Advances in Spacecraft Technologies Part 11 pot

6 Advances in Spacecraft Technologies It can be easily verified that the above expression satisfies the quaternion constraint (Show & Juang, 2003) q T e (q, t )q e (q, t )=(q T q + q 2 4 )(q T d (t)q d (t)+q d4 (t) 2 )=1. (13) Let ω r (t) : [0, ∞) → R 3 be the prescribed angular velocity vector of D relative to I expressed in B. The quaternion error kinematical differential equations are given by ˙q e =  ˙ q e ˙ q e4  = 1 2  q × e + q e4 I 3×3 −q T e  ω e (14) where ω e := ω − ω r (t). The reference angular velocity vector ω r (t) satisfies ˙ ω r (t)=−J −1 ω × r (t)Jω r (t). (15) Therefore, ˙ ω e = ˙ ω − ˙ ω r (t) (16) = −J −1 ω × Jω + J −1 ω × r (t)Jω r (t)+τ. (17) A scalar attitude deviation norm measure function φ : [−1, 1] → [0, 1] is defined as φ (q e4 )=1 − q 2 e4 (18) and the control objective is to enforce the servo-constraint φ (q e4 ) ≡ 0. (19) From (13), the same servo-constraint requirement can also be written as q e ≡ 0 3×1 . (20) The first two time derivatives of φ along the spacecraft error trajectories given by the solutions of (14) and (17) are ˙ φ = q e4 q T e ω e (21) and ¨ φ = 1 2 ω T e  q 2 e4 I 3×3 − q e q T e  ω e + q e4 q T e (−J −1 ω × Jω + J −1 ω × r Jω r + τ). (22) Skew symmetries of the cross product matrices ω × and ω × r imply that the corresponding terms in ¨ φ are zeros. Hence, the expression of (22) reduces to ¨ φ = 1 2 ω T e  q 2 e4 I 3×3 − q e q T e  ω e + q e4 q T e τ. (23) A desired dynamics of φ that leads to asymptotic realization of the servo-constraint given by (19) is described to be stable second-order in the general functional form given by ¨ φ = L(φ, ˙ φ,t) (24) where L is continuous in its arguments. A special choice of L(φ, ˙ φ,t) is L(φ, ˙ φ,t)=−c 1 (t) ˙ φ − c 2 (t)φ (25) 390 Advances in Spacecraft Technologies Inertia-Independent Generalized Dynamic Inversion Control of Spacecraft Attitude Maneuvers 7 where c 1 (t) and c 2 (t) are continuous scalar functions. With this choice of L(φ, ˙ φ,t), the stable attitude deviation servo-constraint dynamics given by (24) becomes linear in the form ¨ φ + c 1 (t) ˙ φ + c 2 (t)φ = 0. (26) With φ, ˙ φ,and ¨ φ given by (18), (21), and (23), it is possible to write (26) in the pointwise-linear form A(q e )τ = B(q e ,ω e ), (27) where the vector valued the controls coefficient function A(q e ) is given by A(q e )=q e4 q T e (28) and the scalar valued controls load function B(q e ,ω e ) is given by B(q e ,ω e )=− 1 2 ω T e  q 2 e4 I 3×3 − q e q T e  ω e − c 1 (t)q e4 q T e ω e − c 2 (t)(1 − q 2 e4 ). (29) The MPGI-based Greville formula is used now to obtain a preliminary form of GDI spacecraft attitude control laws. Proposition 1 (Linearly parameterized attitude control laws) The infinite set of all control laws that globally realize the attitude deviation servo-constraint dynamics given by (26) by the spacecraft equations of motion is parameterized by an arbitrarily chosen null-control vector y ∈ R 3×1 as τ = A + (q e )B(q e ,ω e )+P(q e )y (30) where “ A + ” stands for the MPGI of the controls coefficient (abbreviated as CCGI), and is given by A + (q e )=  A T (q e ) A(q e )A T (q e ) , A(q e ) = 0 1×3 0 3×1 , A(q e )=0 1×3 (31) and P(q e ) is the corresponding controls coefficient nullprojection matrix given by P(q e )=I 3×3 −A + (q e )A(q e ). (32) Proof 1 Multiplying both sides of (30) by A(q e ) recovers the algebraic system given by (27). Therefore, τ enforces the attitude deviation servo-constraint dynamics given by (26) for all A(q e ) = 0 1×3 . The controls coefficient nullprojector P(q e ) projects the null-control vector y onto the nullspace of the controls coefficient A(q e ). Therefore, the choice of y does not affect realizability of the linear attitude deviation norm measure dynamics given by (26). Nevertheless, the choice of y substantially affects transient state response and spacecraft inner stability, i.e., stability of the closed loop dynamical subsystem ˙ ω = −J −1 ω × Jω + A + (q e )B(q e ,ω e )+P(q e )y (33) obtained by substituting (30) in (6). 391 Inertia-Independent Generalized Dynamic Inversion Control of Spacecraft Attitude Maneuvers 8 Advances in Spacecraft Technologies The expression given by (28) for the controls coefficient implies that if the dynamics given by (26) is realizable by spacecraft equations of motion, then lim φ→0 A(q e )=0 1×3 . (34) Accordingly, the discontinuous expression of A + (q e ) given by (31) implies that for any initial condition A(q e ) = 0 1×3 , state trajectories of a continuous closed loop control system in the form given by (5) and (33) must evolve such that lim φ→0 A + (q e )=∞ 3×1 . (35) That is, A + (q e ) must go unbounded as the spacecraft dynamics approaches steady state. This is a source of instability for the closed loop system because it causes the control law expression given by (30) to become unbounded. One solution to this problem is made by switching the value of the CCGI according to (31) to A + (q e )=0 3×1 when the controls coefficient A(q e ) approaches singularity, which implies deactivating the particular part of the control law as the closed loop system reaches steady state, leading to a discontinuous control law (Bajodah, 2006). Alternatively, a solution is made by replacing the Moore-Penrose generalized inverse in (30) by a damped generalized inverse (Bajodah, 2008), resulting in uniformly ultimately bounded trajectory tracking errors, and a tradeoff between generalized inversion stability and steady state tracking performance. A solution to this problem that avoids control law discontinuity and improves singularity avoiding trajectory tracking is presented in (Bajodah, 2010), made by replacing the MPGI in (30) by a growth-controlled dynamically scaled generalized inverse. A generalization of the dynamically scaled generalized inverse is presented in the following section. The dynamically scaled generalized inverse provides the necessary generalized inversion singularity avoidance to the GDI control design. Definition 1 (Dynamically scaled generalized inverse) The DSGI A + s (q e ,ν) is given by A + s (q e ,ν)= A T (q e ) A(q e )A T (q e )+ν (36) where ν satisfies the asymptotically stable dynamics ˙ ν = −aν + ω e  p p , a > 0, p ∈ Z + . (37) The positive integer p is the generalized inversion dynamic scaling index, and . p is the vector p norm. The following properties can be verified by direct evaluation of the CCGI A + (q e ) given by (31) and its dynamic scaling A + s (q e ,ν) given by (36). 392 Advances in Spacecraft Technologies Inertia-Independent Generalized Dynamic Inversion Control of Spacecraft Attitude Maneuvers 9 1. A + s (q e ,ν)A(q e )A + (q e )=A + s (q e ,ν) 2. A + (q e )A(q e )A + s (q e ,ν)=A + s (q e ,ν) 3. (A + s (q e ,ν)A(q e )) T = A + s (q e ,ν)A(q e ) 4. lim ω e  p →0 A + s (q e ,ν)=A + (q e ). The dynamically scaled generalized inverse control law is obtained by replacing the CCGI in the particular part of the expression given by (30) by the DSGI as τ s = A + s (q e ,ν)B(q e ,ω e )+P(q e )y (38) resulting in the following spacecraft closed loop dynamical equations ˙ ω = −J −1 ω × Jω + A + s (q e ,ν)B(q e ,ω e )+P(q e )y. (39) Proposition 2 (Asymptotic Attitude Trajectory Tracking) If the null-control vector y in the control law expression given by (38) is chosen such lim t→∞ ω e = 0 3×1 (40) then lim t→∞ q e = 0 3×1 . (41) Proof 2 Let φ s be a norm measure function of the attitude deviation obtained by applying the control law given by (38) to the spacecraft equations of motion (5) and (6), and let ˙ φ s , ¨ φ s be its first two time derivatives. Therefore, φ s := φ s (q e )=φ(q e ) (42) ˙ φ s := ˙ φ s (q e ,ω e )= ˙ φ (q e ,ω e ) (43) ¨ φ s := ¨ φ s (q e ,ω e ,τ s )= ¨ φ (q e ,ω e ,τ)+A(q e )τ s −A(q e )τ (44) where τ and τ s are given by (30) and (38), respectively. Adding c 1 (t) ˙ φ s + c 2 (t)φ s to both sides of (44) yields ¨ φ s + c 1 (t) ˙ φ s + c 2 (t)φ s = ¨ φ + c 1 (t) ˙ φ + c 2 (t)φ + A(q e )τ s −A(q e )τ (45) = A(q e )[τ s − τ]. (46) Therefore, boundedness of the expr ession of A(q e ) given by (28) in addition to satisfaction of (40) imply that lim t→∞  ¨ φ s + c 1 (t) ˙ φ s + c 2 (t)φ s  = lim t→∞  A(q e )[τ s − τ]  = 0 (47) resulting in lim t→∞ φ s = 0 (48) and therefore, (41) follows for all permissible initial attitude quaternion vectors q 0 ∈ R 3 .Thesame conclusion is obtained by multiplying b oth sides of (38) by A(q e ), resulting in A(q e )τ s = A(q e )A + s (q e ,ν)B(q e ,ω e ) (49) 393 Inertia-Independent Generalized Dynamic Inversion Control of Spacecraft Attitude Maneuvers 10 Advances in Spacecraft Technologies where A(q e )A + s (q e ,ν)= A( q e )A T (q e ) A(q e )A T (q e )+ν . (50) Therefore, 0 < A(q e )A + s (q e ,ν) ≤ 1 (51) and lim ω e →0 3×1 A(q e )A + s (q e ,ν)=1. (52) Dividing both sides of (49) by A(q e )A + s (q e ,ν) yields A(q e ) ¯ τ = B(q e ,ω e ) (53) where A(q e ) and B(q e ,ω e ) are the same controls coefficient and controls load in (27), and ¯ τ = τ s A(q e )A + s (q e ,ν) . (54) Furthermore, (52) implies that lim ω e →0 3×1 ¯ τ = lim ω e →0 3×1 τ s = τ. (55) Therefore, ¯ τ in the algebraic system given by (53) asymptotically converges to τ, recovering the algebraic system given by (27), and resulting in asymptotic convergence of φ s (t) to φ s = φ = 0,andq to q d (t). Proposition 2 states that using the DSGI A + s (q e ,ν) in the attitude control law yields the same attitude convergence property that is obtained by using the CCGI A + (q e ), provided that the condition given by (40) is satisfied. A design of the null-control vector y is made in the next section to guarantee global satisfaction of the condition given by (40). Remark 1 It is well-known that topological obstruction of the attitude rotation matrix precludes the existence of globally stable equilibria for the attitude dynamics (Bhat & Bernstein, 2000). Therefore, although the servo-constraint attitude deviation dynamics given by (26) is globally realizable, there exists no null-control that renders the spacecraft attitude dynamics globally stable. In particular, if q d (t) ≡ 0 3×1 then for any null-control vector y there exists an attitude vector q 0 such that the closed loop system given by (5) and (39) is unstable in the sense of Lyapunov. A Lyapunov-based design of null-control vector y is introduced in this section to enforce spacecraft inner stability. Let y be chosen as y = Kω e (t) (56) where K ∈ R 3×3 is a matrix gain that is to be determined. Hence, a class of control laws that realize the attitude deviation norm measure dynamics given by (26) is obtained by substituting this choice of y in (38) such that τ s = A + s (q e ,ν)B(q e ,ω e )+P(q e )Kω e (t). (57) 394 Advances in Spacecraft Technologies Inertia-Independent Generalized Dynamic Inversion Control of Spacecraft Attitude Maneuvers 11 Consequently, a class of spacecraft closed loop dynamical subsystems that realize the servo-constraint dynamics given by (26) is obtained by substituting the control law given by (57) in (6), and it takes the form ˙ ω = −J −1 ω × Jω + A + s (q e ,ν)B(q e ,ω e )+P(q e )Kω e (t) (58) and the closed loop error dynamics ˙ ω e is obtained from (17) as ˙ ω e = −J −1 ω × Jω + J −1 ω × r Jω r + A + s (q e ,ν)B(q e ,ω e )+P(q e )Kω e . (59) The matrix gain K is synthesized by utilizing the positive-semidefinite control Lyapunov function V (q e ,ω e )=ω e T P(q e )ω e . (60) Evaluating the time derivative of V (q e ,ω e ,) along solution trajectories of the error dynamics given by (59) yields ˙ V (q e ,ω e )=2ω T e P(q e )  − J −1 ω × Jω + J −1 ω × r (t)Jω r (t) + A + s (q e ,ν)B(q e ,ω e )  + 2ω T e P(q e )Kω e + ω T e ˙ P(q e ,ω e )ω e (61) where ˙ P(q e ,ω e ) is obtained by differentiating the elements of P(q e ) along attitude trajectory solutions of the closed loop kinematical subsystem given by (14). Skew symmetry of the cross product matrix [·] × , the nullprojection property of P(q e ),andthesecondpropertyof A + s (q e ,ω e ) imply that the first term in the above equation is zero. Therefore, ˙ V (q e ,ω e )=2ω T e P(q e )Kω e + ω T e ˙ P(q e ,ω e )ω e . (62) Because V (q e ,ω e ) is only positive semidefinite, it is impossible to design a matrix gain K that renders ˙ V (q e ,ω e ) negative definite. Nevertheless, a matrix gain K that renders ˙ V(q e ,ω e ) negative semidefinite guarantees Lyapunov stability of ω e = 0 3×1 if it asymptotically stabilizes ω e = 0 3×1 over the invariant set of q e and ω e values on which V(q e ,ω e )=0. Moreover, the same gain matrix asymptotically stabilizes ω e = 0 3×1 if and only if it asymptotically stabilizes ω e = 0 3×1 over the largest invariant set of q e and ω e values on which ˙ V(q e ,ω e )= 0 (Iqqidr et al., 1996). Proposition 3 Let K = K(q e ,ω e ) be a full-rank normal matrix gain, i.e., KK T = K T K for all t ≥ 0. Then the equilibrium point ω e = 0 3×1 of the closed loop error dynamics given by (59) is asymptotically stable over the invariant set of q e ,andω e values on which V(q e ,ω e )=0. Proof 3 Since the matrix P(q e ) is idempotent, the function V( q e ,ω e ) can be rewritten as V (q e ,ω e )=ω T e P(q e )ω e = ω T e P(q e )P(q e )ω e (63) which implies that V (q e ,ω e )=0 ⇔P(q e )ω e = 0 3×1 . (64) Therefore, V (q e ,ω e )=0 ⇔ ω e ∈N(P(q e )) (65) 395 Inertia-Independent Generalized Dynamic Inversion Control of Spacecraft Attitude Maneuvers 12 Advances in Spacecraft Technologies where N (·) refers to matrix nullspace. Since the matrix K(q e ,ω e ) is normal and of full-rank, it preserves matrix range space and nullspace under multiplication. Accordingly, N (P(q e )) = N (P(q e )K(q e ,ω e )) (66) which implies from (64) t hat V (q e ,ω e )=0 ⇔P(q e )K(q e ,ω e )ω e = 0 3×1 . (67) Therefore, the last term in the closed loop error dynamics given by (59) is the zero vector, and the closed loop error dynamics becomes ˙ ω e = −J −1 ω × Jω + J −1 ω × r (t)Jω r (t)+A + s (q e ,ν)B(q e ,ω e ). (68) On the other hand, since N (P(q e )) = R(A T (q e )) (69) it follows from (65) that V (q e ,ω e )=0 ⇔ ω e ∈R(A T (q e )). (70) Accordingly, V (q e ,ω e )=0 if and only if there exists a continuous scalar function a(t), t ≥ 0, satisfying 0 <| a(t) |< ∞ (71) such that ω e = a(t)A T (q e ). (72) Since the expression of A(q e ) given by (28) is bounded for all values of q e , it follows from (72) that ω e is also bounded. Therefore, the trajectory of ω e must remain in a finite region, and it follows from the Poincare-Bendixon theorem (Slotine & Li, 1991) that the trajectory goes to the equilibrium point ω e = 0 3×1 . Theorem 1 (CCNP Lyapunov control design) Let the nullprojection gain matrix K (q e ,ω e ) be K (q e ,ω e )=− ˙ P(q e ,ω e ) − σ max ( ˙ P(q e ,ω e ))I 3×3 − Q (73) where σ max (·) denotes the maximum singular value, and Q ∈ R 3×3 is arbitrary positive definite. Then the equilibrium point ω e = 0 3×1 of the closed loop error dynamics given by (59) is globally asymptotically stable, and lim t→∞ q e = 0 3×1 . (74) Proof 4 Let Q(q e ,ω) : R 4×1 × R 3×1 → R 3×3 be a positive semidefinite matrix function. Then, a matrix gain K that enforces negative semidefiniteness of ˙ V (q e ,ω e ) is obtained by setting ˙ V (q e ,ω e )=2ω T e P(q e )Kω e + ω T e ˙ P(q e ,ω e )ω e = −2ω T e Q(q e ,ω e )ω e . (75) Hence, K satisfies the following Lyapunov equation 2 P(q e )K + ˙ P(q e ,ω e )+2Q(q e ,ω e )=0 3×3 . (76) Consistency of the above-written nullprojection equation implies that every term maps into P(q e ).The range space of ˙ P(q e ,ω e ) is a subset of the range space of P(q e ). This is shown by writing P(q e )=P(q e )P(q e ) ⇒ ˙ P(q e ,ω e )=2P(q e ) ˙ P(q e ,ω e ) (77) 396 Advances in Spacecraft Technologies Inertia-Independent Generalized Dynamic Inversion Control of Spacecraft Attitude Maneuvers 13 so that R[ ˙ P(q e ,ω e )] = R[P(q e ) ˙ P(q e ,ω e )] ⊆R[P(q e )] (78) where R(·) refers to matrix range space. Moreover, for Q(q e ,ω e ) to map into the range space of P(q e ), then there must exist a positive definite matrix function ¯ Q(q e ,ω e ) : R 4×1 × R 3×1 → R 3×3 such that a polar decomposition of Q(q e ,ω e ) is given by Q(q e ,ω e )=P(q e ) ¯ Q(q e ,ω e ). (79) By substituting the expressions of ˙ P(q e ,ω e ) and Q(q e ,ω e ) given by (77) and (79) in (76), a solution for K that renders ˙ V (q e ,ω e ) negative semidefinite is obtained as K (q e ,ω e )=− ˙ P(q e ,ω e ) − ¯ Q(q e ,ω e ). (80) Furthermore, it follows from Proposition 3 that K guarantees asymptotic stability of ω e = 0 3×1 over the invariant set of q e ,andω e values on which V(q e ,ω e )=0 if K remains nonsingular for all t ≥ 0. This is achieved by choosing ¯ Q(q e ,ω e ) as ¯ Q(q e ,ω e )=σ max ( ˙ P(q e ,ω e ))I 3×3 + Q (81) so that K (q e ,ω e ) remains negative definite. Substituting the above written expression for ¯ Q(q e ,ω e ) in (80) results in the expression of K (q e ,ω e ) given by (73). Therefore, in addition t o rendering ˙ V(q e ,ω e ) negative semidefinite, K(q e ,ω e ) guarantees asymptotic stability of ω e = 0 3×1 over the invariant set of q e and ω e values on which V(q e ,ω e )=0, and Lyapunov stability of ω e = 0 3×1 follows (Iqqidr et al., 1996). Since V (q e ,ω e ) is radially unbounded with respect to ω e , Lyapunov stability of ω e = 0 3×1 is global. Moreover, it is noticed from the expression of ˙ V(q e ,ω e ) given by (61) and from (78) that the largest invariant set of q e and ω e on which ˙ V(q e ,ω e )=0 is the same invariant set on which V (q e ,ω e )=0, implying global asymptotic stability of the equilibrium point ω e = 0 3×1 (Iqqidr et al., 1996). Global asymptotic convergence of the attitude vector q to the desired attitude vector q d (t) follows from Proposition 2 . fi Although the CCNP P(q e ) has bounded elements, dependency of CCNP on the unbounded vector A + (q e ) may cause undesirable behavior of the auxiliary part in the control law τ s during steady state tracking response of time varying trajectories. For this reason, a damped controls coefficient nullprojector (DCCN) P d (q e ,) is used in place of P(q e ) in (57). The DCCN is defined as P d (q e ,) := I 3×3 −A + d (q e ,)A(q e ) (82) where  is a small positive number, and A + d (q e ,) is given by A + d ( q e , ) := A T ( q e ) A(q e ) A T ( q e )+ . (83) Therefore, lim φ→0 A + d (q e ,)=0 3×1 (84) and consequently, lim φ→0 P d (q e ,)=I 3×3 . (85) Hence, the DCCN maps the null-control vector to itself in steady state phase of response, during which the auxiliary part of the control law converges to the null-control vector. 397 Inertia-Independent Generalized Dynamic Inversion Control of Spacecraft Attitude Maneuvers 14 Advances in Spacecraft Technologies Fig. 1. Schematic of GDI spacecraft attitude control system Independency of nullprojection on the attitude state of the spacecraft substantially eliminates unnecessary abrupt behavior of the control vector. Replacing P(q e ) by P d (q e ,) in the control expression given by (57) yields the following form of the GDI control law τ sd = A + s (q e ,ν)B(q e ,ω e )+P d (q e ,)Kω e (t). (86) A schematic of the GDI spacecraft attitude control system is shown in Fig. 1. When the second-order deviation dynamics given by (26) is chosen to be time invariant, then increasing the value of the constant c 1 increases the damping ratio of closed loop spacecraft dynamics. Additionally, increasing the value of c 2 improves steady state trajectory tracking accuracy. Nevertheless, excessively large values of c 1 and c 2 require large control torque inputs and cause large amplitude oscillations of spacecraft body angular velocity components, particularly during the initial phase of response when the state deviation variable φ and its time derivative ˙ φ are at their biggest magnitudes, i.e., when the controls load B(q e ,ω e ) has a large value. Accordingly, to increase damping and to improve steady state tracking with simultaneous avoidance of these drawbacks, the coefficients c 1 (t) and c 2 (t) are chosen to be of the form c 1 (t)=C 1 (1 − e −α 1 t ) and c 2 (t)=C 2 (1 − e −α 2 t ),whereC 1 , C 2 , α 1 ,andα 2 are positive constants. Hence, c 1 (0)=0andc 2 (0)=0, which substantially decreases the magnitude of B(q e ,ω e ). The spacecraft model has inertia scalars I 11 = 200 Kg.m 2 , I 22 = 150 Kg.m 2 , I 33 = 175 Kg.m 2 , I 12 = −100 Kg.m 2 , I 13 = I 23 = 0Kg.m 2 . The first maneuver considered is a rest-to-rest slew maneuver, aiming to reorient the spacecraft at the initial attitude given by q (0)=q 0 to a different attitude given by q d (T),whereT is duration of the maneuver. It is required that the spacecraft quaternion attitude variables follow the trajectories given by the following 398 Advances in Spacecraft Technologies [...]... to zero The aim of the differential game approach is to construct a smaller invariant set S ⊂ Ω, to bring the trajectory x (t) into S , and to maintain it there 2 424 Advances in Spacecraft Technologies Advances in Spacecraft Technologies The idea to use differential game methods to solve stabilization problems with uncertainties is not new (see (Gutman & Leitmann, 1975a; Gutman & Leitmann, 1975b),... 374–386 18 Tracking Control of Spacecraft by Dynamic Output Feedback - Passivity- Based Approach Yuichi Ikeda1, Takashi Kida2 and Tomoyuki Nagashio2 2The 1Shinshu University, University of Electro-Communications, Japan 1 Introduction In this study, we investigate the possibility of capturing an inoperative spacecraft using an orbital servicing vehicle or a space robot in future space infrastructure... 3(e)-(f) have an expanded vertical axis in order to show the change in input) Therefore, by setting the initial state zi (0) as (46) as described in the previous section, the amplitude of the control inputs at an early stage can be reduced without changing the feedback gains On the other hand, the amplitude of the control torque τ c 416 Advances in Spacecraft Technologies does not change This could... control to reduce the control energy required to perform DI The choice of desired stable servo-constraint dynamics has its tangible effect on closed loop system response For instance, choosing the linear servo-constraint dynamics coefficients to 16 400 Advances in Spacecraft Technologies Advances in Spacecraft Technologies 0.8 0.2 q1 qd1 q2 qd2 0.1 0.6 0 0.4 q q 2 1 −0.1 −0.2 0.2 −0.3 0 −0.4 −0.2 0 10 20... coefficient’s MPGI, leading to asymptotic realization of desired servo-constraint stable dynamics Practically stable trajectory tracking control is achieved otherwise qe , q e , q e , q e 4 3 1 2 1 0.5 0 qe1 qe2 qe3 qe4 −0.5 −1 0 50 100 150 t (sec) Fig 5 Quaternion attitude parameters errors vs Time: trajectory tracking maneuver 18 402 Advances in Spacecraft Technologies Advances in Spacecraft Technologies 2... conditions T 418 Advances in Spacecraft Technologies (a) re (c) z1 (e) fc Fig 2 Simulation results (Case1) (b) θ e (d) z2 (f) τ c Tracking Control of Spacecraft by Dynamic Output Feedback - Passivity- Based Approach - (a) re (b) θ e (c) z1 (d) z2 (e) f c (f) τ c Fig 3 Simulation results (Case2) 419 420 Advances in Spacecraft Technologies (a) q e (b) τ c Fig 4 Simulation results at singular point (proposed... be used for tracking a spacecraft with an arbitrary trajectory since the attitude controller has a singular point at which the control input diverges; another instance where the method cannot be used is when the initial state of the control system is restricted In this paper, we propose a new passivity-based control method that involves the use of output feedback for solving the tracking control problem... in section 6 2 Relative equation of motion of spacecraft In this paper, we consider the tracking control problem in which the chaser spacecraft tracks to the target spacecraft that has a broken down actuator and moves in space freely The definition of the coordinate systems and the position vectors are shown in Fig 1 {i} , {c } , and {t} represent the inertial, chaser, and target frame, respectively... generalized inversion stable mode augmentation generalizes the concept of dynamic scaling, and it effectively overcomes controls coefficient generalized inversion singularity If the augmented mode is designed to be very fast, then the delayed DSGI closely approximates the instantaneous DSGI For problems involving time invariant steady state trajectory Inertia-Independent Generalized Dynamic Inversion... Attitude Maneuvers 403 19 Inertia-Independent Generalized Dynamic Inversion Control of Spacecraft Attitude Maneuvers & Applications 4(5): 827–840 Bajodah, A H., Hodges, D H & Chen, Y H (2005) Inverse dynamics of servo-constraints based on the generalized inverse, Nonlinear Dynamics 39(1-2): 179–196 Baker, D R & Wampler II, C W (1988) On inverse kinematics of redundant manipulators, International Journal . approximates the instantaneous DSGI. For problems involving time invariant steady state trajectory 400 Advances in Spacecraft Technologies Inertia-Independent Generalized Dynamic Inversion Control of Spacecraft. response. For instance, choosing the linear servo-constraint dynamics coefficients to 399 Inertia-Independent Generalized Dynamic Inversion Control of Spacecraft Attitude Maneuvers 16 Advances in Spacecraft. A + (q e )B(q e ,ω e )+P(q e )y (33) obtained by substituting (30) in (6). 391 Inertia-Independent Generalized Dynamic Inversion Control of Spacecraft Attitude Maneuvers 8 Advances in Spacecraft Technologies The expression