Sliding mode attitude control using thru

Noname manuscript No (will be inserted by the editor) Sliding Mode Attitude Control using Thrusters and Pulse Modulation for the ASTER Mission Bruno Victorino Sarli · Andr´ e Lu´ıs da Silva · Pedro Paglione Received: date / Accepted: date Abstract ASTER, the first Brazilian mission to the deep space, targets the exploration of the triple asteroid system known as 2001 SN263 The mission requires an attitude controller robust and capable of coping with the non-linearities and the uncertainties present during the exploration phase For such requirements, this paper studies the applicability of two controllers, designed based on the sliding mode control (SMC) technique, one of the controllers include an adaptive law used to compensate for the spacecraft’s inertia variation One application is performed where gain scheduling is used for controlling two different phases: exit from tumbling and track a dynamic reference The actuators of the attitude control loops are impulsive thrusters They are activated by pulse width modulation (PWM) or pulse width pulse frequency modulation (PWPFM) Simulation studies, performed in realistic scenarios, show that the SMC can maintain stability and performance when these modulation techniques are used to approximate B.V Sarli Department of Space and Astronautical Science, The Graduate University for Advanced Studies Sagamihara, 252-5210, Japan E-mail: sarli@ac.jaxa.jp A L da Silva Universidade Federal ABC, Centro de Engenharia, Modelagem e Ciˆ encias Sociais Aplicadas Rua Arcuturus 3, CEP 09.606-070, S˜ ao Bernardo Campo, S˜ ao Paulo, Brazil Tel.: +55-11-23206338 E-mail: andreluis.silva@ufabc.edu.br,taurarm@gmail.com P Paglione Aeronautics Engineering Division, Aeronautical Institute of Technology P¸c Marechal Eduardo Gomes 50, CEP 12.228-900, S˜ ao Jos´ e dos Campos, S˜ ao Paulo, Brazil E-mail: paglione@ita.br the continuous commands It is also shown that PWM can provide better performance, but at a higher control cost In this sense, PWPFM is more appropriate with respect to the fuel consumption and activation times Keywords asteroid mission · nonlinear control · attitude · ASTER · sliding mode · PWM · PWPFM Introduction ASTER, the first deep space Brazilian mission, targets the group of asteroids in the system know as 2001 SN263, which is composed by a central body named Alpha and two satellites, the largest one orbiting further from the system’s center is named Beta, while the smaller asteroid orbiting closer to Alpha is named Gamma The objective of this mission is to explore the system, taking measurements and pictures from all three asteroids and, if possible, finalize the mission by touching down Alpha, [3, 7] One of the many elements for the success of the mission is the ability of correct pointing and stabilization of the spacecraft, which is useful for the orientation of instruments and also for the navigation, which is based on a fixed ion-thruster technology; therefore, in order to change the trajectory for translation within the system, the attitude of the spacecraft needs to be changed A triple asteroid system is highly non-linear due to the nature of the gravity environment generated by the three bodies, allied with the solar radiation pressure Therefore, a control law based on a linear approximation is not suited, neither for the orbit around the system nor the navigation inside it, because the short maneuver time and the long engine operation Furthermore, it is required from the spacecraft to perform large angle maneuvers during its operation within the asteroid system, such tracking cannot be accurately performed by linear controllers, as demonstrated on figure taken from [6], where the track of a large amplitude angle is attempt, resulting in an oscillatory behaviour with high amplitude By the way, some of the tests (in the case of using a reaction wheel) not generate a steady response in the observed time horizon Fig 1: Closed loop spacecraft’s response to step inputs of high amplitudes, [6] Particularly in this work, only the case where the spacecraft orbits the center of the system will be considered That is, a circular orbit around Alpha The attitude control actuators are small thrusters that can provide an anti-symmetrical setting, which means that the attitude can be controlled without inducing translation The design of a feedback control law is performed using continuous control techniques After, the control thrusters are activated by pulse width modulation (PWM) or pulse width pulse frequency modulation (PWPFM), which allows for an approximation of the continuous thrust calculated by the controller A realistic modelling is performed taking into account thrust levels, activation time (on an off times) and time response of practical control thrusters of small spacecraft The control design technique chosen in this study is the sliding mode control (SMC) This formulation is very appealing specially by its invariance to disturbances and model uncertainties, [11, 2, 10] This characteristic is highly desirable for a mission such as ASTER, where many uncertainties in the model of the asteroid system are present The chosen SM formulation has another attractive feature for the mission: an adaptive law that is of great importance when using ion-thrusters, which consumes propellant for long periods of time, making the inertia matrix vary, [13] The focus of this study is to develop the control algorithm; the precise determination of the moment of inertias’ values was not B.V Sarli et al the objective Therefore, the parameters used in the inertia matrix are hypothetical Each controller makes use of two sets of gains (gain schedule) that are tuned for a specific purpose, the first set is applied to take the spacecraft from tumbling and bring it to an equilibrium position, once the equilibrium is achieved, the second set is used and the controller starts to track a time varying attitude angles The determination of a unique set of parameters for different tasks may be very hard and could not determine a suitable performance for both cases The main contribution of the paper, is to show the feasibility of using continuous nonlinear control techniques to activate the control thrusters, via the PWM or PWPFM in the scenario in question For this purpose, the adoption of a robust feedback control is important, in order to cope with the inherent time delays caused by the modulation, and also uncertainties of the thrusters modelling and inertia matrix Particularly, regarding this aspect of perturbations and uncertainties, this paper shows that the gravity gradient torque perturbations generated by the triple asteroid system, and also the solar radiation pressure, are irrelevant when compared with the former effects As it follows, section presents the equations of motion of the spacecraft Section 3, based in the work of [13], presents the formulation of the non-linear controller for the attitude pointing, discussing the sliding surface and control law addressing the problem of chattering, followed by the formulation of the SM adaptive controller, featuring the same important points as in the previous controller Section outlines the implementation of the designed controllers using PWM and PWPFM The arrangement of the control thruster of this work is shown, real design data obtained from similar spacecraft are also discussed Section presents the simulation results, the continuous and pulsed control actions are evaluated and analysed with a detailed set of performance measures Finally, section presents the conclusions Equations of motion Particularly for a circular orbit, or long orbits, the rotational and translational equations of motion of a spacecraft can be treated independently Specifically for the 2001 SN263 system, the main forces acting there are the gravity of the three asteroids and the solar radiation pressure; the gravity of the Sun, due to the distance, has a very small intensity, which can be considered negligible Figure presents an initial evaluation of the magnitude of the accelerations acting on the spacecraft at Sliding Mode Attitude Control using Thrusters and Pulse Modulation for the ASTER Mission Alpha’s equatorial plane (fixed frame at Alpha’s center of mass (cm) corrected for a central system, [4]) For this work, the rotational motion will not affect the −7 x 10 4.5 Acceleration magnitude [m/s2] Alpha Beta Gamma Sun SRP 3.5 2.5 with c3x being its skew-symmetric matrix c3x =  −c33 c32  c33 −c31 , Rj is the distance between the space−c32 c31 craft’s cm and the asteroid’s cm, J is the inertia matrix, j represents the number of the body: Alpha is 1, Beta and Gamma 3, G is the gravitational constant and Mj is the mass of each asteroid The torque due to the solar radiation can be calculated as: FSRP = PSRP CR ArSun (4) 1.5 0.5 20 40 60 80 100 120 Distance from Alpha [km] 140 160 180 Fig 2: Acceleration induced by the disturbances acting on the spacecraft translational motion, because the gravity force of Alpha around the spacecraft will be constant around the circular orbit and the gravity of Beta, Gamma and the solar radiation pressure will be treated as perturbations that are corrected by the orbit control system That is, the perturbing torque or disturbance considered here will come from the gravity gradient torque generated by the three asteroids and the solar radiation pressure Their perturbing torque, denoted by D, can be calculated as: D = {gc }β + {FSRP }β × h (1) where, β denotes the body frame, gc is the gravity gradient, FSRP is the force due to the solar radiation pressure and h is the vector from the spacecraft’s optical pressure center to its geometrical cross-section center The evaluation of the vector from Alpha’s reference into body frame can be made by using the classical quaternion based rotational matrix, 1-3-2 rotation sequence, with q4 being the scalar quaternion:   − 2(q22 + q32 ) 2(q1 q2 + q3 q4 ) 2(q1 q3 − q2 q4 ) Rb0 =  2(q1 q2 − q3 q4 ) − 2(q12 + q32 ) 2(q2 q3 + q1 q4 )  2(q1 q3 + q2 q4 ) 2(q2 q3 − q1 q4 ) − 2(q12 + q22 ) (2) The perturbing torque due to the gravity gradient can be calculated as, [6]: n {gc }β = j=1 GMj c3x Rj3 j Jc3j (3) where, c3 is the third column of the transformation matrix from the inertial frame to the body-fixed frame where, PSRP is the local solar radiation pressure given by PSRP = PSRP /|rSun | (equipotential sphere) where rSun is the position vector Sun-spacecraft in astronomical units, PSRP = 9.15 N/km2 is the value at AU (Astronomical Unit), CR is the reflectivity coefficient considered fixed at 1.14 [12], and A is the cross-section of the spacecraft, assumed to be equal to m2 It is important to point out that the evolution of Beta and Gamma around Alpha were studied in [1] However, no official model has been derived yet Therefore, the ephemeris model used in this work was calculated using the initial conditions from Fang’s work and propagated using an Adams integrator for 20 days from June 1st of 2019 until June 20th of the same year, [7] The rotational equations of motion for the spacecraft as a rigid body can be derived by Euler’s formulation, having this form: ˙ − Ω × (JΩ) + Tb + D J Ω˙ = −JΩ (5) where, Ω is the angular velocity, Tb is the moment due to the actuators and D is the disturbance torque The ASTER spacecraft will have a large ion-thrust for trajectory control and a cluster of small thrusters for attitude control These small thrusters are arranged in such a way to generate three independent control torques Tx , Ty and Tz around the x, y and z axes of the body, respectively, without inducing translation; then, Tb = [Tx Ty Tz ]T The geometrical arrangement of thrusters and technological data are discussed in section 4.1 It is convenient to evaluate the rotational kinematics in quaternions, rather than in Euler angles, in order to avoid singularities The quaternion formulation is: Q= q2 q3 q4  q q1 ¯ T =  q2  = U sin Q q3 T Φ ¯ T q4 = Q T , q4 = cos Φ (6) ¯ is its vectorial part and where Q is the quaternion, Q q4 is its scalar part, as mentioned before U is the rotation vector and Φ is the rotation angle, that describe B.V Sarli et al a rigid body rotation, according to the Euler’s rotation theorem The derivative of the quaternion is:   −q3 q2 1 ¯˙ = ¯ x Ω + q4 Ω, Q ¯ x =  q3 −q1  Q Q 2 −q2 q1 T¯ q˙4 = − Ω Q (7) With Eq and Eq 7, the final form of the spacecraft’s dynamic model can be obtained, where e means an error and d means a desired behaviour, ˙ =1 Q ¯ ex Ωe + qe4 Ωe , q˙e4 = − ΩeT Q ¯e Q e 2 ˙ − Ω × (JΩ) + Tb + D J Ω˙ = −JΩ 3.1 Sliding mode attitude controller design In the development of a SMC, one needs to define a sliding surface This surface is determined in order to obtain a sliding mode (SM) that satisfy the requirements of some application, in this case, the tracking of a desired attitude After the determination of a sliding surface, a reaching condition shall be described, from which the sliding surface can be reached This condition determines the control law that can make the SM possible and, consequently, the desired behaviour, [11, 2] The switching function is defined as: S = P Qe + Ωe (8) where Ωe = Ω − Ωd is an error angular speed with respect to a desired angular speed Ωd and:      qe1 qd4 qd3 −qd2 −qd1 q1  qe2   −qd3 qd4 qd1 −qd2   q2      Qe =  (9)  qe3  =  qd2 −qd1 qd4 −qd3   q3  qe4 qd1 qd2 qd3 qd4 q4 Equation defines an incremental rotation with respect to a desired attitude Qd The desired angular speed and attitude will be considered in the next section, where the problem of attitude tracking control is developed Non linear attitude controller The SMC design adopted in this work came from reference [13] It was derived for the attitude control of a spacecraft with thrusters The approach concerns a nonlinear controller robust with respect to external disturbances and parametric uncertainties A variation of the robust controller is also developed, with an additional adaptive law to estimate the uncertain inertia matrix The complete evaluation of the control scheme can be found in the reference In the following, a synthesis of this approach is presented This particular discussion gives special attention to the meaning of each design parameter on the controls By explaining the role of each element in the control law, the designer can understand the impact of such parameters when tuning a specific control law for some application One application of such concepts is presented in section 5.1, where the continuous control law is determined in order to generate control actions below the physical limits of the thrusters, while still delivering reasonable performance measures (10) where, P is a positive diagonal matrix When S = 0, the switching surface is obtained The path of the system constrained to this surface is the SM This SM shall represent the desired behaviour, which, in this case, is the tracking of the desired attitude So, during the SM, one should expect that the errors in the quaternion and angular velocity Qe and Ωe tend to zero The reference [13] shows that by using Lyapunov T function of the form Ve Qe = kQe Qe , where k is a positive number, the following property is obtained: T T −ckQe P Qe ≥ V˙ e Qe ≥ −kQe P Qe (11) so, with c > the Lyapunov stability theory proves that the SM is stable, in such a way that the origin of the error dynamics is a stable equilibrium point Thus, the tracking error converges to zero during the sliding mode: Qe , Ωe → (03×1 , 03×1 ) , t → (12) This shows that the sliding surface is capable of satisfy the requirement of attitude tracking In this sense, note the meaning of the positive definite matrix P : from Eq 11 the decaying rate of the Lyapunov function depends on the magnitude of P , so, by increasing the magnitude of the elements of this matrix, one can decrease the convergence time to the origin of the error dynamics Once a stable sliding surface is obtained, it is necessary to establish a control law in order to guarantee that the sliding surface is reached, from any initial condition, such that the SM exists That is, the control shall be determined in order to satisfy a reaching condition, [2] The reaching condition can be specified in a variety of ways The Lyapunov function method is chosen by [13] Using a function of the form Vs = 12 S T JS, it is shown Sliding Mode Attitude Control using Thrusters and Pulse Modulation for the ASTER Mission that exponential stability and robust performance can be obtained with the control torque: Tb = − Ks S + J˙0 Ω − J˙0 S + Ω × (J0 Ω) + ¯ ex Ωe + qe4 Ωe − J0 Ω˙ d + Λs − J0 P Q (13) where element Ks S is a feedback of the switching function, Ks = diag ks1 ks2 ks3 is a × positive definite matrix, Λs is a discontinuous control action, responsible for the generation of the SM:   λs1 ¨ d sgn(si ) Λs =  λs2  , λsi = −csi Q, Ω, Qd , Q˙ d , Q s3 (14) ă d are control amplitudes, the terms csi Q, Ω, Qd , Q˙ d , Q sgn(si ) is the sign function:   1, si > 0, sgn(si ) = 0, si = 0, i = 1, 2,  −1, si < 0, (15) T and S = s1 s2 s3 is the sliding surface The remaining elements in the control torque of Eq 13 comprehend the equivalent control, it is responsible for guaranteeing that the SM trajectories are tangent to the sliding surface, [2] Note that it is a continuous nonlinear function that depends on the dynamics of the plant Special attention shall be paid for the meaning of the control gains Ks = diag ks1 ks2 ks3 and the conă d The amplitude trol amplitudes csi Q, Ω, Qd , Q˙ d , Q of the gains ksi are related to the rate of convergence of the reaching mode, in other words, increasing the magnitude of these gains, the time of convergence to the sliding surface is decreased In other way, the control ă d shall be determined amplitudes csi Q, , Qd , Q˙ d , Q in order to guarantee that the reaching condition is satisfied in the presence of the external disturbance and parametric uncertainties There are a variety of ways for satisfy this, one possibility is consider the worst case scenario with a subsequent substitution of the maximal expected value of each variable, [9] Finally, the chattering problem is originated due to the existence of physical limitations in the implementation of the sign function sgn(si ), some problems are delays, dead zones, hysteresis In order to improve the solution and avoid the problem caused by chattering, the sign function can be replaced by a saturation function; in this way, forcing the system to stay within the limits of the boundary layer |Si | < ε (ε represents a positive small scalar value) and no longer exactly on the sliding surface Such change in the control law will be, of course, followed by a reduction in the accuracy of the desired performance, [10] The saturation function is defined as follows:   1, si > ε (16) sat(si , ε) = sεi |si | < ε  −1, si < ε where ε is a constant that shall be chosen as small as possible 3.2 Sliding mode adaptive attitude controller design The variation spacecraft’s inertia cannot be neglected if a particular long phase consumes fuel continuously Because the calculation of the inertia variation is too complex, a very useful solution is to design an adaptive controller that can compensate for the effect of this variation and respective disturbances In short, the adaptive control problem consists in generating a control law and a parameter vector estimation law, such that the tracking error tends exponentially to zero: Given the desired attitude, quaternion Qd , and angular velocity, Ωd , with some of all the parameters unknown of J; Derive a control law for the thrusters torques and an estimation law for the unknown parameters, such that Q and Ω precisely track Qd and Ωd after the initial adaptation process; Let A be a constant × vector containing the unknown parameters and Aˆ (the time-varying parameter vector estimate); with the error A˜ = Aˆ − A Reference [13] shows that, when maintaining the sliding mode surface defined in Eq 10 and using a new Lyapunov function candidate of the form V = 12 S T JS + 12 A˜T Γ −1 A˜ for the reaching condition, where Γ is a positive diagonal matrix, the global stability of the attitude tracking system is guaranteed, provided by choosing a control torque: 1ˆ ˆ Tb =Ω × JΩ − JP Λ − Ka S ¯ ex Ωe + qe4 Ωe + JˆΩ˙ d + Q (17) where, Ka = diag ka1 ka2 ka3 is a × positive defiT nite matrix and Λ = λ1 λ2 λ3 is a vector of discontinuous functions These terms are analogous to that in Eq 13, but note that the nominal matrix J was changed ˆ by the estimated matrix J In [13], it is shown that this controller also provide ¯ e and Ωe and paramethe convergence of the states Q ˜ ter estimation error A The demonstration involves the Lyapunov stability theory and the Barbalat’s lemma This proves that the adaptive controller can also solve the attitude tracking problem Again, special attention shall be paid for the meaning of the control gains Ka = diag ka1 ka2 ka3 , the control amplitudes ki and the elements of Γ The amplitude of the gains kai are related to the rate of convergence of the reaching mode, in other words, by increasing the magnitude of these gains, the time of convergence to the sliding surface is decreased In other way, the control amplitudes ki shall be determined in order to guarantee that the reaching condition is satisfied in the presence of the disturbance D, which is related to the external perturbations and the uncertainties in the inertia matrix Finally, one shall note that the elements of Γ are related with the convergence rate of the parameter estimation law Regarding chattering, in the same way of the standard controller, the sign function should be changed by a saturation function in order to mitigate its effects B.V Sarli et al from the cm is 0.5 [m] which results in a maximum torque of 0.5[m] × 0.5[N ] + 0.5[m] × 0.5[N ] = 0.5[N m] Another critical issue of pulse modulation is the minimal “on” and “off” times of the thrusters These times strongly influence the quality of the pulsed approximation Theoretically, smaller modulation times generate finer controls However, control hardware have limitations as described in the table Fig 3: Illustration of control thrusters arrangement This scheme repeats in the other sides Control Implementation with Thrusters 4.1 Thrusters arrangement and technology The control design of section assumes that a pure torque can be generated by the attitude control thrusters without inducing translation, this effect can be obtained by using a configuration of anti-symmetric thrusters Figure illustrates one of the many possible configurations that generate the desired torque with no translation Naturally, this scheme shall be repeated for all the axis of rotation, resulting in 12 control thrusters Note that a thruster, in fact, generate a force, the torque results from the respective arm until the spacecraft’s cm In practice, the control torques can assume only three levels: zero, minimum negative and maximum positive The force, in fact, can be only zero or positive along the thruster axis; for a specific control axis, positive torque is generated by choosing one pair of thrusters, negative torque is generated by the other pair On the other hand, a feedback control law assumes continuous torques, but they can be approximated by pulse modulation as described in the section 5.2, according a suggestion made by [5] In order to obtain an appropriate result, the magnitude of the torques are required to be within the limits of the force that can be delivered by typical actuator used in small and light spacecraft like ASTER’s Some examples are described at table Based on nominal thrust values presented on table It is assumed that each thruster can provide 0.5 [N] and their distance Table 1: Attitude control thrusters for small spacecrafts 1N MonoH MR-103C 1N Gas Type Catalyst Catalyst Nominal thrust [N] Pressure [MPa] ISP [s] 0.32 ∼ 0.95 0.22 ∼ 1.02 Nitrogen (N2 ) 0.1 ∼ 1.0 0.69 ∼ 2.75 0.62 ∼ 2.76 1.51 ∼ 1.80 221 (2.41MPa) 20 209 ∼ 224 15 >56 (pulse) 30 40 - -