6. Identification of the unknown non-linear system by the neural network

First, it is easy to see that <sup>e</sup>,W^ � � <sup>¼</sup> 0 is a state of equilibrium (equilibrium point). Previous, so we propose to demonstrate that this point of equilibrium is asymptotically stable; for this, be:

$$\tilde{u}\_p = -\gamma \left(\frac{1}{2} + \frac{1}{2} \left\| \left| \hat{W} \right| \right\|^2 L\_\phi^2 \right) e\_p \tag{33}$$

We will show, the feedback system is asymptotically stable. Replacing (36) in (35)

$$aD\_t^a e\_p = \mathbf{A}(e\_p) - \tilde{W}\Gamma\_\mathbf{z}[\mathbf{x}\_n(t-\tau)] + \hat{W}\mathcal{O}(e\_p(t-\tau)) - \gamma \left(\frac{1}{2} + \frac{1}{2}||\hat{W}||^2 L\_\phi^2\right) e\_p \tag{34}$$

We will show, the new state ep is asymptotically stable, and the equilibrium point is ep ! 0, when <sup>W</sup>^ <sup>σ</sup>ð Þ¼ xnð Þ <sup>t</sup> � <sup>τ</sup> 0, as an external disturbance.

Trajectory Tracking Using Adaptive Fractional PID Control of Biped Robots with Time-Delay… DOI: http://dx.doi.org/10.5772/intechopen.90901

Let V be, the next candidate Lyapunov function as

$$\begin{split} V &= \frac{1}{2} \left( e\_p^{\;T}, w^T \right) \left( e\_p, w \right)^T + \frac{1}{2a} tr \left\{ \dot{W}^T \ddot{W} \right\} \\ &+ \frac{1}{a} \int\_{t-\tau}^t \left[ \mathcal{O}\_\sigma^T(s) \hat{W}^T \dot{W} \mathcal{O}\_\sigma(s) \right] ds \end{split} \tag{35}$$

Then, (35) is reduced to

$$aD\_t^a V \le \frac{-1}{a} \left(\lambda - 1 + K\_p\right) \left(e\_p^{\;T}\right) \left(e\_p\right) - \frac{(\gamma - 1)}{a} \left(\frac{1}{2} + \frac{1}{2} ||\hat{W}||^2 L\_\phi^2\right) \left(e\_p^{\;T}\right) \left(e\_p\right) < 0 \quad \text{(36)}$$

The previous inequality guarantees that the identification of the non-linear system is satisfied, that is, the approach error converges to zero asymptotically

$$\begin{aligned} u\_p &= \Delta^\dagger \left[ \dot{W} \Gamma z(\mathbf{x}\_r(t-\tau)) - \dot{W} \Gamma \Big( \mathbf{z}(\mathbf{x}\_n(t-\tau)) - \mathbf{z} \Big( \mathbf{x}\_p(t-\tau) \Big) \right) \\ &- (A+I) \left( \mathbf{x} - \mathbf{x}\_p \right) + K\_p e + K\_i a D\_t^{-a} e + K\_v a D\_t^a e \\ &- \Gamma \Big( \frac{1}{2} + \frac{1}{2} ||\dot{W}||^2 L\_\phi^2 \Big) e\_n - \Gamma \Big( \frac{1}{2} + \frac{1}{2} ||\dot{W}||^2 L\_\phi^2 \Big) e\_p + f\_r(\mathbf{x}\_r, u\_r) \\ &- f\_p \left( \mathbf{x}\_p \right) + A \mathbf{x}\_p - A \mathbf{x}\_r + \dot{W} \Gamma\_\mathbf{z} \left( \mathbf{x}\_p \right) - \mathbf{x}\_r + \mathbf{x}\_p \Big] \end{aligned} \tag{37}$$

#### 7. Simulation

The mathematical model, which describes the movement dynamics of the bipedal robot, is obtained using the Euler-Lagrange equations [1, 13] (Figure 2).

$$D(q(t)\ddot{q}(t) + \mathcal{C}(q(t), \dot{q}(t))\dot{q}(t) + \mathcal{G}(q(t)) = B\tau(t)$$

where q tðÞ¼ <sup>q</sup>31ð Þ<sup>t</sup> <sup>q</sup>32ð Þ<sup>t</sup> <sup>q</sup>41ð Þ<sup>t</sup> <sup>q</sup>42ð Þ<sup>t</sup> � �<sup>T</sup> , is the generalized coordinates vector. As usual, Dqt ð ð Þ) is the inertia matrix, bounded and positive definite, and Cqt ð Þ ð Þ, q t \_ð Þ is the matrix of Coriolis and centripetal forces. Gqt ð Þ ð Þ represents a matrix of gravitational effects and B defines the input matrix. The vector τðÞ¼ t ½ � <sup>τ</sup>31ð Þ<sup>t</sup> <sup>τ</sup>32ð Þ<sup>t</sup> <sup>τ</sup>41ð Þ<sup>t</sup> <sup>τ</sup>42ð Þ<sup>t</sup> <sup>T</sup>, defines the applied joint torques of the robot.

To illustrate the theoretical results obtained, we propose an example, which, as can be seen in the simulations, trajectory tracking is guaranteed.

The neural network is described by:

aD<sup>α</sup> <sup>t</sup> xp <sup>¼</sup> <sup>A</sup>ð Þþ <sup>x</sup> <sup>W</sup><sup>∗</sup> <sup>Γ</sup>zð Þþ x tð Þ � <sup>τ</sup> <sup>Ω</sup>u, with <sup>τ</sup> = 25 s, <sup>A</sup> ¼ �20I,<sup>I</sup> <sup>∈</sup> <sup>4</sup><sup>x</sup>4, and, W<sup>∗</sup> is estimated using the learning law given in (28).

$$\begin{aligned} \Gamma\_{\mathbf{z}}(\mathbf{x}(t-\tau)) &= \left(\tanh\left(\mathbf{x}\_{1}(t-\tau)\right), \tanh\left(\mathbf{x}\_{2}(t-\tau)\right), \dots, \tanh\left(\mathbf{x}\_{n}(t-\tau)\right)\right)^{\mathrm{T}},\\ \Omega &= \begin{pmatrix} \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} \end{pmatrix}^{\mathrm{T}} \text{ and the } u \text{ is obtained using (33).} \end{aligned}$$

and the reference signal that they have to follow, both the non-linear system and the neural network is given by the Duffing equation [14].

Figure 2. Dynamic model of biped robot.

#### Figure 3.

Time evolution for the angular position Leg 1 and Leg 2 (rad) of link 1.

$$\ddot{\mathbf{x}} - \mathbf{x} + \mathbf{x}^3 = \mathbf{0.114} \cos \left( \mathbf{1.1t} \right) : \mathbf{x}(\mathbf{0}) = \mathbf{1}, \dot{\mathbf{x}}(\mathbf{0}) = \mathbf{0.114}$$

$$\frac{\mathbf{x}(t)}{dt} = \mathbf{y}(t)$$

$$\frac{\mathbf{y}(t)}{dt} = \mathbf{x}(t) - \mathbf{x}^3(t) - a\mathbf{y}(t) + \delta \cos \left( at \right)$$

Trajectory Tracking Using Adaptive Fractional PID Control of Biped Robots with Time-Delay… DOI: http://dx.doi.org/10.5772/intechopen.90901

Here, the conventional derivatives are replaced by the fractional derivatives as follows:

$$aD\_t^a \mathfrak{x}(t) = \mathfrak{y}(t)$$

$$aD\_t^a \mathfrak{x}(t) = \mathfrak{x}(t) - \mathfrak{x}^\mathfrak{J}(t) - a\mathfrak{y}(t) + \delta \cos\left(at\right)$$

where α, ω, δ, are the parameters of the Duffing differential equation, which we will use as a reference trajectory, that the non-linear system and the neural network have to follow.

As can be seen in Figures 3–6, the tracking of trajectories in the states of the system are performed with satisfaction, Figure 7 shows the phase plane of the

Figure 4. Time evolution for the angular position Leg 1 and Leg 2 (rad) of link 2.

Figure 5. Time evolution for the angular position Leg 1 and Leg 2 (rad) of link 1.

Figure 6. Time evolution for the angular position Leg 1 and Leg 2 (rad) of link 2.

Duffing equation, while Figure 8 shows the plane phase of the same fractional order differential equation.

Figures 9–12 show the torques applied to the ends of the bipedal robot.

Parameter values of the fractional order, alpha (0.001) and beta (0.0001) are included.

$$a = \mathbf{1}, \quad \beta = \mathbf{1}$$

$$a = \mathbf{0}.\mathbf{0}\mathbf{0}\mathbf{1}, \quad \beta = \mathbf{0}.\mathbf{0}\mathbf{1}$$

#### 8. Conclusions

In this chapter we study the mathematical model and control of non-linear systems, which are modeled by differential equations of fractional order, where it is Trajectory Tracking Using Adaptive Fractional PID Control of Biped Robots with Time-Delay… DOI: http://dx.doi.org/10.5772/intechopen.90901

Figure 8. A phase space trajectory of Duffing equation.

Figure 9. Torque (Nm) applied to Leg 1 and Leg 2 of link 1.

observed that these systems have a better performance than the systems modeled by ordinary differential equations, those of fractional order they produce responses, solutions at simulation level, softer, by varying the order of the derivative.

The magnitude of the fractional order systems are smaller than the responses of the systems of ordinary differential equations, and with smaller control signals, which implies, less energy in the control process.

In this research work, conditions have been obtained in the parameters of the adaptive recurrent neural network, as well as laws of control and laws of neuronal adaptation, which, together, guarantee that the tracking error of trajectories between the non-linear system and the reference signal converges asymptotically to zero, so that trajectory tracking is develops with satisfaction.

Figure 10. Torque (Nm) applied to Leg 1 and Leg 2 of link 2.

Figure 11. Torque (Nm) applied to Leg 1 and Leg 2 of link 1.

Figure 12. Torque (Nm) applied to Leg and Leg 2 of link 2.

Trajectory Tracking Using Adaptive Fractional PID Control of Biped Robots with Time-Delay… DOI: http://dx.doi.org/10.5772/intechopen.90901
