is singular if and only if the group in which failure occurs has at most 3 struts.
i M
i
The following lemma is necessary to prove the Theorem.
?#&&3'@)' For any m $ n matrix, M ,
( T
rank M M ) = rank( M ).
(44)
Proof of lemma: Clearly,
( T
rank M M ) % rank( M ).
(45)
Let M TMx = 0 for
n
x #R .
Then,
2
T
9 M Mx, x: = 9 Mx, Mx: = Mx = 0.
(46)
Hence, Mx = 0 .
#
*%$$+'$+'!"#$%#&, Suppose that M T
is singular. Then,
i M
i
rank (
T
M M ) % 5 .
i
i
Proposition 7 in (Aphale, 2006) determines the rank of M for an OGSP, having p groups
of struts:
64
Parallel Manipulators, Towards New Applications
p
rank( M ) = min( r
' ank( M ),6)
(47)
p
1
where M denotes the inverse Jacobian matrix of the th
p group.
p
In the context of failures, this proposition directly implies
p
rank( M ) = min( r
' ank( p M ),6)
(48)
i
f
1
where p M denotes the inverse Jacobian matrix of the th
p group having f strut failures
f
within the group. That is, ' f = i .
Applying Lemma 8 to equation (48), we have
p
rank(
T
M M ) = min( r
' ank( p T p
M M ),6).
(49)
i
i
f
f
1
The nominal OGSP under consideration consists of two groups of struts. Hence,
2
(
T
) =
['
( e
T e
rank M M
min
rank M
M ),6]
(50)
i
i
f
f
e
e
e=1
where f . f = i . Theorem 1 in (Aphale, 2006) establishes that the maximum rank of the
1
2
Jacobian matrix of a group of struts forming an OGSP is 3 . Therefore, M T
is singular if
i M
i
the group in which any failure occurs has at most 3 struts. The converse is immediate.
#
:#&3%;,' 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 4 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
Quantifying and Optimizing Failure Tolerance of a Class of Parallel Manipulators
65
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
66
Parallel Manipulators, Towards New Applications
scenarios and optimized values of relative manipulability can be obtained based on the
result derived in Theorem 1. 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.
7. References
Aphale, S. (2006). 6%'#/"#"/* +&(7+/+"28* /+9/7:'(%;2&(* <82(,+&.'* ;#(7* &+=9'(* ,298(* (+8%&2">%, ProQuest / UMI, ISBN-10: 0542313596, Ph.D. Dissertation 0* University of Wyoming,
Laramie, WY.
Baillieul, J. (1996). Avoiding obstacles and resolving redundancy, ?&+>%%@#"/'* +,* ABBB*
A"(%&"2(#+"28*C+",%&%">%*+"*D+=+(#>'*2"@*59(+.2(#+" , pp. 1698 – 1703, San Francisco, CA., April 1990.
Hiller, M. & Schneider, M. (1997). Modeling, simulation and control of flexible
manipulators, B9&+<%2"*E+9&"28*+,*F%>72"#>' , vol. 16, 1997, page numbers 127-150.
Hollerbach, J. M. & Suh, K. C. (1987). Redundancy resolution of manipulators through
torque optimization, ABBB* E+9&"28* +,* D+=+(#>'* 2"@* 59(+.2(#+" , vol. RA-3, no. 4,
August 1987, page numbers 308-316.
Kerr, J. & Roth, B. (1986). Analysis of multifingered hands, A"(%&"2(#+"28* E+9&"28* +,* D+=+(#>*
D%'%2&>7, vol. 4, no. 4, 1986, page numbers 3-17.
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67
Kim H. W. ;Lee J. H. ; Yi, B. J & Suh I. H. (2004). Singularity-free load distribution algorithms
for a 6 dof parallel haptic device, ?&+>%%@#"/'* +,* ABBB* A"(%&"2(#+"28* C+",%&%">%* +"*
D+=+(#>'*2"@*59(+.2(#+" , pp. 298-304, New Orleans, LA., May 2004.
Kock, S. & Schumacher W. (1998). A parallel x-y manipulator with actuation redundancy
for high speed and active stiffness applications, ?&+>%%@#"/'* +,* ABBB* A"(%&"2(#+"28*
C+",%&%">%*+"*D+=+(#>'*2"@*59(+.2(#+" , , pp. 2295-2300, Leuven, Belgium, May 1998.
Lebret, G.; Liu, K. & Lewis, F. L. Dynamic analysis and control of a stewart platform
manipulator, Jo 9&"28*+,*D+=+(#>*4)'(%.' , vol. 10, no. 5, 1993, page numbers 629-655.
Lewis, C. L. & Maciejewski, A. A. (1992). Dexterity optimization of kinematically redundant
manipulators in presence of faults, ?&+>%%@#"/'*+,*G+9&(7*A"(%&"2(#+"28*4).<+'#9.*+"*
D+=+(#>'*2"@*F2"9,2>(9&#"/, pp. 279-284, Santa Fe, NM., November 1992.
Maciejewski, A. A. (1990). Fault tolerant properties of kinematically redundant
manipulators, ?&+>%%@#"/'*+,*ABBB*A"(%&"2(#+"28*C+",%&%">%*+"*D+=+(#>'*2"@*59(+.2(#+" , pp. 638-642, Cincinnati, OH., May 1990.
McInroy, J. E. ; O’Brien, J. F. & Neat, G. W. (1999). Precise, fault-tolerant pointing using a
stewart platform, ABBBH54FB* I&2"'2>(#+"'* +"* F%>72(&+"#>' , vol. 4, no. 1, March 1999, page numbers 91-95.
McInroy, J. E. & Jafari, F. (2006). Finding symmetric orthogonal gough-stewart platforms,
ABBB* I&2"'2>(#+"'* +"* D+=+(#>'* 2"@* 59(+.2(#+" , vol. 22, no. 5, October 2006, page numbers 880-889.
Paredis, C. J. J. ; Au, W. K. F. & Khosla, P. K. (1994). Kinematic design of fault tolerant
manipulators, C+.<9(%&'*B8%>(&#>28*B"//30 vol. 20, no. 3, 1994, page numbers 211-220.
Roberts, R. G. & Maciejewski, A. A. (1996). A local measure of fault tolerance for
kinematically redundant manipulators, ABBB* I&2"'2>(#+"'* +"* D+=+(#>'* 2"@*
59(+.2(#+" , vol. 12, no. 4, August 1996, page numbers 543-552.
Roberts, R. G.; Yu, H. G. & Maciejewski, A. A. (2007). Characterizing Optimally Fault-
Tolerant Manipulators Based on Relative Manipulability Indices, *JKKL*A"(%&"2(#+"28*
C+",%&%">%*+"*A"(%88#/%"(*D+=+('*2"@*4)'(%.'*MADN4*JKKLO0 pp. 3925-3930, San Diego,
CA., Oct. 29 - Nov. 2, 2007.
Stewart, D. (1966). A platform with six degrees of freedom, ?&+>%%@#"/'* +,* A"'(#(9(#+"* +,*
F%>72"#>28*B"/#"%%&' , Part 1, vol. 180, no. 15, 1966, page numbers 371-378.
Ting, Y. ; Tosunoglu, S. & Tesar, D. (1993). A control structure for fault-tolerant operation of
robotic manipulators, ?&+>%%@#"/'* +,* ABBB* A"(%&"2(#+"28* C+",%&%">%* +"* D+=+(#>'* 2"@*
59(+.2(#+" , , pp. 684-690, Atlanta, GA., May 1993.
Ukidve, C. S. ; McInroy, J. E. & Jafari, F. (2006). Orthogonal Gough-Stewart Platforms with
optimal fault tolerant manipulability, ?&+>%%@#"/'*+,*ABBB*A"(%&"2(#+"28*C+",%&%">%*+"*
D+=+(#>'*2"@*59(+.2(#+" , pp. 3801-3806, Orlando, FL., May 2006.
Wapler, M. ; Urban, V.; Weisener, T.; Stallkamp, J. ; Durr, M. & Hiller, A. (2003). A stewart
platform for precision surgery, I&2"'2>(#+"'* +,* (7%* A"'(#(9(%* +,* F%2'9&%.%"(* 2"@*
C+"(&+8, vol. 25, no. 4, 2003, page numbers 329-334.
68
Parallel Manipulators, Towards New Applications
Wen, J. T.-Y. & Wilfinger, L. S. (1999). Kinematic manipulability of general constrained rigid
multibody systems, ABBB*I&2"'2>(#+"'*+"*D+=+(#>'*2"@*59(+.2(#+" , vol. 15, no. 3, June
1999, page numbers 558-567.
Yoshikawa, T. (1985). Manipulability of robotic mechanisms, A"(%&"2(#+"28*E+9&"28*+,*D+=+(#>'*
D%'%2&>7, vol. 4, no. 2, 1985, page numbers 3-9.
4
Dynamic Model of a 6-dof Parallel Manipulator
Using the Generalized Momentum Approach
António M. Lopes and Fernando Almeida
UISPA – Unidade de Integração de Sistemas e Processos Automatizados,
Universidade do Porto, Faculdade de Engenharia
Portugal
1. Introduction
The dynamic model of a mechanical system relates the time evolution of its configuration
(position, velocity and acceleration) with the forces and torques acting upon it. The inverse
dynamic model is important for system control while the direct model is used for system
simulation.
Serial structure manipulator dynamic modelling is a well established subject. So, recent
developments have been oriented towards the improvement of numerical efficiency
enabling their use in real-time control algorithms (Lilly, 1993; Naudet, 2003; Mata, 2002; Lee,
2005; Featherstone, 2000). Parallel structure manipulators present a more complex problem,
and, usually, the model algorithms cannot be generalized. When used in a real-time control
framework the resulting models must be simplified as they usually demand a very high
computational effort.
The dynamic model of a parallel manipulator when operated in free space can be
mathematically represented, in the Cartesian space, by a system of nonlinear differential
equations that may be written in matrix form as
I(x)⋅ x + V(x x
, )⋅ x + G(x) = f
(1)
I(x) being the inertia matrix, V(x x
, ) the Coriolis and centripetal terms matrix, G(x) a vector
of gravitational generalized forces, x the generalized position of the mobile platform or end-
effector and f the controlled generalized force applied on the end-effector:
f = J T (x)⋅ τ
(2)
where τ is the generalized force developed by the actuators and J(x) is a jacobian matrix.
The dynamic model of a parallel manipulator is usually developed following one of two
approaches (Callegari, 2006): the Newton-Euler or the Lagrange methods. The Newton-
Euler approach uses the free body diagrams of the rigid bodies. The Newton-Euler equation
is applied to each single body and all forces and torques acting on it are obtained. Do and
Yang, and Reboulet and Berthomieu use this method on the dynamic modelling of a Stewart
platform (Do & Yang, 1988; Reboulet & Berthomieu, 1991). They achieve their result
introducing some simplifications on the legs models. Ji (Ji, 1994) presents a study on the
70
Parallel Manipulators, Towards New Applications
influence of leg inertia on the dynamic model of a Stewart platform. Mouly (Mouly, 1993)
presents a simplified model for a variation of the Stewart platform, only taking into account
the mobile platform. Dasgupta and Mruthyunjaya used the Newton-Euler approach to
develop a closed-form dynamic model of the Stewart platform (Dasgupta & Mruthyunjaya,
1998). This method was also used by other researchers (Dasgupta & Choudhury, 1999;
Khalil & Ibrahim, 2007; Riebe & Ulbrich, 2003; Khalil & Guegan, 2004; Guo & Li, 2006;
Carvalho & Ceccarelli, 2001).
The Lagrange method describes the dynamics of a mechanical system from the concepts of
work and energy. This method enables a systematic approach to the motion equations of
any mechanical system. Nguyen and Pooran use this method to model a Stewart platform,
modelling the legs as point masses (Nguyen & Pooran, 1989). Other researchers follow an
approach similar to the one used by Nguyen and Pooran, but trying to increase the physical
meaning of the obtained mathematical expressions (Liu et al., 1993; Lebret et al., 1993). Geng
and co-authors (Geng et al., 1992) used the Lagrange’s method to develop the equations of
motion for a class of Stewart platforms. Some simplifying assumptions regarding the
manipulator geometry and inertia distribution were considered. Lagrange’s method was
also used by others (Bhattacharya et al., 1998; Gregório & Parenti-Castelli, 2004; Caccavale et
al., 2003).
Unfortunately the dynamic models obtained from these classical approaches usually present
high computational loads. Therefore, alternative methods have been searched, namely the
ones based on the principle of virtual work (Wang & Gosselin, 1998; Tsai, 2000; Li & Xu,
2005; Staicu et al., 2007), and screw theory (Gallardo et al., 2003).
In this paper the authors present a new approach to the problem of obtaining the dynamic
model of a six degrees-of-freedom (dof) parallel manipulator: the use of the generalized
momentum concept.
The manipulator under study may be seen as a variation of the Stewart platform, with the
uniqueness of having all its actuators fixed to the base platform and only moving in a
direction perpendicular to that base (Merlet & Gosselin, 1991). A prototype of this
manipulator, the Robotic Controlled Impedance Device (RCID), was developed aiming a
broad set of force-impedance control tasks. The obtained dynamic model requires a
considerably lower computational effort than the one resulting from the use of classical
Lagrange method.
This paper is organized as follows. Section 2 describes the RCID parallel manipulator.
Section 3 presents the manipulator dynamic model using the generalized momentum
approach. In section 4 the computa