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.
- attitude tracking control
- discrete-time nonlinear control
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 . Although our research group has proposed a sampled-data control method using backstepping  and a discrete-time control method based on sliding mode control  for spacecraft control problem, these methods are disadvantageous because control input amplitude depends on the sampling period as the control law is of the form .
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 effectiveness of proposed control method is verified by numerical simulations.
The following notations are used throughout the chapter. Let and denote the real and the integer numbers. Rn and are the sets of real vectors and matrices. For real vector , is the vector transpose, denotes the Euclidean norm, and is the skew symmetric matrix
derived from vector . For real symmetric matrix , means the positive definite matrix. The identity matrix of size is denoted by . and are the maximal and the minimal eigenvalues of a matrix , respectively.
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 , the modified Rodrigues parameters (MRPs)  are used. The rotational motion equations of the spacecraft?s body-fixed frame are given by the following equations:
where Eq. (1) is the kinematics that represents the attitude of with respect to the , Eq. (2) is the rotation dynamics, [−] is the MRPs, [rad/s] is the angular velocity, [Nm] is the control torque (input), [Nm] is the disturbance input, and [kg m2] is the moment of inertia.
We consider a control problem in which a spacecraft tracks a desired attitude (MRPs) and angular velocity in fixed frame . The MRPs of the relative attitude and the relative angular velocity in the frame are given by
where is the direction cosine matrix from to that expresses the following Eq. :
Hereafter, we assume that the variables of spacecraft and are directly measurable and is known. In addition, regarding the desired states , , , and the disturbance , the following assumption is made.
Assumption 1: the desired states , , and are uniformly continuous and bounded . The disturbance is uniformly bounded .
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 satisfies the following equation:
where is the class KL function and is the class K function. To this end, assume that is the virtual input to subsystem (10), and derive the stabilizing function that is asymptotic convergence to zero. Then, derive the control input that closed-loop system becomes ISS. Here, regarding the variable , the following assumption is made.
Assumption 2:lies in the region that satisfies the following equation:
Remark 1: from the relational expression
where and are the quaternion . Assumption 2 is equivalent to .
In addition, Lemmas when using the derivation of the control law are shown below.
Lemma 1: for all , the following equations hold :
Lemma 2: when the quadratic equation
has two distinct real roots , if , then the solution of the quadratic inequality
3.1. Derivation of virtual input
Assume that is the virtual input to subsystem (10), and define the stabilizing function such that
where is the feedback gain. The candidate Lyapunov function for (10) is defined as
From Lemma 1, the difference of Eq. (13) along the trajectories of the closed-loop system is given by
From Lemma 2, becomes negative, i.e., the range of that holds the following equation
is obtained as
In addition, since under Assumption 2, the range of that holds Eq. (15) is obtained as
Therefore, if satisfies Eq. (17) and , then .
3.2. Derivation of control input
The error variable between the state and is defined as
The control input that makes the closed-loop system becomes ISS is derived.
the discrete-time equation with respect to is
Now, by setting to
Eq. (20) becomes
As Eq. (14) is given by
In Eq. (23), if , then
where is the minimum eigenvalue of and the condition of ISS holds . Hereafter, conditions of and which the matrix holds are derived under Assumption 2.
From Schur complement, condition is equivalent to the following equations:
and the following Eq.
and the following Eq.
must hold in order to have the real number. As under Assumption 2, must satisfy the condition
In addition, since
under Assumption 2, the condition (29) is given by
under Assumption 2, the condition (26) is given by.
Summarizing the above, the following theorem can be obtained.
Then, we show that the pair is semiglobal practical asymptotic (SPA) stabilizing pair for the Euler approximate systems (10) and (11). Hereafter, suppose that sampling period and feedback gains and satisfy Eqs. (31), (32), and (33) under Assumption 2. By using the following coordinate transformation
Lyapunov function and its difference can be rewritten as
Since and , and satisfy following equations:
where is a positive constant. Furthermore, also satisfies the following equation for all with :
where is a positive constant. Therefore, from Eqs. (35) to (38), Lyapunov function and control input satisfied Eqs. (A8)–(A11) in Definition 2 under Assumptions 1 and 2, and the pair becomes SPA stabilizing pair for the Euler approximate systems (10) and (11). Then, the following theorem can be obtained by Theorem A.1 in Appendix.
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 is from . The initial values correspond to Euler angles of 1–2-3 system of . The feedback gains and satisfy Eqs. (25) and (28) for all cases of . The desired states , , and in this simulation are the switching maneuver as shown in Figure 1.
The results of the numerical simulation are shown in Figures 2–5. The relative attitude and relative angular velocity converge to the neighborhood of , and the control input amplitude does not depend on the sampling period although there is a slight difference in the maximal value of .
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 . The effectiveness of proposed control method is verified by numerical simulations. Extension to the guarantee of stability as sampled-data control system will be subject to future work.
Let us consider the following nonlinear system:
where is the state variable and is the control input. The function in Eq. (A1) is assumed to be such that, for each initial condition and each constant control input, there exists a unique solution defined on some intervals of .
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 a sampling period. In addition, assume that the state variable
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:
Definition 1: consider the following discrete-time nonlinear system:
where is the state variable and is a control input. The family of controllers SPA stabilizes the system (A6) if there exists a class KL function such that for any strictly positive real numbers , there exists , and such that for all and all initial state with , the solution of the system satisfies
Definition 2: let be given, and for each , let functions and be defined. The pair of families is a SPA stabilizing pair for the system (A7) if there exist a class functions , , and such that for any pair of strictly positive real numbers , there exists a triple of strictly positive real numbers such that for all with, and :
In addition, if there exists such that Eqs. (A8)–(A11) with hold for all and , then the pair is globally asymptotic (GA) stabilizing pair for the system (A6).
Theorem A.1: if the pair is SPA stabilizing for , then is SPA stabilizing for.
Hence, if we can find a family of pairs of that is a GA or SPA stabilizing pair for , then the controller will stabilize the exact model .