Notes on Hybrid neural network fraction integral terminal sliding mode control of an Inchworm robot manipulator

Author:- Uddesh Tople

Tags : sliding mode neural network chattering hybrid control

Brief Outline

This paper propose a novel hybrid control technique for chattering reduction. Adaptive neural network(ANN) can be used to estimate the unknown disturbances. Chattering phenomena can be reduced by using neural network.

Abstract

• The control scheme proposed is based on fractional integral terminal sliding mode control(FITSMC) and adaptive neural network.
• Fraction integral terminal sliding mode control is applied to obtain initial stability while adaptive neural network method is adopted to approximate system uncertainties and unknown disturbances to reduce chattering phenomena.
• the ITSMC is proposed to remove reaching stage for increasing robustness of the fractional-order system.

Kinematic Description

Distance travelled by snake per cycle is given by

$x=2l(1-cos\alpha )$

where,

$l$ is length of link and

$\alpha$ is the gait angle. The main motion pattern can be categorized into four sub-mechanisms as shown in the table.

considering angles formed by link in the global frame. The geometric constraints of the system can be defined as

$\sum _{i=1}^{n}sin{\varphi }_i=0$ where

$n$ and

${\varphi }_i$ can be defined as no.of joints and angle of

$i^{th}$ link with positive x axis.

Dynamic Modelling of inchworm robot

Velocity of centroid:-

$\overline{v_p}=v_p+\frac{l_p}{2}\dot{{\varphi }_p}(-sin{\varphi }_p\hat{i}+cos\varphi \hat{j})$

Height of centroid point for

$p^{th}$ link :-

$\overline{h_p}=\sum _{j=1}^p(l_{j}sin{\varphi }_j)-\frac{l_p}{2}sin{\varphi }_p$

Total Kinetic Energy :-

$T=\frac{1}{2}\sum _{p=1}^3(m_p{\overline{v_p}}^2+I_p{\dot{{\varphi }_p}}^2)$

$T=\frac{1}{24}m{l}^2(28{\dot{{\varphi }_1}}^2+16{\dot{{\varphi }_2}}^2+4{\dot{{\varphi }_3}}^2+36\dot{{\varphi }_1}\dot{{\varphi }_2}{C}_{12}+12\dot{{\varphi }_2}\dot{{\varphi }_3}{C}_{23}+12\dot{{\varphi }_3}\dot{{\varphi }_1}{C}_{31})$

Total gravitational potential energy is given by :-

$V=\sum _{p=1}^3{m}_{p}g\overline{h_p}=\frac{1}{2}mgl(5sin{\varphi }_1+3sin{\varphi }_2+sin{\varphi }_3)$

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Virtual Work

Consider

$\tau$ as joint torque exerted to the links.

$F_r$ and

$F_f$ denote the normal and frictional force exerted at the tip of last link. Virtual work done due to allnon conservative forces is given by:-

where

$\lambda p=l_p(cos{\varphi }_p+\overline{\mu }sin\varphi )$ and

$\overline{\mu }=\mu sgn(v_{tip}.e_t)$

motion equations can be obtained by applying Euler-Lagrange relation,

$\frac{d}{dt}(\frac{\partial T}{\partial \dot{{\varphi }_x}})-\frac{\partial T}{\partial {\varphi }_x}+\frac{\partial V}{\partial {\varphi }_x}=Q_x$

Subsituting values of

$T,Vand{Q}_x$ from above equations for particular mechanism, we will get equations in the form:-

$M(\theta )\ddot{\theta }+C(\theta ){\dot{\theta }}^2+G(\theta )=D\tau +\lambda (\theta )F_R$

$M(\theta )$ is the mass matrix of manipulator,

$C(\theta )$ is the centrifugal coefficient matrix,

$G(\theta )$ is the gravity vector.

$\ddot{\theta }=-Y{\dot{\theta }}^2-PG+Q\tau +R{F}_r$
…where

$Y=M^{-1}(\theta )C(\theta ),$

$P=M^{-1}(\theta ),$

$Q=M^{-1}(\theta )D,$ and

$R=M^{-1}(\theta )\lambda$

$u(t)=\tau$ is the control vector.

$\mathrm{\Delta }Y$,

$\mathrm{\Delta }P$,

$\mathrm{\Delta }Q$,

$\mathrm{\Delta }F$ presents uncertainties of parameter variations.

$\dot{\theta }=-(Y+\mathrm{\Delta }Y){\theta }^2-(P+\mathrm{\Delta }P)G(\theta )+(Q+\mathrm{\Delta }Q)u(t)+(R+\mathrm{\Delta }R)F_r$

We can also express dynamic model in terms of state variable as:

$\dot{x}=f(x)+g(x)u+d_u$

$y=x$ …where

$y$ is controller output.

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Sliding mode control

Sliding surface is given by

$s=\dot{e}+\beta e$

where

$\beta$ =

$diag[{\beta }_1{\beta }_2{\beta }_3]$ is a known vector of slopes , named bandwidth of sliding mode control and

$e=\theta -{\theta }_d$ and

$\dot{e}=\dot{\theta }-\dot{{\theta }_d}$

For desired performance

$\dot{s}(t)=0$ with respect to uncertainty

$d(t)=0$

$\dot{s}=\ddot{\theta }-{\ddot{\theta }}_r$

$\dot{s}=-Y{\dot{\theta }}^2-PG(\theta )+Qu(t)+R{F}_r-{\ddot{\theta }}_r$

$u_{eq}=Q^{-1}[{\ddot{\theta }}_r+Y{\dot{\theta }}^2+PG(\theta )-R{F}_r]$

The secondary control endeavor deal with reaching control endeavor by

$u_s(t)$.In this issue, the Lyapunov function defined as:

$V(t)=\frac{1}{2}{s}^T(t)s(t)$
With V(0)=0 and V(t)>0 for

$s(t)\ne 0$. The reaching phase which guarantee the trajectory tracking of position error can be referred to reaching phase, which can be written as:-

$\dot{V}=s^T(t)s(t)<0$,

$s(t)\ne 0$
The equivalent control term thus can be written as :-

$u(t)=u_{eq}(t)+u_s(t)$
where

$u_s$ is reaching control and

$u_{e}q$ is sliding control term

According to equation of

$V$ above we have:-

Reaching control can be determined by :-

$u_s=K_{s}sign[s(t)]$
The SMC method displays the undesirable chattering phenomenon due to discontinuous part of controller. This can be reduced by replacing

$sgn(s)$ with

$sat(s,\xi )$

Integral Sliding mode control

Integral Sliding mode function is defined as:

$s(t)=e_x(t)+\lambda {\dot{e}}_{ix}(t)$

${\dot{e}}_{ix}(t)=sign(e_x(t))$

where

$\lambda$ >0;

$e_{ix}(t)$ is the integration of

$sign(e_x(t))$ and has intitial value

$-e_x(0)/\lambda$
therefore we can write above equation as:-

$s(t)=e_x(t)+\lambda {\int }_0^{t}sign(e_x(\tau ))dx$

If

$s(t)$ is maintained at zero such that

$e_x(t)=-\lambda e_{ix}(t)$, then with subsitution we have

${\dot{e}}_{ix}(t)=-sign(\lambda e_{ix}(t))$

However

$e_{ix}$ converge in finite time

$T_s=|e_{ix}(0)|=\frac{|e_x(0)|}{\lambda }$

Hybrid neural network fraction integral terminal sliding mode control(FITSMC)

FITSMC function is defined as;

$s(t)=e_x(t)+\lambda e_{ix}(t)$

${\dot{e}}_{ix}(t)=e_x^{\frac{p}{q}}(t)$

therefore, sliding function can be written as:

$s(t)=e_x(t)+\lambda {\int }_0^t{e}_x^{\frac{q}{p}}(\tau )d\tau$
on the surface s(t)=0, the integrator is defined as:

${\dot{e}}_{ix}(t)=-\lambda \frac{q}{p}{e}_{ix}^{\frac{q}{p}}(t)$

From solving error dynamic, the convergence time of

$e_{ix}(t)$ is obtained as:

$T_f=\frac{|e_{ix}(0){|}^{1-\frac{q}{p}}}{{\lambda }^{\frac{q}{p}}(1-\frac{q}{p})}=\frac{|e_x(0){|}^{1-\frac{q}{p}}}{\lambda (1-\frac{q}{p})}$