Robust Nonlinear Control of a Mobile Robot

Abstract

A robust control intended for a nonholonomic mobile robot is considered to guarantee good tracking a desired trajectory. The main drawbacks of the mobile robot model are the existence of nonholonomic constraints, uncertain system parameters and un-modeled dynamics. in order to overcome these drawbacks, we propose a robust control based on Lyapunov theory associated with sliding-mode control, this solution shows good robustness with respect to parameter variations, measurement errors, noise and guarantees position and velocity tracking. The global asymptotic stability of the overall system is proven theoretically. The simulation results largely confirm the effectiveness of the proposed control.

1. Introduction

One of the basic issues in the field of mobile robotics is the running path. The trajectory tracking is to guide the robot through intermediate points to arrive at the final destination. This guide is done under a time constraint, ie, the robot must reach the goal within a predefined time. In the literature, the problem is treated as the continuation of a robot reference (virtual processor) which moves to the desired trajectory with a certain pace. The real robot must follow this virtual robot accurately and try to minimize the error in distance, varying its linear and angular velocities [1-9].

There are lots of works on its tracking control. Their aims are mainly kinematic models; one method for dynamic models has been suggested [1]. In this case generally use linear and angular velocities of the robot (Fierro & Lewis, 1997; Fukao et al., 2000) or torques (Rajagopalan & Barakat , 1997; Topalov et al., 1998) as an input control vector [2]. The most authors determine the problem of mobile robot stability using nonlinear backstepping algorithm (Tanner & Kyriakopoulos, 2003) with steady parameters (Fierro & Lewis, 1997), or with the known functions (Oriollo et al., 2002) [1-6]. Other goals at the control architectures, the hybrid of the kinematic control, and the dynamic controller, the neural network controller, is proposed as some trajectory tracking methods [3].

In this paper, first, a kinematic controller is introduced to the WMR. Second, the dynamic controller, PI then Lyapunov theory associated to a sliding mode control, is proposed to make the actual velocity of the mobile robot to reach the wheel velocity control desired.

 

2. Kinematic Model

Fig. 1 shows the typical model of a nonholonomic wheeled mobile robot. This last is operated by two independent wheels and with a passive wheel ensuring its stability. The posture of the WMR can be represented as

Fig. 1.Error posture of a nonholonomic WMR

where the (x, y) is the center of mass (COM) position of the WMR in the world X − Y coordinate, and θ is the included angle between the X-axis and X' -axis representing the WMR [5, 7].

Know the derivative of the posture control V = (v w)T is easy. A simple geometric consideration gives

What is written in matrix form

whith

The mobile robot is nonholonomic, this signify the wheels roll without slipping, ie [3]

 

3. Dynamic Model

The dynamic equation of the WMR with n-generalized coordinates q ∈ Rn×1 , and inputs r = n − m , can be expressed as [1, 3, 7]

where M(q) ∈ 𝓡n×m is a positive symmetric definite inertia matrix, is the centripetal and coriolis matrix, denotes the gravitational vector, τn ∈ 𝓡n×1 is bounded unknown distrurbance, B(q) ∈ 𝓡n×(n−m) denotes the input transformation matrix, τ ∈ 𝓡(n−m)×1 is the control input vector, A ∈ 𝓡m×n is a matrix associated with the constraints.

The parameters in (6) are given as

I and m are the moment of inertia and the mass of the WMR, respectively, the motors torques τr and τl act on the right and left wheels respectively [1, 7]. r and R are the radius of the wheel and the distances between the two driving wheels, respectively.

Substituting (2) and its derivative in Eq. (5) pre-multiplied by JT(q) , and without considering uncertainties and disturbances the Eq. (5) can be written as the following [1, 3, 7, 8]

The dynamic model of a WMR unicycle type is simplified and given by

where

 

4. Tracking Controller Design

The problem can be interpreted as consisting of the slave robot to robot reference, whose trajectory is given by t → [xr(t), yr(t)]. It is then desired to control the zero error vector [xr(t) − x(t), yr − y(t)] , where [xr(t)yr(t)θr(t)]T denotes the coordinate vector generalized robot reference and [x(t)y(t)θ(t)]T the vector of generalized coordinates of the real robot [1-6].

For the tracking control problem, a time-varying reference mobile robot model is given as [7]

where and denote the reference velocity and posture of the WMR, J(θr) is the Jacobean defined in the Eq. (4).

We define the error between the desired positions and orientations, and actual by

Fig. 2.Characterization of the trajectory tracking.

We define qe as following

Te is called the matrix

The vector of error variations can be expressed as :

 

5. Design of Hierarchical Controller

Fig. 3 illustrates the architecture of the controller designed to control a WMR.

Fig. 3.Architecture robot controller

 

6. Kinematic Controller

Proposition : Let the Lyapunov function candidate

where ; the derivative L0 is given by

Eq. (11) gives

Then

By substitution in the Eq. (14)

The control law for the system to be stable is , so we choose

Therefore the Eq. (14) can be rewritten as

For must k1, k2 and

We have opted for the following values k1 = 10, k2 = 5 and k3 = 4 .

 

7. Dynamic Controller

We used in the present work two control technicals, first one is simple, it is a PI controller, the second is based on the nonlinear method.

Fig. 4.The robot trajectory in X-Y plane

7.1 PI control in dynamic level

To ensure that the movement of the WMR can follow the desired velocity controller generated by the kinematic, two dynamic controllers are introduced in this section. Both controllers are PI controllers responsible for providing left and right torques capable of powering the left and right wheels.

The parameters of both controllers are kp = ki = 2 .

7.2 Simulation results

To show the effectiveness of the proposed controller, simulations were performed in Matlab-Simulink. The examples chosen is that the tracking of a circular path.

The parameters of the robot used in the simulation are: m = 4 kg , I = 2.5 kg m2 , R = 0.15 m and r = 0.03 m .

The actual initial posture of the mobile robot is q(0) = [3 1 180°]T . The initial posture of the robot is defined by reference : qr(0) = [0 0 0]T .

PI controller gave us satisfactory results. Figs. 5, 6 and 7 show significant oscillations errors, dynamic errors, Figs. 10 and 11, are large. to reduce its peaks and errors, we replaced the PI power controller by Lyapunov controller.

Fig. 5.Tracking errors trajectory e1, e2 and e3

Fig. 6.The tracking errors in X-coordinate

Fig. 7.The tracking errors in Y-coordinate

Fig. 8.The tracking θr and θ

Fig. 9.The tracking error torque (τr − τl) (N*m)

Fig. 10.Linear velocities

Fig. 11.Angular velocities

7.3 Lyapunov controller associated to a sliding mode control in dynamic level

In this section, we replaced the PI controller with a Lyapunov controller associated to a sliding mode control to improve the results to previous results.

The sliding surface is defined by selected

where

For a good pursuit of velocity, it is important to make the invariant surface Ṡ(t) = 0 and attractive ST Ṡ < 0

where Δf (t) present a parameter uncertainties and external disturbances.

Proposition:

Let the Lyapunov function candidate

where ; the derivative is given by:

Eq. (8) gives

Eq. (19) becomes

If we put

We find

The Lyapunov condition is satisfied for ka < 0 and kb < 0 , where ka = −100 and kb = −1000

From the Eq. (17)

The derivative of Lyapunov becomes

If we put

We find

Substitution Eq. (23) in Eq. (27) gives

So for , it is necessary that

And , where kc = −10 and kd = −0,5 .

7.4 Simulation results

The same initial conditions and the same parameters of the WMR are used in this section. To test the robustness of the proposed approach, we have proceeded to two tests. Initially, a large variation in the mass m was introduced, a 50% increase between t=30s → 50s. The second test is the robustness of the controller against the change of the radius r, a 10% reduction of r is applied between t=60s → 70s. The following figures illustrate these results.

Fig. 12.Variation parameters: (a) Mass of the WMR; (b) Radius of the wheel

Figs. 13-20 shows clearly that the proposed control is widely robust against the parameters variations. By comparison, these results with previous results, we note that the oscillations of e1 , e2 and e3 are reduced, which gives us a good improvement of dynamic and static errors, the same for the linear velocities and angular velocities. Tracking in y-coordinate is better and the dynamic error decreases, Fig. 17 shows the θ follows θr with less oscillations. Employing nonlinear method for controlling the WMR gives a good trajectory tracking and velocity, and confirms the robustness of the control.

Fig. 13.The robot trajectory in X-Y plane

Fig. 14.Tracking errors trajectory e1, e2 and e3

Fig. 15.The tracking errors in X-coordinate

Fig. 16.The tracking errors in Y-coordinate

Fig. 17.The tracking θr and θ

Fig. 18.The tracking error torque (N*m)

Fig. 19.Linear velocities

Fig. 20.Angular velocities

 

8. Conclusion

This paper focuses on the design of a nonlinear tracking controller for a nonholonomic mobile robot with unknown parameters, we proposed a controller based on Lyapunov theory associated with the sliding mode control. The stability of the system was proved by Lyapunov theory which satisfies a good performance tracking position control. Simulation results demonstrate that the proposed controller is effective.