturbance observer is proposed, which ensures the control precision, robustness, and transient-state performance of the system, and the buffeting phenomenon is weakened.

[0095] FIG. 12 is a diagram of an architecture of a system for the finite time prescribed performance control of trajectory tracking for the vehicle-manipulator coupling system according to an embodiment of the disclosure. As shown in FIG. 12, the control system includes the following.

[0096] A parameter obtain unit 1210 is used to obtain the present motion state and the desired trajectory of the vehicle-manipulator coupling system, so as to calculate a difference between the present motion state and the desired trajectory to obtain a trajectory tracking error.

[0097] A performance function design unit 1220 is used to design a finite time performance function to constrain the trajectory tracking error so that when the trajectory tracking error converges to a preset convergence boundary, the vehicle-manipulator coupling system reaches a steady state, and when an operation time of the vehicle-manipulator coupling system exceeds a preset convergence time, a gradient of the finite time performance function is not zero, so as to avoid generating a singularity in a calculation of the state of the vehicle-manipulator coupling system and to ensure that a controller of the vehicle-manipulator coupling system does not diverge.

[0098] An error transformation unit 1230 is used to convert the trajectory tracking error to obtain the corresponding transformed error in a case that constraint conditions corresponding to the finite time performance function are satisfied.

[0099] A control input design unit 1240 is used to design the sliding mode surface of the vehicle-manipulator coupling system based on the transformed error to control the transformed error to converge in a finite time, and to observe the external disturbance of the vehicle-manipulator coupling system based on the non-linear disturbance observer and the sliding mode surface; and to design the control input of the vehicle-manipulator coupling system based on the sliding mode surface and external disturbance observed, so that the vehicle-manipulator coupling system is controlled to operate according to the desired trajectory.

[0100] It should be understood that the control system is used to execute the method in the embodiments, and corresponding program units in the control system have implementation principles and technical effects similar to the contents described in the method. For the working process of the control system, reference may be made to corresponding processes in the method described above, and details will not be repeated here.

[0101] Based on the method according to the embodiments, an electronic device is provided according to an embodiment of the disclosure. The device may include at least one storage device for storing programs and at least one processor for executing the programs stored in the storage device, in which when the program stored in the storage device is executed, the processor is used to execute the method according to the embodiments.

[0102] Based on the method according to the embodiments, a computer-readable storage medium is provided according to an embodiment of the disclosure. The computer-readable storage medium stores a computer program, in which when the computer program is run on the processor, the processor executes the method according to the embodiments.

[0103] Based on the method according to the embodiments, a computer program product is provided according to an embodiment of the disclosure. When the computer program product is run on the processor, the processor executes the method according to the embodiments.

[0104] It may be understood that the processor in the embodiments of the disclosure may be a central processing unit (CPU), or other general-purpose processor, digital signal processor (DSP), application specific integrated circuit (ASIC), field programmable gate array (FPGA), or other programmable logic devices, transistor logic devices, hardware components, or any combination thereof. The general-purpose processor may be a microprocessor or any regular processor.

[0105] The steps in the method according to the embodiments of the disclosure may be implemented by hardware or by the processor executing software commands. The software commands may comprise corresponding software modules, and the software modules may be stored in random access memory (RAM), flash memory, read-only memory (ROM), programmable rom (PROM), erasable PROM (EPROM), electrically EPROM (EEPROM), register, hard disk, portable storage device, CD-ROM, or any other forms of storage medium well known in the art. An exemplary storage medium is coupled to a processor so that the processor may read information from the storage medium and write information to the storage medium. Certainly, the storage medium may also be an integral part of the processor. The processor and the storage medium may be located in the ASIC.

[0106] The above embodiments may be implemented in whole or in part by software, hardware, firmware, or any combination thereof. When implemented using software, the disclosure may be implemented in whole or in part in the form of a computer program product. The computer program product includes one or more computer commands. When the computer program commands are loaded and executed on a computer, the processes or functions described in the embodiments of the disclosure are generated in whole or in part. The computer may be a general-purpose computer, a special-purpose computer, a computer network, or other programmable devices. The computer command may be stored in the computer-readable storage medium or transmitted over the computer-readable storage medium. The computer command may be transmitted from a website, computer, server, or data center via wired (such as coaxial cable, optical fiber, digital subscriber line (DSL)) or wireless (such as infrared, wireless, microwave) transmission to another website, computer, server, or data center. The computer-readable storage medium may be any available medium accessible by a computer, or a data storage device such as and integrated server comprising one or more available media, for example, a server or a data center. The available medium may be a magnetic medium (for example, a floppy disk, a hard disk, a magnetic tape), an optical medium (for example, a DVD), or a semiconductor medium (for example, a solid state disk (SSD)).

[0107] It may be understood that the various reference numerals involved in the embodiments of the disclosure are merely for convenience of description and are not used to limit the scope of the embodiments of the disclosure.

[0108] It may be understood for persons skilled in the art that the embodiments are merely some preferred embodiments of the disclosure and the embodiments are not intended to limit the disclosure. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the disclosure shall be regarded as should be included within the protection scope of the disclosure.