Generalised dynamic inversion spacecraft control design methodologies

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Aug 12, 2008 - Abstract: Generalised dynamic inversion control design methodologies for realisation of linear spacecraft attitude servo-constraint dynamics is ...
www.ietdl.org Published in IET Control Theory and Applications Received on 3rd February 2008 Revised on 12th August 2008 doi: 10.1049/iet-cta:20080044

ISSN 1751-8644

Generalised dynamic inversion spacecraft control design methodologies A.H. Bajodah Aeronautical Engineering Department, King Abdulaziz University, Jeddah, P.O. Box 80204, Saudi Arabia 21589 E-mail: [email protected]

Abstract: Generalised dynamic inversion control design methodologies for realisation of linear spacecraft attitude servo-constraint dynamics is introduced. A prescribed stable linear second-order time-invariant ordinary differential equation in a spacecraft attitude deviation norm measure is evaluated along solution trajectories of the spacecraft equations of motion, yielding a linear relation in the control variables. Generalised inversion of the relation results in a control law that consists of particular and auxiliary parts. The particular part works to drive the spacecraft attitude variables in order to nullify the attitude deviation norm measure, and the auxiliary part provides the necessary spacecraft internal stability by proper design of the involved null-control vector. Two constructions of the null-control vector are made, one by solving a state-dependent Lyapunov equation, yielding global spacecraft internal stability. The other is globally perturbed feedback linearising, but yields local stability of the spacecraft internal dynamics. The control designs utilise a damped generalised inverse to limit the growth of the controls coefficient Moore-Penrose generalised inverse as the steady-state response is approached. Both designs guarantee uniformly ultimately bounded attitude trajectory tracking errors. The null-control vector design freedom furnishes an advantage of the approach over classical inverse dynamics, because it can be used to reduce dynamic inversion control signal magnitude. The design methodologies are illustrated by two spacecraft slew and trajectory tracking manoeuvres.

1

Introduction

Generalised inversion is a well-known control system design methodology, with numerous applications in the fields of aerospace engineering and robotics. A fundamental advantage of generalised inversion is that it overcomes the limitations of dimensionality and rank that are related to the notion of inversion, which makes the inversion process capable of solving underdetermined problems where requirements can be satisfied in more than one course of action, as well as solving for best approximating solutions to overdetermined problems, when the requirements cannot be satisfied. Another fundamental property of generalised inversion is the characterisation of solution non-uniqueness. This is depicted by the Greville formula [1], which uses the Moore-Penrose generalised matrix inverse (MPGI) [2, 3] to obtain the general solution of a linear algebraic system equations and parameterises the corresponding linear algebraic system IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239 – 251 doi: 10.1049/iet-cta:20080044

matrix nullspace via a free nullvector that appears explicitly in the solution expression. The generalised inversion feature of nullspace parametrisation has been utilised in engineering analysis and design for the purpose of modelling, control and optimization. At the level of analysis, utilisation of this feature in the field of analytical dynamics was made by deriving the Udwadia-Kalaba equations of motion for constrained dynamical systems [4]. The corresponding free nullvector was chosen in order to optimise acceleration energy of the constrained system, yielding its natural accelerations, that is, those obeying Gauss’ principle of least constraints [5], or equivalently yielding constraint forces that satisfy D’Alembert’s principle of virtual work [6]. At the design level, nullspace parametrisation has been insightful in viewing the active perspective of servoconstrained motion, where the design requirements are 239

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www.ietdl.org formulated in terms of constraints on the system dynamics that are to be imposed using the available control authority, and the control forces are possibly obtained by utilising the same fundamental variational principles of mechanics that are used in modelling passively constrained dynamical systems. This has been used in attempts to synthesise the most natural control laws that realise servo-constraints [7, 8] and to optimise control laws for multibody systems in the context of the operational space framework [9, 10]. By observing that the control problem of a controllable dynamical system is a problem of non-uniqueness, and based on generalised inversion null-space parametrisation provided by the Greville formula, the concept of controls coefficient generalised inversion has been introduced in [11] for spacecraft attitude control, considering nullifying the deviation from desired spacecraft attitude trajectories to be a desired servo-constraint that is to be realised. The control design procedure begins by defining a norm measure function of the spacecraft’s attitude variables deviations from their desired values, and prespecifying a stable second-order linear differential equation in the measure function, resembling the desired servo-constraint dynamics. The differential equation is then transformed to a relation that is linear in the control vector by differentiating the norm measure function along the trajectories defined by the solution of the spacecraft’s state-space mathematical model. The Greville formula is utilised thereafter to invert this relation for the control law required to realise the desired stable linear servo-constraint dynamics. A controls coefficient generalised inversion law consists of auxiliary and particular parts, residing in the null-space of the controls coefficient and the range space of its generalised inverse, respectively. The free null-control vector in the auxiliary part is projected onto the controls coefficient nullspace by a null-projection matrix. Therefore the choice of the null-control vector does not affect dynamics of the deviation measure function, and it furnishes a design freedom in constructing the control law that is capable of realising that dynamics. Generalised inversion of the controls coefficient guarantees global realisation of feedback linearised attitude deviation dynamics. To fulfill the internal stability requirement, and inspired by the control law’s affinity in the null-control vector, the later has been chosen in [11] to be linear in the angular velocity vector, resulting in a stable perturbed feedback linearisation of the spacecraft internal dynamics. The control design provides simultaneous perturbed feedback linearising transformation of the spacecraft global dynamics and realisation of the prescribed linear servoconstraint dynamics. Based on the null-control vector design, this paper presents two spacecraft control design methodologies, following the controls coefficient generalised inversion paradigm. The first 240 & The Institution of Engineering and Technology 2009

methodology is Lyapunov based, and the null-control vector is chosen to be proportional to the spacecraft body angular velocity vector error from a reference angular velocity vector that is derived from the desired attitude trajectory. The proportionality matrix is obtained by solving a novel statedependent Lyapunov equation type that involves the controls coefficient null-projector (CCNP). The control design results in global internal dynamics stability. The second methodology enhances the concept of perturbed feedback linearisation originally introduced in [11]. Although providing local stability of the internal dynamics, the corresponding domain of attraction for internal stability can be arbitrarily enlarged by increasing the design controller constant matrix gain. The analysis provides explicit sufficiency bound on the required gain for a prescribed stability region. Feedback linearisation by dynamic inversion is known to be unfavourable compared to other types of nonlinear control methodologies because of its high actuating energy consumption. This reflects control system blindness to the type of dynamical system nonlinearity. For instance, high control signal magnitude may be required to eliminate plant nonlinearities that can be useful for the control process. The generalised dynamic inversion control introduced in this work is shown to be capable of avoiding this disadvantage by properly designing the null-control vector. The present algebraic controls coefficient generalised inversion approach complements the geometric space decomposition approach implemented in [12, 13]. The feedback control design utilises the concept of generalised inversion damping to limit the growth of the Moore-Penrose generalised inverse, resulting in desired linear attitude deviation dynamics realisation with uniformly ultimately bounded (UUB) trajectory tracking errors. The methodology reveals a tradeoff between trajectory tracking accuracy and damped generalised inverse stability based on the size of the generalised inversion damping factor. The prime contribution of this article is introducing theoretical foundations of the controls coefficient generalised inversion design and presenting two spacecraft attitude control system methodologies within its framework, exhibiting different closed-loop qualitative characteristics. A conclusion drawn from the article is that allowing for Lyapunov direct method to interfere with the inverse dynamics design is substantially economical to the control process.

2

Spacecraft mathematical model

The spacecraft mathematical model is given by the following system of kinematical and dynamical differential equations

r_ ¼ G(r)v,

r(0) ¼ r0

v_ ¼ J 1 v J v þ t,

v(0) ¼ v0

(1) (2)

IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239– 251 doi: 10.1049/iet-cta:20080044

www.ietdl.org where r [ R31 is the spacecraft vector of modified Rodrigues attitude parameters (MRPs) [14], v [ R31 is the vector of spacecraft angular velocity components in its body reference frame, J [ R33 is a diagonal matrix containing spacecraft’s body principle moments of inertia and t :¼ J 1 u [ R31 is the vector of scaled control torques, where u [ R31 contains the applied gas jet actuator torque components about the spacecraft’s principle axes. The cross product matrix x , which corresponds to a vector x [ R31 , is skew symmetric of the form 2

0 x ¼ 4 x3 x2

x3 0 x1

3 x2 x1 5 0

(9) in the pointwise-linear form A(r, t)t ¼ B(r, v, t)

where the vector-valued function A(r, t) : R31  [0, 1) ! R13 is given by A(r, t) ¼ 2zT (r, t)G(r)

B(r, v, t) ¼ 2[G(r)v  r_ r (t)]T [G(r)v  r_ r (t)] _ r, v)v þ G(r)J 1 v J v  r€ r (t)]  2zT (r, t)[G(  2c1 zT (r, t)[G(r)v  r_ r (t)]  c2 kz(r, t)k2 (12)

! (3)

3 Attitude deviation norm measure dynamics Let rr (t) [ R31 be a prescribed desired spacecraft attitude vector such that rr (t) is twice continuously differentiable in t. The spacecraft attitude deviation vector from rr (t) is defined as z(r, t) :¼ r  rr (t)

(4)

Consequently, the scalar attitude deviation norm measure function f : R31  [0, 1) ! R is defined to be the squared norm of z(r, t)

f ¼ kz(r, t)k2 ¼ kr  rr (t)k2

(5)

The first two time derivatives of f along the spacecraft trajectories given by the solution of (1) and (2) are @f @f f_ ¼ G(r)v þ @r @t ¼ 2zT (r, t)[G(r)v  r_ r (t)]

(6) (7)

and  T   f€ ¼ 2 G(r)v  r_ r (t) G(r)v  r_ r (t) þ 2zT (r, t) _ r, v)v þ G(r)[ J 1 v J v þ t]  r€ r (t)]  [G(

(8)

_ r, v) is the time derivative of G(r) obtained by where G( differentiating the individual elements of G(r) along the kinematical subsystem given by (1). We prespecify a desired stable linear second-order dynamics of f in the form

f€ þ c1 f_ þ c2 f ¼ 0,

(11)

and the scalar-valued function B(r, v, t) : R31  R31  [0, 1) ! R is

and the matrix valued function G(r) : R31 ! R33 is given by 1 1  rT r I33  r þ rrT G(r) ¼ 2 2

(10)

c1 , c2 . 0

4

Reference internal dynamics

Invertibility of the matrix G(r) makes it possible to solve explicitly for the angular velocity vector v, which takes the form

v ¼ G 1 (r)_r

(13)

Therefore a reference vector of dynamic variables vr (t) can be obtained from (13) by substituting the desired vector of kinematic variables rr (t) and its time derivative r_ r (t) in place of r and r_ , respectively, such that

vr (t) ¼ G 1 (rr (t))_rr (t)

(14)

5 Linearly parameterised attitude control laws The quasi-linear form given by (10) makes it feasible to assess realisability of the linear attitude deviation norm measure dynamics given by (9) in a pointwise manner.

Definition 1 (realisability of linear attitude deviation dynamics): For a given desired spacecraft attitude vector rr (t), the linear attitude deviation norm measure dynamics given by (9) is said to be realisable by the spacecraft equations of motion (1) and (2) at specific values of r and t if there exists a control vector t that solves (10) for these values of r and t. If this is true for all r and t such that z(r, t) = 031 , then the linear attitude deviation norm measure dynamics is said to be globally realisable by the spacecraft equations of motion.

(9)

With f, f_ and f€ given by (5), (7) and (8), it is possible to write IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239 – 251 doi: 10.1049/iet-cta:20080044

The row vector function A(r, t) is the controls coefficient of the attitude deviation norm measure dynamics given by (9) along the spacecraft trajectories, and the scalar function B(r, v, t) is the corresponding controls load.

Remark 1: The notion of global attitude deviation realisability does not imply global stability in the sense of Lyapunov. 241

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www.ietdl.org Rotational dynamics is not globally stabilisable because of the topological obstruction of the attitude rotation matrix, which is known to preclude the existence of globally stable equilibria for attitude dynamics [15]. For the proof of the following proposition, the reader is referred to [11].

Proposition 1 (linearly parameterised attitude control laws): For any desired spacecraft attitude vector rr (t), the linear attitude deviation norm measure dynamics given by (9) is globally realisable by the spacecraft equations of motion (1) and (2). Furthermore, the infinite set of all control laws realising that dynamics by the spacecraft equations of motion is parameterised by an arbitrarily chosen null-control vector y [ R31 as

t ¼ Aþ (r, t)B(r, v, t) þ P(r, t)y

(15)

where ‘Aþ ’ stands for the MPGI of the controls coefficient given by

Aþ (r, t) ¼

AT (r, t) , kA(r, t)k2

A(r, t) = 013

(16)

6 Perturbed controls coefficient null-projector Definition 2 (Perturbed controls coefficient nullP(r, d, t) is defined as projector): The perturbed CCNP e e P(r, d, t) :¼ I33  h(d)Aþ (r, t)A(r, t)

(20)

where h(d) : R11 ! R11 is any continuous function such that h(d) ¼ 1

if and only if

d¼0

6.1 Properties of the perturbed controls coefficient null-projector The following properties of the perturbed CCNP are utilised in the present development of the controls coefficient generalised inversion-based attitude tracking control. 1. The perturbed CCNP e P(r, d, t) is of full rank for all d = 0. 2. The CCNP P(r, t) commutes with the perturbed CCNP e P(r, d, t) for all d [ R. Furthermore, their matrix multiplication yields the CCNP itself, that is

and P(r, t) [ R33 is the CCNP given by P(r, d, t) ¼ e P(r, d, t)P(r, t) ¼ P(r, t) P(r, t)e þ

P(r, t) ¼ I33  A (r, t)A(r, t)

(17)

Any choice of the null-control vector y in the control law expression given by (15) yields a solution to (10). Therefore the choice of y does not affect realisability of the linear attitude deviation norm measure dynamics given by (9). Nevertheless, the choice of y substantially affects the spacecraft internal state response [8]. In particular, an inadequate choice of y can destabilise the spacecraft internal dynamics given by (2) or causes unsatisfactory closed-loop performance. Substituting the control laws expressions given by (15) in the spacecraft’s equations of motion (1) and (2) yields the following parametrisation of the infinite set of spacecraft closedloop systems equations that realise the dynamics given by (9)

r_ ¼ G(r)v,

r(0) ¼ r0

(18)

3. The CCNP P(r, t) commutes with its inverted perturbation e P 1 (r, d, t) for all d = 0. Furthermore, their matrix multiplication yields the CCNP itself, that is e P 1 (r, d, t) ¼ P(r, t) (22) P 1 (r, d, t)P(r, t) ¼ P(r, t)e Proofs of the first and third properties are found in [11]. The second property is verified by direct evaluation of P(r, t) and e P(r, d, t) expressions given by (17) and (20).

7 Damped controls coefficient generalized inverse The expression of Aþ (r, t) given by (16) implies that lim

A(r,t)!013

v_ ¼ J 1 v J v þ Aþ (r, t)B(r, v, t) þ P(r, t)y v(0) ¼ v0 (19) 242 & The Institution of Engineering and Technology 2009

(21)

Aþ (r, t) ¼ 131

(23)

Implications of the controls coefficient singularity on closed-loop stability is depicted by the following singularity analysis. IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239– 251 doi: 10.1049/iet-cta:20080044

www.ietdl.org 7.1 Controls coefficient singularity analysis

7.2 Damped controls coefficient generalised inverse

The definition of A(r, t) given by (11) implies that

For the purpose of controlling the growth of the CCGI Aþ (r, t) in the control law given by (15), the damped CCGI Aþ d (r, b, t) is introduced.

lim

z(r,t)!031

A(r, t) ¼ 013

(24)

for all finite values of r [ R31 . Therefore a control law t globally realises the linear attitude deviation norm measure dynamics of (9) by the spacecraft equations of motion (1) and (2) only if lim A(r, t) ¼ 013

(25)

t!1

lim

8 T A (r, t) > > > < kA(r, t)k2 : kA(r, t)k  b Aþ d (r, b, t) :¼ > > AT (r, t) > : : kA(r, t)k , b b2

kAþ (r, t)k ¼ 1

(26)

The above definition implies that kAþ d (r, b, t)k 

However, a fundamental property of the matrix G(r) is [16]

smin (G(r)) ¼ smax (G(r)) ¼ s(G(r)) 

1 4

(27)

Therefore

z(r,t)!031

kG T (r)z(r, t)k ¼ s(G(r))kz(r, t)k

kAþ d (r, b, t)k ¼

2 kG T (r)z(r, t)k ¼ 0 z(r,t)!031 b2 lim

(39) þ and that Aþ d (r, b, t) pointwise converges to A (r, t) as b vanishes.

(29)

and that kAþ (r, t)zT (r, t)k  kAþ (r, t)kkz(r, t)k

(38)

(28)

and the definition of A(r, t) given by (11) implies that 1 2s(G(r))kz(r, t)k

1 b

and that lim

kAþ (r, t)k ¼

(37)

where the scalar b is a positive ‘generalised inverse damping factor’.

which implies from (23) that z(r,t)!031

Definition 3 (damped controls coefficient generalised inverse): The damped CCGI is defined as

(30)

7.3 Damped controls coefficient null-projector The damped controls coefficient null-projector is a modified controls coefficient nullprojector with vanishing dependency on the steady-state attitude variables.

¼

1 kz(r, t)k 2s(G(r))kz(r, t)k

(31)

Definition 4 (damped controls coefficient nullprojector): The damped CCNP P d (r, b, t) is defined as

¼

1 2 2s(G(r))

(32)

P d (r, b, t) :¼ I33  Aþ d (r, b, t)A(r, t)

(40)

where Aþ d (r, b, t) is given by (37).

and that

The above definition implies that

kAþ (r, t)zT (r, t)z(r, t)k  kAþ (r, t)zT (r, t)kkz(r, t)k  2kz(r, t)k

(33) (34)

Inequalities (32) and (34) imply that lim

z(r,t)!031

kAþ (r, t)zT (r, t)k  2

(35)

and lim

z(r,t)!031

kAþ (r, t)zT (r, t)z(r, t)k ¼ 0

IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239 – 251 doi: 10.1049/iet-cta:20080044

(36)

P d (r, b, t) 8 I33 > > ! > > T > >  (G ( r )z( r , t) > > : > > > zT (r, t)G(r)) > > > < kG T (r)z(r, t)k2 ¼ > I33 > > ! > > T > >  (4G (r)z(r, t) > > : > > zT (r, t)G(r)) > > > : b2

kA(r, t)k T

¼ k2G (r)z(r, t)k  b

kA(r, t)k ¼ k2G (r)z(r, t)k , b T

(41) 243

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www.ietdl.org and consequently lim

z(r,t)!031

P d (r, b, t) ¼ I33

(42)

Hence, the damped CCNP maps the null-control vector to itself during steady-state phase of response, during which the auxiliary part of the control law converges to the nullcontrol vector.

8 Controls coefficient generalised inversion control For convenience, let the control law expression given by (15) be written as

t ¼ H1 (r, v, t)v þ H2 (r, t) þ P(r, t)y

(43)

where _ r, v) þ G(r)J 1 v J H1 (r, v, t) ¼ 2Aþ (r, t)zT (r, t)[G( þ c1 G(r)]  2Aþ (r, t)[G(r)v  r_ r (t)]T G(r) (44) and H2 (r, t) ¼ c2 Aþ (r, t)zT (r, t)z(r, t) þ 2Aþ (r, t)zT (r, t)  [€rr (t) þ c1 r_ r (t)]  2Aþ (r, t)k_rr (t)k22

(45)

To avoid closed-loop instability because of the CCGI unstable dynamics described by (26), we define H1d (r, v, b, t) and H2d (r, b, t) by replacing Aþ (r, t) in the last terms of the H1 (r, v, t) and H2 (r, t) expressions given by (44) and (45) with the damped CCGI Aþ d (r, b, t), such that _ r, v) H1d (r, v, b, t) ¼ 2A (r, t)z (r, t)[G( þ

Theorem 1 (implication on attitude stability): Let the control law td be given by (48), where the null-control vector y is arbitrary. Then the desired attitude deviation dynamics given by (9) is realised by the spacecraft equations of motion (1) and (2) for all values of f in the domain Df given by Df : f 

b2 4s2 (G(r))

(49)

and the resulting attitude trajectory tracking errors are UUB.

Proof: Let fd be a norm measure function of the attitude deviation obtained by applying the control law given by (48) to the spacecraft equations of motion (1) and (2), and let f_ d , f€ d be its first two time derivatives. Hence fd :¼ fd (r, t) ¼ f(r, t)

(50)

f_ d :¼ f_ d (r, v, t) ¼ f_ (r, v, t)

(51)

f€ d :¼ f€ d (r, v, td , t) ¼ f€ (r, v, t, t) þ A(r, t)td  A(r, t) (52) where t and td are given by (43) and (48), respectively. Adding c1 f_ d þ c2 fd to both sides of (52) yields

f€ d þ c1 f_ d þ c2 fd ¼ f€ þ c1 f_ þ c2 f þ A(r, t)td  A(r, t)t (53) ¼ A(r, t)[td  t] (54) Let the state vector Fd [ R21 be defined as Fd ¼ [fd

f_ d ]T

(55)

T

The attitude deviation norm measure closed-loop dynamics can be written in the state-space form as

þ G(r)J 1 v J þ c1 G(r)]  2Aþ _ r (t)]T G(r) d (r, b, t)[G(r)v  r (46)

_ ¼ L F þ L (r, v, b, t)v þ D (r, b, t) F d 11 d 12 1

(56)

where the strictly stable system matrix L11 [ R22 is

and



H2d (r, b, t) ¼ c2 Aþ (r, t)zT (r, t)z(r, t) þ 2Aþ (r, t)zT  (r, t)[€rr (t) þ c1 r_ r (t)]

rr (t)k2  2Aþ d (r, b, t)k_

(47)

The other terms in H1 (r, v, t) and H2 (r, t) involving the CCGI Aþ (r, t) are left unaltered, because they remain bounded according to inequality (35) and (36) as the closedloop system reaches steady state. The control vector td is defined as

td ¼ H1d (r, v, b, t)v þ H2d (r, b, t) þ P d (r, b, t)y (48) 244 & The Institution of Engineering and Technology 2009

L11

0 ¼ c2

1 c1

 (57)

and the matrix-valued function L12 (r, v, b, t) : R31  R31  (0, 1)  [0, 1) ! R23 is 2 L12 (r, v, b, t) ¼ 4

3

013

5

A(r, t)[H1d (r, v, b, t)  H1 (r, v, t)] (58) and the matrix-valued function D1 (r, b, t) : R31  IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239– 251 doi: 10.1049/iet-cta:20080044

www.ietdl.org (0, 1)  [0, 1) ! R21 is 2 6 D1 (r, b, t) ¼ 6 4

0

Consequently, a class of closed-loop dynamical subsystems realising the dynamics given by (9) is obtained by substituting the control law given by (63) in (2), and it takes the form

3

7 7 A(r, t)[H2d (r, b, t)  H2 (r, t) 5 þP d (r, b, t)  P(r, t)]

(59)

v_ ¼ [J 1 v J þ H1d (r, v, b, t) þ P d (r, b, t)K ]v þ H2d (r, b, t)  P d (r, b, t)K vr (t)

(64)

þ

The expression of A (r, t) given by (16) is identical to the expression of Aþ d (r, b, t) given by (37) for values of r and t that satisfy kA(r, t)k  b. These values of r and t can be described in terms of f by noticing that the singular values of the matrix G(r) are repeated, so that kA(r, t)k ¼ k2zT (r, t)G(r)k ¼ 2s(G(r))kz(r, t)k

The last two terms in the above closed-loop dynamical subsystem are forcing terms that drive the closed-loop internal dynamics of the spacecraft to realise the desired attitude deviation norm measure dynamics.

(60)

Theorem 2 (globally internally stable CCGI trajectory tracking control): Let the control law td be given by (48),

which implies from the definition of f that the corresponding set Df of f values is given by (49). Therefore for values of f in Df , L12 (r, v, b, t) ¼ 023 and D1 (r, b, t) ¼ 021 , expressions (48) for td and (43) for t are identical, and the desired linear attitude deviation dynamics given by (9) is realised. Nevertheless, for values of f in the bounded open complement set given by

where the null-control vector y is given by (62). Then the gain matrix K can be designed such that closed-loop dynamical subsystem given by (2) is GUUB.

Df c : f ,

b2 4s2 (G(r))

(61)

the definition of Aþ (r, t) is different from the definition of Aþ d (r, b, t), and the desired attitude dynamics is not realised. Instead, the dynamics given by (54) is the one that is realised over this bounded domain, resulting in uniformly ultimately bounded attitude trajectory tracking errors rather than in asymptotic attitude tracking. A Stability of internal dynamics is the most important factor to be considered in designing the null-control vector y. The structure of the control law td has a special feature, namely the affinity of its auxiliary part in y, which provides pointwise-linear parametrisation to the nonlinear control law. In the following two sections, we provide two different designs of the null-control vector, each one serves a different objective that is related to the closed-loop system performance.

8.1 Globally internally stabilising null-control vector Let the null-control vector y be chosen as y ¼ K (v  vr (t))

(62)

where vr (t) is given by (14), and K [ R33 is a matrix gain that is to be determined. Hence, a class of control laws that realise the attitude deviation norm measure dynamics given by (9) is obtained by substituting this choice of y in (48) such that

td ¼ [H1d (r, v, b, t) þ P d (r, b, t)K ]v þ H2d (r, b, t)  P d (r, b, t)K vr (t) IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239 – 251 doi: 10.1049/iet-cta:20080044

(63)

Proof: The closed-loop spacecraft internal system is obtained by omitting the two forcing terms in the dynamical subsystem given by (64) such that   v_ ¼ J 1 v J þ H1d (r, v, b, t) þ P d (r, b, t)K v

(65)

A stabilising matrix gain K for the spacecraft internal dynamics can be synthesised by utilising the following squared Euclidean norm of the spacecraft angular velocity vector as a control Lyapunov function V ¼ kvk2

(66)

Differentiating V along the trajectories of the system given by (65) and noticing skew-symmetry of v yields   V_ ¼ 2vT J 1 v J þ H1d (r, v, b, t) þ P d (r, b, t)K v ¼ vT [H1d (r, v, b, t) þ HT1d (r, v, b, t) þ P d (r, b, t)K þ K P d (r, b, t)]v where the matrix gain K is chosen to be symmetric. Global exponential stability of the equilibrium point v ¼ 031 is guaranteed if V_ remains negative-definite as the spacecraft dynamics evolves in time, which implies the existence of a positive-definite constant matrix Q [ R33 such that the Lyapunov equation H1d (r, v, b, t) þ HT1d (r, v, b, t) þ P d (r, b, t)K þ K P d (r, b, t) þ Q ¼ 0

(67)

is satisfied for all t  0. Due to rank deficiency of the damped CCNP P d (r, b, t), the above written Lyapunov equation does not have a unique stabilising solution for the matrix gain K. Nevertheless, the second perturbed CCNP property given by (21) implies that over the domain given by 245

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www.ietdl.org (49), the closed-loop dynamical subsystem given by (72) can be written as

(49), (67) can be written as P(r, d, t)P d (r, b, t)K H1d (r, v, b, t) þ HT1d (r, v, b, t) þ e P(r, d, t) þ Q ¼ 0 þ K P d (r, b, t)e

e1 (r, d, t)][ J 1 v J v v_ ¼ [I  P d (r, b, t)P þ H1d (r, v, b, t)v þ H2d (r, b, t)]

(68)

P 1 (r, d, t)k(v  vr (t)) þ P d (r, b, t)e

where d is any non-zero real scalar. Therefore the unique solution for the gain matrix damped null-projection P d (r, b, t)K that exponentially stabilises the dynamical subsystem given by (65) is

Therefore the closed-loop dynamical subsystem given by (72) is a perturbationfrom the linear system

P d (r, b, t)K ¼ vec1 {[I33  e P(r, d, t) þ e P(r, d, t) I33 ]1

v_ ¼ k(v  vr (t))

 vec[H1d (r, v, b, t)þ HT1d (r, v, b, t) þ Q]} P(r, d, t) e P(r, d, t)] ¼ vec {[e 1

1

 vec[H1d (r, v, b, t)þ HT1d (r, v, b, t) þ Q]} (69) where  and  are, respectively, the kronecker product and sum of matrices, and vec is the matrix vectorising operator [17, p. 251]. Since d is arbitrary, the control law td 1 given by 



td1 ¼ H1d (r, v, b, t) þ P d (r, b, t)K v

(70)

renders the equilibrium point v ¼ 031 of the spacecraft dynamical subsystem given by (2) globally exponentially stable for any real value of d. Global exponential stability of the internal dynamical subsystem given by (65) in addition to global boundedness of the two forcing terms that drive the closed-loop dynamical subsystem given by (64) imply GUUB of the closed-loop dynamical subsystem given by (64). A

8.2 Perturbed feedback-linearising null-control vector

(73)

(74)

that is obtained by perturbing the CCNP P(r, t) in (73) in the manner given by (20). The dynamical system given by (74) together with the kinematics given by (9) constitute a feedback linearising transformation of the global spacecraft closed-loop dynamics over the domain given by A(r, t)  b, realised up to a perturbation e P(r, d, t) from the CCNP.

Theorem 3 (perturbed feedback linearisation CCGI trajectory tracking control): Let the control law td be given by (48), where the null-control vector y is given by (71). Then for any initial condition of the spacecraft systems (1) and (2), there exists a constant gain matrix k such that closed-loop dynamical subsystem given by (2) is UUB.

Proof: The closed-loop dynamical subsystem given by (72) can be rewritten as v_ ¼ h1 (r, v, t)v þ h2 (r, t)v þ D2 (r, b, t)

(75)

h1 (r, v, t) ¼ [I  P d (r, b, t)][J 1 v J þ H1d (r, v, b, t)] þ k

(76)

h2 (r, t) ¼ [I  P d (r, b, t)]k

(77)

where

and

Let the null-control vector y be chosen as y ¼ J 1 v J v  H1d (r, v, b, t)v  H2d (r, b, t) þ k(v  vr (t))

(71)

where k [ R33 is a strictly stable constant matrix gain, and vr (t) is given by (14). Hence, another class of control laws that realise the attitude deviation norm measure dynamics given by (9) is obtained by substituting this choice of y in (48). The corresponding closed-loop dynamical subsystems realising the dynamics given by (9) are obtained by substituting this class of control laws in (2), resulting in 1

(72)

Nevertheless, the third property of the perturbed CCNP e P(r, d, t) given by (22) implies that over the domain given by 246 & The Institution of Engineering and Technology 2009

D2 (r, t) ¼ [I  P d (r, b, t)]H2d (r, b, t)  P d (r, b, t)kvr (t)

(78)

Stability of the first part of the system (75) given by

v_ ¼ h1 (r, v, t)v

(79)

can be analysed by Lyapunov indirect method by verifying that its Jacobian at v ¼ 031 is given by



v_ ¼ [I  P d (r, b, t)][J v J v þ H1d (r, v, b, t)v þ H2d (r, b, t)] þ P d (r, b, t)k(v  vr (t))

and

@[h1 (r, v, t)v] jv¼031 ¼ h1 (r, 031 , t) ¼ k @v

(80)

which is strictly stable. Therefore for any strictly stable k [ R33 , for any bounded trajectory r(t) and t . 0, there IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239– 251 doi: 10.1049/iet-cta:20080044

www.ietdl.org exists a domain of attraction Dv , R31 of v ¼ 031 such that the system given by (79) is locally exponentially stable. Moreover, it follows from Lyapunov direct method that for all v [ Dv and for any positive-definite constant matrix Qv , there exists a positive-definite constant matrix Pv [ R33 , such that the following inequality is satisfied

vT [hT1 (r, v, t)Pv þ Pv h1 (r, v, t)]v , vT Qv v

(81)

Next, we consider stability of the first and second parts of the system (75) given by

v_ ¼ h1 (r, v, t)v þ h2 (r, t)v

(82)

In viewing h2 (r, t) as a perturbation from h1 (r, v, t), (77) implies that

decoupled system

j_ ¼ F 1 (r, v, t)j

(90)

For values of the matrix gain k that are strictly stable, the equilibrium point j ¼ 051 of the system (90) is exponentially stable for all j [ Dj , where Dj : R 2  D v

(91)

Therefore for any positive-definite constant matrix Qj [ R55 , for any bounded r(t) [ R3 and t . 0, and for all j [ R51 in the domain of attraction Dj of j ¼ 051 , there exists a Lyapunov function [18, p. 167] V j ( j) ¼ jT Pj j

(92)

kh2 (r, t)vk  kI  P d (r, b, t)kkkkkvk ¼ smax (k)kvk (83) where Pj [ R55 is a positive-definite constant matrix, such that the time derivative of Vj along the trajectories of the system given by (90) is negative definite, resulting in

On the other hand, (42) implies that lim h2 (r, t) ¼ 033

(84)

t!1

The conditions given by (83) and (84) imply that the equilibrium point v ¼ 031 of the system given by (82) is exponentially stable over Dv . Finally, stability of the forced closed-loop dynamical subsystem of (75) and the corresponding spacecraft closed-loop stability are analysed by augmenting the dynamical subsystem with the kinematical state-space model given by (56). The augmented state-space model takes the form

j_ ¼ F 1 (r, v, t)j þ F 2 (r, v, b, t)j þ D(r, b, t)

jT [F T1 (r, v, t)Pj þ Pj F 1 (r, v, t)]j , jT Qj j

for all j [ Dj . The first and second parts of the augmented state-space model of (85) is the system

j_ ¼ F 1 (r, v, t)j þ F 2 (r, v, b, t)j

(85)

 @Vj (j)  F 1 (r, v, t)j þ F 2 (r, v, b, t)j @j ¼ 2jT Pj [F 1 (r, v, t)j þ F 2 (r, v, b, t)j]

(86)

¼ jT [F T1 (r, v, t)Pj þ Pj F 1 (r, v, t)]j

and the block-diagonal matrix F 1 (r, v, t) is  F 1 (r, v, t) ¼

023 L11 032 h1 (r, v, t) þ h2 (r, t)

þ 2jT Pj F 2 (r, v, b, t)j

 (87)

L12 (r, v, b, t) 0 F 2 (r, v, b, t) ¼ 22 032 033

D(r, b, t) ¼

D1 (r, b, t) D2 (r, b, t)

A norm bound on the last term in (95) is

(96)

 (88) where

indicating a unidirectional dynamic coupling, that is, the internal dynamics is independent of the attitude deviation dynamics, and the driving input vector D(r, b, t) is 

(95)

2jT Pj F 2 (r, v, b, t)j  2lmax (Pj )kF 2 (r, v, b, t)kkjk2

and the cross-coupling matrix F 2 (r, v, b, t) is 

(94)

Evaluating the time derivative of Vj along the trajectories of the system given by (94) yields

where the augmented state is

j ¼ [FTd vT ]T

(93)

¼ kA(r, t)[H1d (r, v, b, t)  H1 (r, v, t)]k ¼ 2k(A(r, t)Aþ d (r, b, t)  1)

 (89)

The first part of the augmented state-space model is the IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239 – 251 doi: 10.1049/iet-cta:20080044

kF 2 (r, v, b, t)k ¼ kL12 (r, b, t)k

 [G(r)v  r_ r (t)]T G(r)k

(97)

Therefore a bound on the expression of (95) is obtained from 247

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www.ietdl.org given by (85) is

(93) and (96) as  @Vj (j)  F 1 (r, v, t)j þ F 2 (r, v, b, t)j @j

(98)

V_ j (r, j, b, t) ¼ 2jT Pj [F 1 (r, v, t)j þ F 2 (r, v, b, t)j] þ 2jT Pj D(r, b, t)

, lmin (Qj )kjk2 þ 2lmax (Pj )kF 2 (r, v, b, t)kkjk2 ¼ [ lmin (Qj ) þ 4lmax (Pj )

A norm bound on the second term in (108) is

_ r (t)]T G(r)k]kjk2  k(A(r, t)Aþ d (r, b, t)  1)[G(r)v  r  ej kjk

(108)

2

2jT Pj D(r, b, t)  2lmax (Pj )kD(r, b, t)kkjk

(99)

(109)

Therefore (99) and (109) imply that

where

V_ j (r, j, b, t)  ej kjk2 þ 2lmax (Pj )kD(r, b, t)kkjk

ej ¼ lmin (Qj ) þ 4lmax (Pj )   2 T kG (r)z(r, t)k þ 1 (s(G(r))kvk  b þ k_rr (t)k)s(G(r))

(110)

for all j [ Dj . Rewriting ej as (100)

If Qj is chosen such that

ej ¼ (1  u)ej þ uej , u [ (0, 1)

(111)

then (110) can be written as



 2 T lmin (Qj ) . 4lmax (Pj ) kG (r)z(r, t)k þ 1 (s(G(r))kvk b þ k_rr (t)k)s(G(r))

V_ j (r, j, b, t)  (1  u)ej kjk2 þ uej kjk2 þ 2lmax (Pj )kD(r, b, t)kkjk

(101)

then ej is negative, which implies that the equilibrium point j ¼ 051 of the nonlinear system of (94) is locally exponential stable over the domain of attraction Dj . An upper bound on Qj is obtained from (93) by evaluating F 1 (r, v, t) at v ¼ 031 , such that

lmin (Qj ) , 2kF T1 (r, v ¼ 031 , t)kkPj k

(102)

(112)

Hence, if the stable matrix gain k is chosen such that (106) is satisfied, then for all j such that kjk .

2lmax (Pj ) kD(r, b, t)k uej

(113)

the following inequality holds

¼ 2kL11 kkh1 (r, v ¼ 031 , t) þ h2 (r, t)kkPj k V_ (r, v, b, t) , (1  u)ej kjk2

(103)  4smax (L11 )smax (k)smax (Pj )

(104)

¼ 4smax (L11 )smax (k)lmax (Pj )

(105)

Inequalities (101) and (105) are used to obtain the following sufficient condition on the matrix gain k for local exponential stability of the system given by (94) over Dj   s(G(r)) s(G(r))kv(0)k þ sup k_rr (t)k smax (L11 ) t   2 s(G(r))kz(031 , 0)k þ 1  b

smax (k) .

(106)

(114)

which implies that trajectories of the system given by (85) are UUB over Dj , the corresponding j trajectories enter the complement domain of the domain given by (113) in finite time, the complement domain becomes an invariant set for j, and uniform ultimate boundedness of v follows. Norm bounds on the components of the driving vector D(r, b, t) given by (59) and (77) are

rd k2 kD1 (r, b, t)k  2k  A(r, t)Aþ d (r, b, t) þ 1kk_ þ kA(r, t)P d (r, b, t)k

(115)

and where   r ¼ arg max {kr(0)k, sup krr (t)k}

kD2 (r, b, t)k  2c2 kz(r, t)k þ 4k€rr (t) þ c1 r_ r (t)k (107)

t

and v(0) [ Dv according to (81). The time derivative of Vj along the trajectories of the augmented state-space model 248 & The Institution of Engineering and Technology 2009

2 þ k_rr (t)k2 þ smax (k)kvr (t)k b

(116)

A IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239– 251 doi: 10.1049/iet-cta:20080044

www.ietdl.org 9 Control system design procedure The procedures for designing CCGI attitude tracking control systems are summarised in the following steps 1. A desired spacecraft attitude trajectory rr (t) is prescribed, where rr is at least twice differentiable in t. The desired angular velocity vector vr (t) is accordingly defined by (14). 2. The coefficients c1 and c2 in the attitude deviation norm measure dynamics (9) are chosen such that the dynamics of f is stable. This implies that both c1 and c2 are strictly positive. 3. The expressions given by (11) and (12) for A(r, t) and B(r, v, t) are obtained, where G(r) and z(r, t) are given by (3) and (4), respectively. 4. The CCGI Aþ (r, t) given by (16) is modified in the manner of (37), and Aþ d (r, b, t) is used to define the expressions of P d (r, b, t), H1d (r, v, b, t) and H2d (r, b, t) according to (40), (46) and (47), respectively. 5. The control law td given by (48) is applied. Two choices of the null-control vector y are made to achieve two different purposes:

(a) Global internal stability: Null-control vector is taken to be linear in the difference between the spacecraft angular velocity vector v and the desired angular velocity vector vr (t) according to (62). Projection of the proportionality gain matrix K on the damped controls coefficient nullspace given by P d (r, b, t)K is given by the expression of (69). The involved perturbed CCNP e P(r, d, t) is given by (20), and the constant matrix Q is arbitrary but positive definite. The control law takes the form of (63). (b) Perturbed feedback linearisation: Null-control vector is given by (71), where the constant matrix gain k [ R33 is strictly stable and satisfies (106) for guaranteed internal stability, and v(0) [ Dv . For a given value of the matrix gain k, a specific value of v is in Dv if (81) is satisfied for some positive-definite constant matrices Pv , Qv [ R33 . The domain Dv can be arbitrarily enlarged by decreasing lmax (k) so that (81) remains satisfied.

constants in (9) are chosen to be c1 ¼ 0:3 and c2 ¼ 0:1. The first spacecraft manoeuvre considered is a rest-to-rest slew type, aiming to reorient the spacecraft from its initial attitude r(0) to a different attitude r(T ), where T is the duration of the manoeuvre. To ensure smooth transition to the new state, it is required that the spacecraft attitude follows the MRPs trajectories shown in Fig. 1 and given by the following fifth-order polynomial [19]   3  t 4  t 5  t rr (t) ¼ r0 þ 10  15 þ6 [r(T )  r0 ] T T T (117) where r0 ¼ [0 0 0]T . With r(0) ¼ [0:25 0:20 0:10]T , r(T ) ¼ [ 0:70 0:80 1:20]T , T ¼ 180 sec, and generalised inversion damping factor b ¼ 0:006, Figs. 2 and 3 show r1 and v1 trajectories, respectively. The two figures compare the globally internally stabilising (GIS) and the perturbed feedback linearising (PFL) generalised dynamic inversion control designs with the classical feedback linearisation (FL) dynamic inversion control design [19] that is made by means of the vectorial form alternative to the one given by (9) and having the same values of c1 and c2 . The PFL design gain matrix is chosen to be k ¼ 0:2I33 , and the selected GIS Lyapunov design matrix is Q ¼ 0:3I33 . The three designs reflect similar performance, despite of the asymptotic convergence behaviour of the FL design. The advantage of the present approach is revealed by the required scaled control torque magnitude shown in Fig. 4, which is approximately half the magnitude required by the FL control design about spacecraft principal axis 1 at the manoeuvre initial time. Similar plots for the remaining MRPs and body angular velocity components are obtained, and plots with the same characteristics of the required scaled control torques magnitudes about the remaining two principal axes are obtained, but are not shown. The second manoeuvre is tracking the reference trajectory defined by rr i (t) ¼ cos 0:1t, i ¼ 1, 2, 3. With the initial conditions r(0) ¼ [0:42 0:35 0:26]T and v(0) ¼ [0:30 0:61 0:30]T , Figs. 5 and 6 show plots of r1 and v1 for the GIS and PFL generalised dynamic inversion

6. Integrate (1) and (2) to obtain the trajectories of r(t) and v(t), where u ¼ J t. The resulting trajectory tracking errors are UUB according to (61).

10

Numerical simulations

The spacecraft model used for numerical simulations has principal inertia parameters I1 ¼ 200 kg-m2, I2 ¼ 150 kgm2, I3 ¼ 175 kg-m2, and the servo-constraint dynamics IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239 – 251 doi: 10.1049/iet-cta:20080044

Figure 1 Rest-to-rest slew manoeuvre: MRPs reference attitude trajectories 249

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Figure 2 Rest-to-rest slew manoeuvre: response of MRP r1

Figure 5 Sinusoidal response of MRP r1

trajectory

tracking

manoeuvre:

Figure 3 Rest-to-rest slew manoeuvre: angular velocity about spacecraft axis 1 Figure 6 Sinusoidal trajectory tracking manoeuvre: angular velocity about spacecraft axis 1 control designs in addition to the FL dynamic inversion control design. The PFL design matrix is taken to be k ¼ 0:7I33 , the GIS Lyapunov design matrix Q is selected to be the identity I33 , and the generalised inversion damping factor value is b ¼ 0:20. The tracking performance is tangibly in favour of the FL control design because of the UUB trajectory tracking errors behaviour that corresponds to the two other designs. However, the GIS required scaled control torque magnitude shown in Fig. 7 is order of magnitude smaller than that corresponding to the FL control design about spacecraft principal axis 1, which is extremely advantageous and

implying that the additional design degree of freedom that is provided by the null-control vector construction has the potential to reduce the control signal magnitude, and therefore to eliminate the main disadvantage of feedback linearisation via dynamic inversion. Similar plots for the remaining MRPs and body angular velocity components are obtained, and plots with the same characteristics of the required scaled control torques magnitudes about the remaining two principal axes are obtained, but are not shown.

Figure 4 Rest-to-rest slew manoeuvre: scaled control torque 1

Figure 7 Sinusoidal trajectory tracking manoeuvre: scaled control torque 1

250 & The Institution of Engineering and Technology 2009

IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239– 251 doi: 10.1049/iet-cta:20080044

www.ietdl.org 11

Conclusion

Two controls coefficient generalised inversion design methodologies for spacecraft attitude control are presented. The two control designs are based on two constructions of the null-control vector in the generalised inversion-based Greville formula for general solution of linear algebraic system equations. The closed-loop attitude dynamics depends on the predetermined attitude deviation linear servo-constraint dynamics and is invariant under the choice of the null-control vector except during time instants at which the generalised inversion damping is activated. However, the internal dynamics is substantially affected by the null-control vector. Accordingly, one construction of the null-control vector is made in order to globally stabilise internal dynamics. The other construction is made in order to globally linearise internal dynamics, forming together with the predetermined linear attitude servo-constraint dynamics a feedback linearisation transformation of the spacecraft global dynamics and a feedback linearisation of the attitude deviation dynamics, realised up to a perturbation from the controls coefficient null-projector. The globally internally stabilising controller exhibits favourable control signal behaviour, as it avoids the blind cancellation of dynamical system nonlinearity that is involved in classical inverse dynamics control. Altering the value of generalised inversion damping factor reveals the tradeoff between trajectory tracking accuracy and damped generalised inverse stability.

12

References

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[7] UDWADIA F.E.: ‘A new perspective on the tracking control of nonlinear structural and mechanical systems’, Proc. R. Soc. Lond. A, 2003, 459, pp. 1783 – 1800 [8] BAJODAH A.H., HODGES D.H., CHEN Y.-H.: ‘Inverse dynamics of servoconstraints based on the generalized inverse’, Nonlinear Dyn., 2005, 39, (1), pp. 179– 196 [9] DE SAPIO V., KHATIB O., DELP S.: ‘Task-level approaches for the control of constrained multibody systems’, Multibody Syst. Dyn., 2006, 16, (1), pp. 73 – 102 [10] DE SAPIO V., KHATIB O., DELP S.: ‘Least action principles and their application to constrained and task-level problems in robotics and biomechanics’, Multibody Syst. Dyn., 2008, 19, (3), pp. 303– 322 [11] BAJODAH A.H.: ‘Singularly perturbed feedback linearization with linear attitude deviation dynamics realization’, Nonlinear Dyn., 2008, 53, pp. 321–343 [12] BLAJER W. , KOLODZIEJCZYK K. : ‘A geometric approach to solving problems of control constraints: theory and a DAE framework’, Multibody Syst. Dyn., 2004, 11, pp. 343– 364 [13] BLAJER W., KOLODZIEJCZYK K.: ‘Control of underactuated mechanical systems with servo-constraints’, Nonlinear Dyn., 2007, 50, pp. 781– 791 [14] SHUSTER M.D. : ‘A survey of attitude representation’, J. Astronaut. Sci., 1993, 41, (4), pp. 439– 517 [15] BHAT S. , BERNSTEIN D.: ‘A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon’, Syst. Control Lett., 2000, 39, (1), pp. 63– 70 [16] TSIOTRAS P.: ‘A passivity approach to attitude stabilization using nonredundant sets of kinematic parameters’. Proc. of the 34th Conf. on Decision and Control, New Orleans, LA, USA, 1995, pp. 515– 520 [17] BERNSTEIN D.: ‘Matrix mathematics: theory, facts, and formulas with application to linear system theory’ (Princeton University Press, 2005)

[5] GAUSS C.F.: ‘Ueber ein neues algemeines grundgesetz der mechanik’, Zeitschrift Fuer die reine und angewandte Mathematik, 1829, 4, pp. 232– 235

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[19] MCINNES C.R.: ‘Satellite attitude slew manoeuvres using inverse control’, Aeronaut. J., 1998, 102, pp. 259 – 265

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