CN105045271B - A kind of spacecraft position motor-driven method under the conditions of drive lacking - Google Patents

Abstract

The invention provides a kind of motor-driven method in spacecraft position under the conditions of drive lacking, step is as follows：(1) relative distance of the velocity error of site error and aircraft between present speed and target velocity under carrier coordinate system between aircraft current location and target location, calculating aircraft and target location is obtained；(2) line of sight rate of aircraft relative target position under carrier coordinate system is calculated；(3) angle of OZ axles and the angle or OX axles and line of sight rate vector of site error vector is calculated；(4) calculation error quaternary number；(5) posture and track of control aircraft, adjust the thrust direction of precise tracking, aircraft is moved to target location from current location.The present invention is directed to the small space aircraft using the dynamical system in aircraft longitudinal axis direction non-control force, and based on appearance rail coupling control method, the change of the locus of aircraft is completed under conditions of not increasing precise tracking or changing dynamical system layout.

Description

A kind of spacecraft position motor-driven method under the conditions of drive lacking

Technical field

The present invention relates to a kind of motor-driven method in spacecraft position, the space under the conditions of more particularly to a kind of drive lacking flies The motor-driven method in row device position.

Background technology

For a class independent of external information input possess sense of autonomy knowledge not, it is autonomous motor-driven, ability of making decisions on one's own small For type spacecraft, space exploration, obstacle avoidance can be performed in the way of pose adjustment, orbit maneuver, and (e.g., space is broken Piece) etc. task.Usual this kind of small space aircraft uses pulsed motor as executing agency, and for the hair of orbits controlling Motivation is only distributed in the plane for perpendicular to aircraft longitudinal axis and passing through barycenter, and such as Fig. 1 sends out for a kind of common 6 attitude controls that possess The dynamical system layout of the spacecraft of motivation and 4 precise trackings, the dynamical system is laid out on carrier axial direction There is no power output, it is impossible to directly the position of axial direction and speed are adjusted by precise tracking, i.e., type of drive is present Shortcoming.The spacecraft being laid out although with the dynamical system is easy to Project Realization in topology layout, but in some spies In different space tasks, the spacecraft being laid out using the dynamical system is not required nothing more than can carry out simple orbital exponent, and And requiring that it can be motor-driven to the spatial domain specified by orbits controlling, this is just proposed more to the rail control algorithm of such aircraft High requirement.

The dynamical system layout of conventional aerospace device makes in gesture stability and orbits controlling direct decoupling, position mobile process Gesture stability and orbits controlling can be with independent designs.And the position motor-driven adjustment for needing posture is aided under the conditions of drive lacking, institute Be not suitable for the position under the conditions of above-mentioned drive lacking in the method for traditional rail control independent design to control, it is necessary to design appearance rail coupling The control method of conjunction completes the motor-driven task in position.

The content of the invention

Present invention solves the technical problem that being：The present invention is directed to using the dynamical system in aircraft longitudinal axis direction non-control force There is provided a kind of motor-driven method in spacecraft position under the conditions of drive lacking, this hair for the small space aircraft of system distribution form It is bright to be based on appearance rail coupling control method, overcome the deficiencies in the prior art, do not increasing precise tracking or changing dynamical system cloth The change of the locus of aircraft is completed under conditions of office.

The present invention technical solution be：

A kind of motor-driven method in spacecraft position under the conditions of drive lacking, comprises the following steps：

(1) site error under carrier coordinate system between aircraft current location and target location is obtainedAnd fly Velocity error of the row device between present speed and target velocityCalculating aircraft and target location it is relative away from From r；

Described carrier coordinate system O-XYZ origin O is located at the barycenter of aircraft, and OX axles exist along aircraft longitudinal direction, OY axles Aircraft is pointed up in longitudinally asymmetric face, and OZ axles constitute right-handed coordinate system with OX axles and OY axles；

(2) line of sight rate of aircraft relative target position under carrier coordinate system is calculated Wherein,

(3) the line of sight rate ω of aircraft relative target position and the line of sight rate threshold value controlled for position are compared Ω₁Size：If ω<Ω₁, then the angle theta of carrier coordinate system OZ axles and site error vector is calculated, step (4) is gone to；

Wherein, EPS is data computational accuracy；

If ω >=Ω₁, then the angle β of carrier coordinate system OX axles and line of sight rate vector is calculated, step (6) is gone to；

(4) if θ≤pi/2, and r>R, then according to equation below calculation error quaternary number q_e0,q_e1,q_e2,q_e3：

If θ>Pi/2, and r>R, then according to equation below calculation error quaternary number q_e0,q_e1,q_e2,q_e3：

If r≤R, step-up error quaternary number q_e0=1, q_e1=0, q_e2=0, q_e3=0；Wherein, R is stopping track control The safe distance of system；

(5) posture for adjusting aircraft according to error quaternion makes the folder between carrier coordinate system OZ axles and site error vector Angle θ is intended to zero；Meanwhile, the track of aircraft is adjusted, makes the position between carrier coordinate system OZ direction of principal axis aircraft and target ErrorIt is intended to zero, and exits；

(6) if β≤pi/2, according to equation below calculation error quaternary number q_e0,q_e1,q_e2,q_e3：

If β>Pi/2, then according to equation below calculation error quaternary number q_e0,q_e1,q_e2,q_e3：

(7) posture for adjusting aircraft according to error quaternion makes between carrier coordinate system OX axles and line of sight rate vector Angle β is intended to zero；Meanwhile, the track of aircraft is adjusted, makes the position between aircraft and target in carrier coordinate system YOZ planes Put errorIt is intended to zero, or makes the velocity error in carrier coordinate system YOZ planes between aircraft and target It is intended to zero.

The specific method of the track of adjustment aircraft is as follows in the step (5)：Perpendicular to carrier aircraft OX axles simultaneously Do not worked, controlled according to θ by configuring the precise tracking on precise tracking, OY axles on the OY axles and OZ axles in the plane of barycenter Precise tracking on OZ axles processed：If Θ≤θ≤π-Θ, the precise tracking on control OZ axles makes carrier coordinate system OZ axles Site error between direction aircraft and targetIt is intended to zero；If θ<Θ or θ>π-Θ, the then rail control on OZ axles is started Machine does not work；Wherein, Θ is the threshold value related to attitude angle control accuracy.

The track specific method of adjustment aircraft is as follows in the step (7)：Perpendicular to carrier aircraft OX axles and leading to Cross on the OY axles in the plane of barycenter and OZ axles and configure precise tracking；If the rail on Θ≤β≤π-Θ, OY axles and OZ axles Control engine does not work；If β<Θ or β>π-Θ, then according to the amplitude ω of line of sight rate and the sight for speed control Angular speed threshold value Ω₂Size：If ω<Ω₂, then the precise tracking on OY axles and OZ axles is controlled to put down carrier coordinate system YOZ Site error in face between aircraft and targetIt is intended to zero；If ω >=Ω₂, then the rail on OY axles and OZ axles is controlled Control engine makes the velocity error in carrier coordinate system YOZ planes between aircraft and targetIt is intended to zero.

Described is used for the line of sight rate threshold value Ω that position is controlled₁=0.1 Ω~0.5 Ω, wherein Ω=a/V_max, a is The acceleration that pulsed motor for orbits controlling is produced, V_maxFly for aircraft along with the maximum in target location line direction Scanning frequency degree.

The safe distance R=1.2V of described stopping orbits controlling_min/ Ω, V_minFor velocity control accuracy.

The described line of sight rate threshold value Ω for speed control₂=0.5 Ω~0.8 Ω, wherein Ω=a/V_max, a is The acceleration that pulsed motor for orbits controlling is produced, V_maxFly for aircraft along with the maximum in target location line direction Scanning frequency degree.

Described threshold value Θ takes 1.5~2 times of attitude angle control accuracy.

Described data computational accuracy EPS takes 10^-7。

The advantage of the present invention compared with prior art is：

(1) present invention overcomes the method for conventional aerospace device rail control independent design to be not suitable for the position under the conditions of drive lacking The defect of control so that spacecraft can realize that position is motor-driven under the conditions of undershooting, is extended using in y direction The function of the small space aircraft of the dynamical system layout of non-control force.

(2) position control method that the present invention is coupled using appearance rail so that spacecraft is not increasing precise tracking Or change the change for the locus that aircraft can be realized under conditions of dynamical system is laid out, improve position control method Stability, reduces cost.

(3) motor-driven method of the invention can be widely applied in small space aircraft so that spacecraft can Complete various space tasks, such as space exploration, obstacle avoidance is simple and be easy to Project Realization, with important application value.

Brief description of the drawings

Fig. 1 is the dynamical system schematic layout pattern of attitude control engine of the present invention and precise tracking.

Embodiment

The input of the inventive method is the site error under referential, the flying speed under referential, the posture of aircraft And attitude angular velocity, it is therefore an objective to the locus for controlling to make aircraft move to target by posture and position, the coordination of speed. Referential is selected according to specific aerial mission.

A kind of motor-driven method in spacecraft position under the conditions of drive lacking, including step are as follows：

(1) site error under carrier coordinate system between aircraft current location and target location is obtainedAnd fly Velocity error of the row device between present speed and target velocityComprise the following steps that：

(1.1) site error under referential between aircraft current location and target location is calculated；

Wherein,For site error of the aircraft under referential,It is aircraft under referential Current location,For target location of the aircraft under referential.

(1.2) site error and speed between aircraft current location and target location under carrier coordinate system is calculated to miss Difference；

Calculating aircraft and the relative distance r of target location:

WhereinFor flying speed of the aircraft under referential,For with reference to the side for being tied to carrier coordinate system To cosine matrix, computational methods depend on used attitude description mode, and such as quaternary number or Eulerian angles may be referred to inertia and lead The coherent reference book of boat, such as《Inertial navigation》, Qin Yongyuan, Science Press, 2006.

Described carrier coordinate system O-XYZ origin O is located at the barycenter of aircraft, and OX axles exist along aircraft longitudinal direction, OY axles Aircraft is pointed up in longitudinally asymmetric face, and OZ axles constitute right-handed coordinate system with OX axles and OY axles；Pulse for orbits controlling Engine (abbreviation precise tracking) be distributed in perpendicular to aircraft OX axles and pass through barycenter plane in OY axles and OZ axles on, I.e. pulsed motor only has orbital exponent ability to Y passages and Z passages.Pulsed motor (abbreviation appearance for gesture stability Control engine) it is distributed in the plane of aircraft OX axles.Referential is selected according to specific aerial mission.As shown in Figure 1 The distribution form of precise tracking be one embodiment of the present of invention, wherein GK1-GK4 is precise tracking, and ZK1-ZK6 is Attitude control engine.

(2) line of sight rate of aircraft relative target position under carrier coordinate system is calculated Wherein,

(3) the line of sight rate ω of aircraft relative target position and the line of sight rate threshold value controlled for position are compared Ω₁Size：If ω<Ω₁, then the angle theta of carrier coordinate system OZ axles and site error vector is calculated, step (4) is gone to；

Wherein, EPS is data computational accuracy, takes 10^-7；

If ω >=Ω₁, then the angle β of carrier coordinate system OX axles and line of sight rate vector is calculated, step (6) is gone to；

(4) if θ≤pi/2, and r>R, then according to equation below calculation error quaternary number q_e0,q_e1,q_e2,q_e3：

If θ>Pi/2, and r>R, then according to equation below calculation error quaternary number q_e0,q_e1,q_e2,q_e3：

If r≤R, step-up error quaternary number q_e0=1, q_e1=0, q_e2=0, q_e3=0.Wherein, R is stopping track control The safe distance of system.

(5) instructed according to the gesture stability of error quaternion application posture adjustment law generation aircraft, and then form attitude control hair The switching logic of motivation, the posture of adjustment aircraft makes the angle theta between its carrier OZ axles and site error vector be intended to zero；Together When, according to the angle theta of carrier coordinate system OZ axles and site error vector, the track of aircraft is adjusted, makes carrier coordinate system OZ axles Site error between direction aircraft and targetIt is intended to zero, and exits；

According to the specific method of the track of the angle theta of carrier coordinate system OZ axles and site error vector adjustment aircraft such as Under：Perpendicular to carrier aircraft OX axles and pass through barycenter plane in OY axles and OZ axles on configure precise tracking, OY axles On precise tracking do not work, according to θ control OZ axles on precise tracking：If Θ≤θ≤π-Θ, using Z passages Position control law generate the position control instruction of aircraft, and then form the switching logic of precise tracking, control OZ axles On precise tracking make the site error between carrier coordinate system OZ direction of principal axis aircraft and targetIt is intended to zero；If θ< Θ or θ>π-Θ, then the precise tracking on OZ axles do not work；Wherein, Θ is the threshold value related to attitude angle control accuracy, is taken 1.5~2 times of attitude angle control accuracy.

(6) if β≤pi/2, according to equation below calculation error quaternary number q_e0,q_e1,q_e2,q_e3：

If β>Pi/2, then according to equation below calculation error quaternary number q_e0,q_e1,q_e2,q_e3：

(7) instructed according to the gesture stability of error quaternion application posture adjustment law generation aircraft, and then form attitude control hair The switching logic of motivation, the posture of adjustment aircraft tends to the angle β between carrier coordinate system OX axles and line of sight rate vector In zero；Meanwhile, according to carrier coordinate system OX axles and the angle β of line of sight rate vector, adjust the track of aircraft, specific method It is as follows：

Perpendicular to carrier aircraft OX axles and pass through barycenter plane in OY axles and OZ axles on configure precise tracking； If the precise tracking on Θ≤β≤π-Θ, OY axles and OZ axles does not work；If β<Θ or β>π-Θ, then according to sight The amplitude ω of the angular speed and line of sight rate threshold value Ω for speed control₂Size selection Y passages and Z passages controlling party Formula：If ω<Ω₂, then the position control instruction of the position control law generation aircraft of application Y passages and Z passages, and then shape Into the switching logic of precise tracking, the precise tracking on control OY axles and OZ axles makes flight in carrier coordinate system YOZ planes Site error between device and targetIt is intended to zero；If ω >=Ω₂, then using Y passages and the speed control of Z passages The rate control instruction of law generation aircraft, and then formed on the switching logic of precise tracking, control OY axles and OZ axles Precise tracking makes the velocity error between aircraft and target in carrier coordinate system YOZ planesIt is intended to zero.

Wherein, Ω₁=0.1 Ω~0.5 Ω, Ω₂=0.5 Ω~0.8 Ω, wherein Ω=a/V_max, a is for orbits controlling Pulsed motor produce acceleration, V_maxFor aircraft along the maximum flying speed with target location line direction, the maximum Flying speed is limited by factors such as fuel, it is necessary to be designed according to specific aerial mission；R is the peace for stopping orbits controlling Full distance, R=1.2V_min/ Ω, V_minFor velocity control accuracy.

Posture adjustment rule uses the control law based on error quaternion form, its calculation formula and control parameter in the present invention Design method bibliography《Quaternion Feedback Regulator for Spacecraft Eigenaxis Rotation》(B.Wie,H.Weiss and A.Arapostathis,Journal of Guidance,Control and Dynamics, Vol.12, No.3, May~June, 1989).Posture adjustment rule act as in the present invention：According to error quaternion By the state of the pose adjustment of aircraft to target.

Y channel positions control law, Z channel positions control law and Y channel speeds control law, Z that the present invention is used Channel speed control law uses the control law of Traditional PID form, inputs as position error signal and speed error signal, Its calculation formula and the reference of control parameter design method《Advanced PID control MATLAB is emulated》, Liu Jinkun, electronics industry publication Society, 2011 or other reference books related to Automatic Control Theory.Single channel (Y passages or Z passages) position is controlled in the present invention The site error for acting as reducing the passage of rule processed, the acting as of single channel (Y passages or Z passages) speed control rule subtracts The velocity error of the small passage.

The content not being described in detail in description of the invention belongs to the known technology of those skilled in the art.

Claims (8)

1. a kind of motor-driven method in spacecraft position under the conditions of drive lacking, it is characterised in that comprise the following steps：

(1) site error under carrier coordinate system between aircraft current location and target location is obtainedAnd aircraft Velocity error between present speed and target velocityCalculating aircraft and the relative distance r of target location；

$r=\sqrt{{\left(r_{ex}^b\right)}^2+{\left(r_{ey}^b\right)}^2+{\left(r_{ez}^b\right)}^2}$

Described carrier coordinate system O-XYZ origin O is located at the barycenter of aircraft, and OX axles are along aircraft longitudinal direction, and OY axles are in flight Device is pointed up in longitudinally asymmetric face, and OZ axles constitute right-handed coordinate system with OX axles and OY axles；

For the site error under carrier coordinate system between aircraft current location and target location x-axis direction component； For the site error under carrier coordinate system between aircraft current location and target location y-axis direction component；For carrier The component of site error under coordinate system between aircraft current location and target location in z-axis direction；

Component of the velocity error for being aircraft under carrier coordinate system between present speed and target velocity in x-axis direction；Component of the velocity error for being aircraft under carrier coordinate system between present speed and target velocity in y-axis direction；For Component of velocity error of the aircraft between present speed and target velocity in z-axis direction under carrier coordinate system；

(2) line of sight rate of aircraft relative target position under carrier coordinate system is calculated Wherein,

$\left\{\begin{array}{c}{\mathrm{\&omega;}}_x=(r_{ey}^b{v}_{ez}^b-r_{ez}^b{v}_{ey}^b)/r^2\\ {\mathrm{\&omega;}}_y=(r_{ez}^b{v}_{ex}^b-r_{ex}^b{v}_{ez}^b)/r^2\\ {\mathrm{\&omega;}}_z=(r_{ex}^b{v}_{ey}^b-r_{ey}^b{v}_{ex}^b)/r^2\end{array}\right.;$

ω_xFor aircraft relative target position under carrier coordinate system line of sight rate x-axis direction component；ω_ySat for carrier Component of the line of sight rate of the lower aircraft relative target position of mark system in y-axis direction；ω_zFor aircraft phase under carrier coordinate system To the line of sight rate of target location z-axis direction component；

(3) the line of sight rate ω of aircraft relative target position and the line of sight rate threshold value Ω controlled for position are compared₁'s Size：If ω<Ω₁, then the angle theta of carrier coordinate system OZ axles and site error vector is calculated, step (4) is gone to；

$\mathrm{\&theta;}=\left\{\begin{array}{cc}0,& r<EPS\\ arccos(r_{ez}^b/r),& r\&GreaterEqual;EPS\end{array}\right.$

Wherein, EPS is data computational accuracy；

If ω >=Ω₁, then the angle β of carrier coordinate system OX axles and line of sight rate vector is calculated, step (6) is gone to；

$\mathrm{\&beta;}=\left\{\begin{array}{cc}0,& \mathrm{\&omega;}<EPS\\ arccos({\mathrm{\&omega;}}_x/\mathrm{\&omega;}),& \mathrm{\&omega;}\&GreaterEqual;EPS\end{array}\right.$

(4) if θ≤pi/2, and r>R, then according to equation below calculation error quaternary number q_e0,q_e1,q_e2,q_e3：

$\left\{\begin{array}{c}{q}_{e0}=\sqrt{0.5\&CenterDot;(1+r_{zx}^b/r)}\\ q_{e1}=-0.5\&CenterDot;r_{ey}^b/{\mathrm{rq}}_{e0}\\ q_{e2}=0.5\&CenterDot;r_{ex}^b/{\mathrm{rq}}_{e0}\\ q_{e3}=0\end{array}\right.$

If θ>Pi/2, and r>R, then according to equation below calculation error quaternary number q_e0,q_e1,q_e2,q_e3：

$\left\{\begin{array}{c}{q}_{e0}=\sqrt{0.5\&CenterDot;(1+r_{zx}^b/r)}\\ q_{e1}=0.5\&CenterDot;r_{ey}^b/{\mathrm{rq}}_{e0}\\ q_{e2}=-0.5\&CenterDot;r_{ex}^b/{\mathrm{rq}}_{e0}\\ q_{e3}=0\end{array}\right.$

If r≤R, step-up error quaternary number q_e0=1, q_e1=0, q_e2=0, q_e3=0；Wherein, R is stopping orbits controlling Safe distance；

(5) posture for adjusting aircraft according to error quaternion makes the angle theta between carrier coordinate system OZ axles and site error vector It is intended to zero；Meanwhile, the track of aircraft is adjusted, makes the site error between carrier coordinate system OZ direction of principal axis aircraft and targetIt is intended to zero, and exits；

(6) if β≤pi/2, according to equation below calculation error quaternary number q_e0,q_e1,q_e2,q_e3：

$\left\{\begin{array}{c}{q}_{e0}=\sqrt{0.5\&CenterDot;(1+{\mathrm{\&omega;}}_x/\mathrm{\&omega;})}\\ q_{e1}=0\\ q_{e2}=0.5\&CenterDot;{\mathrm{\&omega;}}_z/\mathrm{\&omega;}{q}_{e0}\\ q_{e3}=-0.5\&CenterDot;{\mathrm{\&omega;}}_y/{\mathrm{\&omega;q}}_{e0}\end{array}\right.$

If β>Pi/2, then according to equation below calculation error quaternary number q_e0,q_e1,q_e2,q_e3：

$\left\{\begin{array}{c}{q}_{e0}=\sqrt{0.5\&CenterDot;(1+{\mathrm{\&omega;}}_x/\mathrm{\&omega;})}\\ q_{e1}=0\\ q_{e2}=-0.5\&CenterDot;{\mathrm{\&omega;}}_z/\mathrm{\&omega;}{q}_{e0}\\ q_{e3}=0.5\&CenterDot;{\mathrm{\&omega;}}_y/{\mathrm{\&omega;q}}_{e0}\end{array}\right.$

(7) posture for adjusting aircraft according to error quaternion makes the angle between carrier coordinate system OX axles and line of sight rate vector β is intended to zero；Meanwhile, the track of aircraft is adjusted, the position in carrier coordinate system YOZ planes between aircraft and target is missed DifferenceIt is intended to zero, or makes the velocity error in carrier coordinate system YOZ planes between aircraft and targetTend to In zero.

2. the motor-driven method in spacecraft position under the conditions of a kind of drive lacking according to claim 1, it is characterised in that： The specific method of the track of adjustment aircraft is as follows in the step (5)：Perpendicular to carrier aircraft OX axles and passing through barycenter Plane in OY axles and OZ axles on the precise tracking that configures on precise tracking, OY axles do not work, controlled according to θ on OZ axles Precise tracking：If Θ≤θ≤π-Θ, the precise tracking on control OZ axles makes carrier coordinate system OZ direction of principal axis fly Site error between device and targetIt is intended to zero；If θ<Θ or θ>π-Θ, then the precise tracking on OZ axles do not work； Wherein, Θ is the threshold value related to attitude angle control accuracy.

3. the motor-driven method in spacecraft position under the conditions of a kind of drive lacking according to claim 1, it is characterised in that： The track specific method of adjustment aircraft is as follows in the step (7)：Perpendicular to carrier aircraft OX axles and passing through barycenter Precise tracking is configured on OY axles and OZ axles in plane；If the precise tracking on Θ≤β≤π-Θ, OY axles and OZ axles Do not work；If β<Θ or β>π-Θ, then according to the amplitude ω of line of sight rate and the line of sight rate threshold for speed control Value Ω₂Size：If ω<Ω₂, then the precise tracking on OY axles and OZ axles is controlled to make flight in carrier coordinate system YOZ planes Site error between device and targetIt is intended to zero；If ω >=Ω₂, then the precise tracking on OY axles and OZ axles is controlled Make the velocity error between aircraft and target in carrier coordinate system YOZ planesIt is intended to zero.

4. the motor-driven method in spacecraft position under the conditions of a kind of drive lacking according to claim 1 or 2 or 3, it is special Levy and be：Described is used for the line of sight rate threshold value Ω that position is controlled₁=0.1 Ω~0.5 Ω, wherein Ω=a/V_max, a is The acceleration that pulsed motor for orbits controlling is produced, V_maxFly for aircraft along with the maximum in target location line direction Scanning frequency degree.

5. the motor-driven method in spacecraft position under the conditions of a kind of drive lacking according to claim 4, it is characterised in that： The safe distance R=1.2V of described stopping orbits controlling_min/ Ω, V_minFor velocity control accuracy.

6. the motor-driven method in spacecraft position under the conditions of a kind of drive lacking according to claim 3, it is characterised in that： The described line of sight rate threshold value Ω for speed control₂=0.5 Ω~0.8 Ω, wherein Ω=a/V_max, a is for track The acceleration that the pulsed motor of control is produced, V_maxFor aircraft along the maximum flying speed with target location line direction.

7. the motor-driven method in spacecraft position under the conditions of a kind of drive lacking according to Claims 2 or 3, its feature exists In：Described threshold value Θ takes 1.5~2 times of attitude angle control accuracy.

8. the motor-driven method in spacecraft position under the conditions of a kind of drive lacking according to claim 1, it is characterised in that： Described data computational accuracy EPS takes 10-⁷。