Finite time sliding sector control for spacecraft atmospheric entry guidance

Accepted Manuscript Finite time sliding sector control for spacecraft atmospheric entry guidance Biao Xu, Jun Sun, Shuang Li, Tao Cao PII:

S0094-5765(18)31316-X

DOI:

https://doi.org/10.1016/j.actaastro.2018.12.008

Reference:

AA 7230

To appear in:

Acta Astronautica

Received Date: 31 July 2018 Revised Date:

17 November 2018

Accepted Date: 5 December 2018

Please cite this article as: B. Xu, J. Sun, S. Li, T. Cao, Finite time sliding sector control for spacecraft atmospheric entry guidance, Acta Astronautica (2019), doi: https://doi.org/10.1016/ j.actaastro.2018.12.008. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Finite time sliding sector control for spacecraft atmospheric entry guidance Biao Xu1, Jun Sun2, Shuang Li1,* and Tao Cao2 College of Astronautics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China 2

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1

Shanghai Aerospace Control Technology Institute, Shanghai, 201109, China

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Abstract: The paper presents a novel variable structure control with finite time sliding sector for spacecraft atmospheric entry guidance. The finite time convergence of tracking error in the presence of

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system uncertainty can be guaranteed. In contrast with the normally used notion of asymptotic stability in conventional sliding sector, a finite time sliding sector for the entry guidance is designed as a subset of state space in which the Lyapunov function candidate satisfies the finite time stability condition. Then, the finite time sliding sector controller is designed to guarantee the tracking error to converge to

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a small region of zero. The chattering existing on the boundary of the sliding sector is further reduced by introducing inner sector and outer sector which are the subsets of the proposed sector. Finally,

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numerical simulation results are given to demonstrate the effectiveness of the proposed method.

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Keywords: spacecraft, entry guidance, sliding sector, finite time stability

1. Introduction

As the neighbor of Earth, Mars has attracted more attention than other planets. In order to obtain the scientific data of Mars topography and chemical composition, landing a vehicle on the surface of Mars and performing in situ exploration is a prerequisite. Mars landing exploration missions started since the 1970s. So far, more than two-thirds of the Mars missions ended in failure, and only a few spacecrafts

*

Corresponding author. Email Address: [email protected]

ACCEPTED MANUSCRIPT successfully landed on the surface of Mars [1]. Future Mars missions require advanced entry, descent and landing (EDL) technology [2-3]. Precision EDL technology is needed in multiple fields, including sample return, human exploration, landing on the specified sites and so on [4]. The atmospheric entry

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guidance is a challenging problem due to the atmospheric models and the vehicle aerodynamics are uncertain even under the perfect approach navigation. Generally, there are two strategies of entry guidance: the predictor-corrector guidance and the reference trajectory guidance. A three-dimensional

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predictor–corrector entry guidance algorithm is proposed by using the developed solutions for the flight

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path angle and velocity, so the order of the motion equations is reduced [5]. A predictor-corrector guidance method is proposed for the entry mission with waypoints and no-fly zones in [6]. The trajectory prediction is obtained by numerical integration while the corrector is derived based on fuzzy logic. To deal with flight constraints, a predictor–corrector guidance method based on fuzzy logic in

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designed in [7]. A numerical multi-constrained predictor-corrector guidance algorithm is proposed in [8], which focuses on real-time longitudinal trajectory generation, constraint management, and lateral guidance improvement. These predictor-corrector guidance algorithms can provide a high landing

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accuracy, but fast on-board computation, accurate models of aerodynamics and atmospheric are

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required [9]. Guidance algorithms based on a reference trajectory optimizes a trajectory in advance. The reference trajectory is tracked by the control system during the entry phase. The reference trajectory guidance algorithms are less elastic to adapt to various missions, but they do not need fast on-board computation [10].

The reference trajectory entry guidance adopted by Gemini and Apollo missions were the early successful mission with the guided entry. Since advanced entry guidance systems of spacecraft should have the capacity to generate optimal flight trajectories onboard for the latest mission requirements, a

ACCEPTED MANUSCRIPT sequential convex programming method is developed for three-dimensional planetary-entry trajectory optimization problems [11]. Furthermore, based on the results of entry trajectory optimization using convex optimization, an online entry guidance algorithm is proposed to satisfy the demands of highly

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autonomous entry guidance systems [12]. The entry guidance system determines steering commands to guide a spacecraft from its current location to the designed location. In order to reduce the dependence on aerodynamic and atmospheric models, Apollo entry guidance used a reference trajectory algorithm

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based on drag acceleration. During the past several decades, a large number of drag tracking control

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methods have been applied to entry guidance problems [13]. A probability-based hazard avoidance guidance method for landing on planets is designed in [14]. Simulation results show that the proposed probability-based method is reliable under uncertainty.

The sliding mode control method is widely adopted to design guidance laws in Mars hypersonic

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entry guidance [15-17]. A robust entry guidance law based on terminal sliding mode and second-order differentiator is designed for trajectory tracking [18]. To improve the performance of Mars landing, a multiple sliding surface guidance law is proposed for tracking the reference trajectory [19]. An adaptive,

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disturbance-based sliding mode controller is proposed for hypersonic entry vehicles [20]. The scheme

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is based on high-order sliding-mode theory [21], and is able to estimate the combination of known and unknown perturbations acting on the system [22]. The control system of the hypersonic entry vehicles in the terminal phase for maximum target penetration is designed by the adaptive double-layer continuous higher order sliding mode approach [23]. In the presence of system uncertainty, a super-twisting sliding mode control law is developed for the trajectory tracking guidance [24]. Due to the switching delay or high switching frequency, the entry guidance law is suffering the chattering phenomena. Sliding sector control is one of the branches of variable structure control. Instead of sliding

ACCEPTED MANUSCRIPT mode, a linear time invariance sliding sector for a linear time invariant (LTI) system has been proposed to reduce chattering [25]. For the nonlinear time varying (NTV) system with system uncertainty, the forward integration of state dependent differential Riccati equation (SDDRE) is used to design an NTV

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sliding sector [26]. It is shown that the proposed sliding sector control method provides better performance with reduced chattering and high control accuracy in comparison with the sliding mode control. However, the system states just enjoy asymptotic stability or exponential stability inside the

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sliding sector. Based on finite time stability theory [27], a finite time sliding sector control method is

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proposed in [28], but it does not consider unmatched uncertainty or parameter perturbation. In this paper, a novel robust control algorithm with finite time sliding sector for spacecraft atmospheric entry guidance is proposed based on the finite time stability theory. In contrast to the conventional sliding sector, the finite time stability is guaranteed and unmatched parameter

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perturbation is considered inside the sector. The rest of this paper is organized as follows. In “Problem Formulation” section, the entry guidance dynamic is formulated. In “Finite Time Sliding Sector Guidance” section, the finite time sliding sector is proposed and the new guidance law is designed

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using the variable structure control with finite time sliding sector. Numerical simulation results are

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shown in “Numerical Results” section, and conclusions are made in the last section.

2. Problem formulation The standard simplified nonlinear equations for the entry guidance are [4]

θ&=

V cos γ cosψ r cos φ

(1)

V cos γ sinψ r

(2)

φ&=

r&= V sin γ

(3)

V&= − D − g sin γ

(4)

ACCEPTED MANUSCRIPT γ&=

  1 V2   L cos σ −  g −  cos γ  V r   

ψ&= −

(5)

L sin σ V − cos γ cosψ tan φ V cos γ r

(6)

where θ and φ are the longitude and latitude, respectively; r is the radial distance; V is the

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velocity; γ and ψ are the velocity elevation angle and the velocity heading angle, respectively; σ is the bank angle; g = µ / r 2 is the gravitational acceleration, where µ is the gravitational parameter.

1 CL ρV 2 S 2m

D=

1 CD ρV 2 S 2m

(7) (8)

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L=

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The lift and drag accelerations are expressed as

where ρ is the atmospheric density; S is the vehicle reference surface area; m is the mass of the spacecraft; CD and C L are the drag and lift coefficients, respectively. The model of the atmospheric density is expressed as [4, 13]

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ρ = ρ0 e

−

r − rs hs

(9)

where ρ 0 is the density at the reference radius rs and hs is the constant scale height.

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The specific energy for entry guidance of spacecraft is defined by

1 µ E = V2 − 2 r

(10)

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Considering Eqs. (1)-(6), we obtain the energy rate as E&= − DV

(11)

where D ≥ 0 and V ≥ 0 , it is obvious that E& is always negative. Energy is a convenient independent variable in entry guidance dynamics, the velocity V in the equations of motion can be calculated by

V = 2E + 2 Then, the number of equations of motion is reduced.

µ r

(12)

ACCEPTED MANUSCRIPT Since energy is regarded as independent variable for trajectory tracking, the trajectory range only depends on the drag profile. Using (1)-(6), we obtain the derivative of drag acceleration as

S S D&= CDV 2 ρ&+ CD ρVV& m 2m − S CDV 2 ρ 0 e 2hs m

=−

1 2D DV sin γ − ( D + g sin γ ) hs V

r&+

S CD ρV ( − D − g sin γ ) m

(13)

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r − rs hs

=−

Define the error system states as

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x1 = D − Dr , x2 = D&− D&r

(14)

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where Dr is the tracking reference. The nonlinear dynamics of drag tracking error can be expressed in the form of state-dependent linear time-variant (SDLTV) system as x&1 = x2

x&2 = a0 + a1 ( x, t ) x1 + a2 ( x, t ) x2 + b ( x, t ) u x2 ] , T

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where u = cos σ , x = [ x1

& D&  2 Dg D   2 Dg a0 =  2 +  g cos 2 γ −  2 +  V sin γ hs  hs   V  V

2 ( D + g sin γ ) 1 ( D + g sin γ ) sin γ − hs V2

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a1 =

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 2g 1  4g V2 + sin 2 γ −  2 +  cos 2 γ r hs  r V

a2 = −

4D V

 2g 1  b ( x , t ) = − DL cos γ  2 +  hs  V

(15) (16)

(17)

2

(18)

(19)

(20)

Since Eqs. (1)-(6) are simplified nonlinear equations for spacecraft entry guidance. Considering system modeling uncertainties, external disturbances, we can rewrite the system model into the standard form as

x&= ( A + ∆A) x + B ( u + d )

(21)

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0 A=  a1

1 0 , B=   a2  b 

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d is the bounded disturbance and ∆A is the mismatched uncertainty. The parameter a0 in Eq. (16) can be viewed as disturbance terms. It should be noticed that the SDLTV form of the nonlinear drag dynamics is not unique. We need to choose a proper one to make the following calculation of sliding

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sector simple. The unknown disturbances can be comprised of uncertainties, nonlinearities of the system, and external disturbances. The uncertainties of Drag and lift coefficients brings serious

bounded and can be written as [29]

∆A = MHN

(22)

HT H ≤ I

(23)

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with the unknown matrix H satisfying

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challenges for the guidance law design. The mismatched uncertainty ∆A is assumed to be norm

where M and N are known matrices. For a bounded uncertainty, we can always choose two large

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enough matrices M and N to make sure the matrix H satisfying Eq. (23).

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3. Finite time sliding sector guidance 3.1 Finite time sliding sector In this paper, the following finite time sliding sector is utilized to design the entry guidance law

base on the finite time stability theory. In order to design a finite time sliding sector guidance law, some results about finite time sliding sector [26-28] are introduced here. Definition 1: A finite time sliding sector for the nonlinear time varying system is defined as

{

S ( x , t ) = x ϕ ( x, t ) ≤ δ ( x , t )

}

(24)

ACCEPTED MANUSCRIPT inside which there exists a positive Lyapunov function VL ( x, t ) > 0 satisfies the finite time stability condition α V& L ( x , t ) + cVL ( x , t ) ≤ 0

(25)

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where the real numbers c > 0 and 0 < α < 1 . The switching function φ ( x,t ) and the sliding sector parameter δ ( x,t ) are designed as

ϕ ( x, t ) = S F ( x, t ) x

(26)

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δ ( x, t ) = xT ∆( x, t ) x

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where ∆ ( x, t ) ≥ 0 .

(27)

The relation between switching function and sliding sector is shown in Fig. 1. In contrast to the conventional sliding sector, the finite time stability is guaranteed inside introduced sliding sector. As shown in Fig. 1, the sliding sector is a subset around the sliding surface ϕ =0 . Its boundary is

x1

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determined by the design parameters ϕ and δ .

x2

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ϕ = δ

ϕ =0 Fig. 1. Sliding Sector.

An SDDRE is used to determine the sliding sector

− P&= AT P + PA −

1

β

2

PBBT P + λ PMM T P +

1

λ

NT N +Q

(28)

where P and Q are positive definite matrices, λ is positive constant. The solution of the SDDRE

ACCEPTED MANUSCRIPT can be obtained by a forward integration with initial condition P0 = P0T > 0 . The SDDRE function defined in Eq.(28) requires a stronger condition compared with the SDDRE in [26] such that the sliding sector designed by Eq. (28) robust against the system uncertainty ∆A . The parameters of the sliding

S F ( x, t ) =

1

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sector are designed as

BT P

β

(29)

∆( x,t ) = Q − R

(30)

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where β > 0 and R ≥ 0 . The positive constants β and λ are usually chosen in the range of 0.1 ∼ 10. A

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large Q makes sure the system states converge fast, and a large R ensures robustness of the

sliding sector. 3.2 Guidance law design

The design of the guidance law is divided into two steps. One is the guidance law inside the sliding

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sector, and the other is the guidance law outside the sliding sector.

Step 1. Inside the sliding sector, the guidance law is designed as

(

u = − k1 + k2 ϕ

2α1 −1

) sgn (ϕ )

(31)

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where k1 > d and k 2 > 0 .

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Define a Lyapunov function candidate

V1 = xT Px

(32)

The derivative of the Lyapunov function V1 is

(

)

T T V& P&+ ( A + ∆A ) P + P ( A + ∆A ) x + 2 BT P ( u + d ) 1 = x

=x

T

( P&+ A

T

(

)

P + PA x + 2 x PMHNx

+2ϕ −k1 sgn (ϕ ) − k2 ϕ For any positive constant λ , we have

T

2α −1

sgn (ϕ ) + d

(33)

)

ACCEPTED MANUSCRIPT 2 x T PMHNx ≤ λ x T PMM T Px + ≤ λ x T PMM T Px +

1

λ 1

λ

x T N T H T HNx

(34) x T N T Nx

where Eq. (23) is used. Substitution of Eqs. (28) and (34) into Eq. (33) yields

(

+2ϕ − k1 sgn (ϕ ) − k2 ϕ

2α1 −1

sgn (ϕ ) + d

 1  2α ≤ x  2 PBBT P − Q  x − 2k2 ϕ 1 β   T

2α1

− x T Rx

(35)

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≤ ϕ 2 − δ 2 − 2k 2 ϕ

)

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 1 T  1 V& PBBT P − λ PMM T P − N T N − Q  x + 2 x T PMHNx 1 = x  2 β λ  

Inside the sliding sector, we substitute Eq. (24) into Eq. (33), then we obtain

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2α1 V& 1 ≤ −2k 2 ϕ

(36)

Taking the property of singular value into account, we obtain

ϕ 2 ≥ λmin ( S FT S F ) x V1 ≤ λmax ( P ) x

2

2

(37) (38)

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Let the positive constants c1 > 0 and 0.5 ≤ α1 ≤ 1 . We choose the parameter k 2 as

k2 ≥

c1λmax ( P )

2λmin ( S FT SF )

(39)

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Substitution of Eq. (39) into (36) yields

λmin ( S FT S )

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V& 1 ≤ −2k2

λmax ( P )

V1α1

(40)

α1

≤ −c1V1

Therefore, the finite time stability is guaranteed inside the sliding sector. Since V1 is positive definite, the solution of Eq. (40) is

V11−α1 ( x , t ) ≤ V11−α1 ( x0 , 0 ) − c1 (1 − α1 ) t , 0 ≤ t ≤ Tr   V1 ( x , t ) = 0, t ≥ Tr

(41)

where Tr is the settling time. If the system state x is kept inside the sliding sector, the settling time is given by

ACCEPTED MANUSCRIPT Tr ≤

V11−α1 ( x0 , 0 )

(42)

c1 (1 − α1 )

The settling time is relevant to the parameters c1 and α1 . We can control the convergence rate by

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adjusting these parameters.

Remark 1: In the sliding sector control [26, 28], there only exist matched parameter uncertainties and external disturbances. In the proposed sliding sector guidance, unmatched parameter uncertainties are

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considered. Step 2. The control input outside the sliding sector is designed as u = −Gu−1 ( Lu + k3 sgn (ϕ ) )

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(43)

where

Gu = BT PB + x T P

∂B B ∂x

(44)

∂B ∂B   Lu =  BT P&+ x T P A + BT PA  x + x T P ∂x ∂t  

{

we

can

ensure

the

system

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Since δ ≥ 0 ,

states

come

into

(45) the

sliding

sector

}

S ( x , t ) = x ϕ ( x, t ) ≤ δ ( x , t ) , if ϕ decrease to zero. Define the Lyapunov candidate V2

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V2 = β 2ϕ 2

(46)

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The derivative of the Lyapunov function V2 satisfies

(

)

T & T &T & V& 2 = 2ϕ B Px + B Px + B Px

= 2ϕ ( Lu + Gu ( u + d ) )

(47)

Substitution of Eq. (43) into Eq. (47) yields V& 2 = 2ϕ ( − k 3 sgn (ϕ ) + Gd )

(48)

k3 ≥ Gd

(49)

The positive gain k3 is chosen as

Then, it follows that

ACCEPTED MANUSCRIPT V& 2 <0

(50)

Remark 2: With the assumption that the upper bound of the disturbance can be estimated a priori, a finite time sliding sector guidance law is designed based on the proposed sliding sector. Under the

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finite time sliding sector guidance law, the system states can move into sliding sector from any initial states outside the sector. In conventional sliding sector, the system states only enjoy asymptotic stability or exponential stability, which means the drag command will be tracked in infinite time and it

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is undesirable. In the proposed sliding sector, the drag command can be tracked in finite time which

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ensures the high tracking performance.

In order to remove the chattering existing on the boundary of the sliding sector, an inner sector S i ( x , t ) and an outer sector S o ( x, t ) are introduced

{

S i ( x , t ) = x ϕ ( x, t ) ≤ ε δ ( x, t )

{

}

}

(52)

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S o ( x , t ) = x ε δ ( x , t ) < ϕ ( x , t ) ≤ δ ( x, t )

(51)

where the parameter ε satisfies 0 < ε < 1 . It can be found that S ( x , t ) == S i ( x , t ) ∪ S o ( x , t ) and

0, x ∈ ϕi   hd (ϕ , δ ) = unchanged , x ∈ ϕo  1, x ∉ ϕo 

(53)

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S i ( x , t ) ∩ S o ( x , t ) is the null set. A hysteresis dead-zone function on ϕ and δ is defined as

4. Simulation results

In this section, an entry guidance problem is investigated by numerical simulation. The parameters

are taken from the Mars Science Laboratory mission [4]. For convenience, the normalized energy E is introduced

E=

E − Ei E f − Ei

(54)

ACCEPTED MANUSCRIPT where Ei is the initial energy and E f is the final energy. Therefore, E is zero at the initial state and

E equals to 1 at the final phase of the entry guidance. The initial and target conditions are shown in Table 1 and Table 2, respectively. A 100-run Monte Carlo simulation is performed to evaluate the

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proposed method. The drag and lift coefficient dispersions are modeled as random Gaussian distributions and their dispersions are ±30% . The density dispersion is modeled as random multiplier factors, and the density dispersion is ±40% . The matrices P0 , Q and R are chosen as

218.6894   40 0  , Q=   , R = 0.2Q 400.1803   0 1

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 230.1226 P0 =   218.6894

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The other parameters are chosen as β = 0.9 , λ = 0.5 , M = diag ([20, 4]) , N = I , α1 = 0.6 k1 = 0.2 , k 2 = 0.1 and k3 = 0.9 . The proposed method is compared with the NTVSS method [26, 28].

Table 1 Initial conditions State variables

-90.072 deg

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θ0

Value

-43.898 deg

r0

3520.76 km

ψ0

4.99 deg

V0

5505 m/s

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φ0

γ0

-14.15 deg

Table 2 Target conditions State variables

Value

θf

-74.7 deg

φ

-41.8 deg

ACCEPTED MANUSCRIPT hf

8 km

90 Reference FTSS NTVSS

80

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70 60 50 40

22

18

20

16 10

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20

30

0.92 0.94 0.96 0.98 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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0

1

Norminalized Energy

Fig. 2. Drag Tracking Performance.

140

Reference FTSS NTVSS

120

80

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100

8.5 8

60

7.5

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40

241

242

243

244

20

0

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0

50

100

150

200

Time s

Fig. 3. Altitude Tracking Performance.

250

1

ACCEPTED MANUSCRIPT 80 Reference FTSS NTVSS

60 40 20 0

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-20 -40 -60 -80 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig 4. Bank Angle.

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-41.5

-41.8

-42

-41.85 -42.5 -41.9 -75

-74.8 -74.6 -74.4 -74.2

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-43

-43.5

-44 -92

-90

1

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Norminalized Energy

-88

-86

-84

Reference FTSS NTVSS

-82

-80

-78

-76

-74

Longitude deg

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Fig 5. Ground Tracking Performance.

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4

FTSS NTVSS

3

2

1

0

-1

-2 0

50

100

150

Time

Fig 6. Sliding sector

200

250

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-41.2 -41.25 -41.3

10km

-41.35 -41.4

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5 km

-41.45 -41.5 -41.55 -41.6

-41.7 -73.7

-73.6

-73.5

-73.4

-73.3

-73.2

-73.1

-73

-72.9

-72.8

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Longtitude,deg

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-41.65

Fig 7. Deployment positions based on FTSS

-41.2

-41.3 -41.35 -41.4 -41.45

10 km

5 km

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-41.5

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-41.25

-41.55 -41.6

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-41.65

-41.7 -73.7

-73.6

-73.5

-73.4

-73.3

-73.2

-73.1

-73

-72.9

-72.8

Longtitude,deg

Fig 8. Deployment positions based on NTVSS

The tracking performances of drag acceleration tracking, altitude, bank angle and ground tracking are shown in Figure 2-5, respectively (double line for tracking command, solid line for proposed method and dotted line for NTVSS, respectively). The corresponding logic about inner and outer

ACCEPTED MANUSCRIPT control is shown in Figure 6. In the presence of system uncertainty, the finite time sliding sector (FTSS) controller ensures the tracking error to converge to the tracking command in finite time. However, the system states under NTVSS converge to the tracking command at the end of the guidance

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process. Therefore, the guidance precision under the FTSS law is higher than that under the NTVSS law. Variable structure control method is well known for its robustness properties and variable structure control with sliding sector control law provides better control performance with reduced chattering and

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high control accuracy in comparison with the sliding mode control law even in the presence of system

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uncertainty. The proposed FTSS controller employs a much stronger condition such that the finite time stability is guaranteed inside the sliding sector. It was demonstrated that finite time stable systems might enjoy not only faster convergence but also better robustness and disturbance rejection properties. Therefore, the proposed approach ensures a high guidance precision.

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Fig.s 7 and 8 show the dispersions on entry longitude and latitude. The results in Fig. 7 show that almost 85% of the cases based on FTSS have range from target within 5 km, while 95% are within 10 km. Figure 8 indicates that about 75% of the case based on NTVSS have range from target within 5 km,

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and 90% are within 10 km. As a result, simulations show that the proposed FTSS method is robust

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against the uncertainty and can achieve precision landing requirement.

5. Conclusions

In this paper, a novel FTSS method is proposed for spacecraft entry guidance in the presence of system uncertainties and disturbances. Based on the finite time stability theory, a robust finite time sliding sector is designed. The finite time stability is guaranteed inside the sector. It can provide better control performance with reduced chattering and high control accuracy in comparison with the sliding mode control law. Numerical simulation results show that the proposed approach is able to provide a

ACCEPTED MANUSCRIPT high guidance precision in the presence of system uncertainties and disturbances.

Acknowledgements The work described in this paper was supported by the National Natural Science Foundation of

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China (Grant No. 61603183 and 11672126) and Aeronautical Science Fund of China (Grant No. 20160152002 and 20160112002). The authors fully appreciate their financial supports.

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hypersonic vehicles". Proceedings of the IMechE, Part G: Journal of Aerospace Engineering, 2017 (in press). R. Y. Jits and G. D. Walberg, "Blended control, predictor-corrector guidance algorithm: an enabling technology for mars aerocapture." Acta Astronuatica. Vol. 54, No. 6, 2004, pp. 385-398.

[10] C. A. Kluever, "Entry guidance performance for mars precision landing." Journal of Guidance, Control, and Dynamics. Vol. 17, No. 6, 2008, pp. 1350-1356.

[11] Z. Wang and M. J. Grant. "Constrained trajectory optimization for planetary entry via sequential convex programming", Journal of Guidance, Control, and Dynamics, Vol. 40, No. 10, 2017, pp. 2603-2615 [12] Z. Wang and M. J. Grant. Constrained trajectory optimization for planetary Entry via Sequential Convex Programming, Journal of Spacecraft and Rockets, Vol. 55, No. 4, 2018, pp. 993-1006. [13] J. B. Manrique, "Advance in spacecraft atmospheric entry guidance." Doctoral thesis, Irvine Mechanical and Aerospace Engineering Dept., University of California, 2010.

ACCEPTED MANUSCRIPT [14] X. Yuan, Z Yu, P Cui, et al, "Probability-based hazard avoidance guidance for planetary landing." Acta Astronautica, Vol 144, 2018, pp. 12-22. [15] X Jiang, S Li. Anti-windup Terminal Sliding Mode Control for Mars Entry with Super-twisting Sliding Mode Disturbance Observer. Journal of Aerospace Engineering, 2018, 31(5): 06018002. [16] S Li, X Jiang. RBF Neural Network based Second-Order Sliding Mode Guidance for Mars Entry under Uncertainties. Aerospace Science and Technology, 2015, 43: 226-235.

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[17] S Li and Y Peng. Neural Networks Based Sliding Mode Variable Structure Control for Mars Entry. Proceedings of the Institution of Mechanical Engineers, Part G, Journal of Aerospace Engineering, 2012, 226(11): 1373-1386.

[18] J. Dai, A Gao, Y Xia, "Mars atmospheric entry guidance for reference trajectory tracking based on robust nonlinear compound controller.” Acta Astronautica, Vol. 132, 2017, pp. 221-229.

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[19] R. Furfaro and D. R. Wibben, "Mars atmospheric entry guidance via multiple sliding surface guidance for reference trajectory tracking." AIAA/AAS Astrodynamics Specialist Conference, 2012, pp. 2012–4435.

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[20] M. Sagliano, E. Mooij, and S. Theil. "Adaptive Disturbance-Based High-Order Sliding-Mode Control for Hypersonic-Entry Vehicles", Journal of Guidance, Control, and Dynamics, Vol. 40, No. 3, 2017, pp. 521-536.

[21] C. Edwards and Y. Shtessel. "Adaptive continuous higher order sliding mode control", Automatica, Vol. 65, 2016, pp. 183-190

[22] S. Talole, J. Benito, and K. Mease. "Sliding Mode Observer for Drag Tracking in Entry Guidance", AIAA Guidance, Navigation and Control Conference and Exhibit, Guidance, Navigation, and Control and

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Co-located Conferences, Hilton Head, South Carolina, August,20-23, 2007

[23] P. Yu, Y. B. Shtessel, and C. Edwards. "Adaptive Continuous Higher Order Sliding Mode Control of Air Breathing Hypersonic Missile for Maximum Target Penetration", AIAA Guidance, Navigation, and Control Conference, AIAA SciTech Forum, Kissimmee, Florida, 2003

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[24] Z. Zhao, J. Yang, S. Li and et al., "Drag-based composite super-twisting sliding mode control law design for Mars entry guidance." Advances in Space Research, Vol. 57, No. 12, 2016, pp. 2508–2518. [25] K. Furuta and Y. Pan, "Variable structure control with sliding sector." Automatica, Vol. 36, 2000, pp.

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211–228.

[26] Y. Pan, K. D. Kumar, G. Liu and et al., "Design of variable structure control system with nonlinear time-varying sliding sector." IEEE Transactions on Automatic Control, Vol. 54, No. 8, 2009, pp. 1981–1986.

[27] Y. Hong, "Finite-time stabilization and stabilizability of a class of controllable systems." Systems & Control Letters, Vol. 46, No. 4, 2002, pp. 231-236.

[28] B. Xu, D. Zhou and S. Sun, "Finite time sliding sector guidance law with acceleration saturation constraint." IET Control Theory & Applications, Vol. 10, No. 7, 2016, pp. 789–799. [29] K. Zhou and J. C. Doyle. Essentials of robust control.Upper Saddle River, NJ: Prentice Hall, 1999, pp.14–54.

ACCEPTED MANUSCRIPT Title: Finite time sliding sector control for spacecraft atmospheric entry guidance Highlights: Finite time stability is guaranteed inside the proposed sector.

The proposed method can deal with unmatched parameter uncertainties.

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Chattering can be further reduced by introducing inner sector and outer sector.

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1. 2. 3.