CN104932271A - Neural network full order slip form control method for mechanical arm servo system - Google Patents

Abstract

The invention discloses a neural network full order slip form control method for a mechanical arm servo system, which, aiming at the mechanical arm servo system containing a dynamic execution mechanism, is designed by utilizing the full-order slip form control method and combining with the neural network. The full order slip form is designed to guarantee the finite time convergence of the system and can eliminate buffeting and singular problems by avoiding the differential item in a practical control system. Besides, the neural network is used for approaching unknown non-linearity of the system and non-determinacy of internal-external disturbance. The neural network full order slip form control method for a mechanical arm servo system realizes provides a method to eliminate the buffeting and the singularity problem of the surface of the slip form, effectively complements the unknown linearity and the control method of the internal-external disturbance, and realizes fast and stable control of the system.

Description

A kind of neural network full-order sliding mode control method of mechanical arm servo-drive system

Technical field

The present invention relates to a kind of neural network full-order sliding mode control method of mechanical arm servo-drive system, particularly with the mechanical arm servo system control method of the unknown dynamic parameter of Dynamic Execution mechanism and system.

Background technology

Mechanical arm servo-drive system, as the increasingly automated equipment of one, is widely used in robot, the contour performance system of aviation aircraft, and the accurate fast control how realizing mechanical arm servo-drive system has become a hot issue.But unknown dynamic parameter and external disturbance are extensively present in mechanical arm servo-drive system, the efficiency of control system is often caused to reduce or even lost efficacy.For the control problem of mechanical arm servo-drive system, there is a lot of control method, such as PID controls, adaptive control, sliding formwork control etc.

Sliding formwork controls to be considered to an effective robust control method in and external disturbance uncertain at resolution system.The advantages such as sliding-mode control has that algorithm is simple, fast response time, to external world noise and Parameter Perturbation strong robustness.Therefore, sliding-mode control is widely used in every field.Contrast conventional linear sliding formwork controls, and the superiority of TSM control is that his finite time is received.But TSM control discontinuous switching characteristic in itself will cause the buffeting of system, becomes the obstacle that TSM control is applied in systems in practice.In order to address this problem, the method for many improvement is suggested in succession, such as high_order sliding mode control method, observer control method.In these methods, choosing of sliding-mode surface all obtains according to idealized system parameter depression of order.Recently, a kind of full-order sliding mode control method is suggested, and this method well avoids buffeting problem and makes system input signal more level and smooth in the response of system.

But in most of method of above-mentioned proposition, the dynamic model parameters of mechanical system all must be known in advance.Therefore, when system exists uncertain factor, the method for above-mentioned proposition can not directly apply to the control to mechanical arm.As everyone knows, because neural network approaches the ability of any smooth function in an arbitrary accuracy compacted, therefore it has been widely used in the non-intellectual of disposal system and nonlinear problem.For these reasons, many adaptive neural network control methods are used to the mechanical arm system controlling nonlinearity.

Summary of the invention

In order to the unknown nonlinear problem and sliding formwork that overcome the existence of existing mechanical arm servo-drive system control the deficiency of buffeting problem, the invention provides a kind of mechanical arm servo-drive system neural network full-order sliding mode control method containing Dynamic Execution mechanism, eliminate buffeting problem and the singular problem of system, ensure the convergence of system fast and stable.

In order to the technical scheme solving the problems of the technologies described above proposition is as follows:

A kind of mechanical arm servo-drive system neural network full-order sliding mode control method, comprises the following steps:

Step 1, sets up the dynamic model of mechanical arm servo-drive system, initialization system state, sampling time and controling parameters, and process is as follows:

The dynamic model expression-form of 1.1 mechanical arm servo-drive systems is:

$M\left(q\right)\stackrel{\·\·}{q}+C(q,\stackrel{\·}{q})\stackrel{\·}{q}+D\stackrel{\·}{q}+G\left(q\right)=\mathrm{\τ}---\left(1\right)$

Wherein, q, with be respectively the position of joint of mechanical arm, speed and acceleration, M (q), and D represents the symmetric positive definite inertial matrix in each joint respectively, the diagonal angle positive definite matrix of centrifugal Coriolis matrix and damping friction coefficient; G (q) represents gravity item; τ represents the torque input vector in joint;

Formula (1), when considering Dynamic Execution mechanism, is expressed as by 1.2 again:

$M_H\left(q\right)\stackrel{\·\·}{q}+C_H(q,\stackrel{\·}{q})\stackrel{\·}{q}+D_{H}q+G_{H}q=u---\left(2\right)$

Wherein, $M_H=K_{\mathrm{\τ}}^{-1}(N^{-1}M+{\mathrm{NJ}}_m),C_H=K_{\mathrm{\τ}}^{-1}{N}^{-1}C,D_H=K_{\mathrm{\τ}}^{-1}(N^{-1}D+{\mathrm{ND}}_m),$ $G_H=K_{\mathrm{\τ}}^{-1}{N}^{-1}G;u=K_{\mathrm{\τ}}^{-1}{\mathrm{\τ}}_m$ It is the vector of an armature voltage input; represent electromagnetic torque vector, wherein, J _mand D _mrepresent inertia diagonal matrix respectively and reverse ratio of damping; K _τ=diag (K _{τ 1}, K _{τ 2}..., K _{τ n}) be then the torque constant of diagonal matrix; q _mwhat represent is motor Angular position vector; τ _lrepresentative be the vector of electric motor load torque; represent the diagonal matrix of n joint transmission gear;

1.3 owing to existing measurement noises, the impact of load variations and external disturbance, and the systematic parameter in formula (2) can not obtain accurately; So, again the systematic parameter of reality is rewritten as:

$M_H\left(q\right)={\hat{M}}_H\left(q\right)+{\mathrm{\ΔM}}_H\left(q\right)$

$C_H(q,\stackrel{\·}{q})={\hat{C}}_H(q,\stackrel{\·}{q})+{\mathrm{\ΔC}}_H(q,\stackrel{\·}{q})$

$D_H={\hat{D}}_H+{\mathrm{\ΔD}}_H$

$G_H\left(q\right)={\hat{G}}_H\left(q\right)+{\mathrm{\ΔG}}_H\left(q\right)---\left(3\right)$

Wherein, estimated value and represent known portions; Δ M _h(q), Δ D _hand Δ G _h(q) representative system unknown term;

Step 2, based on the mechanical arm servo-drive system with unknown parameter, the neural network of design, process is as follows:

Definition θ ^*for ideal weight matrix of coefficients, so nonlinear uncertain function f approached for:

f＝θ ^*Tφ(x)+ε (4)

Wherein, represent input vector; φ (x)=[φ ₁(x), φ ₂(x) ... φ _m(x)] ^tit is the basis function of neural network; ε represents the approximate error of neural network and meets || ε || and≤ε _n, ε _nit is then a positive constant; φ _ix () is taken as following Gaussian function:

${\mathrm{\φ}}_i\left(x\right)=\exp \[-\frac{\left|\right|x-c_i\left|\right|}{2{\mathrm{\σ}}_i^2}\],i=1,2,...,n---\left(5\right)$

Wherein, c _irepresent the nuclear parameter of Gaussian function; σ _ithen illustrate the width of Gaussian function;

Step 3, calculating control system tracking error, design full-order sliding mode face, process is as follows:

3.1 define system tracking errors are:

e＝q _d-q (6)

Wherein, q _dfor second order can lead desired trajectory; So the first differential of formula (6) and second-order differential are represented as following form:

$\stackrel{\·}{e}={\stackrel{\·}{q}}_d-\stackrel{\·}{q}---\left(7\right)$

$\stackrel{\·\·}{e}={\stackrel{\·\·}{q}}_d-\stackrel{\·\·}{q}---\left(8\right)$

3.2 so full-order sliding mode face will be defined as:

$s=\stackrel{\·\·}{e}+c_{2}sgn\left(\stackrel{\·}{e}\right)|\stackrel{\·}{e}{|}^{{\mathrm{\α}}_2}+c_{1}sgn\left(e\right)|e{|}^{{\mathrm{\α}}_1}---\left(9\right)$

Wherein, c ₁and c ₂be a positive constant, its selection ensures polynomial expression p ²+ c ₂whole characteristic roots of p+c in the left-half of complex plane to ensure system stability; α ₁and α ₂choose, be by following polynomial expression:

$\left\{\begin{array}{c}{\mathrm{\α}}_1=\mathrm{\α},n=1\\ {\mathrm{\α}}_{i-1}=\frac{{\mathrm{\α}}_i{\mathrm{\α}}_{i+1}}{2{\mathrm{\α}}_{i+1}-{\mathrm{\α}}_i},i=2,...,n,\∀n\≥2\end{array}\right.---\left(10\right)$

Wherein, α _n+1=1, α _n=α, α ∈ (1-ε, 1) and ε ∈ (0,1);

Step 4, based on the mechanical arm system containing Dynamic Execution mechanism, according to full-order sliding mode and neural network theory, design neural network full-order sliding mode controller, process is as follows:

4.1 consider formula (2), and neural network full-order sliding mode controller is designed to:

$\begin{array}{c}u={\hat{M}}_H\left(q\right)({\stackrel{\·\·}{q}}_d+c_2\mathrm{sgn}(\stackrel{\·}{e})|\stackrel{\·}{e}{|}^{{\mathrm{\α}}_2}+c_1\mathrm{sgn}\left(e\right)|e{|}^{{\mathrm{\α}}_1}+u_0)\\ +{\hat{C}}_H(q,\stackrel{\·}{q})\stackrel{\·}{q}+{\hat{D}}_H\stackrel{\·}{q}+{\hat{G}}_H\left(q\right)\end{array}---\left(11\right)$

$u_0={\widehat{\mathrm{\θ}}}^T\mathrm{\φ}\left(x\right)-{{\hat{M}}_H}^{-1}\left(q\right)u_n---\left(12\right)$

${\stackrel{\·}{u}}_n+{\mathrm{Tu}}_n=v---\left(13\right)$

v＝-(k _d+k _T+η)sgn(s) (14)

Wherein, c _iand α _iconstant, i=1,2, be defined in formula (9); k _d, k _tall constant with η;

The Rule adjusting of 4.2 design neural network weight coefficient matrix:

$\stackrel{\·}{\widehat{\mathrm{\θ}}}=\mathrm{\Γ}\mathrm{\φ}\left(x\right)s^T---\left(15\right)$

Wherein, Γ is the diagonal matrix of a positive definite;

Formula (11) to be brought in formula (2) and is obtained following equation by 4.3:

$s={\mathrm{\θ}}^{{*}^T}\mathrm{\φ}\left(x\right)+\mathrm{\ϵ}-{\widehat{\mathrm{\θ}}}^T\mathrm{\φ}\left(x\right)+u_n={\widetilde{\mathrm{\θ}}}^T\mathrm{\φ}\left(x\right)+\mathrm{\ϵ}+u_n=d(q,t)+u_n---\left(16\right)$

Wherein, represent the weight evaluated error of neural network; representative system disturbance term, and be bounded, so suppose d (q, t)≤l _dand wherein l _dit is the constant of a bounded; K _tto choose be meet k when K > 0 _t>=Tl _d;

4.4 through types (2), formula (9), formula (11)-(14) and formula (16), full-order sliding mode face is written to following equation:

s＝d(q,t)+u _n(17)

Formula (14) to be brought in formula (13) and is obtained by 4.5:

$\begin{array}{c}{u}_n\left(t\right)=\left(u_n\right(t_0)+(1/T)(k_d+k_T+\mathrm{\η})\mathrm{sgn}\left(s\right))e^{t-t_0}\\ -(1/T)(k_d+k_T+\mathrm{\η})\mathrm{sgn}\left(s\right)\end{array}---\left(18\right)$

At u _n(0), when=0, following equation is obtained:

k _T≥Tl _d≥T|u _n(t)| _max≥T|u _n(t)| (19)

4.6 design Lyapunov functions:

$V=\frac{1}{2}{s}^{T}s---\left(20\right)$

Carry out differentiate to formula (9) to obtain:

$\stackrel{\·}{s}=\stackrel{\·}{d}(q,t)+{\stackrel{\·}{u}}_n=\stackrel{\·}{d}(q,t)+{\stackrel{\·}{u}}_n+{\mathrm{Tu}}_n-{\mathrm{Tu}}_n=\stackrel{\·}{d}(q,t)+v-{\mathrm{Tu}}_n---\left(21\right)$

Formula (13) is brought in formula (21) and obtains:

$\stackrel{\·}{s}=\stackrel{\·}{d}(q,t)-(k_d+k_T+\mathrm{\η})\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n---\left(22\right)$

Carry out differential to formula (20) to obtain:

$\stackrel{\·}{V}=s^T\stackrel{\·}{s}=\stackrel{\·}{d}(q,t)s^T-(k_d+k_T+\mathrm{\η})s^T\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n{s}^T---\left(23\right)$

Formula (19) is brought in formula (23), if then decision-making system is stable.

The present invention is based on unknown nonlinear factor, full-order sliding mode and neural network, the neural network full-order sliding mode control method of design mechanical arm servo-drive system, realizes system stability and controls, eliminate the buffeting that sliding formwork controls, and ensures the convergence of system fast and stable.

Technical conceive of the present invention is: for the mechanical arm servo-drive system containing Dynamic Execution mechanism, utilize full-order sliding mode control method, then in conjunction with neural network, the arm servo-drive system that designs a mechanism neural network full-order sliding mode control method.The design in full-order sliding mode face is the fast and stable convergence in order to ensure system, and by avoiding occurring differential term to eliminate buffeting and singular problem in the control system of reality.In addition, neural network is used to the unknown nonlinear of the system of approaching and the uncertainty of inside and outside disturbance.The invention provides a kind of buffeting problem and the singular problem that can eliminate sliding-mode surface, and the control method of the unknown dynamic parameter of energy effective compensation system and inside and outside disturbance, the fast and stable realizing system controls.

Advantage of the present invention is: eliminate and buffet, the non-dynamic parameter of bucking-out system and inside and outside uncertain disturbance item, realizes fast and stable convergence.

Accompanying drawing explanation

Position tracking effect schematic diagram when Fig. 1 is k=10 of the present invention, wherein, (a) represents joint 1, and (b) represents joint 2.

Position tracking error schematic diagram when Fig. 2 is k=10 of the present invention, wherein, (a) represents joint 1, and (b) represents joint 2.

Speed tracing schematic diagram when Fig. 3 is k=10 of the present invention, wherein, (a) represents joint 1, and (b) represents joint 2.

Controller input schematic diagram when Fig. 4 is k=10 of the present invention, wherein, (a) represents joint 1, and (b) represents joint 2.

Position tracking effect schematic diagram when Fig. 5 is k=40 of the present invention, wherein, (a) represents joint 1, and (b) represents joint 2.

Position tracking error schematic diagram when Fig. 6 is k=40 of the present invention, wherein, (a) represents joint 1, and (b) represents joint 2.

Speed tracing schematic diagram when Fig. 7 is k=40 of the present invention, wherein, (a) represents joint 1, and (b) represents joint 2.

Controller input schematic diagram when Fig. 8 is k=40 of the present invention, wherein, (a) represents joint 1, and (b) represents joint 2.

Fig. 9 is control flow schematic diagram of the present invention.

Embodiment

Below in conjunction with accompanying drawing, the present invention will be further described.

With reference to Fig. 1-Fig. 9, a kind of neural network full-order sliding mode control method of mechanical arm servo-drive system, comprises the following steps:

Step 1, sets up the dynamic model of mechanical arm servo-drive system, initialization system state, sampling time and controling parameters, and process is as follows:

The dynamic model expression-form of 1.1 mechanical arm servo-drive systems is:

$M\left(q\right)\stackrel{\·\·}{q}+C(q,\stackrel{\·}{q})\stackrel{\·}{q}+D\stackrel{\·}{q}+G\left(q\right)=\mathrm{\τ}---\left(1\right)$

Wherein q, with be respectively the position of joint of mechanical arm, speed and acceleration, M (q), and D represents the symmetric positive definite inertial matrix in each joint respectively, the diagonal angle positive definite matrix of centrifugal Coriolis matrix and damping friction coefficient; G (q) represents gravity item; τ represents the torque input vector in joint;

Formula (1), when considering Dynamic Execution mechanism, is expressed as by 1.2 again:

$M_H\left(q\right)\stackrel{\·\·}{q}+C_H(q,\stackrel{\·}{q})\stackrel{\·}{q}+D_{H}q+G_{H}q=u---\left(2\right)$

Wherein, $M_H=K_{\mathrm{\τ}}^{-1}(N^{-1}M+{\mathrm{NJ}}_m),C_H=K_{\mathrm{\τ}}^{-1}{N}^{-1}C,D_H=K_{\mathrm{\τ}}^{-1}(N^{-1}D+{\mathrm{ND}}_m),$ $G_H=K_{\mathrm{\τ}}^{-1}{N}^{-1}G;u=K_{\mathrm{\τ}}^{-1}{\mathrm{\τ}}_m$ It is the vector of an armature voltage input; represent electromagnetic torque vector, wherein, J _mand D _mrepresent inertia diagonal matrix respectively and reverse ratio of damping; K _τ=diag (K _{τ 1}, K _{τ 2}..., K _{τ n}) be then the torque constant of diagonal matrix; q _mwhat represent is motor Angular position vector; τ _lrepresentative be the vector of electric motor load torque; represent the diagonal matrix of n joint transmission gear;

1.3 owing to existing measurement noises, the impact of load variations and external disturbance, and the systematic parameter in formula (2) can not obtain accurately; So, again the systematic parameter of reality is rewritten as:

$M_H\left(q\right)={\hat{M}}_H\left(q\right)+{\mathrm{\ΔM}}_H\left(q\right)$

$C_H(q,\stackrel{\·}{q})={\hat{C}}_H(q,\stackrel{\·}{q})+{\mathrm{\ΔC}}_H(q,\stackrel{\·}{q})$

$D_H={\hat{D}}_H+{\mathrm{\ΔD}}_H$

$G_H\left(q\right)={\hat{G}}_H\left(q\right)+{\mathrm{\ΔG}}_H\left(q\right)---\left(3\right)$

Wherein, estimated value and represent known portions; Δ M _h(q), Δ D _hand Δ G _h(q) representative system unknown term;

Step 2, based on the mechanical arm servo-drive system with unknown parameter, the neural network of design, process is as follows:

2.1 definition θ ^*for ideal weight matrix of coefficients, so nonlinear uncertain function f approached for:

f＝θ ^*Tφ(x)+ε (4)

Wherein, represent input vector; φ (x)=[φ ₁(x), φ ₂(x) ... φ _m(x)] ^tit is the basis function of neural network; ε represents the approximate error of neural network and meets || ε || and≤ε _n, ε _nit is then a positive constant; φ _ix () is taken as following Gaussian function usually:

${\mathrm{\φ}}_i\left(x\right)=\exp \[-\frac{\left|\right|x-c_i\left|\right|}{2{\mathrm{\σ}}_i^2}\],i=1,2,\mathrm{...},n---\left(5\right)$

Wherein, c _irepresent the nuclear parameter of Gaussian function; σ _ithen illustrate the width of Gaussian function;

Step 3, calculating control system tracking error, design full-order sliding mode face;

3.1 define system tracking errors are:

e＝q _d-q (6)

Wherein, q _dfor second order can lead desired trajectory; So the first differential of formula (6) and second-order differential are represented as following form:

$\stackrel{\·}{e}={\stackrel{\·}{q}}_d-\stackrel{\·}{q}---\left(7\right)$

$\stackrel{\·\·}{e}={\stackrel{\·\·}{q}}_d-\stackrel{\·\·}{q}---\left(8\right)$

3.2 so full-order sliding mode face will be defined as:

$s=\stackrel{\·\·}{e}+c_{2}sgn\left(\stackrel{\·}{e}\right)|\stackrel{\·}{e}{|}^{{\mathrm{\α}}_2}+c_{1}sgn\left(e\right)|e{|}^{{\mathrm{\α}}_1}---\left(9\right)$

Wherein, c ₁and c ₂be a positive constant, its selection ensures polynomial expression p ²+ c ₂whole characteristic roots of p+c in the left-half of complex plane to ensure system stability; α ₁and α ₂choose, be by following polynomial expression:

$\left\{\begin{array}{c}{\mathrm{\α}}_1=\mathrm{\α},n=1\\ {\mathrm{\α}}_{i-1}=\frac{{\mathrm{\α}}_i{\mathrm{\α}}_{i+1}}{2{\mathrm{\α}}_{i+1}-{\mathrm{\α}}_i},i=2,\mathrm{...},n,\∀n\≥2\end{array}\right.---\left(10\right)$

Wherein, α _n+1=1, α _n=α, α ∈ (1-ε, 1) and ε ∈ (0,1);

Step 4, based on the mechanical arm servo-drive system containing Dynamic Execution mechanism, according to full-order sliding mode and neural network theory, design neural network full-order sliding mode controller, process is as follows:

4.1 consider formula (2), and neural network full-order sliding mode controller is designed to:

$\begin{array}{c}u={\hat{M}}_H\left(q\right)({\stackrel{\·\·}{q}}_d+c_2\mathrm{sgn}(\stackrel{\·}{e})|\stackrel{\·}{e}{|}^{{\mathrm{\α}}_2}+c_1\mathrm{sgn}\left(e\right)|e{|}^{{\mathrm{\α}}_1}+u_0)\\ +{\hat{C}}_H(q,\stackrel{\·}{q})\stackrel{\·}{q}+{\hat{D}}_H\stackrel{\·}{q}+{\hat{G}}_H\left(q\right)\end{array}---\left(11\right)$

$u_0={\widehat{\mathrm{\θ}}}^T\mathrm{\φ}\left(x\right)-{{\hat{M}}_H}^{-1}\left(q\right)u_n---\left(12\right)$

${\stackrel{\·}{u}}_n+{\mathrm{Tu}}_n=v---\left(13\right)$

v＝-(k _d+k _T+η)sgn(s) (14)

Wherein, c _iand α _i(i=1,2) are constants, are defined in formula (9); k _d, k _tbe all constant with η, and will be described afterwards;

The Rule adjusting of 4.2 design neural network weight coefficient matrix:

$\stackrel{\·}{\widehat{\mathrm{\θ}}}=\mathrm{\Γ}\mathrm{\φ}\left(x\right)s^T---\left(15\right)$

Wherein, Γ is the diagonal matrix of a positive definite;

Formula (11) to be brought in formula (2) and is obtained following equation by 4.3:

$s={\mathrm{\θ}}^{{*}^T}\mathrm{\φ}\left(x\right)+\mathrm{\ϵ}-{\widehat{\mathrm{\θ}}}^T\mathrm{\φ}\left(x\right)+u_n={\widetilde{\mathrm{\θ}}}^T\mathrm{\φ}\left(x\right)+\mathrm{\ϵ}+u_n=d(q,t)+u_n---\left(16\right)$

Wherein, represent the weight evaluated error of neural network; representative system disturbance term, and be bounded, so we suppose d (q, t)≤l _dand wherein l _dit is the constant of a bounded; K _tto choose be meet k when K > 0 _t>=Tl _d;

4.4 through types (2), formula (9), formula (11)-(14) and formula (16), full-order sliding mode face is written to following equation:

s＝d(q,t)+u _n(17)

Formula (14) to be brought in formula (13) and is obtained by 4.5:

$\begin{array}{c}{u}_n\left(t\right)=\left(u_n\right(t_0)+(1/T)(k_d+k_T+\mathrm{\η})\mathrm{sgn}\left(s\right))e^{t-t_0}\\ -(1/T)(k_d+k_T+\mathrm{\η})\mathrm{sgn}\left(s\right)\end{array}---\left(18\right)$

At u _n(0), when=0, following equation is obtained:

k _T≥Tl _d≥T|u _n(t)| _max≥T|u _n(t)| (19)

4.6 design Lyapunov functions:

$V=\frac{1}{2}{s}^{T}s---\left(20\right)$

Carry out differentiate to formula (9) to obtain:

$\stackrel{\·}{s}=\stackrel{\·}{d}(q,t)+{\stackrel{\·}{u}}_n=\stackrel{\·}{d}(q,t)+{\stackrel{\·}{u}}_n+{\mathrm{Tu}}_n-{\mathrm{Tu}}_n=\stackrel{\·}{d}(q,t)+v-{\mathrm{Tu}}_n---\left(21\right)$

Formula (13) is brought in formula (21) and obtains:

$\stackrel{\·}{s}=\stackrel{\·}{d}(q,t)-(k_d+k_T+\mathrm{\η})\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n---\left(22\right)$

Carry out differential to formula (20) to obtain:

$\stackrel{\·}{V}=s^T\stackrel{\·}{s}=\stackrel{\·}{d}(q,t)s^T-(k_d+k_T+\mathrm{\η})s^T\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n{s}^T---\left(23\right)$

Formula (19) is brought in formula (23), if then decision-making system is stable.

In order to obtain the corresponding system parameter value in formula (2), we provide the mechanical arm servo-drive system expression formula in following two joints:

$\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{12}& a_{22}\end{array}\right]\left[\begin{array}{c}{\stackrel{\·\·}{q}}_1\\ {\stackrel{\·\·}{q}}_2\end{array}\right]+\left[\begin{array}{cc}-b_{12}{\stackrel{\·}{q}}_1& -2b_{12}{\stackrel{\·}{q}}_2\\ 0& b_{12}{\stackrel{\·}{q}}_2\end{array}\right]\left[\begin{array}{c}{\stackrel{\·}{q}}_1\\ {\stackrel{\·}{q}}_2\end{array}\right]+\left[\begin{array}{c}{c}_{1}g\\ c_{2}g\end{array}\right]=\left[\begin{array}{c}{\mathrm{\τ}}_1\\ {\mathrm{\τ}}_2\end{array}\right]+\left[\begin{array}{c}{\mathrm{\τ}}_{d_1}\\ {\mathrm{\τ}}_{d_2}\end{array}\right]---\left(24\right)$

Wherein, $a_{11}=(m_1+m_2)r_1^2+m_2{r}_2^2+2m_2{r}_1{r}_{2}cos\left(q_2\right)+J,$ a ₁₂＝m ₂r ₂+m ₂r ₁r ₂cos(q ₂)， b ₁₂＝m ₂r ₁r ₂sin(q ₂)，c ₁＝(m ₁+m ₂)r ₁cos(q ₂)+m ₂r ₂cos(q ₁+q ₂)，c ₂＝m ₂r ₂cos(q ₁+q ₂)， ${\mathrm{\τ}}_d=\left[\begin{array}{c}{\mathrm{\τ}}_{d_1}\\ {\mathrm{\τ}}_{d_2}\end{array}\right].$

For the validity of checking institute extracting method, The present invention gives neural network finite-time control (Neuralnetwork-based finite time control, NNFTC) method and neural network full-order sliding mode control the contrast of (Chattering-free neural sliding mode control, CFNSMC) method:

Contrast in order to more effective, all parameters of system are all consistent, that is: q ₁(0)=0.5, q ₂(0)=0.5, J _m=diag (0.67 × 10 ^-4, 0.42 × 10 ^-4), D _m=diag (0.21,0.15), N=diag (9,1), and given system disturbance is: system control signal parameter is: K _τ=diag (19/40,19/80), Γ=diag (50,50), α ₁=13/27, α ₂=13/27, c ₁=100, c ₂=40, T=1; Mechanical arm actual parameter is: r ₁=0.2, r ₂=0.18, m ₁=2.3, m ₂=0.6, J ₁=0.02, J ₂=0.003, g=9.8.In addition, we make k=k _d+ k _t+ η, and in two kinds of different value of K situations, two kinds of control methods are contrasted respectively.

Situation one: k=10

The mechanical arm system followed the tracks of due to us is two joints, so we follow the tracks of q _d1=sin (2 π t) and q _d2=sin (2 π t).From Fig. 1 and Fig. 2, we find out, CFNSMC method and NNFTC method have approximate tracking effect for the moment in tracking joint; But CFNSMC method has better tracking effect when following the tracks of joint 2 than NNFTC method.Can obviously find out from Fig. 3 tracking velocity schematic diagram, the curve that the tracking velocity curve of CFNSMC method obtains than NNFTC method is more level and smooth.In addition in the diagram, NNFTC method has obvious chattering phenomenon, but CFNSMC method then eliminates this chattering phenomenon.

Situation two: k=40

From Fig. 5 and Fig. 6, we find out that NNFTC method even has better tracking effect when following the tracks of joint 1 than CFNSMC method; But, when following the tracks of joint 2, but not as CFNSMC method.In addition, from Fig. 6 and Fig. 7, we obtain, and on control inputs signal and tracking velocity curve, CFNSMC method is all smoothly more many than NNFTC method.Further, comparison diagram 1 and Fig. 5, we clearly find out that CFNSMC method has stronger robustness than NNFTC method on different gain k.

In sum, contrast NNFTC method, CFNSMC method is more insensitive to different ride gain k, namely has more strong robustness; And in control signal and speed tracing signal, have the ability better eliminated and buffet.

What more than set forth is the excellent effect of optimization that an embodiment that the present invention provides shows, obvious the present invention is not just limited to above-described embodiment, do not depart from essence spirit of the present invention and do not exceed scope involved by flesh and blood of the present invention prerequisite under can do all distortion to it and implemented.

Claims (1)

1. a mechanical arm servo-drive system neural network full-order sliding mode control method, is characterized in that: described control method comprises the following steps:

Step 1, sets up the dynamic model of mechanical arm servo-drive system, initialization system state, sampling time and controling parameters, and process is as follows:

The dynamic model expression-form of 1.1 mechanical arm servo-drive systems is:

$M\left(q\right)\stackrel{\·\·}{q}+C(q,\stackrel{\·}{q})\stackrel{\·}{q}+D\stackrel{\·}{q}+G\left(q\right)=\mathrm{\τ}---\left(1\right)$

Wherein, q, with be respectively the position of joint of mechanical arm, speed and acceleration, M (q), and D represents the symmetric positive definite inertial matrix in each joint respectively, the diagonal angle positive definite matrix of centrifugal Coriolis matrix and damping friction coefficient; G (q) represents gravity item; τ represents the torque input vector in joint;

Formula (1), when considering Dynamic Execution mechanism, is expressed as by 1.2 again:

$M_H\left(q\right)\stackrel{\·\·}{q}+C_H(q,\stackrel{\·}{q})\stackrel{\·}{q}+D_{H}q+G_{H}q=u---\left(2\right)$

Wherein, $M_H=K_{\mathrm{\τ}}^{-1}(N^{-1}M+{\mathrm{NJ}}_m),C_H=K_{\mathrm{\τ}}^{-1}{N}^{-1}C,D_H=K_{\mathrm{\τ}}^{-1}(N^{-1}D+{\mathrm{ND}}_m),$ it is the vector of an armature voltage input; represent electromagnetic torque vector, wherein, J _mand D _mrepresent inertia diagonal matrix respectively and reverse ratio of damping; K _τ=diag (K _{τ 1}, K _{τ 2}..., K _{τ n}) be then the torque constant of diagonal matrix; q _mwhat represent is motor Angular position vector; τ _lrepresentative be the vector of electric motor load torque; represent the diagonal matrix of n joint transmission gear;

1.3 owing to existing measurement noises, the impact of load variations and external disturbance, and the systematic parameter in formula (2) can not obtain accurately; So, again the systematic parameter of reality is rewritten as:

$M_H\left(q\right)={\hat{M}}_H\left(q\right)+{\mathrm{\ΔM}}_H\left(q\right)$

$C_H(q,\stackrel{\·}{q})={\hat{C}}_H(q,\stackrel{\·}{q})+{\mathrm{\ΔC}}_H(q,\stackrel{\·}{q})$

$D_H={\hat{D}}_H+{\mathrm{\ΔD}}_H$

$G_H\left(q\right)={\hat{G}}_H\left(q\right)+{\mathrm{\ΔG}}_H\left(q\right)---\left(3\right)$

Wherein, estimated value and represent known portions; Δ M _h(q), Δ D _hand Δ G _h(q) representative system unknown term;

Step 2, based on the mechanical arm servo-drive system with unknown parameter, the neural network of design, process is as follows:

Definition θ ^*for ideal weight matrix of coefficients, so nonlinear uncertain function f approached for:

f＝θ ^*Tφ(x)+ε (4)

Wherein, represent input vector; φ (x)=[φ ₁(x), φ ₂(x) ... φ _m(x)] ^tit is the basis function of neural network; ε represents the approximate error of neural network and meets || ε || and≤ε _n, ε _nit is then a positive constant; φ _ix () is taken as following Gaussian function:

${\mathrm{\φ}}_i\left(x\right)=\exp \[-\frac{||x-c_i||}{2{\mathrm{\σ}}_i^2}\],i=1,2,...,n---\left(5\right)$

Wherein, c _irepresent the nuclear parameter of Gaussian function; σ _ithen illustrate the width of Gaussian function;

Step 3, calculating control system tracking error, design full-order sliding mode face, process is as follows:

3.1 define system tracking errors are:

e＝q _d-q (6)

Wherein, q _dfor second order can lead desired trajectory; So the first differential of formula (6) and second-order differential are represented as following form:

$\stackrel{\·}{e}={\stackrel{\·}{q}}_d-\stackrel{\·}{q}---\left(7\right)$

$\stackrel{\·\·}{e}={\stackrel{\·\·}{q}}_d-\stackrel{\·\·}{q}--\left(8\right)$

3.2 so full-order sliding mode face will be defined as:

$s=\stackrel{\·\·}{e}+c_{2}sgn\left(\stackrel{\·}{e}\right){\left|\stackrel{\·}{e}\right|}^{{\mathrm{\α}}_2}+c_{1}sgn\left(e\right){\left|e\right|}^{{\mathrm{\α}}_1}---\left(9\right)$

Wherein, c ₁and c ₂be a positive constant, its selection ensures polynomial expression p ²+ c ₂whole characteristic roots of p+c in the left-half of complex plane to ensure system stability; α ₁and α ₂choose, be by following polynomial expression:

$\left\{\begin{array}{c}{\mathrm{\α}}_1=\mathrm{\α},n=1\\ {\mathrm{\α}}_{i-1}=\frac{{\mathrm{\α}}_i{\mathrm{\α}}_{i+1}}{2{\mathrm{\α}}_{i+1}-{\mathrm{\α}}_i},i=2,...,n,\∀n\≥2\end{array}\right.---\left(10\right)$

Wherein, α _n+1=1, α _n=α, α ∈ (1-ε, 1) and ε ∈ (0,1);

Step 4, based on the mechanical arm system containing Dynamic Execution mechanism, according to full-order sliding mode and neural network theory, design neural network full-order sliding mode controller, process is as follows:

4.1 consider formula (2), and neural network full-order sliding mode controller is designed to:

$\begin{array}{c}u={\hat{M}}_H\left(q\right)\left({\stackrel{\·\·}{q}}_d+c_2\mathrm{sgn}\left(\stackrel{\·}{e}\right){\left|\stackrel{\·}{e}\right|}^{{\mathrm{\α}}_2}+c_1\mathrm{sgn}\left(e\right){\left|e\right|}^{{\mathrm{\α}}_1}+u_0\right)\\ +{\hat{C}}_H\left(q,\stackrel{\·}{q}\right)\stackrel{\·}{q}+{\hat{D}}_H\stackrel{\·}{q}+{\hat{G}}_H\left(q\right)\end{array}---\left(11\right)$

$u_0={\widehat{\mathrm{\θ}}}^T\mathrm{\φ}\left(x\right)-{{\hat{M}}_H}^{-1}\left(q\right)u_n---\left(12\right)$

${\stackrel{\·}{u}}_n+{\mathrm{Tu}}_n=v---\left(13\right)$

v＝-(k _d+k _T+η)sgn(s) (14)

Wherein, c _iand α _iconstant, i=1,2, be defined in formula (9); k _d, k _tall constant with η;

The Rule adjusting of 4.2 design neural network weight coefficient matrix:

$\stackrel{\·}{\widehat{\mathrm{\θ}}}=\mathrm{\Γ}\mathrm{\φ}\left(x\right)s^T---\left(15\right)$

Wherein, Γ is the diagonal matrix of a positive definite;

Formula (11) to be brought in formula (2) and is obtained following equation by 4.3:

$s={\mathrm{\θ}}^{{*}^T}\mathrm{\φ}\left(x\right)+\mathrm{\ϵ}-{\widehat{\mathrm{\θ}}}^T\mathrm{\φ}\left(x\right)+u_n={\widetilde{\mathrm{\θ}}}^T\mathrm{\φ}\left(x\right)+\mathrm{\ϵ}+u_n=d(q,t)+u_n---\left(16\right)$

Wherein, represent the weight evaluated error of neural network; representative system disturbance term, and be bounded, so suppose d (q, t)≤l _dand wherein l _dit is the constant of a bounded; K _tto choose be meet k when K > 0 _t>=Tl _d;

4.4 through types (2), formula (9), formula (11)-(14) and formula (16), full-order sliding mode face is written to following equation:

s＝d(q,t)+u _n(17)

Formula (14) to be brought in formula (13) and is obtained by 4.5:

$\begin{array}{c}{u}_n\left(t\right)=\left(u_n\right(t_0)+(1/T)(k_d+k_T+\mathrm{\η})\mathrm{sgn}\left(s\right))e^{t-t_0}\\ -\left(1/T\right)\left(k_d+k_T+\mathrm{\η}\right)\mathrm{sgn}\left(s\right)\end{array}---\left(18\right)$

At u _n(0), when=0, following equation is obtained:

k _T≥Tl _d≥T|u _n(t)| _max≥T|u _n(t)| (19)

4.6 design Lyapunov functions:

$V=\frac{1}{2}{s}^{T}s---\left(20\right)$

Carry out differentiate to formula (9) to obtain:

$\stackrel{\·}{s}=\stackrel{\·}{d}(q,t)+{\stackrel{\·}{u}}_n=\stackrel{\·}{d}(q,t)+{\stackrel{\·}{u}}_n+{\mathrm{Tu}}_n-{\mathrm{Tu}}_n=\stackrel{\·}{d}(q,t)+v-{\mathrm{Tu}}_n---\left(21\right)$

Formula (13) is brought in formula (21) and obtains:

$\stackrel{\·}{s}=\stackrel{\·}{d}(q,t)-(k_d+k_T+\mathrm{\η})\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n---\left(22\right)$

Carry out differential to formula (20) to obtain:

$\stackrel{\·}{V}=s^T\stackrel{\·}{s}=\stackrel{\·}{d}(q,t)s^T-(k_d+k_T+\mathrm{\η})s^T\mathrm{sgn}\left(s\right)-{\mathrm{Tu}}_n{s}^T---\left(23\right)$

Formula (19) is brought in formula (23), if then decision-making system is stable.