Neural network based adaptive non linear PID controller for non-holonomic mobile robot

Abhishek Singh, Garima Bisht and Prabin Kumar Padhy Department of Electronics and Communication Engineering

PDPM Indian Institute of Information Technology, Design and Manufacturing Jabalpur – 482005 (M.P.), India abhishek10007@iiitdmj.ac.in, garima10229@iiitdmj.ac.in, prabin16@iiitdmj.ac.in.

Abstract—this paper presents an approach for velocity and orientation tracking control of a nonholonomic mobile robot based on an adaptive controller. The developed PID controller is based on analog neural networks. The feed forward neural networks controller is trained on-line to find the inverse kinematical model, which controls the outputs of the mobile robot system. The proposed controller has a better performance because of its capability of continuous online learning due to neural network. The simulation results for a differentially driven nonholonomic mobile robot are presented to establish the better performance of the proposed adaptive controller.

Keywords—Neural network; Nonlinear PID; Nonholonomic mobile robot; Adaptive controller

I. INTRODUCTION

Wheeled Mobile Robots (WMRs) have been used extensively in various industrial and service applications. These applications demand WMRs with various mobility configurations (wheel number and type, their location and actuation, single or multibody vehicle structure). WMRs constitute a class of mechanical systems called non holonomic systems [1]. Non-holonomic system is a mechanical system that is subject to nonholonomic constraints. They are the constraints on the velocity of the system which cannot be integrated into position constraints that can be used to reduce the number of generalized coordinates. Mobile robots constitute a typical example of nonholonomic systems.

The nonholonomic behaviour in robotic systems is particularly interesting, because it implies that the mechanism can be completely controlled with a reduced number of actuators. From a real-time point of view, the nonholonomic kinematic controller is to be integrated with the dynamics of a mobile robot as the perfect velocity tracking assumption may not hold good in practice [2]. The adaptive control methods proposed so far for nonholonomic mobile robots do not consider the model with unknown parameters in its kinematic part. Another intensive area of research has been neural-network (NN) applications in closed-loop control. In

contrast to classification applications, in feedback control the NN becomes part of the closed-loop system. Mobile robot navigation can be classified into three basic problems [3] tracking a reference trajectory, following a path, and point stabilization. Some nonlinear feedback controllers have been proposed for solving these problems [7]-[10].

Therefore, it is desirable to have a NN control with on-line learning algorithms that do not require preliminary offline tuning. Neural networks offer exciting advantages such as adaptive learning, fault tolerance and generalization. Fierro and Lewis [2] developed an artificial neural network based controller by combining the feedback velocity control technique and torque controller, using a multilayer feed forward neural network. Here, it motivates us to combine a neural network with PID control. It is anticipated that the combination will take advantage of the simplicity of PID controllers and the powerful capability of learning, adaptability and tackling nonlinearity of neural networks. Based on these reasons an adaptive controller of a nonlinear PID based on analog neural network (ANN) is developed for the velocity and orientation tracking control of nonholonomic mobile robots in this paper.

II. NON HOLONOMIC MOBILE ROBOT MODEL

A mobile robot featuring two differentially driven rear wheels and an omni-directional self-adjusted passive supporting wheel (a castor front wheel), carrying the mechanical structure is shown in figure 1. Both wheels have the same radius denoted by r and two driving wheels are separated by 2R. An assumption is made that the given mobile robot is made up of a rigid frame equipped with non-deformable wheels and it moves in a horizontal plane.

The pose of the robot in the global coordinate frame {O, X, Y} is completely specified by the vector P = [x, y, θ]T, where x and y are the coordinates of the COM in the global coordinate frame and θ is the robotic orientation measured from X-axis in anti-clockwise direction. At any

given time, three generalized coordinates can describe the robot’s configuration (pose).

q = [x, y, θ]T (1)

For the given mobile robot system, pure rolling and non-slipping non-holonomic conditions are assumed. This subjects the motion of robot to a constraint that it can only move in the direction normal to the axis of the driving wheels. The dynamic model representation of the above non-holonomic wheeled-mobile robot for control purposes is given by [1], q = S(q)V(t) (2)

The matrix V = T includes the translational velocity ( ) and rotational velocity ( ) of the mobile robot. The matrix S(q) is given by,

S(q) = cos θ 0sin θ 00 1 (3) MV + CfmV + τd = Bτ (4)

where, M = STMS, Cfm = ST(MS+Cfm S), τd = STτd, B = STB.

The various matrices used in equation (4) for the non holonomic platform can be described as,

= 00

= (5)

fm = 0 00 0 = [ rw lw ]T

where, m is the mass of the mobile robot and I is the moment of inertia of the mobile robot about the vertical axis through C (centre of mass); wr and wl represents right and left wheel torques, respectively.

Here, matrix M(q)ER3x3 is a positive symmetric definite inertia matrix; Cfm( , )ER3x3 is the centripetal and Coriolis force matrix; dER3x1 denotes bounded unknown disturbances; B(q)ER3x2 is the input transformation matrix.

Remark 1: In this dynamic representation of the robot, influence of the passive self-adjusted supporting wheel is not taken into consideration to reduce the complexity of

the model for the feedback controller design. We can make this assumption since it is a free wheel. However, the free wheel may be a source of substantial distortion, particularly in the case of changing its movement direction. If we consider the small velocity of the robot this effect is reduced.

Remark 2: Equation (2) describes the behaviour of the nonholonomic system in a new set of local coordinates, i.e. S(q) is a Jacobian matrix that transforms velocities in mobile base coordinates to velocities in Cartesian coordinates .

Fig 1. Nonholonomic mobile robot configuration

III. ADAPTIVE CONTROL OF MOBILE ROBOT

The conventional PID controller with fixed parameters may usually deteriorate the control performance. Various types of modified PID controllers have been developed in the existing literature. However, if severe nonlinearity is involved in the controlled process, a nonlinear control scheme will be more useful, particularly in the case of high nonlinearity of mobile robots. Nowadays, neural networks have been proved to be a promising approach to solve complex nonlinear control problems and there are two kinds of neural networks for control applications, i.e., digital and analog neural networks [4].

The structure of the newly proposed control algorithm of nonlinear PID-based analog neural networks is shown in figure 2. This control algorithm has the characteristics such as simple structure, little computation time, and continuous auto-tuning method of the neural network controller. Two direct controllers of the PID-based analog neural networks are composed of the orientation and velocity control of the mobile robot. Robot motion can be controlled by its linear velocity v and orientation angle θ.

}

2R Driving Wheels

vSupporting Wheel

Y

O X

2r θ

Here, the control goal is to design two ANN controllers that force the tracking error e = [ep1, ep2]T to zero, where ep1 = vr-v, ep2 = θr-θ, and vr and θr are the desired translational (linear) velocity and the desired orientation angle, respectively [4].

The trajectory tracking problem can be stated according to the equation given below [1],

qr(t) = [xr(t) yr(t) θr(t)]T

generated by a reference robot whose equations of motion are xr = vr cos θr yr = vr sin θr (6)

θr = ωr

with vr > 0 for all t, determine a smooth velocity control law vc = c (Epv, vr , K) for (11) such that lim ∞ q t = qr t

where, Epv, vr and K are the tracking position error, the reference velocity and control gain, respectively.

A control input p can be obtained from the following equation [5],

pv = f(x) and pθ = f(x) (7)

Here x is the input of sigmoid function f(·), which is explained in equation (6), and f(·) is the sigmoid function which has a nonlinear relationship as presented in the following equation, f x =

.γ γ .γ (8)

where, γ is the parameter, which determines the shape of sigmoid function. The sigmoid function f(x) becomes linear function when becomes zero.

The block diagram of neural network is shown in figure 2. Here, Kpv, Kiv, Kdv, epv, eiv and edv are proportional, integral and derivative gain, system error between desired reference velocity and output of the ANN based PID controller designed for velocity, integral of the system error and the difference of the system error, respectively.

Analogously gain values and error values are calculated for the angle parameter also using the second ANN based PID controller designed for the angle parameter.

In figure 2, the input signal of the sigmoid function for velocity calculation becomes [5]

xv(k) = Kpv(k)epv(k)+Kiv(k)eiv(k)+Kdv(k)edv(k) (9)

where,

epv(k) = vr(k)-v(k) eiv(k) = ∑ epv(n)∆T edv(k) = (epv(k)-epv(k-1))/ ∆T

Similarly for angle calculation the input signal for sigmoid function is

xθ(k) = Kpθ(k)epθ(k)+Kiθ(k)eiθ(k)+Kdθ(k)edθ(k) (10)

where,

epθ(k) = θr(k)-θ(k) eiθ(k) = ∑ epθ(n)∆T edθ(k) = (epθ(k)-epθ(k-1))/ ∆T ∆T is the sampling time.

To tune the gains of PID controller for velocity control is, the steepest descent method using the following equation was applied:

Kpv(k+1) = Kpv(k)-ηpv EK

Kiv(k+1) = Kiv(k)-ηiv EK (11)

Kdv(k+1) = Kdv(k)-ηdv EK

where, pv, iv , dv are learning rates determining the convergence speed, and Ev(k) is the error in velocity calculation defined by the following equation:

Ev(k)= (vr(k)- v(k))2 (12)

Fig 2. Block diagram of PID based neural network

}

Kpv(k) epv(k)

edv(k) p1(k)=f(x) Kdv(k)

f(x)

eiv(k)Kiv(k)

}

From equation (11), using the chain rule, the following equations are derived:

EK = E K EK = E K (13) EK = E K

The following equations can also be derived by using equations (7), (9) and (12):

E = -(vr(k) – v(k)) = -epv(k)

= f ′ x k

K = epv(k) (14)

K = eiv(k)

K = edv(k)

And the following expression can be derived from equations (12) and (13):

EK = E K

= - epv(k) f ′ x k epv(k)

= f ′ x k (epv(k))2 EK = E K

= - epv(k) f ′ x k eiv(k) (15)

= f ′ x k eiv(k)epv(k) EK = E K

= - epv(k) f ′ x k edv(k)

= f ′ x k edv(k)epv(k)

and f ′ x 4 . . (16)

As done by Yamada and Yabuta [6], for convenience,

We assume 1 Kpv(k+1) = Kpv(k)+ ηpv (epv(k))2 . . Kiv(k+1) =Kiv(k)+ηiv epv(k)eiv(k)

. . (17)

Kdv(k+1) =Kdv(k)+ηdvepv(k)edv(k). .

Similarly, for angle control

Kpθ(k+1) = Kpθ(k)+ηpθ(epθ(k))2 . . Kiθ(k+1) = Kiθ(k)+ηivepθ(k)eiθ(k)

. . (18)

Kdθ(k+1) = Kdθ (k)+ηdvepθ(k)edθ(k). .

Fig 3. Block diagram of mobile robot control system

IV. RESULTS

In this section, we perform a computer simulation on the dynamic model of a mobile robot by using the adaptive tracking controller which was designed in the previous section. For this purpose, two controllers have been implemented and simulated in MATLAB: 1) A controller that assumes “perfect velocity tracking” 2) A controller that assumes “perfect orientation tracking”

Experiments were carried out with respect to circular path as reference input and the path tracing performance by proposed nonlinear PID controller was demonstrated.

The neural controller designed by this method does not require any knowledge of dynamics. It uses only the input and output of the plant for the adaptation of control parameters and can tune the parameters by means of a continuously online learning process. The purpose of the simulation is to show the effectiveness of the proposed neural control algorithm.

For the simulation purpose, a circular path of radius 1 unit centred at the origin is taken as the reference motion trajectory. The mobile robot starts from any specified point on the proposed trajectory with the initial velocity v(0) = 0 for the initial conditions. The gain parameters of

}

}- epv Vr

PID Based ANN

s

edv p1

eiv 1

}

} 1/s

τrw

1 v

Mobile Robot

θ

epv τlw 1

θr -1

PID Based ANN

-s

edv

eiv p2 1/s

two ANN controllers are chosen as, p1 = 600, ηi1 = 500 and ηd1 = 1.5; p2 = 800, ηi2 = 5 and ηd2 = 800.

Fig 4. Error curves for circular path

Fig 5. Trajectory of mobile robot on circular path

Since external disturbances in any real time path are inevitable, a disturbance matrix

τd = [0.1 sin θ, 0.1 sin θ]T

is added to the mobile robot system to demonstrate that the proposed online neural controller is robust.

In this simulation, physical parameters of the robot are taken as m = 9kg, I = 5 kgm2 2R = 0.306 m and r = 0.052m.

It is observed from the figures that the model estimation errors for each loop are in acceptable range indicating the accuracy of the identification.

The estimation errors of model identification velocity parameter and θ parameter of the circular path are shown in figure 4.

The method used in this paper was compared to another method [11]. In this approach, the mobile robot, including the actuator dynamics is identified by a linear model using recursive least square method. The identified model is then used to design the PID controller to set the target linear and rotational velocities. The output graphs are shown in figure 6.

Fig 6. Error curves for circular path using 2nd method

As can be observed from the experimental results of figure 5 the response of the proposed controller is in good agreement with that of reference path. The mobile robot immediately heads towards the desired velocity and

0 100 200 300 400 500 600 7000

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orientation angle of the mathematically defined path. The tracking control is reasonably accurate since the ANNs provide with adaptive capability and quick online learning of the parameters. The simulation results demonstrate that the adaptive PID-based nonlinear ANN controller for the mobile robot is capable of better tracking performance in real time and is robust to disturbances, since even after introducing disturbing elements the robot manages to trace the path.

V. CONCLUSION

In this paper a stable control algorithm capable of dealing with the two basic non holonomic navigation problems, and that does not require knowledge of the cart dynamics has been derived using an adaptive neural network back-stepping approach.

The proposed direct control method for the mobile robot is capable of real-time control and has a better tracking performance. Simulation results demonstrate the effectiveness of the proposed control algorithm, better dynamic property, and strong robustness, and it was suitable for the control of nonholonomic mobile robots.

There are several features of the proposed controller:

-The controller had an adaptive control capability and the control parameters were optimized via the back propagation algorithm.

-The controller designed by this method does not need any training procedure in advance, but it uses only the input and output of the plant for the adaptation of control parameter and can tune the parameters iteratively.

- Actually, the proposed neural controller does not require the dynamics model of mobile robots, which is needed only in the simulation for this paper

-The proposed control algorithm presented in this study was online control with simple structure and had better dynamic property, strong robustness and it was suitable for the control of various plants, including linear and nonlinear process, compared to the conventional PID controller.

VI. REFERENCES

[1] T. Das, I. N. Kar, S. Chaudhury, Simple neuron-based adaptive controller for a nonholonomic mobile robot including actuator dynamics, Neurocomputing 69 (2006).

[2] R. Fierro, F.L. Lewis, Control of a nonholonomic mobile robot using neural networks, IEEE Trans. Neural Networks 9 (1998).

[3] C. Canudas de Wit, H. Khennouf, C. Samson, and O. J. Sordalen, Nonlinear control design for mobile robots, in Recent Trends in Mobile Robots, Y. F. Zheng, Ed. Singapore: World Scientific (1993).

[4] Jun Ye, Adaptive control of nonlinear PID-based analog neural networks for a nonholonomic mobile robot, Neurocomputing 71 (2008).

[5] T.D.C. Thanh, K.K. Ahn, Nonlinear PID control to improve the control performance of 2 axes pneumatic artificial muscle manipulator using neural network, Mechatronics 16 (2006).

[6] T Yamada, T Yabuta, Neural network controller using autotuning method for nonlinear functions, IEEE Trans. Neural Networks (1992).

[7] A. M. Bloch, M. Reyhanoglu, N. H. McClamroch, Control and stabilization of nonholonomic dynamic systems, IEEE Trans. Automat. Contr. (1992).

[8] G. Campion, B. d’Andr´ea-Novel, G. Bastin, Controllability and state feedback stabilizability of nonholonomic mechanical systems, in Lecture Notes in Control and Information Science, C. Canudas de Wit, Ed. Berlin, Germany: Springer-Verlag (1991).

[9] C. Canudas de Wit, H. Khennouf, C. Samson, and O. J. Sordalen, Nonlinear control design for mobile robots, in Recent Trends in Mobile Robots, Y. F. Zheng, Ed. Singapore: World Scientific (1993)

[10] Y. Kanayama, Y. Kimura, F. Miyazaki, T. Noguchi, A stable tracking control method for an autonomous mobile robot, in Proc. IEEE Int. Conf. Robot Automat. (1990).

[11] P. K. Padhy, Takeshi Sasaki, Sousuke Nakamura, Hideki Hashimoto, Modelling and Position Control of Mobile Robot in the 11th IEEE International Workshop on Advanced Motion Control, Japan (2010).