Stability of representative

## Abstract

This chapter demonstrates analysis and control of the attitude motion of a gravity-gradient stabilized spacecraft in eccentric orbit. The attitude motion is modeled by nonlinear planar pitch dynamics with periodic coefficients and additionally subjected to external periodic excitation. Consequently, using system state augmentation, Lyapunov-Floquet (L-F) transformation, and normal form simplification, we convert the unwieldy attitude dynamics into relatively more amenable schemes for motion analysis and control law development. We analyze the dynamical system’s periodicity, stability, resonance, and chaos via numerous nonlinear dynamic theory techniques facilitated by intuitive system state augmentation and Lyapunov-Floquet transformation. Versal deformation of the normal forms is constructed to investigate the bifurcation behavior of the dynamical system. Outcome from the analysis indicates that the motion is quasi-periodic, chaotic, librational, and undergoing a Hopf bifurcation in the small neighborhood of the critical point-engendering locally stable limit cycles. Consequently, we demonstrate the implementation of linear and nonlinear control laws (i.e., bifurcation and sliding mode control laws) on the relatively acquiescent transformed attitude dynamics. By employing a two-pronged approach, the quasiperiodic planar motion is independently shown to be stabilizable via the nonlinear control approaches.

### Keywords

- gravity gradient
- nonlinear attitude control
- sliding mode
- Lyapunov-Floquet transformation
- normal forms

## 1. Introduction

Ever since the launch of *Sputnik*—the first artificial satellite put into earth’s orbit in 1957—mankind has increasingly become dependent on space-based technology in many areas of our daily lives. For instance, space-based technology performs an indispensable role in telecommunications, navigation, personal entertainment, weather forecasting, farming, security, defense, scientific exploration, research, innovation, etc. Undoubtedly, the prominence of space technology in shaping humanity’s future is unequivocal. The success or failure of a given space mission is largely contingent upon the complex system analysis and design methodologies exerted in converting the initial idea into an elaborate functioning enterprise [1]. It is for this reason that reliable and efficacious methodologies and tools are consistently utilized in space mission formulation and implementation. Thus, there is a need to continuously examine the effectiveness of prevailing space mission analysis and design methodologies. This is in order to improve prevailing tools and approaches that shall expedite relatively simpler, more reliable, and accurate mission modeling and analysis.

Space systems are required to function nominally in their designated orbital locations, maintain appropriate orientation, and conform to planned trajectories despite the ambient perturbing space environment. Strict mission pointing requirements normally constrain spacecraft in orbit around a large body to maintain a fixed stable orbital position and orientation during operation. However, perturbing space-environment torques act to dislodge positioned spacecraft and disorient stabilized ones [2, 3].

Modeling, analyzing, and controlling dynamics of space systems are therefore a crucial component of space mission design. The quest for relatively simpler, more accurate, and more reliable analytical methodologies and tools to represent, scrutinize, and manipulate the dynamics of space systems is therefore a worthwhile undertaking.

Inopportunely, dynamics of space systems tend to be commonly represented by coupled analytical models that possess complex structures encompassing nonlinearity, parameter-variant coefficients, and periodic external excitation terms [4, 5, 6, 7, 8, 9]. The requisite analysis essential to fathom such motion is not a trivial undertaking—except for few special cases, the general solution for such dynamical systems cannot be found. The complex structures of the motion’s analytical models characteristically point to nondeterministic and potentially chaotic systems over a range of initial conditions and system parameters. Therefore, to analyze dynamical space systems, we often have to be content with nonautonomous, nonlinear, and periodic differential equations [10, 11]. This presents an immense analysis challenge. For instance, time-varying eigenvalues of the periodic linear system matrix cannot determine the system stability. Consequently, methods such as linearization [12], averaging [13], and perturbation techniques [14, 15] have been consistently used to analyze such complex nonlinear, periodic motion. However, the two latter approaches tend to be limited to minimally excited systems (parameter multiplying the periodic terms is small), while linearization is restricted to small domains about the operating point. Further, such methods are inclined to be relatively cumbersome and normally augmented with numerical approaches to analyze dynamical systems and accomplish real-life applications [16, 17, 18].

The presence of perturbing torques in the ambient space environment tends to disorient an already stabilized spacecraft and further alters the orbital motion [2, 3]. This is contrary to the prevailing strict pointing requirements that constrain the satellite in orbit to maintain a stable orbital motion and fixed orientation during mission operation. A number of strategies are employed to stabilize spacecraft attitude motion and maintain a desired orientation despite the presence of perturbing torques in the space environment. The most common attitude control and stabilization approaches are three-axis stabilization, spin stabilization, and gravity-gradient stabilization. To provide the determined control input required to offset undesired attitude deviations, these methods employ either active control systems (e.g., thrusters, magnetic torquers, reaction wheels) or passive control systems (e.g., booms). Unlike passive controllers, active controllers utilize an external source of energy to drive the attitude control actuators [17, 19].

Among the stated attitude stabilization methods, gravity-gradient stabilization of spacecraft attitude is attractive due to its relatively intrinsic simplicity, reliability, and low cost [20]. However, it is mostly feasible in low earth orbit due to its principle of operation as discussed in Section 2.

The motion about COM of a rigid gravity-gradient stabilized spacecraft is libratory about the pitch axis. This axis is normal to the orbital plane in an inverse-square gravity field. The satellite will oscillate about a position of stable relative equilibrium if the work done by external perturbing forces is greater than the rotational kinetic energy. The sufficient conditions for stability of relative equilibrium are explained in Section 2. The complete formulation of COM motion for a gravity-gradient stabilized satellite in eccentric orbit consists of six coupled, nonlinear second-order differential equations. This system of equations is considered analytically unsolvable in closed form [5, 7, 21, 22, 23].

This chapter aims to first investigate the periodicity, quasi-periodicity, and chaotic behavior of the gravity-gradient stabilized attitude motion. Moreover, the motion stability, resonances and bifurcation behavior will also be examined. Subsequently in Section 4, we synthesize suitable controllers to adequately offset the attitude perturbations experienced by the gravity-gradient stabilized spacecraft in an eccentric orbit. Requisite assumptions made to facilitate the attitude motion analyses will be explicitly stated and qualified.

To model, analyze, and control the nonlinear motion with parameter-variant coefficients and periodic forcing terms, we intend to use approaches based on:

System state augmentation

Lyapunov-Floquet (L-F) transformations

Normal form (NF) theory

The fitting use of the aforementioned transformations and techniques enables dynamical system analysis and control law development in transformed, parameter-invariant, and more tractable coordinates that preserve the original system Lyapunov stability properties [24, 25]. Consequently, we intend to exploit this propitious attribute in our investigation. Applications of L-F transformations in spacecraft dynamics have been previously investigated by authors such as [8, 26]. The former demonstrates how L-F theory enhances the representation of relative spacecraft dynamics in elliptical orbits, while the latter further proposes an orbit control law based on L-F theory. On the other hand, this chapter focuses on the dynamics of a rigid body about its COM while in elliptical orbit around a central large mass.

## 2. Gravity-gradient attitude stabilization in eccentric orbit

A gravity-gradient stabilized spacecraft attains a state of stable relative equilibrium when its

If the work done by external perturbing torques is greater than the rotational kinetic energy of the spacecraft about its COM, motion of the spacecraft in an elliptical orbit will be libratory as illustrated in Figure 1b. Equations representing the spacecraft orbital motion are identical with those of a point mass in an inverse-square law force field. To analyze the attitude dynamics, the spacecraft orbital motion (motion of COM) can be reasonably assumed to be independent of the spacecraft attitude motion (motion about COM). This assumption is justifiable because the satellite is small compared to the dimensions of the orbit. Under this assumption, the spacecraft’s orbital motion can hence transfer energy to the attitude motion, but the converse is assumed not to be possible. Thus, orbital parameters as determined functions of time are considered in analyzing attitude motion [7, 27].

When the spacecraft is considered as a rigid body in an inverse-square force field along an elliptical orbit, a complete formulation of equations of motion can be derived [5, 7, 21]. The resulting six second-order differential equations of motion are nonlinear and coupled. These equations of motion cannot be solved analytically in this exact form.

Ignoring other torques such as aerodynamic, magnetic, thermal bending, and solar radiation pressure, we can derive the equations of spacecraft attitude motion under the influence of inverse-square force field in an elliptical orbit. Additional assumptions are an ideal, perfect sphere earth without oblateness; largest spacecraft dimensions are extremely small compared to the orbit radius, and the spacecraft mass is negligible compared to the mass of the central body [27].

We further assume that the exact equations of motion can be linearized in small-angle motion characterization. Subsequently, the attitude dynamic models may be considered to consist of two equations with coupled roll-yaw angles and a third uncoupled equation describing the pitch angle dynamics. The pitch motion equation is hence independent of roll-yaw motion. The coupled roll-yaw

Consequently, the exact problem is reduced to the equation of pitch motion with orbital parameters as functions of time and spacecraft mass parameters shown in Eq. (1):

where

To analyze attitude motion in eccentric orbit, we substitute time with true anomaly, where *f* is the independent variable. Moreover, the COM will obey the following Keplerian orbit relations:

Thus

Substituting Eqs. (5) and (6) into Eq. (1) yields

This is the well-known equation of plane pitch angle libratory motion in elliptical orbit [5, 7, 21, 27]. The primes indicate differentiation with respect to *f*. The planar pitch attitude motion equation is hence nonlinear with periodic coefficients in *f*. Analysis of this motion and subsequent synthesis of a fitting controller is not a trivial task. We hence intend to analyze this motion and synthesize suitable controllers to stabilize the system.

## 3. Attitude motion analysis

In general, L-F transformation techniques facilitate obtaining solutions of dynamical systems with periodic coefficients, evaluate periodically forced responses, and design feedback control laws. We shall augment these capabilities with normal form techniques (simplifies nonlinearity) and state augmentation (converts nonautonomous to autonomous system). Subsequently, the emanating synergies serve to accomplish the objectives of this chapter.

Floquet theory enables stability and response analysis of linear systems with periodic coefficients, i.e.,

Further, the the Lyapunov-Floquet (L-F) transformation

Time-independent normal form (TINF) simplification facilitates construction of relatively lesser complex but qualitatively equivalent models of the original nonlinear dynamical systems. On the other hand, time-dependent normal form (TDNF) simplification considers original nonlinear dynamical systems with periodic coefficients by utilizing Lyapunov-Floquet transformation. This simplification is generally implemented on equations arising from Taylor series expansion via successive application of nonlinear near-identity transformations. Such a transformation entails preservation of the original system’s stability and bifurcation characteristics by the transformed models. The fundamental concept behind normal forms methodology is to simplify the system by eliminating as many nonlinear terms as possible. This is accomplished via application of successive series of near-identity transformations on the original system. The near-identity coordinate transformations are nonlinear and local. The reader is directed to the well-documented literature on normal form theory found in works by authors such as [32, 33, 34, 35].

To normalize nonlinear systems subjected to external periodic excitation, several authors such as [34, 35, 36, 37] either utilize approaches that introduce equation variables and/or detuning parameters or incorporate a bookkeeping parameter in their methodology. However, the augmenting parameters involved seemingly lack a uniform explicit connection to the terms in the dynamic equations under consideration. Consequently, we shall utilize a relatively more straightforward and intuitive approach that involves augmenting the system states by converting the periodic external excitation into a system state. The state augmentation approach here is intuitive because the augmented states directly emanate from the periodic forcing term(s); hence, they are neither ad hoc nor arbitrary. Moreover, neither a detuning parameter nor a bookkeeping parameter is required.

Strictly speaking the behavior of the attitude motion as shown in Eq. (7) is characterized in terms variation in true anomaly, *f*. However, the true anomaly similarly varies with time; hence, the reference as implicit time history is preferred. To demonstrate implicit time history behavior, we initially select

Both the motion in original coordinates, Eq. (7) and the corresponding state augmented system, will be scrutinized. Figure 2 shows the implicit time history behavior of the motion in original coordinates.

Similarly, we shall scrutinize the history behavior of the system in Eq. (7) after augmenting the system states. In accordance with binomial expansion theorem [38], since

Let

After further substituting the trigonometric product term with its series approximation given in Eq. (13) to the 7th order, the motion in Eq. (9) can be expressed as

Therefore, the augmented system state-space representation with

Figure 3 shows the augmented state system history behavior. Allowing for the expected minor discrepancies due to series and binomial expansion approximations, the system state response is comparable to that of the system in original coordinates shown in Figure 2.

From Figures 2 and 3a**and**b, the attitude motion is quasiperiodic as characterized by the absence of closed trajectory attractors in the phase space. The pitch angle librates roughly between

### 3.1 Stability and chaos

Stability analysis is preceded by computation of the dynamical system’s state transition matrix,

The pitch angle trigonometric product term in Eq. (7) can be represented as a product of the respective Taylor series expansion of sine and cosine as shown in Eq. (13):

where

We can ignore the terms of order greater than 7 in Eq. (13) without significant loss of accuracy because the follow-on terms have relatively small successive coefficients that rapidly approach zero. For instance, the 9th order term has the coefficient

To normalize the principal period, let

After substituting

It is clear that Eq. (16) is of the form

#### 3.1.1 Floquet multipliers and exponents

To facilitate computation of STM, FTM, and L-F transformation matrices using Chebyshev polynomials, we utilized the Chebfun software package on MATLAB™ [39]. Summarily, Chebfun applies piecewise Chebyshev polynomial interpolation to construct smooth functions over the interval

The computed Floquet multipliers are critical since they lie on the unit circle with values of

The motion is hence stable in the sense of Lyapunov, but the inherent oscillations are disruptively significant to jeopardize nominal execution of the spacecraft mission.

#### 3.1.2 Poincaré map

Figure 5 shows the constructed Poincaré section of the flow. There is a discernible main cluster of points in close proximity to the origin but restricted to the positive side of

The Poincaré section composition suggests two possible flow behaviors. The groupings suggest a quasiperiodic trajectory. On the other hand, the scanty random points devoid of clustering could be due to transient behavior or chaos. A chaotic motion can briefly dwell on a near periodic trajectory before changing to a disparate trajectory with a period that is a multiple of the preceding motion. Consequently, it is needed to further investigate the presence of chaos in the attitude motion dynamics.

#### 3.1.3 Chaos

We define chaos as a bounded aperiodic steady-state motion behavior that is not in equilibrium and is sensitive to initial conditions. A minuscule divergence in the input rapidly grows to spawn an overwhelming difference in the system response. We begin investigating chaos in the attitude motion by plotting the system implicit time history with a minute divergence to the initial condition of the state,

#### 3.1.4 Lyapunov exponents

To determine the average rate of divergence between the initially neighboring trajectories defined locally in the state space, we shall scrutinize the dynamic behavior of the motion’s Lyapunov exponents. Lyapunov exponent stability analysis affords a means of quantifiable expression for initial conditions sensitivity and dependence (i.e., chaos), by describing the exponential rate of growth or decay of a perturbation to a trajectory as time progresses [40]. Lyapunov exponent

where

The computed Lyapunov exponents are equal in magnitude but opposite in sign with increasing periods because the flow in Eq. (16) is nonautonomous Hamiltonian. Hamiltonian systems are conservative. Therefore, the magnitude of

#### 3.1.5 Stability charts

The orbit eccentricity and spacecraft’s ratios of principal moments of inertia are, respectively, defined as

Even though we can only construct transition curves associated with the linear term of the attitude motion equation, the outcome provides insightful perusal into the behavior of the whole equilibrium solution. The complete solution behavior can be arrived at by augmenting the evaluated linear periodic stability behavior with the combined effect of the nonlinear term

The darker regions constitute stable points while the lighter regions are unstable. The stability chart has a slightly larger stable region than the unstable region. Unstable solutions appear to be dominated by two regions approximately defined by (i) *e* plus (ii)

For instance, we have previously considered the pair

Stability region | e | σ | Floquet multipliers ( | ||
---|---|---|---|---|---|

0.82 | 0.6 | 1.06 | 0.94 | ||

Unstable | 0.6 | 0.1 | 0.20 | 4.96 | |

0.8 | 0.14 | 0.10 | 9.79 | ||

Marginal | 0.2 | 0.5 | 1 | 1 | |

0.4 | 0.4 | 1 | 1 | ||

Stable | 0.6 | 0.5 | 1 | 1 | |

0.8 | 0.9 | 1 | 1 |

From Table 1, we note that both selected marginal and stable regions are associated with critical Floquet multipliers. This implies that the pertinent

### 3.2 Resonance

The attitude dynamics are dominated by the linear and forcing terms delineated in Eq. (17). This is because if we consider the motion composed of only the linear and forcing terms, i.e.,

Moreover, if we consider the L-F transformed motion, Floquet exponents (eigenvalues of

Bifurcation triggers the system’s equilibrium solutions to transition between the disparate regions of the stability chart. The orbit eccentricity, *e*, is the bifurcation parameter for the attitude motion (see Eq. (16)). This is because the general on-orbit spacecraft mass properties represented by

### 3.3 Versal deformation of the normal form and bifurcation analysis

Versal deformation refers to embedding the system in a parameterized family of systems containing all possible dynamics that can occur near the bifurcation point. Moreover, the family of systems should be transverse to the bifurcation surface with the number of parameters equal to the codimension of the bifurcation [41]. The attitude motion undergoes a codimension-one bifurcation because only one parameter *e* is responsible for the loss of stability (for gravity-gradient stabilization to be maintained,

#### 3.3.1 State-augmented system

To normalize the state-augmented system, we first apply the modal transformation

This system now possesses 4th order nonlinearity.

where *r*. The state-augmented system is independent of periodic coefficients; hence, we solely obtain the TINF given in Eq. (22):

When the external forcing term is augmented as a system state, the magnitude of the external forcing frequency appears as solitary, linear, imaginary conjugate coefficients in the normal form, i.e., 1 (see Eq. (9)). Eigenvalues of the linear matrix in Eq. (12) constitute the conjugate coefficients in the linear terms of the reduced nonlinearity normal forms (i.e.,

Moreover, after obtaining the straightforward solutions of

where *C*_{1} and *C*_{2} are the integration constants originating from the analytical solutions of

We shall investigate the bifurcation of Eq. (12) via its normal form given above. Because the periodic system maintains the same general structure, we may treat the respective limit cycles as equilibria and study their bifurcations. We utilize the versal deformation of the normal form to investigate the change in the stability structure of the dynamics in the neighborhood of the critical point of the bifurcation parameter. Essentially, construction of versal deformation of the normal form facilitates characterization of system dynamics at the critical point and its small neighborhood. Therefore, we handily gain complete understanding of the qualitative phase space dynamics of the dynamical system in the neighborhood of the critical point.

We define the normal form versal deformation parameter as

where

By defining small increments on the bifurcation parameter as

The closed-form analytical solutions for

After solving Eq. (24), we utilize the results to complete the closed-form analytical solution of Eq. (23) as given in Eq. (24):

Similarly,

The back transformed

#### 3.3.2 L-F transformed system

Here, we also demonstrate analysis of bifurcation behavior subject to different values of

The system likewise possesses critical Floquet multipliers that lie on the unit circle of values (

The computed periodic L-F transformation

We consider the L-F transformed dynamics in Eq. (27) up to the cubic nonlinearity. After applying the L-F transformation given in Eq. (19) on the attitude motion in Eq. (16), we obtain the system in Eq. (27):

To normalize this externally excited motion, the system states are augmented to convert the system from nonautonomous to autonomous. We define additional states shown in Eq. (28) and plotted in Figure 13:

After substituting the above augmented states into Eq. (27), we obtain the system shown in Eq. (29)—whose order of nonlinearity increases to four. The transformed denominator term

Apart from raising the order of nonlinearity, state augmentation further introduces periodic linear terms,

The computed parameters corresponding to the second L-F transformation are as follows: critical Floquet multipliers,

The computed second periodic L-F transformation matrix,

We designate the second L-F transformation as

Applying the modal transformation

where

and

We first evaluate the TDNF and then average out the periodic terms to obtain the simplified TINF. The closed-form analytical solutions for

The Floquet exponents are conjugate coefficients in the linear terms of the normal forms before being multiplied by the substituted constant values equal to

Similarly, we define the normal form versal deformation parameter as

where

After defining small increments on the bifurcation parameter again as

The closed-form analytical solutions for

We solve Eq. (36) and use the results to obtain the remaining analytical solutions of Eq. (35) as shown in Eq. (37):

where

Solutions of the versal deformation equations enable investigation of the post-bifurcation steady-state behavior in the small neighborhood of the bifurcation point. However, as observed by [42], this method is only useful for local analysis. This is because minor errors introduced by back transformation close to the bifurcation points significantly grow as you move further away.

## 4. Attitude motion feedback control

After setting

Motion controller design shall be undertaken on augmented state system, L-F transformed, and near-identity transformed coordinates. We shall hence first transform the system dynamics into these more amenable but topologically equivalent dynamical structures that retain the Lyapunov stability and bifurcation properties of the original system. Augmentation of the attitude dynamic states has been conducted in Section 3. Control law development will be first considered in L-F transformed coordinates and then followed by near-identity transformed coordinates of Eq. (16).

### 4.1 Lyapunov-Floquet transformation

Prior to computing the L-F transformation matrix

After applying the L-F transformation,

where

The Lyapunov stability properties are preserved in the new coordinates after the system is transformed by the L-F transformation matrix. The L-F transformation theory guarantees that a suitable controller realized in the L-F transformed coordinates will be correspondingly efficacious after back transformation into the original system coordinates. Consequently, we shall endeavor to systematically synthesize suitable controllers to stabilize the motion in the transformed coordinates. Our control synthesis strategy will first consider linear control laws before exploring nonlinear control strategies.

In order to formulate appropriate control laws that would stabilize the quasiperiodic motion analyzed in Section 3, we introduce a control input

Using Eqs. (5) and (6), we perform a change of independent variable from time (*t*) to true anomaly (*f*). The closed-loop attitude dynamics hence will be

The control action

### 4.2 Linear control

Though linear control law principles are conventionally intended for controlling linear parameter invariant systems [43], we initially consider them to control our nonlinear dynamics as an initial analysis step. Since most linear control methods tend to be relatively simpler to analyze and implement compared to nonlinear control methods, it is prudent to ascertain the suitability of linear control prior to embarking on relatively more complicated techniques. To implement linear control, we shall consider pole-placement approach to determine the negative feedback gain required to stabilize the system.

#### 4.2.1 State-augmented system

The autonomous state-augmented system in Eq. (12) can be represented in abbreviated form as shown in Eq. (43):

where

The linear state feedback controller is of the form

Consequently, a linear state feedback controller cannot stabilize the system dynamics associated with the

#### 4.2.2 L-F transformed system

In this case, the parameter-invariant linear state feedback controller is similarly of the form

Therefore, the system closed-loop dynamics subjected to a linear control law will be of the form

We initially place poles at

In this analysis, the system response in the original coordinates is realized via the back transformation

### 4.3 Nonlinear control

Nonlinear control appears more suitable than linear control to stabilize the attitude motion. However, conventional nonlinear techniques are not readily amenable to dynamics with periodic coefficients and periodic external excitation. Hence, requisite system state augmentation, L-F, or near-identity transformations will be undertaken prior to controller design. We shall first consider sliding mode control (SMC), and then bifurcation control will be implemented on the marginally stable system to stabilize post-bifurcation response.

#### 4.3.1 Sliding mode control

Sliding mode control is a robust nonlinear feedback control methodology that is suitable for achieving accurate tracking for a class of nonlinear systems. SMC methodology is based on variable structure control law that results in the state trajectory “sliding” along a discontinuity surface in the state space [44, 45]. Though SMC is deterministic, nonlinear, and robust, its implementation is prone to undesirable “chattering” along the sliding surface [46]. Design of SMC involves (i) selection of the switching function (stable hyperplane in the state space on which the dynamics will be restricted) and (ii) control law synthesis.

#### 4.3.1.1 State-augmented system

Here, we implement a SMC that tracks a desired null pitch angle via a negative rate of growth. Dynamics in the original coordinates possess periodic coefficients rendering the dynamics unwieldy and unfavorable to synthesize a SMC. Therefore, we develop the SMC law based on the augmented state dynamics—which are liberated from periodic coefficients. To design a sliding mode controller for the state augmented systems, we designate the switching function as given in Eq. (48). The switching function represents the actual system state (i.e., attitude pitch angle) reference error (difference between desired and actual pitch angle) that the controller desires to maintain at zero. Therefore, when

Subsequently, the closed-loop dynamics of the controlled system are similar to Eq. (42) as shown in Eq. (49):

where

Setting

where

A sigmoid function,

Figure 18 shows the sliding mode controlled system response.

It is observable that the sliding mode controller in the state-augmented system achieves stabilization of the motion throughout any number of orbits. Both

#### 4.3.1.2 L-F transformed system

In this case, we similarly assume a control input

It then follows from Eq. (40) that the controlled L-F transformed system is as shown in Eq. (55):

where

After obtaining the derivative of the sliding function, we substitute for

From Eq. (57), we set

where

To reduce chattering around the sliding surface typical of sliding mode controllers due to fast switching of the signum function, a sigmoid function is similarly preferred. We again apply the direct Lyapunov approach to analyze the sliding mode controller stability by selecting

Figure 19 shows the sliding mode controlled system in L-F transformed coordinates. The response in Figure 19 is back transformed via the inverse L-F transformation resulting in controlled states in the original coordinates shown in Figure 20.

We observe that, similar to the state augmented case, the sliding mode controller stabilizes the L-F transformed motion as well by invariably confining the states to zero as desired. Though specific values for the

#### 4.3.2 Bifurcation control

The critical Floquet multipliers corresponding to purely imaginary Floquet exponents indicate that the system is in the stability boundary. Consequently, it is essential to stabilize the system post bifurcation apart from modifying other motion characteristics such as rate of growth. To achieve these objectives, we engage nonlinear bifurcation control with full state feedback. Synthesis of such a controller is facilitated by the normalized dynamics which are relatively more tractable compared to the dynamics as represented in the original coordinates. Periodic coefficients and complexity in structure of the dynamic equations in the original coordinates drastically convolute synthesis of bifurcation control law. Dynamics of the states in the original coordinates will eventually be obtained via back transformation of the normal form, modal, and L-F transformations. Location of the complex Floquet multipliers on the unit circle (Figure 4) indicates that the pitch attitude motion is undergoing a Hopf bifurcation with a limit cycle attractor of controllable radius. Therefore, the structure of the normal form will also verify a Hopf bifurcation occurring in the neighborhood of the critical point of the bifurcation parameter (i.e., orbit eccentricity).

To illustrate the intended approach, we shall formulate the normal forms of nonlinearities up to the cubic order in Eqs. (12) and (16). Normalization of dynamics with higher order nonlinearities can be accomplished through the same techniques. Similar to the preceding cases, we’ll consider the augmented states and L-F transformed systems separately.

#### 4.3.2.1 State-augmented system

In Section 3.1.1, we demonstrated how to obtain the TINF of the state-augmented system—shown in Eq. (23). Obtaining the closed-form analytical solutions for

where *C*_{1} and *C*_{2} are the integration constants obtained when solving for

We solve Eq. (61) and then utilize the results to complete the closed-form analytical solution of Eq. (23). The closed-form solutions of

To synthesize a bifurcation control law of the normal form, we first add a control input in Eq. (62):

Let the scaling matrix and control input, respectively, be of the form

Back transformation of the

The libratory amplitude of the quasiperiodic pitch angle motion is tremendously stabilized and confined to a significantly diminished limit cycle attractor as illustrated in Figure 21.

#### 4.3.2.2 L-F transformed system

As already indicated in Section 3.3.2, in addition to synthesizing bifurcation control law via L-F transformed dynamics, we shall also demonstrate analysis of the spacecraft attitude dynamics due to different values of *e* and *e* and

In Eq. (34), the closed-form analytical solutions for

The Floquet exponents are conjugate coefficients in the linear terms of the normal forms before being multiplied by the substituted constant values equal to

To obtain

Results from Eq. (64) which is easier to solve are then used to obtain the closed-form analytical solutions to Eq. (34). Then, *p*.

The normal form in Eq. (34) verifies that this is a system undergoing a codimension-one Hopf bifurcation. To synthesize a bifurcation control law, a control input is added to Eq. (34) as shown below:

Let the scaling matrix and control input be of the form

The proportional gains are custom-tuned to

The oscillating amplitude of the quasiperiodic pitch angle motion is tremendously stabilized relative to the initial behavior illustrated in Figure 23. This hence demonstrates successful control of the post-bifurcation attitude dynamics about the spacecraft center of mass.

Bifurcation control is a nonlinear control technique that affects the behavior of the closed-loop system by modifying nonlinearity and post-bifurcation behavior. Therefore, the location of Floquet multipliers (exponents) is generally preserved post-bifurcation control. Figure 26 shows the preserved locations of the Floquet multipliers before and after bifurcation control (

## 5. Conclusions

In this chapter, we illustrated techniques for analyzing and stabilizing the attitude motion of a gravity-gradient stabilized spacecraft. The motion dynamics are shown to be nonlinear with periodic coefficients and subjected to external periodic excitation. Methodologies employed here utilize state augmentation, Lyapunov-Floquet transformation theory, and normal forms to realize relatively more tractable dynamical systems that are amenable to conventional controller synthesis techniques. Floquet theory was used to investigate system stability. State augmentation facilitated analysis via normal forms by transforming the dynamical system from nonautonomous to an autonomous one.

Outcome from the analysis showed that the attitude motion is quasiperiodic, chaotic, and stable in the sense of Lyapunov for the particular

Both outcomes of the twofold versal deformation analyses (disparate values of

The quasiperiodic, nonlinear, and periodically forced pitch attitude motion is challenging to control. The synthesized linear controller served as starting point for developing more adept control laws. Not surprisingly, the linear controller failed to stabilize the complexly structured nonlinear dynamical system. As stated, in general the “region of application” of linear control for nonlinear systems is dependent on magnitude of nonlinearity and initial conditions. Many times, linear control may stabilize nonlinear systems locally, but this is not guaranteed.

On-orbit perturbations cause disturbing torques that bifurcate the attitude motion; it is hence imperative to stabilize the system attitude dynamics in the small neighborhood of the bifurcation parameter’s critical point. Local nonlinear bifurcation control law implemented on the attitude motion undergoing a Hopf bifurcation was shown to stabilize the attitude motion. The bifurcation controller which modifies the nonlinearity and post-bifurcation behavior further prevents the attitude motion from becoming chaotic because bifurcation is the path to chaos. Implemented in the TINF, the bifurcation control law would subsequently stabilize the secular and periodic attitude perturbations experienced by a spacecraft in elliptical orbit about its nominal operating point.

Sliding mode control law was based on driving both system states to zero on the sliding surface when the sliding surface reference error is equal to zero. The SMC law was similarly shown to be successful by invariably restricting the pitch angle to zero.

Future work will consider torques generated by sources such as magnetism and oblateness of the earth, atmospheric drag, solar radiation pressure, thermal bending, etc. Further, nonlinearities beyond the cubic term in the L-F transformation and TDNF case of near identity transformation would also be analyzed. In addition, physical implementation of the controllers and derivation of TDNF-based control laws requires future scrutiny. As demonstrated, all the control effort inputs are single torques per unit moment of inertia which for instance can be implemented via thrusters. Consequently, sizing and implementation of the control effort are an essential task.

## Acknowledgments

Financial support for this research was partially provided by the Interplanetary Initiative of Arizona State University.

## Conflict of interest

The authors declare no conflict of interest.

## Nomenclature

*e*

eccentricity

*f*

true anomaly, *radians*

*r*

orbit radius, m

center of mass

identity matrix

principal moment of inertia about the roll, pitch, and yaw axes, respectively, kgm^{2}

local-vertical/local-horizontal

*P*

semilatus rectum

Lyapunov-Floquet transformation matrix (LFT)

time-dependent normal forms

time-independent normal forms

earth gravitational parameter,

spacecraft angular velocity, rad/s

spacecraft pitch angle,

spacecraft roll angle,

spacecraft yaw angle,

state transition matrix (STM)

Floquet transition matrix (FTM)

## Notes

- These authors contributed equally.