CHAOS 010, 3rd Chaotic Modeling and Simulation 1-4 June 010 Space flying vehicles orbital motion control system synthesis: power invariants Alexander A. Kolesnikov (PhD) Taganrog Institute of Technology Southern Federal University, Dept. of Synergetics and Control Processes 44, Nekrasovky str., Taganrog, 34798, Russia; anatoly.kolesnikov@gmail.com Abstract: We solve an applied problem of low-trust space vehicles control by using system s law of gravity. Control laws synthesis procedure as well as simulation results are provided. Keywords: modeling, space vehicle, nonlinear dynamics, nonlinear systems, nonlinear control. Introduction The problems of low thrust spacecraft control are very important in the modern science. These problems include the tasks of high precision and power saving orbital motion control of a spacecraft [1 6]. It is known that Newton's gravitation law describes stationary motion in a weak gravitational field. It is used to calculate the space vehicle trajectories. However, in practice, it is necessary to solve the tasks of navigation and optimal trajectory stabilization of a spacecraft under the conditions of essential external disturbances. At the present time there is an important problem of synthesis of such control in a form of feedbacks that would allow to form the necessary character of the nonlinear space motion of a spacecraft. This is directly connected to the creation of a new class of space systems that satisfy the high requirements concerning accuracy and power saving. This problem is not yet solved and is to be addressed using the new methods of modern control theory. This is confirmed by the following known fact: more that 30 years ago two spacecrafts Voyager-1 and Voyager- were launched to boundary of our galaxy. Astrophysics were surprised that they have deflected away form the calculated trajectory by more that 300 thousand kilometers. This fact calls for the new approaches to the problems of spacecraft control. This report presents the synergetic approach to synthesis of spacecraft control systems basing on the known method of analytical design of aggregated regulators ADAR [7]. According to it, the desirable invariant manifolds are introduced into the space state of synthesized systems. In our case these manifold are the energy invariants of motion. Such an approach
CHAOS 010, 3rd Chaotic Modeling and Simulation 1-4 June 010 allows to get the feedbacks analytically the feedbacks that would give the required character of the spacecraft motion. Let s present the main features of the ADAR method in application to the problem of system synthesis. Firstly, the synthesized system goals are changed dramatically, secondly, the natural qualities of the nonlinear objects are directly incorporated into the synthesis processes, and thirdly, a new mechanism of control laws generation is formed [7]. 1. Control Laws Synthesis Equations describing the orbital motion of a spacecraft have the following form [1, ]: where: r, θ polar coordinates; 1 h r &( t) = Vr, V& r( t) = Vθ r r + U r, (1) p & θ t = V r 1, V& t = Vr V r 1 + U, () ( ) θ θ( ) θ V, r θ θ V radial and transverse speed components; U r, U θ thrust vector components; θ = χ+ γ, χ natural anomaly, γ angular component that determines the angle between the line of apsides and OX axis (Fig. 1). Polar coordinates r and θ and Cartesian coordinates x and y coordinates are connected by the following expressions x= r cosθ and y= r sinθ. The orbit s parameters are connected by the following expressions: π 1 e p h h = ; = GM, ( 1 e ) T p where p focal radius; G gravitation constant; M mass of attracting center; T rotation time; e ellipse eccentricity. Controls U r and U θ are 1/ h U r + U θ <<. considered to me small if they satisfy the condition ( ) pr max Fig. 1. Coordinate system of an earth s artificial satellite In the ADAR method the key stage is the selection of invariant attracting manifolds that should be adequate to the physical essence of the synthesized control systems. The spacecraft equations (1) and () with
CHAOS 010, 3rd Chaotic Modeling and Simulation 1-4 June 010 U r U = 0 describe the interaction of two bodies according to the = θ Newton s law of gravitation. In this case (1), () is a conservative oscillating system. It is known that for such systems the most characteristic feature is preservation of energy. This means that for synthesis of control laws for conservative systems the best choice is the energy integrals of motion. Let s consider the task of synthesis of control laws U r and the ADAR method [7]. The invariant manifolds are selected as follows: and h ( ) ( 1 e ) h + V + 0 U θ using ψ 0,5 V r θ = (3) pr p ( t) h= rv 0 = r θ θ h= ψ &. (4) Invariant manifold ψ 0 (3) is an energy integral of the spacecraft s ( ) steady state motion, where = < 0 1 e h E full energy; and manifold p ψ = 0 (4) momentum preservation law for the system «Earth - Spacecraft». According to the ADAR method [7], we introduce the following invariant relations: ψ& 1 ( t) + Φψ1( t) = 0 (5) and ψ& ( t) + Φψ ( t) = 0, (6) The manifolds (3) and (4) should satisfy these relations. As a result of a joint solution of the equations (5) and (6) together with the spacecraft s equations (1), () we find the following control laws: Vθ ψ Φ Ur = ψ 1 (7) r Vr and U = ψ θ Φ. (8) r In order to ensure asymptotic stability of the equations (5) and (6) with respect to invariant manifolds ψ 0 (3) and ψ = 0 (4) the function Φ should be definitely positive and be inverse to time in units. So let s select this function as follows: V Φ = k r, (9) h where k unitless coefficient. At the intersection of the invariant manifolds ψ 0 (3) and ψ = 0 (4) the laws (7), (8) are zeroed and therefore the spacecraft will stably move
CHAOS 010, 3rd Chaotic Modeling and Simulation 1-4 June 010 along the set orbit finite cycle. In this case the spacecraft s orbit will be described by the ellipse equation first Kepler's law ω = r 1+ ecosθ p. (10) ( ) 0 If in ψ 0 (3) we substitute the expressions ψ 0 (4) and ω 0 (10) as a result we get the motion integral ω& ( t) 0, i.e. = eh V rs = sinθ. (11) p If we substitute function sin θ from (10) into (11) we get the radial h V e r p pr r rs =, that naturally coincides speed of orbital motion ( ) with V rs at the intersection of invariant manifolds ψ 0 (3) and ψ = 0 (4). Expression (11) can be considered to be an additional dynamic invariant. Using it we can get the known Newton's gravitation law. Where ( t) 1 & ω = for ω 0 and 0 1 ψ ψ. This means that ( t) 0 = ω& is also and 1 = energy invariant that ensues from ψ 0 (3). It is interesting to note the important quality of self-likeness in the system (1), looped with the control laws U r (7) and U θ (8). These laws move the representing point of the system (1), (7), (8) to the intersection of the invariant manifolds ψ 0 (3) and ψ = 0 (4). This means that energy invariant of motion is satisfied. Then the system inevitably gets to the invariant (11), where it will stay for any amount of time until the new disturbances come out. After the system gets to the mentioned intersection, the controls U (7) and U (8) are zeroed (1) and the system satisfies the r θ energy invariant of motion ψ 0 (3). In other words as a result of a kind of ψ ψ & ω (11) a dynamic relay-race of attractors = = ( t) 0 1 0 decomposition of the system is performed [7]. From the synergetic point of view, this is the place where the self-likeness of the processes in the closedloop system is being revealed. Basing on the control laws U r (7) and function (9) by setting the desired parameters = e 0 < 1 U θ (8) accounting for the e and p= p0 a full or partial changing of the spacecraft orbital motion elements can be performed. e 1 h0 E 0 < and p ( ) In the process, the desired values of energy = 0 parameter h 0 will be achieved. They determine the spacecraft s orbit. Setting e 0 = 1 and, therefore, E 0 = 0, we get the solution of the task of spacecraft s acceleration to the parabolic speed. For e 0 > 1 and E 0 > 0 we get the spacecraft s motion along the hyperbolic trajectory etc. 0
CHAOS 010, 3rd Chaotic Modeling and Simulation 1-4 June 010 So basing on the natural gravitational laws (1) and () we ve synthesized system control laws (7) and (8) guarantying the spacecraft s asymptotically stable motion. As we ve shown above, these laws are zeroed on the invariant manifolds ψ 0 (3) and ψ = 0 (4). As a result the spacecraft will move along the desirable orbit determined by it s energy of gravitational interaction with the central body Earth in our case.. Modeling Results As an example, Fig. 10 present the modeling results for a spaceraft (1), (7). (8) with the following orbit s parameters: e = 0, p = 36000 km, πp T = 4 hours, h=, k = 1 and disturbances in the form of initial T conditions: r 0 = 40000 km, r& 0 = 0, 5, θ 0 = 0. In our case the orbit is a circle with the radius of r= p. Under of controls U r (7) and U θ (8) spacecraft returns to the desirable orbit and moves along it. In the process the invariants ψ 1 0 (3), 0 ψ (4) and ( t) 0 ω& (11) are satisfied. 1 Fig.. r ( t) changing graphics Fig. 3. r( t) = V ( t) & changing graphics r Fig. 4. V θ ( t) changing graphics Fig. 5. ( t) θ changing graphics
CHAOS 010, 3rd Chaotic Modeling and Simulation 1-4 June 010 Fig. 6. U r ( t) changing graphics Fig. 7. ( t) U θ changing graphics Fig. 8. Phase portrait Fig. 9. ( t) ψ changing graphics 1 Fig. 10. ( t) ψ changing graphics Likewise Fig. 11 19 present the modeling results for the spacecraft control system with the following parameters of elliptical orbit: e = 0, 5, πab p = 36000 km, T = 4 hours, h=, k = 1 and disturbed initial T conditions: r 0 = 40000 km, r& 0 = 0, 5, θ 0 = 0. Unlike the previous case r ( t ) r & t = V are now the periodic functions of time. The system approaches and ( ) r
CHAOS 010, 3rd Chaotic Modeling and Simulation 1-4 June 010 the invariant ( t) 0 takes place. ω& (11) where the steady state motion of the spacecraft Fig. 11. r ( t) changing graphics & r changing graphics Fig. 1. r( t) = V ( t) Fig. 13. V θ ( t) changing graphics Fig. 14. ( t) θ changing graphics Fig. 15. U r ( t) changing graphics Fig. 16. ( t) U θ changing graphics
CHAOS 010, 3rd Chaotic Modeling and Simulation 1-4 June 010 Fig. 17. Phase portrait Fig. 18. ( t) ψ changing graphics 1 Fig. 19. ( t) ψ changing graphics На Fig. 0 3 present the modeling result for a spacecraft maneuver that is its transition form the circular orbit with parameters: e = 0, πp p = 36000 km, T = 4 hours, h=, k = 1 and initial conditions: T r 0 = 40000 km, r& 0 = 0, 5, θ 0 = 0 to the lower elliptical Earth s orbit with parameters: e = 0, 1, p = 4000 km, T = 4 hours, k = 1. In the control process at certain moments of time the parameters e and p were changed in the laws U r (7) and U θ (8). As you can see from the modeling results, the spacecraft successfully implements the mentioned space maneuver going to a new orbit. On the whole, the presented modeling results fully confirm the proposed basic scientific statements.
CHAOS 010, 3rd Chaotic Modeling and Simulation 1-4 June 010 & r changing graphics Fig. 0. r ( t) changing graphics Fig. 1. r( t) = V ( t) Fig.. V θ ( t) changing graphics Fig. 3. Phase portrait Conclusion Let s summarize. In the report we've synthesized new control laws (7), (8) for a spacecraft s orbital motion, reflecting the natural character of gravitational interaction of two bodies - spacecraft and a planet, e.g. Earth. The mentioned laws allow implementing the energy saving control of the spacecraft's orbital motion by means of setting the desirable parameters of energy invariants for the corresponding orbit. Unlike the known control laws, these ones don t have singularities. Besides, they are built as functions of spacecraft's state coordinates. After it gets to a new orbit, the laws (7) and (8) are zeroed and the spacecraft continues its motion along the orbit according to the Newton s gravitation law. The synthesized control laws (7) and (8) allow to implement the space motion of a spacecraft along a parabolic ( e = 1) and hyperbolic ( e > 1 ) trajectories. Generally, from our point of view the report presents the solution of an important task of a spacecraft control allowing to improve the navigational and energy-saving characteristics of various space systems.
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