## Abstract

Recent space programs require agile and large-angle attitude maneuvers for applications in various fields such as observational astronomy. To achieve agility and large-angle attitude maneuvers, it will be required to design an attitude control system that takes into account nonlinear motion because agile and large-angle rotational motion of a spacecraft in such missions represents a nonlinear system. Considerable research has been done about the nonlinear attitude tracking control of spacecraft, and these methods involve a continuous-time control framework. However, since a computer, which is a digital device, is employed as a spacecraft controller, the control method should have discrete-time control or sampled-data control framework. This chapter considers discrete-time nonlinear attitude tracking control problem of spacecraft. To this end, a Euler approximation system with respect to tracking error is first derived. Then, we design a discrete-time nonlinear attitude tracking controller so that the closed-loop system consisting of the Euler approximation system becomes input-to-state stable (ISS). Furthermore, the exact discrete-time system with a derived controller is indicated semiglobal practical asymptotic (SPA) stable. Finally, the effectiveness of proposed control method is verified by numerical simulations.

### Keywords

- spacecraft
- attitude tracking control
- discrete-time nonlinear control

## 1. Introduction

Recent space programs require agile and large-angle attitude maneuvers for applications in various fields such as observational astronomy [1, 2, 3]. To achieve agility and large-angle attitude maneuvers, it will be required to design an attitude control system that takes into account nonlinear motion because agile and large-angle rotational motion of a spacecraft in such missions represents a nonlinear system.

Considerable research has been done about the nonlinear attitude tracking control of spacecraft [4, 5, 6, 7, 8, 9, 10, 11, 12], and these methods involve a continuous-time control framework. However, since a computer, which is a digital device, is employed as a spacecraft controller, the control method should have discrete-time control or sampled-data control framework.

Although a sampled-data control method for nonlinear system did not advance because it is difficult to discretize a nonlinear system, a control method based on the Euler approximate model has been proposed in recent years [13, 14] and is applied to ship control [15]. Although our research group has proposed a sampled-data control method using backstepping [16] and a discrete-time control method based on sliding mode control [17] for spacecraft control problem, these methods are disadvantageous because control input amplitude depends on the sampling period

For these facts, about the spacecraft attitude control problem that requires agile and large-angle attitude maneuvers, this chapter proposed a discrete-time nonlinear attitude tracking control in which the control input amplitude is independent of the sampling period

The following notations are used throughout the chapter. Let

derived from vector

## 2. Relative equation of motion and discrete-time model for spacecraft

In this chapter, as the kinematics represents the attitude of the spacecraft with respect to the inertia frame

where Eq. (1) is the kinematics that represents the attitude of ^{2}] is the moment of inertia.

We consider a control problem in which a spacecraft tracks a desired attitude (MRPs)

where

Substituting Eqs. (3) and (4) into Eqs. (1) and (2) using the identity

Hereafter, we assume that the variables of spacecraft

**Assumption 1**: the desired states

From Eqs. (A4) and (A5) in Appendix, the exact discrete-time model of relative motion equations is obtained as

and the Euler approximate model of relative motion equations are obtained as

## 3. Discrete-time nonlinear attitude tracking control

We derive a controller based on the backstepping approach that makes the closed-loop system consisting of the Euler approximate modes (10) and (11) become input-to-state stable (ISS), i.e., the state variable of closed-loop system

where

**Assumption 2:**

**Remark 1:** from the relational expression

where

In addition, Lemmas when using the derivation of the control law are shown below.

**Lemma 1:** for all

**Lemma 2:** when the quadratic equation

has two distinct real roots

is

### 3.1. Derivation of virtual input α k

Assume that

where

From Lemma 1, the difference of Eq. (13) along the trajectories of the closed-loop system is given by

From Lemma 2,

is obtained as

In addition, since

Therefore, if

### 3.2. Derivation of control input u k

The error variable between the state

The control input

From Eq. (18), subsystem (10) becomes

From Eqs. (18) and (19) and the following equation

the discrete-time equation with respect to

Now, by setting

Eq. (20) becomes

where

As Eq. (14) is given by

from Eq. (18), by using completing square, the difference of Eq. (22) along the trajectories of the closed-loop system is given by

where

In Eq. (23), if

where

From Schur complement, condition

Condition (24) is the same as Eq. (15), and assume that Eq. (24) holds. From Lemma 2, the range of

and the following Eq.

must hold true in order to obtain a real number. As the denominator of Eq. (27) is the same as Eq. (24), the following equation must hold

in order to hold Eq. (27). From Lemma 2, the range of

and the following Eq.

must hold in order to have the real number. As

In addition, since

under Assumption 2, the condition (29) is given by

Therefore, if

under Assumption 2, the condition (26) is given by.

Therefore, if

Summarizing the above, the following theorem can be obtained.

**Theorem 1:** if sampling period

becomes ISS.

Then, we show that the pair

Lyapunov function

Since

In addition,

where

where

**Theorem 2:** control input (34) is SPA stabilizing for exact discrete-time systems (8) and (9).

## 4. Numerical simulation

The properties of the proposed method are discussed in the numerical study. For this purpose, parameter setting of simulation is as follows:

The moment of inertia **Figure 1**.

The results of the numerical simulation are shown in **Figures 2**–**5**. The relative attitude

## 5. Conclusion

This chapter considers the spacecraft attitude tracking control problem that requires agile and large-angle attitude maneuvers and proposed a discrete-time nonlinear attitude tracking control that the amplitude of the control input does not depend on the sampling period

This section shows preliminary results for nonlinear sampled-data control [13, 14, 19].

Let us consider the following nonlinear system:

where

The nonlinear system (A1) is assumed to be between a sampler (A/D converter) and zero-order hold (D/A converter), and the control signal is assumed to be piecewise constant, that is,

where

is measurable at each sampling instance. The exact discrete-time model and Euler approximate model of the nonlinear sampled-data systems (A1)–(A3) are expressed as follows, respectively:

where we abbreviate

**Definition 1:** consider the following discrete-time nonlinear system:

where

**Definition 2:** let

In addition, if there exists

Using the above definitions, the following theorem is obtained by literatures [13, 14, 19].

**Theorem A.1:** if the pair

Hence, if we can find a family of pairs of