阅读说明：本技术 一种考虑无陀螺仪的非合作航天器姿态和参数估计方法 (Non-cooperative spacecraft attitude and parameter estimation method considering no gyroscope ) 是由 胡庆雷 陈航 郑建英 董宏洋 于 2019-08-01 设计创作，主要内容包括：本发明公开了一种考虑无陀螺仪的非合作航天器姿态和参数估计方法,包括如下步骤：考虑依据光学敏感器建立的目标非合作航天器相对坐标系与其本体坐标系不重合的问题,建立目标非合作航天器相对坐标系下的空间快速翻滚的非合作航天器的姿态运动学与动力学模型；选取目标非合作航天器姿态四元数矢量部分、三轴旋转角速度和转动惯量比的误差作为滤波变量,推导线性化系统的离散状态方程和观测方程；设计乘性拓展卡尔曼滤波器,并改进设计五步预测,以提高滤波估计精度和跟随快速性,解决了光学敏感器采样频率过低导致滤波误差增大的问题。(The invention discloses a non-cooperative spacecraft attitude and parameter estimation method considering no gyroscope, which comprises the following steps: considering the problem that a relative coordinate system of the target non-cooperative spacecraft established according to the optical sensor is not coincident with a body coordinate system of the target non-cooperative spacecraft, establishing a posture kinematics and dynamics model of the non-cooperative spacecraft, which rolls quickly in space, under the relative coordinate system of the target non-cooperative spacecraft; selecting errors of a target non-cooperative spacecraft attitude quaternion vector part, a three-axis rotation angular velocity and a rotational inertia ratio as filter variables, and deriving a discrete state equation and an observation equation of a linearization system; a multiplicative expansion Kalman filter is designed, five-step prediction is improved, so that the filtering estimation precision and the following rapidity are improved, and the problem that the filtering error is increased due to the fact that the sampling frequency of the optical sensor is too low is solved.)

1. a method for non-cooperative spacecraft attitude and parameter estimation considering no gyroscopes, comprising the steps of:

S1: considering the problem that a relative coordinate system of the target non-cooperative spacecraft established according to the optical sensor is not coincident with a body coordinate system of the target non-cooperative spacecraft, establishing a posture kinematics and dynamics model of the non-cooperative spacecraft, which rolls quickly in space, under the relative coordinate system of the target non-cooperative spacecraft;

S2: based on the attitude kinematics and dynamics model of the non-cooperative spacecraft established in the step S1, selecting errors of a target non-cooperative spacecraft attitude quaternion vector part, a triaxial rotation angular velocity and a rotational inertia ratio as filter variables, and deriving a discrete state equation and an observation equation of a linearization system;

s3: based on the discrete state equation and the observation equation of the linearized system derived in step S2, a multiplicative extended kalman filter is designed, and five-step prediction is improved and designed to improve the filtering estimation accuracy and the following rapidity.

2. The method according to claim 1, wherein in step S1, the relative coordinate system of the target non-cooperative spacecraft is established according to the optical sensor as follows:

When 3 non-collinear feature points P on the target non-cooperative spacecraft are observed simultaneously₁，P₂，P₃defining the origin o of the target uncooperative spacecraft relative coordinate system_Tis P₁vector ofIn the direction of x_Taxial direction, x_Tunit vector of axis is

Definition of z_TThe axis being a vectorSum vectorthe unit vector of the normal vector of the plane is

Determining y from the right hand rule_Tunit vector r of axis_yIs composed of

r_y＝r_z×r_x (3)

Then the coordinate system o_Tx_Ty_Tz_TNamely a relative coordinate system of the target non-cooperative spacecraft;

the attitude dynamics equation of the non-cooperative spacecraft is as follows:

Wherein J is a rotational inertia matrix of the target non-cooperative spacecraft under a relative coordinate system, and is a symmetric matrix of 3 multiplied by 3:

wherein, the diagonal element J_xx，J_yy，J_zzIs the rotational inertia of the main shaft, and the off-diagonal element J_xy，J_xy，J_yzis the product of inertia; omega ═ omega_xω_y ω_z]^TThree-axis angular velocity of rotation for the target non-cooperative spacecraft;Three-axis angular acceleration of rotation for the target non-cooperative spacecraft; tau is external disturbance moment and is modeled as Gaussian white noise; omega^×Is a skew antisymmetric matrix, which is specifically formed as follows:

The attitude kinematics equation of the non-cooperative spacecraft is described by quaternion, provides global non-singular motion attitude representation for the non-cooperative spacecraft,

Defining the quaternion of the attitude of the non-cooperative spacecraft under the relative coordinate system of the target non-cooperative spacecraft as The quaternion of the attitude kinematics of the non-cooperative spacecraft is then expressed as:

In the formula (I), the compound is shown in the specification,The matrix xi (q) is defined as follows:

wherein I is an identity matrix.

3. the method according to claim 2, wherein step S2 is implemented as follows:

defining an attitude error quaternion as

In the formula (I), the compound is shown in the specification,multiplication is carried out for quaternion; superscript "^" represents an estimated value; the inverse of quaternion Satisfies the following conditions:

The error of each variable is defined as follows:Wherein, δ q_vA vector part of an attitude error quaternion defined for the target non-cooperative spacecraft in a relative coordinate system; delta omega is error three-axis rotation angular velocity; δ J' is the error moment of inertia ratio;

Defining the state variables:

wherein the content of the first and second substances,

The dynamic equation of the vector part of the attitude error quaternion is linearized to obtain:

The attitude dynamics equation of the non-cooperative spacecraft is linearized as follows:

wherein the content of the first and second substances,is a partial differential symbolNumber;Are Jacobian matrices;

The discrete state equation and the observation equation of the linearized system are

x_k+1＝Φ_kx_k+ε_k (14)

y_k＝Hx_k+υ_k (15)

In the formula, subscript "k" represents the value at time k; x is the number of_kIs a state variable at the moment k; epsilon_kis process noise; upsilon is_kto observe noise; y is_kIs an observed value; phi_kIs a state transition matrix; h is an observation matrix in which the process noise ε_kand observing the noise upsilon_kIs uncorrelated white noise with a mean value of zero,

Φ_k＝e^AΔT≈I_11×11+AΔT (16)

Wherein Δ T is a discrete time interval; i represents an identity matrix;

The observation equation is given according to the output of the optical sensor, and the observed quantity is only an attitude error quaternion and is irrelevant to the three-axis rotation angular velocity and the rotational inertia ratio, namely Obtaining an observation matrix H ═ I_3×3 0_3×3 0_3×5]。

4. The method of claim 3, wherein: in step S3, the building of the multiplicative expansion kalman filter includes four parts, namely, state one-step prediction, filtering observation calculation, filtering update, and attitude correction:

1) state one-step prediction

The attitude dynamics equation (4) of the non-cooperative spacecraft and the attitude kinematics equation (7) of the non-cooperative spacecraft are used for obtainingAndCalculating a pre-estimated value of the attitude quaternion:

Wherein the content of the first and second substances,the estimated value of the three-axis rotation angular acceleration at the moment k is obtained;The estimated value of the attitude quaternion vector part at the moment k;is the estimated value of the time derivative of the attitude quaternion vector part at the moment k;

2) Filtering observation calculation

Calculating attitude error quaternion according to observation quaternion and prediction quaternion operation of optical sensor

3) filter update

recursive calculation is performed according to a Kalman equation:

P_k|k-1＝ΦP_k-1Φ^T+Q_k-1 (20)

K_k＝P_k|k-1H^T(HP_k|k-1H^T+R_k)^-1 (21)

x_k＝x_k-1+K_k(δq_v-Hx_k-1) (23)

wherein, P_k|k-1Estimating error covariance matrix for optimal prediction at the k moment; p_k-1the error covariance matrix of the optimal filtering value at the k-1 moment is obtained; k_kIs a gain matrix at time k; q_k-1The noise variance matrix of the system at the k-1 moment is obtained; r_kmeasuring a variance matrix;

4) attitude correction

updating the angular velocity:δ ω is the state variable x_kError three-axis angular velocity of rotation; and keeping the quaternion normalization of attitude error:

Circulating the steps 1) -4), and outputting a relative state quaternion, a triaxial rotation angular velocity and a rotational inertia ratio matrix of the target non-cooperative spacecraft;

The multiplicative extended Kalman filtering system uses the vector part of the attitude error quaternion for filtering updating, and the complete quaternion for global nonsingular pose recursion, so that the attitude quaternion is corrected in each step of filtering updating by using the formula (24) to carry out attitude recursion,

Considering that the frequency of the non-cooperative spacecraft carrying the optical imaging sensor cannot meet the requirements of high precision and quick tracking of attitude estimation, five hour intervals are divided in a discrete time interval delta T, five times of prediction are carried out, and one time of measurement updating is carried out, namely, five sections of broken lines are used for fitting the numerical solution of a spacecraft attitude nonlinear equation in the delta T.

Technical Field

The invention belongs to the field of spacecraft navigation, and particularly relates to a non-cooperative spacecraft attitude and parameter estimation method considering no gyroscope, which can identify the rotational inertia ratio of a space non-cooperative spacecraft and realize high-precision and real-time relative attitude motion estimation.

background

the spacecraft attitude determination is to determine the attitude of the main body under the condition of having a star sensor and a gyroscope, but at present, most tasks such as invalid spacecraft maintenance, uncontrolled spacecraft capture and the like are encountered, and the targeted research object is a space non-cooperative uncontrolled spacecraft. Such spacecraft often roll rapidly in space out of control, and therefore the motion state and kinetic parameters of the non-cooperative spacecraft need to be acquired under the condition that the target appearance characteristics are unknown, no response exists and no identification exists. Specifically, in order to implement tasks such as capture and control of a space non-cooperative spacecraft, motion states and parameters such as attitude, angular velocity, inertia ratio and the like are required to be estimated and identified.

aiming at the problem of inertia ratio identification while attitude determination of the spacecraft is carried out, the Chinese patent CN 102620886B utilizes a two-step on-orbit identification system parameter method, firstly an EKF filter is established to obtain a rotational inertia ratio, and then a control moment is applied to the combined spacecraft to obtain the least square estimation of the rotational inertia. The method belongs to combination attitude determination, and after a target spacecraft and an active spacecraft are combined, a gyroscope on the active spacecraft provides angular velocity measurement information. However, the method is not suitable for relative navigation without gyroscope measurement information, and the method of applying the control moment to solve is also not suitable for attitude determination of the space non-cooperative uncontrolled rolling spacecraft.

chinese patent CN 107607737 a proposes a navigation technique considering gyro-less angular velocity measurement, which estimates an attitude change matrix by matching the correspondence between two star maps at two shooting times based on star light vectors. However, the method can only fix the posture of the body and is not suitable for relative navigation of non-cooperative spacecrafts.

disclosure of Invention

in order to solve the defects of the prior art, the invention provides a non-cooperative spacecraft attitude and parameter estimation method considering no gyroscope. On the premise of only adopting an optical imaging system as a sensor to acquire attitude information of a target spacecraft, the invention provides a filtering estimation algorithm based on non-gyroscope information, which can realize high-precision estimation and rotational inertia ratio identification of attitude and angular speed of a space non-cooperative spacecraft.

According to an aspect of the invention, there is provided a non-cooperative spacecraft attitude and parameter estimation method considering no gyroscope, comprising the steps of:

S1: considering the problem that a relative coordinate system of the target non-cooperative spacecraft established according to the optical sensor is not coincident with a body coordinate system of the target non-cooperative spacecraft, establishing a posture kinematics and dynamics model of the non-cooperative spacecraft, which rolls quickly in space, under the relative coordinate system of the target non-cooperative spacecraft;

S2: based on the attitude kinematics and dynamics model of the non-cooperative spacecraft established in the step S1, selecting errors of a target non-cooperative spacecraft attitude quaternion vector part, a triaxial rotation angular velocity and a rotational inertia ratio as filter variables, and deriving a discrete state equation and an observation equation of a linearization system;

s3: based on the discrete state equation and the observation equation of the linearized system derived in step S2, a multiplicative extended kalman filter is designed, and five-step prediction is improved and designed to improve the filtering estimation accuracy and the following rapidity.

Further, in step S1, the relative coordinate system of the target non-cooperative spacecraft is established according to the optical sensor as follows:

when 3 non-collinear feature points P on the target non-cooperative spacecraft are observed simultaneously₁，P₂，P₃Defining the origin o of the target uncooperative spacecraft relative coordinate system_TIs P₁Vector ofin the direction of x_TAxial direction, x_TUnit vector of axis is

Definition of z_TThe axis being a vectorSum vectorThe unit vector of the normal vector of the plane is

Determining y from the right hand rule_TUnit vector r of axis_yis composed of

r_y＝r_z×r_x (3)

then the coordinate system o_Tx_Ty_Tz_Tnamely a relative coordinate system of the target non-cooperative spacecraft;

The attitude dynamics equation of the non-cooperative spacecraft is as follows:

Wherein J is a rotational inertia matrix of the target non-cooperative spacecraft under a relative coordinate system, and is a symmetric matrix of 3 multiplied by 3:

wherein, the diagonal element J_xx，J_yy，J_zzIs the rotational inertia of the main shaft, and the off-diagonal element J_xy，J_xz，J_yzIs the product of inertia; omega ═ omega_x ω_y ω_z]^TThree-axis angular velocity of rotation for the target non-cooperative spacecraft;Three-axis angular acceleration of rotation for the target non-cooperative spacecraft; tau is external disturbance moment and is modeled as Gaussian white noise; omega^×Is a skew antisymmetric matrix, which is specifically formed as follows:

the attitude kinematics equation of the non-cooperative spacecraft is described by quaternion, provides global non-singular motion attitude representation for the non-cooperative spacecraft,

defining the attitude four of the non-cooperative spacecraft under the relative coordinate system of the target non-cooperative spacecraftThe element number is the quaternion of the attitude kinematics of the non-cooperative spacecraft is then expressed as:

In the formula (I), the compound is shown in the specification,The matrix xi (q) is defined as follows:

Further, the step S2 specifically includes the following steps:

Defining an attitude error quaternion as

in the formula (I), the compound is shown in the specification,Multiplication is carried out for quaternion; superscript "^" represents an estimated value; the inverse of quaternion Satisfies the following conditions:

the error of each variable is defined as follows：wherein, δ q_vA vector part of an attitude error quaternion defined for the target non-cooperative spacecraft in a relative coordinate system; delta omega is error three-axis rotation angular velocity; δ J' is the error moment of inertia ratio;

defining the state variables:

Wherein the content of the first and second substances,

The dynamic equation of the vector part of the attitude error quaternion is linearized to obtain:

The attitude dynamics equation of the non-cooperative spacecraft is linearized as follows:

Wherein the content of the first and second substances,Is a partial differential sign;are Jacobian matrices;

The discrete state equation and the observation equation of the linearized system are

x_k+1＝Φ_kx_k+ε_k (14)

y_k＝Hx_k+υ_k (15)

In the formula, subscript "k" represents the value at time k; x is the number of_kis a state variable at the moment k; epsilon_kis process noise; upsilon is_kto observe noise; y is_kIs an observed value; phi_kIs a state transition matrix; h is an observation matrix in which the process noise ε_kand observing the noise upsilon_kIs uncorrelated white noise with a mean value of zero,

Φ_k＝e^AΔT≈I_11×11+AΔT (16)

wherein Δ T is a discrete time interval; i represents an identity matrix;

The observation equation is given according to the output of the optical sensor, and the observed quantity is only an attitude error quaternion and is irrelevant to the three-axis rotation angular velocity and the rotational inertia ratio, namely Obtaining an observation matrix H ═ I_3×3 0_3×3 0_3×5]。

Further, in step S3, the building of the multiplicative expansion kalman filter includes four parts, namely, state one-step prediction, filtering observation calculation, filtering update, and attitude correction:

1) State one-step prediction

the attitude dynamics equation (4) of the non-cooperative spacecraft and the attitude kinematics equation (7) of the non-cooperative spacecraft are used for obtainingAndcalculating a pre-estimated value of the attitude quaternion:

wherein the content of the first and second substances,the estimated value of the three-axis rotation angular acceleration at the moment k is obtained;The estimated value of the attitude quaternion vector part at the moment k;is the estimated value of the time derivative of the attitude quaternion vector part at the moment k;

2) Filtering observation calculation

Calculating attitude error quaternion according to observation quaternion and prediction quaternion operation of optical sensor

3) filter update

recursive calculation is performed according to a Kalman equation:

P_k|k-1＝ΦP_k-1Φ^T+Q_k-1 (20)

K_k＝P_k|k-1H^T(HP_k|k-1H^T+R_k)^-1 (21)

x_k＝x_k-1+K_k(δq_v-Hx_k-1) (23)

wherein, P_k|k-1Estimating error covariance matrix for optimal prediction at the k moment; p_k-1The error covariance matrix of the optimal filtering value at the k-1 moment is obtained; k_kIs a gain matrix at time k; q_k-1The noise variance matrix of the system at the k-1 moment is obtained; r_kmeasuring a variance matrix;

4) attitude correction

updating the angular velocity:δ ω is the state variable x_kError three-axis angular velocity of rotation; and keeping the quaternion normalization of attitude error:

Circulating the steps 1) -4), and outputting a relative state quaternion, a triaxial rotation angular velocity and a rotational inertia ratio matrix of the target non-cooperative spacecraft;

The multiplicative extended Kalman filtering system uses the vector part of the attitude error quaternion for filtering updating, and the complete quaternion for global nonsingular pose recursion, so that the attitude quaternion is corrected in each step of filtering updating by using the formula (24) to carry out attitude recursion,

considering that the frequency of the non-cooperative spacecraft carrying the optical imaging sensor cannot meet the requirements of high precision and quick tracking of attitude estimation, five hour intervals are divided in a discrete time interval delta T, five times of prediction are carried out, and one time of measurement updating is carried out, namely, five sections of broken lines are used for fitting the numerical solution of a spacecraft attitude nonlinear equation in the delta T.

The invention has the beneficial effects that:

1) according to the method for the multiplicative expansion Kalman filtering, the error quaternion vector part is used as the state variable, so that the state singularity is avoided, and the filtering convergence speed can be increased; in addition, complete and quick identification can be carried out on the rotational inertia matrix with the inertia product, and the attitude estimation precision is further improved while the target system parameters are obtained.

2) The actual non-cooperative spacecraft lacks angular velocity observation information provided by a gyroscope, the attitude information of the target non-cooperative spacecraft is acquired only by adopting the optical imaging sensor, and the equipment is simple and is suitable for being applied to actual relative navigation.

3) different from the conventional built filtering prediction, the method adopts a five-step prediction link, overcomes the problem of increased filtering error caused by too low sampling frequency of the optical sensor, can reduce estimation error generated by nonlinear function recursion, and improves estimation precision.

Drawings

FIG. 1 is a flow chart of steps of a method of estimating attitude and parameters of a non-cooperative spacecraft of the present invention, taking into account the lack of gyroscopes;

FIG. 2 is a Kalman filtering flow chart for attitude determination of the present invention;

FIG. 3 is a block diagram of a Kalman filtering update module of the present invention;

FIG. 4 is a multi-step prediction error effect display diagram of the present invention;

FIG. 5 is an error plot of the relative attitude of the target non-cooperative spacecraft of the present invention;

FIG. 6 is an error plot of relative angular velocity for a target non-cooperative spacecraft of the present invention;

FIG. 7 is an error plot of the target non-cooperative spacecraft moment of inertia ratio of the present invention.

Detailed Description

The invention is further described below with reference to the accompanying drawings.

as shown in FIG. 1, the method of the present invention for considering non-cooperative spacecraft attitude and parameter estimation without gyroscopes comprises: firstly, considering the problem that a target coordinate system is not coincident with a body coordinate system of the target non-cooperative spacecraft, establishing a posture kinematics and dynamics model of the non-cooperative spacecraft which rolls rapidly in space under the target non-cooperative spacecraft coordinate system; then, selecting errors of a relative attitude quaternion vector part, a triaxial rotation angular velocity and a rotational inertia ratio as filtering variables, and deducing a discrete state equation and an observation equation of the linear system; and finally, designing a multiplicative expansion Kalman filter, improving and designing five-step prediction to improve the filtering estimation precision and following rapidity, and solving the problem of filtering error increase caused by too low sampling frequency of the optical sensor.

the method comprises the following concrete implementation steps:

s1: and considering the problem that the coordinate system of the target non-cooperative spacecraft established according to the optical sensor is not coincident with the coordinate system of the body of the target non-cooperative spacecraft, and establishing a posture kinematics and dynamics model of the non-cooperative spacecraft, in which the space of the target non-cooperative spacecraft rolls rapidly, under the relative coordinate system.

the relative coordinate system of the target spacecraft established by the invention is obviously different from the coordinate system established in the attitude determination of the traditional spacecraft: the traditional spacecraft generally fixes the self attitude, the body information is determined, and three coordinate axes are superposed with a main axis of rotational inertia; aiming at the space non-cooperative spacecraft, the method has the advantages that the target non-cooperative spacecraft lacks prior information, does not communicate with the target, and cannot acquire related attitude information, so that a target relative coordinate system established by depending on an optical sensor (such as a camera) is often not coincident with a target body system, and inertia of a rotational inertia matrix occurs. Establishing a target relative coordinate system according to the optical sensor as follows:

when 3 non-collinear feature points P on the non-cooperative spacecraft are observed simultaneously₁，P₂，P₃defining the origin o of the relative coordinate system_TIs P₁vector ofdirection x_TDirection of axis with unit vector of

Definition of z_Tthe axis being a vectorSum vectorThe unit vector of the normal vector of the plane is

Y can be determined from the right-hand rule_TUnit vector of axis r_y

r_y＝r_z×r_x (3)

then the coordinate system o_Tx_Ty_Tz_Tnamely the target relative coordinate system.

the attitude dynamics equation for considering a non-cooperative spacecraft is:

Wherein J is a rotational inertia matrix of the non-cooperative spacecraft in a relative coordinate system, and is a symmetric matrix of 3 multiplied by 3:

in this example, setWherein, the diagonal element J_xx，J_yy，J_zzis the rotational inertia of the main shaft, and the off-diagonal element J_xy，J_xz，J_yzIs the product of inertia; omega is the three-axis rotation angular velocity of the non-cooperative spacecraft

ω＝[ω_x ω_y ω_z]^T＝[0.2 0.05 0.05]^Trad/s；Three-axis angular acceleration of rotation for a non-cooperative spacecraft; tau is external disturbance moment, including gravity gradient moment, magnetic moment, pneumatic torque, solar radiation moment and the like, and can be modeled as white Gaussian noise; omega^×is a skew antisymmetric matrix, which is specifically formed as follows:

The attitude kinematics equation of the non-cooperative spacecraft is described by quaternion, and can provide a global non-singular motion attitude representation for the non-cooperative spacecraft. Three-axis rotation angular velocity under target relative coordinate systemdegree is omega, and the quaternion of the attitude of the non-cooperative spacecraft under the relative coordinate system of the target is defined as Its initial value is set to [ 0.99430.09910.02740.0274]^TThen the quaternion kinematic equation of the non-cooperative spacecraft is

In the formula (I), the compound is shown in the specification,The matrix xi (q) is defined as follows:

s2: and selecting errors of a quaternion vector part of the relative attitude, the three-axis rotation angular velocity and the rotational inertia ratio as filtering variables, and deriving a discrete state equation and an observation equation of the linear system.

The multiplicative extended Kalman filtering is more advantageous than the classical extended Kalman filtering in attitude estimation described by quaternions, and the normalization constraint of quaternions causes one of four parameters to be redundant, so that in the attitude estimation based on EKF, the establishment of a state equation by using all four parameters can cause an unobservable state, which is shown in that a state error covariance matrix generates singularity in the filtering process:Therefore, multiplicative extended Kalman filtering is selected, an unconstrained attitude error state quaternion vector part is estimated, normalization constraint of quaternion is kept in a filtering updating prediction period, and global nonsingular attitude description is provided for the spacecraft. The following is derived from equation (4):

wherein the content of the first and second substances,

Without loss of generality, assume J_xxThe maximum element in the rotational inertia matrix is obtained by dividing the two sides of the spacecraft dynamics equation by J_xx＝1000：

Wherein

The second term on the right side of the medium sign in the formula (4-3) can be modeled as white gaussian noise, so that the true value of inertia does not need to be identified in attitude filtering estimation, only the relative ratio between the rotational inertia needs to be utilized, and the attitude determination and estimation of the spacecraft are not influenced.

from the above derivation, the attitude error quaternion is defined as:

In the formulaIs quaternion multiplication, the upper mark ^ represents an estimated value, and the inverse of the quaternion is satisfies the following conditions:

The error of each variable is defined as follows:In the formula, δ q_vA vector part of an attitude error quaternion defined under a target relative coordinate system; delta omega is error three-axis rotation angular velocity; δ J' is the error moment of inertia ratio. The error variable of each variable is selected as the state of the filter, so that the filter convergence speed is greatly superior to that of the filter convergence speed by directly using the variable.

Defining the state variables:

wherein the content of the first and second substances,

And (3) linearizing the kinetic equation of the vector part of the attitude error quaternion according to a small-angle approximate condition to obtain:

linearizing the attitude dynamics equation of the non-cooperative spacecraft

wherein the content of the first and second substances,Is a partial differential sign;Are Jacobian matrices.

the discrete state equation and the observation equation of the linearized system are

x_k+1＝Φ_kx_k+ε_k (14)

y_k＝Hδx_k+υ_k (15)

In the formula, subscript "k" represents the value at time k; x is the number of_kIs a state variable at the moment k; epsilon_kIs process noise; upsilon is_kTo observe noise; y is_kIs an observed value; phi_kIs a state transition matrix; h is an observation matrix in which the process noise ε_kAnd observing the noise upsilon_kIs uncorrelated white noise with a mean value of zero.

Φ_k＝e^AΔT≈I_11×11+AΔT (16)

Where Δ T is the discrete time interval, i.e., the sampling interval of the optical sensor, in this example taken as 1 s. The observation equation is given based on the output of the optical sensor. Since the observed quantity is only a quaternion of attitude error and is independent of the three-axis rotation angular velocity and the rotational inertia ratio, i.e. The observation matrix is obtained as: h ═ I_3×3 0_3×3 0_3×5]。

S3: based on the discrete state equation and the observation equation of the linearized system derived in step S2, a multiplicative extended kalman filter is designed, and five-step prediction is improved and designed to improve the filtering estimation accuracy and the following rapidity.

the establishment of the multiplicative expansion Kalman filter is divided into four parts of prediction, calculation, updating and correction, which are described in detail as follows:

1) state one-step prediction

Obtained from formula (4) and formula (7)AndCalculating a pre-estimated value of the attitude quaternion:

Wherein the content of the first and second substances,The estimated value of the three-axis rotation angular acceleration at the moment k is obtained;Is the estimated value of the time derivative of the attitude quaternion vector part at the moment k;Is an estimate of the attitude quaternion vector component at time k.

2) filtering observation calculation

And calculating an attitude error quaternion according to the observed quaternion and the predicted quaternion of the optical sensor:

3) filter update

Recursive calculation is performed according to a Kalman equation:

P_k|k-1＝ΦP_k-1Φ^T+Q_k-1 (20)

K_k＝P_k|k-1H^T(HP_k|k-1H^T+R_k)^-1 (21)

x_k＝x_k-1+K_k(δq_v-Hx_k-1) (23)

wherein, P_k|k-1estimating error covariance matrix for optimal prediction at the k moment; p_k-1The error covariance matrix of the optimal filtering value at the k-1 moment is obtained; k_kIs a gain matrix at time k; q_k-1the noise variance matrix of the system at the k-1 moment is obtained; r_kMeasuring a variance matrix;

4) attitude correction

Updating the angular velocity:δ ω is the state variable x_kerror three-axis angular velocity of rotation; and keeping the quaternion normalization of attitude error:

And (4) circulating the steps 1) -4), outputting a relative state quaternion, a triaxial rotation angular velocity and a moment of inertia ratio matrix of the target spacecraft. The filtering flow and the update block diagram are shown in fig. 2 and 3. The multiplicative expansion Kalman filtering system adopts the vector part of the error quaternion for filtering updating, and the complete quaternion is used for global nonsingular pose recursion. Therefore, the attitude recursion is carried out by using the equation (24) to correct the attitude quaternion in each step of filtering updating, which is the most obvious characteristic of multiplicative expansion of Kalman.

in addition, the filtering prediction link of the method is different from the traditional five-step prediction for expanding Kalman. The one-step prediction in the filtering process is essentially according to the formula (4) and the formula (7), and then the prediction is carried out by a Newton gradient descent method:the frequency of the non-cooperative spacecraft carrying the optical imaging sensor is considered to be 1Hz, namely, the filtering result is corrected once within 1s, and the requirements of high precision and quick tracking of attitude estimation cannot be met. When the sampling time reaches 1s, the angular speed and the estimated attitude deviate from the true value more.Therefore, five small time intervals delta T are divided into five small time intervals delta T of 0.2s within delta T of 1s, five times of prediction are carried out, one-time measurement updating is carried out, namely five sections of broken lines are used for fitting the numerical solution of the spacecraft attitude nonlinear equation within the 1s, therefore, the filtering estimation precision and the following rapidity are improved, and the problem that the filtering error is increased due to the fact that the sampling frequency of the optical sensor is too low is solved. The multi-step prediction error effect of the present invention is shown in fig. 4.

By utilizing the multiplicative extended Kalman filter provided above, the gyro-free acquisition of the rotational inertia parameters of the space non-cooperative spacecraft can be realized, and the high-precision and real-time motion characteristic estimation can be realized. Finally, to verify the effectiveness of the present invention, a practical example is provided in the MATLAB/Simulink platform environment to illustrate the present invention, but the present invention is not limited by the example. The simulation results are as follows:

TABLE 1 Filter accuracy and Convergence time results for various State variables

The error of the relative attitude of the target spacecraft is shown in figure 5; the error of the target spacecraft relative to the angular velocity is shown in fig. 6; the error in the target spacecraft moment of inertia ratio is shown in figure 7.

the simulation results fully show that under the condition of no gyroscope information, the method for estimating the attitude and the parameters of the non-cooperative spacecraft, which is designed by the invention, can meet the requirements of the accuracy and the real-time performance of the motion estimation of the uncontrolled rolling spacecraft and realize the high-accuracy estimation of the relative attitude of the moving target.

Those skilled in the art will appreciate that the invention may be practiced without these specific details. The above description is only an example of the present invention, and is not intended to limit the present invention. Other modifications, substitutions and the like are intended to be within the spirit and scope of the invention.