Composite Immersion and Invariance Based Adaptive Attitude Control of Asteroid-Orbiting Spacecraft in Elliptic Orbit

. This paper proposes a new composite noncertainty-equivalence adaptive (CNCEA) control system for the attitude (roll, pitch, and yaw angle) control of a spacecraft in an orbit around a uniformly rotating asteroid based on the immersion and invariance (I&I) theory. For the design, it is assumed that the asteroid’s gravitational parameters and the spacecraft’s inertia matrix are not known. In contrast to certainty-equivalence adaptive (CEA) or noncertainty-equivalence adaptive (NCEA) systems, the CNCEA attitude control system’s composite identiﬁer uses the attitude angle tracking error, a nonlinear state-dependent vector function, and model prediction error for parameter estimation. The Lyapunov analysis shows that in the closed-loop system, the Euler angles asymptotically track the reference attitude trajectories. Interestingly, there exist two parameter error-dependent attractive manifolds, to which the closed-loop system’s trajectories converge. Moreover, the composite identiﬁer using two types of error signals provides stronger stability properties in the closed-loop system. Simulation results are presented for the attitude control of a spacecraft orbiting in the vicinity of the asteroid 433 Eros. These results show precise nadir pointing attitude regulation, despite uncertainties in the system. using two types of error signals achieves stronger stability properties in the closed-loop system. Third, simulation is done for the control of a spacecraft in an orbit around the asteroid 433 Eros. These simulated responses show that the CNCEA control law achieves precise roll, pitch and yaw angle trajectory control on elliptic orbits, despite parameter uncertainties and large angle rotational maneuvers.


Attitude Dynamics of Asteroid-orbiting spacecraft
Consider a spacecraft on a planar elliptic orbit in the asteroid's equatorial plane (see Fig. 1). The asteroid is rotating about Z I with a constant angular velocity Ω. The inertial (X I , Y I , Z I ), orbital (X 0 , Y 0 , Z 0 ), and body-fixed (X B , Y B , Z B ) coordinates systems are shown in Fig. 1. The orbital frame is obtained by a rotation of the inertial frame through η (the true anomaly) about the Z I axis. The satellite's body-fixed frame is obtained from the orbital frame by three independent successive rotations θ 3 (yaw), θ 2 (pitch), and θ 1 (roll).
The Euler's equations for the rotational motion are: where J = diag{J 1 , J 2 , J 3 } is the diagonal principal inertia matrix of the satellite, M g = [M g1 , M g2 , M g3 ] T is the gravity gradient torque vector, u = [u 1 , u 2 , u 2 ] T is the vector of control torques, and S(ω) denotes the skew symmetric matrix: The gravitational potential U of any arbitrary primary can be expressed as: where µ is the asteroid's gravitational parameter, R is the distance of the orbiting particle from the center of primary, r 0 is the characteristic length, and δ and λ are the latitude and longitude of the orbiting particle expressed in an asteroid-fixed frame (not shown in Fig. 1). Define: where the longitude of the center of mass of the satellite is given by: where R c is the distance between the centers of mass of the asteroid and satellite. In Eq. (6), the plus and minus signs are used for retrograde and prograde (direct) orbits, respectively.
Based on the potential function (3), the gravity gradient torque components M gi are given by [14,15]: .
where nonlinear vectors ψ T sj ∈ R 9 , s ∈ {a, b, c}, j = 1, are defined in equations (7) to (9); and p is a constant parameter vector given by: Define: Then, the gravity gradient torque vector can be written in a compact form as: Now, using Eq. (11) in (2) gives: It is assumed that ψ 1 (θ, t) is known, but the parameter vector p is not known.
Suppose that θ r (t) ∈ R 3 is a given smooth bounded reference trajectory. We are interested in deriving an adaptive control law for the system (1) and (12), such that in the closed-loop system, the attitude angle errorθ = θ − θ r asymptotically converges to zero, despite uncertainties in the parameter vector p. For nadir pointing attitude control, the selected reference trajectory θ r (t) converges to zero.
First, the derivation of a filtered control input is described briefly. The derivation process is similar to that of Ref [30]. (For the details of derivation, readers may refer to [30].) However, it must be noted that even though the derived filtered input is similar, the actual control input u for the CNCEA system (computed later) will differ from the control signal derived in Ref. [30] for the NCEA system. The design of the filtered control torque is completed in two steps of a backstepping design procedure.
Step 2: Consider the angular velocity error dynamics given by: Now, the objective is to choose u to force ω e andθ to zero. Adding and subtracting J[k 2 ω e + k 3 (s + α)(A T 1θ )] in equation (16) gives: where k 2 , k 3 , and α are positive design parameters, and s denotes the Laplace variable. Defining: and Ψ (θ, ω, t) = ψ 1 + ψ 2 , Eq. (17) can be written as: Consider the filtered signals generated by the following equations: The signal (Ψ f , ω ef , u f ) can be obtained by filtering (Ψ, ω e , u) through a stable transfer function H(s) where: Of course, Eq. (22) will not be implemented. Filtering signals of Eq. (19), and ignoring decaying signal ω e (0)exp(−αt) because it will not affect asymptotic stability properties, one obtains: Let an estimate of p bep =p Ig + β(ω ef , Ψ f ), and the parameter error be z =p − p. Later,p Ig and the algebraic vector function β will be obtained. Now, in view of Eq. (24), the filtered control signal u f is selected as: Using Eq. (25) in (24) gives:ω For the stability analysis, consider a Lyapunov function: Then, the derivative of V s along the solution of Eqs. (15) and (26) can be written as: Using ω e =ω ef + αω ef from equation (21), selecting the design parameters to satisfy k 3 + k 2 − α = 0, and after performing some manipulations, one shows that (see [30]): In view of equation (29), it easily follows that if Ψ f z = 0, thenθ andω ef will be bounded. The expression for the control input u f given in Eq. (25) will be used later. However, the parameter estimatep is not known yet.

Composite Identifier and CNCEA System's Stability
In this section, first an estimator, based on a prediction model obtained from Eq. (12), is designed. This will be followed by the design of a composite parameter identifier.
For the derivation of a gradient-based update rule, consider a prediction model based on Eq. (33) as : Define the prediction error as:ũ Then, subtracting Eq. (33) from (34) gives the prediction error: where z =p − p. An adaptation law can be derived by minimizing a performance index P = [ũ T fũ f ]/2. Its gradient with respect top is: An adaptation law is chosen as negative of the gradient to minimize P [33]. Thus, the update rule is: For examining the stability properties, consider a Lyapunov function V e = z T z/2. Then, its derivative is: This implies that z is bounded andũ f is square integrable. The model prediction error-based integral adaptation law (37) will be used in the next subsection to obtain a composite identifier for the attitude control of the spacecraft.

Composite Adaptation Law
In this subsection, a composite adaptation law is derived. Let the full parameter vector estimatep of the actual parameter vector p be: Here,p Ig is the integral component and β is the nonlinear algebraic part of the full estimate. The vector function β(ω ef , Ψ f ) has been judiciously selected based on the I&I principle. Its role will be seen when the derivative of the parameter vector z =p − p will be computed. For the derivation of the integral partp Ig , it is necessary to obtain the dynamics of the parameter error z. Differentiating z and using Eq. (38) gives: Substituting for the derivatives of Ψ f and ω ef from Eqs. (20) and (26) gives: The adaptation law forp Ig is selected as: where, for synthesis, one setsũ f =û f − u f = Wp + Ψ fp (see Eqs. (25), (34), and (35)) in Eq. (41). We note that the adaptation gain γ-dependent terms of Eq. (41) form the integral component of the I&I-based identifier. Thus, the integral parameter adaptation law in Eq. (41) combines the I&I-based adaptation law, as well as the γ g -dependent gradient-based law derived in Eq. (37). Of course, the net parameter estimate iŝ p =p gI + β. Substituting Eq. (41) in (40) gives: Next, consider a Lyapunov function: Now, its derivative will be computed along the solution of Eq. (42) of the composite identifier. Its derivative is:V where λ min (.) denotes the minimum eigenvalue of J −1 . This implies that the equilibrium point z = 0 of Eq. (42) is uniformly stable.

CNCEA System's Stability
Now, for analyzing the stability of the CNCEA attitude control system (including control input Eq. (25) and composite estimation law (41)), consider a Lyapunov function: Then, using Eqs. (29) and (43) gives: (45) Because ν is an arbitrary positive number, certainly there exists l * > 0, such that for a sufficiently large value of ν, one obtains: For such a choice of ν, Eq (45) yields: In the following analysis, it will be assumed that the closed-loop system's trajectories remain away from θ 2 = π/2, where Eq. (2) is not defined. Furthermore, it is assumed that θ r (t) is a smooth bounded function converging to zero and remains away from ±π/2. Thus, the stability properties established here will be valid only locally around θ = 0. The function V c is a positive definite function ofθ, ω ef , and z. SinceV c is negative semi-definite function, it follows that V c has a finite limit. Therefore,θ, ω ef , and z are bounded. Because ω ef is bounded, Eq. (21) implies that ω e is bounded. Thus, all the signals in the closed-loop system will be bounded. Becauseθ, ω ef , ω e , ψ f , and z are differentiable, it follows thatV s exists. Then, using Brabalat's lemma [33], one concludes thatθ, ω ef , Ψ f z, and W z tend to zero, as t tends to ∞. Of course, as ω ef tends to zero, ω e tends to zero. This establishes the convergence of the attitude angles to zero for the choice of θ r converging to zero. Thus, the nadir pointing attitude is attained. (It will be seen later that even for large attitude perturbations, desired attitude regulation can be accomplished.) Remark 1: It should be pointed out that the CNCEA attitude control law provides two attractive manifolds Ω a , and Ω b , defined by: to which the the trajectories of the closed-loop system converge asymptotically. For the closed-loop NCEA system, obtained by setting the adaptation gain γ g to zero, there exists a single attractive manifold Ω a . Of course, for CEA systems, there does not exist any z-dependent attractive manifold. This clearly shows a special feature of the designed composite identifier. Remark 2: It is interesting to note that the Lyapunov derivative in Eq. (46) includes two z-dependent functions −||Ψ f z|| and −||W z||. For this reason V c decays faster. Thus, it causes a faster convergence of the system's trajectories.

CNCEA System's Control Input Torque u
For implementation, it is necessary to determine the actual control torque vector u from the filtered signal u f . The control law Eq. (25) provides a filtered signal: In view of Eqs. (20) to (22), u can be written as: Now, one computes the derivative ofp =p Ig + β. The derivative of β is: Using Eq. (49) and the derivative ofp Ig from Eq. (41) gives: Finally, substituting equation (50) in (48) gives: where for implementation, one uses: in Eq. (51). It is interesting to note that the control law u in Eq. (51) utilizes both the tracking errorθ, as well as the model prediction errorũ f . This control signal differs from the control input computed for the NCEA law in Ref. [30] for the control of Euler angles. This completes the control law derivation.

Simulation Results
This section presents numerical results. The satellite is assumed to be in an orbit around asteroid 433 Eros. The spacecraft's principal moments of inertia are (J x , J y , J z ) = (110, 115, 100) [Kg·m 2 ], and its mass is 600 [kg]. The remaining parameters of the model given in [14] are used for simulation. These are: r 0 = 9.933 [km], C 20 = −0.0878, and C 22 = 0.0439. The asteroid's gravitational parameter and its rotation rate used for simulation are µ = 4.4650 × 10 −4 [km 3 /s 2 ] and Ω = 3.312 × 10 −4 [rad/s], respectively. Assuming that satellite is in an elliptical orbit, its radial distance from the asteroid's center of mass is: where a is the semi-major axis, and e is the eccentricity. The orbital rate is: where the semilatus rectum is p h = a(1 − e 2 ).
The controller gains were set as k i =0.1 (i = 1, 2, 3), and the adaptation gain was assumed to be γ = 1. The parameter α of the filter was chosen to satisfy k 2 + k 3 − α = 0. Different values of γ g ∈ {0, 10 3 , 10 4 } will be considered for simulation to examine the effect of the gradient-based integral update law on the responses. These design parameters were selected by the observation of simulated responses. The closed-loop system including the attitude dynamics (Eqs. (1) and (2)), the control law (Eq. (51)), and the adaptation law (Eq. (41)) was simulated. First, the adaptation gain γ g was chosen to be 10 3 . The responses are shown in Fig. 2. It can be seen that the attitude angles θ i , (i = 1, 2, 3), tracking error vectorθ, and ω ef converge to zero. The maximum control torque is 0.01 (Nm). The response time is of the order of three hours. It can also be seen that Ψ f z and W z converge to zero, as expected.
5.2. Composite adaptive attitude control in a prograde elliptic orbit: γ g = 10 4 , e = 0.2, a = 30 [km] To examine the effect of adaptation gain γ g , it was set as γ g = 10 4 . The remaining parameters and initial conditions of Case 5.1 were retained. Selected responses are shown in Fig. 3. Again, the attitude angles converged to zero. One observes that the larger value of γ g causes a drastic reduction in the transient period in which the attitude angle trajectories undergo oscillations. Of course, the control magnitude is larger because the response time is shorter compared to Case 5.1. Additionally, one observes faster convergence of the trajectories to the manifolds Ω a and Ω b .
5.3. Attitude Control using NCEA law in a prograde elliptic orbit: γ = 0, e = 0.2, a = 30 Finally, the closed-loop system, including the simplified adaptation law obtained by using γ g = 0, was simulated. It is noted that by setting γ g = 0, the control law becomes an NCEA law, designed based on the I&I principle. Therefore, the gradient algorithm based term γ g W Tũ f is not used in the integral update law Eg. (41). The remaining parameters of Case 5.1 were retained. Fig. 4 show the responses. Although the attitude angles converge to zero, the response time has increased. Of course, the NCEA system of Case 5.3 does not have the attractive manifold Ω b to which the trajectories of the closed-loop system can converge.
The performance characteristics of these three cases are summarized in Table 1. It is seen in Table 1 that the control law with γ g > 0 requires a larger peak value of the control magnitude, but the maximum tracking error in the steady state (beyond 4 hours) is smaller, compared with the control law using γ g = 0. Thus, the composite NCEA law achieves a smaller settling time, compared with the simplified control law with γ g = 0. In addition, the composite control system gives smaller peak values of the tracking error e max and ||Ψ f z||. Apparently, the composite NCEA system has additional flexibility in achieving desirable trade-offs among the response characteristics.

Conclusion
In this paper, based on the I&I theory, the design of a composite noncertainty-equivalence adaptive (CNCEA) attitude control law for an asteroid-orbiting spacecraft was considered. It was assumed that the spacecraft's inertia matrix and asteroid's gravitational parameters were not known. A backstepping design method was used to derive the CNCEA law. The composite adaptation law developed here provides parameter estimates based on the information on the tracking error and the model prediction error. Of course, the full parameter estimate includes a nonlinear algebraic vector function, as well as an integral component. The Lyapunov stability analysis established asymptotic convergence of roll, pitch, and yaw angles to the origin. Simulation results were presented for the attitude control of a spacecraft orbiting in the vicinity of the asteroid 433 Eros. These results showed precise nadir pointing attitude regulation, despite uncertainties in the parameters and large angle rotational maneuvers. Furthermore, simulation results confirmed better performance of the CNCEA law, compared with the NCEA law. Additionally, it was seen that adaptation gains of the composite identifier play important role in shaping the responses.