NASA Technical Reports Server (NTRS) 20080000843: Corrigendum: New Form of Kane's Equations of Motion for Constrained Systems

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Corrigendum: New Form of Kane’s Equations of
Motion for Constrained Systems
Carlos M. Roithmayr∗
NASA Langley Research Center, Hampton, Virginia 23681
Abdulrahman H. Bajodah†
King Abdulaziz University, Jeddah, Saudi Arabia 21589
Dewey H. Hodges‡, and Ye-Hwa Chen§
Georgia Institute of Technology, Atlanta, Georgia 30332
Introduction
Acorrectiontothepreviouslypublishedarticle“NewFormofKane’sEquationsofMotionfor
Constrained Systems”1 is presented. Misuse of the transformation matrix between time rates of
changeofthegeneralizedcoordinatesandgeneralizedspeeds(sometimescalledmotionvariables)
resulted in a false conclusion concerning the symmetry of the generalized inertia matrix. The
generalized inertia matrix (sometimes referred to as the mass matrix) is in fact symmetric and
usually positive definite when one forms nonminimal Kane’s equations for holonomic or simple
nonholonomic systems, systems subject to nonlinear nonholonomic constraints, and holonomic
or simple nonholonomic systems subject to impulsive constraints according to Refs. 1, 2, and
3, respectively. The mass matrix is of course symmetric when one forms minimal equations for
holonomicorsimplenonholonomicsystemsusingKane’smethodassetforthinRef.4.
∗AerospaceEngineer,SpaceMissionAnalysisBranch,MailStop462,E-mail: [email protected]. Se-
niorMember,AIAA.
†AssistantProfessor,AeronauticalEngineeringDepartment,E-mail: [email protected].
‡Professor,SchoolofAerospaceEngineering,E-mail: [email protected]. Fellow,AIAA.
§Professor,GeorgeW.WoodruffSchoolofMechanicalEngineering,E-mail: [email protected].
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Symmetry of the Generalized Inertia Matrix
LetS beasimplenonholonomicdynamicalsystem,andletRbeaninertialframeofreference
in which the configuration of S is described by a set of n generalized coordinates q ,...,q . The
1 n
velocityofagenericparticleP ofthesystemrelativetoRcanbewrittenas
n
X
RvP = RvP(q,t)u +RvP(q,t) (1)
r r t
r=1
where the generalized speeds u ,...,u are scalar variables satisfying some nonholonomic con-
1 n
straint relations linear in u ,...,u , and RvP,...,RvP are the corresponding holonomic partial
1 n 1 n
velocitiesofthesystem. Thegeneralizedspeedsalsosatisfythekinematicaldifferentialequations
q˙ = C(q,t)u+D(q,t) (2)
Thesystem’skineticenergyrelativetoRisgivenby1
1
K = uTM(q,t)u+N(q,t)u+R(q,t) (3)
2
where M ∈ Rn×n is symmetric, N ∈ R1×n, and R ∈ R. In most cases M is positive definite;
it can be nonnegative definite (positive semidefinite) when, for example, one neglects a central
principalmomentofinertiaofarigidbodybelongingtoS,suchasaslenderrod,orwhenoneuses
a particle with no mass to represent a point whose motion must be known. The rth nonholonomic
generalizedinertiaforceF˜? canbewrittenintermsoftheholonomicgeneralizedinertiaforcesF?
r r
andintermsofK accordingtoEqs.(4.11.4)and(5.6.6)respectivelyofRef.4,
n−p
X
F˜? = F? + F? A
r r p+s sr
s=1
!
n (cid:20) (cid:18) (cid:19) (cid:21) n−p
X d ∂K ∂K X
= − − W + W A (r = 1,...,p) (4)
sr s,p+k kr
dt ∂q˙ ∂q
s s
s=1 k=1
wheretheelementsofW havethesamemeaningsasinEqs.(2.14.5)ofRef.4,
q˙ = Wu+X (5)
The matrix W is thus in fact identical to C(q,t); it is wrongly taken to be C−1(q,t) following Eq.
(27) in Ref. 1, leading to errors in Eqs. (31), (32), (35), and (36), and to the incorrect conclusion
that the generalized inertia matrix Q can be asymmetric. A rectification of the errors is provided
inwhatfollows.
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The observation that generalized inertia forces are linear in u˙ is made in Ref. 1 and will be
revisited shortly. Consequently F?, a column matrix whose elements are F?, can be written in the
r
form
F? = −Q(q,t)u˙ −L(q,u,t) (6)
In Ref. 1 the nature of matrix Q is considered by using the kinetic energy of the system. Upon
appealingtoEqs.(3)and(5)itcanbeseenthat
∂K ∂K ∂u
(cid:2) (cid:3)
= = uTM +N W−1 (7)
∂q˙ ∂u ∂q˙
Therefore,definingthematrixA asinEq.(16)ofRef.1
2
h i
A = I AT (8)
2
yieldsthefollowingmatrixrepresentationofEqs.(4)
(cid:18) (cid:19)
d
(cid:2) (cid:3)
A F? = −A WT W−TMu+W−TNT −K T
2 2 q
dt
(cid:18) (cid:19)
d d
(cid:2) (cid:3) (cid:2) (cid:3)
= −A WT W−TMu˙ + W−TM u+ W−TNT −K T (9)
2 q
dt dt
where
∂K
K = bK ...K c = (10)
q q1 qn ∂q
IfEq.(6)ismultipliedbyA andcomparedwithEq.(9),weobtain
2
Q =4 WTW−TM = M (11)
(cid:18) (cid:19)
d d
L =4 WT (cid:2)W−TM(cid:3) u+ (cid:2)W−TNT(cid:3)−K T (12)
q
dt dt
Equations (11) and (12) correct Eqs. (35) and (36) of Ref. 1. The foregoing analysis shows that
the generalized inertia matrix Q is symmetric and, as noted previously, usually positive definite.
InadditiontoitsroleinRef.1,QplaysacentralpartinformingnonminimalKane’sequationsfor
systemssubjecttotheconstraintstreatedinRefs.2and3.
Alternative Demonstration of Symmetry
It is noted in Ref. 4 that frequently it is inefficient to use K to construct F? or F˜?; likewise,
r r
readers of Ref. 1 are made aware that the use of K to form Q or L is undesirable. The definition
of F? given in Eqs. (4.11.2) and (4.11.3) of Ref. 4 provides alternative means for showing that Q
r
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isidenticaltoM,andforconstructingM andL. WhenS ismadeupofparticlesP ,...,P ,
1 ν
ν
F? =4 −Xm RvPi · RaPi (r = 1,...,n) (13)
r i r
i=1
wherem isthemassofP ,andwheretheaccelerationofP inR,denotedby RaPi,canbewritten
i i i
in terms of partial velocities by differentiating Eq. (1) with respect to t in R as shown in Refs. 5
and2.
" ! #
Xn Rd Rd
RaPi = RvPiu˙ + RvPi u + RvPi
r r dt r r dt t
r=1
n
X
= RvPiu˙ + RaPi (i = 1,...,ν) (14)
r r t
r=1
wheretheremaindertermoftheaccelerationisdefinedas
!
RaPi =4 Xn Rd RvPi u +Rd RvPi (i = 1,...,ν) (15)
t dt r r dt t
r=1
SubstitutionfromEqs.(14)into(13)yields
!
ν n
X X
F? = − m RvPi · RvPiu˙ + RaPi
r i r s s t
i=1 s=1
!
n ν ν
X X X
= − m RvPi · RvPi u˙ − m RvPi · RaPi (r = 1,...,n) (16)
i r s s i r t
s=1 i=1 i=1
It is immediately clear that F? is linear in u˙ ,...,u˙ ; each time derivative of a generalized
r 1 n
speedismultipliedbyaninertiacoefficientdefinedas
ν
m =4 Xm RvPi · RvPi = m (r,s = 1,...,n) (17)
rs i r s sr
i=1
A matrix M whose elements are m is referred to as a generalized inertia matrix or mass matrix.
rs
Taken together, Eqs. (16) and (17) show conclusively that M is symmetric by definition when
obtainedwithKane’smethodassetforthinRef.4. Althoughthepresentdemonstrationinvolvesa
holonomic system, the result is readily shown to apply also to a simple nonholonomic system; the
interested reader is referred to Eqs. (5.5.3) and (5.5.4), and Problem 10.11 in Ref. 4. The form of
m isdependentonone’schoiceofgeneralizedspeedsbecausethepartialvelocitiesaredependent
rs
onthischoice;however,M issymmetricforanyvalidchoicewhatsoever. ThematrixM isinfact
identicaltoQthatappearsinEq.(6)andthusasymmetricmassmatrixappearsinthenonminimal
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Kane’s equations presented in Refs. 1, 2, and 3. The property of symmetry does not necessarily
extendtoamassmatrixobtainedwithothervariantsofKane’smethod,suchasthelowertriangular
matrixobtainedwiththemethodpresentedinRef.6.
It is important to note that not all methods of deriving dynamical equations guarantee a sym-
metric mass matrix. For example, it is well known that Newton-Euler methods do not guarantee
symmetry. Although Lagrange’s equations always produce a symmetric mass matrix, the same
cannotbesaidingeneralfortheircousins,theBoltzmann-Hamelequations.7
As indicated on p. 150 of Ref. 4, M is the matrix in Eq. (3) associated with the portion K of
2
kineticenergythatisquadraticinthegeneralizedspeeds
n n
1 1 XX
K = uTMu = m u u (18)
2 rs r s
2 2
r=1 s=1
4
A comparison of Eqs. (16) and (6) reveals that the elements of matrix L are given by L =
r
Pν m RvPi · RaPi (r = 1,...,n). As pointed out in Ref. 1, it is advisable to construct F?
i=1 i r t r
according to the procedure set forth in Sec. 4.11 of Ref. 4 rather than by forming the matrices M
andL.
References
1Bajodah,A.H.,Hodges,D.H.,andChen,Y.-H.,“ANewFormofKane’sEquationsofMotionforConstrained
Systems,”JournalofGuidance,Control,andDynamics,Vol.26,No.1,2003,pp.79–88.
2Bajodah, A. H., Hodges, D. H., and Chen, Y.-H., “Nonminimal Kane’s Equations of Motion for Multibody
Dynamical Systems Subject to Nonlinear Nonholonomic Constraints,” Multibody System Dynamics, Vol. 14, No. 2,
2005,pp.155–187.
3Bajodah,A.H.,Hodges,D.H.,andChen,Y.-H.,“NonminimalGeneralizedKane’sImpulse-MomentumRela-
tions,”JournalofGuidance,Control,andDynamics,Vol.27,No.6,2004,pp.1088–1092.
4Kane,T.R.andLevinson,D.A.,Dynamics:TheoryandApplications,McGraw-HillBookCompany,NewYork,
1985,pp.46,124–129,150,151,153,319.
5Rosenthal,D.E.,“OrdernFormulationforRoboticSystems,”TheJournaloftheAstronauticalSciences,Vol.38,
No.4,1990,pp.511–529.
6Rosenthal,D.E.,“TriangularizationofEquationsofMotionforRoboticSystems,”JournalofGuidance,Control,
andDynamics,Vol.11,No.3,1988,pp.278–281.
7Meirovitch,L.,MethodsofAnalyticalDynamics,McGraw-HillBookCompany,NewYork,1970,pp.162–163.
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