asymptotic generalised dynamic inversion

Published in IET Control Theory and ApplicationsReceived on 6th January 2009Revised on 19th July 2009doi: 10.1049/iet-cta.2009.0008

ISSN 1751-8644

Asymptotic generalised dynamic inversionattitude controlA.H. BajodahAeronautical Engineering Department, King Abdulaziz University, Jeddah 21589, Saudi ArabiaE-mail: [email protected]

Abstract: This study introduces a generalised dynamic inversion control methodology for asymptotic spacecraftattitude trajectory tracking. An asymptotically stable second-order servo-constraint attitude deviation dynamics isevaluated along spacecraft equations of motion, resulting in a linear relation in the control vector. A control lawthat enforces the servo-constraint is derived by generalised inversion of the relation using the Greville formula.The generalised inverse in the particular part of the control law is scaled by a decaying dynamic factor thatdepends on desired attitude trajectories and body angular velocity components. The scaled generalisedinverse uniformly converges to the standard Moore –Penrose generalised inverse, causing the particular partto converge uniformly to its projection on the range space of the controls coefficient generalised inverse, anddriving spacecraft attitude variables to nullify attitude deviation. The auxiliary part of the control law acts onthe controls coefficient nullspace, and it provides the spacecraft internal stability with the aid of the null-control vector. The null-control vector construction is made by means of novel semidefinite nullprojectioncontrol Lyapunov function and state-dependent null-projected Lyapunov equation. The generalised dynamicinversion control signal is multiplied by an exponential factor during transient closed-loop response toenhance the control signal in terms of magnitude and rate of change. Illustrating examples show efficacy ofthe methodology.

1 IntroductionNon-linear dynamic inversion (NDI) is a transformationfrom a non-linear system to an equivalent linear system,performed by means of a change of variables and throughfeedback. The theory of NDI was initially formalised by Su[1] and Hunt et al. [2], and its first reported application tospacecraft attitude control problem is due to Dwyer [3].The methodology is widely accepted among control systempractitioners because it substantially facilitates controlsystem design. Additionally, it preserves the non-linearnature of plant’s dynamics and thus it avoids limitations oflinearising approximations.

Classical NDI is based on constructing inverse mapping ofthe controlled plant and augmenting it within the feedbackcontrol system. Therefore the linearising transformationdepends heavily on the nature of the plant, and it becomesdifficult or impossible as complexity of the plant increases.For this reason it may become necessary to introduce

simplifying approximations to the plant’s mathematicalmodel in order to obtain the NDI linearisingtransformation, which adversely affects closed-loop controlsystem stability and performance characteristics in realimplementations of the transformation. Additionally, NDIin particular situations must be local in state space, as it isthe case for spacecraft attitude dynamics [3].

A paradigm shift was made to NDI by Paielli and Bach in[4] in the context of spacecraft attitude control. Theirapproach aims to impose a prescribed dynamics on theerrors of spacecraft attitude variables from their desiredtrajectory values. Rather than inverting the mathematicalmodel of the spacecraft, the desired attitude error dynamicsis inverted for the control variables that realise thedynamics. The transformation is global and does notinvolve deriving inverse equations of motion. It involvessimple mathematical inversions of terms that includemotion variables and control system design parameters, andtherefore it is easier and more systematic than its counterpart.

IET Control Theory Appl., 2010, Vol. 4, Iss. 5, pp. 827–840 827doi: 10.1049/iet-cta.2009.0008 & The Institution of Engineering and Technology 2010

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Nevertheless, a common feature between the above-mentionedNDI approaches is that the linearising transformation eliminatesnon-linearities from the transformed closed-loop systemdynamics without distinguishing between types of non-linearities. For instance, a non-linearity may cause the spacecraftat a particular time instant to accelerate in a manner that is infavour of the control objective, for example, in performing adesired manoeuvre. Yet a needless control effort is made toeliminate that non-linearity, and an additional control effort ismade to satisfy the control objective. This can be extremelydisadvantageous as large control signals may cause actuatorsaturation and control system’s failure.

It is therefore desirable to come up with a dynamicinversion control design methodology that provides a globallinearising transformation, gets around the difficulty ofplant’s mathematical model inversion and requires lesscontrol effort to perform the inversion by avoiding blindcancelation of dynamical system’s non-linearity. Thesefeatures are offered by generalised non-linear dynamicinversion (GNDI) control. Some basic elements of GNDIwere introduced by the author in [5, 6], together withparticular GNDI control designs. Every design exhibitsdifferent characteristics in terms of closed-loop systemstability, performance and control signal behaviour.

The GNDI methodologies add the flexibility of non-squareinversion to the simplicity of NDI by observing that althoughthe inverse mathematical model of a plant is unique, theinverse dynamics problem is a problem with non-uniquesolution, that is, there exist infinite sets of values for the stateand control variables that satisfy a specific dynamics at aspecific time instant. Therefore the original philosophy ofdynamic inversion is quite restrictive, and there must existinfinite generalised inversion control laws that realise aservo-constraint dynamics, that is, the differential equationin system’s state variables that has its steady-state solutionsatisfies the control design objective.

A GNDI spacecraft control design begins by defining a normmeasure function of attitude error from desired attitude trajectory.An asymptotically stable second-order linear differential equationin the norm function is prescribed, resembling the desiredservo-constraint dynamics. The differential equation is thentransformed to a relation that is linear in the control vector bydifferentiating the norm measure function along the trajectoriesdefined by solution of the spacecraft’s state-space mathematicalmodel. The Greville formula [7] is utilised thereafter to invertthis relation for the control law required to realise desired stablelinear servo-constraint dynamics.

The Greville generalised inversion formula exhibits usefulgeometrical features of generalised inversion. It consists ofauxiliary and particular parts, residing in the nullspace ofthe inverted matrix and the complementary orthogonalrange space of its transpose, respectively. The particularpart involves the standard Moore–Penrose generalisedinverse (MPGI) [8, 9], and the auxiliary part involves a free

null-vector that is projected onto nullspace of the invertedmatrix by means of a nullprojection matrix.

The Greville formula is capable of modelling solutionnon-uniqueness to problems where requirements can besatisfied in more than one course of action. For that reason, theformula had remarkable contributions towards advancementsin science and engineering. In the arena of robotics, it has beenextensively used in analysis and design of kinematicallyredundant manipulators [10]. Utilisation of the formula in thefield of analytical dynamics was made by deriving theUdwadia–Kalaba equations of motion for constraineddynamical systems [11]. Other applications include theevolving subject of pointwise optimal control in the senseof Gauss’ principle of least constraint [12], see for example,[13–15].

However, a fundamental shortcoming of the Grevilleformula for matrices containing dynamic elements is MPGIsingularity. This problem is well known in applications ofthe formula, and it has been thoroughly investigated inthe subject of inverse kinematics, for example, [16]. Thereason for MPGI singularity is that a matrix withcontinuous function elements has discontinuous MPGIfunction elements. These discontinuities occur whenever theinverted matrix changes rank. Moreover, these discontinuouselements approach infinite values at discontinuities.Accordingly, the corresponding solutions provided by theGreville formula must also be discontinuous and unbounded.

The MPGI singularity forms an obstacle in the way ofutilising the Greville formula in engineering solutions.Several remedies for the problem of generalised inversioninstability due to MPGI singularity have been offered inthe literature of robotics and control moment gyroscopicdevices, in what has become known as the singularityavoidance problem. Remedies are either nullspaceparametrisation-based, made by proper choices of the null-vector in the auxiliary part of the Greville formula, forexample, [17–19] or approximation-based, made bymodifying the definition of the generalised inverse itself inthe particular part of the formula, for example, [20–22].

A few solutions to the generalised inversion instabilityproblem have been provided in the context of GNDIcontrol. One solution is made by deactivating the particularpart of the Greville formula-based control law in thevicinity of singularity, resulting in discontinuous controllaws [23]. Another solution is presented in [5], made bymodifying the definition of MPGI by means of a dampingfactor, resulting in uniformly ultimately bounded attitudetrajectory tracking and a trade-off between generalisedinversion stability and closed-loop system performance.

This paper introduces a novel concept of generalisedinversion by which the Greville formula is modified forguaranteed generalised inversion stability and asymptotictracking. The concept is based on replacing the MPGI

828 IET Control Theory Appl., 2010, Vol. 4, Iss. 5, pp. 827–840

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matrix in the Greville formula by a growth-controlleddynamically scaled generalised inverse (DSGI) matrix, suchthat the DSGI matrix elements converge uniformly to thestandard MPGI matrix elements, resulting in uniformasymptotic convergence of spacecraft attitude parameters totheir desired values in time.

The procedure begins by defining a reference angularvelocity vector that depends on desired attitude trajectoryvector function and its time derivative. The reference angularvelocity vector has the property that its convergence tospacecraft’s body angular velocity vector implies spacecraft’sattitude vector convergence to desired attitude vector.

The DSGI is constructed by adding a dynamic scalingfactor to the controls coefficient’s squared norm in thedenominators of MPGI’s elements. The dynamic scalingfactor is the pth integer power of the vector p norm of errorbetween spacecraft body angular velocity and referenceangular velocity. The null-control vector in the auxiliarypart of the control law is designed to nullify the dynamicscaling factor such that the DSGI asymptotically recoversthe structure of the MPGI. This causes the particular partof the control law to converge to its projection on the rangespace of the controls coefficient’s MPGI, which drives theattitude variables to satisfy desired servo-constraint stabledynamics, resulting in global asymptotic attitude trajectorytracking.

The GNDI paradigm makes it possible to merge dynamicinversion with other control system design methodologies toenhance control system design features. This is achievedthrough construction of the null-control vector that appearsexplicitly in the auxiliary part of the control law. The null-control vector provides by its affine parametrisation ofcontrols coefficient’s nullspace a convenient way to stabiliseinternal dynamics of the closed-loop control systemwithout affecting servo-constraint realisation.

In particular, Lyapunov control design can be augmentedwith GNDI to reduce control effort required by dynamicinversion. Lyapunov control design has a successful historyin spacecraft control, and is well known to consume lessenergy than dynamic inversion. This fact is verified in thecontext of GNDI in [6], and it makes merging the twomethodologies quite promising for control engineeringpractice.

This paper utilises some attractive geometrical features ofgeneralised inversion with a novel type of positivesemidefinite control Lyapunov functions [24, 25] for null-control vector design. The control Lyapunov functioninvolves the controls coefficient nullprojection matrix, andapplying Lyapunov direct method [26] yields a controlscoefficient null-projected Lyapunov equation. The equationis solved to obtain a simple control law for globalasymptotic stability of internal spacecraft dynamics.

Despite the excellent performance of GNDI control design,heavy controls load at initial closed-loop control time causesrelatively big initial control signal magnitude compared tocontrol signal magnitudes at later stages of closed-loopcontrol times. Moreover, for the control system to enforcedesired servo-constraint dynamics, a rapid decrease ofcontrol signal magnitude follows. This causes undesirablehigh-frequency behaviour of spacecraft angular velocity andmay excite un-modelled structural modes, which adverselyaffects spacecraft closed-loop dynamics. In order to avoid thepossible corrupting of spacecraft functionality, the GNDIcontrol signal is multiplied by an exponential factor duringtransient closed-loop control system response, resulting insubstantial enhancement of control signal behaviour in termsof initial magnitude and rate of change.

The contribution of this article is two-fold. First, theGNDI methodology is modified for global asymptoticattitude trajectory tracking and globally asymptoticallystable internal dynamics. Second, the control design isenhanced to reduce control signal magnitude and rateduring transient closed-loop control system response.

2 Spacecraft mathematical modelThe spacecraft mathematical model is given by thefollowing system of kinematical and dynamical differentialequations

r = G(r)v, r(0) = r0 (1)

v = J−1v×Jv+ t, v(0) = v0 (2)

where r [ R3×1 is the spacecraft vector of modified Rodriguesattitude parameters (MRPs) [27], v [ R3×1 is the vector ofspacecraft angular velocity components in its body referenceframe, J [ R3×3 is the spacecraft’s symmetric bodymoments of inertia matrix and t := J−1u [ R3×1 is thevector of scaled control torques, where u [ R3×1 containsthe applied gas jet actuator torque components about thespacecraft’s principal axes. The cross product matrix x×

which corresponds to a vector x [ R3×1 is skew symmetricof the form

x× =0 x3 −x2

−x3 0 x1

x2 −x1 0

⎡⎣ ⎤⎦and the matrix valued function G(r) : R3×1 � R3×3

is finite and invertible for any value of r [ R3×1, and isgiven by

G(r) = 1

2

1 − rTr

2I3×3 − r× + rrT

( )(3)


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3 Servo-constraint attitudedeviation dynamicsLet rr(t) [ R3×1 be a prescribed desired spacecraft attitudevector such that rr(t) is twice continuously differentiablein t. The spacecraft attitude error vector from rr(t) isdefined as

r(r, t) := r(t) − rr(t) (4)

Consequently, the scalar attitude deviation norm measurefunction f : R3×1 × [0, 1) � [0, 1) is defined to be thesquared norm of r

f(r, t) = ‖r(r, t)‖2 (5)

Therefore a servo-constraint on the attitude dynamics thatrepresent the control design objective is given by

f(r, t) ; 0 (6)

The first two time derivatives of f along the spacecrafttrajectories given by the solutions of (1) and (2) are

f = ∂f

∂rG(r)v+ ∂f

∂t(7)

= 2rT(r, t)[G(r)v− rr(t)] (8)

and

f = 2[G(r)v− rr(t)]T[G(r)v− rr(t)]

+ 2rT(r, t)[G(r, v)v+ G(r)[ J−1v×Jv+ t] − rr(t)]

(9)

where G(r, v) is the time derivative of G(r) obtained bydifferentiating the individual elements of G(r) along thekinematical subsystem given by (1). The desireddynamics of f that leads to asymptotic realisationof the servo-constraint given by (6) is described tobe stable second-order in the general functional formgiven by

f = L(f, f, t) (10)

where L is continuous in its arguments. With f, f andf given by (5), (8) and (9), it is possible to write (10) inthe pointwise-linear form

A(r, t)t = B(r, v, t) (11)

where the vector valued function A(r, t) : R3×1×[0, 1) � R1×3 is given by

A(r, t) = 2rT(r, t)G(r) (12)

and the scalar valued function B(r, v, t) : R3×1× R3×1 ×[0, 1) � R is given by

B(r, v, t) = −2[G(r)v− rr(t)]T[G(r)v− rr(t)]

− 2rT(r, t)[G(r, v)v+ G(r)J−1v×Jv− rr(t)]

+L(f(r, t), f(r, v, t), t)

(13)

The row vector function A(r, t) is the controls coefficientof the attitude deviation norm measure dynamics given by(10) along the spacecraft trajectories, and the scalarfunction B(r, v, t) is the corresponding controls load.

3.1 Linear attitude deviation dynamics

A special choice of L(f, f, t) is

L(f, f, t) = −c1f− c2f (14)

where c1 and c2 are positive scalars. With this choice ofL(f, f, t), the stable attitude deviation servo-constraintdynamics given by (10) becomes linear in the form

f+ c1f+ c2f = 0 (15)

The corresponding controls load B(r, v, t) given by (13)becomes

B(r, v, t) =−2[G(r)v− rr(t)]T[G(r)v− rr(t)]

− 2rT(r, t)[G(r, v)v+G(r)J−1v×Jv− rr(t)]

− 2c1rT(r, t)[G(r)v− rr(t)]− c2‖r(r, t)‖2

(16)

4 Realisability of attitudedeviation dynamicsRealisability of attitude deviation dynamics is a pointwiseassessment of the control system for the ability to enforce aservo-constraint on the controlled spacecraft.

Definition 1 (Realisability of attitude deviationservo-constraint dynamics): For a given desiredspacecraft attitude vector rr(t), the linear attitude deviationnorm measure dynamics given by (10) is said to berealisable by spacecraft equations of motion (1) and (2) atspecific values of r and t if there exists a control vector t

that solves (11) for these values of r and t. If this is truefor all r and t such that r(r, t) = 03×1, then the linearattitude deviation norm measure dynamics is said to beglobally realisable by the spacecraft equations of motion.

Realisability of a prescribed attitude dynamics judges onthe existence of control vector values that enforce thatdynamics for every attitude state and at every time instant.



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This distinguishes the notion from the equivalent notion ofcontrollability, which evaluates the ability of driving a statefrom one point to another in the state space. The algebraicform of (11) is a substitute for the differential form of (10),and realisability of the dynamics given by (10) by spacecraftequations of motion (1) and (2) for a given desiredspacecraft attitude vector rr(t) at specific values of r and tis equivalent to the existence of a control vector t thatsolves (11) for these values of r and t.

Proposition 1 (Global realisability of servo-constraintdynamics): The attitude deviation servo-constraintdynamics given by (10) is globally realisable by spacecraftequations of motion (1) and (2).

Proof: The existence of a vector t that solves (11) at specificvalues rw, vw and tw is equivalent to the fact thatB(rw, vw, tw) is in the range space of A(rw, tw). This ispossible for any value that B(rw, vw, tw) may take, providedthat not all elements of A(rw, tw) vanish, for which theequation is said to be consistent. Since G(r) is of full rank forallr [ R3×1, the expression given by (12) forA(r, t) implies that

A(r, t) = 01×3 ⇔ r(r, t) = 03×1 (17)

which proves realisability of (10) for all r and t such thatr(r, t) = 03×1. A

5 Reference angular velocityand accelerationInvertibility of the matrix G(r) in (1) makes it possible tosolve explicitly for the vector v, which takes the form

v = G−1(r)r (18)

Therefore a vector of reference angular velocity vr(t) isobtained by substituting the desired vector of attitudevariables rr(t) and its time derivative rr(t) in place of r andr, respectively, in (18), such that

vr(t) = G−1(rr(t))rr(t) (19)

A vector of reference angular acceleration vr(t) att = t(r, vr , t) is obtained from (2) by substituting vr(t) inplace of v such that

vr(t) = J−1v×r (t)Jvr(t) + t(r, vr , t) (20)

6 Generalised dynamic inversionattitude controlThe MPGI-based Greville formula is used now to obtain apreliminary form of GNDI spacecraft attitude control laws.

Proposition 2 (Linearly parameterised attitudecontrol laws): The infinite set of all control laws that

globally realise the attitude deviation servo-constraintdynamics given by (10) by the spacecraft equations ofmotion is parameterised by an arbitrarily chosen null-control vector y [ R3×1 as

t = A+(r, t)B(r, v, t) + P(r, t)y (21)

where ‘A+’ stands for the MPGI of the controls coefficient(abbreviated as CCGI), and is given by

A+(r, t) =AT(r, t)

A(r, t)AT(r, t), A(r, t) = 01×3

03×1, A(r, t) = 01×3

⎧⎪⎨⎪⎩ (22)

and P(r, t) [ R3×3 is the corresponding controls coefficientnullprojector (CCNP), given by

P(r, t) = I3×3 −A+(r, t)A(r, t) (23)

Proof: Multiplying both sides of (21) by A(r, t) recovers thealgebraic system given by (11). Therefore t enforces theattitude deviation servo-constraint dynamics given by (10)for all A(r, t) = 01×3. A

The controls coefficient nullprojector P(r, t) projects thenull-control vector y onto the nullspace of the controlscoefficient A(r, t). Therefore the choice of y does notaffect realisability of the linear attitude deviation normmeasure dynamics given by (10). Nevertheless, the choiceof y substantially affects transient state response andspacecraft internal stability, that is, stability of the closed-loop dynamical subsystem

v = J−1v×Jv+A+(r, t)B(r, v, t) + P(r, t)y (24)

obtained by substituting (21) into (2) [28].

7 Perturbed controls coefficientnullprojectorDefinition 2 (Perturbed controls coefficientnullprojector): The perturbed CCNP P(r, d, t) is given by

P(r, d, t) := I3×3 − h(d)A+(r, t)A(r, t) (25)

where h(d) : R1×1 � R1×1 is any continuous function suchthat

h(d) = 1 if and only if d = 0

7.1 Properties of perturbed controlscoefficient nullprojector

The first and second properties below are proven in [5]. Thethird property is verified by direct evaluation of P(r, t) andP(r, d, t) expressions given by (23) and (25).

1. P(r, d, t) is of full rank for all d = 0.


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2. P−1(r, d, t)P(r, t) = P(r, t)P−1(r, d, t) = P(r, t).

3. P(r, t)P(r, d, t) = P(r, d, t)P(r, t) = P(r, t).

8 Generalised inverse instabilityThe expression given by (12) for the controls coefficientimplies that if the dynamics given by (10) is globallyrealisable by the spacecraft equations of motion, then

limf�0

A(r, t) = 01×3 (26)

Accordingly, the expression of A+(r, t) given by (22) impliesthat for any initial condition r0 = rr(0), state trajectories ofthe closed-loop control system given by (1) and (24) mustevolve such that

limf�0

A+(r, t) = 13×1 (27)

That is, A+(r, t) must go unbounded as the spacecraftdynamics approaches steady state. This is a source ofinstability for the closed-loop system because it causes thecontrol law expression given by (21) to become unbounded.One solution to this problem is made by switching thevalue of the CCGI according to (22) to A+(r, t) = 03×1

when the controls coefficient A(r, t) approaches singularity,which implies deactivating the particular part of the controllaw as the closed-loop system reaches steady state, leading toa discontinuous control law [23].

Alternatively, a solution is made by replacing the MPGI in(21) by a damped generalised inverse [5], resulting inuniformly ultimately bounded trajectory tracking errors, anda trade-off between generalised inversion stability andsteady-state tracking performance. A solution to thisproblem that avoids control law discontinuity and providesasymptotic attitude tracking is made by replacing theMPGI in (21) by the growth-controlled DSGI introducedin the next section.

9 Dynamically scaled generalisedinverseThe notion of dynamically scaled generalised inversion iscrucial for internally stable asymptotic generalised inverseattitude control.

Definition 3 (DSGI): The DSGI A+s (r, v, t) : R3×1×

R3×1 × [0, 1) � R3×1 is given by

A+s (r, v, t) = AT(r, t)

A(r, t)AT(r, t) + ‖v− vr(t)‖pp

(28)

where the positive integer p is the generalised inversiondynamic scaling index, and ‖·‖p is the vector p norm.

9.1 Properties of DSGI

The following properties can be verified by direct evaluationof the CCGI A+(r, t) given by (22) and its dynamic scalingA+

s (r, v, t) given by (28).

1. A+s (r, v, t)A(r, t)A+(r, t) = A+

s (r, v, t).

2. A+(r, t)A(r, t)A+s (r, v, t) = A+

s (r, v, t).

3. (A+s (r, v, t)A(r, t))T = A+

s (r, v, t)A(r, t).

4. lim‖v−vr (t)‖p�0 A+s (r, v, t) = A+(r, t).

10 DSGI controlThe DSGI control law is obtained by replacing the CCGI inthe particular part of the expression given by (21) by theDSGI as

ts = A+s (r, v, t)B(r, v, t) + P(r, t)y (29)

resulting in the following spacecraft closed-loop systemequations

r = G(r)v, r(0) = r0 (30)

v = J−1v×Jv+A+s (r, v, t)B(r, v, t)

+ P(r, t)y, v(0) = v0

(31)

Proposition 3 (Bounded trajectory tracking error): Ifthe null-control vector y in the control law expression givenby (29) is chosen such that the angular velocity vector v ofthe closed-loop system given by (30) and (31) satisfies

‖v− vr(t)‖p , 1, ∀ t ≥ 0 (32)

then the resulting closed-loop attitude trajectory error vectorr(r, t) is bounded.

Proof: Since the matrix G(r) has finite elements for anyattitude vector r, then it is evident from the expression ofthe controls coefficient A(r, t) given by (12) that A(r, t) isbounded if and only if r(r, t) is bounded. Thereforeassuming on the contrary that there exists a null-controlvector y that causes the closed-loop angular velocity vectorv to satisfy (32) such that

limt�1

r(r, t) = 1 (33)

then it follows that

limt�1

A(r, t) = 1 (34)

which implies from (28) and (32) that

limt�1

A+s (r, v, t) = A+(r, t) (35)



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It accordingly follows from the expression of ts given by(29) that

limt�1

ts = t (36)

where t is given by (21), causing the closed-loop systemtrajectories to asymptotically satisfy the stable servo-constraint dynamics given by (15), and resulting in

limt�1

f = 0 (37)

which contradicts (33). Therefore the control law ts given by(29) must yield

r(r, t) , 1 (38)

and must yield bounded elements of A(r, t). A

Proposition 4 (Asymptotic attitude trajectorytracking): If the null-control vector y in the control lawexpression given by (29) is chosen such that the angularvelocity vector v of the closed-loop system given by (30)and (31) satisfies (32) and such that

limt�1

v = vr(t) (39)

then the attitude vector r asymptotically converges to thedesired attitude vector rr(t).

Proof: Let fs be a norm measure function of theattitude deviation obtained by applying the control lawgiven by (29) to the spacecraft equations of motion (1)and (2), and let fs, fs be its first two time derivatives.Therefore

fs := fs(r, t) = f(r, t) (40)

fs := fs(r, v, t) = f(r, v, t) (41)

fs := fs(r,v, ts, t)= f(r,v, t, t)+A(r, t)ts −A(r, t)t (42)

where t and ts are given by (21) and (29), respectively.Adding c1fs + c2fs to both sides of (42) yields

fs + c1fs + c2fs = f+ c1f+ c2f+A(r, t)ts −A(r, t)t (43)

= A(r, t)[ts − t] (44)

Therefore the boundedness of A(r, t) inferred fromProposition 3 in addition to satisfaction of (39) imply that

limt�1

[fs + c1fs + c2fs] = limt�1

[A(r, t)[ts − t]] = 0 (45)

resulting in

limt�1

fs = 0 (46)

and therefore

limt�1

r = rr(t) (47)

for all r0 [ R3. The same conclusion is obtained bymultiplying both sides of (29) by A(r, t), resulting in

A(r, t)ts = A(r, t)A+s (r, v, t)B(r, v, t) (48)

where

A(r, t)A+s (r,v, t)= A(r, t)AT(r, t)

A(r, t)AT(r, t)+‖v−vr(t)‖pp

(49)

Therefore

0 , A(r, t)A+s (r, v, t) ≤ 1 (50)

and

limv�vr (t)

A(r, t)A+s (r, v, t) = 1 (51)

Dividing (48) by A(r, t)A+s (r, v, t) yields

A(r, t) �t = B(r, v, t) (52)

where A(r, t) and B(r, v, t) are the same controls coefficientand controls load in (11), and

�t =ts

A(r, t)A+s (r, v, t)

(53)

Furthermore, (51) implies that

limv�vr (t)

�t = limv�vr (t)

ts = t (54)

Therefore �t in the algebraic system given by (52)asymptotically converges to t, recovering the algebraicsystem given by (11), and resulting in asymptoticconvergence of fs(t) to fs = f = 0, and r to rr(t). A

Proposition 4 states that employing the DSGI A+s (r, v, t)

in the attitude control law yields the same attitudeconvergence property that is obtained by employing theCCGI A+(r, t), provided that the conditions given by (32)and (39) are satisfied. A design of the null-control vector yis made in the next section to guarantee global satisfactionof the conditions given by (32) and (39).

Remark 1: It is well known that topological obstruction ofthe attitude rotation matrix precludes the existence of globallystable equilibria for the attitude dynamics [29]. Thereforealthough the servo-constraint attitude deviation dynamics


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given by (10) is globally realisable, there exists no null-controlthat renders the spacecraft attitude dynamics globally stable.In particular, if rr(t) ; 03×1 then for any null-controlvector y there exists an attitude vector r0 such that theclosed-loop system given by (30) and (31) is unstable in thesense of Lyapunov.

11 Nullprojection Lyapunovcontrol designA Lyapunov-based design of null-control vector y isintroduced in this section to enforce spacecraft internalstability. Let y be chosen as

y = K (v− vr(t)) (55)

where vr(t) is given by (19), and K [ R3×3 is a matrix gainthat is to be determined. Hence, a class of control laws thatrealise the attitude deviation norm measure dynamics givenby (10) is obtained by substituting this choice of y in (29)such that

ts = A+s (r, v, t)B(r, v, t) + P(r, t)K (v− vr(t)) (56)

Consequently, a class of spacecraft closed-loop controlsystems that realise the servo-constraint dynamics given by(10) is obtained by substituting the control law given by(56) in (2), and it takes the form

r = G(r)v, r(0) = r0 (57)

v = J−1v×Jv+A+s (r, v, t)B(r, v, t)

+ P(r, t)K (v− vr(t)), v(0) = v0

(58)

The closed-loop reference angular acceleration vector vr(t) isobtained by replacing v by vr(t) in (58), resulting in

vr(r, t) = J−1v×r (t)Jvr(t) +A+

s (r, vr(t), t)B(r, vr(t), t)

+ P(r, t)K (vr(t) − vr(t))

(59)

= J−1v×r (t)Jvr(t) +A+(r, t)B(r, vr(t), t) (60)

By introducing the angular velocity error variablev(t) ; v(t) − vr(t), the error dynamics ˙v is obtained from(58) and (60) as

˙v = J−1v×Jv− J−1v×r (t)Jvr(t)

+A+s (r, v, t)B(r, v, t)

−A+(r, t)B(r, vr(t), t) + P(r, t)K v (61)

The matrix gain K is synthesised by utilising the positive-semidefinite control Lyapunov function

V (r, v, t) = vTP(r, t)v (62)

Evaluating the time derivative of V (r, v, t) along solutiontrajectories of the error dynamics given by (61) yields

V (r, v, t) = 2vTP(r, t)[ J−1v×Jv− J−1v×r (t)Jvr(t)

+A+s (r, v, t)B(r, v, t)−A+(r, t)B(r, vr(t), t)]

+ 2vTP(r, t)K v+ vTP(r, v, t)v (63)

Skew symmetry of the cross product matrix [.]×, thenullprojection property of P(r, t), and the second propertyof A+

s (r, v, t) imply that the first term in the aboveequation is the zero matrix. Therefore the equation can bewritten in the symmetrical form

V (r, v, t) = vT[P(r, t)K + K TP(r, t)

+ P(r, v, t)]v (64)

Because V (r, v, t) is only positive semidefinite, it isimpossible to design a matrix gain K that rendersV (r, v, t) negative definite. Nevertheless, a matrix gain Kthat renders V (r, v, t) negative semidefinite guaranteesglobal Lyapunov stability of v = 03×1 if it asymptoticallystabilises v = 03×1 over the invariant set of r, v, and tvalues on which V (r, v, t) = 0. Moreover, the same gainmatrix globally asymptotically stabilises v = 03×1 if andonly if it asymptotically stabilises v = 03×1 over the largestinvariant set of r, v, and t values on which V (r, v, t) = 0[24].

Proposition 5: Let K = K (r, v, t) be a full-ranknormal matrix gain, that is, KK T = K TK for all t ≥ 0.Then the equilibrium point v = 03×1 of the closed-looperror dynamics given by (61) is asymptotically stable overthe invariant set of r, v, and t values on whichV (r, v, t) = 0.

Proof: Since the matrix P(r, t) is idempotent, the functionV (r, v, t) can be rewritten as

V (r, v, t) = vTP(r, t)v = vTP(r, t)P(r, t)v (65)

which implies that

V (r, v, t) = 0 ⇔ P(r, t)v = 03×1 (66)

Therefore

V (r, v, t) = 0 ⇔ v [ N (P(r, t)) (67)

where N (·) refers to matrix nullspace. Since the matrixK (r, v, t) is normal and of full-rank, it preservesmatrix range space and nullspace under multiplication.Accordingly,

N (P(r, t)) = N (P(r, t)K (r, v, t)) (68)



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which implies from (66) that

V (r, v, t) = 0 ⇔ P(r, t)K (r, v, t)v = 03×1 (69)

Therefore the last term in the closed-loop error dynamicsgiven by (61) is the zero vector, and the closed-loop errordynamics becomes

˙v = J−1v×Jv− J−1v×r (t)Jvr(t)

+A+s (r, v, t)B(r, v, t)

−A+(r, t)B(r, vr(t), t) (70)

On the other hand, since [30, 11]

N (P(r, t)) = R(AT(r, t)) (71)

it follows from (67) that

V (r, v, t) = 0 ⇔ v [ R(AT(r, t)) (72)

Accordingly, V (r, v, t) = 0 if and only if there exists acontinuous scalar function a(t), t ≥ 0, satisfying

0 , |a(t)| , 1 (73)

such that

v = a(t)AT(r, t) (74)

Therefore assuming that v goes unbounded, then AT(r, t)also goes unbounded, both expressions of A+(r, t) andA+

s (r, t) given by (22) and (28) must go to zero, and theclosed-loop error dynamics given by (70) approaches theLyapunov-stable uncontrolled dynamics

˙v = J−1v×Jv− J−1v×r (t)Jvr(t) (75)

implying boundedness of v, in contradiction with theoriginal argument. Therefore, the trajectory of v mustremain in a finite region, and it follows from the Poincare–Bendixon theorem [31] that the trajectory goes to theequilibrium point v = 03×1.

Theorem 1 (CCNP Lyapunov control design): Let thecontrols coefficient nullspace-projected gain matrix be

P(r, t)K (r, v, t) = −vec−1{[P(r, d1, t) ⊕ P(r, d1, t)]−1

vec[P(r, v, t) + P(r, t)U(r, v, t)]}

(76)

where the operation ⊕ is the kronecker sum of matrices,‘vec’ and ‘vec21’ are the matrix vectorising and inversevectorising operators [30, p. 251], P(r, v, t) is obtained bydifferentiating the elements of P(r, t) along attitudetrajectory solutions of the closed-loop kinematical

subsystem given by (57), d1 = 0 is an arbitrary scalar, andU(r, v, t) : R3×1 × R3×1 × [0, 1) � R3×3 is orthogonaland positive-definite. Then the equilibrium point v = 03×1

of the closed-loop error dynamics given by (61) is globallyasymptotically stable, and the attitude vector r of closed-loop system equations (57) and (58) is globallyasymptotically convergent to rr(t).

Proof: The existence of a matrix gain K that renders theexpression of V (r, v, t) given by (64) negative semidefinite isguaranteed because the range space of P(r, v, t) is a subsetfrom the range space of P(r, t). This is shown by writing

P(r, t) = P(r, t)P(r, t) ⇒ P(r, v, t)

= 2P(r, t)P(r, v, t) (77)

so that

R[P(r, v, t)] = R[P(r, t)P(r, v, t)] # R[P(r, t)] (78)

where R(·) refers to matrix range space. Negativesemidefiniteness of V (r, v, t) is equivalent to the existence ofa positive-semidefinite matrix function Q(r, v, t) : R3×1×R3×1 × [0, 1) � R3×3 such that

P(r, t)K +KP(r, t)+ P(r, v, t)+Q(r, v, t) = 03×3 (79)

Furthermore, if the above equation is consistent then every termin the equation must map into the range space of P(r, t), whichimplies that a polar decomposition of Q(r, v, t) is given by

Q(r, v, t) = P(r, t)U(r, v, t) (80)

where U(r, v, t) is orthogonal and positive definite.Substituting (80) into (79) and using the third property ofP(r, d, t) yields

P(r, d1, t)P(r, t)K + KP(r, t)P(r, d1, t)

+ P(r, v, t) + P(r, t)U(r, v, t) = 03×3 (81)

The unique solution of the above equation for the gain matrixnullprojection P(r, t)K is given by [30]

P(r, t)K (r, v, t) = −vec−1{[I3×3 ⊗ P(r, d1, t)

+ P(r, d1, t) ⊗ I3×3]−1

× vec[P(r, v, t) + P(r, t)U(r, v, t)]}

(82)

where ⊗ denotes the kronecker product of matrices.Equation (82) can be written in the compact form givenby (76). Solution uniqueness of (81) implies that thesymmetric matrix gain K (r, v, t) remains non-singular forall t ≥ 0. Accordingly, K (r, v, t) satisfies the condition ofProposition 5. Hence, in addition to rendering V (r, v, t)negative semidefinite, K (r, v, t) guarantees asymptotic


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stability of v over the invariant set of r, v and t values on whichV (r, v, t) = 0, and Lyapunov stability of v = 03×1 follows[24]. Since V (r, v, t) is radially unbounded with respect tov, Lyapunov stability of v = 03×1 is global. Moreover, it isnoticed from the expression of V (r, v, t) given by (63) andfrom (78) that the largest invariant set of r, v and t onwhich V (r, v, t) = 0 is the same invariant set on whichV (r, v, t) = 0, implying global asymptotic stability of theequilibrium point v = 03×1 [24]. Global asymptoticconvergence of the attitude vector r to the desired attitudevector rr(t) follows from Proposition 4. A

Since the orthogonal positive-definite matrix functionU(r, v, t) is arbitrary, it follows from (77) that there is noloss of generality in writing

P(r, v, t) + P(r, t)U(r, v, t) = P(r, t)Q (83)

where Q [ R3×3 is an arbitrary orthogonal and positive-definite constant matrix. Accordingly, the gain matrixnullprojection P(r, t)K given by (82) can be rewritten as

PK (r, t) = −vec−1{[P(r, d1, t)

⊕ P(r, d1, t)]−1vec[P(r, t)Q]} (84)

Increasing the magnitude of Q causes faster convergence of vto the origin, at the attendant cost of increasing the relativemagnitude of the auxiliary part in the control law.

12 Damped controls coefficientnullprojectorAlthough the CCNP has bounded elements, dependency ofCCNP on the unbounded vector A+(r, t) may causeundesirable behaviour of the auxiliary part in the controllaw ts during steady-state tracking response of time-varyingtrajectories. For this reason, a damped controls coefficientnullprojector (DCCN) Pd (r, e, t) is used in place ofP(r, t) in (56). The DCCN is defined as

Pd (r, e, t) := I3×3 −A+d (r, e, t)A(r, t) (85)

where e is a small positive number, and A+d (r, e, t) is given by

A+d (r, e, t) := AT(r, t)

A(r, t)AT(r, t) + e(86)

Therefore

limf�0

A+d (r, e, t) = 03×1 (87)

and consequently

limf�0

Pd (r, e, t) = I3×3 (88)

Hence, the DCCN maps the null-control vector to itself in

steady-state phase of response, during which the auxiliarypart of the control law converges to the null-control vector.Independency of nullprojection on the attitude state of thespacecraft substantially eliminates unnecessary abruptbehaviour of the control vector.

13 Exponentially factored GNDIcontrolThe GNDI control design does not guarantee an acceptablebehaviour of the GNDI control signal. If the initial value ofcontrols load B(r0, v0, 0) is big, then the magnitude of initialcontrol signal must be big also. Moreover, continuation ofservo-constraint dynamics enforcement causes a rapid decayof control signal by one or more orders of magnitude in avery short time compared to transient response time. Suchcontrol signal characteristics are extremely undesirable fromseveral technical aspects. For instance, if the controlactuator’s magnitude or rate limits are met, then the controlsystem fails to provide the required GNDI control signal. Inorder to avoid unnecessarily big GNDI control signalmagnitude and rate of change, the GNDI control signal isexponentially factored during early closed-loop control time as

te = [1 − exp(−ht)]ts (89)

The control signal te starts at zero values at t ¼ 0 and thenconverges to ts given by (56), where h . 0 determines therate of convergence.

Increasing the magnitude of the parameter h causes moresuppression of the control signal in the very early stage ofclosed-loop response, but it causes a rapid decay of theexponential term, which limits the benefit of theexponentially factored design as the control load B(r, v, t)may remain big after the exponential term vanishes. On theother hand, excessive decrease in magnitude of theparameter h causes the closed-loop control signal to remainsmall for a long time, which adversely affect the trackingperformance of the control system by delaying steady-statephase of response. A suitable compromise for the value ofh is in the order of 0.1.

To ensure continuity of the control signal, theexponentially factored signal te is applied for a sufficientlybig time interval before the GNDI signal ts takes over.

14 Control system designprocedureThe GNDI control system design methodology for trackingsmooth attitude trajectories is summarised in the followingsteps:

1. A desired spacecraft attitude trajectory rr(t) is prescribed,where rr is at least twice differentiable in t. The desiredangular velocity vector vr(t) is given by (19), where G(r) isgiven by (3).



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2. The attitude deviation norm measure dynamics given by(10) is specified such that the dynamics of f isasymptotically stable. A special choice is given by (15),where both c1 and c2 are strictly positive.

3. The expressions given by (12) and (13) for A(r, t) andB(r, v, t) are obtained, where r(r, t) is given by (4).

4. The control law ts given by (56) is obtained, whereA+

s (r, v, t) is given by (28), the controls coefficient null-projected gain matrix P(r, t)K is given by the expression of(84), the perturbed CCNP P(r, d, t) is given by (25) andthe constant matrix Q is arbitrary but orthogonal andpositive definite.

5. The exponentially factored control law te given by (89) isused in (2) until the exponential term vanishes, and thecontrol law ts is applied afterwards.

6. Equations (1) and (2) are integrated to obtain the closed-loop trajectories of r(t) and v(t).

15 Numerical simulationsThe first manoeuvre considered is a rest-to-rest slewmanoeuvre, aiming to reorient a spacecraft at an initialattitude given by r(0) to a different attitude given by rr(T ),where T is the duration of the manoeuvre. The spacecrafthas principal moments of inertia I11 ¼ 200 kg m2,I22 ¼ 150 kg m2, I33 ¼ 175 kg m2 and the control torqueactuators are mounted along principal axes. To insuresmooth transition to the new state, it is required that thespacecraft attitude variables follow the MRPs trajectoriesgiven by the following fifth-order polynomial [32]

rr(t) = rr(0) + 10t

T

( )3

−15t

T

( )4

+6t

T

( )5[ ]

× [rr(T ) − rr(0)] (90)

where rr(0) = [0 0 0]T, rr(T ) = [−0.70 0.80 −1.20]T, andT ¼ 180 s. To avoid oscillatory closed-loop state responseinduced by underdamped servo-constraint dynamics, it ispreferred to choose values of c1 and c2 that yieldoverdamped second-order servo-constraint dynamics.Additionally, damping ratio of the servo-constraintdynamics should be sufficiently big to produce a relativelyshort duration of the transient response, which can beobtained by choosing c1 to be of an order of magnitudelarger than c2. Values of second-order attitude deviationdynamics constants are chosen to be c1 = 1.0 and c2 = 0.1,resulting in an overdamped servo-constraint dynamics withdamping ratio equals to 1.58. The orthogonal positive-definite Lyapunov matrix Q is selected to be the identitymatrix, and the function h(d) is taken to be

h(d) = 1

1 + d(91)

where a design value d = 0.1 is chosen. Withr(0) = [0.25 −0.20 0.10]T and a dynamic scaling indexp ¼ 2, Figs. 1 and 2 show r and v trajectories, respectively.Fig. 3 shows time history of the GNDI control variablests1

and ts2, and compares with time history of the

exponentially factored GNDI control variables te1and te2

,where h ¼ 0.1 is selected. The control vector ts takesinitial value ts(0) = [−7.41 × 10−2 −2.47 × 10−2 4.94×10−2]T (not shown), and drops rapidly in magnitude to theorder of 1025 in less than 10 s. The exponential factoringlasts for the first 40 s, and it has a minor effect onspacecraft state response. However, it substantially improves

Figure 2 Rest-to-rest slew manoeuvre: angular velocitycomponents against t

Figure 1 Rest-to-rest slew manoeuvre: MRPs attitudeparameters against t

Figure 3 Rest-to-rest slew manoeuvre: control torquesagainst t


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the control signal by reducing its magnitude and rate ofchange. No discontinuities in the control variables att ¼ 40 s are tangible because te converges to ts in shortertime. Similar plots of ts3

and te3are obtained, but are not

shown.

The second manoeuvre considered is a trajectory trackingmanoeuvre. The desired attitude trajectory is defined byreference attitude parameters

rri(t) = cos 0.1t, i = 1, 2, 3 (92)

The spacecraft has principal moments of inertiaI11 ¼ 100 kg m2, I22 ¼ 60 kg m2, I33 ¼ 75 kg m2. Forcontrol torque actuators mounted along principal axes,dynamic scaling index p ¼ 4, d = 0.1, h(d) given by (91),e = 10−5, Q = I3×3, r(0) = [−0.5 0.4 0.8]T and v(0) =[0.2 −0.7 0.4]T. Figs. 4 and 5 show time response of thecomponents of attitude and angular velocity vectors r andv, respectively. Transient and asymptotic behaviours ofattitude and angular velocity variables indicate excellentclosed-loop control system performance. Fig. 6 shows timehistory of attitude deviation scalar f for different values ofdynamic scaling factor p. It is observed that increasing phas a favourable influence on attitude deviation andtracking performance. This is because the expression given

by (28) provides better approximations of the controlscoefficient’s MPGI for higher values of p as steady-stateresponse is approached, that is, as the controls coefficientsquared norm in the denominator of the expressionvanishes. However, it is noticed from Fig. 6 that no furtherimprovement of tracking performance is achieved byincreases p over 4, and that the value of f starts to oscillatewithin the order of e. Accordingly, more trackingperformance enhancement requires decreasing e. Finally,Fig. 7 shows time history of GNDI control variables ts1and ts2

and exponentially factored GNDI control variableste1

and te2, where h = 0.1 is selected. The ts signal suffers

from a high initial magnitude and a rapid rate of decay.The exponential factoring lasts for the first 5 s, and itsubstantially improves control signal by reducing itsmagnitude and rate of change with no major effect onspacecraft state response or tangible discontinuities in thecontrol vector components. Similar plots of ts3

and te3are

obtained, but is not shown.

16 ConclusionDriven by the advantages that generalised dynamic inversionprovide over classical dynamic inversion, this paper buildson the recently developed GNDI control paradigm byimproving the control law to yield asymptotic tracking ofdesired smooth trajectories. The DSGI in the particular

Figure 4 Trajectory tracking manoeuvre: MRPs attitudeparameters against t

Figure 5 Trajectory tracking manoeuvre: angular velocitycomponents against t

Figure 6 Trajectory tracking manoeuvre: f against t,e ¼ 1025

Figure 7 Trajectory tracking manoeuvre: control torquesagainst t



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part of the control law is capable of overcoming controlscoefficient generalised inversion singularity, and itconverges to the standard MPGI as the dynamic scalingfactor decays and closed-loop steady-state responseapproaches. The null-control vector in the auxiliary part ofthe control law is designed by means of novel semidefinitecontrol Lyapunov function and nullprojected controlLyapunov equation that utilise geometric features of theGNDI control law’s structure. The exponentially factoredGNDI control signal replaces the GNDI control signal inearly stage of the control time to improve control signalquality. Perfect plant mathematical modelling and an idealcontrol environment are assumed for the purpose ofconstructing the null-control vector. The presentconstruction of the null-control vector can be used as abase for designing adaptive and robust GNDI control lawsin the presence of input disturbances, measurement noisesand modelling uncertainties.

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