Multifarious Chaotic Attractors and Its Control in Rigid Body Attitude Dynamical System

The Euler dynamical equation which describes the attitude motion of a rigid body will exhibit very complex dynamic behaviors under the action of diﬀerent external torques. Many special types of new chaotic attractors are presented, including hidden attractors, double-body-double-core chaotic attractors, and single-body-three-core-tree-wing chaotic attractors. The position of equilibrium points in several typical cases of the Euler dynamic equation is solved, and the stability of linearized equation at each equilibrium point and its inﬂuence on the formation of the chaotic attractor are analyzed. An improved nonlinear relay control law based on Euler angle feedback is developed to stabilize a new chaotic spacecraft attitude motion to an appointed equilibrium point or a periodic orbit.


Introduction
e detailed exploration of the chaotic attitude motion of spacecraft and satellites still remains one of the main problems of rigid body attitude dynamics. A number of investigators have demonstrated that spinning satellite [1], dual-spin spacecraft (DSSC) [2][3][4], multispin spacecraft (MSSC) [5], gyrostat satellite [6][7][8][9][10][11][12], tethered satellite, and other complicated satellites would exhibit chaotic attitude motion in gravitational fields [13], geomagnetic field [14,15], and sunlight flux. Beletsky et al [14] discussed the chaotic motion of a magnetic spacecraft in circular polar orbit without damping and gravitational torque. Chen and Liu [15,16] investigated chaotic attitude motion of a magnetic rigid spacecraft with internal damping in a circular orbit near the equatorial plane of the earth. ese works mainly analyzed the dynamics of different mathematical models of rigid body attitude motion from a different point of view. e attitude motion of a rigid body could be described by the Eulerian dynamic equations.
is nonlinear equation will present very complex dynamic behaviors, including various forms of chaotic motion. Leipnik and Newton [17] found two strange attractors from this system. In the previous paper [18,19], we introduced several new chaotic attractors but did not analyze the influence of the properties of equilibrium point on the formation of chaotic attractors. Different control techniques have been employed to suppress or manipulate chaotic attitude motion of spacecraft and satellites, including sliding mode variable structure control [20,21], time-delayed feedback control [22,23], observer-based control [24,25], impulsive control, adaptive control, and open-plus-closed-loop control. Alban and Antonia [26] made use of three methods to control a sixdimensional chaotic system that describes the attitude dynamics of a rigid body spacecraft subjected to deterministic external perturbations that induce chaotic motion when no control is acted. e three techniques are a simple delayed feedback control method, the Otani-Jones technique, and a higher dimensional variation of the OGY method. Other control methods were employed to suppress chaos by Meehan and Asokanthan [1] and Awad [20]. Generally, the angular velocity of the spacecraft ω was chosen as feedback control variable. However, ω can be measured directly in practice.
Different spacecraft have different mission requirements and need different control methods. A spinning spacecraft usually is required to rotate along one axis with constant rotation rate and not to rotate along the other two axes. When a spinning spacecraft is disturbed and produces unexpected rotation, the nozzle thruster is usually used to control it. Due to the coupling effect of the three channels, the single-channel dead zone relay control method cannot effectively suppress a chaotic motion of spinning spacecraft. e innovation of this paper lies in the following: (1) Many special types of new chaotic attractors are presented, including hidden attractors, doublebody-double-core chaotic attractors, and singlebody-three-core-tree-wing chaotic attractors. (2) e position of equilibrium points in several typical cases of the Euler dynamic equation is solved, and the stability of linearized equation at each equilibrium point and its influence on the formation of chaotic attractor are analyzed. (3) A more practicable and efficient control techniques are utilized to suppress chaos and control state of the system to an appointed fixed point or a periodic orbit.

The Attitude Motion Equations of a Rigid Body
e attitude motion of a rigid body could be described by the Eulerian equations, which consists of kinematic equations and dynamic equations. e attitude orientation of spacecraft at a given point can be locally described in terms of three Eulerian angles φ, θ, and ψ which are successive clockwise rotations about inertial axes X, Y, and Z, respectively. ese successive rotations transform the inertially fixed set of orthonormal axes X, Y, and Z (regarded as initially instantaneously coincident with the body axes) into the axes x, y, and z fixed in the body. e kinematic equation of the rigid body can be expressed as [21] or the following form � ω x sin ψ + ω y cos ψ, Here, ω � (ω x , ω y , ω z ) are the angular velocities of a rigid body. e dynamical motion equations of a spacecraft with principle axes at the center of mass are [21] Here, I x , I y , and I z are the principal moments of inertia with respect to body axes x, y, and z; u x , u y , and u z are the component of the control torques; and M x , M y , and M z are the perturbing torques. e total disturbing torque M which is put on a spacecraft can be written as Here, M g and M m are the gravitational and magnetic torque; M d is the internal damping torque which is proportional to the angular velocity of the body with coefficient c; M c and M f are atmosphere resistance torque and solar pressure torque. ey all have relation to the attitude angles φ, θ, and ψ, the attitude angular velocities ω, and the orbital angular velocities ω 0 . For example, the component of gravitational torque M gz on Z-axis can be written as e magnetic torque about Z-axis can be obtained as Here, k is a coefficient with respect to the magnetic moment constant of the earth and the magnetic moment of the spacecraft and the distance between spacecraft and the earth. α is the angle of inclination of orbital plane. β is the argument of perigee. c is the true anomaly of spacecraft as the angular coordinate measured from perigee.
Equation (3) can be rewritten as Here, a x � (I y − I z )/I x , a y � (I z − I x )/I y , a z � (I x − is a perturbing frequency matrix.

Multifarious Chaotic Attractors
Equation (7) is a more generalized 3-dimensional nonlinear system which will exhibit complex (periodic, quasiperiodic, or chaotic) dynamic behaviors under the action of different external torques.
Earlier papers [1] have taken ese torques are chosen to be sufficiently large to induce chaotic motion and are comparable in magnitude with the available thruster torques. e dynamics of the satellite will then exhibit chaotic motion. We constructed the chaotic attractor of this system shown in Figure 1.

A Type of Hidden Attractors.
In the study of the chaotic motion of a nonlinear system, it is often concerned with the chaotic attractors near the equilibrium point. Some hidden attractors exist widely in the greater region or the smaller region of the phase space.
Let us take Leipnik-Newton system (9) as an example. ere are five equilibriums in equation (9). e origin of coordinates S 0 : (0, 0, 0) is one of five equilibriums. e other four equilibriums are composed of the proper combination of the following three values: , , Example 1. Let a � 0.4 and b � 0.3, and the four equilibriums are e eigenvalues of the Jacobian linearization matrix at four equilibriums are the same: S 1 , S 2 , S 3 , S 4 : λ 1 � −0.8, λ 2,3 � 0.15 ± 1.1419i. In the phase space, there exist one chaotic attractor near S 1 and S 2 , and one periodic attractor near S 3 and S 4 , and one big hidden attractor far away from equilibriums. e phase trajectories of the three attractors are shown in Figure 2. e starting point of the small periodic attractor is selected at (−0.0459, 0.0802, 0.2871). e starting point of the big hidden attractor is selected at (−74.2294, 52.3574, −29.0246).

Another Type of Double-Body-Double-Core Chaotic
Attractors. Investigating the reduced form of equation (7), ere are five equilibriums in equations (12). Obviously, the origin of coordinates S 0 : (0, 0, 0) is one of five equilibriums. e other four equilibriums are composed of the proper combination of the following three values: When the feedback coefficients a 11 , a 22 , and a 33 make 0 ≥ μ k ∈ R, (k � x, y, z), equation (12) degenerates to the single equilibrium point system. Here, let us just discuss the case of 0 ≤ μ k ∈ R, (k � x, y, z).
Let us assume I x > I y > I z > 0, a x > 0, a y < 0, a z > 0, and there are two cases of combination forms.
Case A: a x > 0, a y < 0, a z > 0, a 22 > 0, a 11 < 0, a 33 < 0. e other four equilibriums are expressed as Case B: a x > 0, a y < 0, a z > 0, a 22 < 0, a 11 > 0, a 33 > 0. e other four equilibriums are expressed as Equation (12) is linearized at equilibrium point S 1 : (μ x , μ y , μ z ), and the Jacobian matrix is e characteristic equation of the Jacobian linearization system is e eigenvalues can be calculated by the following steps: Here, Simulation studies indicate that system (12) will exhibit complex dynamic behaviors under the different feedback coefficients.
ere exist double-body-double-core chaotic attractors in the phase space. e phase trajectories of the two attractors are shown in Figure 3.

A Type of Single-Body-ree-Core Chaotic Attractors.
Investigating another reduced form of equation (7), ere are five equilibriums in equation (13). e origin of coordinates S 0 : (0, 0, 0) is one of five equilibriums. e other four equilibriums are composed of the proper combination of the following values: If a y and a 22 are the opposite sign, then a z and a 33 must be the opposite sign. When ω xe > 0, ω ye and ω ze must be the same sign. When ω xe < 0, ω ye and ω ze must be the opposite sign.
Obviously, S 1 , S 2 , S 3 : Re(λ 2,3 ) > 0 but S 4 : Re(λ 2,3 ) < 0, S 4 is stable equilibrium point, and S 1 , S 2 , S 3 are unstable. ere is a small attraction basin around S 4 (shown in Figure 4(a)) and a single-body-three-core-tree-wing chaotic attractor which are formed by S 1 , S 2 , and S 3 in the larger domain. e phase trajectories of the attractors and the attraction basin (red point) are shown in  Mathematical Problems in Engineering

Multifarious Chaotic Attractors.
Keeping ω u � 0, changing parameter value of a x , a y , and a z and matrix A, system (7) will exhibit chaotic attitude motion, and a lot of new chaotic attractors are found in state space, as follows:

Analysis of the Properties of New Chaotic Attractors
In order to analyze the structure of new chaotic attractors, the concept of attractive plane is introduced from the linear system: where → has one real eigenvalue λ 1 and a pair of complex conjugate eigenvalues λ 2,3 � α ± βi, the solution of equation (16) can be written as where C ij (i, j � 1, 2, 3) are coefficients in relation to structure parameters of system (23) and initial values. k i ″ (t, x i0 ) is the partial solution of (23).

(25)
Proof. it is assumed that the motion trajectory of the partial solution k i ″ (t, x → 0 ) is inside a plane across coordinate origin (0, 0, 0) in the phase space, and the normal vector of this plan is P → � p 1 , p 2 , p 3 . e point-norm form equation of plane can be expressed as Now, let us prove this plane exists, and its normal vector P → is constant and has nothing to do with the initial conditions. Put formulas (25) into (26): 3 C 32 e αt cos βt + p 1 C 13 + p 2 C 23 + p 3 C 33 e αt sin βt � 0.

(29)
As long as let (p 1 a 11 + p 2 a 21 + p 3 a 31 )/p 1 � (p 1 a 12 + p 2 a 22 + p 3 a 32 )/p 2 � (p 1 a 13 + p 2 a 23 + p 3 a 33 )/p 3 � k, it can be derived as follows: where k is the ratio of vectors module. Arbitrarily choose one vector coordinate, for example, p 1 � 1, and other coordinates (p 1 , p 2 , p 3 ) could be solved. Obviously, it has nothing to do with the initial conditions and the eigenvalue λ. Proof ends. □ Definition 1. Assume the matrix B → has one real eigenvalue λ 1 and a pair of complex conjugate eigenvalues λ 2,3 � α ± βi. When λ 1 < 0 (or λ 1 > 0), the fixed plane is named attractive plane (or repulsive plane) in which the partial solution

Structural Properties of Chaotic Attractors
Properties 1. e size of various attractors in equation (7) is determined by the relative positions of all equilibrium points. In other words, the relative distances of all equilibrium points are magnified k times, and the sizes of various attractors are magnified 1/k times synchronously, and the structure and shape remain unchanged. e parameters a x , a y , and a z in equation (7) are simultaneously magnified k times, and the sizes of various attractors are magnified 1/k times, and the structure and shape remain unchanged.

Properties 2.
ere is an attractive plane in the neighborhood of each equilibrium point in the chaotic system. e nonlinear motion mode near an equilibrium point of a Mathematical Problems in Engineering 7 complicated chaotic system can be approximately described with its linearized equation. Each chaotic motion comprises at least two nonlinear motion modes. e essential reason to form chaotic attractor is nonperiodic and asynchronous switch between two modes.

Control of Chaotic Attitude Motion in a Spinning Spacecraft
A spinning spacecraft usually is required to rotate along one axis with constant rotation rate and not rotate along the other two axes. e goal of control is to stabilize the state of the system at the desired equilibrium point (ω x , ω y , ω z ) � (c, 0, 0). e angular velocity ω in system (7) cannot be measured directly in practice and ω y ≠ θ .
, and _ ψ of the spacecraft using attitude sensors such as earth sensors, solar sensors, and gyroscopes. Compared with the desired attitude angle and angular velocity, if there is a deviation, the switch control instruction is formed by the controller to suppress the unwanted rotation of the spacecraft.
As all know, the nonlinear relay control law which is based on position and velocity feedback with dead band peculiarity is suitable for the nonchaotic system when A → � 0 in system (7). However, the situation is completely different for chaotic systems, because there are multiple equilibrium points in the system, and the control torques applied to the three axes are interacting. So, the attitude angle change of other direction must be considered into the design when the control law for one direction was designed.
Let us control six-dimensional chaotic system (21) (its chaotic attractor is shown in Figure 5(a)): � ω x sin ψ + ω y cos ψ,   Mathematical Problems in Engineering Adopting single-channel dead zone relay control law based on position and speed feedback, where M is control torque to be generated by nozzle thrusters. φ 1 , θ 1 , ψ 1 is angle position error of dead band corresponding to the driving signal when the nozzle baffle is opened. ere are five equilibriums in equations (21), and the point S 1 : (−0.34, 0.44, 0.34) is one of five equilibriums. e control action may drive any state to approach a nearest equilibrium point from an initial state. An equilibrium state is not the desired state for a spinning spacecraft though the chaotic motion is suppressed. Figure 6 shows the time history of ω y and θ under the single-channel dead zone relay control action, starting from initial state (0.2, 0.1, 0.2), beginning to control on 1500 s, and ending in the equilibrium point S 1 : (−0.34, 0.44, 0.34). Now, considering the mutual effect among ω x , ω y , and ω z , changing the dual-channel control law to be

Conclusion
e Euler dynamical equation which describes the attitude motion of a rigid body will exhibit very complex dynamic behaviors and include many special types of chaotic attractors. In the same phase space, beyond chaotic attractors, there may be a very large hidden attractor or very small hidden attractors. Even if the system is stable near a certain equilibrium point, chaotic attractors may still be generated in a larger space. From the simulation analysis, we can see that the stability of the linearized equation near the equilibrium point determines the shape and size of the chaotic attractor. e single-channel dead zone relay control method cannot effectively suppress a chaotic motion of spinning spacecraft. e simulation results confirm that the single-channel dead zone relay control method cannot effectively suppress a chaotic motion of spinning spacecraft. e dual-channel control method proposed in this paper can control a chaotic system to an appointed equilibrium point or a periodic orbit.

Data Availability
All the data generated or analyzed during this study are included in this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.