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Robotics » Robotics and Automation » "Parallel Manipulators, towards New Applications", book edited by Huapeng Wu, ISBN 978-3-902613-40-0, Published: April 1, 2008 under CC BY-NC-SA 3.0 license. © The Author(s).

Chapter 3

Quantifying and Optimizing Failure Tolerance of a Class of Parallel Manipulators

By Chinmay S. Ukidve, John E. McInroy and Farhad Jafari
DOI: 10.5772/5425

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Overview

Gough-Stewart Platform
Figure 1. Gough-Stewart Platform
A Flexure Jointed Hexapod at the University of Wyoming
Figure 2. A Flexure Jointed Hexapod at the University of Wyoming
[4 4] cylindrical OGSP with optimal fault tolerant manipulability
Figure 3. [4 4] cylindrical OGSP with optimal fault tolerant manipulability
[4 4] cylindrical OGSP with one failure in group 1.
Figure 4. [4 4] cylindrical OGSP with one failure in group 1.
[4 4] cylindrical OGSP with one failure in group 2.
Figure 5. [4 4] cylindrical OGSP with one failure in group 2.
Variation of the relative manipulability under a single failure, for various two group geometries
Figure 6. Variation of the relative manipulability under a single failure, for various two group geometries

Quantifying and Optimizing Failure Tolerance of a Class of Parallel Manipulators

Chinmay S. Ukidve1, John E. McInroy1 and Farhad Jafari1

1. Introduction

For any robotic system, fault tolerance is a desirable property. This work uses a comparative approach to investigate fault tolerance and the associated problem of reduced manipulability of robots. An important result in combinatorial matrix theory is first obtained. Its consequent modifications are then applied to the theory of fault tolerance of robotic manipulators.

It is shown that for a certain class of parallel manipulators, the mean squared relative manipulability over all possible cases of a given number of actuator failures is always constant irrespective of the geometry of the manipulator. A theorem formulates the value of the mean squared relative manipulability. It is shown that this value depends only upon the number of simultaneous joint failures, the nominal number of joint degrees of freedom and the nominal task degrees of freedom. It is difficult to predict specific failures at the design stage and as such failure of any actuator is considered equally likely. In this context, optimal fault tolerant manipulability is quantified. The theory is applied to a special class of parallel manipulators called Orthogonal Gough-Stewart Platforms (Orthogonal GSPs or OGSPs). A class of two-group symmetric OGSPs which inherently provide for optimal fault tolerant manipulability under a single failure is developed.

2. Background

Robotic manipulators have become popular in numerous applications. They have been employed in automation of industrial processes, underwater exploration, space exploration and innovative defence technologies. The nature of some of these applications makes human presence near manipulators difficult and in many cases, impossible. This is especially true for robots employed in remote and hazardous environments. The repair and maintenance tasks for such robots are extremely difficult. In such cases, operational reliability is of prime importance. Therefore, it is imperative to incorporate failure tolerance in system design. Under the occurrence of failures, fault tolerance enables the robotic system to maintain critical functioning with a reduced level of performance.

Current research efforts are focussed on developing techniques for designing fault tolerant manipulators and robotic vision systems.

Redundant manipulators are rapidly becoming a focus of research due to a multitude of potential advantages they provide. In serial robots, kinematic redundancy has been employed for obstacle avoidance (Baillieul, 1990), dexterity optimization (Lewis & Maciejewski, 1992) and torque minimization (Hollerbach & Suh, 1987). In parallel manipulators, singularity avoidance (Kim et al., 2004) and stiffness improvement (Kock & Schumacher, 1998) are broad areas where kinematic redundancy has proven useful. Another significant attribute of redundancy that has come under recent investigation is fault tolerance.

Kinematic failures commonly occur in manipulators. The effect of such failures on manipulator performance depends upon the nature of failure and the nominal kinematic design of the manipulator. For instance, loss of an actuator can render a serial manipulator completely unmanipulable. On the other hand, parallel manipulators can be designed to retain kinematic stability under loss of actuators. Locking of actuators is another commonly observed failure phenomenon. In any failure scenario a robotic system loses partial or complete manipulability. Fault tolerance and the consequent problem of reduced manipulability have been studied by a number of researchers.

Maciejewski (Maciejewski, 1990) associates the concept of fault tolerance to manipulator configuration. Dexterity index is used as a measure of fault tolerance. Optimal fault tolerant configurations are defined using this measure.

Roberts and Maciejewski (Roberts & Maciejewski, 1996) propose a local measure to quantify fault tolerance of a manipulator pose in terms of a manipulability index. Their approach uses the singular value decomposition of the manipulator Jacobian matrix. They describe a direct relation between relative manipulability and the null-space of the Jacobian matrix. They propose relative manipulability index as a measure of fault tolerance.

Paredis, Au and Khosla (Paredis, Au & Khosla, 1994) consider fault tolerance with respect to manipulator workspace and reach. They define fault tolerant workspace of a manipulator and suggest task based design of manipulators. Their approach uses iterative techniques to design manipulators.

Ting, Tosunoglu and Tesar (Ting, Tosunoglu & Tesar, 1993) explore control algorithms for fault tolerant operation of manipulators.

McInroy, O’Brien and Neat (McInroy, O’Brien & Neat, 1999) propose a fault tolerant precision pointing strategy using a class of parallel manipulators called Gough-Stewart Platforms (GSPs) (Stewart, 1966). A GSP is used as a pointing platform to reject vibrations from a noisy spacecraft bus over all frequencies. At low frequencies two-axis or three-axis pointing method is used, while at high frequencies six-axis vibration isolation is employed. The benefits of this approach include broadband pointing stability without a high-bandwidth pointing sensor or destabilizing excitation of the high frequency structural modes. To incorporate fault tolerance, they propose a reconfiguration algorithm to compute a decoupling matrix which allows motion in ` off degrees of freedom' to compensate for failures.

This work uses a comparative approach to investigate fault tolerance and the associated problem of reduced manipulability. Following is a description of the main contributions.

An important result in combinatorial matrix theory is first obtained. Its consequent modifications are then applied to the theory of fault tolerance of robotic manipulators.

It is shown that for a certain class of parallel manipulators, the mean squared relative manipulability over all possible cases of a given number of actuator failures is always constant irrespective of the geometry of the manipulators. This work uses the manipulability index suggested by Yoshikawa (Yoshikawa, 1985) and the resulting relative manipulability indices proposed by Roberts and Maciejewski (Roberts & Maciejewski, 1996).

A theorem formulates the value of the mean squared relative manipulability. It is shown that this constant depends only upon the number of simultaneous failures, the nominal number of joint degrees of freedom and the nominal task degrees of freedom.

It is difficult to predict specific failures at the design stage and as such failure of any actuator is considered equally likely. From this perspective optimal fault tolerant manipulability for a given number of faults has been defined by Roberts and Maciejewski in (Roberts & Maciejewski, 1996). This work quantifies optimal fault tolerant manipulability based only upon the number of simultaneous failures, the nominal number of joint degrees of freedom and the nominal task degrees of freedom.

For parallel manipulators employed in micromanipulation, the workspace is very small. For such manipulators, the definition of optimal fault tolerant manipulability for a given number of faults carries a different interpretation. Such manipulators can be assumed to operate only at a particular pose and therefore the the same definition can be applied to the manipulator in general rather than to a specific pose. As an illustration, the concept of optimal fault tolerant manipulability is applied to a special class of parallel manipulators called Orthogonal Gough-Stewart Platforms (Orthogonal GSPs or OGSPs). This work develops a class of two-group symmetric OGSPs (McInroy & Jafari, 2006) which inherently provide for optimal fault tolerant manipulability under a single failure.

3. Quantifying optimal fault tolerant manipulability

3.1. Manipulability index

In the robotics standard, the Jacobian matrix mapping joint velocities, θ˙ , to generalized end effector velocities, V , is denoted by J

A number of researchers have proposed different measures that quantify the manipulability of a manipulator. One such manipulability index, based on a matrix determinant, was proposed by Yoshikawa (Yoshikawa, 1985):

w ( J ) = det ( JJT )
(2)

Failures in manipulators can occur in various ways. In this work, only mechanical failures that cause a manipulator to lose an actuator are considered. The impact of such failures varies with different classes of parallel manipulators. In a wide class of parallel mechanisms, under a joint failure, the resulting manipulator Jacobian matrix, referred to as reduced Jacobian matrix, is given by the original Jacobian matrix except that the column corresponding to the failed joint is removed. Gough Stewart Platforms are a classic example of manipulators belonging to this class. On the contrary, in multi-fingered grasps, the impact of failures on the kinematic representation is a function of composite manipulability Jacobian matrix, described in (Wen & Wilfinger, 1999) and cannot be directly derived by elimination of rows or columns.

The mechanisms treated in this work belong to the former class, which will be characterized by JΩ . Consider a nominal Jacobian matrix, J , with n actuators. A reduced Jacobian matrix with i simultaneous actuator failures will be denoted by iJ . Note that this iJ is not unique since there may be multiple ways in which i struts may fail. In order to identify all reduced Jacobian matrices uniquely, a subscript j will be used. Therefore, 1Jj j {1,2,...,nCi } , will describe the alternative i strut failure schemes. This representation will be more clear from the following example.

Suppose J denotes a manipulator with 3 actuators. Then 1J denotes a reduced Jacobian matrix with 1 actuator failure. There are 3C1 ways in which 1 actuator can fail at a time from 3 actuators. Here, fCg denotes the usual combinatorial notation. Therefore, 1Jj with j {1,2,3} completely represents all reduced Jacobian matrices.

To analyze the post fault performance of a manipulator the relative manipulability index, irj , will be used:

irj = det ( iJjiJjT ) det ( JJT )
(3)

Clearly this index is normalized and the scaled translational and rotational components of the manipulator Jacobian matrix do not affect this value.

3.2. Optimal fault tolerant manipulability

For serial manipulators, (Roberts & Maciejewski, 1996) define optimally fault tolerant configurations have been defined as those J in which the relative manipulability index irj remains constant over all possible j , for a given i . A rigorous method is provided in (Roberts & Maciejewski, 1996) to calculate irj over all j for a given i . However, that does not allow direct determination of optimal irj over all j for a given i . The following theorem shows that for a given n and i the sum of squares of irj over all j is invariant. Consequently, a formulation is developed that determines optimal irj over all j (Ukidve, McInroy & Jafari, 2006).

Theorem 1. Let J be an m×n n m Jacobian matrix representing the operating configuration of any manipulator having n actuators such that JΩ . Then the mean squared relative manipulability over all possible failures j given that i ( i ( nm ) ) actuators fail at a time is constant and is given by
j =1nCiirj2nCi = 1nCij =1nCidet ( iJjiJjT ) det ( JJT ) = ( ni ) CmnCm
(4)
Proof: Note: Following identities in combinatorics are used in the proof.
fCg = f ! g ! ( fg ) !
(5)
fCg = fC ( fg )
(6)

Case 1:

Note
j =1nCiirj2 = j =1nC ( nm ) irj2 = j =1nCmirj2 =1
(8)

This equality follows from the Binet-Cauchy Theorem, because for i = ( nm )

det ( JJT ) = j =1nCmdet ( iJjiJjT )
(9)

Case 2:

Let ckRm , denote the kth column of J k =1 to n Then,
J = [c1c2...cn]
(11)
and
JJT = c1c1T+c2c2T++cncnT
(12)
Let
( nm ) = t
(13)

By Binet-Cauchy Theorem,

det ( JJT ) = j =1nCmdet ( tJjtJjT )
(14)

Suppose i actuators fail at a time from n actuators ( i t ) . This leads to a reduced Jacobian matrix iJj with j {1,2,...,nCi } . Without loss of generality, we may assume that the actuators corresponding to last i columns fail. This says that all the reduced Jacobian matrices can be completely expressed by iJj with j {1,2,...,niCm } . Note that although this loss of generality argument is not applicable to the reduced Jacobian matrix, equation (7) makes it clear that the argument is valid for the product of the reduced Jacobian matrix and its transpose. The number of actuators remaining is ( ni ) . The number of ways in which the failures can happen at a time are nCi . In other words, nCi = nC ( ni ) is the number of ways in which ( ni ) un-failed actuators can be chosen from n actuators. So, there will be nCi such reduced Jacobian matrices, iJj , where

j {1,2,...,nCi }
(15)

Apply the Binet-Cauchy Theorem to a representative reduced Jacobian matrix iJj , denoted by iJj1

Then, we have

det ( iJj1iJj1T ) = j =1 ( ni ) Cmdet ( tJj1tJj1T )
(16)

Similarly,

det ( iJj2iJj2T ) = j =1 ( ni ) Cmdet ( tJj2tJj2T )
(17)

In general,

det ( iJjnCiiJjnCiT ) = j =1 ( ni ) Cmdet ( tJjnCitJjnCiT )
(18)

Note that all of the ( ni ) Cm terms appearing on the R.H.S. of equation (9) are exactly those terms on the R.H.S. of equation (8) which do not have the eliminated i columns in their sub-matrices tJj . Moreover, each equation from ((9) - (11)) has one term on the L.H.S and ( ni ) Cm terms on the R.H.S and there are mCi such equations.

Using equations ((9) - (11)) to add all possible nCi reduced Jacobian matrices iJj2 iJj3 iJjnCi i.e. taking the summation of all possible nCi reduced Jacobian matrices,

det ( iJj1iJj1T ) +det ( iJj2iJj2T ) ++det ( iJjnCiiJjnCiT ) = j =1 ( ni ) Cmdet ( tJj1tJj1T ) +j =1 ( ni ) Cmdet ( tJj2tJj2T ) ++j =1 ( ni ) Cmdet ( tJjnCitJjnCiT )
(19)

Now comparing equation (8) with equation(12), each term in the R.H.S. of (12) is a term on the R.H.S. of (8). Choose a particular term on the R.H.S. of (8), for example det ( tJ1tJ1T ) This term will occur in only those detiJjpiJjpT ( p =1 to nCi ) for which iJjp contains exactly those n columns in tJ1 . For n given columns, the number of columns left to choose for iJj ; with i columns eliminated is ( nm ) ; from which we choose the remaining ( nmi ) columns. Therefore, the number of occurrences for det ( tJ1tJ1T ) will be

( nm ) C ( nmi )
(20)

This is true for each term on the R.H.S. of(8). Furthermore, we also know that each term from the ( ni ) Cm terms appearing on the R.H.S. of mCi equations ((9) - (11)) occurs in the R.H.S. of equation (8).

Dividing equation (12) by (8), we have

det ( iJj1iJj1T ) det ( JJT ) +det ( iJj2iJj2T ) det ( JJT ) ++det ( iJjnCiiJjnCiT ) det ( JJT ) = ( nm ) C ( nmi )
(21)

Therefore,

j =1nCiirj2 = j =1nCidet ( iJjiJjT ) det ( JJT ) = ( nm ) C ( nmi )
(22)

Dividing both sides by nCi to take the mean and noting that,

( nm ) C ( nmi ) = ( ni ) CmnCmnCi
(23)

the result follows.

This proof leads to the definition of optimal fault tolerant manipulability. Definition: A manipulator operating about a single point in the workspace is said to be optimally fault tolerant for a given number of failures

i if for all j {1,2nCi }

where c is a constant.

It is clear from Theorem that if post-fault relative manipulability for certain cases of failure are higher than the optimal fault tolerant manipulability value, then for other worst cases of failure, post-fault relative manipulabilities are extremely low. This has precisely been the motivation for developing optimally fault tolerant manipulators and the above definition arises as a direct consequence.

Corollary 2. A manipulator characterized by JΩ and operating about a single point in the workspace is optimally fault tolerant to i faults if
irj = ( ni ) CmnCm
(26)

for all

j {1,2,...,nCi }
(27)

Proof: Equation (16) defines manipulators with optimal fault tolerant manipulability for i faults. By Theorem , if each irj is constant, c is given by equation

Roberts and Maciejewski (Roberts & Maciejewski, 1996) present a singular value decomposition approach to identify fault tolerant configurations for serial manipulators and describes a rigorous method to determine whether a given nominal configuration possesses optimal fault tolerant manipulability. The above theorem states a formulation which directly gives the value of optimal manipulability under a given number of failures, for any manipulator with a given number of actuators. In fact, the theorem proves that irrespective of the operating configuration, all manipulators having the same number of actuators, have the same value of optimal manipulability under a given number of failures. This new idea plays a key role in the design of fault tolerant serial manipulators. The following example illustrates this point.

Suppose a fault tolerant serial manipulator is to be designed such that it has 3 degrees of freedom ( m =3) and it is desired to have an optimal fault tolerant manipulability of 0.5 ( irj =0.5) under a single-actuator failure ( i =1) . This implies that the manipulator operating in the nominal configuration, should be able to sustain the failure of any actuator and retain half of its manipulability. Substituting all known values in equation , we get n =4 . This means that the manipulator has to have 4 actuators in order to have optimal fault tolerant manipulability for single failure.

Another implication of this theorem is significant in terms of understanding post-fault behavior of any manipulator. From equation (14), it is clear that if the value of irj is quite close to ( nm ) C ( nmi ) for some j then the reduction in manipulability is far more pronounced if a possible failure combination corresponding to some other j were to occur. Taking this into account, it is possible to assemble the actuators in such a way that actuators which are more likely to fail (for example, actuators with actuators that are more likely to have manufacturing defects) correspond to those j which give

irj ( ni ) CmnCm
(29)

This idea has a greater impact on the design of parallel manipulators. While actuator failures may cause a serial manipulator to stop functioning, actuator failures have a comparatively smaller effect on redundant parallel manipulators because they can retain kinematic stability. Therefore, the two consequences that can be applied to the design of serial manipulators are applicable to parallel manipulators as well.

The most significant area of investigation where the above results influence parallel manipulator design is the choice of geometry. This area will be explored in the Section 5.

3.3 Examples

Some specific examples provide more insight to understanding this concept of optimal fault tolerant manipulability.

Number of Nominal Actuators n Number of Failures i Sum of all possible irj2 Optimal Fault Tolerant Manipulability
7110.377
8120.500
8220.189
9130.577
9230.288
9330.109

Table 1.

Optimal Fault-tolerant manipulability of 6-dof redundant manipulators

It is clear from Table 1 that as the number of failures increases, the optimal fault tolerant manipulability decreases drastically, regardless of the geometry. Consider the example of any 8-actuator manipulator suffering from 2 simultaneous failures. The sum of squares of relative manipulabilities is 2 for 28 failure possibilities. This means that if some post-fault relative manipulabilities are more than the optimal fault tolerant manipulability (0.189), then the worst case values are far less than 0.189. Therefore, irrespective of the geometry of the 8-actuator manipulator, for worst cases of failure the relative manipulabilities will have negligible values. Hence, it is important to design manipulators that are optimally fault tolerant to a given number of failures.

Table 1 also provides another important inference which is significant from the design perspective. Any redundant manipulator gives very low optimal fault tolerant manipulability values for more than one failures, and these values decrease drastically with number of failures. For example, for two failures in an octopod the optimal fault tolerant manipulability is 0.189 and, for two and three failures in a nanopod the optimal fault tolerant manipulabilities are 0.288 and 0.109 respectively. This means that under the hypothesis of equal probability of failure for each actuator, it is not practical to design manipulators optimally fault tolerant to more than one fault.

4. Symmetric orthogonal Gough Stewart platforms

4.1. Gough Stewart platforms

A Gough-Stewart Platform (GSP) is a parallel manipulator consisting of a base, a moving platform (or payload) and struts. The length of struts is controlled by actuators. The struts have spherical joints at the payload end and U joints at the base. To provide six degrees of freedom, six struts are commonly used. Figure 1 is a diagrammatic representation of a GSP. Payload attachment points and base attachment points are represented by pi and qi i {1,2,3,4,5,6} respectively.

media/image154.png

Figure 1.

Gough-Stewart Platform

OGSPs are a special class of GSPs that provide kinematic and dynamic decoupled control. Therefore, OGSPs are being widely used in commercial, military and space applications. Scientists at Northrop Grumman Space Technologies (NGST) are currently experimenting with an 8-strut OGSP. More recent applications of OGSPs include laser tracking and pointing, ultra-precise manipulation (McInroy & Jafari, 2006) and robotic surgery (Wapler et al., 2003). The very nature of these applications makes maintenance or repair of manipulators very difficult. Moreover, a single failure may compromise the fulfilment of objective or cause costly downtime. As a consequence, it is desirable to design OGSPs which can sustain failures, while retaining an acceptable level of manipulability. Figure 2 shows one of the flexure jointed hexapods at the University of Wyoming. It has a mutually orthogonal geometry.

media/image155.png

Figure 2.

A Flexure Jointed Hexapod at the University of Wyoming

Recent research has shown that symmetric groups of struts can be used to generate OGSPs having desired properties at their home position (McInroy & Jafari, 2006) and several new results have been obtained.

The following part of this section recapitulates important results from (McInroy & Jafari, 2006).

4.2. Kinematics of symmetric OGSPs

The inverse Jacobian, M , of a GSP maps the generalized velocity of the payload to the corresponding joint velocities of each strut ( θ˙ = MV ). It has the form:

M = [u1Tv1TulTvlT]
(30)

where uivi3 vi = pi×ui ui is the unit vector along strut i and piR3 is the moving platform attachment point of strut i . Please refer to Figure 1. Note that, even though M is called the inverse Jacobian to comply with the robotics standard, its computation does not require inversion, thus it is well defined for all GSP.

MMl×6 ( R ) . Write

where U VM3×l ( R ) . We say M GSP, M is a Gough-Stewart Platform, if:

We say M is a Weighted Orthogonal Gough-Stewart Platform, M w-OGSP, if M GSP and:

  • MTKM
is a diagonal matrix for a diagonal K

Where K = I these matrices become the Orthogonal Gough-Stewart Platforms.

media/image180.png

Figure 3.

[4 4] cylindrical OGSP with optimal fault tolerant manipulability

(McInroy & Jafari, 2006) develops properties and designs of symmetrical weighted OGSPs. Struts that are geometrically symmetrical are treated together, so the entire OGSP is decomposed into m different groups, with the ith group having ni struts. Then

n = [n1n2...nm]T
(34)

is a vector of positive integers describing the number of struts in each group. The total number of struts in the GSP is then l = j =1mnj . Let uijvij3 correspond to the ith strut in group j . Let U = [ u11u21un11u12unmm ] and V = [ v11v21vn11v12vnmm ] . A GSP can then be found for these struts by letting M = [ UTVT ]

Following is the summary of results in (McInroy & Jafari, 2006).

Proposition 3. Conditions (a) and (b) in the GSP definition are satisfied if
uij = [SφijCβijSφijSβijCφij]vij = xijvxij+yijvyij
(35)

where Sx = sinx Cx = cosx

φijβijxijyij
(36)
and

vxij = [SβijCβij0]vyij = [CφijCβijCφijSβijSφij]
(37)

Conversely, if M GSP, then M may be represented by a parameterization given by (19) and (20).

Theorem 4. Let all groups contain more than two struts, i.e. minjnj 2 . Then M w-OGSP if

where

x = [x1x2xm]y = [y1y2ym]φ = [φ1φ2φm]
(43)
k = [k1k2km]β = [β1β2βm]
(44)
Ax = [ k1n1Sφ1k2n2Sφ2kmnmSφm ],
(45)
Ay = [ k1n1S2φ1k2n2S2φ2kmnmS2φm ]
(46)
φiβjkj may be freely chosen. x and yRm may be freely chosen to satisfy (F). Furthermore, if σi2 denotes the ith diagonal element of MTKM , then
σ12 = σ22 = 12j =1mnjkjSφij2
(47)
σ32 = j =1mnjkj2σ12
(48)
σ42 = σ52 = 12j =1mnjkj ( xj2+yj2Cφij2 )
(49)
σ62 = j =1mnjkjyj2Sφij2
(50)

In (Aphale, 2006) robust fault tolerance is defined as the property by which the rank of M equals 6 or the number of struts remaining after failures, whichever is minimum. Not all geometric designs of OGSPs are robustly fault tolerant. In fact, it has been proved that [3 3 2] geometry gives the only robustly fault tolerant design for 8- strut (octopod) OGSPs. This means that [3 3 2] geometry is the only one wherein, if any two struts fail, the rank of M remains 6. While robust fault tolerance guarantees motion in 6 degrees of freedom for a n - strut platform under any nm failures ( m ( n6) ) , experiments made on the University of Wyoming octopod clearly show that robustly fault tolerant designs suffer from serious post-fault stability problems due to poor conditioning. On the other hand, in many cases the design specifications may require a single failure tolerant architecture. For instance, in a typical case, it would be better to design an 8-strut OGSP which gives an optimal fault tolerant manipulability of 0.5 for a single failure, instead of designing a robustly fault tolerant 8-strut OGSP. This argument will be clearer from the example explained in the next section where a class of symmetric OGSPs having optimal fault tolerant manipulability is proposed.

5. Fault tolerant Gough Stewart platforms

5.1. Design

For parallel manipulators, the problem of inverse kinematics is easier to solve. Therefore, in most literature on parallel manipulators, the inverse Jacobian, M , is used for study.

Remark: In this work, it is assumed that the Jacobian relating joint and Cartesian motion is constant. This is equivalent to considering that the operation is about a single point, rather than across a workspace. The rationale for making this assumption is that there are several high precision OGSP applications which demand operation over a very small workspace. These include high precision motion control for telescopes, scanning microscopes, integrated circuit fabrication, stiffness, precision pointing and vibration isolation.

As mentioned in Section 3, [4 4] redundant OGSPs are currently under investigation by a number of researchers. This section develops a more general class of symmetric OGSPs with optimal fault tolerant manipulability under one fault.

A key characteristic of symmetric OGSPs is rotational invariance. Rotational invariance of groups of struts can be clearly understood with the help of Figure 3, Figure 4 and Figure 5.

Figure 3 represents a symmetric 8-strut OGSP, having M given as,

[0.86600.00000.50000.13690.59690.23720.00000.86600.50000.59690.13690.23720.86600.00000.50000.13690.59690.23720.00000.86600.50000.59690.13690.23720.00000.50000.86601.03380.23720.13690.50000.00000.86600.23721.03380.13690.00000.50000.86601.03380.23720.13690.50000.00000.86600.23721.03380.1369]
(51)

It can be clearly seen that a strut failure in group 1 (Figure 4) or a strut failure in group 2 (Figure 5) causes the same effective change in manipulability.

media/image238.png

Figure 4.

[4 4] cylindrical OGSP with one failure in group 1.

This prominent feature provides symmetric OGSPs with inherent optimal fault tolerant manipulability under the occurrence of a failure. Furthermore, for symmetric OGSPs it is possible to estimate post-fault reduction in manipulability by knowing the geometry. This is explained in the following theorem.

Theorem 5. For a [p q] (p 3,q 3 or q 3,p 3) geometry, satisfying (A)- (F) in Theorem , the relative manipulability after a single failure in group [p] is given by 1rj where 1rj is the optimal fault tolerant manipulability under one fault for an OGSP with [p p] geometry. For the remaining cases of failure i.e. those corresponding to group [q], the relative manipulability is given by 1rj where 1rj is the optimal fault tolerant manipulability under one fault for an OGSP with [q q] geometry.Proof: Consider a manipulator with [p q] (p 3,q 3 or q 3,p 3) geometry. Let Mp and Mq denote the inverse Jacobian corresponding to each group. Then the composite inverse Jacobian matrix M is given by

Consider the case that a single link in group [p] fails. Then from rank one perturbation of a matrix, we have

det ( MTM ) = det ( MTM ) (1+pf ( MTM ) 1pfT )
(53)

where pf represents the row of Mp corresponding to the link failure and M represents the inverse Jacobian matrix after failure. Then,

det ( MTM ) det ( MTM ) = 1 (1+pf ( MTM ) 1pfT )
(54)

Using the Matrix Inversion Lemma for the expression on the R.H.S. of equation (29)

det ( MTM ) det ( MTM ) =1 ( pf ( MTM+pfTpf ) 1pfT )
(55)

Using the formulation as in equation (7), we have

det ( MTM ) det ( MTM ) =1 ( pf ( MTM ) 1pfT )
(56)

Using conditions (A)- (F) given in Theorem , for a [p q] geometry with equal strut stiffness, we have

φi1 = φpφi2 = φqxi1 = xpxi2 = xqyi1 = ypyi2 = yq
(57)
and
( MTM ) 1 = diag [ 1σ121σ221σ321σ421σ521σ62 ]
(58)

Note that pf also has a trigonometric parametrization given by Proposition .

pfT = [SφpCβijSφpSβijCφpSβij+CφpCβijCβij+CφpSβijSφp]
(59)

and substituting equation (32) in equations ((23)-(26)), we get

σ12 = σ22 = 12 ( pSφp2+qSφq2 ),
(60)
σ32 = ( p+q ) 2σ12
(61)
σ42 = σ52 = 12 ( p ( xp2+yp2Cφp2 ) +q ( xq2+yq2Cφq2 ) )
(62)
and
σ62 = ( pyp2Sφp2+qyq2Sφq2 )
(63)

Substituting equations ((35)-(38)) into equation(33), we get ( MTM ) 1 in terms of design parameters. Using this formulation of ( MTM ) 1 into equation (31), then substituting equation (34) in equation (31) and simplifying the complicated trigonometric expression, we get

det ( MTM ) det ( MTM ) =1 ( pf ( MTM ) 1pfT ) =13p
(64)

It is important to note that this expression does not depend upon q or the particular geometric parameters φij xij yij and

Note that the optimal fault tolerant manipulability for any [p p] manipulator is given by equation in Theorem . Hence,

1rj = (2p1) C62pC6 = 13p
(66)

Since the choice of p does not cause any loss of generality, we have

1rj = (2q1) C62qC6 = 13q
(67)

Results from this Theorem are plotted in Figure 6. Figure 6 depicts the change in values of the relative manipulability, for different geometries, under the occurrence of one failure.

This Theorem proves the independence of the manipulability contributions of each symmetric group of a two-group OGSPs which may have different number of struts in each group. It is shown that within the group, any failure will give the same manipulability reduction even in any two-group OGSPs. Figure 6 depicts the change in relative manipulability under on failure, for symmetric OGSPs with different two–group geometrical designs.

media/image281.png

Figure 5.

[4 4] cylindrical OGSP with one failure in group 2.

Looking at Figure 6 it is now possible to estimate the level of post fault reduction in manipulability of symmetric OGSPs. Corollary 6 proves that all two-group OGSPs ( i.e. with [m m] (m 3) geometries ) possess optimal fault tolerant manipulability.

Corollary 6. Any 2s-strut OGSP with [s s] (s 3) geometry generated by Theorem possesses optimal fault tolerant manipulability under one fault and its value is given by,
1rj = (2s1) C62sC6
(69)

for all

j {1,2,...,2sC1 }
(70)

Proof: Consider a manipulator with [p q] (p 3,q 3 or q 3,p 3) geometry. Substitute q=p=s. Using Theorem ,
1rj = (2s1) C62sC6
(71)

for all

j {1,2,...,2sC1 }
(72)

media/image293.png

Figure 6.

Variation of the relative manipulability under a single failure, for various two group geometries

For the particular case of a symmetric 8-strut OGSP introduced at the beginning of this section,

1rj =0.5 for all
j {1,2,...,8}
(74)

This inherent property possessed by symmetric OGSPs can be put to a significant advantage in design. Therorem and Corollary (6) allow freedom of designing symmetric OGSPs with a high value of nominal manipulability. For example, by Corollary (6) it is seen that an 8-strut OGSP sustains any single-strut failure while retaining half of its nominal manipulability. The optimal fault tolerant manipulability of symmetric OGSPs makes them a suitable choice for critical applications where failure tolerance is necessary.

5.2. Singularities

While designing OGSPs with optimal fault tolerant manipulability, it is important to identify symmetric OGSPs which may be rendered singular under the occurrence of one fault. At the onset of singularity, unexpected motions are possible and the manipulator cannot be controlled. This is highly undesirable and potentially destructive. The following Theorem develops the necessary and sufficient condition to identify optimal fault tolerant OGSPs with potential singularity problems.

Theorem 7. Let M be the inverse Jacobian matrix of an OGSP with two groups. Then, iMTiM is singular if and only if the group in which failure occurs has at most 3 struts.

The following lemma is necessary to prove the Theorem.

Lemma 8. For any m×n matrix, M
rank ( MTM ) = rank ( M )
(75)
Proof of lemma: Clearly,
rank ( MTM ) rank ( M )
(76)
Let MTMx =0 for

Then,

MTMxx = MxMx = Mx2 =0
(78)

Hence,

Proof of Theorem: Suppose that iMTiM is singular. Then,rank
( iMTiM ) 5
(81)

Proposition 7 in (Aphale, 2006) determines the rank of M for an OGSP, having p groups of struts:

rank ( M ) = min ( 1prank ( Mp ),6)
(82)

where Mp denotes the inverse Jacobian matrix of the pth group.

In the context of failures, this proposition directly implies

rank ( iM ) = min ( 1prank ( fpM ),6)
(83)

where fpM denotes the inverse Jacobian matrix of the pth group having f strut failures within the group. That is, f = i

Applying Lemma to equation (48), we have

rank ( iMTiM ) = min ( 1prank ( fpMTfpM ),6)
(84)

The nominal OGSP under consideration consists of two groups of struts. Hence,

rank ( iMTiM ) = min [ e =12rank ( feeMTfeeM ),6]
(85)

where f1+f2 = i . Theorem 1 in (Aphale, 2006) establishes that the maximum rank of the Jacobian matrix of a group of struts forming an OGSP is 3 . Therefore, iMTiM is singular if the group in which any failure occurs has at most 3 struts. The converse is immediate.

Remark: It is worthwhile to note that unitarily equivalent Jacobian matrices (and inverse Jacobian matrices) have the same manipulability, and it may be readily checked that all single failure reduced inverse Jacobian matrices of a 2s OGSP with an [s s] geometry generated by Theorem are unitarily equivalent. This observation highlights the fact that these designs produce manipulators with optimal fault tolerant manipulability.

5.3. Application example: air borne laser (ABL)

Currently, feasibility of missile defense using an aircraft equipped with a high energy laser is being explored. At the concept level, the system uses a mirror inside the fuselage which focusses a beam from a megawatt-class chemical laser. Optic and beam control systems keeps the beam locked on a small supersonic target hundreds of kilometers away. It is believed that ABL can destroy hostile theater ballistic missiles while they are still in the highly vulnerable boost phase of flight before separation of the warheads. ABL can operate above the clouds, where it is possible to autonomously detect and track missiles as they are launched, using an onboard surveillance system. The defense system acquires the target, then accurately points and fires the laser with sufficient energy to destroy the missile. Airborne optical or electro-optical systems may be too large for all elements to be mounted on a single integrating structure, other than the aircraft fuselage itself. An eight-legged six-DOF OGSP (Octopod) is a perfect candidate to maintain the required alignment between elements. However the various smaller integrating structures (benches) must still be isolated from high-frequency airframe disturbances that could excite resonances outside the bandwidth of the alignment control system. The combined active alignment and vibration isolation functions must be performed by flight-weight components, which may have to operate in a vacuum. The platform used must be able to perform the dual functions of low-frequency alignment and high-frequency isolation (Keinholz, 1999).

The manipulability requirements for OGSPs intended for such an application are very demanding and Aphale (Aphale, 2006) describes them in detail. It is also shown (Apahle, 2005) that OGSPs are capable of meeting the manipulability requirements, making them suitable for the ABL application. Failure tolerance is imperative for this missile defense application. Furthermore, it is difficult to predict specific failures at the design stage and as such failure of any actuator is considered equally likely. If an equal reduction of manipulability is desired under a failure of any strut, an OGSP with optimal fault tolerant manipulability is an excellent choice.

6. Conclusions and future work

6.1. Conclusions

This work proves that for a certain class of parallel manipulators functioning about a single point in its workspace, the mean squared relative manipulability over all possible cases of a given number of actuator failures is always constant irrespective of the geometry of the manipulator. In this context, optimal fault tolerant manipulability is defined and quantified using a simple algebraic formulation. The definition is more suited to parallel manipulators since they can retain kinematic stability under failures which constitute loss of actuators.

For micromanipulation, symmetric OGSPs can be designed to possess optimal manipulability under actuator failures. OGSP geometries that may be rendered singular due to faults can be identified and avoided. OGSPs with optimal fault tolerant manipulability are highly suitable for critical applications since they retain a reasonable and equal fault tolerant performance if any actuator fails. For example, Figure 3 illustrates a cylindrical [4 4] OGSP that can be used in aerospace applications with ABL. These OGSPs will provide operational reliability critical to the application.

6.2. Future work

Currently most OGSPs are seen to have a very small range of motion in the joint space. In such scenarios, the assumption that the Jacobian matrix remains constant with respect to time, is valid. Recent applications demand OGSPs with a larger range of motion. The assumption of the Jacobian being constant does not hold validity in such cases. Investigating the fault tolerant characteristics of a manipulator Jacobian which will take into account the change with respect to time can be of great practical importance. It has recently been shown (Roberts, Yu & Maciejewski, 2007) that, regardless of a manipulator's geometry or the amount of kinematic redundancy present in a manipulator, no fully spatial manipulator Jacobian can be equally fault tolerant to three or more joint failures. Due to these constraints in generalization, it would be useful to formulate manipulator Jacobian matrices that possess equal fault tolerance to specified scenarios involving multiple failures. In particular, weights can be assigned to relative manipulability indices corresponding to multiple failure scenarios and optimized values of relative manipulability can be obtained based on the result derived in Theorem . Exploring the application of design and control techniques devised for OGSPs in areas of medical robotics and haptic interfaces can be considered. Robotics holds promise in standardized surgical procedures like eye surgery, knee surgery, etc. The theory developed thus far can be applied efficiently in medical applications where principles of robotics and computer vision combine towards a single objective. Multiple finger grasp mechanisms and other parallel manipulators have been considered for such applications. In these applications there is a need to withstand failures with almost no degradation in performance. It is possible to transfer many theories and techniques related to parallel manipulators to the analysis of multiple finger grasps with some modification.

It would be worthwhile to consider optimizing control for grasps such that fault tolerance can be achieved. Internal force calculations have been done for parallel mechanisms like multi-finger grasp mechanisms (Kerr & Roth, 1986). Internal force issues in other forms of parallel manipulators have also been explored (Lebret, Liu & Lewis, 1993) (Hiller and Schneider, 1997). Literature on the internal forces generated in GSPs is limited. OGSPs being a very recently defined class haven't been explored with respect to the internal forces they generate and need to withstand. With redundancy comes more number of actuators than the required minimum and a large number of constraints associated with them. Under failures, internal forces will be a major factor in the dynamics and control of OGSPs. Generating OGSPs that provide equal tolerance to failures with respect to the dynamic manipulability index seems feasible.

Finally, it is most important to recognize that the main contribution of this work is a combinatorial result in linear algebra. Numerous systems in various disciplines can be modeled by matrices. For instance, matrices are used to model power transmission and distribution systems. In matrix models where failures amount to elimination of rows and (or) columns, the theory of fault tolerance developed thus far would be useful and worthwhile extending.

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