Robust Linear Quadratic Regulator via Sliding Mode Guidance for Spacecraft Orbiting a Tumbling Asteroid

Aiming to ensure the stability of the spacecraft with multiuncertainties and mitigate the threat of the initial actuator saturation, a Robust Linear Quadratic Regulator (RLQR) via sliding mode guidance (SMG) for orbiting a tumbling asteroid is proposed in this paper. The orbital motion of the spacecraft near a tumbling asteroid is modelled in the body-fixed frame considering the sunrelative effects, and the orbiting control problem is formulated as a stabilization of a nonlinear time-varying system. RLQR based on the adaptive feedback linearization is proposed to stabilize the spacecraft orbiting with the uncertainties of the asteroid’s rotation and gravitational field. In order to avoid the initial actuator saturation, SMG is applied to generate the transition process trajectory of the closed-loop system. The effectiveness of the proposed control scheme is verified by the simulations of orbiting the asteroid Toutatis 4179.


Introduction
The exploration of asteroids has become a hot topic of interests.Equilibria and periodic orbits near asteroids are usually highly unstable due to the irregular shapes and the complex rotation of the asteroids.Hence, the control of a spacecraft in closed proximity of an asteroid is among the most challenging technical problems in the exploration of asteroids [1].
Many researches have paid great attention to the orbital control of the spacecrafts near asteroids.Sawai et al. [2] presented a control based on the one-dimensional altimetry measurements to stabilize a hovering spacecraft.Broschart and Scheeres [3] investigated the stability of realistic hovering control in the body-fixed and the inertial reference frames, respectively.Then, Broschart and Scheeres [4] proposed the sufficient conditions for a dead-band controller to bound spacecraft hovering motion in time-invariant Lagrangian dynamical systems.Wie [5] presented the dynamic modeling and control analysis of multiple gravity tractors in halo orbits.Furfaro et al. [6] investigated a novel closed-loop autonomous guidance law based on multiple sliding surfaces for the soft landing of the spacecraft on the designated point on the asteroid.Liu et al. [7] presented orbital control law for the spacecraft, which consists of PD controller and a nonsingular terminal sliding mode controller, to track the soft landing trajectory.Guelman [8] investigated a simple three-dimensional guidance law for the orbit transfer to a quasi-circular orbit about a rotating small celestial body using continuous thrust.In these previous contributions, asteroids are assumed to be a pure rotation configuration about the axis with its maximum inertia.However, the rotation of asteroids is very complex in practice and lots of them are time-varying [9].The asteroid with nonprincipal axis rotational motion is so-called "tumbling asteroid" [10].Nazari et al. [11] investigated the observer based body-frame hovering control over a tumbling asteroid, which is based on the time-varying LQR or the Lyapunov-Floquet transformation and timeinvariant LQR, respectively.However, the rotation of the asteroid needs to be modelled in his control, which will cost a huge amount of telescope observing time to be determined [12], and the effect of the gravitational fields uncertainty is not analyzed.Hence, designing a robust control law for the spacecraft under the multiuncertainties of the asteroid's gravitational field and rotation is urgent.
The initial actuator saturation problem also needs to be considered.The output of the controller may have a high peak at the start, whose magnitude depends on the size of the initial error.As the continuous adjustable thrust is small, the control output may rise over the limitation of the thrust and threaten the stability of the system [13].Aiming to avoid this problem, arranging the transition process is a good idea to acquire an acceptable controlled quantity.In many researches of the active disturbance rejection controllers, Tracking Differentiator (TD) is applied to arrange the transition process to avoid much larger control outputs [14][15][16][17].However, the stability of the TD is difficult to be approved and the magnitude of the control output via TD is hardly adjustable.The sliding mode control is an important improvement of the control theory [18].Based on it, the sliding mode guidance (SMG) is confirmed due to its good performance in the orbital control of the spacecraft near asteroids [19].We apply the SMG to arrange the transition process of the controller.By doing this, the max magnitude of the control output becomes adjustable and the threat of the initial actuator saturation problem is reduced.
In this paper, we proposed RLQR via the SMG for stabilizing the orbit of the spacecraft around a tumbling asteroid, which does not need to model the rotation of the asteroid.Firstly, the spacecraft orbital motion near a tumbling asteroid is modeled as a restricted three-body problem.RLQR based on the adaptive feedback linearization is proposed.The feedback linearization consists of a feedforward control, which is based on the spheric harmonic coefficient model of the asteroids gravitational field, and an adaptive compensator to ensure the robust stability against model uncertainties.Aiming to mitigate the threat of the initial actuator saturation, the SMG is applied to arrange the transition process of the RLQR.Simulations of orbiting the Toutatis 4769 are performed to verify the effectiveness of the proposed controller.The results of applying the RLQR with and without the SMG are compared to show the advantages.
The rest of the paper is organized in the following form.In Section 2, the orbital dynamic of spacecraft orbital motion is modeled and the problem formulation for control is proposed.In Section 3, the RLQR based on the adaptive feedback linearization is proposed.The SMG is applied to arrange the transition process of the proposed RLQR.In Section 4, simulations of orbiting the Toutatis 4769 are performed.Conclusions are drawn in Section 5.

Problem Formulation
For the relative orbital motion of a spacecraft near an asteroid, the dynamic model includes the nonspherical gravity field of the asteroid and the solar radiation pressure (SRP) [1].In most previous researches, the orbital dynamic of the spacecraft was formulated into two regimes: the gravity dominated regime, in which the effects on the spacecraft from the sun were ignored, and  the solar dominated regime, in which the gravity of the asteroid was out of consideration.Few researches considered both of the effects from the sun and the asteroid [20].In this section, the orbital dynamic of the spacecraft near a tumbling asteroid is modeled as a restricted three-body problem with the joint perturbations.

Orbital Dynamic of the Spacecraft near
Tumbling Asteroids.Before the modeling, two relative frames need to be defined.As shown in Figure 1, the inertial frame  −       centered at the mass center of the asteroid.The   -axis is parallel to the sun-line.The   -axis is coinciding with the direction of the velocity vector of the asteroid.The   -,   -, and   -axis compose the right-handed coordinate system.The body-fixed frame − fixes on asteroid with the origin coinciding with the mass center of the asteroid.The , , and  are coinciding with the axis of the asteroids maximum, minimum, and intermediate moment of inertia, respectively.
The orbital motion of spacecraft in the inertial frame can be expressed as where Ω   is the orbital angular velocity vector of the asteroid, r   is the vector of the sun to the asteroid, r   is the vector of the sun to the probe, r  is the vector of the asteroid to the probe, r  = r   − r   ,   is the gravitational parameter of the sun, g  is the gravitational acceleration of the asteroid, p   is the solar radiation pressure on the probe, and u  is the vector of control.
Assume that the asteroid orbits the sun in a circle; (1) can be simplified as Transform ( 2) into the body-fixed frame; it can be expressed as where Ω  is the rotation vector of the asteroid and f  is the sun-relative effect on the spacecraft.

Sun-Relative Effects.
Define Ω   = [0, 0,   ]  , and the following equation can be established according to the assumption of asteroids circle orbit: Apply the Taylor expansion; an approximate linearization of (4) can be expressed as Considering that the main effect of the SRP arises from its first-order perturbations [21], p   can be defined as where  is a defined constant parameter of SRP.Define the transformation matrix from the inertial frame to the body-fixed frame as () = {ℎ  } 3×3 and the rotation vector as Ω  = [  ,   ,   ]  .Then, (6) can be rearranged as where As usually   ≪   ≪ 1,  = , , , and the elements of transformation matrix ℎ  ≤ 1, ,  = 1, 2, 3, hence (9) can be simplified as 2.1.2.Asteroid's Gravitational Field.The gravitational acceleration of the asteroid can be modelled by the gradient of the gravitational potential function of the asteroid as Different from the regular large objects, the bodies of asteroids are very irregular.Several methods have been investigated to approach the gravitational field of asteroids, such as the Spherical Harmonic Expansion Model (SHEM) [22], the polyhedron model [23], and the ellipsoidal harmonic expansion model [24].The second-order SHEM, which models the asteroid's perturbation potential with the most significant gravity coefficients and is widely used in the previous researches, is selected to formulate the nominal gravitational field in this paper.It can be expressed as where   is the gravitational parameter of the asteroid,  0 is the normalizing radius,  20 and  22 are the spherical harmonic gravity coefficient, and , , and  are the radius, latitude, and longitude of the field point, respectively.Regard the asteroid as a three-axis ellipsoid with three main axes  1 ≥  2 ≥  3 , and the spherical harmonic gravity coefficients  20 and  20 can be estimated by the three axes of the asteroid as ) . ( 4 Mathematical Problems in Engineering Then, the real gravity of the asteroid can be formulated as where Δg is the uncertainty of the asteroids gravitational model.
Hypothesis 1. Δg is bounded and the constant Γ exists, which satisfy the following conditions: 2.2.Problem Formulation for Controller Design.The stable hovering in the inertial frame is equivalent to a stable orbiting in the body-fixed frame.Hence, the stable orbiting condition in the body-fixed frame is expressed as Define the error vector as Substitute ( 3) into (17); the time derivative of the error state is expressed as Then, the stable orbiting control problem can be formulated as the stabilization of the nonlinear system equation (18).
The rotation of the asteroid is difficult to be modelled before the close proximity orbiting observation.Hence, the following assumptions need to be declared before the controller design: (i) The rotation vector Ω  () is unmodeled, but its upper bound is known.
(ii) The current state of the rotation can be observed, so that the error vector e can be applied to the feedback.
(iii) The position r and the velocity ṙ of the spacecraft in the body-fixed frame are provided by the navigation system.

Robust Linear Quadratic Regulator via Sliding Mode Guidance
3.1.Robust Linear Quadratic Regulator.The nonlinear system equation ( 18) can be transformed into (19).Consider Because the rotation vector Ω  () is unmodeled, the timevarying matrix () is uncertain.However, the upper bound of Ω  () is known.Hence, we can define the upper bound matrix of () as Design RLQR for the stabilization of the nonlinear system equation (19) as u ff is a feedforward control as u 1 is a linear quadratic regulator, which minimizes the cost function as (25).It can be expressed as where  and  are positive weight matrices and  is a positive symmetric matrix, which is the solution of the Riccati equation as u  is an adaptive compensator, which can be expressed as where  is the adaptive parameter and  is an adjustable field in the error space that contains the origin.
Theorem 1. Considering the spacecraft orbiting problem which is formulated as the stabilization of a nonlinear system as (19), define the error state as (18) and apply the robust LQR as (24).If Hypothesis 1 is satisfied, the closed-loop system is globally uniformly stable.
Proof.Define a Lyapunov function candidate as where  * is a constant, which make all e ∉  satisfy the condition as where the function sign(e) is defined as [sign(  ), sign(  ), sign(  )]  and the function || is defined as As , , and  are positive matrices, for all e ∉ , V 1 < 0. According to the Lyapunov stability theorem, the nonlinear system as (18) under the control as ( 22) is uniformly stable.The bound of the steady state error depends on the adjustable field .

RLQR via the Sliding Mode
Guidance.The proposed RLQR could ensure the robust stability of the orbiting.However, the proposed controller has the potential threat from the actuator saturation.Firstly, depending on the error state, the control output may have a high peak at the start.Secondly, the gain of the adaptive compensator may be too conservative, which is much larger than necessary.As the adjustable continuous thrusters only could provide a small thrust, the saturation of the thrust may threaten the safety of the spacecraft.To avoid the initial actuator saturation problem of the RLQR, the sliding mode guidance is applied to arrange the transition process of the controller.

The Sliding Mode Guidance.
Define the outputs of the sliding mode guidance as  1 and  2 , and their initial values are The sliding mode guidance is expressed as where s is the sliding mode function as s =  2 +  1  1 and  1 and  2 are selected parameters of the SMG.
Define a Lyapunov function candidate as Then, the time derivative of  2 is expressed as Obviously, V 2 = 0 only when s = 0.According to the Lyapunov stability theorem, the SMG is globally asymptotically stable.When  → +∞,  1 → 0 and  2 → 0.

The RLQR Based on the Virtual Error Vector. Define the virtual error vector as
The time derivative of the virtual error  can be expressed as As the SMG is globally asymptotically stable and the matrix () is bounded the constant Γ  exists, which satisfy the condition as Then, design the RLQR based on the virtual error as Theorem 2. Considering the spacecraft orbiting problem which is formulated as the stabilization of a nonlinear system as (18), define the sliding mode guidance as (32) and the virtual error vector as ( 17) and ( 35), and apply the robust LQR based on the virtual error as (38).If Hypothesis 1 is satisfied, the closedloop system is globally uniformly stable.
Proof.Define a Lyapunov function candidate as where   is a constant, which make all  ∉  satisfy the condition as min {       ()      } ≥ Γ + Γ  ,  = , , ,  () = . (41) Then, the time derivative of  3 is represented as As , , and  are positive matrices, for all  ∉ , V 3 < 0. Hence, when  → +∞,  1 → 0, for all e ∉ , V 3 < 0. According to the Lyapunov stability theorem, the RLQR via SMG could make the system as (18) uniformly stable.The bound of the steady state error is .

Simulations
We specifically use the numerical data of the asteroid Toutatis 4179 in the simulation.To make the simulation more realistic, the gravitational field of the asteroid is modeled by the polyhedron method [23] based on Hudson's 33996-face shape data [25], which can be expressed as (43).The second-order SHEM is applied as the nominal model in the controller in the same time: where  is the gravitational constant,  is the constant density of the asteroid, ]  is a vector from the field point to an arbitrary point on each edge, ]  is a vector from the field point to an arbitrary point on each face, E  is a dyad defined in terms of the face and edge normal vectors associated with each edge, F  is the outer product of face normal vectors,   is a logarithmic term expressing the potential of 1D straight wire, and   is the solid angle subtended by a face when viewed from the field point.According to the shape data, the maximum, intermediate, and minimum major axes of Toutatis 4179 are ( 1 ,  2 ,  3 ) = (4.5, 2.4, 1.9) km.The rotation of Toutatis in two periods is shown in Figure 2, which starts at Ω  (0) = (4.81,0, −12.53) × 10 −6 rad/s.The initial attitude of the asteroid expressed by 3-2-1 Euler angle is (0, /2, /3).The initial transformation matrix determined is by the initial attitude and the rotation  of the asteroid.The gravitational parameters of Toutatis are shown in Table 1.As illustrated in Figure 3, the error vector under RLQR control converges to zero quickly.Then, the errors keep near to the zero axes during the remaining times of the simulation and the system becomes stable.The outputs of the RLQR are shown in Figure 4.In the first several seconds, the control acceleration has a high peak, and the adaptive gain  grows quickly (as shown in Figure 5).Then, the control acceleration decreases and becomes very small that mainly restrains the effect of the sun and the gravity of the asteroid.The adaptive gain  is almost stable at 0.2213.In the simulation, the control cost of the steady-stable orbiting is about 0.0228 m/s per hour, which is acceptable for the deep space missions lasting months.Through these simulations, the effectiveness of the RLQR has been verified.

4.2.
The Orbiting under the RLQR via the SMG.The parameters of the SMG are  1 = 0.003 and  2 = 0.005.The adjustable field of the virtual error state is  :    ≤ 1 × 10 −6 .The error states under the control of the RLQR via SMG are shown in Figure 6.The convergence of the error states takes more time than the ones in Figure 3.The reduction of convergence speed is the price of the decrease of the control magnitude.As illustrated in Figure 7, the max value of the control acceleration is 0.046 m/s 2 , which is much less than the one in Figure 4. What is more, the max magnitude of the control acceleration can be adjusted by the parameters  1 and  2 .Hence, the risk of the initial actuator saturation problem caused by the large initial error is mitigated.The gain of the adaptive compensator with the SMG is much smaller than the one without the SMG (see Figure 8).The control cost of the steady-stable orbiting is similar to the one without the SMG.Through the simulations in Figures 6-8, it was verified that the RLQR via the SMG could stabilize the orbiting and limit the magnitude of the control output.The threat of the initial actuator saturation has been mitigated.

The Proposed Controller under Different Magnitudes of Uncertainty.
To analyze the effects of the gravity model's uncertainties on the proposed algorithm, we did more simulations which apply no gravity model, the  SHEM, the fourth-order SHEM, and the polyhedron model (no uncertainty), respectively.Adding the order of the SHEM to the fourth, the gravitational potential of the asteroid can be expressed as  The spheric harmonic coefficient can be determined as In the simulation, the initial position of the spacecraft is [8,10,11]    As shown in Figures 9-11, the errors in three axes are converged near zero no matter which model applied.Also, the error curves controlled with different uncertainties have little difference.As shown in Figures 12-14, the control outputs are similar too.
The reason of these results may be shown in Figures 15-17.The asteroid's gravitational acceleration is too small, which is in the magnitude of 10 −5 ∼ 10 −6 .Although we limited the control output, it is still much larger than the asteroid's gravity.
As a conclusion, the uncertainty of the asteroid's gravity has little effect on the proposed control algorithm.The proposed controller can be used in the orbiting around the asteroids which are not well mapped in terms of gravity field.

Conclusion
In this paper, RLQR via SMG is proposed to ensure the robust stability of the spacecraft orbiting a tumbling asteroid.
Based on the adaptive feedback linearization, the controller is able to suppress the effects of the multiuncertainties in the asteroids rotation and gravity model.Applying the SMG to arrange the transition process, the max magnitude of the control output is limited and threatening of the initial actuator saturation of the controller is mitigated.Also, the control cost of the steady-stable orbiting is acceptable for the deep space missions lasting months.As a consequence, the proposed control scheme can be properly selected to

Figure 1 :
Figure 1: Two relative frames of the orbiting.

Figure 7 :
Figure 7: The control acceleration of the RLQR with SMG.

Figure 8 :
Figure 8: The comparison of the adaptive parameter in two kinds of controls.

Figure 9 :Figure 10 :
Figure 9: The error curve in the -axis.
km in the body-fixed frame.The initial velocity of the spacecraft is [21.3, −8.67, 9.62] m/s in the body-fixed frame.The simulation results are shown as Figures 9-14 .

Figure 13 :Figure 14 :
Figure 13: The control acceleration in the -axis.

Table 1 :
The gravitational parameters of the Toutatis.