CN115359095B - Universal motion platform tracking and guiding calculation method - Google Patents
Universal motion platform tracking and guiding calculation method Download PDFInfo
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- CN115359095B CN115359095B CN202211277344.8A CN202211277344A CN115359095B CN 115359095 B CN115359095 B CN 115359095B CN 202211277344 A CN202211277344 A CN 202211277344A CN 115359095 B CN115359095 B CN 115359095B
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Abstract
The invention relates to a general tracking, guiding and calculating method for a moving platform, which belongs to the technical field of photoelectric tracking, and is used for acquiring the ground-fixed coordinates of a target and the geodetic coordinates of the moving platform in real time or acquiring the azimuth pitching coordinates of the target relative to a horizontal coordinate system of the moving platform in real time and calculating the horizontal coordinates of the target at the moving platform; the method comprises the steps of obtaining the three-axis attitude of the motion platform in real time, calibrating the three-axis installation attitude angle of an observation system coordinate system relative to the motion platform coordinate system, and resolving the spherical coordinate of the target in the observation system coordinate system based on the horizontal coordinate of the target at the motion platform to obtain tracking guidance data.
Description
Technical Field
The invention belongs to the technical field of photoelectric tracking, and particularly relates to a general tracking guidance calculation method for a motion platform.
Background
In a photoelectric observation system on a motion platform such as a vehicle, a ship, a machine and the like, operations such as tracking guidance, capturing, tracking closed loop, target identification and the like are often required to be carried out on an observation object, wherein the tracking guidance is a starting point of the whole link, and a guidance method and precision are crucial to the target capturing efficiency. Because the platform is moving and the position and the posture of the platform change in real time, a calculation model needs to be established to complete tracking and guiding calculation. Due to various installation modes of the photoelectric observation system, such as vertical type, horizontal type and nacelle type, different calculation models are caused, and the universality is lacked.
Disclosure of Invention
In order to solve the above problems, a general motion platform tracking and guiding calculation method is proposed.
In order to achieve the purpose, the invention provides the following technical scheme:
a general motion platform tracking and guiding calculation method comprises the following steps:
s100, acquiring the ground-fixed coordinate of a target and the geodetic coordinate of a moving platform in real time, or acquiring the azimuth pitching coordinate of the target relative to a ground coordinate system of the moving platform in real time, and calculating the ground coordinate of the target at the moving platform;
s200, acquiring the three-axis attitude of the motion platform in real time, calibrating the three-axis installation attitude angle of an observation system coordinate system relative to the motion platform coordinate system, and resolving the spherical coordinates of the target in the observation system coordinate system based on the horizontal coordinates of the target at the motion platform to obtain tracking guidance data.
The invention is further configured that, in step S100, when the target is an unnatural celestial body, calculating the horizon coordinate of the target at the motion platform, including the following steps:
s101, acquiring geodetic coordinates of a motion platform in real time, and resolving the geodetic coordinates of the motion platform based on the geodetic coordinates of the motion platform;
s102, acquiring the earth-fixed coordinates of a target in real time, and calculating the station center earth-fixed coordinates of the target at the motion platform based on the earth-fixed coordinates of the motion platform;
s103, calculating the horizon coordinate of the target at the moving platform based on the station center earth-fixed coordinate of the target and the geodetic coordinate of the moving platform.
The invention is further arranged that in step S101, the geodetic coordinates of the motion platform obtained in real time are
Resolving the geodetic coordinates of the motion platform into
The method specifically comprises the following steps:
wherein, the first and the second end of the pipe are connected with each other,
a represents the equatorial radius of the earth, and
f represents the oblateness of the earth, and
l represents longitude, B represents latitude, and H represents altitude.
The invention is further configured that in step S102, the real-time acquired earth-fixed coordinates of the target are
Based on the ground-fixed coordinates of the motion platform
Resolving the station center earth-fixed coordinates of the target at the motion platform
The method specifically comprises the following steps:
the invention is further arranged that, in step S103, the station center earth-fixed coordinates based on the object are determined
And geodetic coordinates of said motion platform
Calculating the horizon coordinate of the target at the moving platform
The method specifically comprises the following steps:
wherein, the first and the second end of the pipe are connected with each other,
a coordinate transformation matrix representing a rotation by an angle θ about an x-axis or a y-axis or a z-axis, axis = x or y or z, representing an axis about which the coordinates are rotated;
depending on the axis about which the coordinates are rotated,
the specific expressions of (a) are respectively:
the invention further provides that, in step S100, when the target is a planet, the horizon coordinate of the target at the motion platform is calculated, specifically:
acquiring azimuth and elevation coordinates (A, E) of a target relative to a horizon coordinate system at a moving platform in real time, wherein the horizon coordinate of the target at the moving platform is
And is made of
A represents an azimuth angle, and E represents a pitch angle.
The invention is further arranged that step S200 comprises the steps of:
s201, acquiring a three-axis attitude of the motion platform in real time, and resolving a rectangular coordinate of the target in a coordinate of the motion platform based on a horizontal coordinate of the target at the motion platform;
s202, calibrating a three-axis installation attitude angle of a coordinate system of an observation system relative to a coordinate system of a motion platform, and resolving a rectangular coordinate of the target in the observation system based on the rectangular coordinate of the target in the coordinate of the motion platform;
s203, resolving the spherical coordinates of the target in the coordinate system of the observation system based on the rectangular coordinates of the target in the observation system to obtain tracking guidance data.
The invention is further configured that, in step S201, the three-axis posture of the motion platform is obtained in real time as
Calculating rectangular coordinates of the object within the motion platform coordinates based on the horizon coordinates of the object at the motion platform
The method specifically comprises the following steps:
The invention is further arranged that in step S202, the three-axis installation attitude angle of the coordinate system of the observation system relative to the coordinate system of the motion platform is calibrated
Solving the rectangular coordinate of the target in the observation system based on the rectangular coordinate of the target in the coordinates of the motion platform
The method specifically comprises the following steps:
The present invention is further configured that, in step S203, the global coordinate (α, e, R) of the target in the observation system coordinate system is calculated based on the rectangular coordinate of the target in the observation system, specifically:
in the case of a vertical type observation system,
；
for a horizontal-type observation system, the observation angle is set to be zero,
；
in the case of a pod-type observation system,
；
where α represents an azimuth angle, e represents a pitch angle, and R represents a pitch distance.
The invention is further arranged that in step S200, the three-axis installation attitude angle of the coordinate system of the observation system relative to the coordinate system of the motion platform
The calibration method comprises the following steps:
fixedly mounting an observation system on a motion platform, and acquiring the geodetic coordinates and the three-dimensional attitude of the motion platform in real time
；
Selecting more than three points with known positions in different directions of space as calibration points, and acquiring the earth-fixed coordinates of the calibration points;
obtaining the horizon coordinate of the target at the motion platform based on the step S100, and resolving the position of the calibration point relative to the horizon coordinate system of the observation system
；
Adjusting the three-axis installation attitude angle of the observation system to observe the calibration points to obtain the observation positions of the calibration points
Resolving three-axis installation attitude angle of coordinate system of observation system relative to coordinate system of motion platform
。
The invention is further arranged that the horizon coordinate of the target at the motion platform is obtained based on the step S100
And resolving the position of the calibration point relative to the horizon coordinate system of the observation system
The method specifically comprises the following steps:
The invention is further arranged that the observation position of each calibration point is obtained
Resolving three-axis installation attitude angle of coordinate system of observation system relative to coordinate system of motion platform
The method specifically comprises the following steps:
the method comprises the steps of obtaining a heading angle of a motion platform, obtaining an attitude elevation angle of the motion platform, obtaining a roll angle of the motion platform, obtaining an azimuth angle, obtaining a roll angle of the motion platform, and obtaining a pitch angle.
The beneficial effects of the invention are:
1. by adopting a coordinate system transformation method, a general motion platform tracking guidance calculation method is established, real-time calculation of target tracking guidance data is realized, and the method has a good application prospect in the field of motion platform photoelectric observation.
2. And aiming at observation systems in different installation modes such as vertical type, horizontal type and pod type, a calculation method of corresponding differences is provided.
3. And aiming at different types of targets, a difference calculation method corresponding to the targets is provided.
4. The coordinate system of the observation system is an abstract concept relative to the three-axis installation attitude angle of the coordinate system of the motion platform, the abstract concept is invisible in physical space and cannot be obtained through tool measurement, accurate data can be obtained through a calibration method of the coordinate system of the observation system relative to the three-axis installation attitude angle of the coordinate system of the motion platform, and accuracy of target tracking guide data is improved.
Drawings
FIG. 1 is a flow diagram of a general motion platform tracking guidance calculation method;
FIG. 2 is a schematic diagram of an observation system coordinate system, a motion platform coordinate system, and a horizon coordinate system;
FIG. 3 is a schematic view of a vision system in a different installation;
fig. 3 (a) is a schematic view of a vertical observation system, fig. 3 (b) is a schematic view of a horizontal observation system, and fig. 3 (c) is a schematic view of a pod-type observation system.
Detailed Description
In order to make the technical solutions of the present invention better understood, the following description of the technical solutions of the present invention with reference to the accompanying drawings of the present invention is made clearly and completely, and other similar embodiments obtained by a person of ordinary skill in the art without any creative effort based on the embodiments in the present application shall fall within the protection scope of the present application. In addition, directional terms such as "upper", "lower", "left", "right", etc. mentioned in the following embodiments are directions with reference to the drawings only, and thus, the directional terms used are intended to illustrate rather than limit the inventive concept.
The first embodiment is as follows:
as shown in fig. 1, a general motion platform tracking guidance calculation method includes the following steps:
s100, acquiring the earth-fixed coordinates of the target and the geodetic coordinates of the moving platform in real time, or acquiring the azimuth pitching coordinates of the target relative to the horizon coordinate system of the moving platform in real time, and resolving the horizon coordinates of the target at the moving platform.
The earth-fixed coordinate system relates to an earth-fixed coordinate system, and the earth-fixed coordinate system is a coordinate system fixed on the earth and rotating together with the earth. The geodetic coordinates refer to a geodetic coordinate system which is established in geodetic surveying with reference to an ellipsoid as a reference surface, and the positions of ground points are expressed in terms of longitude, latitude, and height (altitude). The horizontal coordinate system at the moving platform takes the location of the moving platform as a central point, and the horizontal line of the location of the moving platform is taken as a base plane.
Preferably, when the target is an unnatural celestial body, including a ground fixed point target, an aerial rotor unmanned aerial vehicle target, a floating balloon, an airplane, a missile, a satellite and the like, the method for calculating the horizon coordinate of the target at the moving platform comprises the following steps:
s101, acquiring the geodetic coordinates of the motion platform in real time, and resolving the geodetic coordinates of the motion platform based on the geodetic coordinates of the motion platform.
Preferably, the geodetic coordinates of the motion platform acquired in real time are
Resolving the geodetic coordinates of the motion platform into
The method specifically comprises the following steps:
wherein, the first and the second end of the pipe are connected with each other,
a represents the equatorial radius of the earth, and
f represents the oblateness of the earth, an
L represents longitude, B represents latitude, and H represents altitude.
S102, acquiring the earth-fixed coordinates of the target in real time, and calculating the station center earth-fixed coordinates of the target at the moving platform based on the earth-fixed coordinates of the moving platform.
Preferably, the real-time acquired ground-fixed coordinate of the target is
Based on the ground-fixed coordinates of the motion platform
Resolving the station center earth-fixed coordinates of the target at the motion platform
The method specifically comprises the following steps:
s103, calculating the horizon coordinate of the target at the moving platform based on the station center earth-fixed coordinate of the target and the geodetic coordinate of the moving platform.
Preferably, the station center earth-fixed coordinate based on the target
And geodetic coordinates of the motion platform
Calculating the horizon coordinate of the target at the motion platform
The method specifically comprises the following steps:
wherein the content of the first and second substances,
a coordinate transformation matrix representing a rotation by an angle θ about an x-axis or a y-axis or a z-axis, axis = x or y or z, representing an axis about which the coordinates are rotated;
depending on the axis about which the coordinates are rotated,
the specific expressions of (a) are respectively:
preferably, when the target is a planet, the method includes resolving a horizon coordinate of the target at the motion platform, specifically:
acquiring azimuth and elevation coordinates (A, E) of a target relative to a horizon coordinate system at a moving platform in real time, wherein the horizon coordinate of the target at the moving platform is
And is and
a represents an azimuth angle, and E represents a pitch angle.
The planets mainly refer to natural celestial bodies such as heaven, moon, star and the like, and for such targets, the earth-center-earth-fixed coordinates of the planets cannot be obtained generally
And only the azimuth and elevation data (A, E) of the horizon coordinate system at the position of the target relative to the moving platform can be obtained, the target distance R =1 is implied, R is reduced in the process of calculating the tracking guide angle, so that the R setting has little influence, and the processing skill is often used in the tracking guide calculation of the remote target.
S200, acquiring the three-axis attitude of the motion platform in real time, calibrating the three-axis installation attitude angle of an observation system coordinate system relative to the motion platform coordinate system, and resolving the spherical coordinates of the target in the observation system coordinate system based on the horizontal coordinates of the target at the motion platform to obtain tracking guidance data.
Preferably, the platform coordinate system refers to a coordinate system defined on a moving platform, such as an airplane, a vehicle, a ship, a satellite, etc. Taking an airplane as an example, the coordinate origin may be defined at any position on the airplane, where the x-axis points to the right wing direction, the y-axis points to the nose direction, and the z-axis points to the sky direction. In the tracking and guiding calculation process of the motion platform, the real-time position and the attitude of the motion platform need to be introduced, the position of the motion platform can be measured by instruments such as a GPS (global positioning system) and the like or can be obtained by astronomical calculation, and the attitude can be measured by angle sensors (a gyroscope, inertial navigation and the like) and comprises a course angle alpha, a pitch angle beta and a roll angle gamma. In view of the fact that the pose change of the motion platform coordinate system needs to be reflected in real time by means of measurement equipment such as inertial navigation equipment, the motion platform coordinate system is generally represented by an installed inertial navigation coordinate system, the origin of the motion platform coordinate system is in the inertial navigation center, and the three-axis direction is the three edge directions of an inertial navigation product.
The attitude angle has definite definition and value range: the heading angle can be defined as the included angle between the heading angle and the true north direction, the true north is 0 DEG, the heading angle is counted clockwise and takes the value
(ii) a The pitch angle is defined as the angle between the head direction and the horizontal plane, 0 degree is in the horizontal plane, positive is above the plane and negative is below the plane, and the value is taken
(ii) a The roll angle is defined as the angle between the lateral direction and the horizontal plane, the downward pressure is positive, the upward pressure is negative, and the value is taken
. The three attitude angles are rotation angles around the z, x, y axes of the motion platform coordinate system, respectively.
The observation system coordinate system refers to a coordinate system defined on the observation system, and is generally characterized by two angles, such as an observation azimuth angle and an observation pitch angle. The definition depends on the rotary joint of the observation system and is marked by appointing an angle zero position and an angle increasing and decreasing direction. The zero position includes an equipment zero position and a space zero position, the equipment zero position is a zero scale mark on an angle sensor carried by the observation system, the space zero position refers to a defined zero position in a specific occasion, for example, the north direction is generally defined as an azimuth angle zero position, and the horizontal plane is defined as a pitch angle zero position.
For a static and flat ground observation platform, the unification of the zero position of equipment and the zero position of space is a work which is often done, and the work is called orientation and pitching zero position calibration and is generally determined by observing the Polaris. For observation systems on motion platforms such as vehicles, ships and machines, generally, zero position unified work is not carried out, but the pose relation of an observation system coordinate system relative to a motion platform coordinate system is calibrated, as shown in fig. 2. Under the condition of not considering the relative offset of the origin of the two coordinate systems, the relative relationship of the two coordinate systems can be represented by three attitude angles, namely a heading angle u, a pitch angle v and a rolling angle w of the coordinate system of the observation system relative to the coordinate system of the motion platform, and the three angles are calibrated or an attitude rotation matrix combined with the three angles is calibrated
. Since these three angles function in the form of three attitude rotation matrices, they are generally used together as a composite attitude matrix, namely:
specifically, step S200 includes the following steps:
s201, acquiring the three-axis attitude of the motion platform in real time, and resolving the rectangular coordinate of the target in the motion platform coordinate based on the horizontal coordinate of the target at the motion platform.
Preferably, the three-axis attitude of the motion platform is acquired in real time as
Calculating rectangular coordinates of the object within the motion platform coordinates based on the horizon coordinates of the object at the motion platform
The method specifically comprises the following steps:
S202, calibrating a three-axis installation attitude angle of a coordinate system of the observation system relative to a coordinate system of the motion platform, and resolving a rectangular coordinate of the target in the observation system based on the rectangular coordinate of the target in the coordinate of the motion platform.
Preferably, the three-axis installation attitude angle of the coordinate system of the observation system relative to the coordinate system of the motion platform is calibrated
Solving the rectangular coordinate of the target in the observation system based on the rectangular coordinate of the target in the coordinates of the motion platform
The method specifically comprises the following steps:
S203, resolving the spherical coordinates of the target in the coordinate system of the observation system based on the rectangular coordinates of the target in the observation system to obtain tracking guidance data.
Preferably, as shown in fig. 3, the calculating the object spherical coordinate (a, e, R) in the observation system coordinate system based on the object rectangular coordinate in the observation system specifically includes:
in the case of a vertical observation system,
；
for a horizontal-type observation system, the observation system,
；
in the case of a pod-type observation system,
；
where α represents an azimuth angle, e represents a pitch angle, and R represents a pitch distance.
The vertical observation system, the horizontal observation system and the pod observation system are mature prior art and do not belong to the invention point of the invention, so the installation mode is explained by adopting the existing photos.
The above calculation formulas for the horizontal observation system and the pod observation system are not simple and depend on the coordinate definition of the actual system, including the zero position and the increasing and decreasing direction of the angle a and e. The definition of azimuth angle and pitch angle in a vertical observation system is basically defined in a common way: the true north is an azimuth zero line and is counted clockwise; the horizontal plane is a pitching zero-position plane and is up-right and down-negative.
In step S200, the coordinate system of the observation system is corresponding to the three-axis installation attitude angle of the coordinate system of the motion platform
The calibration method comprises the following steps:
fixedly mounting an observation system on a motion platform, and acquiring geodetic coordinates and three-dimensional postures of the motion platform in real time
。
Selecting more than three points with known positions in different directions of the space as calibration points, and acquiring the earth-fixed coordinates of the calibration points.
Obtaining the horizon coordinate of the target at the motion platform based on the step S100, and resolving the position of the calibration point relative to the horizon coordinate system of the observation system
。
Specifically, the horizontal coordinate of the target at the motion platform is obtained based on step S100
And resolving the position of the calibration point relative to the horizon coordinate system of the observation system
The method specifically comprises the following steps:
adjusting the three-axis installation attitude angle of the observation system to observe the calibration points to obtain the observation positions of the calibration points
Resolving the three-axis installation attitude angle of the coordinate system of the observation system relative to the coordinate system of the motion platform
。
Preferably, the obtaining of the observed position of each calibration point
Resolving the three-axis installation attitude angle of the coordinate system of the observation system relative to the coordinate system of the motion platform
The method specifically comprises the following steps:
wherein, theta represents the course angle of the moving platform, beta represents the attitude elevation angle of the moving platform, and gamma represents the roll angle of the moving platform.
When the target is a star, the number is almost infinite, with millions or more. In most cases, the observation system does not predict which fixed star will be seen, which causes no pertinence in preparation of tracking guidance data, but real-time calculation of all fixed stars is impractical and unnecessary, and the invention does not describe the tracking guidance calculation of fixed stars in detail.
While the invention has been described in detail in connection with what is presently considered to be the most practical and preferred embodiment, it is to be understood that the invention is not to be limited to the disclosed embodiment, but on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.
Claims (9)
1. A general tracking and guiding calculation method for a motion platform is characterized by comprising the following steps:
acquiring the earth-fixed coordinates of a target and the geodetic coordinates of a moving platform in real time, or acquiring the azimuth pitching coordinates of the target relative to the horizon coordinate system of the moving platform in real time, and calculating the horizon coordinates of the target at the moving platform;
acquiring the three-axis attitude of the motion platform in real time, calibrating a three-axis installation attitude angle of an observation system coordinate system relative to a motion platform coordinate system, and resolving the spherical coordinate of the target in the observation system coordinate system based on the horizontal coordinate of the target in the motion platform to obtain tracking guidance data;
the calibration method of the three-axis installation attitude angle of the coordinate system of the observation system relative to the coordinate system of the motion platform specifically comprises the following steps:
fixedly installing an observation system on a motion platform, and acquiring geodetic coordinates and three-dimensional postures of the motion platform in real time;
selecting more than three points with known positions in different directions of space as calibration points, and acquiring the ground-fixed coordinates of the calibration points;
based on the horizon coordinate of the target at the motion platform, calculating the position of a coordinate point relative to the horizon coordinate system of the observation system;
and adjusting the three-axis installation attitude angle of the observation system to observe the calibration points, obtaining the observation position of each calibration point, and resolving the three-axis installation attitude angle of the coordinate system of the observation system relative to the coordinate system of the motion platform.
2. The method of claim 1, wherein when the target is an unnatural celestial body, calculating the horizon coordinate of the target at the motion platform comprises the following steps:
acquiring geodetic coordinates of a motion platform in real time, and resolving the geodetic coordinates of the motion platform based on the geodetic coordinates of the motion platform;
acquiring the ground-fixed coordinates of a target in real time, and calculating the station center ground-fixed coordinates of the target at the motion platform based on the ground-fixed coordinates of the motion platform;
and calculating the horizon coordinate of the target at the moving platform based on the station center earth-fixed coordinate of the target and the geodetic coordinate of the moving platform.
3. The method according to claim 1, wherein when the target is a planet, the method calculates the horizon coordinate of the target at the motion platform, specifically:
acquiring azimuth and elevation coordinates (A, E) of an object relative to a horizon coordinate system at a moving platform in real time, wherein the horizon coordinate of the object at the moving platform is (x) 2 ，y 2 ，z 2 ) And is and
a represents an azimuth angle, and E represents a pitch angle.
4. The method according to claim 1, wherein the method comprises the steps of obtaining three-axis postures of the motion platform in real time, calibrating three-axis installation posture angles of an observation system coordinate system relative to a motion platform coordinate system, and calculating a spherical coordinate of the target in the observation system coordinate system based on a horizon coordinate of the target at the motion platform to obtain tracking guidance data, and comprises the following steps:
acquiring the three-axis attitude of the motion platform in real time, and resolving the rectangular coordinate of the target in the motion platform coordinate based on the horizontal coordinate of the target at the motion platform;
calibrating a three-axis installation attitude angle of an observation system coordinate system relative to a motion platform coordinate system, and resolving a rectangular coordinate of the target in the observation system based on the rectangular coordinate of the target in the motion platform coordinate;
and resolving the spherical coordinates of the target in the coordinate system of the observation system based on the rectangular coordinates of the target in the observation system to obtain tracking guidance data.
5. The method according to claim 4, wherein the three-axis attitude of the motion platform is obtained in real time as (α, β, γ), and the three-axis attitude is based on the horizon coordinate (x) of the target at the motion platform 2 ，y 2 ，z 2 ) Solving for rectangular coordinates (x) of the target within the motion platform coordinates 3 ，y 3 ，z 3 ) The method specifically comprises the following steps:
6. The method according to claim 5, wherein three-axis installation attitude angles (u, v, w) of an observation system coordinate system relative to a motion platform coordinate system are calibrated, and rectangular coordinates (x, v, w) of the target in the observation system are calculated based on rectangular coordinates of the target in the motion platform coordinates 4 ，y 4 ，z 4 ) The method specifically comprises the following steps:
7. The method of claim 6, wherein the calculation is based on the rectangular coordinates (x) of the target in the observation system 4 ，y 4 ，z 4 ) Calculating a spherical coordinate (alpha, e, R) of the target in an observation system coordinate system, specifically:
in the case of a vertical observation system,
for a horizontal-type observation system, the observation system,
in the case of a pod-type observation system,
where α represents an azimuth angle, e represents a pitch angle, and R represents a pitch distance.
8. The method according to any one of claims 1 to 7, wherein the object has a horizon coordinate (x) at the motion platform 2 ，y 2 ，z 2 ) And resolving the position (A, E) of the calibration point relative to the horizon coordinate system of the observation system, which specifically comprises the following steps:
9. The method according to claim 8, wherein the obtaining of the observed position (a, e) of each calibration point is performed by calculating a three-axis installation attitude angle (u, v, w) of an observation system coordinate system relative to a motion platform coordinate system, specifically:
the method comprises the following steps that theta represents a course angle of a motion platform, beta represents an attitude elevation angle of the motion platform, gamma represents a roll angle of the motion platform, alpha represents an azimuth angle, and e represents a pitch angle.
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