Angular velocity determination of spinning solar sails using only a sun sensor

Zhai Kun

School of Aerospace Engineering,Tsinghua University,Beijing 100084,China

Angular velocity determination of spinning solar sails using only a sun sensor

Zhai Kun

School of Aerospace Engineering,Tsinghua University,Beijing 100084,China

Condition number;Least-squares method;Solar sail;Spinning;Velocity determination

The direction of the sun is the easiest and most reliable observation vector for a solar sail running in deep space exploration.This paper presents a new method using only raw measurements of the sun direction vector to estimate angular velocity for a spinning solar sail.In cases with a constant spin angular velocity,the estimation equation is formed based on the kinematic model for the apparent motion of the sun direction vector;the least-squares solution is then easily calculated.A performance criterion is defined and used to analyze estimation accuracy.In cases with a variable spin angular velocity,the estimation equation is developed based on the kinematic model for the apparent motion of the sun direction vector and the attitude dynamics equation.Simulation results show that the proposed method can quickly yield high-precision angular velocity estimates that are insensitive to certain measurement noises and modeling errors.

1.Introduction

Unlike spacecraft traveling around the earth,solar sails aimed at exploring deep space have difficulty finding references other than the sun.Spin stabilization with a known angle between spin axis and sun direction is the most feasible flying mode for these solar sails.1,2The first solar sail demonstration spacecraft, ‘IKAROS,” precisely adopted this spin stabilization mode.3,4

To achieve spin stabilization using an active control,the sun incidence angle and angular velocity of the solar sail should be known.A typical attitude determination system(such as that in Table 1 from Ref.3)for solar sails is equipped with a sun sensor and rate gyros.The sun sensor measures the sun direction vector to determine sun incidence angle,and rate gyros are used to determine angular velocity.In their current state of technological development,rate gyros are characterized by low reliability.The probability of failure becomes very high when rate gyros are used in the solar sails with a duty cycle of several decades.So,it is very necessary to seek a gyro-less angular velocity estimation method for spinning solar sails.

In recent years,the rapid development of small,inexpensive spacecraft has led to an increased interest in the search for gyro-less angular velocity estimation methods.5–10Many such methods have been proposed.In Ref.11,methods for gyroless angular velocity estimation are classified as either derivative or estimation approaches.The former requires that the attitude be known,and filters are adopted to suppress thehigh-frequency noise induced by differentiating.The latter applies an estimator directly to the raw measurements to avoid differentiating.Among these methods,at least two noncollinear vectors are often assumed to be observed simultaneously.

Considering the miniaturization requirements of spacecraft,some methods attempt to design a single vector angular velocity determination algorithm;among these,the three-axis magnetometer(TAM)isoftenused.BasedonTAM-only measurement data,a Kalman filter was designed to estimate attitude,attitude rate,and constant disturbance torque based on attitude kinematics and dynamics equations.12Furthermore,a globally self-initializing attitude determination filter was developed in Ref.13.For a spinning spacecraft employing wire booms,a magnetometer-based filter and smoother have been presented for estimating attitude,rate,and boom orientations.14A deterministic magnetometer-only attitude and rate determination(DADMOD)method and a real-time sequential filter(RTSF)method were proposed and used successfully to obtain attitude and angular velocity when an earth radiation budget satellite(ERBS)experienced an in-orbit uncontrolled tumble.15These methods can also be used to estimate rates of rapidly spinning gyro-less spacecraft.16In Ref.17,the RTSF method was improved by directly applying filtering equations to both attitude dynamics and the first-time derivative of measured magnetic field vectors.

Based on only one observation vector,a two-stage angular velocity estimation method has also been presented for spacecraft in high angular velocity scenarios.18Under the assumption that the inertial vector measurement is nearly constant in inertial frame during a short sampling time,the two-stage method consists of a deterministic algorithm and an extended Kalman filter(EKF).This two-stage approach is improved in Ref.19with the deterministic algorithm using a global nonlinear least-squares solver to determine the unknown angular momentum component along the magnetic field direction and an EKF to estimate the attitude rate vector and corrections to five of the six inertia matrix elements.The EKF proposed in Ref.20uses analytic propagation to achieve better accuracy and reduce computational burden.It has been found that the angular velocity estimated by the deterministic algorithm has large estimation error and is susceptible to measurement noises or modeling errors.The main reason for poor estimation results is that the recursive matrix is formed with only a few measurements and is badly conditioned.

Based on the idea of using a single vector to estimate angular velocity,an angular velocity estimation method for spinning solar sails that uses only raw measurements of the sun direction vector is proposed by combining the kinematic model for the apparent motion of the sun direction vector and attitude dynamics.This paper is outlined as follows:Section 2 introduces the mathematical model used to develop the estimation method;Details of the proposed angular velocity estimation method are presented in Section 3;In Section 4,numerical simulations are implemented to demonstrate performance of the proposed method;Section 5 contains a summary and conclusions.

2.Mathematical model

For the purpose of this paper,the solar sail will be assumed to be a rigid body with a uniform,flat,square sail.The spinning axis is parallel to the normal direction of the solar sail,as shown in Fig.1.The solar body-fixed frameObXbYbZbis located at the sail’s center of mass.The axes ofObXbandObYbare aligned to a boom withObZbpointing in the normal direction of the solar sail.β is the incidence angle,and ωsis the angular velocity rotating aboutObZb.

According to Euler’s equations,the angular dynamics can be expressed as

where ω =[ωx，ωy，ωz]Tis the angular velocity vector.When the spinning axis is parallel to the normal direction of the solar sail,ωx= ωy=0,and ωz= ωs.For the solar sail whose shape is similar to a flat plate,the moment of inertia matrix can be approximately defined as J=diag(JT，JT，2JT),withJTthe principle moment of inertia about any transverse principal axis;H=Jω is the total solar sail angular momentum;Tcis the control torque;Tdis the external disturbance torque.Obviously,H×ω=0,and the angular dynamics can be written as

The kinematic model for the apparent motion of the sun direction vector can be expressed as

where S is the unit vector of sun direction represented in the solar sail body fixed coordinate system.(d/dt)S is the total temporal derivative of S,and˙S=(∂/∂t)S is the local derivative of S in the body-fixed coordinate system.

According to the assumptions of Ref.18,the left-hand side of Eq.(3)is very small relative to both terms on the righthand side and can be treated as a noise,which yields

3.Angular velocity determination

Since[S×]is not invertible,the least-squares solution^ω(also the optimal approximation solution)of Eq.(4)can be calculated by

where[S×]#is the pseudoinverse of[S×].Because[S×]is not a full column rank matrix,the least-squares solution has large estimation error and is susceptible to measurement noises and modeling errors.

3.1.Least-squares estimation algorithm

When the spin angular velocity is constant andmgroups of sun direction data are considered,then

Whenm≥ 2,Als，mmay be treated as a full column rank matrix.The least-squares solution^ω of Eq.(8)can be calculated by

The algorithm of Eqs.(7)and(9)is named the least-squares estimation algorithm(LSEA).Estimation error can be calculated by solving Eq.(8)in terms of bls，mand substituting the result into Eq.(9),which yields

Considering the measurement noise of the sun sensor,ns(its variance is σ2s),Eq.(10)can be rewritten as

where ΔAls，mand Δbls，mare the noise induced perturbations of Als，mand bls，m,respectively,and

where Δtis the sample interval and can be expressed as(tf-t0)/m,witht0andtfthe starting time and ending time;ns，0is the noise at timet0.

For in-orbit application,the recursive least-squares algorithm can be adopted to solve Eq.(7).Then,a recursive least-squares estimation algorithm(RLSEA)is obtained.The RLSEA has good tracking performance and can be applied to cases with a variable spin angular velocity.

3.2.Improved least-squares estimation algorithm

Because of the design process of LSEA and RLSEA,they cannot get high-precision estimation results when spin angular velocity is variable.Here,an improved method is proposed to estimate variable angular velocity.

When the spin angular velocity is variable,Eq.(7)can be rewritten as

According to Eq.(2),the angular velocity has the below propagating form:

Substituting Eq.(15)into Eq.(14)gives where

The least-squares solution^ωmof Eq.(16)can be calculated by

The algorithm of Eqs.(7),(17)and(19)is named the improved least-squares estimation algorithm(ILSEA).The estimation error can be measured by the value of κ.When the measurement noise nsof the sun sensor is considered,then

Obviously,this ILSEA is also suitable for cases with constant spin angular velocity.

4.Numerical simulations

Some simulations are performed to demonstrate performance of the presented angular velocity determination algorithm.Detailed simulation parameters are listed in Table 1,where Teisthe errorbetween valuesofTdand Tc,and Te～N(0，(‖Td‖/10)2).

To quantify the error between the true angular velocity and the estimated angular velocity,a single figure of merit(FM)is introduced:

4.1.Analyzing rules of κ

Whent0=0 s,andtf=40 s,values of 1/κ for β from 0°to 90°and ωsfrom 1(°)/s to 30(°)/s are shown in Fig.2.The largest values of 1/κ for ωsfrom 1(°)/s to 30(°)/s are displayed in Fig.3.For purposes of comparison,results fortf=60 s andtf=90 s are also simulated.Values of β corresponding to the largest values of 1/κ are displayed in Fig.4.

According to these simulations,some conclusions can be drawn:

(1)The largest values of 1/κ increase along with the growth of tf.

(2)Most of largest values of 1/κ appear in situations where the incidence angle is around 55°.Following along with the growth of ωs,this phenomenon becomes more obvious.So,it is a good choice to design the incidence angle β to be about 55°from the angle determining the angular velocity.

4.2.Constant spinning angular velocity

The first simulation focuses on the influence of different values of β on estimation accuracy of the angular velocity.For this simulation,tf=40 s，ωs=6(°)/s,and β are equal to 15°,35.26°(the angle at which the maximum transverse force can be produced)1and 55°,respectively.

Fig.5 shows values of κ for different incidence angles.The case of β =55°has the smallest value of κ.It agrees with the second conclusion from Fig.4.

Values of FM from using LSEA and ILSEA are displayed in Fig.6.Obviously,the two algorithms have the same estimation result.Estimation accuracy is better than 10%after 5 s.The case with the smallest value of κ,β =55°,has better estimation accuracy,which at 40 s is about 0.8%.

The second simulation focuses on the influence of different values of ωson estimation accuracy of the angular velocity.For this simulation,tf=40 s, β =35.26°,and ωsequals to 2(°)/s,6(°)/s and 10(°)/s,respectively.

Fig.7 shows values of κ for different angular velocities.The case of ωs=10(°)/s has the smallest value of κ.Values of FM from using LSEA and ILSEA are displayed in Fig.8.Again,the two algorithms have the same estimation result.Estimation accuracy is better than 10%after 5 s.The case with the smallest value of κ,ωs=10(°)/s,has the best estimation accuracy,which at 40 s is about 0.7%.

4.3.Variable spinning angular velocity

The previous simulations focused on the performance of ILSEA.For this simulation,tf=40 s, β =35.26°, ωs=6(°)/s,and Tc=Td+Te+0， 0， 10sin(0.1t)[

]T.

The true spin velocity is shown in Fig.9.Values of FM from using RLSEA and ILSEA are displayed in Fig.10.Obviously,ILSEA has better estimation accuracy,which at 40 s is about 0.7%.

ILSEA depends on the accuracy of the inertia matrix to calculate the velocity increment δk.Because of deformations,such as wrinkles,the moment of inertia is not equal to(and is in fact smaller than)the known nominal value.In this paper,the real inertia matrix is written as the sum of the nominal inertia matrix and an uncertain matrix:

When-0.1J is chosen as ΔJ,estimation results are shown in Fig.11.ILSEA has good estimation accuracy,which at 40 s is about 1.4%.Obviously,performance has decreased due to the inertia uncertainty,but not by much.Estimation results are acceptable for in-orbit application.

5.Conclusions

(1)A new angular velocity estimation method was developed for spinning solar sails based only on measurements of sun direction.The kinematic model for apparent motion of the sun direction vector and the attitude dynamics equation are utilized to establish the estimation equation.

(2)A performance criterion similar to condition number is defined to measure estimation accuracy and resilience to measurement noises and modeling errors.

(3)The smallest value of the performance criterion appears at the incidence angle of about 55°and increases along with the growth of tf.This phenomenon becomes more obvious as spin angular velocity increases.Designing the incidence angle to be about 55°from the angle determining the angular velocity is a good choice.

(4)Simulations show that the proposed method can yield highly-precise estimation results for angular velocity and is insensitive to measurement noises and modeling errors.

Acknowledgement

This work was supported by the National Natural Science Foundation of China(No.11302113).

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18 May 2016;revised 22 August 2016;accepted 28 September 2016

Available online 21 December 2016

Ⓒ2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

E-mail address:zhaikun@tsinghua.edu.cn

Peer review under responsibility of Editorial Committee of CJA.