Hopf bifurcation analysis for liquid-filled Gyrostat chaotic system and design of a novel technique to control slosh in spacecrafts

1 Now, for discussing its dynamics around originIntroduction

In 1852, Leon Foucault invented a device, used for investigating rotation of the earth, named as gyrostat. This device worked in a manner that plane of rotation depends on latitude of its location. Later on, gyrostat is used in aerial vehicle to detect path and attitude control. There are various shapes of gyrostat that has many advantages in aerial vehicles [1], satellites and robots [2]. The first governing equation for discussing dynamics of gyrostat was introduced by Wittenburg [3] in 1977, whereas Chatys and Koruba [4] used mini gyro as controller in mini aerial vehicles for the first time. A dual-spin spacecraft [5] is established in the presence of damping factor with the aid of Lie–Poison structures. A gyrostat is a rigid body consisting of rotors to a main platform of the body. In Figure 1, one can observe that this frame is further categorized into three axisymmetric rotors, which are rotating about their respective axis with respect to body frame. Euler dynamical equations of rotational dynamics for a disturbed gyrostat system given by ref. [6] are as follows:

Now, by shifting inertial terms I x , I y and I z to the right side of system (1), we get:

Throughout this article, the authors have used the following parameter values for simplicity:

Hence, substituting parameter values (3) in system (2), we get the following system:

where X = [ x y z ] T is the angular velocity vector; I x , I y and I z are the principal moments of inertia of the gyrostat in the body axis frame; h x , h y and h z are constants of total angular momentum, whereas L x , L y and L z are external torques applied on gyrostat. System (4) shows unpredictability in trajectories for I → = [ 0.85 0.45 0.2 ] T , μ → = [ 6 6.43 5.8 ] T , h → = [ 0 0.57416 2.38 ] T and L → = [ 0 0 22.8 ] T .

Figure 1

Satellite attached with partially filled fuel tank and three rotors.

A phenomenon, in which given dynamical system changes its structure with the variation in parameter values is famous as bifurcation [7]. Hopf bifurcation [8] is one of the important types of bifurcation, in which one can seek appearance and disappearance of oscillatory solutions for various values of parameter around bifurcation point. A variety of research exist for observing oscillatory solutions in dynamical systems. Recently, Marwan et al. have considered forced Chen system and discussed existence of Hopf bifurcation [9] for non-zero equilibria. In 2020, Marwan and Ahmad [10] discussed Bogdanov–Takens bifurcation for energy and resource transportation model between two cities in China. Existence of oscillatory solutions is not only limited to mechanical systems but also has a variety of impact on biological models. In 2019, Mondal and Samanta [11] utilized MATCONT for observing Hopf bifurcation in the predator–prey model, when predator is harvested at a constant rate. The same biological model is analysed in the presence of fear effect [12] for the existence of oscillatory solutions, whereas two different food chain models are considered by Sahoo and Samanta in their work [13], with the effect of delay parameter, to investigate switching of stability through Hopf bifurcation. In 2020, several bifurcations including generalized Hopf and codimension 2 bifurcations are studied in the predator–prey model under the assumption that prey population is infected by a disease [14]. Apart from importance of Hopf bifurcation in biological models, it has great impact on spacecraft models. A specific case of gyrostat-based satellite moving on circular path is examined for two different cases with the help of local bifurcation by Sontos and Melicio [15]. Similarly, various shaped gyrostats, such as spheroid gyrostat [16] and axial gyrostat [17], are studied for the existence of bifurcations. There are many ways to prove the existence of chaos in dynamical system, such as monotone, co-existing attractors, hidden attractors and self-existed attractors [18], but the basic way of obtaining chaos is bringing negligible changes in initial conditions and bifurcation parameters in given dynamical system. Qi and Yang [6] introduced chaotic gyrostat model for the first time and determined unpredictability of angular momentum for fixed parameter values. Qi et al. also discussed its dynamics with the aid of external and internal torques, forces and energies.

The term chaos has applications in numerous engineering-based systems, but unpredictability in gyrostat proceeds to disturbance in attitude control and communication loss. Therefore, controllability [19,20] in such systems seek attention of many researchers. Chai considered six degrees of freedom vehicle [21] and designed its attitude control. In 2012, Aslanov and Yudintsev considered free gyrostat chaotic system [22] along with three-dimensional inertial components and asymmetric rotors. A chaotic gyrostat dynamical system is synchronized with identical derive system with the help of newly designed controller [23], named as variable substitution and feedback controller (VSFC) for achieving global stability. In 2018, Chai et al. [24] introduced receding horizen control for spacecraft path planning. They have also decreased computational cost with the aid of of two nested gradient methods by embedding in control scheme. A new three-layer hybrid control [25] is designed for space vehicles based on stochastic algorithm to enhance global stability. Abtahi derived a gyrostat model [26] based on the Hamiltonian method for orbital motion of spacecraft with the aid of extended Deprit transformation and showed its chaotic effect using the Melnikov technique. Doroshin [27] generalized case of V. A. Stekloff’s rigid body motion and investigated its angular motion, whereas Wang et al. [28] worked on its controllability using Fuzzy controller based on fuzzy logic. Several researchers worked on attitude dynamics and control of spacecraft models [29,30,31].

We have studied dynamical systems derived for spacecrafts using Hamiltonian mechanics [32] and devices attached within it [33,34]. We have noticed that several biological models are taken into account for oscillatory solutions [35,36] in the aforementioned citations, but impact of seeking local oscillatory solutions using Andronov–Hopf bifurcation for spacecraft with attached fuel tank is never discussed. Moreover, the review work of Chai [37], in which authors have discussed path planning and controllability of spacecraft models using different techniques in detail, has motivated us to work on the design of controller for bringing stability in sloshing effect which arises inside coupled tank and leads to havoc in dynamics of gyrostat.

This article is organized as follows: in Section 2.1, the existence of Hopf bifurcation is shown using singularity condition. Moreover, Section 3 consists of our proposed controller; M chaotic and its algorithm, which is applied to gyrostat chaotic system in Section 4. Finally, numerical simulation, discussion of Hopf bifurcation and M chaotic controller are provided in Section 5.

2 Dynamical analysis

3 Algorithm and limitations of M chaotic controller

In this section, we used the concept of Lyapunov stability based on the classical control theory to design algorithm for M chaotic controller. This controller works in a manner that estimated parameters start approaching towards their original values. Then, unpredictable trajectories start moving to their desired values as t approaches to infinity. It is important to mention that we have selected stable equilibria of system (4), for the sake of convergence, instead of any desired values.

Algorithm 1. Algorithm for M chaotic controller

1. Suppose a closed loop system with M = M A + M W :

   (17) y ˙ = g ( y , M )

   for an open loop system:

   (18) x ˙ = f ( x ) .

2. Add a controller; M to the right side of open loop system (18), where, number of controllers depends on the nature of chaotic system.

3. Derive error dynamical system:

   (19) e ˙ = y ˙ − x ˙ ,

   using error term between the state variables of systems (18) and (17).

4. As M = M A + M W given in step 1. Therefore, select M A on the basis of active control technique [18], in which all terms, excluding error terms, are eliminated.

5. For M W , consider the remaining parameters involved in error dynamical system (19) as unknown or anonymous.

6. Select a positive definite quadratic energy function in terms of error trajectories and estimated parameters:

   (20) V ( e ) = ∑ i = 1 n e i 2 + e Φ i 2

   for finding anonymous parameters using updated parameter law, where Φ i shows estimated parameters.

7. If V ˙ is less than zero, then we are done, else go to step 1 again.

Algorithm 1 is utilized in Section (4) for Gyrostat chaotic system (4) to achieve global stability in its velocity vectors. We also have the following limitation for our designed controller:

1. Chaos in dynamical systems occurs due to negligible change in initial solution or bifurcation parameter and sometimes due to both reasons. But, if chaos occurs due to initial conditions only, then our designed controller fails to be implemented in such case.

2. If estimated parameters existed in any chaotic system are in complicated way, then it may be difficult to design a suitable controller.

4 M chaotic controller for gyro chaotic system

5 Results and discussion

In this section, we have verified analytical results given in Sections 2–4 with the aid of simulations. Moreover, we have divided this portion into two sub-sections.

5.1 Hopf bifurcation

Gyrostat is a rigid body, used for detecting path and helps in stabilizing movement of spacecrafts [6]. Here, we have proved the existence of Hopf bifurcation for two parameters involved in system (4).

In Figure 2(a)–(d), we have noticed the existence of oscillation which changes its shape by adjusting parameter values given in equation (13). We have selected different values to observe trajectories for confirmation of Hopf bifurcation. In Figure 2(a) and (b), one can observe if damping constant is less than Hopf bifurcation point; h y = μ x 2 2 I x , it shows if disturbance in torques of gyro and components of total angular momentum along y -axis are small enough. Then, there exists a stable spiral which starts shrinking into origin with selection of value equals to Hopf bifurcation point, which can be seen in Figure 2(c). These trajectories change into an unstable spiral, given in Figure 2(d), surrounding stable limit cycle for parameter value greater than h y . Hence, qualitative behaviour of such trajectories confirms the existence of supercritical Hopf bifurcation.

Figure 2

Parametric study of gyrostat chaotic system for seeking oscillatory solutions: (a) h x y < < B H , (b) h x y < B H , (c) h x y = B H , (d) h x y > B H .

Further working on studying various aspects of dynamical behaviour of the gyro chaotic system, in our another paper, it leads us to the point that there also exists Hopf bifurcation for our considered system, with different parametric values.

In Figure 3, one can observe damping of oscillation in trajectories of gyrostat chaotic system. From Figure 4(a)–(c), the existence of subcritical Hopf bifurcation is confirmed. In Figure 4(a), we have selected μ x as bifurcation parameter. For value less than bifurcation point, unstable spiral can be observed in Figure 4(a), which vanishes in Figure 4(b) and emerges again with stable spiral in Figure 4(c).

Figure 3

Time history of system (4) indicating damping oscillation.

Figure 4

Existence of Hopf bifurcation from both ends: (a) μ x x < B H , (b) μ x x = B H , (c) μ x x > B H .

5.2 M chaotic controller

We have already discussed analytical results for our proposed control technique in Sections 3 and 4. Therefore, we have considered system (4), for applying M chaotic controller, which is chaotic for the aforementioned fixed parameter values. Figure 5(a)–(c) is phase portraits and Figure 5(d) is time history for the existence of chaos in gyrostat system. An attached spherical cylinder with spacecraft can be seen in Figure 1. Therefore, negligible disturbance in movement of spacecraft can bring sloshing around walls inside fuel tank. Due to this effect, one can observe frequent unstable changes in centre of mass and exert pressure on walls of attached tank. Hence, gyro device is attached within spacecrafts for continuous stability and path planning for spacecrafts. Moreover, chaos in gyrostat can lead to create disconnection between satellite and spacecraft and can lose its position in space before reaching to its desired destination.

Figure 5

(a)–(c) Phase portraits and (d) time history of system (4): (a) x y -axis, (b) y z -axis, (c) x z -axis, (d) time history.

Therefore, we have introduced new control function for surpassing unpredictability in system (4). However, many researchers have proved that chaotic trajectories start converging towards desired point as estimated parameters approach their original values for time approaches to infinity. But, after introducing control technique given in this article for chaotic systems, we have observed that the same result of controllability is achieved with chaotic updated law. This means, when the control inputs designed in this article are applied to chaotic system (4) using Algorithm 1, then randomness in trajectories, which can be seen in Figure 5(d), is controlled and can be observed in Figure 6 at t = 1.2 .

Figure 6

Controlled state variables of gyro chaotic system.

In Figure 6, we can see that solutions of angular velocities begin with oscillations but with advancement in time, our proposed control inputs start its function at t = 0.5 . Hence, chaos in gyrostat dynamical system is controlled. Interestingly, Figure 7 is not any other chaotic system, but is numerical simulation of updated law given in equation (33). Moreover, the results achieved in Figure 7 are negation of the fundamental updated law, especially, used in adaptive control technique. Hence, in our case, as estimated parameters start approaching towards their original values in random way. Then, angular velocities of all coordinates start moving in controlled way after some oscillation.

Figure 7

Chaotic updated law (33).

The authors have included almost all phase portraits Figure 8(a)–(l) for chaotic updated law to clarify their new findings. Moreover, in comparison with various other techniques, it has been observed that, in 2019, Marwan et al. [40] used three controllers for double convection model in which adaptive controller is activated at t = 2 , sliding control technique works at t = 5.5 and backstepping technique acquire the same result for t = 6 . Similarly, in the work of Moysis et al. [18] active controller is used for hyperjerk system and chaos is surpassed at t = 3 . But, controller designed in our work is activated at t = 1 (can be seen in Figure 6), which shows our proposed controller give better results in comparison.

Figure 8

Chaotic behaviour of updated law for controlling chaos in system (4): (a) 3D Phase portrait of μ x x − h x z − h x y , (b) 3D phase portrait of h y z − μ y y − h z y , (c) 3D Phase portrait of μ y y − h z y − μ z z , (d) μ x x h x z -axis , (e) μ x x h x z -axis , (f) μ y y h z y -axis , (g) h x z h x y -axis , (h) μ y y h z y -axis , (i) h z y μ z z -axis , (j) μ x x h x y -axis , (k) h y z h z y -axis , and (l) μ y y μ z z -axis .

6 Conclusion

In this work, normal form theory was utilized to show oscillatory solutions in mechanical model coupled with fuel filled tank. It was observed that Hopf bifurcation was not only limited to biological models but had great impact in spacecraft models. Furthermore, appearance and disappearance of negligible limit cycle around stable spiral were observed with changes in parameter values. On the other hand, a new controller based on active control technique and estimated parameter law was designed to bring stability in the considered model. Moreover, these results were also compared with other latest techniques and observed that our proposed technique took less time in comparison.