Passive Fault-Tolerant Control of an Underactuated Re-Entry Capsule

AbstrAct: The design of a passive fault-tolerant control for an underactuated re-entry capsule is considered in this paper; however, the control input of the capsule is failed. At first, kinematics and dynamics of the capsule are studied and modeled, and adaptive control law is used to design a passive controller for the control of the capsule. The guidance law for the capsule is designed based on the guidance law which is used in Apollo. A simulation is performed based on Apollo capsule in order to assess the controller. The result shows good control authority of the controller in the presence of failure in roll and yaw control channels. It is also shown that the guidance law is not credible in the presence of yaw channel input failure.

Passive Fault-Tolerant Control of an Underactuated Re-Entry Capsule (DoF).These systems are complicated, and commonly control methods are not usable for compensation.So the control of these systems is seriously faced with critical challenges (Crouch 1984).
In this paper, the control problem of the underactuated spacecraft in the atmospheric reentry phase is studied.It is aimed to study the fault-tolerant control problem of spacecraft attitude control in the re-entry phase.
Commonly, FTCS are summarized into 2 categories: passive (PFTCS) and active (AFTCS) (Jiang and Yu 2012).In PFTCS, controllers are designed to be robust against a class of presumed faults and so they have a fixed structure.This approach does not need FDD subsystem and does not have controller reconfiguration.A list of potential defects likely already prepared at the design stage as well as failure modes and normal operating conditions are considered in the design phase of this approach.In other words, the passive controller is a controller with a fixed structure, so it can be used for limited situations.In AFTCS, the system component failures are actively controlled by reconfiguring control actions so that the stability and acceptable performance of the entire system can be maintained.Therefore, the main purpose of a FTCS is to design a controller to ensure stability and performance in a good level.This ability exists not only when all control subsystems are in healthy situation, but also in cases when there are failures in control system components (sensors, actuators, etc.).
Statistical studies have shown that about 24% of crashes in military and civilian spacecraft attitude control systems have been related to thruster defects (Tafazoli 2009).In the literature, a survey of fault-tolerant control of the spacecraft has founded numerous papers in orbital motion phase, but there is no report for the re-entry one.Next, there is a review of some of them.
An underactuated spacecraft was stabilized by using 2 momentum wheel actuators (Krishnan et al. 1995), and a proposed control law reduced the control effort required for a non-smooth time invariant feedback control of an underactuated spacecraft (Tsiotras and Luo 1997).
In another study, it was found a sliding mode control designed for stabilization of the angular velocity of a rigid body.The system is supposed to have only 2 control torques and to be subjected to external disturbances (Floquet et al. 2000).In Eshaghi and Wang (2001), Lyapunov direct method was used to control and stabilize a rigid satellite, and one of the input channels has lost their control.This method does not need to identify the failure and switching to the new control law.Morin (1996) showed that most proposed controls in the past are not robust to errors in actuators' location.This author proposed a robust asymptotically-stabilizing feedback control law.
An adaptive asymptotic stabilization of angular velocities of a rigid body with 2 control inputs in the presence of uncertainty in dynamic parameters was presented in Wang and Tao (2007).Hamiltonian control approach for the stabilization of a rigid body system that is controlled by 2 torques was presented by Aguilar-Ibanez et al. (2008).In this stabilization, the closedloop system was forced to be globally asymptotically-stable by solving a feasible matching condition.
In Zheng and Ge (2008), a backstepping control method was used to stabilize an underactuated spacecraft which has only 2 control inputs.In Bajodah (2009), it was presented a novel concept of feedback linearization which is introduced for smooth asymptotic stabilization of underactuated spacecraft equipped with 1 or 2 degrees of actuation, and the concept is based on generalized inversion.
In the study of Pong (2009), a Model Predictive Control (MPC) was proposed as a control technique to solve thruster failure recovery.A method of online MPC was described, implemented, and tested on the Experimental Satellites (SPHERES) testbed at the Massachusetts Institute of Technology (MIT) and on the International Space Station (ISS).These results show that the proposed control improved the performance of underactuated spacecraft.Jin and Xu (2010) presented the angular velocity stabilization and attitude stabilization control for an underactuated spacecraft by using 2 single gimbal control moment gyros (SGCMGs) as actuators.In Wang et al. (2013), sufficient and necessary condition of controllability of underactuated spacecraft was illustrated on the basis of Lie Algebra Rank Condition (LARC).
In Mirshams et al. (2014), angular and 3-axis stabilization control laws were designed for the case of 2 existing thrusters by Lyapunov method and LaSalle invariant theorem.Sliding mode control was used for passive fault-tolerant control of a flexible spacecraft with faulty thrusters.Tube-based MPC approach was used in attitude control of an underactuated spacecraft by Mirshams and Khosrojerdi (2016).
The rest of this paper is organized as follows: the next section presents kinematics and dynamics modelling of the re-entry capsule.In the subsequent section, guidance law of the re-entry capsule is presented.Then, passive fault-tolerant control based on adaptive control method is illustrated.The simulation results and its analysis are provided in the section "Simulation" and, fi nally, conclusions and recommendations are presented.

ModELLInG oF tHE rE-EntrY cAPSuLE
Kinematics and dynamics of the re-entry capsule are studied in this section, and the relative equations are derived.

KINeMAtIcs
S BE is the position vector representing the center of body coordinate with respect to the Earth center Earth fi xed (ECEF) coordinate system.Th e velocity vector is derived by diff erentiating the position vector: Th e expression of Eq. 2 is carried out in the geographic coordinate, so it can be shown as follows: where D E is the derivative operator and v E B is the velocity of the body coordinate, both in relation to the ECEF coordinate system.
By using the well-known operator equation acting on the position vector, Eq. 1 is rewritten as: where D G is the derivative operator in the geographic coordinate (G), i.e. the origin at the center of mass of the spacecraft , x-axis toward the North Pole, y-axis toward the east, and z-axis toward the Earth's center (Fig. 1); ω GE is the angular velocity of the geographic coordinate relative to the ECEF coordinate system.
Figure 1.Geographic coordinate system with its vectors g 1 , g 2 , and g 3 .
Green wi ch M er id ia n E q u a t o r where h represents the distance of the capsule from the Earth; R e is the Earth's radius, thus: Th e velocity of the capsule with respect to ECEF in the wind coordinate system, i.e. a coordinate system which is defi ned in relation to the craft 's velocity V, is: where W is the wind coordinate.
Using a transformation matrix, Eq. 7 can be expressed in the geographic coordinate as: where T G W is the transformation matrix from wind to Geographic coordinate.
Th e angular velocity of the geographic coordinate with respect to ECEF can be represented by: where λ and l are latitude and longitude of the capsule position, respectively; {y 2 } E is the second unit vector of ECEF coordinate; {x 3 } l is the third unit vector of the Earth's center inertial (ECI) coordinate. (1) Passive Fault-Tolerant Control of an Underactuated Re-Entry Capsule Th e angular velocity can be expressed in the geographic coordinate by transformation matrixes: In addition: where T G E and T G l are transformation matrixes from ECEF and ECI to the geographic coordinate, respectively.
By substituting Eqs.6 -10 in Eq. 3, the following relation can be derived for kinematics of the re-entry capsule: where: {g 3 } G is the third unit vector of the geographic coordinate.where S BI represents the position of the center in body coordinate system with respect to the ECI coordinate: where S EI is the position of the center in ECEF coordinate system with respect to the ECI coordinate system.Since the centers of ECEF and ECI are coincident, one has: By substituting Eq. 15 in Eq. 13, and using vector derivative rules, Th us, the derivative is 0: Equation 17 can be rewritten as: On the other hand, the summation of external exerted forces is: where: f a , f g , and f u are aerodynamic force, gravity force, and motor force, respectively.In this modelling, f u is considering to be 0.
The aerodynamic forces are commonly resolved into 2 components: drag (D) is the force component parallel to the direction of the relative motion, and lift (L) is the force component perpendicular to the direction of the relative motion.Th en, these forces in the wind coordinate system are calculated according to Roskam (2001): where ω BI is the angular velocity of the body coordinate with respect to the inertial coordinate: Th us, Eq. 25 can be rewritten as follows: Th e above equation can be explained in the wind coordinate system: where and and the angular velocity of wind coordinate with respect to ECEF is defi ned as: where ø is the roll angle; γ is the path angle; ξ is the heading angle; w 1 is the fi rst unit vector of the wind coordinate system; v 2 is the second unit vector of the velocity coordinate system; z 3 is the third unit vector of the body coordinate system.By substituting Eq. 29 in Eq. 28, the translational dynamic equation of the re-entry capsule can be expressed as follows: So Eq. 32 is rewritten as follows: By substituting Eq. 34 in Eq. 31: Th is equation can be explained in the body coordinate: Th e angular velocity of the geographic coordinate with respect to the inertial coordinate is obtained as follows: And M is the summation of external exerted moments: where M a is the aerodynamic moment; M u is the control moment.
Th e aerodynamic moments for pitch, roll, and yaw channels are calculated by (Roskam 2001): where C l , C m and C n are roll and pitch, and yaw moment coefficients, respectively, which can be calculated by the following formulas: Since the capsule has no wings and tail, the coeffi cients are simplifi ed as follows: Passive Fault-Tolerant Control of an Underactuated Re-Entry Capsule

GuIdAncE LAW oF tHE rE-EntrY
A GNC system is formed by 3 main subsystems: guidance, navigation, and control.Th e task of the navigation is to process the sensor outputs and produce an estimation of the attitude and position of the spacecraft .Guidance and control functions work in parallel.Th e guidance unit aft er receiving an estimated value of attitude and position from the navigation one will produce reference signals for the control unit, which must trace the guidance signals.
One of the common shapes for re-entry vehicles is the blunt bodies (similar to the Apollo spacecraft).This configuration has small and nearly constant lift-to-drag ratio (L/D) and can be considered and designed to fly at a constant angle of attack.
In these vehicles, the tracking of the trajectories has been controlled by modulation of the bank angle.Altitude and direction of the vehicle are controlled by the bank angle magnitude and sign, respectively.A conceptual schematics of this method is illustrated in Fig. 2. Modulation of bank angle, energy, and range for the target of the vehicle are managed in order to achieve the precise landing.
of bank angle modulation to manage energy and range for the target, rigorously tested, verifi ed, and fl own.
In the re-entry of the capsule, the target point is the recovery position where parachute must be opened.Its guidance scheme is divided into diff erent phases, which are selected depending on the current state of the vehicle (Guerreiro 2011): • Initial Roll.

•
Targeting.The first phase of the guidance scheme is the Initial Roll.In this phase, the lift vector is rotated downwards by banking the capsule for 180 degrees in order to maximize the force that pulls down the vehicle and ensures that it is captured.This phase lasts until a maximum altitude rate is reached, at a point where a significant amount of energy has been dissipated in order to avoid skipping, while enough energy remains so that the vehicle is still able to reach the final target.
The following optional phases -Constant Drag, Up Control, and Ballistic -allow the additional dissipation of energy and correction of the range until the predicted miss (that is, the distance between the landing site and the predicted landing point) is less than a pre-specific value.These are only used to adjust the vehicle's position and energy to the values required for the trigger of the targeting phase.In the Constant Drag phase, a constant drag value is tracked.The Up Control and Ballistic phases allow performing a skip-entry, where the Reentry Vehicle is controlled in an upward trajectory and projected out of the atmosphere with a certain velocity as well as flight path angle.The Ballistic phase is activated until the RV returns again to the atmospheric environment, expectedly at a satisfactory range from the target.
When the RV is at approximately the desired range from the target and has an acceptable energy level, the Targeting phase is activated.In this final phase, the RV aims at the targeting position.It is done by tracking a reference trajectory generated online, using a gain-scheduled PID controller whose gains are the derivatives of range with respect to altitude rate F 2 (V), drag F 1 (V), and L/D F 3 (V), also stored on-board.The expression used to predict the range in this final phase is (Guerreiro 2011):

Li vector
Li vector

Initial bank angle (41)
A re-entry guidance algorithm must be consisted of (Bairstow 2006): • Capture into the atmosphere.

•
Manage energy by removing excess of velocity through drag management.
• Steer to a target.In this paper, the algorithm will be based on the Apollo re-entry guidance algorithm, in which it is used the principle (42) where S is the range; h represents altitude; D is the drag; the subscripts "pred" and "ref " stand for "predicted" and "reference".Note that the reference values are all taken from the entry profi le stored on-board and are mapped with respect to the velocity V. Th e predicted range thus computed is plugged into the computation of the desired L/D ratio (Guerreiro 2011): robust controls; model predictive control; variable structure and sliding mode control; linear quadratic; pseudo-inverse/ control mixer; gain scheduling/linear parameter varying; feedback linearization or dynamic inversion; generalized internal model control; (model reference) adaptive control/ model following; eigenstructure assignment; multiple-model; intelligent control using expert systems, neural networks, fuzzy logic, and learning methodologies.Four main criteria are used for classification of these methods: (1) mathematical design tools; (2) design approaches; (3) reconfiguration mechanisms; and (4) type of systems to be dealt with.
Selecting the appropriate method among the proposed ones depends on the error compensation methods in the presence of failures in real time.For this purpose, a new control structure configuration must be proactive in terms of bondage with the least try and error and the selected method as tangible even if the answer is not optimized.
One of the proposed methods is the adaptive control, which is robust to dynamic models and parameter uncertainties.This method, according to the classification, is considered as PFTCS.In order to control the defective dynamic system, there is no need for the FDD module, but it has a small working range and cannot cover any type of crashes.The important fact about this method is that, despite the changes in the parameters of the dynamic coefficients, the presence of update law for control coefficients ensures the pursuit of sustainability and tracking guidance law, so spacecraft can continue the mission.Therefore, it can be anticipated that, in this method, there is no fault detector unit, but it has a good robustness against the malfunctioning of the actuators.
Th e revision and selection of other methods of passive and active fault-tolerant control are presented next.In this paper, Model Reference Adaptive Control (MRAC, also known as Model Reference Adaptive System -MRAS) is considered.Th is approach is one of the adaptive control methods.Th e general concept behind MRAC is to construct a closed-loop dynamic system with updatable parameters that change the system's response.Th e response of the actual system is compared with a response of the reference model.
Th e control parameters are updated according to an adaptive law based on these errors.Th e purpose of the adaptive law is: the undefi ned parameters should converge to values that make the dynamic system reaction match the response of the reference model.
In the Apollo guidance scheme, the Lateral Logic function is called to ensure the lateral guidance control in all phases.Th e lateral control is performed by reacting the horizontal component of L about the vertical axis, either to the right or to the left , which therefore takes the RV into any of the 2 directions via determination of a bank angle control command.Th is is done by the expression (Guerreiro 2011): where K 1 and K 2 are indicators for switching the bank angle sign; L/D max is the maximum lift -to-drag ratio.
Note that this maximum value is a constant and that the argument of the arccos function comes from the result of Eq. 43 in the fi nal phase of the entry guidance algorithm.Th e orientation of the lift reaction, that is, the values of K 1 and K 2 , is determined based on the predicted cross range miss: if it exceeds a predetermined fraction of the vehicle cross range capability, the direction of the lift vector is changed.

PASSIVE FAuLt-toLErAnt controL dESIGn
Due to real-time requirements and the nature of the dynamics of the system, there is usually a very limited amount of time available to carry out the post-fault model construction and control reconfiguration actions.The tradeoff between various design objectives and the interaction between different subsystems have to be carried out online in real time.However, for AFTCS, there are several additional challenges besides those in the conventional control systems, such as redundancy management, integration of FDD and reconfigurable controller, safety and reliability design purposes.The existing fault-tolerant control methods are categorized into one of the following manners: H∞ and other (43) (44) For the design of the MRAC controller, the following steps must be taken (Astrom and Wittenmark 1994): 1. Determine the structure of the controller.
2. Get the error equation.
3. Select an appropriate Lyapunov function that tend the error to 0 and provide update law for parameters.Th e dynamic model of the system is as follows: By substituting Eq. 54 in Eq. 53, one has: where  is the system matrix, b is the input matrix and u is the input vector.
And the reference model is considered as: Now, the Lyapunov candidate is selected as follows: where L is the adaptive law gain that must be selected.where Ѳ 1 and Ѳ 2 are gain controller; u r is the reference signal.
Th e goal of the controller is to track the reference signal, then: Th e ideal value of gains is obtained as: and, in order to minimize the error, using error equation, there are: Th e error between the ideal and estimated gains is introduced as follows:

SIMuLAtIon
Th is study refers to a re-entry space vehicle without wings, similarly to Apollo in features, as it is intended (Fig. 4).Hence, to carry out a simulation, relevant information of this spacecraft is used, and some of its mass-volumetric features have been used in the simulation shown in Table 1.In order to simulate and illustrate the operation of the controller, various scenarios will be considered to represent the behavior of the closed-loop system in the state of being healthy and in the malfunctioning of the actuators.These scenarios are in accordance with Table 2.
Th e initial conditions for the simulation are considered as:   have been brought up together to be easily compared with each other.As can be observed, the lack of control forces on the pitch or roll channels does not affect the actual path, and the bank angle is satisfied by the other control input.Therefore, the pursuit of latitude and longitude coordinates of the path does not change.However, with the failure on the yaw channel actuators, the ability to prosecute the guidance law is lost and caused deviation from the desired path.
In order to analyze the results, the equation of spacecraft re-entry mechanism is determined.Assuming that the cross momentum of inertia is 0, Euler equation (Eq.36) can be written as: where m c i is the control input; m A i is aerodynamic moment for each channel.
According to Eqs. 39 and 41, as well as Table 3, Eq.62 is rewritten as: As it is seen, in the yaw channel, C n β is very low and β (side slip angle) is trimmed to 0, so the aerodynamic moment does not have any eff ect, and then the failure of actuators in this channel has most eff ort on attitude control of the spacecraft .However, if the actuators of roll and pitch channels are corrupted, the aerodynamic moment is still exerted.Besides, according to dependency of aerodynamic moment to α and β angles, as well as coupling of angles, the bank angle can be controlled by roll and yaw channels separately.
Simulation results confirm this subject, and the important point of this simulation is: the adaptive control method has this ability in such a way that the actuators' failure, considered as the uncertainty in the input control and other healthy inputs, helps to compensate the existing failure.

concLuSIon
In this paper, it is considered the design of a passive fault-tolerant control of an underactuated re-entry capsule which has lost one of the control inputs.At first, kinematics and dynamics models of the re-entry capsule are obtained.The study illustrated fault-tolerant control manner, and adaptive control as a passive manner is chosen and designed to control the re-entry capsule.The simulation result shows good ability of the designed controller to tolerate input failures in the roll and pitch channels, but it cannot track guidance law if the yaw input control is failed.So it can be concluded that, if the re-entry capsule is underactuated in roll or pitch channels, adaptive control can be considered as a passive fault-tolerant control.
of the capsule in the geographic coordinate can be defi ned as follows: R is defi ned as: trANsLAtIoNAL MotIoN dyNAMIcs From the Newton's second law, one has: where a B I is the acceleration and m B is mass of body Th e acceleration of the capsule is derived by the second derivative of the position: where q -= 1/2ρv 2 ,C D = C D 0 + C D α α and C L = C L 0 + C L α α.The gravity force in the geographic coordinate can be calculated by: .21 and 22 in Eq. 12, one has:Alikhani A Th e fi rst term of Eq. 25 can be written as: Euler's law is expressed as follows:I BBI is the angular momentum so the fi rst term of Eq. 31 is:
Figure3shows the block diagram of the designed controller.

Figure 3 .
Figure 3. Block diagram of the designed controller.

table 3 .
Stability derivatives parameters of the simulated spacecraft.

table 1 .
Mass-volumetric features of Apollo.