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V L Krishnan, P M Pathak, S C Jain and A K SamantarayReconfiguration of four-legged walking robot for actuator faults

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Reconfiguration of four-legged walkingrobot for actuator faultsV L Krishnan*1, P M Pathak1, S C Jain1, and A K Samantaray2

1Robotics and Control Laboratory Mechanical and Industrial Engineering Department, Indian Institute of Technology,

Roorkee, India2Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, India

The manuscript was received on 9 December 2010 and was accepted after revision for publication on 21 April 2011.

DOI: 10.1177/0959651811410257

Abstract: This paper presents fault accommodation through the reconfiguration of a fourlegged (quadruped) walking robot. It is assumed that the walking robot is moving using abounding gait. A bond graph model to represent the robot’s locomotion in the sagittal plane isdeveloped. A novel and simple attitude control device based on a moving appendage is pro-posed. This appendage device serves as redundant hardware and is activated only when therobot experiences a locked joint failure. The bond graph model of the system is used to gener-ate the analytical redundancy relations which are then evaluated with actual measurements togenerate residuals. These residuals are used to perform structural fault isolation. Once thefault list is updated in the equipment availability database, an automaton selects the bestoption to reconfigure the system such that the given control objectives are achieved. Thedeveloped methodology is validated by considering the failure of a joint’s actuator.

Keywords: walking robot, locked joint failure, fault accommodation, reconfiguration, attitude

control, moving appendage

1 INTRODUCTION

Robots are increasingly being deployed in prefer-

ence to humans in situations that are extremely

complex and/or hazardous such as nuclear power

plant accidents, military operations, and space

exploration. To improve the probability of mission

success the tasked robot must be able to autono-

mously perform fault detection and isolation activi-

ties and take corrective/preventive maintenance

action in real-time. In the case of multi-legged walk-

ing robots the possible failure situations may be

broadly categorized as either a failure in the kine-

matic part of the leg or a communications failure

between the controller and the leg.

The failure in the kinematic part of the leg can be

one of two types:

(a) a free-swinging failure: in this failure the actua-

tor torque of a leg joint is lost and the joint

swings freely and thus it cannot support the

body of the robot;

(b) a locked joint failure: in this failure the joint is

locked in place. Consequently, the workspace/

reach of the leg tip is restricted; however, it can

still support the body of the robot. Thus, the

failed leg still partially contributes towards the

movement of the robot.

The locked joint failure is the more common of

these failure types due to its numerous causes

including the effects of an increased friction level in

bearings, clutch slippage, and erroneous application

of the brakes.

These failures can be rectified using either fault-

tolerant control or fault accommodation through

reconfiguration [1, 2]. In fault-tolerant control the

control laws and their associated control problem

are altered in order to maintain the desired output.

Fault accommodation through reconfiguration is

*Corresponding author: Robotics and Control Laboratory

Mechanical and Industrial Engineering Department, Indian

Institute of Technology,247667, Roorkee, India.

email: [email protected]

1

Proc. IMechE Vol. 225 Part I: J. Systems and Control Engineering



accomplished by deploying alternate standby devices

called hardware redundancies in place of the faulty

components and suitable modifications of the con-

trol laws.

Most of the work done pertaining to fault-tolerant

control in walking robots [3–6] is based on the

assumption that the system does not have redun-

dancy and hence is concerned with the generation of

fault-tolerant gait patterns for the walking robot.

Jakimovski et al. [7] attempted to create fault tolerance

through reconfiguration using a swarm-intelligence-

based approach that did not require additional inverse

kinematics modelling. This concept is applicable to

robots with more then two legs provided that they are

spatially configured in a circle.

Recently, Abdi and Nahavandi [8] used the con-

cept of actuator fault accommodation through

reconfiguration in the case of a five-degree-of-

freedom robotic manipulator. Chi et al. [9] pre-

sented a method for the real-time diagnosis and

accommodation of sensor faults in a flight control

system using analytical redundancy. However, fault

accommodation through reconfiguration for quad-

ruped robots has not yet been reported in the litera-

ture except for an initial report on part of this work

that was presented at a recent conference [10]. The

present work proposes a scheme for fault accommo-

dation through reconfiguration for a quadruped

walking robot that experiences a locked joint failure.

This work proposes a novel attitude control device

for walking robots that allows the attitude of the

walking robot to be controlled with the help of a

moving appendage. The attitude control device acts

as a piece of redundant hardware in the system, that

is, it is not used in nominal or fault-free operation.

The scheme has been validated through simulations

carried out using Symbols [11], a bond graph mod-

elling and simulation software.

2 MODELLING OF THE WALKING ROBOT

The modelling of the quadruped robot consists in

modelling the translational and angular dynamics of

the robot’s legs and body. Figure 1 is a schematic

diagram of the robot. Each leg of the robot has been

modelled as an open chain manipulator comprising

of two rigid links connected through revolute joints.

Each leg of the walking robot has two degrees of

freedom and hence the robot has 14 degrees of free-

dom (eight degrees of freedom for four legs and six

degrees of freedom for the body). In Fig. 1, frame

{A} is an inertial frame of reference and {V} is a

body-fixed frame. A coordinate frame is also

attached to each link. The link frames are labelled

using a number that is based on the link to which

they are attached, i.e. frame {i} is rigidly attached to

link i. The joint between links i and i + 1 is num-

bered as i + 1. Frame {0} and {1} are coincident,

where frame {0} is the body-fixed frame for the hip

joint. Joint rotation axes are shown in Fig. 1. The

rotational inertia of a body (or link) is defined about

a frame fixed at the centre of gravity (CG) of the

body (or link). The axis of the CG frame is fixed

along the principal directions of the body (or link).

The robot is assumed to be performing movement

using a bounding gait without any slipping of the

legs on a hard surface. Experimental implementa-

tion of the gait has been carried out by de Lasa and

Buehler [12] for walking robots.

The forward direction of motion of the robot was

taken to be along the negative Y-axis. The robot

dynamics were analysed in the sagittal plane of

movement, i.e. the YZ plane. The case was consid-

ered in which the frontal dynamics could be taken

care of using a moving appendage operated in the

transverse direction, i.e. X-axis. The corresponding

representation of the quadruped robot in the sagit-

tal plane of movement is shown schematically in

Fig. 2. The figure shows a view of the robot whose

legs have two degrees of freedom in the sagittal

plane, i.e. the YZ plane.

The position of the front leg tip with respect to

inertial reference frame {A} is given as

YFlt = YCG + rF cos p + fð Þ+ l1 cos p + f + u1Fð Þ+ l2 cos p + f + u1F + u2Fð Þ

(1)

Fig. 1 Schematic diagram of the quadruped robot

2 V L Krishnan, P M Pathak, S C Jain, and A K Samantaray




ZFlt = ZCG + rF sin p + fð Þ+ l1 sin p + f + u1Fð Þ+ l2 sin p + f + u1F + u2Fð Þ (2)

In equations (1) and (2), YCG and ZCG represent

the location of frame {V} with respect to frame {A},

rF is the distance of the {0F} frame from the {V}

frame, f is the attitude of the body, u1F and u2F are

the joint angles of the first and second joints of the

front leg, respectively, and l1 and l2 are the link

lengths of the first and second links. Differentiating

equations (1) and (2) allows the components of the

velocity of the front leg tip in the y and z directions

to be written as

_YFlt = _YCG� rF sin p + fð Þ _f� l1 sin p + f+ u1Fð Þ _f+ _u1F

� �

� l2 sin p + f+u1F +u2Fð Þ _f+ _u1F + _u2F

� �(3)

_ZFlt = _ZCG + rF cos p + fð Þ _f + l1 cos p + f + u1Fð Þ _f + _u1F

� �

+ l2 cos p + f + u1F + u2Fð Þ _f + _u1F + _u2F

� �(4)

If it is assumed that the lengths of the rear and

front legs are equal then the position of the rear leg

tip with respect to the inertial reference frame {A}

can be represented as

YRlt = YCG + (rR cos f) + l1 cos f + u1Rð Þ+ l2 cos f + u1R + u2Rð Þ

(5)

ZRlt = ZCG + (rR sin f) + l1 sin f + u1Rð Þ+ l2 sin f + u1R + u2Rð Þ (6)

where rR is the distance of the {0R} frame from the

{V} frame and u1R and u2R are the joint angles of the

first and second joints of the rear leg, respectively.

The rear leg tip velocities can be evaluated from

equations (5) and (6) as

_YRlt = _YCG � (rR sin f) _f� l1 sin f + u1Rð Þ _f + _u1R

� �

� l2 sin f + u1R + u2Rð Þ _f + _u1R + _u2R

� �(7)

_ZRlt = _ZCG + (rR cos f) _f + l1 cos f + u1Rð Þ _f + _u1R

� �

+ l2 cos f + u1R + u2Rð Þ _f + _u1R + _u2R

� �(8)

A bond graph model of the quadruped walking

robot dynamics in the sagittal plane can be obtained

using equations (3), (4), (7), and (8) as shown in Fig. 3.

In Fig. 3, pad structures have been used to avoid

differential causalities in the bond graph. Pads

appear as 1-C-R structures appended to intermedi-

ate 0-junctions, corresponding to ports where differ-

ential causalities exist. The dynamic effect of the

pad can be reduced by choosing high values of the

corresponding C and R elements.

Figure 4 presents the sub-model of the actuator

in detail. The considered gear ratio is high so as to

create the actuator self-locking property in non-

actuated conditions. Contact dynamics between the

leg tip (toe) and environment (i.e. ground) is mod-

elled through a linear spring–damper system. The

dynamic analysis of the quadruped robot in the

sagittal plane, using the bond graph in Fig. 3, con-

cerns the control of forward motion and body atti-

tude of the robot.

In order for the robot to move in the forward

direction a gait pattern has to be specified. Figure 2

presents the sagittal plane representation of the

quadruped robot. It shows the left side (with respect

to the movement direction) of the walking robot.

Hence, only leg 1 and leg 3 of the robot, which,

respectively, represent the front and rear legs of the

robot, are visible in the figure. The following section

presents the gait pattern of the walking robot.

2.1 The gait pattern

The following steps describe the gait pattern of a

quadruped robot.

Step 1: the joint 2 on each of the front legs is rotated

by 0.2 rad in the clockwise direction about the -x axis.




0.2 rad in the anticlockwise direction about the -x axis.

Step 4: the joint 2 on each of the rear legs is rotated

by 0.2 rad in the clockwise direction about the

x axis.


by 0.1 rad in the clockwise direction about the x axis.

Fig. 2 Sagittal plane representation of the quadrupedrobot

Reconfiguration of four-legged walking robot for actuator faults 3





by 0.1 rad in the anticlockwise direction about the -

x axis.





x axis.



x axis.



x axis.

These steps together form a locomotion cycle. This

cycle is repeated a number of times to obtain the

desired movement in the forward direction. The steps

1 to 10 of the locomotion cycle are shown in Fig. 5

using frames obtained in an animation of simulation

results. The parameters used in the simulation of the

walking robot model are listed in Table 1A (Appendix

2). To take the joint to the specified value of the joint

angles the voltage supplied to the motor is based on

proportional-derivative (PD) control. One locomotion

cycle is assumed to last 2.2 s. The movement of the

robot was simulated for a period of 11 s, i.e. five cycles.

Figures 6(a) and (b) show the rotation of joints 1

and 2 of the front legs with respect to time.

Similarly, Figs 6(c) and (d) show the trajectory of

joints 1 and 2 of the rear legs. From these figures it

can be seen that the desired joint angles are

achieved as per the specified gait pattern.

Figure 7(a) shows the body attitude (f) of the

walking robot. It can be noted from this figure that

initially transients are present which settle down

quickly. These transients are due to the pads used

to remove the differential causality from the model.

Figure 7(b) shows the CG displacement of the body

in the negative Y-direction (i.e. forward direction of

motion). Body movement in the forward direction

can be concluded from this figure. Figure 7(c) shows

the CG displacement of the body in the Z-direction.

Figure 7(d) shows the animation result for robot

Fig. 4 Actuator sub-model

Fig. 3 Bond graph representation of sagittal plane dynamics of the quadruped robot





movement in the forward direction. A definite

advancement of the walking robot can be seen from

the animation. Here only few frames are shown to

have clarity.

Figures 8(a) and (b) show the Y and Z displace-

ments of the front leg tip, while Figs 8(c) and (d)

show the same displacements for the rear leg tip. It

can be noted from the simulation results that the

displacements of the front and rear leg tips are also

along the negative y-axis, i.e. in the direction of

body advancement.

In the next section a novel scheme for attitude

control using a moving appendage is conceptua-

lized. This device is proposed to be used in the

reconfiguration of a robot that has experienced a

failure in its leg joint.

3 ATTITUDE CONTROLLER USING A MOVINGAPPENDAGE

Attitude control of a walking robot is a very impor-

tant issue from the perspective of achieving a

desired walking pattern in terms of both the gait

speed and direction control. In this work a novel

scheme that uses a moving appendage for attitude

control is presented. The system is shown schemati-

cally in Fig. 9. The body whose attitude is to be con-

trolled is shown in the figure along with the moving

appendage arrangement. The body dynamics are

considered to be constrained to the YZ plane.

The body is considered to move with respect to

an inertial reference frame {A} and it can undergo

rotational movements due to the moments caused

by the ground reaction forces. The body’s linear

velocity with respect to the frame {A} is resolved into

vy and vz components. The angular velocity of the

body’s fixed frame {V} with respect to frame {A} is

represented by _f.

The frame {V} is attached to the CG of the body.

The velocity components are determined by the iner-

tia elements attached to the 1-junction corresponding

to vy , vz , and _f in the bond graph shown in Fig. 10.

The body orientation f can be controlled using

the moving appendage which is an inertial element

in the form of a rack that balances the disturbing

moments on the body. The rack can perform only

linear motion along the paths provided in the body.

The moving appendage system consists of a pinion

that is driven by a motor. Rotational motion of the

pinion is then converted into linear motion through

a rack and pinion arrangement. The velocity of the

rack can be expressed as vrz. The velocity of the rack

can be obtained by taking the derivative of the coor-

dinate positions of the rack with respect to frame

{A}. The coordinates of rack can be given as

Zr = ZCG + pup sin f (9)

where p is the pitch circle radius of the pinion, up is

the rotation of the pinion of the attitude controller

with respect to the frame {V}, and YCG and ZCG are

the coordinates of the body’s CG with respect to the

frame {A}.

Hence, the velocity of the rack can be represented as

vrz =d(Zr)

dt= vZ + p _up sin f + pup

_f cos f(10)

where vz = d(ZCG)/dt. The angular velocity of the

pinion with respect to the inertial reference frame

{A} can be expressed as

A _up = A _fV + V _up (11)

where A _fV is the angular velocity of the body with

respect to frame {A} and V _up is the angular velocity

of pinion with respect to the body frame {V}. Using

equations (10) and (11) the bond graph for the body

with the moving appendage can be obtained as

Fig. 5 Frames representing the gait pattern of thequadruped walking robot





Fig. 6 Simulation results of quadruped walking robot: (a) front leg joint 1 angle position (u1F)versus time; (b) front leg joint 2 angle position (u2F) versus time; (c) rear leg joint 1 angleposition (u1R) versus time; and (d) rear leg joint 2 angle position (u2R) versus time

Fig. 7 Simulation results for the quadruped walking robot: (a) body angular displacement (f)versus time; (b) body’s CG displacement along Y-direction (YCG) versus time; (c) body’s CGdisplacement along Z-direction (ZCG) versus time; and (d) animation of walking robot





shown in Fig. 10. The velocity of the moving appen-

dage, i.e. rack can be governed by controlling the

rotation of the pinion which, in turn, can be con-

trolled through the voltage supplied to the motor.

The control law governing the voltage input to the

motor can be written as

Vac = Kpa(fdes � fa) + Kva( _fdes � _fa) (12)

where fdes and fa are respectively the desired and

actual angular displacement of the body with

respect to the inertial reference frame {A}, and Kpa

and Kva are the proportional and derivative gains.

The movement of the rack causes the CG of the

rack to change its position in order to achieve the

desired attitude of the body fdes. Table 2A

(Appendix 2) lists the values of the parameters

assumed in the simulation. The simulation result in

Fig. 11 shows that the body’s attitude is adequately

controlled through the moving appendage to a

desired orientation of 0.2 rad.

4 RECONFIGURATION OF THE WALKING ROBOT

A schematic representation of a walking robot with

the proposed attitude controller is shown in Fig. 12.

The bond graph representation of the sagittal plane

dynamics of this system is shown in Fig. 13.

Fault accommodation in the walking robot using

the reconfiguration approach is carried out as per

the procedure followed in reference [13]. The fault

accommodation procedure consists of the following

stages.

Fig. 8 Simulation results of walking robot: (a) Y displacement front leg tip (YFlt) versus time; (b) Zdisplacement front leg tip (ZFlt) versus time; (c) Y displacement rear leg tip (YRlt) versustime; and (d) Z displacement rear leg tip (ZRlt) versus time

Fig. 9 Conceptualization of the moving appendage forattitude control





4.1 Generation of fault indicators

To generate the fault indicators, the sensors in the

bond graph model shown in Fig. 13 are dualized

into sources and pads are then removed [2]. This

leads to the creation of storage elements with differ-

ential causalities. However, no causal loops are

formed due to these transformations carried out in

the bond graph model. The transformed bond graph

model presented in Fig. 14 is the diagnostic bond

graph (DBG) model [14] of the quadruped robot

equipped with the attitude controller. Note that

causalities in some of the integration blocks are not

inverted because they are not the actual sensors in

the plant; they are in the model so that simulation

results can be plotted. The analytical redundancy

relations (ARRs) are then derived from the DBG, by

following well established algorithms presented in

reference [2]. The ARRs are written in terms of

system parameters and measured quantities. They

are the expressions for the variables measured at

residual sensors (effort variables in bonds connected

to De* elements in Fig. 14).

The DBG model of each leg joint actuator is

shown in Fig. 15. Each leg has two sensors which

lead to two ARRs as follows

(Vm)j � Lm

d(i)j

dt� Rm(i)j � n Ktm( _urot)j = 0 (13)

Rbf ( _urot)j � n Ktm(i)j + (Irot€urot)j + tuj = 0 (14)

where (Vm)j and (i)j are respectively supply voltage

and armature current to the jth joint actuator, ( _urot)j

is the angular velocity of the jth joint actuator rotor,

and tuj is the reaction torque on the jth actuator.

Since each leg has two joint actuators, there are in

total four ARRs per leg. Hence, ARRs 1–16 are asso-

ciated with the four legs in the order starting from

Fig. 10 Bond graph of attitude controller

Fig. 11 Simulation result: body orientation (withrespect to inertial reference frame {A}) versustime for bond graph shown in Fig. 10

Fig. 12 Schematic representation of a walking robotwith the proposed attitude controller





front left leg, to rear left leg, followed by front right

and then rear right leg.

Corresponding to the actuator for attitude control

the ARRs can be written as

ARR 17:

Vac � Rac(i)ac � Kta_up = 0

(15)

ARR 18:

Kta(i)ac � (Mr €yr)(p cos f)� (Mr€zr + Mrg)(p sin f)

�Jp(€up + €f) = 0 (16)

where Vac and iac are the supply voltage and arma-

ture current, respectively, corresponding to the atti-

tude controller actuator, _up is the angular velocity of

the pinion of the attitude controller, and yr and zr

are the rack displacement components in the y and

z directions, respectively.

Furthermore, corresponding to the sensors mea-

suring body motion in the YZ plane and pitch angle

f of the robot body, there are three ARRs as follows

ARR 19:

Mb€YCG + C1( _up) +

X2

j = 1

FFY ( _uj, _Y Fj)

+X2

j = 1

FRY ( _uj, _Y Rj) = 0 (17)

whereP2

j = 1 FFY ( _uj, _Y Fj) andP2

j = 1 FRY ( _uj, _Y Rj) are

the Y components of the reaction forces acting on

Fig. 13 Bond graph model of the sagittal plane dynamics of the quadruped robot equipped withthe proposed attitude controller





the body, due to the combined effect of inertia of

the front and rear leg links and the leg tip’s interac-

tion with the ground. C1( _up) is the reactionary force

exerted by the attitude controller along the Y-axis of

the body CG frame {V} and is equal to Mr €yr

ARR 20:

Mb€ZCG + Mbg + C2( _up) +

X2

j = 1

FFZ ( _uj, _ZFj)

+X2

j = 1

FRZ ( _uj, _ZRj) = 0 (18)

whereP2

j = 1 FFZ ( _uj, _ZFj) andP2

j = 1 FRZ ( _uj, _ZRj) are

the Z components of the reaction forces acting on

Fig. 14 Diagnostic bond graph model of sagittal plane dynamics of quadruped robot equippedwith the attitude controller

Fig. 15 DBG model of the leg joint actuator





the body, due to the combined effect of the inertia

of the front and rear leg links and the leg tip’s inter-

action with the ground. C2( _up) is the reactionary

force exerted by the attitude controller along the

Z-axis of the body CG frame {V}. C2( _up) is equal to

(Mr€zr + Mrg)

ARR 21:

Jb€f+C3( _up)� rF sin(p+f)½ �

X2

j=1

FFY ( _uj, _Y Fj)

+ rF cos(p+f)½ �X2

j=1

FFZ ( _uj, _ZFj)

� (rR sinf)X2

j=1

FRY ( _uj, _Y Rj)+(rR cosf)X2

j=1

FRZ ( _uj, _ZRj)

+tuF( _uj, _Y Fj, _ZFj)+tuR( _uj, _Y Rj, _ZRj)=0 (19)

where tuF( _uj, _YFj, _ZFj) and tuR( _uj, _YRj, _ZRj) are the

components of the reactionary moment acting on

the body due to the combined effect of the inertia of

the front and rear leg links and the leg tip’s interac-

tion with the ground. C3( _up) is the reactionary

moment exerted by the attitude controller about the

X-axis of the robot’s body and it can be expressed as

C3( _up) = pup cos f(Mr€zr + Mrg)� pup sin f(Mr €yr)

+ Jp(€up + €f) (20)

4.2 Fault detection and isolation

Fault detection and isolation is a very important

step from the perspective of fault-tolerant control. A

detection and isolation scheme for actuator faults

and fault accommodation was developed in [14–17].

In that scheme the generated ARRs are structurally

analysed and are then used to generate the fault sig-

nature matrix (FSM) or a binary fault sensitivity

table. If at least one element of a row in the FSM is

non-zero, then the corresponding component can

be monitored. If all the rows of the FSM are unique,

i.e. there is no repetition of rows (unique fault sig-

nature) then the fault is isolatable. Table 1 presents

the FSM corresponding to the ARRs represented by

equations (13) to (19).

4.3 Fault accommodation throughreconfiguration

A reconfigurable system usually has more sensors

and actuators than actually immediately required.

The ARRs and the FSM must be modified every

time a system is reconfigured. The functional ser-

vices offered by the components are organized into

coherent subsets, called operating modes (OMs)

where each OM is associated to a bond graph

model. In the event of a component failure the

equipment availability database is searched and

the reconfiguration automata specify the control

law to be adopted for another OM. Theoretically, a

process can operate in a normal manner as long as

at least one device is available for each basic func-

tion. When a device fails, the branch associated

with it is removed and the system is reconfigured

using the next device, according to the defined

hierarchy.

Figure 16 shows the various desired tasks to be

accomplished by the quadruped robot, corre-

sponding operating modes, and the basic devices

available for performing the task. Usually a quad-

ruped robot may be required to perform any of the

three missions shown in the figure. To perform

these tasks it can rely on any of the various gaits

available i.e. continuous, discontinuous, free, or

turning gait [18–22]. At the lowest level of control,

the leg joint actuators need to be fully functioning

to accomplish these missions. In the event of the

failure of a leg joint actuator the robot can switch

to an alternative gait to accomplish a mission. It

does so by exploiting the actuator which becomes

redundant as a consequence of adopting the alter-

native gait.

Table 1 The FSM

Component (actuator)

Residual

Isolabilityr1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21

FLL joint 1 actuator 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1FLL joint 2 actuator 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1RLL joint 1 actuator 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1RLL joint 2 actuator 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1FRL joint 1 actuator 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1FRL joint 2 actuator 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1RRL joint 1 actuator 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1RRL joint 2 actuator 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1Attitude controller actuator 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1Body 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1





For instance, if the robot while performing a

straight line motion on a flat terrain (with continu-

ous gait) faces the situation of the failure of an

actuator it can switch to a discontinuous gait. In the

discontinuous gait a walking robot uses various

actuators sequentially and unlike continuous gait

not all the leg joint and body attitude actuators are

used simultaneously. Hence, it can use the redun-

dant attitude controller actuator to cover for the

joint actuator fault.

5 SIMULATIONS AND RESULTS

In these studies it was considered that joint 2 of leg 1

fails while the robot is walking. It was assumed that

only one fault event occurred during the movement

and that this fault could not be recovered. The robot

was considered to begin its operation with all its joint

actuators being fully functional. The simulation of the

robot was performed for five complete locomotion

cycles. One locomotion cycle was assumed to take

2.2 s. A fault in joint 2 of leg 1 was simulated after the

third cycle, by ending the supply of power to the cor-

responding actuator joint. If the desired angle of rota-

tion for joint 2 of leg 1 during the movement cycle is

0.2 rad and it fails at another angle, then reconfigura-

tion is obtained using the designated value of the

desired value fdes given by the expression

fdes = � p + fa + u1F + 0:2ð Þ (21)

Fig. 16 Management of operating modes and equip-ment availability

Fig. 17 Simulation results for the reconfigured walking robot (RWR) when the attitude controlleris operational: (a) RWR u1F versus time; (b) RWR u2F versus time; (c) RWR u1R versus time;and (d) RWR u2R versus time





A PD controller was designed to evaluate the vol-

tage supplied to the attitude control motor to bring

the body to the desired value given by equation (21).

The proportional and derivative gain values used in

the simulation were 1.2 and 0.012, respectively. The

walking behaviour of the robot experiencing the con-

sidered joint failure was simulated for two cases:

(a) the attitude controller remains switched ‘OFF’ dur-

ing the entire locomotion cycle, i.e. the robot walks

with a failed joint throughout the movement;

(b) the attitude controller is switched ‘ON’ at cer-

tain point of the locomotion cycle after the

joint fails. Switching on the controller implies

starting the power supply to the motor of the

attitude controller using equation (12).

Figures 17(a) and (b) show the rotation of joints 1

and 2 of the front legs when the attitude controller is

switched ‘ON’ for the purpose of fault accommoda-

tion. Figure 17(b) indicates the failure of the joint 2

actuator in the front leg during a locomotion cycle.

Figures 17(c) and (d) show the rotation of the joints

1 and 2 of the rear legs. From these figures it can be

seen that the desired joint angle displacements are

achieved as per the specified gait pattern in Fig. 5.

Figures 18(a) and (b) present simulation results

for the RWR for the case where the attitude control-

ler (AC) remains switched OFF during the entire

locomotion cycle. Figures 18(c) and (d) show the

simulation results obtained when the AC is switched

ON, to allow fault accommodation, after the joint

actuator fails. Figure 18(a) shows the body angular

displacement (f) of the quadruped robot in the first

case, i.e. the AC remains switched OFF even after

the joint fails. It can be noted from the figure that

initially transients are present which settle down

quickly. These transients are due to the pads used in

the simulation to remove the differential causality

from the model. Figure 18(b) shows the movement

of the body’s CG along the negative Y-axis (i.e. the

desired forward direction), when the AC remains

switched OFF. Figure 18(c) shows the body attitude

(f) of the RWR, i.e. when AC is switched ON.

Figures 18(b) and (d) show the body displacement

in the forward direction for the two considered

cases. It can be seen that the robot moves in the

desired direction satisfactorily even when the AC is

Fig. 18 Simulation results for the RWR: case (i) when the AC is switched OFF (a) f (rad) versustime and (b) YCG versus time and case (ii) when the AC switched ON (c) RWR f (rad) ver-sus time and (d) RWR YCG versus time





switched OFF, but the overall progression achieved

is greater when AC is switched ON.

Figures 19(a) and (b) show the front and rear leg

tips’ Y displacement when the AC remains switched

OFF and Figs 19(c) and (d) show the front and rear

leg tips’ Y displacement when the AC is switched

ON. The leg tip displacements in Figs 19(c) and (d)

are greater than the leg tip displacements shown in

Figs 19(a) and (b). The proposition that the consid-

ered fault event can be accommodated through the

AC (based on the moving appendage) is validated

from these simulation results.

6 CONCLUSIONS

In the present work, bond graph modelling of the

locomotion dynamics in the sagittal plane for a

quadruped robot has been carried out. A novel

scheme for controlling the attitude disturbance of

the walking robot based on a moving appendage

has been proposed. The simulation results demon-

strated successful control of body attitude using the

proposed scheme. An AC has been used for fault

accommodation through reconfiguration for a

quadruped robot that has experienced a locked joint

failure. The simulation results also indicated that

the robot traverses greater distance as compared to

a situation when an AC is not used.

It must be noted that fault accommodation

through reconfiguration is possible only when

redundant components are available. In light of that

point the current work assumes even more signifi-

cance because the AC itself is being used to achieve

fault accommodation, thus optimizing the hardware

redundancy and the corresponding payload. In

future the experimental implementation of the pro-

posed methodology is to be carried out.

� Authors 2011

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Fig. 19 Simulation results for the RWR: case (i) when the AC switched OFF (a) YFlt versus time (b) YRlt

versus time and case (ii) when the AC is switched ON (c) YFlt versus time and (d) YRlt versus time





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APPENDIX 1

Notation

Subscripts

a actual

bf bearing friction

CG centre of gravity

des desired

F front

Flt front leg tip

m motor

p pinion of attitude controller

r rack

R rear

Rlt rear leg tip

rot rotor of Motor

1F front leg first joint

1R rear leg first joint

2F front leg second joint

2R rear leg second joint





APPENDIX 2

Parameters used in simulations

Table 1A Walking robot parameters

ParametersSymbolused Value Units

Body mass Mb 2 kgPolar moment of inertia Jb 0.02 kg.m2

Position of front hip from body CG rF 0.5 mPosition of rear hip from body CG rR 0.5 mLength of link 1 l1 0.2 mLength of link 2 l2 0.4 mMass of link 1 Ml1 0.2 kgMass of link 2 Ml2 0.2 kgPolar moment of inertia of link 1 J1 0.001 kg.m2

Polar moment of inertia of link 2 J2 0.001 kg.m2

Actuator motor inductance Lm 0.001 HActuator motor resistance Rm 0.1 OActuator motor constant Ktm 0.2 Nm/ARotor inertia Irot 0.0001 kg.m2

Gear ratio n 100Bearing frictional resistance Rbf 0.1 N.s/mGround contact stiffness Kg 10000 N/mGround contact damping Rg 10 N.s/mGround friction resistance Rf 10 N.s/m

Table 2A Attitude controller parameters

ParametersSymbolused Value Units

Mass of rack Mr 0.05 kgMass of body Mb 1.00 kgPolar moment of inertia of body Jb 0.0025 kg.m2

Pad stiffness Kpad 1000 N/mPad damping Rpad 10 N.s/mProportional gain Kpa 0.2 V/radDerivative gain Kva 0.6 V.s/radAttitude control motor resistance Rac 0.1 OAttitude control motor constant Kta 0.2 N.m/APitch of the rack and pinion pair p 0.001 mPolar moment of inertia of pinion Jp 0.001 kg.m2

Distance of front tip from body CG rF 0.1 mDistance of rear tip from body CG rR 0.1 m





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