Introduction

The higher demands to achieve more reliable performance in modern robotic systems have necessitated the development of appropriate fault diagnosis methods. The appearance of faults is inevitable in all systems, such as wheeled robots, either because their elements are worn out or because the environment in which they operate, presents unanticipated situations [4]. In a large number of applications, as for example search and rescue, planetary exploration, nuclear waste cleanup or mine decommissioning, the wheeled robots operate in environments where human intervention can be very costly, slow or even impossible. They can move freely in such dynamic environments. It is therefore essential for the robots to monitor their behavior so that faults may be addressed before they result in catastrophic failures. A wheeled mobile robot is usually an embedded control platform, which consists of an on-board computer, power, motor control system, communications, sonars, cameras, laser radar system and sensors such as gyroscope, encoders, accelerometers etc, Fig. 1.

Fault diagnosis and accommodation for wheeled mobile robots is a complex problem due to the large number of faults that can be present such as faults of sensors and actuators [10] - [20]. Model based fault detection and isolation is a method to perform fault diagnosis using a certain model of the system. The goal is to detect faults as early as possible in order to provide a timely warning [8]. The aim of timely handling the fault occurrence is to accommodate their consequences so that the system remains functional. This can be achieved with fault tolerance. In cases where fault could not be tolerated, it is necessary to use redundant hardware. In practice there exist two different approaches for fault tolerance control, static redundancy and dynamic redundancy [8]. In [10] and [16], the research is focused only on the problem of fault detection and identification in a mobile robot and different approaches related to state estimation were introduced. In [9] and [15], the research interest is focused only on the problem of fault detection which is a separate problem in the fault diagnosis domain. The research efforts in [7] and [12] - [14] are primarily intended to detect faults in the sensors of a wheeled robot. Concerning the research area of detection and accommodation on wheeled robots there is also a small number of efforts [18] with different approaches and methodologies. As a fault, it can be considered any unpermitted deviation from the normal behavior of a system. Fault diagnosis is the procedure of determination of the component which is faulty. Consequently, the aim of fault diagnosis is to produce the suitable fault statement regarding the malfunction of a wheeled robot.

Fault diagnosis includes fault detection, which is the indication that something is going wrong in the system and fault isolation, which is the determination of the magnitude of the fault, by evaluating symptoms. Follows fault detection. Fault detection and isolation tasks together are referred to as fault diagnosis (FDI -Fault Detection and Isolation). Among the various methods in the design of a residual generator, only few deal with nonlinear systems. Structural analysis is a technique that provides feasible solutions to the residual generation of nonlinear systems Structural analysis methods are used in research publications [2] and [6]. Paper [3] presents a structural analysis for complex systems such as a ship propulsion benchmark. In [13] and [14] the authors discusses how structural analysis technique is applied to an unmanned ground vehicle for residual generation. In this research, a model based fault diagnosis for a four wheel skid steering mobile robot (SSMR) is presented. The basic idea is to use structural analysis based technique in order to generate residuals. For this purpose we use the kinematic model of the mobile robot that serves to the design of the structural model of the system. This technique provides the parity equations which can be used as residual generators. The advantage of the proposed method is that offers feasible solution to the residual generation of nonlinear systems. Additionally, we a propose a fault accommodation technique based on RLS approximation in order to provide recalculated control inputs in the case that the left or right set of the robot tires becomes flat. The mobile robot is supposed to be equipped with two high resolution optical quadrature shaft encoders mounted on reversible-DC motors which provide rotational speeds of the left and right wheels L ω and R ω respectively and an inertial measurement unit (IMU) which provides the forward linear acceleration and the angular velocity well as the angle θ between the mobile robot axle and the x axis of the mobile robot. The absolute pose (horizontal position and orientation) of the robot is available via a camera system mounted on the workspace of the robot. A distinctive marker is place at the top side of the robot. The paper is organized as follows. We start by presenting the mathematical model of a Pioneer 3-AT mobile robot in section 2. Section 3 describes the fault diagnosis procedure. Section 4 describes the methodology of fault accommodation. In section 5 we present the application results of the proposed method to the robotic platform. Conclusions and directions for future work are presented in Section 6.

Mathematical Model of Pioneer 3-AT Mobile Robot

In this work, the mobile robot Pioneer 3-AT was used as a robotic platform. This robot is a four wheel skid -steering vehicle actuated by two motors, one for the left sided wheels and the other for the right sided wheels. The wheels on the same side are mechanically coupled and thus have the same velocity. Also, they are equipped with encoders and the angular readings are available through routine calls.

The kinematic model describes the motion constrains of the system, as well as the relationship of the sensors measurements with the system states and it is crucial for the fault diagnosis procedure.

Kinematic Model

The geometry of the robot is presented in Fig. 2. To consider the model of the four wheel skid steering mobile robot (SSMR) it is assumed that the robot is placed on a plane surface where ( )

, Ι Ι Χ Υ

is the inertial reference frame and ( )

,

Χ Υ is a local coordinate frame fixed on the robot at its center of mass (COM). The position of the COM is ( )

x, y with respect to the inertial frame and ϑ is the orientation of the local coordinate frame with respect to the inertial frame.

X Ι ϑ 2 R R y v 2c Y Ι 2 x v 1x v 3x v 4 x v COM x v v 1y v 1 v 4 v 4 y v 3 v 3 y v 2 y v 2 v X Y a b y x 2 L R ICR ICR x ICR y Figure 2. Mobile Robot Geometry.

As depicted in Fig. 2, a is the distance between the center of mass and the front wheels axle along X, b is the distance between the center of mass and the rear wheels axle along X, c is half distance between wheels along Y and L R , R R are the radii of left and right wheels respectively. The coordinates of the instantaneous center of rotation (ICR) are ( )

ICR ICR

x ,y . Assuming that the robot moves on a horizontal plane the linear velocity with respect to the local frame is given by 0

x y υ υ     =       υ (1)

and its angular velocity is given by

0 0 ω     =       ω (2)

The state vector with respect to the inertial frame is

q x y ϑ     =      (3)

Proceedings of the 26 th International Workshop on Principles of Diagnosis

The time derivatives of (3) denotes the robot's velocity vector and is given by cos sin 0 sin cos 0 0 0 1

x y x y ϑ ϑ υ ϑ ϑ υ ϑ ω −             =                      (4)

Assuming that longitudinal slip between the wheels and the surface can be neglected we have the following equation,

ix i i R υ ω = (5)

where ix υ is the longitudinal component of the total veloci- ty vector i υ of the i-th wheel expressed with respect to the local frame and i R is the rolling radius of that wheel. If we take into account all wheels (Fig. 2), the following relationships between the wheels can be obtained [11],

1 2 3 4 1 4 2 3 L x x R x x F y y B y y υ υ υ υ υ υ υ υ υ υ υ υ = = = = = = = = (6)

where L υ refers to the longitudinal coordinates of the left wheels velocities, R υ refers to the longitudinal coordinates of the right wheels velocities, F υ refers to the lateral coor- dinates of the front wheels velocities and B υ refers to the lateral coordinates of the rear wheels velocities. Unlike other mobile robots, lateral velocities of the four wheel skid steering mobile robot are generally nonzero since from its mechanical structure the lateral skidding is necessary if the robot changes its orientation. Therefore, in order to complete the kinematic model, the following nonholonomic constrain in Pfaffian form is introduced [ ] ( )

sin cos 0 A q q ICR x x y ϑ ϑ ϑ     − − = =           (7)

Then we have ( )

q q S =  η (8)

where ( )

cos sin sin cos 0 1 S q ICR ICR x x ϑ ϑ ϑ ϑ     = −       (9) x υ ω   =     η (10) ( )

S q is a full rank matrix, whose columns are in the null space of ( )

A q , ( ) ( ) 0 q q T T S A = (11)

It is noted that since ( ) ( )

dim 2 dim 3 = < = η q

, equation ( 8) describes the kinematic of a sub-actuated robot with the nonholonomic constraint given by (7).

We suppose that the mobile robot localization is calculated via the following measurement devices:

• two high resolution optical quadrature shaft encoder mounted on reversible-DC motors which provide rotational speeds of the left and right wheels L ω and R ω respectively,

• an Inertial Measurement Unit (IMU) which provides the forward linear acceleration and the angular velocity as well as the angle ϑ between the mobile robot axle and the x axis of the mobile robot. • A camera system, which calculates the pose of the robot, by tracking a marker placed at the top side of it. In this work we are only interested in abrupt faults which occur in the actuators of the mobile robot and as consequence, we make the following assumptions.

• Assumption 1: When the mobile robot starts functioning all its components are in normal mode. • Assumption 2: The magnitude of the noise is assumed to be significantly smaller than the magnitude of the faults. • Assumption 3: Regarding the wheel radius the following inequalities are satisfied:

0 & 0 R R L L R R R R δ δ + > + >

According to this assumption, faults that result in the complete loss of the wheel are not considered.

Fault Detection and Isolation

Between several techniques for generating residuals, limited number of them concerns nonlinear systems. Such one is structural analysis. Using this method we can extract information about system components that we are not able to measure. Also we can take the parity equations that allow generating residuals. The structure of the mobile robot is described using the following sets of constrains C and variables V { }

1 2 9 , ,..., C c c c = (12) V X K = ∪(13)

X is a subset of the unknown ones and K is a subset of known that are measurements and inputs. The above subsets are

{ }

, , , ,

x y X x y ϑ υ υ =   (14)

{ }

, , , , , , ,

L R K x y x y ϑ ω ω ω =  (15)

The constrain set of the mobile robot is Through the above technique we create the following incidence matrix that describes the robot structure, Table 1.

Table 1. Incidence Matrix KNOWN UNKNOWN x y θ x  y  ω L ω R ω x  y  ϑ  x υ y υ 1 c 1 1 1 1 2 c 1 1 1 1 3 c 1 1 4 c 1 1 5 c 1 1 6 c 1 1 7 c 1 1 1 8 c 1 1 9 c 1 1

Applying matching algorithm [1] to the incidence matrix, we take out the following matched M and unmatched U constrains { } ( ) ( )

1 sin 2 cos 2 cos sin R L R L d r r y dt r xd ϑ ω ω ϑ ω ω t ϑ ϑ  =− + +      + −       +    ∫  (

Fault Accommodation

Fault accommodation is the phase that follows the fault diagnosis. One of the most important issues to consider for the design of fault tolerant control is relative to the performance and functionality of the system under consideration. More specific it should take into consideration, the degree of performance degradation that is acceptable. There are two aspects of system performance, dynamic and steady state. In our approach we take into account the second one. We also use the aforementioned fault diagnosis method to monitor the system. The goal is to have the necessary information about the fault occurrence for timely counteraction. Figure 3 shows the overall structure of the proposed fault tolerant mechanism. It consists of two parts: i) the fault detection module which accepts as inputs the measurement of the linear and angular velocity of the SSMR and decides about the type of fault according to the method described in Section 3, and ii) the fault accommodation module which accepts as inputs the type of fault as well as the measurement of the linear and angular velocity and recalculates accordingly the command inputs in order to compensate for the fault. When a fault occurs the appropriate action is undertaken (e.g. maintenance, repair, reconfiguration, stop operation) in such a way to prevent system failures. In that level the performance degradation that is acceptable is relative to the minimum requirements that ensure the system functionality. There is always the case that the malfunction may cause hazard for the process or the environment, and a decision for stopping the operation is unavoidable.

In this work, we propose a fault accommodation technique which is employed when either the left or the right set of tires becomes flat during the operation of a SSMR. It is obvious that when a flat tire fault occurs, the total nominal radius NOM R (rim and tire) of the fault wheel changes to F R , where

F NOM R R <

. The proposed fault accommoda-tion strategy relays on the online estimation of the new radius F R , in order to correct the commanded rotational speeds of the faulty wheel and compensate for the fault which otherwise will inevitable lead the vehicle to diverge from its nominal course. As explained in [11], the kinematic model of the SSMR can be consider equivalent with the unicycle differential drive one, mainly due to the existence of a single motor drive and a transmission belt for each set of wheels (left and right), which impose the same rotational speed for each set of wheels. According to this assumption we can safely assume that:

1 1 1 2 L x R u c c c υ υ ω       =       −     (35)

where ,

L L L R R R R R υ ω υ ω = =

are the equivalent linear velocities of the left and right wheels respectively in relation to the rotational speeds and radii. If we consider that the fault will occur only at the one set of the wheels (left or right), we may consider only the angular velocity equation for the accommodation. Thus, only the angular velocity measurement is needed. The fault accommodation is based on the online estimation of the new radius F R employing a Recursive Least Squares algorithm. More specific, we may consider the following linear equation for the measurement of the mobile's robot body angular velocity, in case a left side fault occurs:

( ) 0.5 0.5 0, k k k R R k F k k L L k k R H R v c H H c v R ω ω ω − = + = = − − (36)

while in the case of a right side fault:

( ) 0.5 0.5 0, k k k L L k F k k R R k k R H R v c H H c v R ω ω ω + = + = = − (37)

Having defined the measurement model of the robot angular velocity in the body frame, we proceed to the on line estimation of the fault wheel radius employing the following Recursive Least Squares approximation algorithm:

1. Initialize the estimator:

( ) ( ) ( ) 0 0 2 0 ˆF F L R F F R E R R P E R R = = = − (38) where | L R

R is the nominal radius of the left or right wheel set.

Obtain a new measurement k

ω , assuming that it is given by the equation (36), or (37).

3. Update the estimate ˆk F R and the covariance k P of the estimation error sequentially according to:

( ) ( ) ( )

1 1 1 1 1 1 ˆˆk k k T T k k k k k k k F F k k k F k k k k K P H H P H R R R K H R P I K H P ω − − − − − − = + = + − = − (39)

where k ω is the actual measurement of the body an- gular velocity as delivered by the IMU sensor. 4. Using the estimated wheel radius ˆk F R we correct the commanded wheel angular velocity as follows:

_ 2 ˆk k R R L cor F c R R ω ω ω − + = (40) _ 2 ˆk k L L R cor F c R R ω ω ω + = (41)

in case there is a left or a right wheel fault respectively.

Application Results

The proposed method has been implemented and tested experimentally on Pioneer 3-AT mobile robot. All experiments have been performed indoors. We consider a faulty situation where the right wheel set is flat (forward and backward wheels). We apply a command of 5 / L R rad s ω ω = = for both set of wheels. In the nominal situation (no faults) the robot should move (almost) straight forwards without any deviation. The robot starts from the origin of the inertial frame and moves for 2.5m.

The time interval dt between successive IMU measure- ments is 2.5 sec m . The nominal radius of the wheels (proper inflation) is 0.115

L R R R m = =

. In the first experiment (Fig. 4), the fault accommodation algorithm is not enabled and as we can observe from the trajectory of the vehicle, the SSMR significantly diverges from its nominal course to the right. The fault detection algorithm is enabled, and as we can see from Fig. 5 the fault was successfully detected by the proposed structural analysis algorithm. In the second experiment we impose the same control inputs to the SSMR , but this time not only the fault detection but also the proposed fault accommodation algorithm is enabled. As we can see in Fig. 6 the on line estimation algorithm quickly converge to the new radius of the faulty wheel set and consequently the fault accommodation algorithm provides modified inputs to the right wheel set (Fig. 7).

Conclusion

The notion of fault tolerant control for a 4-wheel skid steering mobile robot is an important problem to deal with, since faults appearance is inevitable in such systems. The most significant challenge arises from the complexity of the system. In this paper we have introduced the underlying concepts for our approach to fault tolerant control for mobile robots focusing our attention mainly to control reconfiguration. As concerning the issue of fault diagnosis the structural analysis based technique is used in order to generate residuals. We use the kinematic model of the mobile robot that serves to the development of the structural model of the system. The above technique provides the parity equations which can be used as residual generators since model based fault diagnosis approach is based on residuals. The advantage of the above method is that it can offer a feasible solution to the residual generation of nonlinear systems. The fault accommodation procedure targets in the case where one of the two wheel tire sets becomes flat. The proposed accommodation method is based on a RLS approximation of the new faulty wheel radius and via this information a new control input is calculated in order to compensate for the fault. The efficacy of the proposed method is demonstrated through an extensive experimental procedure using a mobile robot Pioneer 3-AT.