Optimal nonlinear feedback guidance algorithm for Mars powered descent

Aerospace Science and Technology 45 (2015) 359–366

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Aerospace Science and Technology www.elsevier.com/locate/aescte

Optimal nonlinear feedback guidance algorithm for Mars powered descent Yiyu Zheng, Hutao Cui ∗ Deep Space Exploration Research Center, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 11 July 2014 Received in revised form 23 May 2015 Accepted 6 June 2015 Available online 10 June 2015 Keywords: Mars landing Powered descent guidance Optimal feedback control Fuel consumption

a b s t r a c t In this paper, an optimal nonlinear feedback guidance algorithm with complex state and control constraints is developed for Mars powered decent. The analysis of the optimal control problem for Mars powered descent is undertaken firstly. Then based on the real-time sampling optimal feedback control theory, the Mars powered descent guidance (PDG) algorithm is designed and analysed. A practical method is also proposed to solve the problem of the initialization of the PDG algorithm. Numerical simulations are performed to evaluate the effectiveness of the proposed PDG algorithm. The effects of the sampling period and the prediction errors on landing errors are studied in the numerical simulations. The fuel consumption performances of the proposed PDG algorithm and the Apollo guidance algorithm are also studied and compared. The simulation results show that the less fuel consumption is obtained with the proposed PDG algorithm. Monte Carlo simulation verifies the high landing precision of the proposed PDG algorithm. © 2015 Elsevier Masson SAS. All rights reserved.

1. Introduction The process of delivering a Mars lander from the planetary orbit to a stationary position on the Mars surface, which presents a unique set of challenges, may generally be split into three phases: entry, descent, and landing (EDL) for MSL-class landers [1,2]. The future Mars missions, such as the sample return, may target scientifically interesting features that lie in areas far more hazardous. To avoid hazards and land safely and precisely, during the powered decent phase, future landers must have the ability to detect hazards in the landing zone and manoeuvre to a selected safe landing site, which requires autonomous, onboard trajectory planning and execution, with hazard detection sensors in the control loop [3]. A substantial number of papers that examine the trajectory optimization and guidance algorithm design for Mars powered descend have been published. Wong presents a Mars powered decent guidance (PDG) algorithm similar to that used for the Apollo lunar module, using polynomials of time to describe the desired position, velocity, and acceleration profiles [3]. This guidance algorithm is autonomous in nature and satisfies the request of real-time guidance. This guidance algorithm is not an optimal guidance law in that it does not minimize fuel or any other cost functional. Also, the state and control constraints are neglected by Wong in his pa-

*

Corresponding author. E-mail addresses: [email protected] (Y. Zheng), [email protected] (H. Cui).

http://dx.doi.org/10.1016/j.ast.2015.06.008 1270-9638/© 2015 Elsevier Masson SAS. All rights reserved.

per. Topcu derives a solution with maximum–minimum–maximum structure for the minimum-fuel powered descent guidance [4]. However, the state constraints are not considered in [4]. Taking state and control constraints into account, Acikmese [5] presents a convex optimization approach for the fuel-optimal Mars powered descent. Blackmore [6] further develops this method for the case where no feasible pinpoint landing trajectories exist. Guo investigates an optimization approach to generate waypoints in the context of employing the zero-effort-miss/zero-effort-velocity feedback guidance algorithm for the Mars landing problem [7], in which two cases with power-limited and thrust-limited engine are considered respectively. The approaches [4–6] require precise mathematical model of the lander dynamics and are not robust against uncertainties, e.g., the aerodynamic drag and wind, due to the open-loop strategy. This may lead to great errors at the final time. It is shown in [4–6] that the optimal solutions can be efficiently computed numerically using the interior point methods or indirect methods. But the uses of interior point methods and indirect methods in a real-time terminal descent scenario are still an open research issue [8]. Although a closed-loop feedback is adopted to improve the robustness in [7], the complex constraints, e.g., thrust pointing constraints introduced by [5], cannot be handled sufficiently. Using the desensitized optimal control methodology, Shen [9] develops a Mars powered decent guidance law, which aims at reducing the sensitivity of the minimum-fuel powered descent trajectory in the presence of uncertainties and perturbations.

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Sostaric uses a Legendre pseudospectral method to develop a numerical solution to the Mars powered descent [10]. However, numerical optimization algorithms adopted to find approximate optimal solutions in [7,9,10] remain a challenge to approximate the optimal solution precisely with computations as less as possible, which may make these approaches inapplicable for autonomous onboard implementation, especially the real-time guidance. Therefore, to develop real-time and robust Mars PDG algorithms, in which the state and control constraints are sufficiently considered, much more work is needed. In this paper, we investigate a novel nonlinear closed-loop guidance algorithm applicable to the suboptimal-fuel guidance for Mars powered decent. The proposed PDG algorithm uses a sampling optimal feedback approach and is applicable for the autonomous onboard implementation. In addition, the proposed PDG algorithm is robust against uncertainties and unmodeled dynamics. The PDG algorithm is developed with the consideration of the state and control constraints and has its roots in the real-time sampling optimal feedback control theory [11–14]. This paper is organized as follows. In Section 2, we formulate the powered descent guidance problem for Mars pinpoint landing with complex state and control constraints. In Section 3, the analysis of the optimal control problem for Mars powered descent is undertaken firstly. The development of the proposed PDG algorithm goes after the analysis of the optimal control problem. The analysis of the performance of the proposed PDG algorithm is also presented in this section. In Section 4, several numerical simulations are performed to evaluate the effectiveness of the proposed PDG algorithm. The simulation results are discussed and analysed in this section. Finally, a 200-run Monte Carlo simulation is performed considering several uncertainties and navigation errors. 2. Powered decent control problem formulation The powered descent control problem formulation includes the lander translational motion dynamics and several constraints on the states and controls of the lander. The lander translational motion equations are presented firstly. To this end, we assume that n identical thrusters with equal thrust vector T at each time is mounted such that it is canted at an angle φ from the net thrust direction [5,6,8]. The translational motion equations of the lander in this document use a Mars surface-fixed Cartesian coordinate system with x and y axis located in the horizontal plane, z axis pointing upward and completing the right-hand coordinate system. The equations of motion neglect the Coriolis accelerations due to the Mars rotation because the accelerations are too small compared to the thrust acceleration. The equations of the translational motion for the lander are as follows:

r˙ = v



v˙ = g (r ) + T net + F per



 ˙ = −  T net  I sp g e cos φ m

m

(1) (2) (3)

where r = [x, y , z]T and v = [ v x , v y , v z ]T are the position and Mars-relative velocity vectors respectively; g = [0, 0, g m ]T denotes the gravitational acceleration vector on the surface of Mars, where g m = 3.76 m/s2 is the average gravitational acceleration on the surface of Mars; T net = nT cos φ is the net thrust vector; F per is the total perturbing force, accounting for unmodeled or unknown forces, such as the perturbations from the wind and aerodynamic forces; m is the lander mass. I sp is the specific impulse, g e = 9.807 m/s2 is the Earth’s gravitational constant. The complex constraints on the lander motion include inequality and equality constraints. In detail, they are the constraints on the thrust magnitude, boundary conditions, lander mass, path, and

Table 1 Complex constraints on the lander motion. Constraints

Representations

Thrust magnitude constraint Boundary conditions constraint

T min ≤  T net /n cos φ ≤ T max r (t 0 ) = r 0 , v (t 0 )= v 0 r t f = r f , v tf = v f m (t 0 ) = m0 , m t f ≥ mdry  0 ≤ sin θ˜alt ≤ sin θalt = c T r r  ≤ 1  ˜ 0 ≤ sin θcam ≤ sin θcam = c T T net  T net  ≤ 1

Mass constraint Path constraint Net thrust vector direction constraint

net thrust vector direction respectively. The complex constraints are introduced briefly here. For more details about the derivation process of the constraints above, readers may refer to [5,6]. These constraints are summarized in Table 1 for the formulation of the powered descent control problem in this paper. In Table 1, T min and T max are the minimum and maximum value of the thrust magnitude; t 0 and t f are the initial and final time of the power descent; r 0 , v 0 , and m0 are the initial value of the position, velocity, and lander mass; r f and v f are the final value of the position and velocity, which are determined by the landing requirement; mdry is the mass of the lander without the propellant. The mass constraint is used to avoid the lander running out of the fuel during the powered decent for a safely landing. The path constraint is constructed to prevent the subsurface fight, where θ˜alt ∈ [0, π /2 ] is a constant angle and c = [0, 0, 1]T is a unit vector. The thrust vector direction constraint considers a vision-based lander powered decent landing scenario. The lander is equipped with a camera which is required to be directed to the ground during the control process. Consequently, the downward pointing camera imposes a constraint on the lander attitude motion. For a lander with a single net thrust vector, the translational motion is controlled through the attitude manoeuvre of the lander and changing the direction of the net thrust force. To ensure the camera working normally with high imaging qualities, the value of the angle θcam between the net thrust vector and the horizontal plane should have a right scope. In reality, a small θcam can result in a failing imaging. In Table 1, θ˜cam ∈ [0, π /2 ] is a design parameter related to the camera. The translational motion model described by Eqs. (1)–(3) is employed to simulate the power descent of the lander. In the development of the PDG algorithm, the total perturbing force Eq. (2) is neglected. The purpose of the PDG algorithm is to find a command T net in each guidance cycle for delivering the lander to the specified landing point, while simultaneously satisfying the constraints listed in Table 1 above and consuming the fuel as less as possible. 3. Optimal feedback guidance design and analysis 3.1. Analysis of the optimal PDG The analysis of the optimal PDG problem neglects the effects of the total perturbing force and expresses the net thrust vector by

T net = T net, max u α

(4)

where T net,max is the maximum net thrust magnitude, u ∈ [0, 1] is the net thrust ratio, and

α = [cos β cos θ, cos β sin θ, sin β ]T

(5)

is the unit vector of the net thrust direction with the direction angle β and θ defined as in Fig. 1. Thus, we have the equations of the translational motion for the analysis and design of the PDG:

r˙ = v



v˙ = g (r ) + T net, max u α m

(6) (7)

Y. Zheng, H. Cui / Aerospace Science and Technology 45 (2015) 359–366

Fig. 1. Definition of the net thrust vector.



˙ = − T net, max u I sp g e cos φ m

(8)

Construct a performance index as follows

J=

t f

T net, max I sp g e cos φ

(1 − ε ) u + ε u 2 dt

(9)

t0

where ε is an arbitrary parameter, varying from 0 to 1. One can see that ε = 1 and ε = 0 result in a fuel-optimal performance index and an energy-optimal performance index respectively. The Hamiltonian of the system is built as



H = λTr v + λTv g (r ) + T net, max 

+

I sp g e cos φ

T net,max u α



m

(1 − ε ) u + ε u 2

− λm

T net, max I sp g e cos φ (10)

Using Pontryagin’s maximum principle (PMP), one obtains the optimal thrust direction vector and the thrust ratio as follows

α∗ = − u∗ =



λv λ v 

u0 , uε ,

(11)

ε=0

(12)

else

where



uε =

1, s < −ε 0.5 − 0.5s/ε , |s| ≤ ε and u 0 = 0, s>ε



1, u 0 ∈ [0, 1] , 0,

s<0 s=0 s>0

with the switching function

s = 1 − λm −

I sp g e cos φλ v  m

(13)

The costate differential equations are

∂H ˙ ∂H ∂H λ˙ r = − , λv = − , λ˙ m = − ∂r ∂v ∂m

(14)

One can see that u ε is a continuous function of s, while u 0 is a discontinuous function of s because u 0 is not specified by any value at the point s = 0. Generally, the switching function takes the value zero only at finite isolated points. So, u 0 only takes the value of zero or one, which suggests that a bang–bang control may be needed to minimize the fuel consumption. To solve the optimal problem, we may consider the utilization of the indirect method. The indirect method transforms the optimal control problem into a two-point boundary-value problem (TPBVP) [15]. Although indirect method requires less computation cost and ensures hight precision, the initial guess for the TPBVP is difficult to obtain because the optimal solution has small convergence domains and is very sensitive to the initial guess. In addition, the complex constraints given by Table 1 also pose difficulties in finding the optimal solution through the utilization of the indirect method. The other way to approach the optimal solution is the direct method, such as the direct collocation method and the pseudospectral method, which transfers the optimal control problem into

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a static parameters optimization problem [16]. The direct method is widely adopted in engineering practice due to their ease of implementation and large convergence regions [17–19]. Also, the direct method has outstanding advantages in solving the optimal control problems with complex constraints. However, it may fail in finding converging to the optimal solution due to the discontinuity of the bang–bang control. It is well known that, for the bang–bang control problem, the pseudo-spectral method has proved to be an effective tool by adding knots which are placed where the solution is subject to sudden changes [20]. However, adding knots may directly degrade the performance of computing efficiency. In addition, as the optimal solution is usually not known up front, it is difficult to find these points of interest in advance. As discussed above, the possible appearance of the bang–bang control poses numerical difficulties in solving the optimal problem and we need to come up with a compromise method. In this document, to ensure an online guidance and reduce the fuel consumption simultaneously, we let the parameter in the performance index above take a non-zero value. Then we obtain a suboptimal fuel performance index, of which the solution is expected to be more numerically benign. 3.2. Guidance algorithm A novel nonlinear PDG algorithm for Mars powered descent is presented in this subsection. The development of the PDG algorithm employs the recent advancements in real-time sampling optimal feedback control theory [11–14]. The overall objective of the design is to derive a PDG algorithm that 1) satisfies the complex constraints (see Table 1); 2) guarantees high landing accuracy; 3) robust to the uncertainties; 4) consumes less fuel during the powered descent. To this end, we first construct a suboptimal fuel problem Si for the ith sampling time t i as: determine the control inputs u i (t ) and α i (t ) that minimize the performance index

J=

t f

T net, max I sp g e cos φ

(1 − ε ) u i (t ) + ε u 2i (t )dt

(15)

ti

which subjects to the dynamics Eqs. (6)–(8) and the complex constraints (see Table 1). Note that the initial time and states in Table 1 should be replaced by those of the ith sampling. In this document, the direct method is adopted to solve Si for each ith sampling time. One has the equality

t i +1 = t i + τ

(16)

with the fixed sampling period τ . Assume that the optimal solution for any S could be obtained within a sampling period. We also assume that u i (t ) and α i (t ) resulting from Si have been obtained. By denoting t i +1 as the current time t cur , we know that u i +1 (t ) and α i +1 (t ) will be available at the time t i +2 . To undertake the PDG in the current sampling period from the current time t cur = t i +1 to the time t i +2 = t cur + τ , we let the control inputs be

u (t ) = u i (t ) ,

α (t ) = α i (t )

(17)

for t ∈ [t cur , t cur + τ ]. One can see that since the current control inputs of the current sampling period adopt those of the last sampling period, thus the PDG can be undertaken to guide the lander to the landing point and satisfies the complex constrains. Once the optimal solution for Si +1 could not be obtained within the current sampling period, the PDG continues to use the control inputs obtained in the previous sampling period. Hence, the control inputs for the PDG are always available and the PDG has sufficient reliability due to the use of the real-time sampling optimal feedback control theory.

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However, the PDG algorithm above assumes that the control inputs u 0 (t ) and α 0 (t ) resulting from solving S0 at the initial time of the powered descent have been obtained in advance. It indicates that the solutions for S0 can be obtained instantly. This requirement is very difficult to meet due to the computing delay. In this document we propose a practical method to solve the problem of the initialization of the PDG algorithm above. It is known that the powered descent goes after the parachute descent ends [1,2]. Thus the lander states at the beginning of the powered descent can be predicted before the parachute descent ends. Using these predictive states r 0,p , v 0,p , and m0,p , the initialization of the PDG algorithm can be undertaken prior to the powered descent of the lander. In reality, there are some deviations on the predictive states. But the guidance performance ensured by the PDG algorithm should be insensitive to these deviations due to the feedback control of the proposed PDG algorithm. For ease of use, the PDG algorithm is summarized as follows: 1. Predict the states r 0,p , v 0,p , and m0,p before the powered descent begins. 2. Set i = 0 and solve S0 . 3. Apply u 0 (t ) and α 0 (t ) resulting from solving S0 to the PDG for t ∈ [t 0 , t 2 ]. 4. Set i = 2. 5. Take the measurements of the states of the lander at t = t i −1 . 6. Solve Si −1 . 7. Apply u i −1 (t ) and α i −1 (t ) resulting from solving Si −1 to the PDG for t ∈ [t i , t i +1 ]. 8. Set i = i + 1 and go back to step 5.

where  = exp ( A ) is the state transition matrix. The real dynamics employ the control input T net, i −1 = T net, max u i −1 (t )α i −1 (t ) resulting from solving Si −1 at the time domain t ∈ [t i , t i +1 ]. Thus we have the real state vector at t = t i +1

x (t i +1 ) =  (t i +1 − t i ) x (t i ) +

1 m

t i+1  (t i +1 − t ) B T net, i −1 (t ) dt ti

t i+1 +  (t i +1 − t ) B gdt ti

+

1 m

t i+1  (t i +1 − t ) B T per (t ) dt

(21)

ti

The difference between the real state vector and nominal at t = t i +1 is

x (t i +1 ) − xnom (t i +1 )

=

1 m

t i+1    (t i +1 − t ) B T net, i −1 (t ) − T net, i (t ) dt ti

+

1 m

t i+1  (t i +1 − t ) B T per (t ) dt

(22)

ti

Thus, 3.3. Analysis of the performance

x (t i +1 ) − xnom (t i +1 ) 

Defining a state vector as x = [r , v ]T , thus we have the real dynamics of the translational motion

x˙ = Ax + B



T net + F per



m+g

=

(18)

with

A=



03×3 03×3



I 3×3 , 03×3

 B=

03×3 I 3×3

m



3 (t i +1 − t )2 + 2

m

 ×  T net, i −1 (t ) − T net, i (t )  +  T per (t )  dt √ t i+1   3 τ2 + 2  T net, i −1 (t ) − T net, i (t )  +  T per (t )  dt ≤ m

ti

(23) At the time domain t ∈ [t i , t i +1 ], it is acceptable to make the assumption that the total perturbing force is norm-bounded, that is  T per (t )  ≤ c per , where c per is a positive constant. Also we assume that the difference between the control input T net, i −1 and T net, i is norm-bounded at the time domain t ∈ [t i , t i +1 ], that is  T net, i −1 (t ) − T net, i (t )  ≤ c inp , where c inp is also a positive constant. With these assumptions taken into account, Eq. (23) can be rewritten as

    x (t i +1 ) − xnom (t i +1 )  = 3 τ 2 + 2 c inp + c per τ m

ti

ti

1

t i+1 ti

t i+1  (t i +1 − t ) B T net, i (t ) dt

t i+1 +  (t i +1 − t ) B gdt

m

t i+1  (t i +1 − t )  F  B  F  T per (t ) dt



xnom (t i +1 ) =  (t i +1 − t i ) x (t i )

+

1

ti

≤

where I is a unit matrix. Assume the mass to be a constant at the time domain t ∈ [t i , t i +1 ]. The assumption is reasonable because the sampling period is small and the mass changes little in the time domain t ∈ [t i , t i +1 ]. Note that the nominal trajectory xnom (t ) satisfies the all the complex constraints. If the state of the lander follows the nominal trajectory, then the precision landing can be obtained. Thus the difference between the real trajectory x (t ) and the nominal trajectory xnom (t ) should be norm-bounded at each point. Integrating the nominal dynamics Eq. (19) with the control input T net, i = T net, max u i (t )α i (t ) resulting from solving Si yields

1

+

(19)



m

t i+1  (t i +1 − t )  F  B  F  T net, i −1 (t ) − T net, i (t ) dt ti

and the nominal dynamics

x˙ nom = Axnom + B ( T net /m + g )

1

(20)

(24)

Thus the difference between the real trajectory and the nominal trajectory is norm-bounded. We see that a small sampling period helps reduce the difference between the real trajectory and the nominal trajectory and improves the guidance precision finally.

Y. Zheng, H. Cui / Aerospace Science and Technology 45 (2015) 359–366

Fig. 2. Parametric analysis: effects of the sampling period

363

τ on the landing errors.

Fig. 3. Landing trajectories generated by the open-loop simulation: the sampling period is infinite.

4. Simulation results

4.1. Effects of the sampling period on landing errors

The goal of this section is to assess the effectiveness of the proposed PDG algorithm in this paper. All the numerical simulations are performed in the Matlab/Simulink software environment. The software package GPOPS 5.1 [21,22] is employed to solve the suboptimal fuel problem Si for each time-based sampling. GPOPS 5.1 is an open-source optimal control software and implements Gauss and Radau HP-adaptive pseudospectral methods [23]. According to [14,24–26], even implemented in a legacy hardware, the pseudospectral methods are capable of obtaining optimal solutions within fractions of a second. The translational motion model described by Eqs. (1)–(3) is implemented to generate guided landing trajectories. The simulation conditions are given by:

Of specific interest in this subsection is the effects of the sampling period on the landing errors. To this end, the sampling period is first parametrized as τ = kτ0 for k = 1, 2, 3, . . . , 20, and τ0 = 0.2 s. The total perturbing force is assumed to be F per = [100, 100, 0]T N. Fig. 2 shows the three-dimensional guided landing trajectories, the histories of the final distance from the lander to the target point r f , the final speed and the residual mass of the fuel. One can see the final distance and speed basically increase as the sampling period increases. Therefore, increasing the sampling period basically results in a lower landing precision. The final distance and speed are quite small and vary relatively slightly with the sampling periods less than nearly 1 s. The histories of the residual mass are also presented in Fig. 2. The differences in the residual mass for each simulation are small mostly. The residual masses are nearly 151 kg for these simulations. By way of comparison, we set the sampling period to an infinite value, which results in an open-loop guidance. The landing trajectories for such an open-loop guidance are presented in Fig. 3. The x and y directions of the lander position cannot convergence to the desired value of 0. The final values of x and y are 97.7 m and 97.6 m

g = [0, 0, −3.71]T m/s2 , mdry = 1505 kg, m0 = 1905 kg I sp = 225 s, T¯ = 3.1 kN, T min = 0.3 T¯ , T max = 0.8 T¯ n = 6, φ = 27 deg, θ˜alt = 14 deg, θ˜cam = 45 deg r 0 = [1.0, 1.0, 3.1]T km, v 0 = [−25, −20, −50]T m/s r f = [0, 0, 10]T m, v 0 = [0, 0, 0]T m/s

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The final distances and the final speeds can be further reduced by decreasing the sampling period. 4.3. Fuel consumption comparison of two guidance algorithms

Fig. 4. Final distances varying with the parameter k1 .

Fig. 5. Final speeds varying with the parameter k1 .

respectively. The final distance and speed are nearly 138.10 m and 4.71 m/s respectively, which mean a relative low landing precision. 4.2. Effects of the prediction errors of initial conditions on landing errors In this subsection, aiming to verify that the landing precision is insensitive to the prediction errors of the initial position and velocity (see Section 3.2), the effects on the landing errors due to the prediction errors are studied through a set of numerical simulations. In these simulations, the sampling periods are all specified by a value of 1 s. The total perturbing force is given by F per = [100, 100, 0]T N. The initial position and velocity prediction errors are parametric as k1 e¯ r and k1 e¯ v respectively, where k1 = 1, 2, 3, . . . , 11, and e¯ r = [2, 2, 2]T m and e¯ v = [0.2, 0.2, 0.2]T m/s. The final distances and the final speeds varying with the parameter k1 are presented in Fig. 4 and Fig. 5 respectively. As we can see, the final distances and the final speeds change little with the increase of the parameter k1 . In these simulations, the final distances are nearly 1.4 m and the final speeds are nearly 0.6 m/s.

Two simulations are performed in this subsection to compare the performance of the proposed PDG algorithm and the Apollo guidance algorithm [3,27]. The Apollo guidance algorithm was employed by the Apollo lunar module, using polynomials of time to describe the desired position, velocity, and acceleration profiles. Wong [3] further develops this algorithm and applies it to Mars powered descent. Of specific interest is mainly the fuel consumption comparison of the guidance algorithms. Further comparisons between the proposed PDG algorithm and the Apollo guidance algorithm are beyond the scope of the current work. For ease of comparison, the total perturbing force and the prediction errors of the initial position and velocity are assumed to be zeros. Simulation results of Mars powered descent guided by the proposed PDG algorithm and the Apollo guidance algorithm are respectively presented in Fig. 6 and Fig. 7. As we can see, if we ignore the fuel consumption, then both the proposed PDG algorithm and the Apollo guidance algorithm guide the lander to the specified target point with high guidance precision. The histories of the thrust ratio and mass are very different from each other. The thrust ratio of the Apollo guidance algorithm firstly takes the value of nearly 0.5 and decreases from nearly 0.5 to nearly 0.3. The thrust ratio of the proposed PDG algorithm firstly takes the minimum value 0.3 and increases gradually from the minimum value to the maximum value 0.8, which is close to the bang–bang control. Hence, the proposed PDG algorithm ensures a suboptimal fuel consumption, while the Apollo guidance algorithm cannot do that. The histories of the mass also verify the less fuel consumption of the proposed PDG algorithm. One can see that, the values of the mass resulting from the employment of the proposed PDG algorithm are all larger than mdry while the mass of the lander guided by the Apollo algorithm becomes less than mdry after nearly t = 120 s. As we know, mdry is the mass of the lander without the propellant. Hence the lander guided by the Apollo algorithm runs out of the fuel before it reaches the target point. In this sense, the Apollo algorithm fails to guide the lander to the target point because the lander has not enough fuel to reach the target point.

Fig. 6. Results of Mars powered descent guided by the proposed PDG algorithm: positions, velocities, thrust ratio, and mass.

Y. Zheng, H. Cui / Aerospace Science and Technology 45 (2015) 359–366

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Fig. 7. Results of Mars powered descent guided by the Apollo guidance algorithm: positions, velocities, thrust ratio, and mass.

Fig. 8. Three-dimensional landing trajectories.

Fig. 9. Histories of mass.

4.4. Monte Carlo simulation To verify the robustness of the proposed PDG algorithm and its ability to drive the lander to the desired point under an uncertain environment, a 200-run Monte Carlo simulation is implemented in this subsection. The proposed PDG algorithm requires the knowledge of the current lander state, i.e., the mass, position and velocity, which are obtained from the accurate navigation during powered decent. Hence, in this Monte Carlo simulation, several simple statistical models are employed to model the navigation errors. The true mass, position and velocity for each time-based sampling are perturbed using a Gaussian noise with zero mean and 0.1 kg, 0.5 m and 0.1 m/s standard deviation, respectively. The lander initial position and velocity prediction errors are also modelled as Gaussian noises with zero mean and 1.0 m and 0.2 m/s standard deviation. The sampling period is specified by a value of 0.3 s for the Monte Carlo simulation. The total perturbing force is modelled as a Gaussian noise with zero mean and 50 N standard deviation. Modelling the total perturbing force in an accurate way may be required in reality. But this is beyond the scope of the current work. The three-dimensional landing trajectories for the 200-run Monte Carlo simulation are presented in Fig. 8. The histories of the mass are presented in Fig. 9. As we can see, the values of the mass resulting from the Monte Carlo simulation are all larger than mdry . The final value of the masses is nearly 1656 kg. Running out of the fuel does not occur in this Monte Carlo simulation. The final distance and speed statistics are reported in Fig. 10 and Fig. 11 respectively. The proposed PDG algorithm performs very well in Monte Carlo simulation. The mean values of the final distance and

Fig. 10. Final distance statistics.

Fig. 11. Final speed statistics.

final speed are respectively 0.659 m and 0.36 m/s. In this 200-run Monte Carlo simulation, 182 (or 91%) are within the distance of 1.2 m and 186 (or 93%) are within the speed of 0.8 m/s.

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5. Conclusions This paper investigates an optimal nonlinear feedback guidance algorithm with complex state and control constraints for Mars powered decent. The proposed PDG algorithm is based on the realtime sampling optimal feedback control theory. The problem of the initialization of the PDG algorithm is also handled and solved by the practical method presented in Section 3.2. The proposed PDG algorithm is applicable for the autonomous onboard implementation and is robust against uncertainties and unmodeled dynamics. The effects of the sampling period and the prediction errors on landing errors are studied in the numerical simulations. The fuel consumption performances of the proposed PDG algorithm and the Apollo guidance algorithm are also studied and compared. The Monte Carlo simulation shows that the proposed PDG algorithm performs very well in the presence of the dynamic uncertainties, initial condition prediction errors and navigation errors. The good robustness performance and high landing precision can be obtained with the employment of the proposed PDG algorithm. A small sampling period is required to obtain a high landing precision, such as a sampling period of 1 s. The prediction errors of the initial conditions of the lander have little effects on the landing precision. The lander guided by the proposed PDG algorithm consumes much less fuel compared with that guided by the Apollo guidance algorithm. High landing precision can be obtained with the proposed PDG algorithm in the uncertain environment. Future work could include the application and assessment of the proposed PDG algorithm considering the attitude dynamics of landers. Conflict of interest statement The authors declare that there is no confict of interests regarding the publication of this article. Acknowledgements This work was carried out by the Deep Space Exploration Research Center, Harbin Institute of Technology, China. This work was co-supported by the National Basic Research Program of China (No. 2012CB720000) and the National Natural Science Foundation of China (No. 61174201). References [1] R.D. Braun, R.M. Manning, Mars exploration entry, descent, and landing challenges, J. Spacecr. Rockets 44 (2) (2007) 310–323, http://dx.doi.org/10.2514/1.25116. [2] A. Wolf, E. Sklyanskiy, J. Tooley, B. Rush, Mars pinpoint landing systems trades, in: Astrodynamics 2007, Amer Inst Aeronaut & Astronaut; Amer Astronaut Soc, 2008, p. 129, Pts I-iii. [3] E.C. Wong, G. Singh, J.P. Masciarelli, Guidance and control design for hazard avoidance and safe landing on Mars, J. Spacecr. Rockets 43 (2) (2006) 378–384.

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