Trajectory Tracking Using Adaptive Fractional PID Control of Biped Robots with Time-Delay Feedback

1. Introduction

Fractional calculus is a generalization of differential and integral calculus which involves generalized functions. The first to work this new branch of mathematics was Leibniz. Due to the growing interest in the applications of fractional calculation, in this work we obtain conditions that guarantee the tracking of trajectories of nonlinear systems generated by differential equations of fractional order which we will call plants (This term is widely used in engineering), which in our case will be a mechanical arm, a helicopter, a plane or limbs of a humanoid, all of fractional order.

The problem of tracking control of trajectories is very important, since the control function allows the non-linear system to carry out a previously assigned task, work or trajectory, for example, a mechanical arm and its objective is to cut a piece with a previously generated form, or the coupling of two aircraft in space.

In this chapter we use adaptive recurrent neural networks with time delay, since its use allows us to work with systems whose mathematical model is unknown and with the presence of uncertainties, this is a well-known problem of robust control.

We include mathematical models with time delay, since the processing and transmission of information is important in this type of systems, which depending on the delay, these systems can generate undesirable oscillatory or chaotic dynamics, and cause instability in the mathematical model that describes the trajectory tracking error.

The chapter is organized as follows: first, the general mathematical model of non-linear systems is proposed, as a second part, the Neural Network is proposed that will adapt to the non-linear system and the reference signal that both must follow, as a third part obtains the dynamics of the tracking error between the non-linear system and the reference, after obtaining conditions in the laws of adaptation of weights in the Neural Network and obtaining the control law that guarantees that the tracking error converges to zero, so that the non-linear system will follow the indicated reference signal, which is what was wanted to be demonstrated. Finally simulations are presented, which illustrate the theoretical results previously demonstrated. The proposed new control scheme is applied via simulations to control of a 4-DOF Biped Robot [1].

We use the scheme of Figure 1 to indicate the procedure used in the obtaining of the laws of adaptation of weights and the laws of control that guarantee that the tracking error between the non-linear system, the neural network and the reference signal converges to zero.

2. Time-delay adaptive neural network and the reference

3. Tracking error problem

In this part, we will analyze the trajectory tracking problem generated by

are the state space vector, input vector and fr, is a nonlinear vectorial function.

To achieve our goal of trajectory tracking, we propose the error between the plant and the reference as: e=xp−xr=xp−xn+xn−xr=xp−x+x−xr.

Let ep=xp−x, and en=x−xr, be the trajectory tracking error and e=ep+er

The time derivative of the error is:

Eq. (8), can be rewritten as follows, adding and subtracting, the next terms ŴΓzxrxt−τ, αrtŴ, Aeand wper=x−xp, then,

The unknown plant will follow the fractional order reference signal, if:

where

Now, Ŵis part of the approach, given by W∗. Eq. (11) can be expressed as Eq. (12), adding and subtracting the term ŴΓzxt−τand if Γzxt−τ=Γzxt−τ−zxrt−τ

If

And by replacing Eq. (13) in Eq. (12), we have:

And:

So, the result for Ωu1is

and Eq. (14), is simplified:

Taking into account that e=xp−xr, shortening notation a little bit by setting σ=Γz, and defining Øσt−τ=σxpt−τ−σxrt−τ, the equation for aDtαeis

Now, the problem is to find the control law Ωu2, which it stabilizes to the system Eq. (20). The control law, we will obtain using the fractional order Lyapunov-Krasovskii methodology.

4. Study of trajectory tracking error

Our mathematical model of the dynamics in the tracking error is described in (17). In this equation we can see that an equilibrium state of this system is eŴ=0.

Without loss of generality we can assume that the matrix Ais given A=−λI,λ>0, where Iis the identity matrix of order nxn.

For the study of the stability of the tracking error we propose the following PID control law [4], widely used in science and engineering.

We will determine conditions in the parameters that guarantee that the tracking error converges to zero, and we will also use the following control law [5].

We also include the following control law, PIλDα[6]:

Substituting Eq. (18) in Eq. (17):

then

If a=1−Kv, then

And if w=1aKiaDt−αe, then aDtαw=1aKiet, [7], then Eq. (20) we rewrite as:

We will show, the new state enwTis asymptotically stable, and the equilibrium point is enewT=00T, when W∼σxrt−τ=0, as an external disturbance.

Let Vbe, the next candidate Lyapunov function as [8, 9]:

The fractional order time derivative of (22) along the trajectories of Eq. (21) is:

In this part, we select the next learning law from the neural network weights as in [10, 11]:

Then Eq. (24) is reduced to

Next, let us consider the following inequality proved in [12]

Which holds for all matrices X,Y∈Rnxkand Λ∈Rnxnwith Λ=ΛT>0. Applying (27) with Λ=Ito the term eTaŴØσt−τfrom Eq. (26), where

we get

Here, we select 1+Kia=0and Kv=Ki+1, with Kv≥0then Ki≥−1, with this selection of the parameters from Eq. (28) is reduced to:

From the previous inequality, we need to guarantee that Eq. (29) is less than zero, for which we select,

λ−1+Kp>0,a>0, γ−1>0, so that: aDtαV≤0,∀e,w,Ŵ≠0, e≠0, is wanted to be demonstrate.

The control law is given by Eq. (30)

Theorem: The control law Eq. (30) and the neuronal adaptation law given by Eq. (25) guarantee that the fractional tracking error converges to zero, by which the tracking of trajectories of the non-linear system is guaranteed Eq. (5).

Corollary 2: If aDtαV≤−1aλ−1+KpenTen−γ−1a12+12Ŵ2Lϕ2enTen<0, ∀e≠0,∀Ŵ, where Vis decreasing and bounded from below by V0, and:

then we conclude that e,W∼∈L1; this means that the weights remain bounded.

5. Modeling of the time-delay adaptive neural network and the delayed plant

The nonlinear delayed unknown plant and the neural network are given as:

where xp,fp∈Rn,u∈Rm,gp∈Rnxn. And fp, is unknown and gp=I,A=−λI, with ΓLypschitz function, W∗are the fixed weigths but unknown from the neural networks, which minimize the modeling error.

Theorem: We will show that epand entend to zero and therefore etends to zero, that is, the neural network follows the plant.

For this proposal, we first define the modeling error between the neural network and the plant: ep=xp−xn, whose derivative in the time is

Adding and subtracting, to the right hand side from (34) the terms ŴΓzxrxt−τ, αptŴ

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

First, it is easy to see that eŴ=0is a state of equilibrium (equilibrium point). Previous, so we propose to demonstrate that this point of equilibrium is asymptotically stable; for this, be:

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

We will show, the new state epis asymptotically stable, and the equilibrium point is ep→0, when Ŵσxnt−τ=0, as an external disturbance.

Let Vbe, the next candidate Lyapunov function as

Then, (35) is reduced to

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

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 ).

where qt=q31tq32tq41tq42tT, is the generalized coordinates vector. As usual, D(qt) is the inertia matrix, bounded and positive definite, and Cqtq̇tis the matrix of Coriolis and centripetal forces. Gqtrepresents a matrix of gravitational effects and Bdefines the input matrix. The vector τt=τ31tτ32tτ41tτ42tT, 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:

aDtαxp=Ax+W∗Γzxt−τ+Ωu, with τ = 25 s, A=−20I,I∈R4x4, and, W∗is estimated using the learning law given in (28).

Γzxt−τ=tanhx1t−τtanhx2t−τ…tanhxnt−τT, Ω=00100001Tand the uis obtained using (33).

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].

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

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 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.

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 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.

Acknowledgments

Authors thank, Mexican National Science and Technology Council, (CONACyT), Mexico, to the Autonomous University of Nuevo Leon, Dynamical Systems Group of the Department of Physical and Mathematical Sciences, (FCFM-UANL), Centro de Investigación y Estudios de Posgrado, Facultad de Ingeniería, Dr. Manuel Nava #8, Zona Universitaria, C.P. 78290, San Luis Potosí, S.L.P., México The research of V. Ramírez-Rivera has been supported by CONACyT-México under the project Problemas Nacionales (2015-01-786).

Conflict of interest

The first author of the reference manuscript, in his name and that of all authors, declares that there is no potential conflict of interest related to the article.