The precise estimate of a projectile trajectory parameters expected in military applications, space travel, rescue and recovery missions, evacuation warnings, games and hobbies, etc. is a challenging task requiring exact analysis of the three-dimensional projectile flight. Currently existing to it solutions are rather complicated and therefore inconvenient. As cited in U.S. Pat. No. 3,748,440 solutions to two dimensional non-linear differential equations are developed in inertial coordinate systems where integrations are performed to obtain X and Y coordinates associated with Latitude and Longitude. Geometric Line of Sight angles is used in U.S. Pat. No. 6,262,680 B1 to track the target in inertial coordinate system. In U.S. Pat. No. 7,605,747 B1 position and velocity vectors are referenced to a non-inertial reference frame such as Earth Centered Earth Fixed (ECEF) when positional registration bias state vector δX represents the sensor position with respect to the ECEF coordinates. Other publications include introduction of either extra degrees of freedom (Burnett, 1962) in order to describe motion in orthogonal two dimensional planes, or consideration of specific initial conditions (Kashiwagi, 1968), or synchronous geo-satellite (Isaacson et al., 1996) utilizing Kalman Filter algorithm, or algorithm (Siouris, 2004) associated with projectile coordinate along its track. Simple solutions from two technical articles found in the websites are just referring to the Coriolis force to proximate the deviation from estimated locations (https://www.phas.ubc.ca/˜berciu/TEACHING/PHYS206/LECTURES/FILES/coriolis.pdf and http://www.marts100.com/projectile.htm).
Lastly, programmed in MATLAB the process of reverse conversion of Sensor Target Measurements into ECEF coordinates allows predicting Sensor Target Measurements at impact location of projectile and enables the radar to direct its beam to the location (I-295 to I-312, DTFA01-88-c-00042, CDRL-EN25, Change 2, Volume I, 6 Sep. 1991).
General and precise solution to the complex problem of a projectile motion would create the grounds for designing a new device highly efficient for any application related to a projectile flight on any planet.
Proposed a new device allowing easily, quickly and accurately predict dependable on time location and speed of a capsule or it debris on any planet: Earth, Mars and Moon included in this application. The working principle of this device is based on developed unique and efficient method to evaluate precise geographical coordinates and speed of a projectile at any moment of its flight for a given launching conditions which could be input manually or transferred from a radar.
Manual input provides with exact parameters of impact allowing to evaluate planned launch efficiency and estimate possible impact outcome, while use of initial radar data predicts projectile flight ahead of time significantly improving radar functionality by enabling to rapidly redirect the radar beam to estimated impact location ahead of impact time.
Ability to use initial radar report provides device with the option of accounting on the mechanical distortions on a projectile flight such as projectile rotation and planet atmospheric conditions resulting in drag force.
is shown.
The following definitions are useful in understanding the process of precise projectile aiming and tracing for rapidly directing the radar sensor beam to geographical impact location of objects.
When projectile velocity way below speed of light, unique solutions for its motion can be developed through application of classical mechanics:
According to the Second Newton's Law general equation of projectile dynamics in non-inertial frame of reference, which rotates with constant angular velocity Ω, is
where
is gravitational force;
Introduction of radial unit vector u and its derivatives:
as well as taking into consideration that r=r·u and
where altitudinal angular speed
while azimuthal angular speed
modifies Eq.1 as:
r{dot over (θ)}2σ Consequent dot multiplication of this equation first by v, then by w and u correspondingly provides with:
Solution of Eqs.2 and 3 leads to the conclusion that r4 [{dot over (φ)}2+({dot over (θ)}+Ω)2 sin2 φ]=const
This constant is actually squared magnitude of projectile angular momentum:
L2=r4[{dot over (φ)}2+({dot over (θ)}+Ω)2 sin2φ]=const (Eq.5)
which could be defined as
L=r2[{dot over (φ)}w+({dot over (θ)}+Ω)v] (Eq.6)
Altitudinal component Lφ={dot over (φ)}w of angular momentum in not affected by rotational frame of reference and is the same as it would be in inertial frame, while azimuthal component Lθ consists of two coaxial vectors: angular momentum L′θ={dot over (θ)}v projectile would have in inertial frame and additional component LΩ=Ωv due to reference frame rotation. Magnitude of angular momentum remains constant, but its direction is constantly changing so its rotating vector creates conic surface.
Substitution Eq.5 into Eq.4 reveals energy −e conservation in non-inertial frame of reference:
Vector of projectile velocity V contains radial Vr={dot over (r)}u, altitudinal Vφ=r{dot over (φ)}w, and azimuthal Vθ=r({dot over (θ)}+Ω)v components and can be presented as V={dot over (r)}u+rω, where rotational velocity in the projectile plane
Magnitude of ω is defined by rotational angle measured in this plane (
Magnitude of angular momentum L2 and projectile total energy −e could be defined from initial condition as
Here Vro={dot over (r)}o=Vo sin ϑo; Vφ
ro, Vo, and ϑo are launching radial coordinate, velocity and angle correspondingly; β is horizontal direction of launch negative for westward and positive for eastward measurement (
Eq.10 provides with the expression for maximum projectile elevation when its radial velocity is zero:
where flight constant K=γ2 M2−2eL2
Substitution Eq.10 into Eq.7 reveals
with solution for flight time t:
Substitution of Eq.2 solution:
(Ω+{dot over (θ)})·r2·sin2φ=Lθ=const
into Eq.5 leads to
providing with altitudinal coordinate φ:
where
function
and coefficient
indicates altitudinal coordinate increase or decrease depending on the launching direction β.
Substituting expression for time increment dt expressed in terms of Eq.12 into Eq.8 obtain
with solution for rotational angle :
Employment of the latest equation (Eq.8) together with Eqs.6 and 15 reveals:
with the solution for azimuthal coordinate θ:
where coefficient
takes into account azimuthal coordinate increase or decrease depending on the launching direction β.
Eqs.13, 15, 16, and 20 is actually the parametric system of equations defining a projectile trajectory in non-inertial frame of reference and require for their completion equation of hodograph, i.e. velocity vs. time dependence.
Due to energy conservation (Eq.8) projectile speed at any moment of time can be defined as
Introduction of projectile ascending speed Va, speed at maximum elevation Vmin, and descending speed Vd breaks Eq.21 into three parts logically fitting trajectory analysis:
Derived above equations of projectile motion in three-dimensional non-inertial frame of reference indicate that, due to the presence of additional component of angular momentum, projectile coordinates are going to be shifted in westward direction “twisting” the trajectory plane.
The other obvious reasons for coordinates shift is possible projectile rotation, atmospheric conditions, drag force etc. which are actually taken into account by a radar.
Actually, any point of projectile trajectory can be treated as the “launching” (with script 0) one. Next to it position (with script 1) traced by a radar is so close to the initial that both projectile positions can be assumed to be in flat two dimensional plane. Thus, these two radar records can be treated as a projectile moving in inertial frame of reference (Ω=0) under existing deflecting its flight conditions.
Equation of projectile dynamics according to Eq.1 in this case becomes
Introduction (
and taking into consideration that r=r·U;
and L=r2=ro·Vo·cos
o provides with the solution:
which, in turn, reveals the following expressions for two consequent projectile positions:
where, cos 1=cos(φ1−φo)·cos [(θ1−θo)·sin φ1] what is the cosine rule of spherical right triangle centered at the origin of spherical coordinates (
If the set of longitude/latitude radar coordinates is Lo0/La0 and Lo1/La1 then altitudinal and azimuthal angles are defined as follows: θ0=Lo0; θ1=Lo1; φ0=90°−La0; φ1=90°−La1 Application of energy conservation to radar provided positions:
reveals the solution for initial (launching) angle
positive for ascending projectile and negative for descending.
Direction of “launch” can be defined from the sine rule of spherical right triangle (
Establishing “launching” parameters of a projectile flight from two consequent radar measurements allows to further predict projectile real behavior ahead of time.
Developed technique creates the basis of device operation (
Features of proposed device include:
While illustrative embodiment of the invention has been shown and described, numerous variations and alternate embodiments, including eliminating one or more of the steps or elements presented herein, will occur to those skilled in the art. Such variations and alternate embodiments are contemplated and can be made without departing from the spirit and scope of the invention as mentioned in the appended claims.
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4288049 | Alpers | Sep 1981 | A |
5102065 | Couderc | Apr 1992 | A |
5474255 | Levita | Dec 1995 | A |
6262680 | Muto | Jul 2001 | B1 |
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Number | Date | Country | |
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20180356510 A1 | Dec 2018 | US |
Number | Date | Country | |
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Parent | 14984870 | Dec 2015 | US |
Child | 15974614 | US |