TERMINAL ATTITUDE-CONSTRAINED GUIDANCE AND CONTROL FOR LUNAR SOFT LANDING

IAA-AAS-DyCoSS2-14 -02-05 TERMINAL ATTITUDE-CONSTRAINED GUIDANCE AND CONTROL FOR LUNAR SOFT LANDING Zheng-Yu Song, Dang-Jun Zhao, and Xin-Guang Lv This work concentrates on a 3-dimensional guidance and control (G&C) scheme for the terminal landing phase of a lunar spacecraft. In order to cope with terminal attitude constraints, we take the commanded acceleration as an extended state whose terminal value relates to the terminal attitude constraints. Then in optimal control framework, an optimal terminal attitude-constrained feedback guidance law is derived. In order to accommodate various disturbances, a finite-time convergent extended state observer is utilized to estimate the disturbance. Driven by the disturbance observer, an improved sliding mode controller with a disturbance compensation is proposed. The attitude tracking error signals theoretically prove to be ultimately uniformly bounded. The convincing simulation results reveal the excellent performance is achieved by the proposed method during the terminal landing phase. INTRODUCTION Although the Apollo lunar spacecraft has successfully reached to the moon in 1969, 1 lunar soft landing is still a challenging problem since high accuracy (less than 10 meters) is required in the future robotic lunar missions. To achieve pinpoint landing on a desired site of the moon s surface, high performance guidance and control system is necessary in the landing process of a lunar spacecraft, which contains powered decending phase and terminal landing phase. Due to negligible atmosphere around the moon, the deceleration of the lunar spacecraft is only fulfilled by using inverse force of main thruster. Combining with the attitude control thrusters the spacecraft will be guided to reach the preselected site with a low touch-down velocity. Thus, it is crucial to minimize fuel consumption in G&C system design. Moreover, the naturally coupling dynamics between translational motion and rotational motion will substantially undermine the landing accuracy, hence, an integral G&C control scheme which can accommodate various disturbance and system coupling will improve the performance in future lunar missions. Actually, many works on G&C schemes of lunar landing have been reported. 2, 3, 4 As for fuel minimization problem, Reference 2 transforms the optimal soft landing problem in a planar framework into a standard optimal control problem with continuous state inequality constraints, then a numerical solution via the control parameterization is developed. In 3-dimension space, Reference 3 developed a nonlinear optimal feedback control strategy which depends on the off-line optimal Beijing Aerospace Automatic Control Institute (CASC), Post Office Box 142-402, 100854, Beijing, China. (zycalt12@sina.com). School of Aeronautic and Astronautic, Central South University, Lushan South Rd. 932, 410083, Changsha, China. Beijing Aerospace Automatic Control Institute (CASC), Post Office Box 142-402, 100854, Beijing, China. 1

trajectory, this implies the relocation of landing site is not acceptable. Actually, the prominent disadvantage of these methods stemmed from conventional optimal control theory is the large computational burden. In general, the spacecraft should softly achieve to the pre-defined site with vertical attitude in terminal landing phase. A number of guidance laws, such as explicit guidance law, 4 gravity-turn, 5 etc., are derived, however, these guidance laws within a planar framework are hard to achieve the necessary accuracy of 3-dimensional pinpoint landing, especially for the terminal landing phase, in which, the mutually coupled dynamics between the translational motion and the rotational motion is an important issue to be dealt with. This implies that an integrated control strategy, in which, the translational dynamics and rotational dynamics are simultaneously considered, is essentially needed to address the 6-DOF control problem in the terminal landing phase. To this end, Reference 6 presents an integrated guidance and control strategy via a filter-based back-stepping method, which results in an asymptotic performance. However, the time-consuming convergent process means much more fuel consumptions. In this work, we concentrate on a G&C scheme in 3-dimensional space for the terminal landing problem of a lunar spacecraft. An terminal attitude-constrained optimal guidance method is derived to provide commanded attitude. In order to cope with terminal attitude constraints, we take the differential signal of the commanded acceleration as a virtual input variable, which relates to attitude of a spacecraft. Then the commanded acceleration becomes a new state in the new extended system. According to the classical optimal control theory, an optimal feedback guidance law is derived. To accurately track the commanded attitude provided by the guidance law, a finite-time convergent extended state observer is employed to estimate the overall disturbances. The disturbance estimation signal acts as a compensation term in an sliding mode controller. The improved controller can accommodate various disturbances during the landing phase. The uniformly ultimately stable performance is validated by theoretical analysis and numerical simulation results. PROBLEM FORMULATION To facilitate the modeling of terminal landing motion, three right-handed reference frames illustrated in Figure 1 are defined: (1) an inertial reference frame MXY Z with origin located on the center of the moon is further defined with MX locating in the moon s equatorial plane and pointing to the zero-degree-longitude line, while MY points to the arctic pole of the moon; (2) a body frame Bx b y b z b whose origin locates on the mass center of the spacecraft and Bx b is along the symmetrical axis of the spacecraft; and (3) a target coordinate frame Oxyz with origin attached to the preselected landing point O. The axis Ox is along the direction of MO, while Oz located in the MOZ plane is perpendicular to Ox. The transformation matrix from the body frame to the target frame is derived as TB O = cos θ cos ψ cos γ sin ψ + sin θ sin γ cos ψ sin γ sin ψ + sin θ cos γ cos ψ cos θ sin ψ cos γ cos ψ + sin θ sin γ sin ψ sin γ cos ψ + sin θ sin γ cos ψ sin θ cos θ sin γ cos θ cos γ where ψ, θ and γ represent yaw, pitch and roll angles, respectively. (1) In order to simplify the mathematical model, we reasonably assume that, the gravity field is homogenous and the moon s rotation is assumed to be ignorable. 6 2

Figure 1. Reference Frame Definitions of Terminal Landing Translational Model The translational motion of the spacecraft is given by µ r = R MO + r 3 (R MO + r) + F m (2) where µ represent the moon s gravitational constant, m the total mass of the spacecraft, r = R MB R MO the spacecraft s position relative to the target point O. F = [F x, F y, F z ] T R 3 is the propulsion vector in Oxyz frame. In general, we assume the thrust vector F b is aligned with the Bb x direction, i.e., F b = [F, 0, 0] T, then we have F x F F cos θ cos ψ F = F y = TB O 0 = F cos θ sin ψ (3) F z 0 F sin θ Let r = [r x, r y, r z ] T R 3 and v = [v x, v y, v z ] T R 3, rendering the scalar formulation as ṙ x = v x, ṙ y = v y, ṙ z = v z µ v x = R MO + r 3 (R MO + r x ) + F cos θ cos ψ m µ v y = R MO + r 3 r y + F cos θ sin ψ m µ v z = R MO + r 3 r z F m sin θ (4) Attitude Dynamics Attitude motion is often modeled in a body frame. Let Θ = [γ, θ, ψ] T be the attitude vector in the target frame, and ω = [ω x, ω y, ω z ] T be the angular rate vector in the body frame. According to our previous work, 7 the rotational motion of a rigid body is governed by I ω + ω Iω = M d + τ (5) 3

where I = diag(i x, I y, I z ) is the moment of inertia matrix, the external disturbance torque vector is M d = [M dx, M dy, M dz ] T, and the control torque vector is τ = [τ x, τ y, τ z ] T. In light of the rotation relationship between the body frame and the target frame, we have the attitude kinematics as follows 1 tan θ sin γ tan θ cos γ Θ = S(Θ)ω = 0 cos γ sin γ ω (6) 0 sec θ sin γ sec θ cos γ Combining Eqs. (5) and (6) yields the attitude motion equations in scalar form as follows γ = ω x + ω y tan θ sin γ + ω z tan θ cos γ θ = ω y cos γ ω z sin γ ψ = ω y sin γ sec θ + ω z cos γ sec θ ω x = I y I z ω y ω z + M dx + τ x I x I x I x ω y = I z I x ω x ω z + M dy + τ y I y I y I y ω z = I x I y ω x ω y + M dz + τ z I z I z I z (7) Eq. (4) and Eq. (7) mathematically describe the motion of the spacecraft during lunar landing. The control aim is to make the spacecraft accurately landing on a preselected site in a vertical manner. G&C CONTROL SCHEME Terminal Attitude-Constrained Guidance Law During the terminal landing phase, the height of the spacecraft varies from about 2000m to 0m, which is much less than the radius of the moon, hence the term relating to the moon s gravity in Eq. µ (2) is assumed to be a constant, i.e., R MO (R +r 3 MO + r) g,. Consequently, the translational motion Eq. (2) can be rewritten as ṙ = v, v = g + F /m = g + a (8) Then the optimal guidance law is the solution of a standard optimal control problem with the performance index J = 1 2 tf subject to the dynamics Eq. (8) and the boundary conditions as follows t 0 a T adt (9) r(t = t 0 ) = r 0 ; v(t = t 0 ) = v 0 ; r(t = t f ) = r tf ; v(t = t f ) = v tf (10) This problem is well worked out in Reference 8, in which an optimal feedback guidance law with an explicit form is presented, however, the terminal attitude constraints related to control variable a are not considered. Actually, in the conventional optimal control theory, the terminal values of control variable cannot be constrained. 4

In order to constrain the terminal attitude along to a given direction, we make a reasonable assumption that the commanded acceleration vector is continuous and differentiable. Then a virtual variable u = ȧ is introduced thereby a new extended system ṙ = v, v = g + a, ȧ = u (11) Consider the performance index J = t f 2 ut udt to be minimized, which means minimum variation of the commanded accelerations. Then the following Hamiltonian function is constructed: t 0 1 H = 1 2 ut u + p T r v + p T v (g + a) + p T a u (12) where p r, p v and p a, respectively, are the costate vectors related to the position, velocity and acceleration vectors. The boundary conditions contain Eq. (10) and a(t = t 0 ) = a 0 ; a(t = t f ) = a tf (13) where a 0 relates to the current thrust force and the attitude, and a tf depends on the terminal attitude, thus the terminal attitude constraints are involved in the optimal control problem. In accordance with the classic optimal theory, the costate equations are derived as ṗ r = H r = 0, ṗ v = H v = p r, ṗ a = H a = p v, 0 = H u = u + p a (14) It is obvious that p r (t) is constant, i.e., p r (t) = p rf, where the subscript f represents the final value of corresponding variable. Then we have p v (t f ) p v (t) = p vf p v (t) = tf t p rf dτ = p rf (t f t) = p rf t go i.e., p v (t) = p vf + p rf t go (15) where t go = t f t is time-to-go. Similarly, we have p a (t) = p af + p vf t go + 1 2 p rf t 2 go (16) u = p af p vf t go 1 2 p rf t 2 go (17) Note that the terminal value of costate vectors are denoted by p rf, p vf and p af. Substituting Eq. (17) into the extended system Eq. (11) with complicated computations yields the following equations r = r f v f t go + 1 2 (g + a f ) t 2 go + 1 6 p af t 3 go + 1 24 p vf t 4 go + 1 120 p rf t 5 go v = v f (g + a f ) t go 1 2 p af t 2 go 1 6 p vf t 3 go 1 24 p rf t 4 go a = a f + p af t go + 1 2 p vf t 2 go + 1 6 p rf t 3 go (18) 5

Reconsidering the boundary conditions Eq. (10) and Eq. (13) in Eq. (18) we can obtain the expressions of p rf, p vf and p af, which are omitted here for the sake of paper length. Combining these expressions and Eq. (17) yields the optimal virtual control u = [ 60 (r f r) 60v f t go + 36(v f v)t go 6 (g + a f ) t 2 go + 9 (a f a) t 2 go] /t 3 go (19) Without loss of generalities, the desired landing site locates on the origin of the target frame thereby r f = 0, v f = 0. Moreover, the terminal attitude is constrained to be vertical, hence, we can assume a f = g, which guarantees not only the terminal constrained-attitude but also the minimal impact force are achieved. Then the optimal guidance law with a simple form is obtained from Eq. (19) as follows u = [ 60r 36vt go + 9 (a f a) t 2 go] /t 3 go (20) The time-to-go is determined by the transversality condition J t go = H = 0, where t f is free, J is the optimal value of J. Substituting Eq. (20) and the solved costate vectors p r (t), p v (t), p a (t) into Eq. (12) renders an algebraic equation about t go as follows after complicated algebraic computation. 0 =3600r T r + 2880r T vt go + [ 576v T v 720r T (g + a) 1080r T (a f a) ] t 2 go [ 384v T (g + a) + 528v T (a f a) ] t 3 go [ ] 72(a a f ) T (g + a) 81(a f a) T (a f a) t 4 go (21) The time-to-go is determined by the minimum real positive solution of Eq. (21). According to the relationship between the virtual control and the commanded acceleration, we have t a = udt; F = ma (22) t 0 From Eq. (3), the commanded main thrust force is and the commanded attitude is given as Attitude Control Scheme θ c = arcsin ( F z F F = F (23) ) ( ; ψ c = arcsin F y F cos(θ c ) The aim of attitude controller is to accurately track the commanded attitude Θ c = [γ c, θ c, ψ c ] T, where θ c and ψ c are provided by guidance law. Note that the terminal roll angle is given by the terminal attitude constraint, hence, we should regulate the roll angle to a given value, and this is a classic set-point control problem. Consider the rotational dynamics Eqs. (5) and (6) we have Θ = Ṡ(Θ)ω + S(Θ) ω = Ṡ(Θ)S 1 (Θ) Θ + S(Θ) [ I 1 S 1 (Θ) Θ IS 1 (Θ) Θ ] + I 1 M d + I 1 τ = Ṡ(Θ)S 1 (Θ) Θ S(Θ)I 1 S 1 (Θ) Θ IS 1 (Θ) Θ + S(Θ)I 1 M d + S(Θ)I 1 τ = G( Θ, Θ, M d ) + S(Θ)I 1 τ ) (24) (25) 6

Let e = Θ c Θ be the attitude tracking error, thus the error system becomes ë = Θ c G( Θ, Θ, M d ) S(Θ)I 1 τ (26) Then we define a time mapping function Γ(t) = Θ c G( Θ, Θ, M d ) which is defined as a generalized disturbance. 7 In any feasible flight envelop, a reasonable assumption can be made that the generalized disturbance is continuous and differentiable. Define e 1 = e and e 2 = ė 1 = ė, then the error system is written as ė 1 = e 2 ; ė 2 = Γ S(Θ)I 1 τ (27) To reconstruct the generalized disturbance, a finite-time convergent extended state observer proposed by B. Guo and Z. Zhao 9 is utilized here. ê 1 = ê 2 κ 1 sig β 1 (ê 1 e) ê 2 = ˆΓ κ 2 sig β 2 (ê 1 e) S(Θ)I 1 τ ˆΓ = κ 3 sig β 3 (ê 1 e) (28) where κ 1, κ 2 and κ 3 are the diagonal gain matrix to be designed, sig β (x) represents sgn(x) x β, especially, when x = [x 1, x 2,..., x n ] T R n, there has sig β (x) = [sgn(x 1 ) x 1 β, sgn(x 2 ) x 2 β,..., sgn(x n ) x n β] T Note that sgn( ) is a standard signum function. In light of Theorem 2.2 of Reference 9, given appropriate observer gains and β 1 (2/3, 1), β 2 = 2β 1 1, and β 3 = 3β 1 2, the extended state observer Eq. (28) will reconstructed the error state and the generalized disturbance, i.e., ê 1 e < δ 1, ê 2 ė < δ 2 ˆΓ Γ < δ 3, t > T κ (29) where δ 1, δ 2, δ 3 and T κ are sufficiently small positive numbers, respectively. On the basis of the observation signals provided by Eq. (28), an improved sliding mode controller is developed. In general, define a sliding mode s = ė + Λe (30) where Λ is a diagonal positive-definite parameter matrix to be designed. The desired control torques are to make s small enough such that e and ė are bounded in a small region therefore accurately tracking performance. From the definition Eq. (30), there have following inequalities hold. 10 e s, ė s (31) σ min (Λ) Differentiating Eq. (30) with respect to time along the system Eq. (26) yields Then the ideal control torque vector is designed as ṡ = ë + Λė = Γ S(Θ)I 1 τ + Λ(s Λe) (32) τ = IS 1 (Θ) ( Γ Λ 2 e + Ks ) (33) 7

Substituting Eq. (33) into Eq. (32) yields the closed-loop dynamics ṡ + (K Λ)s = 0 (34) Given K Λ > 0 the sliding mode surface s will converge to origin in an exponential manner. However, Γ and ė cannot be directly obtained, thus the ideal controller Eq. (33) is hard to be implemented in real engineering. A practical way is to employ the observation signals provided by Eq. (28), thus we define a new sliding mode surface s = ê 2 + Λe = ė + ẽ 2 + Λe (35) Note that ė = ê 2 ẽ 2 and ê 2 ė = ẽ 2 < δ 2 ( t > T κ ). Again, differentiating Eq. (35) with respect to time along the observer dynamics Eq. (28 renders ṡ = ê 2 + Λė = ˆΓ κ 2 sig β 2 (ê 1 e) S(Θ)I 1 τ + Λ(s Λe ẽ 2 ) (36) Thus, we can design the controller as follows ) τ = IS 1 (Θ) (ˆΓ Λ 2 e + Ks (37) Consequently, the closed-loop dynamics by substituting Eq. (37) to Eq. (36) is given as ṡ + (K Λ)s = κ 2 sig β 2 (ê 1 e) Λẽ 2 (38) Since ê 1 e < δ 1 ( t > T κ ), then the right hand of Eq. (38) satisfies κ 2 sig β 2 (ê 1 e) Λẽ 2 with a small positive number. Given K Λ > 0, there have s Further, from Eq. (35) the following equalities hold., ṡ (39) σ min (K Λ) e s ẽ 2 σ min (Λ) s + ẽ 2 σ min (Λ) σ min (K Λ)σ min (Λ) + δ 2 σ min (Λ) (40) ė s ẽ 2 s + ẽ 2 σ min (K Λ) + δ 2 (41) This implies attitude tracking errors will ultimately uniformly converge into a hyper-ball whose radius can be minimized by maximizing the minimum singularities of Λ and K Λ. SIMULATION RESULTS To validate the proposed method in the last section, a numerical simulation is conducted in this section. The parameters of the lunar spacecraft are same to Reference 6: the inertial matrix I = diag(100, 80, 80)kg m 2, the total mass m = 100kg, the moon s gravitational constant µ = 4902.75km 3 /s 2, and the average radius of the moon R MO = 1738km. The initial states 8

of the spacecraft in Oxyz are given by r 0 = [1500, 2000, 2000] T m, v 0 = [ 10, 15, 20] T m/s, Θ 0 = [π/4, π/4, π/4] T rad, and ω 0 = [0, 0, 0] T rad/s. The extended state observer Eq. (28) parameters are set as: β 1 = 0.7, β 2 = 2β 1 2 = 0.4, β 3 = 3β 1 2 = 0.1, κ 1 = diag(20, 20, 20), κ 2 = diag(40, 40, 40), κ 3 = diag(100, 100, 100). The sliding mode controller Eq. (37) is employed with the parameters that Λ = diag(1, 1, 1) and K = diag(6, 8, 6). To demonstrate the robustness of the proposed method, the disturbance acceleration A d and torque M d are considered in simulations: where ω m is the angular rate of the moon, and A d = A 0 + A s sin(ω m t) + A c cos(ω m t)m/s 2 M d = M 0 + M s sin(ω m t) + M c cos(ω m t)n m A 0 = [0.001, 0.0015, 0.0005] T, A s = [ 0.002, 0.001, 0.002] T, A c = [0.003, 0.002, 0.001] T M 0 = [0.002, 0.0005, 0.00015] T, M s = [0.001, 0.0015, 0.001] T, M c = [ 0.003, 0.003, 0.002] T The normal case and disturbing case are considered in the simulation experiments. In such two cases, the magnitudes of terminal position and velocity vectors, as well as the values of terminal attitude are presented in Table 1, which reveals the almost same landing accuracies in two cases are achieved by the proposed method. It is to note that the terminal attitude does not achieve the ideal zero vector. The reason is that, the guidance law is cutoff before the terminal site in order to avoid the problem aroused by t go = 0. Table 1. Terminal Values of Normal and Disturbing Cases Case r f m v f m/s θ f deg ψ f deg γ f deg Normal 4.79e-5 0.0017 1.65 4.62 1.54 Disturbing 1.72e-4 0.0026 1.68 4.75-1.54 Figure 2. Position History Figure 3. Velocity History Figures 2-8 present the simulation results in the disturbing case. Shown as Figures 2 and 3, the spacecraft accurately reached to the target point in 104.33 seconds. The commanded main thrust demonstrated by Figure 7 is smooth enough. Meanwhile, the terminal attitude is nearly vertical to 9

Figure 4. Pitch Angle History Figure 5. Yaw Angle History Figure 6. Roll Angle Figure 7. Commanded Acceleration F/m the ground surface around the target point, and this is illustrated by Figures 4, 5 and 6. Although the guidance law derived from a simplified motion equation in which external disturbances are not involved, high performance is still achieved in practical model Eq. (4) with additive acceleration disturbances since of the nature of closed-loop guidance. The simulation results validate the proposed guidance scheme is sufficiently effective to constrain the terminal attitudes. The effectiveness Figure 8. Control Torque τ x, τ y, τ z of the improved sliding mode controller is shown by Figures 4-8, in which, the attitude controller can perfectly tracking the commanded attitude, meanwhile, the magnitudes of control torque are bounded in 20Nm (shown in Figure 8). The excellent attitude tracking capacities of the proposed method guarantee the lunar spacecraft accurately landing on the preselected site on the moon. 10

CONCLUSION In this paper, to address the pin-point landing problem during the lunar exploration, we proposed a G&C scheme which consists of a terminal attitude-constrained guidance strategy and an improved sliding mode controller. The differential signal of the commanded acceleration is taken as a virtual input, then the commanded acceleration relating to the attitude becomes an extended state. Thus the terminal attitude constraints can be handled as a state constraint in the conventional optimal control framework, rendering an optimal feedback guidance law. Then we design an improve sliding mode controller with a disturbance compensation provided by a finite-time convergent extended state observer. Finally, the convincing simulation results indicate that the proposed method is effective while dealing with various disturbances. REFERENCES [1] F. V. Bennett, Apollo experience report-mission planning for lunar module descent and ascent, NASA TechReport, 1972. [2] X.-L. Liu, G.-R. Duan, and K.-L. Teo, Optimal soft landing control for moon lander, Automatica, Vol. 44, 2008, pp. 1097 1103. [3] J. Zhou, K. L. Teo, D. Zhou, and G. Zhao, Nonlinear optimal feedback control for lunar module soft landing, Journal of Global Opimization, Vol. 52, 2012, pp. 211 227. [4] C. T. Chomel and R. H. Bishop, Analytical Lunar Descent Guidance Algorithm, Journal of Guidance, Control, and Dynamics, Vol. 32, No. 3, 2009, pp. 915 926. [5] D. Wang, X. Huang, and Y. Guan, GNC system scheme for lunar soft landing spacecraft, Advances in Space Research, Vol. 42, 2008, pp. 379 385. [6] F. Zhang and G.-R. Duan, Integrated translational and rotational control for the terminal landing phase of a lunar module, Aerospace Science and Technology, Vol. 27, 2013, pp. 112 126. [7] D. Zhao, Y. Wang, and L. Liu, Robust Fault-tolerant Control of Launch Vehicle via GPI Observer and Integral Sliding Mode Control, Asian Journal of Control, Vol. 15, No. 2, 2013, pp. 614 623. [8] Y. Guo, M. Hawkins, and B. Wie, Waypoint-Optimized Zero-Effort-Miss/Zero-Effort-Velocity Feedback Guidance for Mars Landing, Journal of Guidance, Control, and Dynamics, Vol. 36, No. 3, 2013, pp. 799 809. [9] B.-Z. Guo and Z.-l. Zhao, On the convergence of an extended state observer for nonlinear systems with uncertainty, Systems & Control Letters, Vol. 60, 2011, pp. 420 430. [10] F. Lewis, S. Jagannathan, and A. Yesildirek, Neural Network Control of Robot Manipulators and Nonlinear Systems. London: Taylor and Francis, 1999. 11