- Email: [email protected]
Work-Space Design for Fault Tolerant Manipulators
Copyright @ 1996 IFAC 13th Trie nni31 World Congress. San Francisco. USA
Ib-09 6
WORK-SPACE DESIGN FOR FAULT TOLERANT MANIPULATORS
Kubilay Kaan A YDlN
Middle East Technical University, Industrial Engineering Department, 06531 Ankara, TURKEY, e-mail: [email protected]
Abstract: Kinematic redundancy can be utilised to increase the reliability of operations performed by manipulators as presented in this paper. Fault tolerance is defined as the ability to continue the performance of a task after total failure or partial immobilisation of a manipulator joint. Fault tolerant work space is defined as a sub-work-space volwne where a fault tolerant redundant manipulator can continue its work even under the failure of one or more of its joints. Design examples are presented for planar positional manipulators showing that fault tolerant work spaces of desired shape and dimensions can be attained by the proper design of redundant manipulators. Keywords: Reliability Analysis, Fault Tolerance, Redundant Manipulators, Simulation
I. INTRODUCTION Fault toleranee is defined as the ability of a system to continue its operation even after the failure of some internal system component. Fault tolerance technology has been developed for a variety of systems including computers, aerospaee systems, nuclear power plants and military applications. It is known that kinematic redundancy can be utilised to increase the reliability of operations performed by manipulators (Maciejewski, 1990; Cleary and Tesar, 1990; Sreevijayan, et al., 1994). In this paper fault toleranee for manipulators is defined as
reconfiguration of the manipulator due to the fault has one less OOF and has different link lengths when compared to the original manipulator. 'Fault tolerant work-space' is defined as a sub-work-space volume where a fault tolerant redundant manipulator can continue its work even under the failure of one or more of its joints. Sinee the kinematic properties of the resultant reconfiguration of the manipulator is different than the original one, the shape and dimensions of the resultant fault tolerdDt work-space is also different (Lewis and Maciejewski, 1994). Fault tolerant work spaees of desired shape and dimensions can be attained by the proper design of redundant manipulators as presented in this paper.
the ability of a redundant manipulator to continue the performanee of a task after total failure or partial immobilisation of one or more of its joints. One of the redundant degrees-of-freedom (DOF) is employed to compensate for one faulty joint. The faulty joint is frozen at an angle at the occurrenee of the fault. The resultant
This paper is organised as follows: Section 2, defines fault tolerance coDeepts. Fault tolerant work-spaee design guidelines are given in Section 3 witb examples. Section 4, is a summary and conclusion.
313
2. FAULT TOLERANT WORK-SPACE A 'fault tolerant manipulator' is defined as a manipulator that is able to continue the performance of a task after total failure or partial immobilisation of one or more of its joints. Fault tolerance property has measures. For example, a four DOF planar manipulator is more fault tolerant than a three DOF planar manipulator. A four DOF planar manipulator can compensate for two faulty joints whereas a three DOF planar manipulator can compensate for only one faulty joint. The measure of fault tolerance can be defined for redundant manipulators as follows: Definition 1: A redundant manipulator is called 'fault
tolerant of order m' (m-FT), if the manipulator can continue the performance of a task after the failure of lilY, m number, of its joints. The combination of joints, containing the m number of faults, must not affect the performance of the task. The manipulator should be able to continue its operation irrespective of which combination of m joints are faulty. If a specific combination of m faulty joints prohibit the execution of the task then the manipulator is not m-Fr.
values. If a specific combination of 8tCi) values prohibit the execution of the task then the manipulator is not m-Fr. This extended definition of fault tolerance will be used when a reference is made to a fault tolerant manipulator for the rest ofthis paper. When a fault tolerant manipulator is re-configured, in order to compensate for a failure in one or more of its joints, the resultant manipulator has different kinematic properties. The two links adjacent to the faulty jOint are assumed to be replaced by a single link. Figure I, illustrates this procedure. A 3-DOF PPM, with link lengths of one unit, is shown in Figure 1. When a fault occurs on joint I of the PPM joint I is frozen at 8j(1). If joint I is frozen links one and two act as a single link. In Figure I the dotted line from joint 0 to joint 2 shows the hypolhetical link that is assumed to replace the two links adjacent to joint 1. Due to the change in kinematic properties of the manipulator the work-space of the manipulator changes. In general the work-space of a manipulator shrinks and changes shape when it is reconfigured in order to compensate for a faulty joint. In Figure 1 the original 3-DOF PPM has the whole half circle, with radius of 3 units, as its work-space.
It must be noted that redundancy of manipulators can be
defined with respect to a task. General purpose operation of manipulators will be considered throughout this paper, so task dependent redundancy will not be considered. In this paper the focus will be on Planar Positional Manipulators (pPM) without considering orientation of the end-effector. Note that a two DOF PPM is sufficient to reach every point in its work-space if no orientation for the end-effector is necessary. When a manipulator joint is immobilised due to a failure the joint is frozen at an angle at the occurrence of the fault. It is assumed that this angle can be measured after the occurrence of the fault. The angle at which the faulty joint is frozen is called as 'fault angle'. Fault angles will be denoted by ej{i), where the indices, i , denotes the faulty joint. When general purpose (non-task dependent) fault tolerance is considered there should be no assumptions on the valne of ej{i), for any joint i. The definition of fault tolerant manipulators can be extended as follows:
y
_joint 2
Work-space of planar manipUlator
_. Nint 1 • _ _.._.. - .- - \Irozcn)
~~~~~----~~~----~--~x Link Lenghts : '.. 2 3
LI - L2=L3=1 unit
joint 0
Fig. I : Work-space oftbree link planar manipulator. Figures 2 and 3 show the faulty manipulator with Elj{ 1)=90 and 6j(1)=±l&O degrees respectively. y
Definition 2: A redundant manipulator is called 'fault tolerant of order m' (m-FT), if the manipulator can
continue the performance of a task after the failure of lilY,
Link Lenghts : LI - L2- L3- 1 unit
m number, of its joints at lilY, ej{i), degrees. The ej{i) values maybe different for each of the m number of faulty joints. The manipulator should be able to continue its operation irrespective of the combination of etCi)
2
3 x
Fig. 2: Work-space of re-configured three link planar manipulator for 6j(1)=9O degrees.
314
The ami within 0.41< r <2.41 in Figure 2 shems the workspace of the re-configured manipulator for 91<1)=90 degrees. In Figure 2 it can be seen that the fault tolerant work-space is a subset of the original work-space of the 3-00F PPM In Figure 3 the work-space of the rc-<:onftgurcd manipulator for e1
."
..
_~int
"
1
" lm~
~~~----L---~~~~------~~x
Link Lenghts :
2
approach it can be shown that a m-FT manipulator with n-OOF can be realised by a «m+1)n)-OOF manipulator. For planar manipulators the work space is \WO dimensional. Therefore a 2-00F manipulator is sufficient for general P1I.I'pOSe operation of a PPM. Following the above approach it has been sho"n that (2(m+ 1»-OOF are necessaIy and sufficient to construct a m-FT PPM (Paredis and Khosla, 1994). The example given in Figure 3 shaM that 3-00F is not sufficient to implement a I-FT PPM. Note that (2(1+1»=4. The PPM in figure 3 has one dimensional work-space which is not acceptable for a planar manipulator. Therefore a I-FT PPM must have four or more OOF. The one dimensional work-space shown in Figure 3 also corresponds to the FTWS of the 3-00F PPM with unit link lengths. The work-spaces corresponding to the n>-eonfigured versions of any 3-00F PPMs can be calculated analytically as follaM:
3
LJ- L2- L3""} unit
Fig. 3: Work-space of re-configured three link planar manipulator for 91(1)= ±IS0 degrees. The work-space of a re-configured manipulator, due to a faulty joint, is called as the 'work-space under fault'. When general p1J.l1lOSe fault tolerance is considered the following definition can be made for the work-space of a fault tolerant manipulator under all possible joint failures that it can compensate for: Definition 3: The intersection of work-spaces for all possible reconfigurations of a m-FT manipulator, due to joint failures, is called as the 'fault tolerant work-spacc' (FIWS) of the manipulator.
In this definition all possible reconfigurations of a m-FT manipulator for -180 ~ 91
3. FAULT TOLERANT WORK-SPACE DESIGN In order to implement fault tolerance of order m, for a given n-OOF manipulator, evel)' OOF of the manipulator must be duplicated m times. The result will be a m-FT manipulator. Whenever a joint fails there would be m number of working joints to compensate for the faulty joint. If the joint that replaced the original faulty joint should also fail there will be (m-I) number of working joints to compensate for the second failure. Using this
i) If the first joint is faulty (-180 ~ 91<1) ~ 180) the work space is the region with inner radius, ri, and outer radius, r2, where rl and r2 are calculated as follo~ :
rl
=
ab.{[«Llj2 + (uj2) - 2(LlXL2Xcos(ISO - 9t<1))]112 - L3 } (I)
r2= ((Llj2 + (L2j2) - 2(LIXUXcos(180 - 81<1))] 112 + L3
(2)
ii) If the second joint is faulty (-ISO S €l1<2) ~ ISO) the work space is the region with inner radius, r3, and outer radius, r4, where r3 and r4 are calculated as follo~:
r3 = abs{((Llj2 + (uj2) - 2(L3XL2Xoos(180 - 81<2»JII2 - Ll} (3) r4=
((L3j2 + (L2j2) - 2(L3XUXcos(ISO - 6t<2»JII2 + Ll
(4)
The FTWS of a 3-00F PPM is the intersection of all workspace. calculated by the use of the above equations (1-4). At fault angles given by 81(1) = ±180 and 01<2) = ±ISO degrees the equations reduee to the following.
rl = ahs{[«Ll)2 + (L2)2) - 2(L1)(L2)(l))112 - L3} = ahs{[(LI- L2)2]1I2 - L3} = L3
(5)
r2 = [«Ll)2 + (L2)2) - 2(LI)(L2)(I)]112 + L3 = [(LI- L2)2JII2 + L3 = L3
(6)
r3 = ahs{[«(L3)2 + (L2)2) - 2(L3)(L2)(1)J112 - Ll) = (7) ahs{[(L3 - L2)1 J II2 - Ll} = Ll r4 = ((L3)2 + (L2)2) - 2(L3)(L2)(1)(112 + Ll = (L3 - L2)1)112 + LI = Ll (8)
315
For unit link lengths (LI = L2 = L3 = I unit) the inner and outer radii of work-spaces are all equal to one unit (rl = r2 = r3 = r4 = I unit). The work-spaces of the re-configured manipulators for fault angles 9t(1) = ±180 and 91(2) = ±ISO degrees give the work-spaces with minimum area. Therefore the work-spaces of the re-configured manipulators with fault angles 81(1) = ±ISO and 81(2) = ±ISO degrees determine the FTWS for this example. The FTWS for the 3-DOF PPM with unit link lengths is the onc dimensional curve with radius of one unit. To construct a m-FT PPM, a «m+ I)n)-DOF manipulator is necessary. The shape and dimensions of the FTWS is determined by the kinematic specifications of the manipulator. It is possible to design a fault tolerant manipulator with FTWS of predefined shape and dimensions. The design of a FTWS of predefined shape and dimensions for a 4-DOF PPM that is I-FT will be presented next. The work-space of a 3-DOF PPM is the circular area with inner radius, rin, and outer radius, rout, (rin and rout are conamtric) where rin and rout are calculated as follows: (9)
rout =LI +L2 + L3 ifLl > L2 + L3 then else ifL2 > Ll + L3 then else ifL3 > Ll + L2 then else
rin = Ll - (L2 + L3)
rin = L2 - (L I + L3) rin =L3 - (LI + L2)
rin =0
(10)
Where Ll, L2, L3 are the three link lengths of the manipulator. Note: that the calculation of the inner radius for the work-space is a conditioual equation with four steps depending on the relative magnitudes of the link lengths. When a fault occurs in onc joint of a I-FT 4-DOF PPM the manipulator loses one DOF. All re-configured versions of a I-FT PPM are 3-DOF manipulators. For the two links that are not adjacent to the faulty joint the link lengths of the original manipulator and the re-configured 3-DOF manipulator are the same. The remaining two links that are adjacent to the Caulty joint / are replaced by a single link, 4, in the re-configured versions of the manipulator. The length of4is calculated as follows : 4= [«L/; + (L j + J; ) - 2(Lj)(Lj + J )(cos( ISO - Si
(11)
Where. L j is the link before joint / and Li+ I is the link after joint /. 91
Four link lengths of the I-FT PPM will be denoted by xl, x2, xl, x4. The following equations give the three link lengths (LI , L2, U) of the re-configured 3-DOF PPMs with minimum work-space area: 81(1)~ degrees: L1=x1 + x2, L2=x3,
(12)
81(2)=0 degrees: Llaxl, L2=x2 + x3 ,
L3=x4
(13)
81(3)=0 degrees: Ll=x I, L2=x2,
L3=xl + x4
(14)
Elj(I)-±lSO degrees: Ll=abs(xl - x2), L2=x3,
L3=x4
(15)
Elj(2)=±180 degrees: LI=xl, L2-abs(x2 - x3), L3=x4
(16)
81(3)=±180 degrees: L1-xI, L2=x2,
L3=abs(xl - x4) (17)
Applying equations (9) and (10) to the link length values given by equations (12) to (17) in order to caleulate the work-spaces with minimum area, result in the following equations: (xl +x2)+xl +x4 > rout
(18)
xl + (x2 + x3) + x4 > rout
(19)
xl + x2 + (x3 + x4) > rout
(20)
316
abs(xl - xl) + x3 + x4 > 'out
(21)
xl + abs(xl - x3) + x4 > 'out
(22)
xl + xl + abs(x3 - x4) > 'out
(23)
if (xl + xl) > x3 + ,,4 then (xl + xl) - (x3 + x4) < 'in else ifx3 > (xl + xl) + x4 then x3 - «(xl + xl) + x4) < 'in else ifx4 > (xl + xl) + x3 then x4 - «xl + xl) + x3) <'in else
no condition on rin value
Solution I:
xl=I, xl=2, x3=I, x4=2
Solution TI:
xl=2, xl=I , x3=I , x4=2
Solution Ill:
xl=2, xl=I, x3=2, x4=1
Solution N :
xl=2, xl=2,x3=2,x4=2
Run 2: 'out = 3 units, 'in = 1 units, maximum link length = 3 units. The software has caleuIated 11 feasible solutions for xl , xl, x3, x4.
ii) Simulation
Hi) Simulation Run 3 : rout = 3 units. rin = 2 units~ maximum link length = 3 units. Tbe software has caleuIated 31 feasible solutions for xl, xl, x3 , x4.
(24)
= 3 units, rin = 1 units, maximum link length = 4 units. The software has caleuIated 21 feasible solutions for xl , x2, x3, x4. vi) Simulation Run 4: rout
The calculation of the inner radius for the work-space is a conditional equation with four steps as given by equation (10). The first link length values for 61<1)=0 degrees, given by equation (12), have been applied to equation (10) resulting in equation (24). There are five more group5 of equations, with four stCp5 similar to equation (24), for each link length value given by equations (13) to (17). These five equations, with four stCp5, will be numbered as (25), (26), (27), (28), (29). It can be seen that equations (18), (19), (20) are the same. Equations (21), (22), (23) are more restrictive than equations (18), (19), (20); therefore equations (18), (19), (20) will nOl be used.
Equations (21), (22), (23), (24), (25), (26), (27), (28), (29) define the work-spaces with minimum area for all possible re-coofigurcd 3-DOF PPMs. The FfWS of the I-FT PPM is the intersection of these work-spaces with minimum area. Therefore equations (21) to (29) can be utilised together to find the FfWS of the I-FT PPM. If link lengths xl, xl, x3, x4 are given the inner and outer radii for the FfWS can be computed. If the inner and outer radii for the FfWS are given a range of values for the link lengths xl, xl, x3, x4 can be computed. A software using equations (21) to (29) to compute the ranges for the link lengths xl, x2, x3, x4 has been developed. The software takes the inner and outer radii for the FTWS and a 'limit number', defining the maximum admissible link length, and computes the ranges for the link lengths xl, xl, x3, x4. Results of five simulation runs are as follows:
v) Simulation Run 5: rout = 3 units, rin = 2 units, maximum link length = 4 units. The software bas caleuIated 73 feasible solutions for xl, xl, x3, x4. The number of feasible solutions for xl, x2, x3, x4 values increase as the allowed link length increases. The number of feasible solutions for xl, xl, x3, x4 values decrease as the FfWS area increases. Tbese two results can be verified intuitively when tbe definition of FTWS is considered. The shape of work-spaces for PPMs are concentric circular regions whose dimensions are determined by an inner and an outer radius. Due to this property the FfWS for all m-FT PPMs are cireuIar regions. This result can also be verified if the self-motion topology of PPMs are investigated. Throughout the paper it is assumed that there are no joint limits and the work-space is not constrained in any way. Wbcn joint limits are introduced the self-motion topology of a given PPM changes (Luck and Lee, 1993). As an extension to this result the shape of the work-space and thus the shape of the FfWS changes. Under jOint limits the shape of the work-space and FTWS for a PPM is the intersection of circular areas that are not concentric. In this paper the case when joint 0 is faulty has not been considered. Researcb for the formulation of the FTWS for the case when joint 0 can also be faulty is still continuing.
4. CONCLUSION In this paper fault tolerance has been defined as the ability
il Simulation Run 1: 'out = 4 units, 'in = 0 units, maximum link length = 2 units. The software has calculated 4 feasible solutions for xl , xl, x3, x4 values as follows:
of a manipulator to continue the performance of a task after total failure or partial immobilisation of one or more manipulator joints. Fault tolerant work-space has been defined as a sub-work-space volume where a fault tolerant redundant manipulator can continue its work even under
317
the failure of anyone or more of its joints at any angle. Kinematic redundancy can be utilised to increase the fault tolerance of manipulators as demonstrated by design examples for planar positional manipulators. A software calculating possible link lcogth values for the design of a I-FT PPM with a FTWS of predefmed shape and dimensions is presented. Simulation runs of the software have shown that the number of feasible solutions for link lcogth values of a m-FT PPM increases when the maximum permissible link lcogth increases and when the desired FTWS area dccrcascs. These rcsults can be extended to all fault tolenmt redundant manipulators. Fault tolerant work spaces of desired shape and dimensions can be attained by the proper design of redundant manipulators. Research on more general redundant manipulator design rules and guidelines to serve this purpose is still continuing.
REFERENCES Cleary, K. and D. Tesar (1990), Incorporating Multiple Criteria in the Operation of Redundant Manipulators, Proc. IEEE Int. Con! on Robotics and Automation, pp. 618-624. Lewis, C. L. and A A Maciejewski (1994), An Example of Failure Tolerant Operation of a Kinematically Redundant Manipulator, Proc. IEEE Int. Con! on Robotics and Automation, pp. 1380-1387. Luck, C. L. and S. Lee (1993), Self-Motion Topology for Redundant Manipulators with Joint Limits, Proc.
IEEE Int. Con! on Robotics and Automation, pp. 626-631. Maciejewski, A A (1990), Fault Tolerant Properties of Kinematically Redundant Manipulators, Proc.
IEEE Int. Con! on Robotics and Automation, pp. 638-642. Paredis, C. and P. K. Khosla (1994), Mapping Tasks into Fault Tolerant Manipulators, Proc. IEEE Int. Con! on RoboticsandAutomation, pp. 6%-703. Sreevijayan D., S. Tosunoglu and D. Tesar (1994), Architectures for Fault-Tolerant Mechanical Systems, Proc. Mediterranean EJectrotechnical Conference, pp. 1029-1033.
318
Ib-09 6
WORK-SPACE DESIGN FOR FAULT TOLERANT MANIPULATORS
Kubilay Kaan A YDlN
Middle East Technical University, Industrial Engineering Department, 06531 Ankara, TURKEY, e-mail: [email protected]
Abstract: Kinematic redundancy can be utilised to increase the reliability of operations performed by manipulators as presented in this paper. Fault tolerance is defined as the ability to continue the performance of a task after total failure or partial immobilisation of a manipulator joint. Fault tolerant work space is defined as a sub-work-space volwne where a fault tolerant redundant manipulator can continue its work even under the failure of one or more of its joints. Design examples are presented for planar positional manipulators showing that fault tolerant work spaces of desired shape and dimensions can be attained by the proper design of redundant manipulators. Keywords: Reliability Analysis, Fault Tolerance, Redundant Manipulators, Simulation
I. INTRODUCTION Fault toleranee is defined as the ability of a system to continue its operation even after the failure of some internal system component. Fault tolerance technology has been developed for a variety of systems including computers, aerospaee systems, nuclear power plants and military applications. It is known that kinematic redundancy can be utilised to increase the reliability of operations performed by manipulators (Maciejewski, 1990; Cleary and Tesar, 1990; Sreevijayan, et al., 1994). In this paper fault toleranee for manipulators is defined as
reconfiguration of the manipulator due to the fault has one less OOF and has different link lengths when compared to the original manipulator. 'Fault tolerant work-space' is defined as a sub-work-space volume where a fault tolerant redundant manipulator can continue its work even under the failure of one or more of its joints. Sinee the kinematic properties of the resultant reconfiguration of the manipulator is different than the original one, the shape and dimensions of the resultant fault tolerdDt work-space is also different (Lewis and Maciejewski, 1994). Fault tolerant work spaees of desired shape and dimensions can be attained by the proper design of redundant manipulators as presented in this paper.
the ability of a redundant manipulator to continue the performanee of a task after total failure or partial immobilisation of one or more of its joints. One of the redundant degrees-of-freedom (DOF) is employed to compensate for one faulty joint. The faulty joint is frozen at an angle at the occurrenee of the fault. The resultant
This paper is organised as follows: Section 2, defines fault tolerance coDeepts. Fault tolerant work-spaee design guidelines are given in Section 3 witb examples. Section 4, is a summary and conclusion.
313
2. FAULT TOLERANT WORK-SPACE A 'fault tolerant manipulator' is defined as a manipulator that is able to continue the performance of a task after total failure or partial immobilisation of one or more of its joints. Fault tolerance property has measures. For example, a four DOF planar manipulator is more fault tolerant than a three DOF planar manipulator. A four DOF planar manipulator can compensate for two faulty joints whereas a three DOF planar manipulator can compensate for only one faulty joint. The measure of fault tolerance can be defined for redundant manipulators as follows: Definition 1: A redundant manipulator is called 'fault
tolerant of order m' (m-FT), if the manipulator can continue the performance of a task after the failure of lilY, m number, of its joints. The combination of joints, containing the m number of faults, must not affect the performance of the task. The manipulator should be able to continue its operation irrespective of which combination of m joints are faulty. If a specific combination of m faulty joints prohibit the execution of the task then the manipulator is not m-Fr.
values. If a specific combination of 8tCi) values prohibit the execution of the task then the manipulator is not m-Fr. This extended definition of fault tolerance will be used when a reference is made to a fault tolerant manipulator for the rest ofthis paper. When a fault tolerant manipulator is re-configured, in order to compensate for a failure in one or more of its joints, the resultant manipulator has different kinematic properties. The two links adjacent to the faulty jOint are assumed to be replaced by a single link. Figure I, illustrates this procedure. A 3-DOF PPM, with link lengths of one unit, is shown in Figure 1. When a fault occurs on joint I of the PPM joint I is frozen at 8j(1). If joint I is frozen links one and two act as a single link. In Figure I the dotted line from joint 0 to joint 2 shows the hypolhetical link that is assumed to replace the two links adjacent to joint 1. Due to the change in kinematic properties of the manipulator the work-space of the manipulator changes. In general the work-space of a manipulator shrinks and changes shape when it is reconfigured in order to compensate for a faulty joint. In Figure 1 the original 3-DOF PPM has the whole half circle, with radius of 3 units, as its work-space.
It must be noted that redundancy of manipulators can be
defined with respect to a task. General purpose operation of manipulators will be considered throughout this paper, so task dependent redundancy will not be considered. In this paper the focus will be on Planar Positional Manipulators (pPM) without considering orientation of the end-effector. Note that a two DOF PPM is sufficient to reach every point in its work-space if no orientation for the end-effector is necessary. When a manipulator joint is immobilised due to a failure the joint is frozen at an angle at the occurrence of the fault. It is assumed that this angle can be measured after the occurrence of the fault. The angle at which the faulty joint is frozen is called as 'fault angle'. Fault angles will be denoted by ej{i), where the indices, i , denotes the faulty joint. When general purpose (non-task dependent) fault tolerance is considered there should be no assumptions on the valne of ej{i), for any joint i. The definition of fault tolerant manipulators can be extended as follows:
y
_joint 2
Work-space of planar manipUlator
_. Nint 1 • _ _.._.. - .- - \Irozcn)
~~~~~----~~~----~--~x Link Lenghts : '.. 2 3
LI - L2=L3=1 unit
joint 0
Fig. I : Work-space oftbree link planar manipulator. Figures 2 and 3 show the faulty manipulator with Elj{ 1)=90 and 6j(1)=±l&O degrees respectively. y
Definition 2: A redundant manipulator is called 'fault tolerant of order m' (m-FT), if the manipulator can
continue the performance of a task after the failure of lilY,
Link Lenghts : LI - L2- L3- 1 unit
m number, of its joints at lilY, ej{i), degrees. The ej{i) values maybe different for each of the m number of faulty joints. The manipulator should be able to continue its operation irrespective of the combination of etCi)
2
3 x
Fig. 2: Work-space of re-configured three link planar manipulator for 6j(1)=9O degrees.
314
The ami within 0.41< r <2.41 in Figure 2 shems the workspace of the re-configured manipulator for 91<1)=90 degrees. In Figure 2 it can be seen that the fault tolerant work-space is a subset of the original work-space of the 3-00F PPM In Figure 3 the work-space of the rc-<:onftgurcd manipulator for e1
."
..
_~int
"
1
" lm~
~~~----L---~~~~------~~x
Link Lenghts :
2
approach it can be shown that a m-FT manipulator with n-OOF can be realised by a «m+1)n)-OOF manipulator. For planar manipulators the work space is \WO dimensional. Therefore a 2-00F manipulator is sufficient for general P1I.I'pOSe operation of a PPM. Following the above approach it has been sho"n that (2(m+ 1»-OOF are necessaIy and sufficient to construct a m-FT PPM (Paredis and Khosla, 1994). The example given in Figure 3 shaM that 3-00F is not sufficient to implement a I-FT PPM. Note that (2(1+1»=4. The PPM in figure 3 has one dimensional work-space which is not acceptable for a planar manipulator. Therefore a I-FT PPM must have four or more OOF. The one dimensional work-space shown in Figure 3 also corresponds to the FTWS of the 3-00F PPM with unit link lengths. The work-spaces corresponding to the n>-eonfigured versions of any 3-00F PPMs can be calculated analytically as follaM:
3
LJ- L2- L3""} unit
Fig. 3: Work-space of re-configured three link planar manipulator for 91(1)= ±IS0 degrees. The work-space of a re-configured manipulator, due to a faulty joint, is called as the 'work-space under fault'. When general p1J.l1lOSe fault tolerance is considered the following definition can be made for the work-space of a fault tolerant manipulator under all possible joint failures that it can compensate for: Definition 3: The intersection of work-spaces for all possible reconfigurations of a m-FT manipulator, due to joint failures, is called as the 'fault tolerant work-spacc' (FIWS) of the manipulator.
In this definition all possible reconfigurations of a m-FT manipulator for -180 ~ 91
3. FAULT TOLERANT WORK-SPACE DESIGN In order to implement fault tolerance of order m, for a given n-OOF manipulator, evel)' OOF of the manipulator must be duplicated m times. The result will be a m-FT manipulator. Whenever a joint fails there would be m number of working joints to compensate for the faulty joint. If the joint that replaced the original faulty joint should also fail there will be (m-I) number of working joints to compensate for the second failure. Using this
i) If the first joint is faulty (-180 ~ 91<1) ~ 180) the work space is the region with inner radius, ri, and outer radius, r2, where rl and r2 are calculated as follo~ :
rl
=
ab.{[«Llj2 + (uj2) - 2(LlXL2Xcos(ISO - 9t<1))]112 - L3 } (I)
r2= ((Llj2 + (L2j2) - 2(LIXUXcos(180 - 81<1))] 112 + L3
(2)
ii) If the second joint is faulty (-ISO S €l1<2) ~ ISO) the work space is the region with inner radius, r3, and outer radius, r4, where r3 and r4 are calculated as follo~:
r3 = abs{((Llj2 + (uj2) - 2(L3XL2Xoos(180 - 81<2»JII2 - Ll} (3) r4=
((L3j2 + (L2j2) - 2(L3XUXcos(ISO - 6t<2»JII2 + Ll
(4)
The FTWS of a 3-00F PPM is the intersection of all workspace. calculated by the use of the above equations (1-4). At fault angles given by 81(1) = ±180 and 01<2) = ±ISO degrees the equations reduee to the following.
rl = ahs{[«Ll)2 + (L2)2) - 2(L1)(L2)(l))112 - L3} = ahs{[(LI- L2)2]1I2 - L3} = L3
(5)
r2 = [«Ll)2 + (L2)2) - 2(LI)(L2)(I)]112 + L3 = [(LI- L2)2JII2 + L3 = L3
(6)
r3 = ahs{[«(L3)2 + (L2)2) - 2(L3)(L2)(1)J112 - Ll) = (7) ahs{[(L3 - L2)1 J II2 - Ll} = Ll r4 = ((L3)2 + (L2)2) - 2(L3)(L2)(1)(112 + Ll = (L3 - L2)1)112 + LI = Ll (8)
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For unit link lengths (LI = L2 = L3 = I unit) the inner and outer radii of work-spaces are all equal to one unit (rl = r2 = r3 = r4 = I unit). The work-spaces of the re-configured manipulators for fault angles 9t(1) = ±180 and 91(2) = ±ISO degrees give the work-spaces with minimum area. Therefore the work-spaces of the re-configured manipulators with fault angles 81(1) = ±ISO and 81(2) = ±ISO degrees determine the FTWS for this example. The FTWS for the 3-DOF PPM with unit link lengths is the onc dimensional curve with radius of one unit. To construct a m-FT PPM, a «m+ I)n)-DOF manipulator is necessary. The shape and dimensions of the FTWS is determined by the kinematic specifications of the manipulator. It is possible to design a fault tolerant manipulator with FTWS of predefined shape and dimensions. The design of a FTWS of predefined shape and dimensions for a 4-DOF PPM that is I-FT will be presented next. The work-space of a 3-DOF PPM is the circular area with inner radius, rin, and outer radius, rout, (rin and rout are conamtric) where rin and rout are calculated as follows: (9)
rout =LI +L2 + L3 ifLl > L2 + L3 then else ifL2 > Ll + L3 then else ifL3 > Ll + L2 then else
rin = Ll - (L2 + L3)
rin = L2 - (L I + L3) rin =L3 - (LI + L2)
rin =0
(10)
Where Ll, L2, L3 are the three link lengths of the manipulator. Note: that the calculation of the inner radius for the work-space is a conditioual equation with four steps depending on the relative magnitudes of the link lengths. When a fault occurs in onc joint of a I-FT 4-DOF PPM the manipulator loses one DOF. All re-configured versions of a I-FT PPM are 3-DOF manipulators. For the two links that are not adjacent to the faulty joint the link lengths of the original manipulator and the re-configured 3-DOF manipulator are the same. The remaining two links that are adjacent to the Caulty joint / are replaced by a single link, 4, in the re-configured versions of the manipulator. The length of4is calculated as follows : 4= [«L/; + (L j + J; ) - 2(Lj)(Lj + J )(cos( ISO - Si
(11)
Where. L j is the link before joint / and Li+ I is the link after joint /. 91
Four link lengths of the I-FT PPM will be denoted by xl, x2, xl, x4. The following equations give the three link lengths (LI , L2, U) of the re-configured 3-DOF PPMs with minimum work-space area: 81(1)~ degrees: L1=x1 + x2, L2=x3,
(12)
81(2)=0 degrees: Llaxl, L2=x2 + x3 ,
L3=x4
(13)
81(3)=0 degrees: Ll=x I, L2=x2,
L3=xl + x4
(14)
Elj(I)-±lSO degrees: Ll=abs(xl - x2), L2=x3,
L3=x4
(15)
Elj(2)=±180 degrees: LI=xl, L2-abs(x2 - x3), L3=x4
(16)
81(3)=±180 degrees: L1-xI, L2=x2,
L3=abs(xl - x4) (17)
Applying equations (9) and (10) to the link length values given by equations (12) to (17) in order to caleulate the work-spaces with minimum area, result in the following equations: (xl +x2)+xl +x4 > rout
(18)
xl + (x2 + x3) + x4 > rout
(19)
xl + x2 + (x3 + x4) > rout
(20)
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abs(xl - xl) + x3 + x4 > 'out
(21)
xl + abs(xl - x3) + x4 > 'out
(22)
xl + xl + abs(x3 - x4) > 'out
(23)
if (xl + xl) > x3 + ,,4 then (xl + xl) - (x3 + x4) < 'in else ifx3 > (xl + xl) + x4 then x3 - «(xl + xl) + x4) < 'in else ifx4 > (xl + xl) + x3 then x4 - «xl + xl) + x3) <'in else
no condition on rin value
Solution I:
xl=I, xl=2, x3=I, x4=2
Solution TI:
xl=2, xl=I , x3=I , x4=2
Solution Ill:
xl=2, xl=I, x3=2, x4=1
Solution N :
xl=2, xl=2,x3=2,x4=2
Run 2: 'out = 3 units, 'in = 1 units, maximum link length = 3 units. The software has caleuIated 11 feasible solutions for xl , xl, x3, x4.
ii) Simulation
Hi) Simulation Run 3 : rout = 3 units. rin = 2 units~ maximum link length = 3 units. Tbe software has caleuIated 31 feasible solutions for xl, xl, x3 , x4.
(24)
= 3 units, rin = 1 units, maximum link length = 4 units. The software has caleuIated 21 feasible solutions for xl , x2, x3, x4. vi) Simulation Run 4: rout
The calculation of the inner radius for the work-space is a conditional equation with four steps as given by equation (10). The first link length values for 61<1)=0 degrees, given by equation (12), have been applied to equation (10) resulting in equation (24). There are five more group5 of equations, with four stCp5 similar to equation (24), for each link length value given by equations (13) to (17). These five equations, with four stCp5, will be numbered as (25), (26), (27), (28), (29). It can be seen that equations (18), (19), (20) are the same. Equations (21), (22), (23) are more restrictive than equations (18), (19), (20); therefore equations (18), (19), (20) will nOl be used.
Equations (21), (22), (23), (24), (25), (26), (27), (28), (29) define the work-spaces with minimum area for all possible re-coofigurcd 3-DOF PPMs. The FfWS of the I-FT PPM is the intersection of these work-spaces with minimum area. Therefore equations (21) to (29) can be utilised together to find the FfWS of the I-FT PPM. If link lengths xl, xl, x3, x4 are given the inner and outer radii for the FfWS can be computed. If the inner and outer radii for the FfWS are given a range of values for the link lengths xl, xl, x3, x4 can be computed. A software using equations (21) to (29) to compute the ranges for the link lengths xl, x2, x3, x4 has been developed. The software takes the inner and outer radii for the FTWS and a 'limit number', defining the maximum admissible link length, and computes the ranges for the link lengths xl, xl, x3, x4. Results of five simulation runs are as follows:
v) Simulation Run 5: rout = 3 units, rin = 2 units, maximum link length = 4 units. The software bas caleuIated 73 feasible solutions for xl, xl, x3, x4. The number of feasible solutions for xl, x2, x3, x4 values increase as the allowed link length increases. The number of feasible solutions for xl, xl, x3, x4 values decrease as the FfWS area increases. Tbese two results can be verified intuitively when tbe definition of FTWS is considered. The shape of work-spaces for PPMs are concentric circular regions whose dimensions are determined by an inner and an outer radius. Due to this property the FfWS for all m-FT PPMs are cireuIar regions. This result can also be verified if the self-motion topology of PPMs are investigated. Throughout the paper it is assumed that there are no joint limits and the work-space is not constrained in any way. Wbcn joint limits are introduced the self-motion topology of a given PPM changes (Luck and Lee, 1993). As an extension to this result the shape of the work-space and thus the shape of the FfWS changes. Under jOint limits the shape of the work-space and FTWS for a PPM is the intersection of circular areas that are not concentric. In this paper the case when joint 0 is faulty has not been considered. Researcb for the formulation of the FTWS for the case when joint 0 can also be faulty is still continuing.
4. CONCLUSION In this paper fault tolerance has been defined as the ability
il Simulation Run 1: 'out = 4 units, 'in = 0 units, maximum link length = 2 units. The software has calculated 4 feasible solutions for xl , xl, x3, x4 values as follows:
of a manipulator to continue the performance of a task after total failure or partial immobilisation of one or more manipulator joints. Fault tolerant work-space has been defined as a sub-work-space volume where a fault tolerant redundant manipulator can continue its work even under
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the failure of anyone or more of its joints at any angle. Kinematic redundancy can be utilised to increase the fault tolerance of manipulators as demonstrated by design examples for planar positional manipulators. A software calculating possible link lcogth values for the design of a I-FT PPM with a FTWS of predefmed shape and dimensions is presented. Simulation runs of the software have shown that the number of feasible solutions for link lcogth values of a m-FT PPM increases when the maximum permissible link lcogth increases and when the desired FTWS area dccrcascs. These rcsults can be extended to all fault tolenmt redundant manipulators. Fault tolerant work spaces of desired shape and dimensions can be attained by the proper design of redundant manipulators. Research on more general redundant manipulator design rules and guidelines to serve this purpose is still continuing.
REFERENCES Cleary, K. and D. Tesar (1990), Incorporating Multiple Criteria in the Operation of Redundant Manipulators, Proc. IEEE Int. Con! on Robotics and Automation, pp. 618-624. Lewis, C. L. and A A Maciejewski (1994), An Example of Failure Tolerant Operation of a Kinematically Redundant Manipulator, Proc. IEEE Int. Con! on Robotics and Automation, pp. 1380-1387. Luck, C. L. and S. Lee (1993), Self-Motion Topology for Redundant Manipulators with Joint Limits, Proc.
IEEE Int. Con! on Robotics and Automation, pp. 626-631. Maciejewski, A A (1990), Fault Tolerant Properties of Kinematically Redundant Manipulators, Proc.
IEEE Int. Con! on Robotics and Automation, pp. 638-642. Paredis, C. and P. K. Khosla (1994), Mapping Tasks into Fault Tolerant Manipulators, Proc. IEEE Int. Con! on RoboticsandAutomation, pp. 6%-703. Sreevijayan D., S. Tosunoglu and D. Tesar (1994), Architectures for Fault-Tolerant Mechanical Systems, Proc. Mediterranean EJectrotechnical Conference, pp. 1029-1033.
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