AIAA 2015-2398 AIAA Aviation 22-26 June 2015, Dallas, TX AIAA Atmospheric Flight Mechanics Conference
Improvements to Entry Terminal Point Controller for Mars Atmospheric Entry Yiyu Zheng∗ , Hutao Cui† and Yang Tian‡
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Harbin Institute of Technology, Harbin, Heilongjiang, 150080, China Although further improvements in approach navigation will reduce the landing ellipse, it is clear that, to achieve landing accuracy on the order of ±1.0 km from a designated landing point, closed-loop entry guidance will be necessary. This paper firstly evaluates the performance of the entry terminal point controller (ETPC) under high atmospheric parameter perturbations. Then, Considering the large uncertainties in the Martian atmosphere, an improved entry terminal point controller (IETPC) is developed. Numerical simulation results show that compared with the ETPC algorithm, IETPC makes considerable improvement in terminal downrange error for all Martian atmospheric parameter perturbation belonging to the investigated range.
Nomenclature CD CL D E g L r V m s Sref x
Aerodynamic drag coefficient Aerodynamic lift coefficient Aerodynamic drag acceleration, m/s2 Specific energy Mars gravitational acceleration, m/s2 Aerodynamic lift acceleration, m/s2 Radial distance form the center of Mars, m Mars-relative velocity, m/s Vehicle mass, kg Downrange, m Reference surface area, m2 State vector
Greek γ Mars-relative flight path angle, deg µ Mars gravitational constant, m3 /s2 ρ Mars atmosphere density, kg/m3 σ Bank angle, deg
I.
Introduction
The Mars Science Laboratory (MSL) is the first Mars mission to perform a guided entry, which consists of three parts: prebank, range control and heading alignment.1, 2 Once the filtered drag acceleration magnitude exceeds 1.96 m/s2 (0.2 Earth g), the MSL entry guidance ceases prebank and begins range control. The range control phase adopts the entry terminal point controller (ETPC) derived from the Apollo final entry phase guidance algorithm,3, 4 which modulates the bank angle to control the range flown. The ETPC algorithm is the guidance for the Mars Science Laboratory mission and controls to a terminal state of downrange at a ∗ Ph.D.
Candidate, Deep Space Exploration Research Center, Harbin, Heilongjiang, 150080;
[email protected]. Deep Space Exploration Research Center, Harbin, Heilongjiang, 150080. ‡ Lecturer, Deep Space Exploration Research Center, Harbin, Heilongjiang, 150080. † Professor,
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particular velocity.5 The predicted range-to-go is calculated as a function of state errors between the current states and the reference states. To improve the robustness of the ETPC algorithm, an overcontrol gain K is used to exaggerate the downrange error correction necessary. Because of slow system and trajectory response to guidance commands, entry performance is empirically improved by using the overcontrol gain K. However, the ETPC only takes current state errors into account, while ignoring downrange errors caused by other uncertainties, especially the large uncertainties in the Martian atmosphere density. During Mars atmospheric entry, the terminal downrange error caused by the deviations in the parameter of the Martian atmosphere may be quite large. In this paper, an improved entry terminal point controller (IETPC) using the bank angle modulation control is developed for low lift-to-drag L/D Mars atmospheric entry vehicles. The IETPC algorithm is analytical and is robust to the great uncertainty which exists in Martian atmosphere density. Theory of sensitivity analysis6 is applied to the reduced-order longitudinal entry dynamics, aiming to compute the terminal downrange error caused by the deviation of Martian atmospheric density parameter. Based on this sensitivity analysis, a Martian atmospheric parameter estimation method is proposed. Simulation results show that the proposed method of parameters estimation is effective and robust. After the parameters estimation method is proposed and evaluated, we derive the IETPC algorithm by using the linear perturbation theory. The notable modifications from the original ETPC algorithm are that the large uncertainty in the Martian atmosphere density has been considered in IETPC. The necessary modification improves the performance and robustness of the entry guidance system for Mars landers.
II.
Entry Guidance Problem Formulation
Neglecting the Coriolis terms, the Mars-relative longitudinal translational motion of the unpowered lander in a stationary can be described by the downrange s, radial distance form center of Mars r, Mars-relative velocity V and flight path angle γ as follows:7, 8 r˙ = V sin γ
(1)
V˙ = −D − g sin γ γ˙ = L cos σ − g − V 2 /r cos γ /V
(2)
s˙ = V cos γ
(3) (4)
where σ is the bank angle, g is the gravitational acceleration , D and L are the aerodynamic drag acceleration and lift acceleration, which are given by (5) D = 0.5ρV 2 Sref CD m (6) L = 0.5ρV 2 Sref CL m 2 g=µ r (7) where µ is the Mars gravitational constant, Sref is the reference surface area; m is the mass of the vehicle, CD and CL are the aerodynamic drag and lift coefficients, ρ is the Mars atmosphere density, which is fitted to an exponential function ρ = ρs exp ( −h/hs ) (8)
where h is the altitude, ρ0 = 0.0158 kg/m3 and h0 = 9354 m are the surface density and the scale height.7 Define a specific energy E as follow E = µ/r − 0.5V 2 (9) Then, the velocity V is determined by V = The differential equation that governs E is
p 2 (µ/r − E)
(10)
E˙ = DV
(11)
Then, the reduced-order longitudinal entry dynamics equations can be rewritten with E as the independent variable: r′ = sin γ/D (12) 2 of 9 American Institute of Aeronautics and Astronautics
s′ = cos γ/D γ ′ = u V 2 − g − V 2 r cos γ DV 2
(14)
s (Ef ) = s∗f
(15)
(13)
where u = (L/D) cos σ. For a precise landing, the typical constraints of entry are that the lander reaches a specified downrange s∗f at a specified altitude rf∗ and velocity Vf∗ . However, if we define a specified energy Ef = µ/rf∗ − 0.5Vf∗ 2 , then the typical constraints of entry can be combined to a single final constraint:9
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III.
Entry Terminal Point Controller and Analysis
Generally, the Mars entry begins at the first contact with the Martian atmosphere and ends with the deployment of the additional aerodynamic deceleration. The entry phase is responsible for delivering an entry vehicle from its initial conditions to the designated terminal conditions safely and accurately, such that the vehicle has sufficient altitude and time to perform the parachute descent and powered descent. The entry terminal point controller (ETPC) is originally derived from the guidance algorithm of the Apollo final entry phase.10, 11 Only the bank angle is commanded to control the range flown in this guidance scheme. The controller gains of ETPC are obtained using influence coefficients with respect to errors about the nominal L/D reference trajectory, which is defined by the range-to-go, drag acceleration, and altitude rate as a function of Mars-relative velocity or the specific energy.2, 10, 12 The desired vertical component of the lift-to-drag ratio u is calculated as the addition of the reference u ¯ plus a function of the difference between the actual s and predicted range-to-go sp :4, 10 u=u ¯+
K (s − sp ) ∂s/∂ (L/D)
(16)
The predicted range-to-go sp is calculated as a function of differences between the current estimates and the reference values of filtered drag acceleration and altitude rate errors sp = s¯ +
∂s ¯ + ∂s (r˙ − ¯r) D−D ˙ ∂D ∂ r˙
(17)
The overcontrol gain K of the ETPC is used to exaggerate the downrange error correction necessary. It’s recognized that the entry performance is empirically improved by using the overcontrol gain K because of slow system and trajectory response to guidance commands.2, 10 From Eq. (17), it can be seen that the ETPC only takes current state errors into account, while ignoring downrange errors caused by other uncertainties, especially the large uncertainties in the Martian atmosphere. The Mars entry guided by the ETPC algorithm simulations have been carried out for different Martian atmospheric parameter (ρ0 ) perturbation varied within the domain: −30% ∼ 30%. The nominal entry states are given by r0 = 3522.02 km, R0 = 0 km, V0 = 6000 m/s, γ0 = −15 deg. The aerodynamic coefficients are assumed to be constants and are given by CD = 1.450 and CL = 0.348. Thus the nominal lift-to-drag ration is 0.24. The aerodynamic reference area is Sref = 15.9 m2 . The nominal mass of the vehicle is also assumed to be a constant and is given by m = 2200 kg. Other simulation conditions are given by: 1. Entry state perturbations: ∆r = 100 m, ∆s = 3 km, ∆V = 50 m/s, ∆γ = 0.1 deg 2. Aerodynamic parameter and mass perturbations: ∆CD = 0.2C¯D , ∆CL = 0.2C¯L , ∆m = 0.2m ¯ The simulation result is presented in figure 1. As we can see, on some level, the overcontrol gain K indeed improves the robustness of the ETPC algorithm, but it is limited. Figure 1 shows that the terminal downrange error increases with the increasing of the magnitude of the Martian atmospheric parameter perturbation. The Martian atmosphere parameter affects the terminal downrange error considerably. This can be explained by uncertainties in the Martian atmosphere density ignored by the ETPC algorithm. Figure 1 also shows that if the Martian atmosphere parameter perturbation is zero, then the terminal downrange error is relatively small. This indicates that the terminal downrange error is caused mainly by the Martian atmosphere parameter perturbation. Therefore, for a precision landing, which requires a safe landing within hundreds of meters to the target of interest, the Mars entry guidance system should be able to null the downrange errors caused by the deviations in the parameter of the Martian atmosphere as well as current state errors. 3 of 9 American Institute of Aeronautics and Astronautics
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Terminal downrange error (km)
6 K=1 K=2 K=5 K = 10
4 2 0 −2 −4 −6 −8 −30
−20
−10 0 10 Perturbation (%)
20
30
Figure 1. Terminal downrange error guided by ETPC: K = 1, 2, 5, and 10.
IV.
Improved Entry Terminal Point Controller Derivation
Theory of sensitivity analysis6 is applied to the reduced-order longitudinal entry dynamics, aiming to compute the downrange deviation caused by the Martian atmosphere density. In this document, such sensitivity analysis enables us to develop a estimation method for the value of the Martian atmosphere density deviation. The proposed estimation algorithm evaluated using several numerical simulations. Finally, we derive the IETPC algorithm based on the proposed estimation method and the the linear perturbation theory. It is shown that the IETPC considers the deviations in the parameter of the Martian atmosphere as well as current state errors. A.
Atmospheric Parameter Sensitivity Analysis and Estimation T
Let x = (r, s, γ) be the reduced-order state vector and α = ρ0 , then reduced-order longitudinal entry dynamics equations can be rewritten with E as the independent variable: x′ = f (x, u, α, E) , x (E1 ) = x1
(18)
and the dynamics with the nominal parameter α = α0 is x′ = f (x, u, α0 , E) , x (E1 ) = x1
(19)
where α0 is the nominal value of parameter ρ0 . The state error induced by the perturbation ∆α = α − α0 can be expressed by ∆x (E, α) = x (E, α) − x (E, α0 ) Using the Taylor formula, one obtains the first-order approximation of ∆x (E, α): ∂x (E, α) ∆x (E, α) ≈ ∆α ∂α α=α0
(20)
(21)
Let’s define the trajectory sensitivity function as:
λ (E, α) =
∂x (E, α) ∂α
(22)
Then, we have ∆x (E, α) ≈ λ (E, α0 ) ∆α
(23)
The first derivative of λ (E, α) versus E is d ∂x ∂x′ dλ = = dE dE ∂α ∂α 4 of 9 American Institute of Aeronautics and Astronautics
(24)
Considering that ∂x′ ∂f (x, α, E) ∂x ∂f (x, α, E) = + ∂α ∂x ∂α ∂α Then, we have the sensitivity equation λ′ (E, α0 ) = As (E, α0 ) λ (E, α0 ) + Bs (E, α0 ) where
(25)
(26)
∂f (x, α, E) As (E, α0 ) = ∂x α=α0 ∂f (x, α, E) Bs (E, α0 ) = ∂α
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α=α0
Because the initial condition x (E1 ) is irrelevant to α, one obtains the initial condition for the sensitivity equation ∂x (E1 ) =0 (27) λ (E1 ) = ∂α Integrating Eq. (26) from E = E1 to E = E2 , we obtain the trajectory sensitivity function λ (E, α0 ) at the energy domain E ∈ [E1 , E2 ]. The state error ∆x (E2 ) caused by the perturbation ∆α can be computed by Eq. (23). Because the downrange error is caused mainly by Martian atmospheric parameter perturbation, for simplicity, we assume that the downrange error is only determined by the Martian atmospheric parameter perturbation ∆α. Using Eq. (23), one obtains ∆s (E2 ) = λ2 (E2 ) ∆α
(28)
where ∆s (E2 ) = s (E2 ) − s˜ (E2 ). s (E2 ) is the actual downrange at E = E2 . s˜ (E2 ) is the nominal downrange at E = E2 , which is computed by integrating Eq. (19), using the same input u and initial condition x (E1 ) with the actual dynamics Eq. (18). λ2 (E2 ) is the second component of λ (E) at E = E2 , which is obtained by integrating Eq. (26). The parameter estimation value can be given by α ˆ = α0 +
s (E2 ) − s˜ (E2 ) λ2 (E2 )
(29)
where E1 is the energy where drag is 0.2 Earth g and the range control begins. E2 is specified as E2 = Ef /10 in this document. Two simulation cases have been have been carried out for different Martian atmospheric parameter (ρ0 ) perturbation varied within the domain: −30% ∼ 30%. Case 1 assumes there exists no other uncertainties, whereas case 2 takes other uncertainties ∆CD , ∆CL , and ∆m listed above into account. The simulation result is shown in figure 2. In the case 1, the maximum absolute relative error of estimation occurs at −30% and 30% parameter perturbation with the magnitude of about 6.0% and 4.0% respectively. In the case 2, the maximum absolute relative error of estimation occurs at the point where parameter perturbation is zero with the magnitude of about 8.0%. The simulation results verify the effectiveness of the estimation method.
B.
Improved Entry Terminal Point Controller
T ¯ = (¯ For a given bank angle profile u ¯ (E), the nominal reference trajectory x r , s¯, γ¯ ) can be determined by integrating Eq. (12)-Eq. (14) easily. Expanding Eq. (18) in a Taylor series about the nominal reference trajectory, one obtains the first-order approximation:
d∆x = A (E) ∆x + Bc (E) ∆u + Bp (E) ∆α dE ¯ and where ∆u = u − u¯, ∆α = α − α0 , ∆x = x − x ∂f (x, u, α, E) A (E) = ∂x x=¯ x,α=α0 ,u=¯ u 5 of 9 American Institute of Aeronautics and Astronautics
(30)
Absolute relative error of estimation (%) Downloaded by HARBIN INSTITUTE OF TECHNOLOGY on August 15, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2398
10 Case 1 Case 2 8 6 4 2 0 −30
−20
−10 0 10 Perturbation (%)
20
30
Figure 2. Absolute relative error of estimation
∂f (x, u, α, E) Bc (E) = ∂u x=¯ x,α=α0 ,u=¯ u ∂f (x, u, α, E) Bp (E) = ∂α x=¯ x,α=α0 ,u=¯ u
Construct a system adjoin to Eq. (30) as follow:
dζ = −AT (E) ζ (E) , ζ (Ef ) = ζf dE For the downrange control, the terminal condition should be specified by h iT ζf = 0 1 0
(31)
(32)
One easily obtain that ζ T (Ef ) ∆x (Ef ) = ζ T (E) ∆x (E) + κc (E) ∆u + κp (E) ∆α where κc (E) =
Z
Ef
Z
Ef
(33)
ζ T (τ ) Bc (τ ) dτ
(34)
ζ T (τ ) Bp (τ ) dτ
(35)
E
κp (E) =
E
Using Eq. (33), we have ∆u =
ζ T (Ef ) ∆x (Ef ) − ζ T (E) ∆x (E) − κp (E) ∆α κc (E)
(36)
Considering r′ = sin γ/D
(37)
D = (µ/r − E) Sref CD ρ0 exp ((rs − r) /hs ) /m
(38)
and using the Taylor formula, one obtains ∂r′ ∂r′ ∂r′ ′ ∆r + ∆γ + ∆α ∆r = ∂r r¯,¯γ ,α0 ,E ∂γ r¯,¯γ ,α0 ,E ∂α r¯,¯γ ,α0 ,E 6 of 9 American Institute of Aeronautics and Astronautics
(39)
∂D ∂D ∆D = ∆r + ∆α ∂r r¯,¯γ ,α0 ,E ∂α r¯,¯γ ,α0 ,E
If define c1 = then
(40)
∂r′ ∂D ∂r′ ∂D ∂r′ , c = , c = , c = , c = 2 3 4 5 ∂r r¯,¯γ ,α0 ,E ∂γ r¯,¯γ ,α0 ,E ∂r r¯,¯γ ,α0 ,E ∂α r¯,¯γ ,α0 ,E ∂α r¯,¯γ ,α0 ,E ∆r = ∆D/c3 − c5 ∆α/c3
(41)
∆γ = ∆r /c2 − c1 ∆D/c2 c3 + ( c1 c5 /c2 c3 − c4 /c2 ) ∆α
(42)
′
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Substituting into Eq. (36) gives 1 (ζs ∆s + ( ζr /c3 − ζγ c1 /c2 c3 ) ∆D + ζγ ∆r′ /c2 + (κp + ζγ ( c1 c5 /c2 c3 − c4 /c2 ) − ζr c5 ∆α/c3 ) ∆α) κp (43) Therefore we have the IETPC algorithm: ∆u = −
u = u¯ + (s − s¯ + f1 ∆D + f2 ∆r′ + f3 ∆α)/f4
(44)
where f1 = ζr /c3 − ζγ c1 /c2 c3 f2 = ζγ /c2 f3 = κp + ζγ ( c1 c5 /c2 c3 − c4 /c2 ) − ζr c5 ∆α/c3 f4 = −κp Further more, the desired vertical component of the lift-to-drag ratio u can be rewritten as u=u ¯ + (s − spre )/f4
(45)
Although IETPC and ETPC have the same form, the predicted range-to-go spre of IETPC is calculated as a function of differences not only between the current estimates and the reference values of filtered drag acceleration and altitude rate errors, but also between the estimation of α and the nominal value α0 : ¯ − f2 (r˙ − ¯r) ˙ − f3 (ˆ α − α0 ) (46) spre = s¯ − f1 D − D
where α ˆ is calculated using Eq. (29). Numerical simulations, which have the same simulation conditions with the previous ones, are carried out to assess the performance and robustness of the IETPC algorithm. The overcontrol gain K of ETPC is chosen as K = 4. Simulation results are presented in figure 3. Compared to the ETPC algorithm, one can see considerable improvement in terminal downrange error for all Martian atmospheric parameter perturbation belonging to the investigated range. For example, at −30% parameter perturbation, the ETPC algorithm has a terminal downrange error about 4 km, while the IETPC algorithm only has a terminal downrange error about 0.1 km.
V.
Monte Carlo Simulations
A 500-run Monte Carlo simulation is carried out in this section to preliminarily evaluate the effectiveness of IETPC and verify its ability to deliver the lander to the desired target under an uncertain environment. The entry state perturbations ∆r, ∆s , ∆V , ∆γ are modeled using Gaussian noise with zero mean and 0.1 km, 1.0 km, 50 m/s and 0.05 deg standard deviation (3σ), respectively. Aerodynamic parameter, mass and density parameter perturbations ∆CD , ∆CL , ∆m, ρ0 are also modeled using Gaussian noise with zero mean and 0.2C¯D , 0.2C¯L , 0.2m, ¯ 0.4ρ¯0 standard deviation (3σ). The terminal downrange error statistics are reported in figure 4 and figure 5 respectively. One can see that, the IETPC performs better than ETPC in this set of simulations. For IETPC, in the 500-run Monte Carlo simulation, as shown in figure 4, 487 (or 97.4%) are within 0.1 km of the target. The rms of the all target miss distances is 0.09 km. For ETPC, in this 500-run Monte Carlo simulation, as shown in figure 5, 271 (or 54.2%) are within 0.1 km of the target and 484 (or 96.8%) are within 5 km of the target. The rms of the all target miss distances is 2.13 km. 7 of 9 American Institute of Aeronautics and Astronautics
6 4 2 0 −2 −4 −30
−20
−10 0 10 Perturbation (%)
20
30
Figure 3. Terminal downrange error guided by ETPC (K = 4) and IETPC
Number of cases
200
150
100
50
0 −0.5
0 0.5 1 Terminal downrange error (km)
1.5
Figure 4. Terminal downrange error statistics, IETPC
70 60
Number of cases
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Terminal downrange error (km)
IETPC ETPC
50 40 30 20 10 0 −5
0 5 10 Terminal downrange error (km)
15
Figure 5. Terminal downrange error statistics, ETPC
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VI.
Conclusions
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In this paper, the weaknesses of the ETPC algorithm for Mars entry guidance have been exposed. It has been shown that the ETPC algorithm only takes current state errors into account, while ignoring downrange errors caused by other uncertainties, especially the large uncertainties in the Martian atmosphere. If a lander is guided by the ETPC algorithm, the terminal downrange error will be affected by the Martian atmospheric parameter perturbation magnitude considerably. The terminal downrange error is caused mainly by Martian atmospheric parameter perturbation. The proposed method of parameters estimation is effective and robust. The IETPC algorithm improves the performance and robustness of the system for Mars landers. Future work could include the application of the proposed IETPC algorithm considering the lateral motion of landers. More numerical simulation cases with 3-DOF and 6-DOF model of the dynamics of the entry vehicle should be performed to evaluate the IETPC algorithm more sufficiently,
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