Embed Size (px)
Citation preview
This article was downloaded by: [University of Tennessee At Martin]On: 04 October 2014, At: 04:56Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
International Journal of ControlPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tcon20
Asymptotic perturbed feedback linearisation ofunderactuated Euler's dynamicsAbdulrahman H. Bajodah aa Aeronautical Engineering Department , King Abdulaziz University , P.O. Box 80204, Jeddah21589, Saudi ArabiaPublished online: 06 Aug 2009.
To cite this article: Abdulrahman H. Bajodah (2009) Asymptotic perturbed feedback linearisation of underactuated Euler'sdynamics, International Journal of Control, 82:10, 1856-1869, DOI: 10.1080/00207170902788613
To link to this article: http://dx.doi.org/10.1080/00207170902788613
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
International Journal of ControlVol. 82, No. 10, October 2009, 1856–1869
Asymptotic perturbed feedback linearisation of underactuated Euler’s dynamics
Abdulrahman H. Bajodah*
Aeronautical Engineering Department, King Abdulaziz University,P.O. Box 80204, Jeddah 21589, Saudi Arabia
(Received 13 March 2008; final version received 29 January 2009)
A unifying methodology is introduced for smooth asymptotic stabilisation of underactuated rigid body dynamicsunder one and two degrees of actuation. The methodology is based on the concept of generalised inversion,and it aims to realise a perturbation from the unrealisable feedback linearising transformation. A desired lineardynamics in a norm measure of the angular velocity components about the unactuated axes is evaluated alongsolution trajectories of Euler’s underactuated dynamical equations resulting in a linear relation in the controlvariables. This relation is used to assess asymptotic stabilisability of underactuated rigid bodies with arbitraryvalues of inertia parameters, and generalised inversion of the relation produces a control law that consistsof particular and auxiliary parts. The generalised inverse in the particular part is scaled by a dynamic factor suchthat it uniformly converges to the Moore–Penrose inverse, and the null-control vector in the auxiliary partis chosen for asymptotically stable perturbed feedback linearisation of the underactuated system.
Keywords: asymptotic stabilisation; perturbed feedback linearisation; underactuated rigid body stabilisation;controls coefficient generalised inversion; dynamic scaling factor; perturbed nullprojection
1. Introduction
Controllability of underactuated rigid bodies underdifferent degrees and types of actuation, initiallyinvestigated in the seminal article by Crouch (1984),has continued to draw attention within control systemscommunities. The cascaded nature of Eulers modelfor angular motion allows to divide the problem ofrigid body analysis and control into two parts. The firstpart, the focus of this article, deals with the dynamicsof the rigid body and aims at stabilising its bodyangular velocity components. The second part dealswith the kinematics of the rigid body and aims atdriving its attitude variables to their desired values.
Research on analysis and control of underactuatedrigid body dynamics has taken different approaches,including differential geometric approaches, e.g.Brockett (1983) and Aeyels (1985), and approachesbased on Lyapunov methods and energy principles,e.g. Aeyels and Szafranski (1988), Sontang andSussmann (1988), Bloch and Marsden (1990), Outbiband Sallet (1992), and Andriano (1993). An extensivesurvey on the corresponding results is found inTsiotras and Doumtchenko (2000).
Complexity of underactuated system control can besubstantially reduced if the system of motion variablescan be transformed to a different coordinate systemin which the control design is easier to carry out;
see, for example, Tsiotras and Longuski (1994) and
Astolfi (1996). A convenient type of coordinate system
transformations that is well known in the control
literature is the feedback linearising transformation.
However, an underlying feature of underactuated
dynamics is that it is uncontrollable via feedback
linearisation; see Isidori (1995; p. 231) and Wichlund,
Sordalen, and Egeland (1995). The reason is that the
complex geometrical structure of underactuated
dynamics is not tolerated by the usual notion of
dynamic inversion due to the dimensionality and rank
limitations that are associated with the notion.Very few extensions of feedback linearisation to
underactuated system dynamics have been reported.
Among these extensions is the non-smooth version
of non-regular feedback linearisation (Sun and Ge
2003), implementable on specific non-holonomic
system structures. Another extension is partial feedback
linearisation (Spong 1994), made by decomposing the
dynamical system into actuated and unactuated por-
tions. The methodology utilises usual square inversion
to linearise the actuated portion, and alternatively
under restrictive assumptions utilises generalised
inversion to linearise the unactuated portion.Nevertheless, the common factor between most
former control engineering applications of generalised
inverses is that the systems are either fully or
*Email: [email protected]
ISSN 0020–7179 print/ISSN 1366–5820 online
� 2009 Taylor & Francis
DOI: 10.1080/00207170902788613
http://www.informaworld.com
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
overactuated, i.e. the numbers of independent controlvariables are either equal or exceed the numbers ofcontrolled system degrees of freedom. In particular,generalised inverses have been utilised in controlsystem design for control variables allocation; seeDurham (1993).
A generalised inversion-based control systemdesign methodology has been introduced in Bajodah(2008). The primary tool used is the controls coefficientgeneralised inverse (CCGI) and the controls coefficientnullspace parametrisation of redundancy in controlauthority (Bajodah, Hodges, and Chen 2005). Thisarticle extends the methodology, in the context of therigid body model, to control underactuated systems.This is motivated by the fact that redundancy incontrol systems is ultimately in the control processitself rather than in the control variables, and thatcontrollable dynamical systems are dynamically redun-dant even if underactuated. That is, there exists nounique strategy to control a controllable dynamicalsystem, regardless of its degree of actuation.
The design procedure begins by partitioning theunderactuated Euler’s system of equations into actu-ated and unactuated subsystems, and a condition isprovided which guarantees capability of the availablecontrol authority to realise a desired linear dynamicsof the unactuated subsystem. The CCGI design isused thereafter to construct the control law. A CCGIcontrol law consists of auxiliary and particular parts,acting, respectively, on the complementary orthogonalsubspaces of controls coefficient nullspace and CCGIrange space. The particular part works to realise aprescribed desired liner dynamics of the unactuatedsubsystem, and the auxiliary part works to stabilisethe actuated subsystem. In particular, the null-controlvector in the auxiliary part is chosen to producean asymptotically stabilising perturbed feedbacklinearisation of the actuated subsystem.
A well-known obstacle in the way of employinggeneralised inverses of matrices having variableelements is singularity of the generalised inverse.In this article, a novel type of generalised inverses isconstructed by scaling the Moore–Penrose generalisedinverse (MPGI) (Moore 1920; Penrose 1955) witha dynamic factor that depends on the vector normof the body angular velocity components about theactuated axes. The dynamic scaling factor vanishes asthese components vanish, such that the dynamicallyscaled generalised inverse uniformly converges to thestandard MPGI, asymptotically realising the desiredstable unactuated dynamics.
The CCGI design methodology requires non-trivialcontrols coefficient nullspace to stabilise the actuateddynamics. This is a source of difficulty when applyingthe methodology to stabilise a rigid body with a single
degree of actuation, because this corresponds to
a controls coefficient nullspace of zero dimension.
The problem is solved in this article by providing
an additional fictitious degree of actuation in order to
increase the controls coefficient nullspace dimension
to two, and constraining the null-control vector in
the auxiliary part of the CCGI law to eliminate the
pseudo-actuation.The contribution of this article is twofold. First, at
the level of analysis, underactuated dynamics is shown
to be feedback linearisable up to a perturbation from
the feedback linearising transformation, and detailed
necessary and sufficient asymptotic stabilisability
conditions are derived for rigid bodies having arbitrary
inertia properties and equipped with one and two
degrees of actuation. Second, at the design level,
a unifying CCGI design methodology is applied to
stabilise rigid body underactuated dynamics with
any stabilisable combination of inertia and actuation.
The involved generalised inverse is modified by a
dynamic scaling factor yielding asymptotic stabilisa-
tion of Euler’s unactuated dynamics, and the null-
control vector is designed for asymptotically stable
perturbed feedback linearisation of Euler’s actuated
dynamics.
2. Partitioned form of underactuated Euler’s
equations of angular motion
The Euler’s model of underactuated rigid body
dynamics is given by the system of differential
equations
_! ¼ Sð!Þ!þ �, !ð0Þ ¼ !0 ð1Þ
where !2R3�1 is the vector of angular velocity
components about the body-fixed axes, S(!)2R3�3 is
given by
Sð!Þ ¼ J�1!�J ð2Þ
such that J2R3�3 is a matrix containing the body
moments of inertia, and is given by
J ¼
J11 �J12 �J13
�J12 J22 �J23
�J13 �J23 J33
264375, ð3Þ
!� is a skew symmetric matrix of the form
!� ¼
0 !3 �!2
�!3 0 !1
!2 �!1 0
264375, ð4Þ
and � 2R3�1 is the scaled control vector. Let d be
the degree of actuation of the body, i.e. number of
International Journal of Control 1857
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
independent control torque actuator pairs. The vectors
! and � can be put in the partitioned forms
! ¼ !Tu !T
a
� �T, � ¼ 0T uT
� �Tð5Þ
where !u2R(3�d)�1 is the vector of angular velocity
components about the unactuated body axes,
!a2Rd�1 is the vector of angular velocity components
about the actuated body axes, u2Rd�1 is the scaled
vector of available control torques, and 02R(3�d)�1
contains zero elements. The matrix S(!) is partitionedcompatibly as
Sð!Þ ¼S11ð!Þ S12ð!Þ
S21ð!Þ S22ð!Þ
� �ð6Þ
where S112R(3�d)�(3�d), S122R
(3�d)�d, S212Rd�(3�d),
and S222Rd�d. Hence, Euler’s underactuated system
given by (1) splits into two coupled subsystems. The
first one is unactuated and given by the equation
_!u ¼ S11ð!Þ!u þ S12ð!Þ!a ð7Þ
and the second one is fully actuated and given by the
equation
_!a ¼ S21ð!Þ!u þ S22ð!Þ!a þ u: ð8Þ
3. Global realisability of prescribed unactuated
rigid body dynamics
The unactuated dynamics given by (7) is affected
indirectly by the control vector u through !a. For the
purpose of analysing unactuated dynamics stabilisa-
bility, it is convenient to set a coordinate transforma-
tion by which u appears explicitly in the transformed
unactuated dynamics. Let �(!u): R(3�d)�1
!R be
a continuous function satisfying
�ð!uÞ ¼ 0, !u ¼ 0ð3�dÞ�1: ð9Þ
The relative degree of � with respect to u that is
inferred from the partitioned dynamics given by (7)
and (8) is two (Slotine and Li 1991). Therefore, a
desired transformation of the unactuated dynamics
must be second-order in �. Such a transformation can
be written in the general functional form
€� ¼ Lð�, _�, tÞ ð10Þ
where � is at least globally twice differentiable in !u,
continuously in the first derivative. The first time
derivative _� of �(!u) along solution trajectories
of underactuated Euler’s equations of motion (1) is
given by
_� ¼ Lf�ð!uÞ ð11Þ
where
Lf�ð!uÞ ¼@�ð!uÞ
@!f ð!Þ ð12Þ
is the first Lie derivative (Khalil 2002) of �(!u) along
f(!) :¼S(!)!. The second time derivative €� of �(!u)
along solution trajectories of underactuated Euler’s
equations of motion (1) is given by
€� ¼@Lf�ð!uÞ
@!½ f ð!Þ þ �� ¼ L2
f�ð!uÞ þ@Lf�ð!uÞ
@!� ð13Þ
¼ L2f�ð!uÞ þ
@Lf�ð!uÞ
@!au ð14Þ
where
L2f�ð!uÞ ¼
@Lf�ð!uÞ
@!f ð!Þ ð15Þ
is the second Lie derivative of �(!u) along f(!). With _�and €� given by (11) and (14), it is possible to write (10)
in the pointwise-linear form
Að!Þu ¼ Bð!Þ, ð16Þ
where A(!)2R1�d is given by
Að!Þ ¼@Lf�ð!uÞ
@!a
ð17Þ
and B(!)2R is given by
Bð!Þ ¼ �L2f�ð!uÞ þ Lð�ð!uÞ,Lf�ð!uÞ, tÞ: ð18Þ
The row vector A(!) is the controls coefficient of the
unactuated dynamics given by (10) relative to �(!u)
along vector field f(!), and the scalar B(!) is the
corresponding controls load.
Definition 3.1: The dynamics given by (10) is said
to be realisable by underactuated Euler’s system of
equations at some value of ! if there exists a control
vector u that solves (16) at that value of !. If this is truefor all !2R
3�16¼ 03�1, then the dynamics given by
(10) is said to be globally realisable by underactuated
Euler’s system of equations.
Proposition 3.2: Let A(!) be the controls coefficient
relative to �(!u) along f(!) of a dynamics given by (10)
that is globally realisable by underactuated Euler’s
system of equations (1). Then
Að!Þ ¼ 01�d , ! ¼ 03�1: ð19Þ
Proof: The existence of a vector u that solves (16) at
a specific value of ! is equivalent to the fact that B is in
the range space of A at that value of !. This is possiblefor any value of B, provided that not all elements of A
vanish at that value of !, for which the equation is said
to be consistent. Therefore, the existence of a vector
1858 A.H. Bajodah
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
!? 6¼ 03�1 such that A(!?)¼ 01�3 implies that the
dynamics given by (10) is not realisable at !?, whichviolates global realisability of the unactuated dynamics
given by (10), proving sufficiency. Necessity follows
from the fact that the elements of f are multivariable
polynomials with only quadratic elements in the
components of ! and that � has a bounded !u
gradient, so that A(03�1)¼ 01�d. œ
Definition 3.3: The zero actuated state Jacobian of
the controls coefficient is defined to be the square
matrix resulting from differentiating the controls
coefficient with respect to !a, evaluated at !a¼ 0d�1
J að!uÞ ¼@Að!Þ
@!a
� �!a¼0d�1
: ð20Þ
Proposition 3.4: The unactuated dynamics given by
(10) is globally realisable by underactuated Euler’s
system of equations (1) if and only if
det J að!uÞ½ � 6¼ 0 8!u 6¼ 0ð3�dÞ�1: ð21Þ
Proof: Taking the derivative of (16) with respect to !a
and evaluating the resulting equation at !a¼ 0d�1 gives
J au ¼@B
@!a
� �!a¼0d�1
: ð22Þ
Invertibility of the zero actuated state Jacobian of the
controls coefficient implies that a control law u can be
constructed for global realisability of the unactuated
dynamics given by (10) as
u ¼ J�1a
@B
@!a
� �!a¼0d�1
ð23Þ
which proves necessity. Now consider the non-linear
time varying system given by the equations
_za ¼ ATðzÞ ð24Þ
where z ¼ ½ zTu zTa �T, zu2R
(3�d)�1, and za2Rd�1. From
Proposition 3.2, global realisability of the unactuated
dynamics given by (10) implies that the origin za¼ 0d�1is the unique equilibrium point of the system (24)
at zu¼ 0(3�d)�1. Furthermore, since AT is a smooth
vector field, it follows from Milnor’s theorem (Milnor
1964; Isidori 1995) that it is also a global diffeomorph-
ism on R3, i.e. it has continuous partial derivatives
and an invertible zero actuated state Jacobian, proving
sufficiency. œ
Theorem 3.5: If the unactuated dynamics given by (10)
has non-singular zero actuated state Jacobian J a(!u)
of the controls coefficient A(!) along f(!) for all
!u 6¼ 0(3�d)�1, then the infinite set of all control laws
that globally realise the unactuated dynamics by under-
actuated Euler’s system of equations is given by
u ¼ Aþð!ÞBð!Þ þ Pð!Þy ð25Þ
where ‘Aþ(!)’ stands for the CCGI, and is given by
Aþð!Þ ¼
ATð!Þ
Að!ÞATð!Þ
, Að!Þ 6¼ 01�d
0d�1, Að!Þ ¼ 01�d
8><>: ð26Þ
and P(!)2Rd�d is the corresponding controls coefficient
nullprojector (CCNP), given by
Pð!Þ ¼ Id�d �Aþð!ÞAð!Þ ð27Þ
where Id�d is the d� d identity matrix, and y2Rd�1 is
an arbitrarily chosen null-control vector.
Proof: Satisfying the condition given by (21) implies
that the unactuated dynamics given by (10) is globally
realisable by underactuated Euler’s equations. From
Proposition 3.2, this global realisability implies that
A(!) 6¼ 01�d 8!u 6¼ 0(3�d)�1, at which infinite number
of solutions for the point-wise linear relation (16) exist.
Multiplying both sides of (25) by A(!) yields
Að!Þu ¼ Að!ÞAþð!ÞBð!Þ þ Að!ÞPð!Þy ð28Þ
¼ Bð!Þ ð29Þ
recovering the system given by (16). Therefore, the
expression of u given by (25) linearly parameterises
all solutions of (16) by the null-control vector y; see
Greville (1959) and Udwadia and Kalaba (1996). œ
The expression given by (25) consists of two parts.
The particular part Aþ(!)B(!) acts on the range space
of Aþ(!). The auxiliary part P(!)y acts on the
orthogonal complement space in R3, the nullspace
of A(!). Substituting control laws expressions given
by (25) in the fully actuated Euler’s subsystem (8)
yields the following parametrisation of the infinite set
of closed loop underactuated Euler’s system equations
that globally realise the dynamics given by (10):
_!u ¼ S11ð!Þ!u þ S12ð!Þ!a ð30Þ
_!a ¼ S21ð!Þ!u þ S22ð!Þ!a þAþð!ÞBð!Þ þ Pð!Þy:
ð31Þ
Any choice of the null-control vector y in the control
laws expressions given by (25) yields solution trajec-
tories that satisfy the unactuated dynamics given
by (10). However, an arbitrary choice of y may
not guarantee stability of closed loop system
equations (30) and (31).
International Journal of Control 1859
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
4. Global asymptotic stabilisability of underactuated
rigid body dynamics
The condition given by (21) can be used to assessglobal asymptotic stabilisability of the angular motionof underactuated rigid bodies. This is made byrestricting the unactuated dynamics in the transformedform given by (10) to have a unique asymptoticallystable equilibrium point at � ¼ _� ¼ 0, so that realisingthe transformed unactuated dynamics is by virtue ofthe condition given by (9) equivalent to asymptoticallystabilising the equilibrium state !u¼ 03�d of theunactuated subsystem equation (7). Expanding theexpression given by (12) for Lf�(!u) yields
Lf�ð!uÞ ¼@�ð!uÞ
@!uS11ð!Þ!u þ S12ð!Þ!að Þ: ð32Þ
Accordingly, the following expression for the controlscoefficient is obtained:
Að!Þ ¼@
@!aLf�ð!uÞ
¼@�ð!uÞ
@!u
@S11ð!Þ
@!a!u þ
@S12ð!Þ
@!a!a þ S12ð!Þ
� �ð33Þ
arriving at the zero actuated state Jacobian
J að!uÞ ¼
�@
@!a
�@�ð!uÞ
@!u
�@S11ð!Þ
@!a!u
þ@S12ð!Þ
@!a!a þ S12ð!Þ
��!a¼0d�1:
ð34Þ
The condition given by (21) becomes
det
�@
@!a
�@�ð!uÞ
@!u
�@S11ð!Þ
@!a!u þ
@S12ð!Þ
@!a!a
þ S12ð!Þ
��!a¼0d�1
6¼ 0 8!u 6¼ 0ð3�dÞ�1: ð35Þ
Nevertheless, the smooth function �(!u) is arbitraryup to the requirement given by (9), and there is no lossof generality in specifying it in (35) to be any functionthat satisfies that requirement, e.g.
�ð!uÞ ¼1
2!Tu!u: ð36Þ
Therefore, the condition given by (35) can be rewrittenin the following form that is independent of �(!u):
det
�@
@!a
�!Tu
�@S11ð!Þ
@!a!u þ
@S12ð!Þ
@!a!a
þ S12ð!Þ
��!a¼0d�1
6¼ 0 8!u 6¼ 0ð3�dÞ�1 ð37Þ
which is necessary and sufficient for global asymptoticstabilisability of underactuated Euler’s dynamics.
This condition is used next to assess all possible inertia
properties of underactuated rigid bodies for global
asymptotic stabilisability.
4.1 Case 1: two degrees of actuation (d^ 2)
The actuated body axes in this case are the ones about
which the angular velocity components are !2 and !3,
and the unactuated body axis is the one about which
the angular velocity component is !1. Therefore,
S112R, S122R1�2, S212R
2�1, S222R2�2, !u¼!1,
!a¼ [!2 !3]T, and � ¼ ½ 0 uT �T, where u2R
2�1.
The condition given by (37) for an arbitrary fully
populated moments of inertia matrix J isn��J12ðJ12J33 þ J13J23Þ þ J22ðJ22J33 � J223Þ
� J13ðJ12J23 þ J13J22Þ � J33ðJ22J33 � J223Þ�2
þ 4�J12ðJ12J23 þ J13J22Þ � J23ðJ22J33 � J223Þ
���J13ðJ12J33 þ J13J23Þ
� J23ðJ22J33 � J223Þ�o w2
1
D26¼ 0 8!1 6¼ 0 ð38Þ
where
D ¼ J13ðJ12J23 þ J13J22Þ þ J23ðJ11J23 þ J12J13Þ
� J33ðJ11J22 � J212Þ:
The condition given by (38) reduces to�J12ðJ12J33 þ J13J23Þ þ J22ðJ22J33 � J223Þ
� J13ðJ12J23 þ J13J22Þ � J33ðJ22J33 � J223Þ�2
� 4�J12ðJ12J23 þ J13J22Þ � J23ðJ22J33 � J223Þ
���J13ðJ12J33 þ J13J23Þ � J23ðJ22J33 � J223Þ
�6¼ 0:
ð39Þ
Therefore, the rigid body is globally asymptotically
stabilisable by two torque actuators that are mounted
in a common body fixed plane or in distinct body fixed
planes if and only if the condition given by (39) is
satisfied. In particular, the rigid body is globally
asymptotically stabilisable by two torque actuators
that are mounted along two axes belonging to
a principal system of axes, i.e. J12¼ J23¼ J13¼ 0,
unless the third (unactuated) principal axis is an axis
of inertial symmetry, i.e. J22¼ J33. This result was first
obtained in Brockett (1983) by means of control
Lyapunov function construction. Independency of the
condition given by (39) of !1 implies that if the under-
actuated rigid body is locally asymptotically stabilisa-
ble then it is also globally asymptotically stabilisable.
1860 A.H. Bajodah
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
4.2 Case 2: one degree of actuation (d^ 1)
The actuated body axis in this case is the one about
which the angular velocity component is !3, and the
unactuated body axes are the ones about which
the angular velocity components are !1 and !2.
Therefore, S112R2�2, S122R
2�1, S212R1�2, S222R,
!u ¼ ½!1 !2 �T, !a¼!3, and � ¼ ½ 0 0 u �T, where
u2R. The condition given by (37) for an arbitrary fully
populated moments of inertia matrix J is
� 2�J13ðJ12J33 þ J13J23Þ � J23ðJ22J33 � J223Þ
�w1
D
� 2�J23ðJ12J33 þ J13J23Þ � J13ðJ11J33 � J213Þ
�w2
D6¼ 0
8!u 6¼ 02�1 ð40Þ
where D is the same constant in (38). The condition
reduces to the following two mutually exclusive
conditions for global asymptotic stabilisability:
J13ðJ12J33 þ J13J23Þ � J23ðJ22J33 � J223Þ 6¼ 0 ð41Þ
and
J23ðJ12J33 þ J13J23Þ � J13ðJ11J33 � J213Þ 6¼ 0: ð42Þ
Independency of the above conditions of !1 and !2
implies that if the underactuated rigid body is locally
asymptotically stabilisable then it is also globally
asymptotically stabilisable. The following facts follow:
(1) The rigid body is not asymptotically stabilisable
by a single torque actuator that is mounted on
a principal axis, i.e. if J13¼ J23¼ 0.(2) If a single torque actuator is mounted in
a principal plane such that J12¼ J13¼ 0, then
the rigid body is globally asymptotically
stabilisable if and only if J23ðJ22J33 � J223Þ 6¼ 0.(3) If a single torque actuator is mounted in
a principal plane such that J12¼ J23¼ 0, then
the rigid body is globally asymptotically
stabilisable if and only if J13ðJ11J33 � J213Þ 6¼ 0.(4) If a single torque actuator is mounted such
that J13¼ 0 and if J223 ¼ J22J33, then the rigid
body is globally asymptotically stabilisable if
and only if J12 6¼ 0.(5) If a single torque actuator is mounted such
that J23¼ 0 and if J213 ¼ J11J33, then the rigid
body is globally asymptotically stabilisable if
and only if J12 6¼ 0.
The argument of item (1) above was obtained in Aeyels
and Szafranski (1988) for the special case that
J12¼ J23¼ J13¼ 0 by noticing that
Hð!Þ ¼J33 � J11
J22!21 �
J22 � J33J11
!22
is a first integral of underactuated Euler’s system,
i.e. the level surfaces defined by constant values of
H(!) are invariant, precluding asymptotic stabilisa-
bility. Moreover, the above items (2) and (3) answer
the question raised in Aeyels and Szafranski (1988)
on asymptotic stabilisability of a rigid body with one
torque perpendicular to a principal axis, i.e. axis 1 for
item (2) and axis 2 for item (3).
5. Dynamically scaled generalised inversion
Let the function �(!u) : R(3�d)�1
!R be globally twice
continuously differentiable and satisfy the condition
given by (9). Proposition 3.2 implies that if the
dynamics given by (10) is globally realisable by
underactuated Euler’s system of equations, then
lim�ð!uÞ!0
Að!Þ ¼ 01�d: ð43Þ
Accordingly, the definition of the Aþ(!) given by (26)
implies that for any non-zero initial condition
!u(0)¼ 0(3�d)�1,
lim�ð!uÞ!0
Aþð!Þ ¼ 1d�1: ð44Þ
That is, Aþ(!) must go unbounded as the body
detumbles. This is a source of instability for the
closed loop system because it causes the control law
expression given by (25) to become unbounded. One
solution to this problem is made by switching the value
of the CCGI according to (26) to Aþ(!)¼ 0d�1 when
the controls coefficient A(!) approaches singularity,
which is equivalent to deactivating the particular
part of the control law as the closed loop system
reaches steady state (Bajodah 2007). To avoid such
a discontinuity in the control law, the growth-
controlled dynamically scaled generalised inverse is
introduced next.
5.1 Dynamically scaled generalised inverse
Definition 5.1: The DSGI Aþs ð!Þ 2 Rd�1 is given by
Aþs ð!Þ ¼
ATð!Þ
Að!ÞATð!Þ þ k!ak
pp
ð45Þ
where k!akp is the vector p norm of !a for some
positive dynamic scaling integer index p.
5.2 Properties of the dynamically scaled
generalised inverse
The following properties can be verified by direct
evaluation of the CCGI Aþ(!) given by (26) and its
International Journal of Control 1861
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
dynamic scaling Aþs ð!Þ given by (45):
(1) Aþs ð!ÞAð!ÞAþð!Þ ¼Aþð!ÞAð!ÞAþs ð!Þ ¼A
þs ð!Þ:
(2) ðAþs ð!ÞAð!ÞÞT¼ A
þs ð!ÞAð!Þ:
(3) limk!akp!0Aþs ð!Þ ¼ A
þð!Þ.
Replacing Aþ(!) by Aþs ð!Þ in the particular part
of the control law given by (25) yields the control law
u ¼ Aþs ð!ÞBð!Þ þ Pð!Þy ð46Þ
and the closed loop actuated subsystem
_!a ¼ S21ð!Þ!u þ S22ð!Þ!a þAþs ð!ÞBð!Þ þ Pð!Þy:
ð47Þ
Therefore, if the null-control vector y is designed
such that
limt!1k!akp ¼ 0 ð48Þ
then property (3) implies that _!a given by (47)
converges to its expressions given by (31), resulting
in asymptotic realisation of the unactuated dynamics
given by (10).
6. Perturbed feedback linearisation
An interesting choice of Lð�, _�, tÞ in (10) is the
feedback linearising
Lð�, _�, tÞ ¼ �c1 _�� c2�, c1, c2 4 0 ð49Þ
resulting in the asymptotically stable linear time-
invariant unactuated dynamics
€�þ c1 _�þ c2� ¼ 0: ð50Þ
The corresponding expression for the controls load
B(!) relative to �(!u) along vector field f(!) is
Bð!Þ ¼ �L2f�ð!uÞ � c1Lf�ð!uÞ � c2�ð!uÞ: ð51Þ
6.1 Perturbed nullprojection
A fundamental property of nullprojection matrices
is that they are rank deficient. To facilitate the present
development, the full rank perturbed controls coeffi-
cient nullprojector is constructed by perturbing the
controls coefficient nullprojection matrix to disencum-
ber its rank deficiency.
Definition 6.1: The perturbed CCNP ePð!, �Þ is
defined as ePð!, �Þ :¼ Id�d � hð�ÞAþð!ÞAð!Þ ð52Þ
where h(�): R!R is any continuous function such that
hð�Þ ¼ 1 if and only if � ¼ 0: ð53Þ
Proposition 6.2: The perturbed CCNP ePð!, �Þ is of
full rank for all � 6¼ 0.
Proof: The singular value of A(!) is given by
�ðAð!ÞÞ ¼ ½Að!ÞATð!Þ�ð
12Þ ¼kAð!Þ k2 : ð54Þ
Therefore, the singular value decomposition of A(!) isgiven by
Að!Þ ¼ Uð!ÞDð!ÞVTð!Þ ð55Þ
where U(!)¼ 1 and
Dð!Þ ¼hk Að!Þ k2 01�ðd�1Þ
ið56Þ
and V(!)2Rd�d is orthonormal. By inspecting the four
conditions identifying the MPGI, it can be easily
verified that it is given for A(!) by
Aþð!Þ ¼ Vð!ÞDþð!Þ ð57Þ
where 'þ(!) is the MPGI of '(!)
Dþð!Þ ¼
1
k Að!Þ k201�ðd�1Þ
� �T: ð58Þ
Therefore,
Aþð!ÞAð!Þ ¼ Vð!ÞDþð!ÞDð!ÞVTð!Þ: ð59Þ
The right-hand side of (59) is a singular value
decomposition of Aþ(!)A(!), where the diagonal
matrix 'þ(!)'(!) contains the singular values of
Aþ(!)A(!) as its diagonal elements
Dþð!Þ�ð!Þ ¼
1 01�ðd�1Þ
0ðd�1Þ�1 0ðd�1Þ�ðd�1Þ
� �: ð60Þ
Consequently,
ePð!, �Þ ¼ Id�d � hð�ÞAþð!ÞAð!Þ ð61Þ
¼ Id�d � hð�ÞVð!ÞDþð!ÞDð!ÞVTð!Þ ð62Þ
¼ Vð!Þ½Id�d � hð�ÞDþð!ÞDð!Þ�VTð!Þ ð63Þ
¼ Vð!Þ1� hð�Þ 01�ðd�1Þ
0ðd�1Þ�1 Iðd�1Þ�ðd�1Þ
" #VTð!Þ ð64Þ
which is of full rank for all � 6¼ 0. œ
Proposition 6.3: The controls coefficient nullprojector
P(!) commutes with its inverted perturbation eP�1ð!, �Þfor all � 6¼ 0. Furthermore, their matrix multiplication
equals to the controls coefficient nullprojector itself, i.e.
Pð!ÞeP�1ð!, �Þ ¼ eP�1ð!, �ÞPð!Þ ¼ Pð!Þ: ð65Þ
1862 A.H. Bajodah
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
Proof: The first part of the identities follows from
the Sherman–Morrison–Woodbury’s matrix inversion
lemma; see Duncan (1944) and Bernstein (2005; p. 45):
ðAþ BCDÞ�1 ¼ A�1 � A�1BðC�1 þDA�1BÞ�1DA�1
ð66Þ
with A¼ Id�d, B¼�h(�)Id�d, C¼ Id�d, and
D¼Aþ(!)A(!). The second part of identities (65) is
obtained by interchanging the definitions of B and D in
the lemma and proceeding in the same manner; see
Bajodah (2008) for a detailed proof. œ
6.2 Perturbed feedback linearisation
Let the null-control vector y be chosen as
y ¼ K!a � S21ð!Þ!u � S22ð!Þ!a �Aþð!ÞBð!Þ ð67Þ
where K2Rd�d is strictly stable. The resulting closed
loop actuated subsystem given by (31) is
_!a ¼ S21ð!Þ!uþS22ð!Þ!aþAþð!ÞBð!Þ
þPð!Þ K!a�S21ð!Þ!u�S22ð!Þ!a�Aþð!ÞBð!Þ
� �:
ð68Þ
Nevertheless, identities (65) imply that (68) can be
rewritten as
_!a ¼S21ð!Þ!uþS22ð!Þ!aþAþð!ÞBð!ÞþPð!ÞeP�1ð!,�Þ
�
�K!a�S21ð!Þ!u�S22ð!Þ!a�A
þð!ÞBð!Þ
�:
ð69Þ
Therefore, the closed loop system in the transformed
form given by (50) and (69) is a perturbation from the
linear system
€�þ c1 _�þ c2� ¼ 0, _!a ¼ K!a ð70Þ
obtained by replacing P(!) by ePð!, �Þ in (69). Linear
transformations like the one given by (70) are
known to be unrealisable by underactuated dynamics
(Brockett 1983).
7. Asymptotic perturbed feedback linearisation
Let the null-control vector y be
y ¼ K!a � S21ð!Þ!u � S22ð!Þ!a �Aþs ð!ÞBð!Þ: ð71Þ
A control law of the form given by (46) becomes
u ¼ Aþs ð!ÞBð!Þ
þ Pð!Þ K!a � S21ð!Þ!u � S22ð!Þ!a �Aþs ð!ÞBð!Þ
� �ð72Þ
and the corresponding closed loop actuated subsystem
equation (47) is
_!a¼S21ð!Þ!uþS22ð!Þ!aþAþs ð!ÞBð!Þ
þPð!Þ K!a�S21ð!Þ!u�S22ð!Þ!a�Aþs ð!ÞBð!Þ
� �:
ð73Þ
Furthermore, identities (65) imply that (73) can be
written as
_!a¼S21ð!Þ!uþS22ð!Þ!aþAþs ð!ÞBð!ÞþPð!Þ
eP�1ð!,�Þ�
�K!a�S21ð!Þ!u�S22ð!Þ!a�A
þs ð!ÞBð!Þ
�:
ð74Þ
Therefore, the closed loop system in the transformed
form given by (50) and (74) is a perturbation from
the linear system given by (70), obtained as
limt!1As(!)¼A(!) by replacing P(!) with ePð!, �Þin (74). Finally, property (1) of the DSGI implies that
the last term in (72) is
�Pð!ÞAþs ð!ÞBð!Þ ¼ Aþs ð!ÞBð!Þ �A
þð!ÞAð!ÞAþs ð!ÞBð!Þ
ð75Þ
¼ Aþs ð!ÞBð!Þ �A
þs ð!ÞBð!Þ ð76Þ
¼ 0d�1: ð77Þ
Therefore, the expression given by (72) for u becomes
u ¼ Aþs ð!ÞBð!Þ þ Pð!Þ K!a � S21ð!Þ!u � S22ð!Þ!a½ �:
ð78Þ
Comparing (78) with (46) yields the perturbed feed-
back linearisation null-control vector
yp ¼ K!a � S21ð!Þ!u � S22ð!Þ!a: ð79Þ
Theorem 7.1: Let �(!u) be globally twice continuously
differentiable and satisfy the condition given by (9),
A(!) and B(!) are, respectively, the controls coefficient
and controls load relative to �(!u) of the linear
unactuated dynamics given by (50) along f(!), and
u ¼ Aþs ð!ÞBð!Þ þ Pð!Þyp ð80Þ
where Aþs ð!Þ is given by (45) for some p norm, P(!) isgiven by (27), and yp is given by (79). If the zero actuated
state Jacobian of A(!) satisfies (21), then for every
neighbourhood D! of the origin !¼ 03�1 there exists
a strictly stable matrix gain K2Rd�d that renders the
closed loop underactuated Euler’s system given by (7)
and (8) asymptotically stable with a domain of
attraction D!.
International Journal of Control 1863
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
Proof: With the feedback control law given by (80),
the closed loop actuated subsystem given by (8)
becomes
_!a ¼ S21ð!Þ!u þ S22ð!Þ!a þAþs ð!ÞBð!Þ þ Pð!Þyp:
ð81Þ
The linearisation of AT(!) about !a¼ 0d�1 via Taylor
series expansion is
ATl ð!Þ ¼
@ATð!Þ
@!a
� �!a¼0d�1
!a ¼ JTa ð!uÞ!a ð82Þ
where the zero actuated state Jacobian J a(!u) is given
by (20). Therefore, the DSGI expression given by (45)
is partially linearisable about !a¼ 0d�1 as
Aþs ð!Þ ¼
ATl ð!Þ
Að!ÞATð!Þ þ k!ak
pp
¼J T
a ð!uÞ!a
Að!ÞATð!Þ þ k!ak
pp
:
ð83Þ
Accordingly, the actuated subsystem given by (81) can
be approximated in the vicinity of !a¼ 0d�1 by
_!a ¼ S21ð!Þ!u þ S22ð!Þ þJ T
a ð!uÞBð!Þ
Að!ÞATð!Þ þ k!ak
pp
" #!a
þ Pð!Þyp: ð84Þ
With yp given by (79), (84) becomes
_!a ¼
�Pð!ÞKþ
J Ta ð!uÞBð!Þ
Að!ÞATð!Þ þ k!ak
pp
þAþð!ÞAð!ÞS22ð!Þ
�� !a þA
þð!ÞAð!ÞS21ð!Þ!u: ð85Þ
Let V(!a)¼k!ak2 be a control Lyapunov function.
Evaluating the time derivative of V(!a) along the first
part of the vector field of (85) given by
_!a ¼
�Pð!ÞKþ
J Ta ð!uÞBð!Þ
Að!ÞATð!Þ þ k!ak
pp
þAþð!ÞAð!ÞS22ð!Þ
�!a ð86Þ
yields
_V1ð!Þ ¼ 2!Ta
�Pð!ÞKþ
J Ta ð!uÞBð!Þ
Að!ÞATð!Þ þ k!ak
pp
þAþð!ÞAð!ÞS22ð!Þ
�!a: ð87Þ
Asymptotic stability of the equilibrium point !a¼ 0d�1for the system given by (86) is guaranteed if the time
derivative _V1ð!Þ of V(!a) along the trajectories of the
system is negative, i.e. if P(!)K satisfies
!Ta ½Pð!ÞK �!a
5 �!Ta
�J T
a ð!uÞBð!Þ
Að!ÞATð!Þþk!ak
pp
þAþð!ÞAð!ÞS22ð!Þ
�!a:
ð88Þ
Furthermore, restricting the eigenvalues of K to bein the open left half complex plane implies from theprojection property of P(�) that
!Ta ½Pð!ÞK�!a 4 �min½Pð!ÞK�k!ak
2 4 �min½K�k!ak2:
ð89Þ
Hence, the following condition is obtained from (88):
�min½K�k!ak2
5 �!Ta
�J T
a ð!uÞBð!Þ
Að!ÞATð!Þþ k!ak
pp
þAþð!ÞAð!ÞS22ð!Þ
�!a
ð90Þ
and a sufficiency condition for asymptotic stability of
the system given by (86) is
�min½K�k!ak2 5 �bð!Þk!ak
2 ð91Þ
where
bð!Þ ¼jBð!Þj
Að!ÞATð!Þ
�max½J að!uÞ� þ �max½S22ð!Þ�: ð92Þ
Therefore, if K is strictly stable and
�min½K�5 �bð!0Þ ð93Þ
then the equilibrium point !a¼ 0d�1 for the systemgiven by (86) is asymptotically stable over a domainof attraction D!a
, and
_V1ð!Þ5 0 8!a 2 D!a: ð94Þ
On the other hand, property (3) of the DSGI impliesthat if the actuated subsystem given by (84) isasymptotically stable then the control function givenby (80) converges to one of the forms given by (25)
made by setting y¼ yp. Consequently, the conditionsgiven by (9) and (21) ensure that
limk!ak!0
k!uk ¼ 0: ð95Þ
The above fact and continuity of !u imply that !u
remains bounded. Evaluating the time derivative of
V(!a) along the trajectories of the system given by (85)yields
_Vð!Þ ¼ _V1ð!Þ þ 2!TaAþð!ÞAð!ÞS21ð!Þ!u ð96Þ
� _V1ð!Þ þ 2�max½S21ð!Þ�k!ukk!ak: ð97Þ
1864 A.H. Bajodah
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
Therefore, inequality (94) implies that _Vð!Þ remains
negative on D!aif
�min
�Pð!ÞKþ
J Ta ð!uÞBð!Þ
Að!ÞATð!Þ þ k!ak
pp
þAþð!ÞAð!ÞS22ð!Þ
�� k!ak4�max½S21ð!Þ�k!uk: ð98Þ
The left-hand side of the above inequality is
bounded as
�min
�Pð!ÞKþ
J Ta ð!uÞBð!Þ
Að!ÞATð!Þ þ k!ak
pp
þAþð!ÞAð!ÞS22ð!Þ
�� k!ak � ½�max½Pð!ÞK� þ bð!Þ�k!ak � ½�max½K�
þ bð!Þ�k!ak ð99Þ
so that
½�max½K� þ bð!Þ�k!ak4 �max½S21ð!Þ�k!uk: ð100Þ
Therefore, if K satisfies (93) then the closed loop
actuated subsystem given by (85) is asymptotically
stable if K additionally satisfies
�max½K�4 �max½S21ð!0Þ�k!uð0Þk
k!að0Þk� bð!0Þ: ð101Þ
Multiplying both sides of (80) by A(!) yields
Að!Þu ¼ Að!ÞAþs ð!ÞBð!Þ þ Að!ÞPð!Þyp ð102Þ
¼ Að!ÞAþs ð!ÞBð!Þ ð103Þ
where 05Að!ÞAþs ð!Þ51. Dividing both sides of
(103) by Að!ÞAþs ð!Þ yields
Að!Þ ~u ¼ Bð!Þ ð104Þ
where
~u ¼u
Að!ÞAþs ð!Þ:
Satisfying the system dynamics constraint given by
(104) in addition to satisfying (95) imply that control-
ling Euler’s underactuated system by the control law
u given by (80) is equivalent to globally realising the
linear asymptotically stable unactuated dynamics
given by (50) by u, assuring asymptotic stability
of the unactuated subsystem. Asymptotic stability of
the two subsystems given by (30) and (85) implies
that the origin !¼ 03�1 of the closed loop under-
actuated system is asymptotically stable if !a(0) is
within D!a. œ
In the case of one degree of actuation, the range
space of the controls coefficient is a scalar, and the
nullprojector given by (27) becomes
Pð!Þ ¼ 1�Að!Þ
Að!Þ¼ 0, ð105Þ
which implies that the nullspace of the controls
coefficient A(!) has the dimension zero, and that the
auxiliary part of the control law given by (25) vanishes.
To maintain a non-trivial controls coefficient nullspace
for which a null-control vector y can be designed
for perturbed feedback linearisation of the actuated
dynamics, an artificially actuated Euler’s system
of equations that has two degrees of actuation is
considered, and a dependency among the designed
null-control vector y that accounts for the non-existing
control torque is created. Therefore, proceeding by
letting �ð!uÞ ¼ !21, the second-order unactuated
dynamics given by (50) and the corresponding point-
wise linear equation (16) are formed as if the rigid body
is equipped by two degrees of actuation. The control
law given by (80) can then be rewritten in the form of
the following two scalar equations:
u1 ¼ Aþsð1,1Þð!ÞBð!Þ þ Pð1,1Þð!Þy1 þ Pð1,2Þð!Þy2 ð106Þ
u2 ¼ Aþsð2,1Þð!ÞBð!Þ þ Pð2,1Þð!Þy1 þ Pð2,2Þð!Þy2 ð107Þ
where u1� 0. Equation (106) is a constraint on the
null-control vector y, and it can further be written as
0 ¼ Aþsð1,1Þ ð!ÞBð!Þ þ �Pð1,1Þð!Þy1
þ ð1� �ÞPð1,1Þð!Þy1 þ Pð1,2Þð!Þy2 108Þ
where the real number � 6¼ 0, 1. Equation (108) can be
written as
y1 ¼ �1� �
�y1 þ C1ð!Þy2 þD1ð!Þ ð109Þ
where
C1ð!Þ ¼ �Pð1,2Þð!Þ
�Pð1,1Þð!Þ, D1ð!Þ ¼ �
Aþsð1,1Þð!Þ
�Pð1,1Þð!ÞBð!Þ:
ð110Þ
Therefore, y can be written as
y ¼ Cð!ÞyþDð!Þ ð111Þ
where
Cð!Þ ¼�ð1� �Þ=� C1ð!Þ
0 1
� �Dð!Þ ¼
D1ð!Þ
0
� �:
ð112Þ
Substituting the expression of y given by (111) in the
control law u given by (80) yields
u ¼ Aþs ð!ÞBð!Þ þ Pð!Þ Cð!ÞyþDð!Þ½ �: ð113Þ
In addition to globally realising the asymptotically
stable unactuated dynamics given by (50), the control
law given by (113) accounts for rigid body single degree
International Journal of Control 1865
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
of actuation via a controls coefficient nullprojectorof a higher dimension by constraining the freedom ofthe corresponding null-control vector y. Proceedingwith the perturbed feedback linearising control design,the null-control vector y given by (79) can easily beshown to render the closed loop fully actuatedsubsystem of (8) globally asymptotically stable and aperturbation from the system given by
_!a ¼ K!a: ð114Þ
Singularity of the nullprojector P(!) causes numericaldifficulties in computing C1 and D1 if P11(!) becomesat or near zero. This can be avoided by replacingthe components of P(!) in (106) and (107) by thecorresponding components of the full rank perturbednullprojector ePð!, �Þ, assuring solution existence anduniqueness for y. Accordingly, the control law givenby (113) is replaced by
u ¼ Aþs ð!ÞBð!Þ þePð!, �Þ Cð!ÞyþDð!Þ½ �: ð115Þ
8. Control design procedure
A unifying control system design procedure forperturbed feedback linearisation of stabilisable under-actuated Euler’s dynamics under one and two degreesof actuation is summarised by the following steps,considering !u¼!1 and !a ¼ ½!2 !3�
T:
(1) The controls coefficient A(!) given by (17) andthe controls load B(!) given by (51) relativeto �ð!1Þ ¼ !
k1 are obtained, where k is even
integer, c1 and c2 are positive scalars.(2) The null-control vector y given by (79) is
constructed, where S212R2�1 and S222R
2�2
are the lower partitions of the matrix S(!)given by (6), K2R
2�2 is a strictly stable matrixthat satisfies (93) and (101).
(3) The control law is given by (46) for two degreesof actuation, and by (115) for one degree ofactuation, where Aþs is given by (45) for somep norm, P(!) is given by
Pð!Þ ¼ I2�2 �Aþð!ÞAð!Þ
andePð!, �Þ ¼ I2�2 � hð�ÞAþð!ÞAð!Þ:
9. Numerical simulations
Example 1: The rigid body under consideration hasa principal plane of inertial symmetry, where the threeprincipal moments of inertia (in kgm2) are 50, 50, and85. The above analysis implies that the rigid body is
not asymptotically stabilisable by two torque actuators
that are mounted along the two principal axes residing
in the principal plane of inertial symmetry, i.e. if J11¼
85 kgm2 and J22¼ J33¼ 50 kgm2. Nevertheless, the
rigid body is asymptotically stabilisable if one of the
two actuators is mounted along the third principal axis
instead, i.e. if J11¼ J22¼ 50 kgm2 and J33¼ 85 kgm2.
The controls coefficient A(!) relative to the function
�(!u) given by
�ð!uÞ ¼ !k1 ð116Þ
is
Að!Þ ¼kðJ22 � J33Þ
J11!k�11 !3
kðJ22 � J33Þ
J11!k�11 !2
� �ð117Þ
and its zero actuated state Jacobian is
J að!1Þ ¼@ATð!Þ
@!2
@ATð!Þ
@!3
� �!2 ¼ 0
!3 ¼ 0
ð118Þ
¼
0kðJ22 � J33Þ
J11!k�11
kðJ22 � J33Þ
J11!k�11 0
26643775ð119Þ
which is non-singular for all !1 6¼ 0. Therefore, (50) is
globally realisable by underactuated Euler’s equations.
Figures 1 and 2 show the resulting angular velocities
about the three principal axes and the required control
variables for k¼ 2, desired linear unactuated dynamics
constants c1¼ 3 and c2¼ 1, and initial body angular
velocity vector !0¼ [2.5 �3.0 1.0]T. The matrix K is
taken to be diagonal with elements �1 and �8, so that
(93) and (101) are satisfied, and a dynamic scaling
index p¼ 6 is chosen. Figure 3 compares the conver-
gence of � corresponding to p¼ 6 with that corre-
sponding to p¼ 2. The value of p is shown to
0 10 20 30 40 50 60−3
−2
−1
0
1
2
3
4
ω (r
ad/s
ec)
ω1
ω3
ω2
t (s)
Figure 1. !1, !2, !3 vs t: two degrees of actuation: p¼ 6.
1866 A.H. Bajodah
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
substantially affect the rate of convergence, and thedesired second-order linear dynamics is approachedby increasing p, at the attended cost of increasing theabrupt behaviour of the control signal as seen from u1in Figure 4. Similar plots are obtained for u2 but arenot shown.
Example 2: The control torque actuator is mountedalong a body-fixed axis about which the moment of
inertia is J33¼ 50 kgm2. The moments of inertia about
two chosen orthogonal axes in a plane normal to the
actuated axis are (in kgm2) J11¼ 40 and J22¼ 35, and
their product of inertia is J12¼ 0 kgm2. The remaining
two products of inertia are J13¼�25 and J23¼ 0 kgm2.
The function � used to assess the rigid body asymptotic
stabilisability is chosen to be
�ð!uÞ ¼ !21 þ !
22, ð120Þ
and the condition given by (42) and (41) implies that
the desired second-order stable unactuated dynamics
given by (50) is globally realisable by the under-
actuated Euler’s model of the rigid body. Nevertheless,
to avoid the trivial nullprojection discussed above, the
function � is chosen to be
�ð!uÞ ¼ !21, ð121Þ
and the control law given by (113) is used to account
for the non-existing control torque. Figures 5 and 6
show the resulting angular velocity components about
the three body-fixed axes and the required control
variable for constants c1¼ 8 and c2¼ 1, �¼ 5, �¼ 10�4,
and !(0)¼ [�0.4 1.5 0.7]T, where the matrix K is
0 10 20 30 40 50 60–1.5
−1
−0.5
0
0.5
1
1.5
t (s)
u (r
ad/s
ec2 )
u1
u2
Figure 2. u1, u2 vs t: two degrees of actuation: p¼ 6.
10−1 100 101 10210−15
10−10
10−5
100 p=2p=6 ≈ 2nd order
t (s)
φ (ω
1)(r
ad2 /s
ec2 )
Figure 3. � vs t: two degrees of actuation: p¼ 2, 6, secondorder.
0 10 20 30 40 50 60−0.5
0
0.5
1
1.5
2
t (s)
p=6
p=2
u 1 (r
ad/se
c2 )
Figure 4. u1 vs t: two degrees of actuation: p¼ 2, 6.
0 50 100 150 200 250 300 350 400−1
−0.5
0
0.5
1
1.5
ω (r
ad/s
ec) ω2
ω3
ω1
t (s)
Figure 5. !1, !2, !3 vs t: one degree of actuation: p¼ 4.
0 50 100 150 200 250 300 350 400−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
t (s)
u 1 (r
ad/se
c2 )
Figure 6. u1 vs t: one degree of actuation: p¼ 4.
International Journal of Control 1867
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
taken to be diagonal with elements �3 and �6 so that(93) and (101) are satisfied, and a dynamic scalingindex p¼ 4 is chosen. The effects of altering thedynamic scaling index p on the convergence character-istics of � and on the control signal u1 are shownin Figures 7 and 8. Similar conclusion as in Example 1is obtained on the trade-off between higher conver-gence rate of � and less rapid control signal behaviour.
10. Conclusion and future work
Necessary and sufficient conditions for asymptoticstabilisability assessment are derived for underactuatedrigid bodies with arbitrary inertia distributions andequipped with one and two degrees of actuation.The controls coefficient generalised inversion metho-dology is applied thereafter to design asymptoticallystabilising underactuated rigid body control laws.The generalised inverse employed in the control lawsis modified by a dynamic scaling factor, and uniformlyconverges to the standard MPGI, asymptoticallyrealising a prescribed linear second-order unactuateddynamics. Increasing the dynamic scaling factor leadsto enhanced approximation of the desired unactuateddynamics, but also implies higher control signal rateof change. The null-control vector in the auxiliary part
of the control law is designed for perturbed feedbacklinearisation of the actuated dynamics. Extending themethodology to attitude stabilisation of underactuatedrigid bodies is an ongoing research effort by theauthor.
References
Aeyels, D. (1985), ‘Stabilization by Smooth Feedback of the
Angular Velocity of a Rigid Body’, Systems & Control
Letters, 5, 59–63.
Aeyels, D., and Szafranski, M. (1988), ‘Comments on the
Stabilisability of the Angular Velocity of a Rigid Body’,
Systems & Control Letters, 10, 35–39.Andriano, V. (1993), ‘Global Feedback Stabilisation of the
Angular Velocity of a Symmetric Rigid Body’, Systems &
Control Letters, 20, 361–364.Astolfi, A. (1996), ‘Discontinuous Control of Nonholonomic
Systems’, Systems & Control Letters, 27, 37–45.Bajodah, A.H., Hodges, D.H., and Chen, Y.-H. (2005),
‘Inverse Dynamics of Servo-constraints Based on the
Generalised Inverse’, Nonlinear Dynamics, 39, 179–196.
Bajodah, A.H. (2007), ‘Stabilization of Underactuated
Spacecraft Dynamics via Singularly Perturbed Feedback
Linearization’, in Proceedings of the 7th IFAC Symposium
on Automatic Control in Aerospace, Vol. 17, Part 1,
Toulouse, France.Bajodah, A.H. (2008), ‘Singularly Perturbed Feedback
Linearization with Linear Attitude Deviation Dynamics
Realization’, Nonlinear Dynamics, 53, 321–343.Bernstein, D. (2005), Matrix Mathematics: Theory, Facts,
and Formulas with Application to Linear System Theory,
Princeton, NJ: Princeton University Press.
Bloch, A.M., and Marsden, J.E. (1990), ‘Stabilization of
Rigid Body Dynamics by the Energy-Casimir Method’,
Systems & Control Letters, 14, 341–346.Brockett, R.W. (1983), ‘Asymptotic Stability and Feedback
Stabilization’, in Differential Geometric Control Theory,
Boston: Birkhauser, pp. 181–191.Crouch, P.E. (1984), ‘Spacecraft Attitude Control:
Applications of Geometric Control Theory to Rigid
Body Models’, IEEE Transactions on Automatic Control,
29, 321–331.Duncan, W. (1944), ‘Some Devices for the Solution of Large
Sets of Simultaneous Linear Equations (with an Appendix
on the Reciprocation of Partitioned Matrices)’, The
London, Edinburgh, and Dublin Philosophical Magazine
and Journal of Science, Seventh Series, 35, 660–670.
Durham, W.C. (1993), ‘Constrained Control Allocation’,
Journal of Guidance, Control, and Dynamics, 16, 717–725.Greville, T.N.E. (1959), ‘The Pseudoinverse of a
Rectangular or Singular Matrix and its Applications to
the Solutions of Systems of Linear Equations’, SIAM
Review, 1, 38–43.Isidori, A. (1995), Nonlinear Control Systems (3rd ed.),
New York, NY: Springer-Verlag.
Khalil, H.K. (2002), Nonlinear Systems (3rd ed.), Upper
Saddle River, NJ: Prentice-Hall.
10−1 100 101 102
10−10
10−5
100
t (s)
p=3
p=5
2nd order
φ(ω
1)(r
ad2 /s
ec2 )
Figure 7. � vs t: one degree of actuation: p¼3, 5, secondorder.
0 50 100 150 200 250 300 350 400–0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
p=5p=3
t (s)
u 1 (r
ad/se
c2 )
Figure 8. u1 vs t: one degree of actuation: p¼3, 5.
1868 A.H. Bajodah
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
Milnor, J.W. (1964), ‘Differentiable Topology’, in Lectures inModern Mathematics (Vol. II), ed. T. Saaty, New York:
Wiley.Moore, E.H. (1920), ‘On the Reciprocal of the GeneralAlgebraic Matrix’, Bulletin of the American MathematicalSociety, 26, 394–395.
Outbib, R., and Sallet, G (1992), ‘Stabilizability of theAngular Velocity of a Rigid Body Revisited’, Systems &Control Letters, 18, 93–98.
Penrose, R. (1955), ‘A Generalised Inverse for Matrices’,Proceedings of the Cambridge Philosophical Society, 51,406–413.
Slotine, J.E., and Li, W. (1991), Applied Nonlinear Control,Englewood Cliffs, NJ: Prentice-Hall.
Sontang, E., and Sussmann, H.J. (1988), ‘Further Commentson the Stabilisability of the Angular Velocity of a Rigid
Body’, Systems & Control Letters, 12, 213–217.Spong, M.W. (1994), ‘Partial Feedback Linearisation ofUnderactuated Mechanical Systems’, IEEE International
Conference on Intelligent Robotics and Systems, 1,314–321.
Sun, Z., and Ge, S.S. (2003), ‘Nonregular FeedbackLinearisation: A Nonsmooth Approach’, IEEETransactions on Automatic Control, 48, 1772–1776.
Tsiotras, P., and Doumtchenko, V. (2000), ‘Control of
Spacecraft Subject to Actuator Failure: State-of-the-artand Open Problems’, Journal of the Astronautical Sciences,48, 337–358.
Tsiotras, P., and Longuski, J.M. (1994), ‘Spin AxisStabilisation of Symmetric Spacecraft with Two ControlTorques’, Systems & Control Letters, 23, 395–402.
Udwadia, F.E., and Kalaba, R.E. (1996), AnalyticalDynamics, A New Approach, New York, NY: CambridgeUniversity Press.
Wichlund, K.Y., Sordalen, O.J., and Egeland, O. (1995),
‘Control Properties of Underactuated Vehicles’, in IEEEInternational Conference on Robotics and Automation,pp. 2009–2014.
International Journal of Control 1869
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
04:
56 0
4 O
ctob
er 2
014
