Well-Conditioned Configurations of Fault-Tolerant ManipulatorsHamid Abdi 1 , Member IEEE Centre for Intelligent Systems Research Deakin University, Waurn Ponds, Victoria 3216 AustraliaSaeid Nahavandi, SMIEEE Centre for Intelligent Systems Research Deakin University, Waurn Ponds, Victoria 3216 AustraliaAbstract: Fault-tolerant motion of redundant manipulators can be obtained by joint velocity reconfiguration. For fault- tolerant manipulators, it is beneficial to determine the configurations that can tolerate the locked-joint failures with a minimum relative joint velocity jump. This is because the manipulator can rapidly reconfigure itself to tolerate the fault. This paper uses the properties of condition numbers to introduce those optimal configurations for serial manipulators. The relationship between the manipulator’s locked-joint failures and the condition number of the Jacobian matrix is indicated by using a matrix perturbation methodology. It is observed that the condition number provides the upper bound of required relative joint velocity change for recovering the faults which leads to define the optimal fault-tolerant configuration from the minimization of the condition number. The optimization problem to obtain the minimum condition number is converted to three standard Eigen value optimization problems. Finally, in order to obtain the optimal fault-tolerant configuration, the proposed method is applied to a 4-DoF planar manipulator. Keywords: Redundant manipulator, Condition number, Optimal fault tolerance, Actuator failure, Isotropic manipulators. I.INTRODUCTION Fault-tolerant manipulators are essential where highly dependable robots are required such as robotic manipulators working in hazardous environments, nuclear disposal and exploring of deep sea or outer space [1-3]. The design or control of a fault-tolerant manipulator aims to maintain the dependability of the manipulator despite partial failures that can occur in the manipulator’s actuators or sensors [4, 5]. The literature surrounding fault-tolerant manipulators focuses on the design or control of the manipulators. Within the literature dealing with design of the manipulators, different structures such as serial [6] or parallel [7, 8] manipulators have been studied or a manipulator with a specific fault-tolerant property has been designed [6, 9]. Within the literature dealing with the control, the fault detection [10], fault isolation and identification [11] and fault recovery [5, 12, 13] are discussed. Various strategies such as model based or artificial intelligence (AI) controllers have been proposed for fault-tolerant manipulators. 1 Corresponding Author: hamid.abdi@deakin. edu.au Serial link manipulators (SLM) have received significant attention in the robotics community because of their wide range of applications. Fault-tolerant design of the SLMs can be achieved by adding extra kinematic redundancy [3] . By this redundancy, manipulators are considered as serial link redundant manipulators (SLRM). It has been shown that SLRMs can maintain their dependability to perform the required [14] or prioritized tasks [15] despite joint failures. It has also been observed that adding kinematic redundancy not only improves the fault tolerance specifications of the manipulators, but also promotes other static or dynamic properties including higher dexterous movements [16], lower maintenance time and repair costs, obstacle avoidance [17] and capability for motion planning and control with multiple constraints[15, 18, 19]. However, it should also be noted that having kinematic redundancy does not guarantee the fault-tolerant operation of redundant manipulators [6], because the kinematic redundancy has to be efficiently used to achieve the fault tolerance. In [20] the number of required joint redundancies is investigated by applying the joint failure probability and total reliability of the manipulator. In [6, 21] fundamental limits of optimal fault-tolerant configuration for SLRMs are studied and the constraints of the optimal fault tolerance are presented. Identifying a fault-tolerant operation of a redundant manipulator in the forward or inverse kinematic domains is hardly possible due to the nonlinearity of the forward or inverse kinematic equations. Therefore, commonly the properties of the Jacobian matrix or null space of the Jacobian matrix are used by the researchers. For example, the jacobian matrix is used to define fault tolerance indices that can be deployed to perform fault tolerance analysis or design of fault-tolerant controllers. In [6, 21, 22], the main properties of Jacobian matrix or null space of the Jacobian matrix have been analyzed. Using the kinematic redundancy, the robotic manipulators can be positioned in an optimal configuration for better performance in fault-tolerant operation. In these configurations, the fault tolerance will be more convenient than other configurations because there will be lower required reconfiguration. For example, if a surgical robot cuts a patient’s body with a surgical knife, then an appropriate configuration can minimize the undesirable behavior of the robot when locked-joint failures occur. This
Hamid Abdi 1, Member IEEECentre for Intelligent Systems Research
Deakin University, Waurn Ponds, Victoria 3216Australia
Saeid Nahavandi, SMIEEECentre for Intelligent Systems Research
Deakin University, Waurn Ponds, Victoria 3216Australia
Abstract: Fault-tolerant motion of redundant manipulatorscan be obtained by joint velocity reconfiguration. For fault-tolerant manipulators, it is beneficial to determine theconfigurations that can tolerate the locked-joint failures with aminimum relative joint velocity jump. This is because themanipulator can rapidly reconfigure itself to tolerate the fault.This paper uses the properties of condition numbers to
introduce those optimal configurations for serial manipulators.The relationship between the manipulator’s locked-jointfailures and the condition number of the Jacobian matrix isindicated by using a matrix perturbation methodology. It isobserved that the condition number provides the upper boundof required relative joint velocity change for recovering thefaults which leads to define the optimal fault-tolerantconfiguration from the minimization of the condition number.The optimization problem to obtain the minimum conditionnumber is converted to three standard Eigen valueoptimization problems. Finally, in order to obtain the optimalfault-tolerant configuration, the proposed method is applied toa 4-DoF planar manipulator.
Fault-tolerant manipulators are essential where highlydependable robots are required such as robotic manipulatorsworking in hazardous environments, nuclear disposal andexploring of deep sea or outer space [1-3]. The design orcontrol of a fault-tolerant manipulator aims to maintain thedependability of the manipulator despite partial failures thatcan occur in the manipulator’s actuators or sensors [4, 5].The literature surrounding fault-tolerant manipulatorsfocuses on the design or control of the manipulators. Withinthe literature dealing with design of the manipulators,different structures such as serial [6] or parallel [7, 8]manipulators have been studied or a manipulator with aspecific fault-tolerant property has been designed [6, 9].Within the literature dealing with the control, the faultdetection [10], fault isolation and identification [11] andfault recovery [5, 12, 13] are discussed. Various strategiessuch as model based or artificial intelligence (AI) controllershave been proposed for fault-tolerant manipulators.
Serial link manipulators (SLM) have received significantattention in the robotics community because of their widerange of applications. Fault-tolerant design of the SLMs can
be achieved by adding extra kinematic redundancy [3] . Bythis redundancy, manipulators are considered as serial linkredundant manipulators (SLRM). It has been shown thatSLRMs can maintain their dependability to perform therequired [14] or prioritized tasks [15] despite joint failures. Ithas also been observed that adding kinematic redundancynot only improves the fault tolerance specifications of themanipulators, but also promotes other static or dynamic
properties including higher dexterous movements [16],lower maintenance time and repair costs, obstacle avoidance[17] and capability for motion planning and control withmultiple constraints[15, 18, 19].
However, it should also be noted that having kinematicredundancy does not guarantee the fault-tolerant operation ofredundant manipulators [6], because the kinematic
redundancy has to be efficiently used to achieve the faulttolerance. In [20] the number of required joint redundanciesis investigated by applying the joint failure probability andtotal reliability of the manipulator. In [6, 21] fundamentallimits of optimal fault-tolerant configuration for SLRMs arestudied and the constraints of the optimal fault tolerance are
presented.
Identifying a fault-tolerant operation of a redundantmanipulator in the forward or inverse kinematic domains ishardly possible due to the nonlinearity of the forward orinverse kinematic equations. Therefore, commonly the
properties of the Jacobian matrix or null space of theJacobian matrix are used by the researchers. For example,the jacobian matrix is used to define fault tolerance indicesthat can be deployed to perform fault tolerance analysis ordesign of fault-tolerant controllers. In [6, 21, 22], the main
properties of Jacobian matrix or null space of the Jacobianmatrix have been analyzed.
Using the kinematic redundancy, the robotic manipulatorscan be positioned in an optimal configuration for better
performance in fault-tolerant operation. In theseconfigurations, the fault tolerance will be more convenientthan other configurations because there will be lowerrequired reconfiguration. For example, if a surgical robotcuts a patient’s body with a surgical knife, then an
appropriate configuration can minimize the undesirable behavior of the robot when locked-joint failures occur. This
will add to the safety of the patient and reliability of therobot.
In the optimal configurations for a fault-tolerantmanipulator, the faulty manipulator can continue its taskwith a minimum relative reconfiguration. This suggests theratio of the norm of the required compensating velocities to
the norm of the joint velocities as an objective function thatcan be minimized. This problem is called optimalconfiguration with a minimum relative joint velocity jump(RJVJ). The answer to this problem is useful for fast faultrecovery of the manipulators.
Problem statement: if a SLRM is capable of performingfault-tolerant motion, then what is the optimal configurationfor the given pose that requires a minimum RJVJ for faulttolerance?
With regard to this problem, the fault-tolerant motion of themanipulators has been addressed in the literature withminimum joint velocity jump (JVJ) and minimum end
effector (EEF) velocity jump. For example, the minimumJVJ for fault recovery from a joint failure has been obtainedfor a given configuration in [23, 24]. Using the minimumEEF velocity jump strategy, the control law for joint velocityreconfiguration with minimum EEF velocity jump andminimum JVJ has been obtained for a class of static nonlinear systems in [5] and for SLRMs in [12, 13]. It is clear tosee that this literature deals with the post failure operation ofthe manipulator because they aim to tolerate the faults thatcan occur in a given configuration of the manipulator. Incontrast, in this paper, the optimal configuration for faultrecovery is investigated which is related to prior to thefailure occurrence time.
The other contribution of this paper is to use the conditionnumber for fault tolerance of manipulators. Note that thecondition number is commonly used as an isotropic dexterityindex of manipulators [25-28] while the relativemanipulability and worst case dexterity are used mostly toobtain the optimal fault-tolerant configurations of themanipulators [6, 9, 29]. There is not a great deal of workusing condition numbers for fault tolerance analysis of themanipulator. For example, the condition number has been
proposed for fault tolerance in [3, 30] but it has not beendeployed.
This work is an extension of the observation for the fault
tolerance property of the condition number in [31] and thederived control law for fault tolerance in [5, 12, 13]. In this
paper, different aspects of the properties of the conditionnumber for SLRMs are discussed. This leads to theintroduction of the optimal configuration for fault recovery.Through a framework that is associated to the conditionnumber, those configurations of the manipulators that canrecover the fault with a minimum RJVJ are defined.
This paper is organized as follows. The Jacobian matrix ofSLRM subjected to locked-joint failures is presented insection II. Then, the fault tolerance indices are reviewed insection III. Following that, the condition number and its
application for fault tolerance are addressed in section IVwhere the optimal configurations for fault recovery of themanipulator are defined by using the optimality of the
condition number. The condition number of a 4-DoF planarmanipulator is studied from the dexterity and the fault-tolerant point of views in section V. Then, in section VI, it isshown that the optimal configuration is the solution of thesingular value optimization problems. The optimalconfigurations for fault recovery are obtained for a casestudy in section VII. Next, the research presented in this
paper is compared with other literature in section VIII.Finally, the conclusion remarks are presented in section IX.
II. K INEMATICS OF FAULTY MANIPULATORS
The forward kinematics of a manipulator relates the jointangles to the EEF position/orientation by
)(qf x (1)
T nqqq ....21q (2)
T m x x x ...21x (3)
where joint variables (2) define the configuration space, position/orientation variables (3) define the work space ofthe manipulator, n is the dimension of the configurationspace and m is the dimension of the workspace
The inverse kinematic is given by
)(1 xf q (4)
The manipulator with n-DoF (Degrees of Freedom) isredundant mn , and the degrees of kinematic redundancy(DoR) is obtained by mn . It is assumed that the
manipulator has only revolute joints.The Jacobian matrix of the manipulators is obtained by
nm R
qf
J (5)
Jacobian matrix relates the EEF translational and orientationvelocities to the joint velocities by
qJx (6)
nk k k j j j j j jJ ...... 1121
(7)
If mk R j is the k-th column of the Jacobian matrix, then
this column indicates the contribution of the k-th joint intothe EEF translation and orientation velocity. The Jacobianmatrix of a redundant manipulator with locked-joint failurescan be obtained from the above Jacobian matrix, because ifthe manipulator’s k-th joint has failed, then this joint will not
be able contribute into the EEF velocity [6, 21]. Therefore,the faulty manipulator Jacobian matrix can be obtainedsimply by replacing a 0 vector instead of the k-th column ofthe original Jacobian matrix that will be
Then the velocity equation for the faulty manipulator subjectto the k-th joint failure is obtained by
qJx k k (9)
where
nk k
k
j j j j jJ ..... 1121 (10)
nk k k qqqqq ..... 1121 q (11)
The Jacobian matrix (10) is called the k-th reduced Jacobianmatrix. There are n reduced Jacobian matrices associated to
the failure of joint n,...,2,1 that are shown by
JJJ n,...,,21 (12)
With a similar column elimination approach, if the
manipulator has f faults, then the reduced Jacobian
matrices are obtained with the permutation of f zerovectors replaced into the original Jacobian matrix showingthe failures of the corresponding joints and then eliminatethe columns. For example for the case of two locked-joint
failures, there are2
)1( nn reduced matrices that are shown
by Jik , where nik ,...,1, and k i represents the faulty joints. This reduced matrix is obtained by eliminating thezero columns of the following matrix
niik k j j j j j j ...0...0... 11111
(13)
The velocity equation of the manipulator when the k-th andi-th joints are failed is then obtained by
qJx ik ik ,, (14)
where
niik k ik j j j j j jJ ......... 11111
,
(15)
niik k ik qqqqqq ......... 11111
, q (16)
III. FAULT TOLERANCE INDICES
The most common local fault tolerance indices ofmanipulators are shown in Table 1. These indices are basedon properties of the Jacobian matrix. The indices such asmanipulability (17), (20), relative manipulability (18) andworst case dexterity (21) have been studied for faulttolerance of the manipulator by different of researchers. Forinstance, the optimal configuration and optimal Jacobianmatrix for fault-tolerant manipulator have been addressed
based on relative manipulability in [6, 21, 22, 32, 33] and based on worst case dexterity in [9, 16, 34].
Condition number (19) has been proposed for the isotropicdexterity of the manipulators in [25-28]. It has also been
proposed for the fault tolerance in [3, 16, 30], but even inthis literature, the authors have selected the worst casedexterity for the analysis of the fault tolerance. There is lackof an extensive study of condition number for fault toleranceof the redundant manipulators.
TABLE I - COMMON LOCAL FAULT -TOLERANT INDICES
Index name Definition Description
Manipulability )det( T k k JJ (17)Jk
is the Jacobian matrixof faulty manipulator.[3]
RelativeManipulability
)det(
)det(T
T k k
JJ
JJ (18)
Ratio of the manipulabilityof the faulty and healthy
manipulators[3]
Condition Number
min
max
(19)
minmax , are max and
min singular values of theJacobian matrix [3]
DynamicManipulability
)(det( 1 k T k k k MMJ
(20)M
k
is mass matrix andJk
is Jacobian of thefaulty manipulator [16]
Worst CaseDexterity min
(21)minimum Singular valueof the Jacobian matrix [3]
Null spacematrix N (22)
Columns of N are theorthogonal bases of nullspace, the norm of the
columns are equivalent torelative manipulability [6]
IV. CONDITION NUMBER FOR FAULT TOLERANCE
A. Singular value decomposition and norms
If nm R J is a Jacobian matrix at a given EEF pose, thenthe singular value decomposition (SVD) of this Jacobianmatrix is
T V UΣJ (23)
where mm R U is an orthogonal matrix,nm R Σ is a
diagonal matrix consisting the singular values of J and
)0,...,0,,...,,( 21 ldiag Σ where ),min( nml .
Also nn R V is an orthogonal matrix and is provided by
the Eigen vectors of JJ T and the vectors of the orthogonal base of the null space of the Jacobian matrix. Note that forSLRMs nm and hence the manipulators are not insingular configuration, therefore ml .
In this paper norm of matrices and vectors are often used.
B. Modeling a fault using a matrix perturbationmethodology
Generally, in the existing literature, the modeling of thelocked-joint failure is performed by using the method thatwas introduced in section II, for example in equations (9)-(11) [6, 21]. But a fault can be also modeled by using a
perturbation method. In this method, any locked joint failure perturbs to the velocity equation because it affects both thefaulty joint velocity and the Jacobian matrix [12, 13, 31].The perturbed velocity equation is given by
xxqqJJ (26)
where indicated the perturbation into the parameter dueto the faults
Then by using (6) and rearranging (26), the change into the
velocity of the EEF of the manipulator is obtained by qqJqJx (27)
If the manipulator is fault-tolerant, then EEF velocity isrequired to remain with no change. Therefore it is essentialto have 0x and as the result
qqJqJ (28)
where q is the change into the joint velocities that aims totolerate the fault
The above equation can be solved to determine the required
joint velocity jump by using pseudo inverse. The pseudoinverse depends to the column rank of the matrix [12, 13,31].
The joint velocity jump to maintain the velocity of the EEFis determined by
qqJJq † (29)
whereT T JJJJ 1† )(
when J is a rank deficient and1† )( T T JJJJ for full rank matrices [12, 13, 31]
C. Condition number
Using Appendix-C and applying the norm inequality to theequation (29) results to
qqJJq † (30)
and consequently
J
JJJ
qq
q † (31)
whereqq
q is the ratio of the required change in q to
tolerate the fault
This showsJ
JJJ † as an upper bound for the required
relative change of q . This upper bound only depends to the properties of the Jacobian matrix. It is also easy to see that
JJ
J
J
qq
q
† (32)
Showing
J
Jqq
q
is bounded by JJ † .
D. Reconfiguration bounds and condition number
The definition of condition number [35] is given by
1~
:~max)( qqqJ
qJq (33)
where qq~
, are two unit norm joint velocity vectors
that are related to the the input directions which result to themaximum and the minimum gain of the Jacobian matrix
It has been shown that max)max(
qJ and
min)~
min( qJ [35]. This results to the common
definition of the condition number using the ratio of the themaximum to the minimum singular values as
min
max)(
q (34)
It is also shown [35] than the condition number is equivalent
to JJ † , therefore JJq †)( .
E. Fault tolerance and condition numberSingular values are used for dexterity analysis of themanipulator [18] because each singular value is the length ofthe corresponding Eigen vector in the image space of theJacobian matrix. Therefore, the condition number iscommonly considered as the isotropic dexterity index of themanipulators. By this concept, minimization of the conditionnumber results in the maximization of the isotropic dexterityof the manipulator [25-28].
The present paper aims to investigate the condition numberfor the fault tolerance of manipulators. It is observed from(31) that condition number can be used for the faulttolerance. The connection between the fault tolerance andthe condition number is achieved by using the perturbation
method that was presented in section IV.C. Based on thisobservation, if the condition number is small, then theJacobian matrix is called a well-conditioned matrix.Therefore, it is logical to name the configuration with well-conditioned configuration. In these configurations through alittle change into the joint velocities, the faulty manipulatorEEF velocity can be maintained as that of the healthymanipulator. In contrast, if the condition number is largewhich is for ill-conditioned Jacobian matrices, then theconfiguration is named ill-conditioned configuration. In ill-conditioned configurations, the fault requires a large changeinto the joint velocities to maintain the EEF velocity. Inconclusion, the configurations with minimum conditionnumber are optimal for fault tolerance, because themanipulator will be able to rapidly reconfigure itself totolerate the locked-joint failures.
It has to be noted that the condition numbers (34) are validwithin the other norm domains such as Frobenius or l
norms. However, the physical interpretation of those
condition numbers and their applicability for fault tolerancerequires further research.
F. Properties of condition number
For a SLRM at a given EEF pose and a given joint velocitiesthe following properties exist.
1. For all configurations 1)( q .
2. )(q is a local index suitable for dexterity analysisand for fault tolerance analysis.
3. The lower value for )(q is the better isotropicdexterity of the manipulators.
4. The lower value for )(q provides a higher faulttolerance property for the configuration. This is
because, if the manipulator is subjected to a locked joint failure, then it is possible to maintain the EEFvelocity with a small relative reconfiguration.
5. The upper bound of the change into joint velocity is
defined by qqJ
Jqq )( (35)
6. This upper bound is general and it is valid for anysingle and multiple joint failures.
7. The bound is valid for other definitions of norms
including lll ,...,, 21 and Frobenius norms.
G. Condition number and single locked-joint failures
The perturbation model of equation (25) requires an input tocompensate the effect of the failure. If the reconfigurationinput is shown by u then
xxuqqJJ (36)
where x is the velocity jump at EEF, J is the Jacobian perturbation, q is the change of joint velocities, and
uq is the total joint velocity change for compensation
of the failure
For the case of the k-th joint failure, the perturbations areobtained by
0......0 k jJ (37)
0......0 k q q (38)
and similar to the approach used to obtain equation (31), theratio of the total joint velocity change is obtained by
J
jq
uqq
uq k )( (39)
Therefore, the upper bound of the relative joint velocity jump for the case of single locked-joint failures is
J
jq k )( .
V. CASE STUDY I- CONDITION NUMBER FORDEXTERITY AND FAULT TOLERANCE
A 4-DoF planar manipulator with the Denavit Hartenberg (DH) parameters presented in Table 2 is used for the casestudies in this paper. The robot is modeled in MatlabRobotics Toolbox [36]. The manipulator in the configurationthat is shown in Figure 1 has the parameters that are
presented in Table 3.
Figure 1. The 4-DoF manipulator of the case study I and II
The EEF of the manipulator is at m x T 588.0334.0
TABLE 2- DH PARAMETERS OF A 4-D OF PLANARMANIPULATOR USED IN THE CASE STUDY I AND II
TABLE 3-CONFIGURATION AND PARAMETERS OF THEMANIPULATOR
Joint q deg q rad/s Parameter of configuration in Figure 1
1 10 0.05 378.0min , 816.0max
165.2)( q
smT /002.0005.0x
mT
558.0334.0x
2 70 0.403 25 0.20
4 65 0.10
A. Condition number and dexterity
In this case study, the dexterity and fault tolerance of themanipulator is studied at the EEF pose shown in Figure 1.In order to do this, we obtained 28000 differentconfigurations where the manipulator remains in its current
pose. Then, for these configurations we calculated thecondition number from the corresponding Jacobian matrix.The condition number associated to the selectedconfigurations is shown in Figure 2. From this figure, it isclearly observed that not all the configurations are well-conditioned, because they do not have a low value condition
number. It also indicates that from the 28000 configurations,there is almost no configuration with a condition numberlower than 1.895. Additionally, the worst ill-conditionedconfiguration has a condition number that is slightly morethan 5.
Designing the manipulator with a low condition number isdiscussed by researchers in order to design isotropicmanipulators. For example, such a design has been presentedin [25, 26] that are providing a minimum condition number.
Figure 2. The condition number of 28000 configurations forthe 4-DoF planar manipulator. All configurations belong to
the manifold of the EEF pose mT 588.0334.0x
Two selected configurations with high and low conditionnumbers are indicated in Figure 3 and Figure 4. Physically,the high conditioning and low condition for these twoconfigurations is meaningful, because intuitively, the motionof the EEF for the manipulator in Figure 3 is much moreconvenient than the motion of the manipulator EEF at theconfiguration shown in Figure 4. Additionally, if theseconfigurations are considered for human arms and hand,
then the configuration in Figure 3 is more dexterous orconvenient than that in Figure 4 because, when we are doing
daily tasks normally we choose the configurations in Figure3, except if there is some sort of limitations or disability.
Figure 3. Configuration with low condition number, the EEF position m588.0334.0x and 895.1)( q
Figure 4. Configuration with high condition number for the
EEF pose mT 588.0334.0x and 095.5)( q
Quantitative general dexterity analysis of manipulators has been studied in [37, 38]. It is also possible to understand theconnection between the well conditioned configurations andhumans arm commonly used configurations. For example,three of the well-conditioned configurations are indicated inFigure 5(a),(b) and (c).
Figure 5(a). A configuration with a low condition number,
the EEF pose mT 588.0334.0x
Figure 5(b). A configuration with a low condition number,
the EEF position at mT 588.0334.0x
Figure 5(c). A configuration with a low condition number,
the EEF position at mT 588.0334.0x
The present paper aims to show that the well-conditionedconfigurations are not only dexterous but also suitable forfault tolerant. The fault tolerance property of the well-conditioned configurations is explained in the followingsubsection.
B. Condition number and fault tolerance
It was mentioned earlier that condition number has been proposed for fault tolerance in [3, 16] , but it has not beendeployed in the literature of fault tolerance. In the previoussection, it was also indicated that the condition number can
be utilized to define the well-conditioned configurations. Inthese configurations, the faults can be tolerated with aminimum relative joint velocity jump. Placing themanipulator in these configurations enhances themanipulator’s fault tolerance property because themanipulator can rapidly reconfigure itself in order to toleratethe fault.
From the framework in (39), if a failure occurs to the one of
the manipulator joints, then it is expected that for well-conditioned configurations such as those in Figure 3 andFigures 5 (a),(b) and (c), the manipulator will require a lowRJVJ to maintain the EEF velocity. However, for ill-conditioned configurations such as that in Figure 4, themanipulator will require high RJVJ to maintain the EEFvelocity. This required RJVJ of the well-conditionedconfiguration of Figure 3 is nearly 60% lower from therequired RJVJ for the ill-conditioned configuration of Figure4.
To quantitatively show this hypothesis, the twoconfigurations in Figure 3 and Figure 4 are used and it isassumed that the manipulator is in a fault-tolerant motionoperation. The joint velocity of the manipulator is
srad T /0.100.200.400.05q . The EEF pose
is mT 558.0334.0x and the EEF velocity at the
fault time is smT /002.0005.0x .
If a locked joint failure occurs in the second joint of themanipulator. The JVJ to compensate this fault has infinitesolution as the faulty manipulator is still redundantmanipulator (a 3-DoF planar manipulator). The control lawto maintain the EEF velocity with a minimum JVJ [5, 13] isused to obtain the JVJ.
In [5, 12, 13] it was shown that, the optimal reconfigurationcommand for fault recovery of the EEF velocity jump withthe minimum joint velocity jump is obtained by
k k k k q jJu † that is for single joint failures. Then u is the
obtained from uk
by inserting a zero in its k-th row.
Using the above reconfiguration law, the JVJ is obtained fortolerating the fault, and the relative joint velocity jump is
obtained fromuqq
uq, where q and uqq
are joint velocity vectors for the healthy and the faultymanipulator.
Using the reconfiguration law, the faulty manipulator’s EEFvelocity will remain the same as that of the healthymanipulator. Table 4 provides the joint velocities prior to thefailure and after the failure for the fault-tolerant motion.
TABLE 4- JOINT VELOCITY JUMP FOR THE WELL -CONDITIONED CONFIGURATION OF FIGURE 3
A similar study has been performed to calculate the JVJ forthe manipulator in the ill-conditioned configuration shown inFigure 4 and its result is indicated in Table 5.
TABLE 5-JOINT VELOCITY JUMP FOR THE ILL - CONDITIONED CONFIGURATION OF FIGURE 4
By comparing the relative reconfiguration for joint velocity jumps for the both well and ill configurations that are shownin Tables 4 and 5, it is observed that the well-conditionedconfiguration requires 55.26% lower RJVJ.
Having a lower RJVJ for well-conditioned configurations isvalid in other EEF poses and different configurations. This
confirms the proposed hypothesis of this paper that the well-conditioned configurations are also suitable for faulttolerance.
This conclusion is used in the next section and the optimalconfigurations for fault tolerance of the manipulators areobtained from the minimum condition numbers.
VI. OPTIMAL FAULT -TOLERANT CONFIGURATIONS
From section IV.G, it is clearly seen that the minimization ofthe condition number can minimize the bound of thereconfiguration for tolerating the locked joint failures.Therefore, in the fault-tolerant operations, it is useful to find
the configurations with low condition numbers as theyrequire less reconfiguration. This section aims to show that
by the minimization of the condition number of the Jacobianmatrix, the optimal configuration for fault tolerance can beobtained.
To optimize the condition number, any of the following costfunctions can be used
JJ † f I (40)
min
max
f I (41)
The cost function in (40) is applicable if the unitinconsistency exists or the Jacobian matrix is non-homogen.This can be the case when both positional and orientationalvelocities are considered.
In this paper, we use (41) because only translationalvelocities are considered and the Jacobian matrix is a
homogen. Then the minimization of (41) can be rewritteninto three standard singular value (Eigen value) optimization problems [39].
A. Singular value optimization-first method
The first method for the optimizing of the equation (41) is toconstrain the maximum singular value of the Jacobianmatrix and then maximize the minimum Eigen value. Thiscase results in the following constrained singular valuemaximization problem
h
q
zmax
min ))(max
(42)
where h z is a constant number
The problem can be converted to
h
q
z
z
z
max
min
)(max
(43)
Considering that for all singular values max
i when
,…,m ,i= 21 and zmin then simply it is obtained
0 zi . In addition, by using the Sylvester theorem
introduced in Appendix-B, the constraints of (43) can bewritten in terms of determinants of the principal sub-
matrices of IJJ zT where I is the identity matrix.Therefore, the maximization of the minimum singular valuecan be obtained by the following standard constrainedoptimization problem.
h
iT
q z
z
,…,m ,i= z
z
max
,
21 0)det(
)(max
IJJ (44)
B. Singular value optimization-second method
The second method is to constrain the lower bound ofminimum singular value and then minimize the maximumsingular value of the Jacobian matrix. Therefore
l
q
zmin
max ))(min
(45)
Similar to the first method and by using the Sylvestercondition, the minimization of the maximum singular value
can be obtained by the following standard constrainedoptimization problem.
h
iT
q
z
,…m ,i= z
z
min
210)det(
)(min
IJJ (46)
C. Singular value optimization-third method
This method converts the optimality problem into a multi-objective Eigen value optimization problem. Note that theJacobian matrix is a positive definite matrix, therefore all theEigen values of the Jacobian matrix and reduced Jacobian
matrix are positive. Hence the minimization of min
results into the maximization of min therefore the multi-
objective are defined by
),(min minmax
q f I (47)
Then using the Sylvester theorem, a multi-objective multi-constrained optimization problem is obtained by
,…,m ,i=
,…,m ,i= z
z
iT
iT
q
210)det(
210)det(
),(max
JJI
IJJ
(48)
Note that in this method there are m2 non equalityconstraints and two objective functions.
D.
Selection between the optimization methodsFrom the concept of the dexterity, the second and thirdmethods are preferred, because in the first optimizationmethod, it is possible to converge to a configuration with avery low condition number and therefore a low totaldexterity. In contrast, in the second and the third methods,the total dexterity is already preserved through the thresholdor maximization of the minimum singular value. Betweenthe last two methods, we selected the third optimizationmethod and implemented the method in Matlab using thefunctions of optimization toolbox.
VII. CASE STUDY II– OPTIMAL FAULT -TOLERANTCONFIGURATIONS FOR THE 4-D OF MANIPULATOR
The 4-DoF manipulator with the DH parameters of Table 2and with the positional parameters in Table 3 was used inthis case study. Figure 6 indicates the singular valuescorresponding to the 28000 configurations of the case studyI. The maximum and minimum singular values are used asthe initial guess for the maximum and minimum singularvalues.
A. Optimization
The optimal configuration was obtained through theoptimization process of (48) and via a multi-objective multi-constraints optimization method. Five results of the optimumconfigurations are indicated in Table 6. These results include
the joint angles and the corresponding condition number forthe five obtained optimal configurations. In this table, the
joint angles are indicated in degree and the conditionnumbers are indicated in last row of the table.
Figure 6. Maximum and minimum singular values fordifferent configuration used for the initial guess for the
From figure 2, it is understood that the optimal configurationis not a unique configuration as there is a whole bunch ofwell-conditioned configurations with similar level ofcondition number. The non uniqueness of the optimalconfiguration is simply observed from the results of Table 6
because the optimization does not converge to a specificsolution. Actually, there are infinite numbers of well-conditioned configurations and the result of the optimization
problem is very sensitive to the initial guess. However, in allthe configurations in the Table 6, the values of the conditionnumbers are consistent.
C. Limitation and unit inconsistency
The dexterity indices are not well physically interpretedwhen they applied for the non homogenous Jacobianmatrices. This is also correct for the condition number. This
problem is general in most of the literature of themanipulators when different units are combined in theJacobian matrix.
In this case we proposed using (40) as the cost functioninstead of the (41), but we haven’t deployed it in this paper.Such a deployment requires further research.
Furthermore, the generalization of the proposed method formanipulators with combined revolute and prismatic joints orthe manipulators with positional and rotational Jacobian
matrix require also further research. But the heuristic of thedexterous configurations for the fault tolerance is physicallyvalid even for those manipulators.
Another limitation is that the proposed method is suitableonly for locked-joint failures. For free swing joint failuresand joint sensor failures, it is possible to consider a
mechanical brake to lock the joints. In this case, the proposed method of this paper will be applicable for othertypes of joint failures.
VIII. COMPARISON OF THIS WORK WITH OTHERWORKS ON FAULT -TOLERANT MOTIONS
From the literature survey we found that condition numberhas been proposed for fault tolerance but has not beendeployed to limit the bound of the RJVJ for fault recovery.This motivated the research that was presented in this paperaimed to promote the tolerating of the faults in the SLRMsusing condition number. Then by using matrix perturbation
methodology, a framework for finding the optimal fault-tolerant configurations is introduced. The main property ofthese configurations is that, they require a lower RJVJ totolerate the locked-joint failures. In comparison to the workof the other researchers, the joints velocity jump and theEEF velocity jump have been addressed in [23, 24] and [12,13, 31] to provide the control law for joint velocityreconfiguration. The reconfiguration laws result to minimumJVJ in [23, 24] or to minimum JVJ and minimum EEFvelocity jump in [12, 13, 31]. The other work in [6, 9, 16,21, 22, 29, 30, 40-42] in the literature, are related to designof the optimal fault tolerant redundant manipulators and inthem the dexterity or relative manipulability indices are
used. These works are also different from the preset paper because the paper uses condition number as the index offault tolerance.
IX. CONCLUSION
Matrix perturbation method was applied for modeling thelocked-joint failure for serial link redundant manipulators.Then the condition number of the Jacobian matrix wasstudied and it was observed that in addition to the commonuse of the condition number for the isotropic dexterity, it can
be useful to define the optimal configurations for faulttolerance. This fact led to introducing well-conditioned
configurations from the optimality of the condition number.In these configurations, reconfiguring the manipulator totolerate the failures requires a low relative reconfiguration.This property was demonstrated for a 4-DoF planarmanipulator. The results showed that the reconfigurationsrequired for fault tolerance in the well-conditionedconfiguration is much lower than that in the ill-conditionedconfigurations.
APPENDICES
Appendix-A
If a matrix is a positive definite then all the Eigen values aregreater than zero. Therefore, if we suppose the singularvalues as ni zii ,...,2,1,,0 then any matrix
by the singular values of
ni z z ii ,...,2,1,0 is a positive definite
matrix.
The square of the singular values of the Jacobian matrix is
the Eigen values of JJ T and 22 zi are Eigen values of
IJJ zT and all the 022 zi for ,…n ,i= 21 then
IJJ zT is a positive define matrix.
Appendix-B
Sylvester condition: A matrix is a positive semi-definitematrix if and only if all the determinants of the principalsub-matrices which are called principal minors are positive[35, 43].
Appendix-C
If A , B and C are three arbitrary matrices satisfyingABC then following inequality is valid for all the norm
definitions including lll ,...,, 21 and Frobenius norms.
BAC
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