Tracking of periodic oscillations in an underactuated system via adaptive neural networks

Abstract

In this paper, the tracking control of periodic oscillations in an underactuated mechanical system is discussed. The proposed scheme is derived from the feedback linearization control technique and adaptive neural networks are used to estimate the unknown dynamics and to compensate uncertainties. The proposed neural network-based controller is applied to the Furuta pendulum, which is a nonlinear and nonminimum phase underactuated mechanical system with two degrees of freedom. The new neural network-based controller is experimentally compared with respect to its model-based version. Results indicated that the proposed neural algorithm performs better than the model-based controller, showing that the real-time adaptation of the neural network weights successfully estimates the unknown dynamics and compensates uncertainties in the experimental platform.

Introduction

The study of oscillatory behavior as well as its control in many physical systems has been an active research topic in recent years. For example, Parmananda et al. ¹ reported numerical and experimental results indicating successful stabilization of unstable steady states and periodic orbits in an electrochemical system. Semenov et al. ² investigated the stabilization of the inverted pendulum with vertically oscillating suspension under hysteretic control. In Olm et al., ³ an analysis of digital controllers under time-varying sampling period for the tracking of periodic oscillations was given. More recently, tracking control of a class of nonlinear systems using oscillation was considered in Wang and Guo. ⁴ Specifically, based on the oscillation functions associated with accessible vibrating components of the system, oscillatory control was designed to track a desired trajectory.

There is a large class of underactuated mechanical systems which arises from problems in robotics and mechatronic systems. The Furuta pendulum is an underactuated electromechanical system of two degrees of freedom (DOF) and has only one actuator. ⁵ It is an inverted pendulum, which is classified into a nonlinear, nonminimum phase and underactuated system. ⁶ One of the most important applications of Furuta and inverted pendulums is to test linear and nonlinear control techniques. Some control techniques developed are bang-bang control,^5,7 PID control,^8,9 energy-based control,^10–12 feedforward control,^13,14 sliding mode control,^15,16 fuzzy logic,^17–19 singular perturbation-based control, ²⁰ as well as hybrid control ²¹ and predictive control. ²² However, we will focus our attention to neural network-based control.^23–25

Artificial neural networks are useful to compensate unknown dynamic systems. There are some works where neural networks were used with this objective as Nelson and Kraft ²⁶ where a full-state feedback optimal controller was developed to control an inverted pendulum in the upright position. In Sazonov et al., ²⁷ a hybrid system controller, incorporating a neural network plus an optimal linear controller applied to inverted pendulum in regulation mode was presented. The work in Shaheed ²⁸ introduced an open-loop control strategy with neural networks for vibration suppression in a flexible manipulator system. In Jung and Cho, ²⁹ a decoupled reference compensation technique with neural networks applied to the control of a two DOF inverted pendulum mounted on a x–y table was presented. In Noh et al., ³⁰ the implementation of position control applied to a mobile inverted pendulum using radial basis function neural networks was presented.

On the other hand, feedback linearization is a known control technique used in nonlinear systems.^31,32 The basic idea of this technique consists in defining an output function (usually linear in terms of the system state), which is derived r times with respect to time until the control input appears. Thus, the controller can be computed so that the solutions of resulting closed-loop system in terms of the output converge to zero. Depending on the class of the underactuated mechanical system under consideration and the proposed output function, the closed-loop system may be composed of an internal dynamics which must be stable in order to accomplish the control task. ³³ The previous literature review shows that only a few works have addressed the problem of tracking control of periodic oscillatory trajectories of the Furuta pendulum.

In this paper, the problem of inducing periodic oscillatory behavior in the Furuta pendulum is addressed by proposing a new adaptive neural network-based controller for the tracking of periodic trajectories. The proposed control algorithm has a structure inspired by a model-based controller, which is designed using the feedback linearization technique with a new output function. The advantage of using adaptive neural networks is that the parameters of the dynamic model are not needed for the design of the control algorithm. The convergence analysis of the output trajectories is shown and conditions for the solutions of the internal dynamics to be bounded are given. An extensive experimental evaluation is performed in a test bed, where the new adaptive neural network controller is compared with respect to its model-based version. The two controllers are tested using two different periodic trajectories, showing that better tracking performance is obtained with new scheme.

Furuta pendulum dynamics and control goal

As mentioned before, the Furuta pendulum is an underactuated system of two DOF. Figure 1 shows a description of the joint positions and applied torque. The dynamic model of the Furuta pendulum in Euler–Lagrange form^6,34–36 is written as

$$M\left(\mathit{q}\right)¨\mathit{q}+C\left(\mathit{q},\dot{}\mathit{q}\right)\dot{}\mathit{q}+{\mathit{g}}_m\left(\mathit{q}\right)+\mathit{f}(\dot{}\mathit{q})=\mathit{u}$$

(1)

with

$$\begin{array}{c}\mathit{q}\mathrm{=}\left[\begin{array}{c}{q}_1\\ q_2\end{array}\right],\mathit{u}\mathrm{=}\left[\begin{array}{c}\tau \\ 0\end{array}\right],\\ M\left(\mathit{q}\right)=\left[\begin{array}{cc}{\theta }_1+{\theta }_2{\sin }^2\left(q_2\right)& {\theta }_3\cos \left(q_2\right)\\ {\theta }_3\cos \left(q_2\right)& {\theta }_4\end{array}\right],\\ C\left(\mathit{q},\dot{}\mathit{q}\right)=\left[\begin{array}{cc}\frac{1}{2}{\theta }_2\dot{}{q}_2\sin \left(2q_2\right)& -{\theta }_3\dot{}{q}_2\sin \left(q_2\right)+\frac{1}{2}{\theta }_2\dot{}{q}_1\sin \left(2q_2\right)\\ -\frac{1}{2}{\theta }_2\dot{}{q}_1\sin \left(2q_2\right)& 0\end{array}\right],\\ {\mathit{g}}_m\left(\mathit{q}\right)=\left[\begin{array}{c}0\\ -{\theta }_5\sin \left(q_2\right)\end{array}\right],\mathit{f}(\dot{}\mathit{q})=\left[\begin{array}{c}{\theta }_6\dot{}{q}_1+{\theta }_8\mathrm{sig}\mathrm{n}\left(\dot{}{q}_1\right)\\ {\theta }_7\dot{}{q}_2+{\theta }_9\mathrm{sig}\mathrm{n}\left(\dot{}{q}_2\right)\end{array}\right]\end{array}$$

where $\mathit{q}\in {\mathbb{R}}^2$ is the vector of joint positions, $u\in {\mathbb{R}}^2$ u∈R2 is the vector of input torques, being $\tau \in \mathbb{R}$ the torque applied to the arm, $M\left(\mathit{q}\right)\in {\mathbb{R}}^{2\times 2}$ is the symmetric positive definite inertia matrix, $C\left(\mathit{q},\dot{}\mathit{q}\right)\dot{}\mathit{q}\in {\mathbb{R}}^2$ is the vector of centripetal and Coriolis torques, ${\mathit{g}}_m\left(\mathit{q}\right)\in {\mathbb{R}}^2$ is the vector of gravitational torques, $f(\dot{}\mathit{q})\in {\mathbb{R}}^2$ is the vector of friction torques, and the model parameters θ_i are positive constants related to the physical properties of the system. Finally, $\mathrm{sign(x)}{=}^1$ , for $\mathrm{x}{>}^0$ , $\mathrm{sign(x)}{=}^{-1}$ , for $\mathrm{x}{<}^0$ , and $\mathrm{sign(x)}{=}^0$ , for $\mathrm{x}{=}^0$ .

Figure 1.

Three-dimensional model of the Furuta pendulum with the relative measurement of the joint positions and applied torque.

It is assumed that the desired trajectory $q_d(t)$ is a bounded, continuous, and twice-differentiable signal, which satisfies

$$\left|\right|q_d\left(t\right)\left|\right|,\left|\right|{\dot{q}}_d\left(t\right)\left|\right|,\left|\right|{\ddot{q}}_d\left(t\right)\left|\right|\le \mu$$

(2)

where μ is a positive constant. Besides, in this study we consider that the desired trajectory for the arm position is a periodic oscillatory signal, that is $q_d\left(t\right)=q_d\left(t+T_s\right)$ with $T_s>0$ being the period of the signal. However, the proposed methodology can be applied for the tracking of any continuous and twice-differentiable signal bounded in the sense of equation (2).

The motivation of introducing oscillatory behavior in pendulum-like systems has been well explained in Canudas-de-Wit et al. ³⁷ and Shiriaev et al. ³⁸ where the authors established that a pendulum is the simplest model of a walking robot leg and introducing periodic behavior in the closed-loop system for underactuated mechanical systems is an approach to solve the motion control problem in walking robots.

The joint position tracking error vector is defined as

$$\mathit{e}=\left[\begin{array}{c}{e}_1\\ e_2\end{array}\right]=\left[\begin{array}{c}{q}_d-q_1\\ -q_2\end{array}\right]\in {\mathbb{R}}^2$$

(3)

Then, the control problem consists in designing a controller $\tau (t)\in \mathbb{R}$ , which does not require the exact knowledge of the Furuta pendulum model, such that the error signal $e(t)$ is uniformly ultimately bounded.

Adaptive neural network controller

The Furuta pendulum model in equation (1) can be rewritten as

$$¨\mathit{q}=-M{\left(\mathit{q}\right)}^{-1}\left[C\left(\mathit{q},\dot{}\mathit{q}\right)+{\mathit{g}}_m\left(\mathit{q}\right)+\mathit{f}(\dot{}\mathit{q})\right]+M{\left(\mathit{q}\right)}^{-1}u=\left[\begin{array}{c}{f}_1\left(\mathit{q},\dot{}\mathit{q}\right)\\ f_2\left(\mathit{q},\dot{}\mathit{q}\right)\end{array}\right]+\left[\begin{array}{c}{g}_1\left(\mathit{q}\right)\\ g_2\left(\mathit{q}\right)\end{array}\right]\tau$$

which at the same time can be expressed in terms of the tracking error in equation (3) in the following form

$$\frac{\text{d}}{\text{d}t}{e}_1={\dot{e}}_1$$

(4)

$$\frac{\text{d}}{\text{d}t}{\dot{e}}_1={\ddot{q}}_d-f_1-g_1\tau$$

(5)

$$\frac{\text{d}}{\text{d}t}{e}_2={\dot{e}}_2$$

(6)

$$\frac{\text{d}}{\text{d}t}{\dot{e}}_2=-f_2-g_2\tau$$

(7)

The control goal is to satisfy with only one input τ(t) the tracking of a desired trajectory for the arm position while the pendulum remains regulated at the upward unstable equilibrium. In other words, both the trajectory tracking error e₁(t) and the regulation error e₂(t) defined in equation (3) must remain bounded in a region around the origin, in spite of the fact that only one actuator is available to control the system. Thus, the designed control action $\tau (t)\in \mathbb{R}$ is given on the basis of an output function in terms of the signals e₁(t) and e₂(t). Inspired by passivity-based control schemes, where output signals with the structure $s=\dot{e}+\Delta e$ are commonly used, we propose the output function $y(t)\in \mathbb{R}$ given by

$$y={\Delta }_1{e}_1+{\dot{e}}_1+{\Delta }_2{e}_2+{\dot{e}}_2$$

(8)

where ${\Delta }_1$ and ${\Delta }_2$ are positive constants.

The time derivative of the output in equation (8) is

$$\dot{y}={\ddot{q}}_d-F-G\tau +{\Delta }_1{\dot{e}}_1+{\Delta }_2{\dot{e}}_2$$

(9)

where

$$F=f_1+f_2$$

(10)

$$G=g_1+g_2$$

(11)

A feedback linearization controller is proposed for the cancellation of nonlinearities through a feedback control law, transforming the nonlinear system in equation (9) into an equivalent linear system whose trajectories converge asymptotically to the origin. The proposed control law is given by

$$\tau =\frac{-F+{\ddot{q}}_d+{\Delta }_1{\dot{e}}_1+{\Delta }_2{\dot{e}}_2+ky+\delta \text{sign}\left(y\right)}{G}$$

(12)

where the proportional term ky, with the positive constant k, ensures the asymptotic convergence of the output trajectories; the nonlinear term $\delta \text{sign}(y)$ , with the positive constant δ, is a term aimed to introduce robustness in the compensation of external disturbances and unmodeled error dynamics; the nonlinear functions F and G are defined in equations (10) and (11), respectively. Then, by substituting equation (12) into equation (9), we obtain

$$\dot{y}=-ky-\delta \text{sign}\left(y\right)$$

(13)

It is possible to show that y(t) converges to zero. A way to prove this claim is computing the time derivative of the following positive definite function

$$U_1=\frac{1}{2}{y}^2$$

(14)

which is given by

$${\dot{U}}_1=y\left[-ky-\delta \text{sign}\left(y\right)\right]=-k{y}^2-\delta \left|y\right|$$

(15)

As can be seen, the time derivative of U₁ given in equation (15) is a negative definite function, proving that the trajectories of y(t) converge to the origin with exponential rate of convergence as time t goes to infinity.

A smooth function $h:{\mathbb{R}}^n\to \mathbb{R}$ can be approximated by a neural network. Then, given a compact set $S\in {\mathbb{R}}^n$ , there exists a two-layer neural network such that^23,39

$$h\left(\mathit{x}\right)=W^{\top }\mathit{\sigma }\left(V^{\top }\overline{\mathit{x}}\right)+ϵ$$

(16)

where $ϵ$ is a constant expressing the approximation error such that ${ϵ}_N>\left|ϵ\right|>0$ , $\overline{\mathit{x}}={[{\mathit{x}}^{\top }1]}^{\top }\in {\mathbb{R}}^{n+1}$ is the vector of input signals containing a unitary element for threshold weights, with the state vector $\mathit{x}\in {\mathbb{R}}^n,V\in {\mathbb{R}}^{(n+1)\times L}$ and $W\in {\mathbb{R}}^L$ are matrices containing values of input and output weights of the neural network, respectively, and $\mathit{\sigma }\in {\mathbb{R}}^L$ is a vector of membership functions. In the study presented in this paper, n = 4 since $\mathit{x}{=}^{\left[q_1{q}_2\dot{}{q}_1\dot{}{q}_2\right]}\in {\mathbb{R}}^4$ .

Then, taking into account the model-based controller in equation (12), the following adaptive neural network controller is proposed

$$\tau =\frac{-\widehat{F}+{\ddot{q}}_d+{\Delta }_1{\dot{e}}_1+{\Delta }_2{\dot{e}}_2+ky+\delta \text{sign}\left(y\right)}{\widehat{G}}$$

(17)

where k and δ are positive constants, $\text{sign}(y)$ is defined as the sign of y, and $\widehat{F}$ and $\widehat{G}$ are neural network-based estimations of the function F and G, respectively, given by

$$\widehat{F}={\widehat{W}}_f^{\top }\mathit{\sigma }\left(V_f^{\top }\overline{\mathit{x}}\right),\widehat{G}={\widehat{W}}_g^{\top }\mathit{\sigma }\left(V_g^{\top }\overline{\mathit{x}}\right)$$

(18)

where the vectors ${\widehat{W}}_f\in {\mathbb{R}}^L$ and ${\widehat{W}}_g\in {\mathbb{R}}^L$ are the estimated output weights of the neural network, $V_f\in {\mathbb{R}}^{5\times L}$ and $V_g\in {\mathbb{R}}^{5\times L}$ are the input weights, in which the threshold values have been taken into account. The values of the matrices V_f and V_g are constant and random, ⁴⁰ $\mathit{\sigma }(\mathit{p})={\left[\tanh ({\mathit{p}}_1)\dots \tanh ({\mathit{p}}_{\mathit{L}})\right]}^{\top }\in {\mathbb{R}}^L$ is a vector of membership functions. Besides, L is the quantity of neurons, n = 4 is the dimension of the state vector of the mechanical system. Finally, the estimated output weights of the neural networks can be obtained with the following adaptation laws

$${\dot{\widehat{W}}}_f=-M\mathit{\sigma }\left(V_f^{\top }\overline{\mathit{x}}\right)y-{\kappa }_{f}M\left|y\right|{\widehat{W}}_f$$

(19)

$${\dot{\widehat{W}}}_g=-N\mathit{\sigma }\left(V_g^{\top }\overline{\mathit{x}}\right)y\tau -{\kappa }_{g}N\left|y\tau \right|{\widehat{W}}_g$$

(20)

where ${\kappa }_f>0,{\kappa }_g>0$ and $M,N\in {\mathbb{R}}^{L\times L}$ are diagonal positive definite matrices. See the Figure 2 for a description of the block diagram implementation of the adaptive neural network controller in equation (17).

Figure 2.

Block diagram implementation of the adaptive neural network controller in equation (17).

The proposed controller in equation (17) has a nonlinear quotient structure with an adaptive neural network function in the numerator and another one in the denominator. Besides, the controller includes linear PD terms in the tracking error $e_1(t)$ and $e_2(t)$ , which results from the output function y(t). The new controller has a robustifying term in order to compensate the modeling errors that remain from the neural network compensation. This controller allows carrying out the tracking of a reference trajectory for the arm while the pendulum in the second joint is regulated at the upright position. The weights of the neural network are updated in real time by means of properly designed adaptation laws.

By using the universal function approximation property of the neural networks in equation (16), which implies that

$$F=W_f\mathit{\sigma }\left(V_f^{\top }\overline{\mathit{x}}\right)+{ϵ}_f,\text{ and }G=W_g\mathit{\sigma }\left(V_g^{\top }\overline{\mathit{x}}\right)+{ϵ}_g$$

with $W_f\in {\mathbb{R}}^L$ and $W_g\in {\mathbb{R}}^L$ being the constant ideal output weights, â_f, ${ϵ}_g\in \mathbb{R}$ being the approximation errors, and with the weight estimation error definitions

$${\tilde{W}}_f=W_f-{\widehat{W}}_f,{\tilde{W}}_g=W_g-{\widehat{W}}_g$$

we have that

$$\dot{y}=-ky-\delta \text{sign}\left(y\right)-{\tilde{W}}_f^{\top }\mathit{\sigma }\left(V_f^{\top }\overline{\mathit{x}}\right)-{\tilde{W}}_g^{\top }\mathit{\sigma }\left(V_g^{\top }\overline{\mathit{x}}\right)\tau -{ϵ}_f-{ϵ}_g\tau$$

(21)

Taking into account that the ideal weights W_f and W_g are constants, equations (19) and (20) can be written in terms of the estimation errors ${\tilde{W}}_f$ and ${\tilde{W}}_g$ , as follows

$${\dot{\tilde{W}}}_f=M\mathit{\sigma }\left(V_f^{\top }\overline{\mathit{x}}\right)y+{\kappa }_{f}M\left|y\right|\left[W_f-{\tilde{W}}_f\right]$$

(22)

$${\dot{\tilde{W}}}_g=N\mathit{\sigma }\left(V_g^{\top }\overline{\mathit{x}}\right)y\tau +{\kappa }_{g}N\left|y\tau \right|\left[W_g-{\tilde{W}}_g\right]$$

(23)

In order to prove that the solution y(t) coming from closed-loop system in equations (21) to (23) converges to zero as time t increases and the output weight estimation errors ${\tilde{W}}_f(t)$ and ${\tilde{W}}_g(t)$ are bounded for all time, the following positive definite function is proposed

$$U_2=\frac{1}{2}{y}^2+\frac{1}{2}\text{Tr}\left({\tilde{W}}_f^{\top }{M}^{-1}{\tilde{W}}_f\right)+\frac{1}{2}\text{Tr}\left({\tilde{W}}_g^{\top }{N}^{-1}{\tilde{W}}_g\right)$$

(24)

where $\text{Tr}(A)$ means the trace of a square matrix $A\in {\mathbb{R}}^{n\times n}$ , i.e. $\text{Tr}(A)={\displaystyle {\sum }_{i=1}^n{a}_{ii}}$ . The time derivative of U₂ in equation (24) is given by

$${\dot{U}}_2=y\dot{y}+\text{Tr}\left({\tilde{W}}_f^{\top }{M}^{-1}{\dot{\tilde{W}}}_f\right)+\text{Tr}\left({\tilde{W}}_g^{\top }{N}^{-1}{\dot{\tilde{W}}}_g\right)$$

(25)

By replacing equations (21) to (23) into equation (25), we obtain

$$\begin{array}{c}{\dot{U}}_2=-k{y}^2-\delta \left|y\right|-{ϵ}_{f}y-{ϵ}_{g}y\tau +{\kappa }_f\left|y\right|\text{Tr}\left({\tilde{W}}_f^{\top }\left[W_f-{\tilde{W}}_f\right]\right)\\ +{\kappa }_g\left|y\tau \right|\text{Tr}\left({\tilde{W}}_g^{\top }\left[W_g-{\tilde{W}}_g\right]\right)+\left|y\right|{\psi }_0\end{array}$$

(26)

$$=-k{y}^2-\delta \left|y\right|-{ϵ}_{f}y-{ϵ}_{g}y\tau -{\kappa }_f\left|y\right|\text{Tr}\left({\tilde{W}}_f^{\top }{\tilde{W}}_f\right)-{\kappa }_g\left|y\tau \right|\text{Tr}\left({\tilde{W}}_g^{\top }{\tilde{W}}_g\right)+\left|y\right|{\psi }_0$$

(27)

where

$${\psi }_0=-{ϵ}_f\text{sign}\left(y\right)-\tau {ϵ}_g\text{sign}\left(y\right)+{\kappa }_f\text{Tr}\left({\tilde{W}}_f^{\top }{W}_f\right)+{\kappa }_g\left|\tau \right|\text{Tr}\left({\tilde{W}}_g^{\top }{W}_g\right)$$

(28)

It will be useful to recall Frobenius norm of a matrix $A\in {\mathbb{R}}^{m\times n}$ , which is given by

$$\left|\right|A|{|}_F=\sqrt{Tr\left(A^{\top }A\right)}=\sqrt{{\displaystyle \sum _{i,j}{a}_{ij}^2}}$$

(29)

Besides, a useful property for matrices $A\in {\mathbb{R}}^{m\times n}$ and $B\in {\mathbb{R}}^{m\times n}$ is

$$\left|Tr\left(A^{\top }B\right)\right|\le \left|\right|A\left|{|}_F\right|\left|B\right|{|}_F$$

(30)

Thus, using the definition in equation (29) and the property in equation (30), the signal ψ₀ in equation (28) satisfies

$$\left|\right|{\psi }_0\left|\right|\le c_1\left|\right|{\tilde{W}}_f|{|}_F+c_2+[c_3\left|\right|{\tilde{W}}_g|{|}_F+c_4]\left|\tau \right|$$

(31)

where

$$\begin{array}{c}{c}_1={\kappa }_f\left|\right|W_f|{|}_F,\\ c_2=\left|{ϵ}_f\right|,\\ c_3={\kappa }_g\left|\right|W_g|{|}_F,\\ c_4=\left|{ϵ}_g\right|\end{array}$$

The definition in equation (30), the fact $\left|y\tau \right|=\left|y\right|\left|\tau \right|$ , and the inequality in equation (31) can be used to write the following upper bound on ${\dot{U}}_2$ as follows

$$\begin{array}{c}{\dot{U}}_2\le -k{y}^2-\delta \left|y\right|-{\kappa }_f\left|y\right|\left|\right|{\tilde{W}}_f|{|}_F^2-{\kappa }_g|y\left|\right|\tau \left|\right|\left|{\tilde{W}}_g\right|{|}_F^2\\ +\left[c_1\left|\right|{\tilde{W}}_f|{|}_F+c_2\right]\left|y\right|+\left[c_3\left|\right|{\tilde{W}}_g|{|}_F+c_4\right]\left|y\right|\left|\tau \right|\end{array}$$

(32)

After some algebra and by completing the square for two set of terms associated to $\left|\right|{\tilde{W}}_f|{|}_F$ and $\left|\right|{\tilde{W}}_g|{|}_F$ , we are able to write

$$\begin{array}{c}{\dot{U}}_2\le -k{y}^2-\left[\delta -c_2-{\kappa }_f\frac{1}{4}\frac{c_1^2}{{\kappa }_f^2}-{\kappa }_g\frac{1}{4}\frac{c_3^2}{{\kappa }_g^2}\left|\tau \right|\right]\left|y\right|\\ -{\kappa }_f\left|y\right|{\left[\left|\right|W_f|{|}_F^2-\frac{1}{2}\frac{c_1}{{\kappa }_f}\right]}^2-{\kappa }_g\left|y\right|\left|\tau \right|{\left[\left|\right|W_g|{|}_F^2-\frac{1}{2}\frac{c_3}{{\kappa }_g}\right]}^2\end{array}$$

(33)

Since the torque τ(t) in equation (17) is bounded for some compact set of the state space $\left[{\mathit{q}}^{\top }{\dot{q}}^{\top }{\widehat{W}}_f^{\top }{\widehat{W}}_g^{\top }\right]\in {\mathbb{R}}^{4+2L}$ and under the assumption that the desired arm trajectory $q_d(t)$ satisfies equation (2), the condition

$$\delta >c_2+{\kappa }_f\frac{1}{4}\frac{c_1^2}{{\kappa }_f^2}+{\kappa }_g\frac{1}{4}\frac{c_3^2}{{\kappa }_g^2}|\tau {|}_{\text{max}},|\tau \left(t\right)|\le |\tau {|}_{\text{max}}$$

guarantees

$${\dot{U}}_2\le -k{y}^2=-\chi \left(y\right)$$

(34)

As ${\dot{U}}_2(t)\le 0$ , which implies that y(t), ${\tilde{W}}_f(t)$ , and ${\tilde{W}}_g(t)$ are bounded. By integrating both sides of equality (34) it can be shown that ${\int }_0^t\chi \left(y\left(\varsigma \right)\right)\text{d}\varsigma \le -{\int }_0^t{\dot{U}}_2\left(\varsigma \right)\text{d}\varsigma \le U_2\left(0\right)-U_2\left(t\right)\le U_2\left(0\right)$ ; therefore, $\underset{t\to \infty }{\lim }{\int }_0^t\chi \left(y\left(\varsigma \right)\right)\text{d}\varsigma$ exists, and it is finite. Moreover, since $\dot{\chi }\left(t\right)$ is bounded, $\chi \left(t\right)$ is uniformly continuous and by Barbalat’s lemma, ³¹ $\underset{t\to \infty }{\lim }\chi \left(t\right)=0$ implying also that $y\left(t\right)\to 0$ as $t\to \infty$ .

By using the transformation

$$\mathit{z}=\left[\begin{array}{c}{\eta }_1\\ {\eta }_2\\ {\eta }_3\\ y\end{array}\right]=H\mathit{x}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -\frac{g_2}{g_1}& 1\\ {\Delta }_1& {\Delta }_2& 1& 1\end{array}\right]\left[\begin{array}{c}{e}_1\\ e_2\\ {\dot{e}}_1\\ {\dot{e}}_2\end{array}\right]$$

(35)

which is invertible, the internal dynamics can be computed as follows

$$\frac{d}{dt}\left[\begin{array}{c}{\eta }_1\\ {\eta }_2\\ {\eta }_3\end{array}\right]=\left[\begin{array}{c}-{\Delta }_1{G}_1{\eta }_1-{\Delta }_2{G}_1{\eta }_2-G_1{\eta }_3+G_{1}y\\ -{\Delta }_1{G}_2{\eta }_1-{\Delta }_2{G}_2{\eta }_2+G_1{\eta }_3+G_{2}y\\ \frac{{\theta }_3}{{\theta }_4}\cos \left({\eta }_2\right){\ddot{q}}_{d1}-\frac{{\theta }_3}{{\theta }_4}\cos \left({\eta }_2\right)f_1-\frac{{\theta }_3}{{\theta }_4}\sin \left({\eta }_2\right){\dot{\eta }}_1{\dot{\eta }}_2-f_2\end{array}\right]$$

(36)

where $G_1=\frac{g_1}{g_1+g_2}$ and $G_2=\frac{g_2}{g_1+g_2}$ .

A local analysis of the system (36) showed that necessary and sufficient conditions to ensure the uniformly ultimately boundedness of the trajectories ${\left[{\eta }_1\left(t\right){\eta }_2\left(t\right){\eta }_3\left(t\right)\right]}^{\top }$ are the following

$${\theta }_3>{\theta }_4$$

(37)

$$0<{\Delta }_1<\frac{{\theta }_5}{{\theta }_7}$$

(38)

$${\Delta }_2>\frac{{\Delta }_1{\theta }_4+{\theta }_7}{{\theta }_3}+\frac{{\Delta }_1{\theta }_5\left({\theta }_3-{\theta }_4\right)}{{\theta }_3\left({\theta }_5-{\Delta }_1{\theta }_7\right)}$$

(39)

where ${\Delta }_1$ and ${\Delta }_2$ are control gains and the Furuta pendulum parameters θ_i are constants. This implies that the error trajectories $\mathit{e}(t)={[e_1(t)e_2(t)]}^{\top }\in {\mathbb{R}}^2$ are also uniformly ultimately bounded. In fact, the inequalities (38) and (39) provide an explicit tuning guideline for the control gains ${\Delta }_1$ and ${\Delta }_2$ . Further details about the local analysis of the system (36) can be found in Moreno-Valenzuela et al.^41,42

Experimental results

In this section, two controllers are experimentally tested: the model-based feedback linearization controller in equation (12), and the new adaptive neural network control law in equation (17). The experimental tests have been carried out in a Furuta pendulum built at the Instituto Politécnico Nacional—CITEDI, as shown in Figure 3. The experimental platform consists of a PC running Windows XP. The controllers are programmed using Matlab, Simulink, and Real-Time Windows Target. The experimental setup is also equipped with a 30A20AC servo amplifier of Advanced Motion Controls, a servomotor which is used as actuator, and a Sensoray 626 data acquisition board. The Furuta pendulum is provided with optical incremental encoders in order to sense the relative joint position. The numerical values of the parameters θ_i of the dynamic model in equation (1) were experimentally identified and are given in Table 1.

Figure 3.

Furuta pendulum built at the Instituto Politécnico Nacional—CITEDI.

Table 1.

Identified parameters of the Furuta pendulum dynamic model in equation (1).

Symbol	Value	Unit
θ 1	0.06196	kg m2 rad
θ 2	0.01492	kg m2 rad
θ 3	0.01855	kg m2 rad
θ 4	0.01318	kg m2 rad
θ 5	0.50767	kg m2 rad
θ 6	0.00837	$\text{N}\text{m}\frac{\text{rad}}{\text{s}}$
θ 7	0.00071	$\text{N}\text{m}\frac{\text{rad}}{\text{s}}$
θ 8	0.01886	$\text{N}\text{m}\frac{\text{rad}}{\text{s}}$
θ 9	0.00871	$\text{N}\text{m}\frac{\text{rad}}{\text{s}}$

The joint velocities are estimated using the following algorithm

$$\dot{\mathit{q}}\left(kT\right)=\frac{\mathit{q}\left(kT\right)-\mathit{q}\left(\left(k-1\right)T\right)}{T}$$

(40)

where T = 0.001 $[s]$ is the sample time and $k=0,1,2,\dots$ is the time index. Initials conditions of the system were set to ${\left[q_1\left(0\right)q_2\left(0\right)\right]}^{\top }={\left[00\right]}^{\top }\left(\text{rad}\right)$ and ${\left[{\dot{q}}_1\left(0\right){\dot{q}}_2\left(0\right)\right]}^{\top }={\left[00\right]}^{\top }\left(\text{rad/s}\right).$

For the model-based controller in equation (12) and adaptive neural network controller in equation (17), the gains

$${\Delta }_1=0.7,{\Delta }_2=3.5,k=2.1,\text{ and }\delta \text{=0}\text{.07}$$

(41)

were used.

For the adaptive neural network controller in equation (17), the number of neurons was L = 10, the matrix $M=0.7I_{10}\in {\mathbb{R}}^{10\times 10}$ , the matrix $N=1.75I_{10}\in {\mathbb{R}}^{10\times 10},I_{10}\in {\mathbb{R}}^{10\times 10}$ is an identity matrix, ${\kappa }_f=0.035$ and ${\kappa }_g=0.175$ .

Periodic oscillation I

A set of experiments by using the desired periodic oscillation

$$q_d\left(t\right)=1.0\sin \left(t\right)\left(\text{rad}\right)$$

(42)

was done under the above-written conditions.

The results of the real-time implementation of the controller in equation (12) are given in Figure 4, which shows the performance of joint positions $q_1(t)$ and q₂(t) and the applied torque τ(t).

Figure 4.

Periodic oscillation I, model-based controller in equation (12). Left-hand side plot: time evolution of (a) q₁(t) and (b) q₂(t). Right-hand side plot: (c) applied torque τ(t).

The new adaptive neural network controller proposed in equation (17), with signals $\widehat{F}(t)$ and $\widehat{G}(t)$ in equation (18), is also implemented in real time. Figure 5 shows the time evolution of q₁(t) and $q_2(t)$ , and the applied control effort τ(t). Finally, the estimated weights ${\widehat{W}}_f(t)$ and ${\widehat{W}}_g(t)$ of the neural network are depicted in Figure 6, which remain bounded during the time that the controller is working, as predicted by theory.

Figure 5.

Periodic oscillation I, adaptive neural network controller in equation (17). Left-hand side plot: time evolution of (a) q₁(t) and (b) q₂(t). Right-hand side plot: (c) applied torque τ(t).

Figure 6.

Periodic oscillation I, adaptive neural network controller in equation (17). Time evolution of the estimated output weights (a) ${\widehat{W}}_f(t)$ and (b) ${\widehat{W}}_g(t)$ .

Periodic oscillation II

In this section, in order to show that the proposed controller is able to track a variety of periodic trajectories, a more complex desired trajectory was selected. By using the control gains in equation (41) in the model-based controller in equation (12) and the adaptive neural network controller in equation (17), we have carried out real-time experiments using the desired periodic oscillation

$$q_d\left(t\right)=1.0\sin \left(t\right)+0.2\cos \left(4t\right)\left(\text{rad}\right)$$

(43)

which exhibits a more complex behavior with respect to the desired periodic oscillation in equation (42), allowing us to test the performance of the new controller.

Results for the model-based controller in equation (12) are given in Figure 7, where the time evolution of joint trajectories q₁(t) and q₂(t), and the applied torque τ(t) are appreciated.

Figure 7.

Periodic oscillation II, model-based controller in equation (12). Left-hand side plot: time evolution of (a) q₁(t) and (b) q₂(t). Right-hand side plot: (c) applied torque τ(t).

The results of the implementation of the adaptive neural network controller in equation (17) are given in Figure 8, which depicts the actual joint position trajectories q₁(t) and q₂(t), and the applied control action τ(t). Figure 9 describes the estimated weights ${\widehat{W}}_f(t)$ and ${\widehat{W}}_g(t)$ obtained with update laws in equations (22) and (23), respectively. Similarly, the output weights remain bounded for all time.

Figure 8.

Periodic oscillation II, adaptive neural network controller in equation (17). Left-hand side plot: time evolution of (a) q₁(t) and (b) q₂(t). Right-hand side plot: (c) applied torque τ(t).

Figure 9.

Periodic oscillation II, adaptive neural network controller in equation (17). Time evolution of the estimated output weights (a) ${\widehat{W}}_f(t)$ and (b) ${\widehat{W}}_g(t)$ .

Note that in all experiments the applied torque τ(t) in Figures 4, 5, 7, and 8 exhibits high frequency components. The reasons of these high frequency components are the quantization noise introduced by the encoders and the discrete estimation of the joint velocity given by equation (40).

Performance comparison

In order to compare the performance of the implemented controllers, Figure 10 depicts the time evolution of errors $e_1(t)$ and $e_2(t)$ for both controllers. In the left-hand side plot, the results by using the periodic oscillation $q_d(t)$ in equation (42) are given, and in the right-hand side plot the comparison of the errors by using the periodic oscillation $q_d(t)$ in equation (43) is illustrated. Visual examination of the plots in Figure 10 suggests that the best performance is obtained with the new adaptive neural network-based controller (17).

Figure 10.

Comparison of the tracking errors (a), (c) e₁(t) and (b), (d) e₂(t) for the implementation of the model-based controller in equation (12) and the adaptive neural network controller in equation (17). Left-hand side plot: comparison of the errors for the periodic oscillation I. Right-hand side plot: comparison of the errors for the periodic oscillation II.

In order to assess the tracking performance of both controllers, the RMS values of the tracking error ${[e_1(t)e_2(t)]}^{\top }$ signal is computed. The results are given in Tables 2 and 3 for the desired periodic oscillation in equations (42) and (43), respectively. The column improvement means the percentage of variation of the new adaptive neural network controller in equation (17) with respect to the model-based controller in equation (12). In order to compare the performance during different time intervals, the RMS value is computed for different time intervals.

Table 2.

Periodic oscillation I: RMS values of the tracking error signals ${[e_1(t)e_2(t)]}^{\top }$ .

Interval (s)	Model based	Adaptive NN	Improvement (%)
20–25	0.2591	0.1720	+33.62
15–25	0.2392	0.1696	+29.10
5–25	0.2594	0.1697	+34.58

NN: neural network; RMS: root mean square.

Table 3.

Periodic oscillation II: RMS values of tracking error signals ${[e_1(t)e_2(t)]}^{\top }$ .

Interval (s)	Model based	Adaptive NN	Improvement (%)
20–25	0.3369	0.2990	+11.25
15–25	0.3303	0.2873	+13.02
5–25	0.3219	0.2788	+13.39

NN: neural network; RMS: root mean square.

The results observed in Figure 10, which shows the time evolution of $e_1(t)$ and $e_2(t)$ for both implementations and the obtained RMS values of the tracking errors in Tables 2 and 3, suggest that the best performance is obtained with the adaptive neural network controller in equation (17).

The improvement can be only explained due to the contribution of the adaptive neural network in the controller, which compensates other types of nonmodeled dynamics in equation (1), such as static friction, switching of the PWM servo amplifier, and vibrations in the mechanical structure of the mechanism.

Finally, in order to verify that the neural network contribution to the total applied torque is significant, Figure 11 shows the torque contributions for the model-based controller in equation (12) and the adaptive neural network controller in equation (17), which were separated as follows

$$\tau ={\tau }_1+{\tau }_2$$

where

$${\tau }_1=\frac{-F}{G},{\tau }_2=\frac{{\ddot{q}}_d+{\Delta }_1{\dot{e}}_1+{\Delta }_2{\dot{e}}_2+ky+\delta \text{sign}\left(y\right)}{G}$$

for the model-based controller in equation (12), and

$$\begin{array}{c}{\tau }_1=\frac{-\widehat{F}}{\widehat{G}},\\ {\tau }_2=\frac{{\ddot{q}}_d+{\Delta }_1{\dot{e}}_1+{\Delta }_2{\dot{e}}_2+ky+\delta \text{sign}\left(y\right)}{\widehat{G}}\end{array}$$

for the adaptive neural network controller in equation (17).

Figure 11.

Torque contributions for the (a) model-based controller in equation (12) and (b) the adaptive neural network-based controller in equation (17).

As can be seen in Figure 11, the right-hand side plot shows that the neural network contribution ${\tau }_1(t)$ is comparable in magnitude with ${\tau }_2(t)$ , which contains the proportional part plus the sign term in y(t). It can also be seen that the signal ${\tau }_2(t)=-\widehat{F}(t)/\widehat{G}(t)$ from adaptive neural-network scheme in the right-hand side of Figure 11 remains bounded and is comparable in magnitude to the signal ${\tau }_2(t)=-F(t)/G(t)$ from the model-based controller in the left-hand side of Figure 11.

Conclusion

The main purpose of this document has been the introduction of a new controller in order to induce periodic oscillations in an underactuated system. The new scheme is based on adaptive neural networks, which has the advantage that the system model is not required to be known.

The new scheme is inspired from the feedback linearization technique and a novel output function y(t). The output function y(t) was chosen as a linear combination of the position errors and velocity errors.

An extensive real-time experimental study has also been presented, where a model-based controller is compared with the new adaptive neural network scheme. Two types of periodic oscillation were used in order to assess the adaptation capabilities of the new scheme. Better results were obtained with the new scheme since this is able to compensate disturbances and unmodeled dynamics.