Abstract
Methods, systems, and devices solving Euler’s rotational rigid body equation of motion, formed within two non-inertial frames of reference, that determine the vector inconstant variables of angular acceleration, velocity, and trajectory using a single piezoresistive accelerometer sensor, an AC coupling algorithm and 1.sup.st and 2.sup.nd running integrals to in-flight acquire rotational inconstants in high-density Terramedia Terraflight and determine a Penetrator’s loading profiles and method to parse vector Terraflight for rotational Pitch and Yaw enabling precision trajectory tracking utilizing three axial facing piezoresistive accelerometers, a differencing algorithm and 1.sup.st and 2.sup.nd running integrals enabling Penetrator flight control and precision guidance.
Claims
1. A method to solve an Euler rotational rigid body vector equation of motion for an in-flight Penetrator thru a high density Terramedia and a determination of an inconstant vector Euler Coriolis acceleration Alpha dynamic ‘g’ loading with a running digitization and an AC coupling algorithm to remove an axial DC component Omega.sup.2 and an axial DC Euler solution “A” from a single axial piezoresistive analog accelerometer and a running 1.sup.st integral of the Euler Coriolis acceleration to determine a lateral angular vector velocity Omega and a running 2.sup.nd integral of the Euler Coriolis acceleration to determine a traversed vector path Theta and an in-flight determination of the Penetrator’s Euler rotational inconstant ‘g’ loading at all physical points on the Penetrator by: Mounting the piezoresistive accelerometer analog electrical sensor on the primary longitudinal axis of the Penetrator ⅘ of the Penetrator body length behind a torque impulse at the Penetrator’s nose; Implement the running digitization of the piezoresistive accelerometer analog electrical sensor output signal and simultaneously run the AC coupling algorithm and remove the Omega.sup.2 and the Euler “A” DC components from the piezoresistive accelerometer sensor’s signal and determine the Euler Coriolis Alpha.sub.x acceleration; Implement the running 1.sup.st integral of the Euler Coriolis Alpha.sub.x acceleration and divide by the distance between the piezoresistive accelerometer analog sensor and the position of the Penetrator’s nose torque impulse obtaining the vector rotational velocity Omega in radians/sec; Implement the running 2.sup.nd integral of the Euler Coriolis Alpha.sub.x acceleration simultaneously with the 1.sup.st and divide by the distance between the piezoresistive analog electrical sensor and the distance to the Penetrator’s nose torque impulse obtaining the vector trajectory angular path Theta traversed in radians.
2. A method to solve an Euler rotational rigid body vector equation of motion of an in-flight Penetrator for an Euler Coriolis scaler acceleration Alpha.sub.y and Alpha.sub.z , a lateral angular velocity Omega.sub.y and Omega.sub.z and a traversed path Theta.sub.y and Theta.sub.z for a Pitch and a Yaw solution for rotation of the Penetrator from the Euler rotational rigid body vector equation of motion for a rigid body using a three axial facing piezoresistive analog acceleration sensor configuration in an “L” pattern with a running digitization, a difference algorithm to remove common modes, a running 1.sup.st integral of the Euler Coriolis acceleration Alpha.sub.y and Alpha.sub.z to determine the lateral angular velocities Omega.sub.y and Omega.sub.z and a running 2.sup.nd integral of the Euler Coriolis acceleration Alpha.sub.y and Alpha.sub.z to determine the traversed vector paths Theta.sub.y and Theta.sub.z for the Penetrator Pitch and the Yaw by: Mounting the three piezoresistive axial facing accelerometer analog electrical sensors on the longitudinal axis of the Penetrator and ⅘ of the Penetrator body length behind a torque impulse on the Penetrator’s nose; Implement the running digitization and the difference algorithm of the piezoresistive “L” accelerometer sensors output signals designating the sensor on the top of the “L” the Pitch difference Alpha.sub.y with respect to a Common accelerometer sensor at the intersection of the vertical and horizontal legs of the “L” and placed on the primary longitudinal axis of the Penetrator and designating the acceleration sensor on the right of the “L” the Yaw difference Alpha.sub.z with the respect to the Common accelerometer sensor; Implement the running 1.sup.st integral of the Pitch difference acceleration signal Alpha.sub.y and divide by the physical distance between the Pitch sensor and the Common sensor and simultaneously run the 1.sup.st integral of the Yaw difference acceleration signal Alpha.sub.z and divide by the physical distance between the Yaw sensor and the Common sensor to obtain the lateral angular velocities Omega.sub.y and Omega.sub.z; Implement the running 2nd integral simultaneously with the 1st integral of the Pitch difference acceleration signal Alpha.sub.y and Yaw difference acceleration signal Alpha.sub.z and divide by the physical distances between the Pitch sensor and the Common sensor and the Yaw sensor and the Common sensor respectively to obtain the traversed paths Theta.sub.y and Theta.sub.z .
Description
BRIEF DESCRIPTION OF DRAWINGS
[0018] The embodiment set forth in the drawings are illustrative and exemplary in nature and not intended to limit the subject matter defined by the claims. The following brief description of the illustrative embodiments can be understood when read in conjunction with the following drawings.
[0019] FIG. 1 depicts the Euler and Poncelet ‘g’ solutions to the Terraflight of a Penetrator in high density Terramedia.
[0020] FIG. 2 depicts the static geometry properties of a Penetrator, its in-flight dynamics and the Euler solution compared to measured data from a gun launched Penetration event.
[0021] FIG. 3 are graphs of the lateral and axial ‘g’ loading of a Penetrator’s measured data profiles and contrasts the measured data to National Laboratory Researcher’s empirical code ‘g’ loading solutions for the same event.
[0022] FIG. 4 shows the ‘g’ loading results from accelerometer sensors in ‘x’, ‘y’ and ‘z’ cartesian body co-ordinates for a gun launched Penetrator and its Terraflight thru a constructed geologic Terramedia.
[0023] FIG. 5 is the running positional tracking solution to Euler’s rotational rigid body vector equation of motion in two non-inertial frames and an algorithm to vector track a Penetrator’s Terraflight thru Terramedia.
[0024] FIG. 6 is the solution to Euler’s rotational rigid body vector equation of motion in a non-inertial Penetrator body frame in a cartesian co-ordinate system for strategically placed accelerometer sensors.
[0025] FIG. 7 is the Euler separated equations difference solutions in a non-inertial “body” Penetrator frame in a cartesian co-ordinate system and defines the dynamic and geometry features of the Penetrator and location of sensors with respect to their centers of rotation for a Penetration event.
[0026] FIG. 8 is the solution to Euler’s rotational rigid body vector equation of motion in a non-inertial “spatial” Penetrator frame in a polar co-ordinate system for strategically placed accelerometer sensors and unmasks the missing terms in the solution of FIG. 6.
[0027] FIG. 9 is the verification of the algorithmic AC coupling method of obtaining the rotational properties of angular acceleration α (Alpha), angular velocity ω (Omega), and angular path Θ (Theta) of an in-flight Penetrator with a singular sensor.
[0028] FIG. 10 is the algorithmic Penetrator tracking solution for Penetrator Terraflight Pitch and Yaw using three axial facing accelerometer sensors.
[0029] FIG. 11 shows the physical method and accelerometer sensor placement to extract the Pitch and Yaw components from a Penetrator in Terraflight and the Pitch and Yaw equations.
[0030] FIG. 12 is the verification of the algorithmic method of extracting the Pitch and Yaw components from a Penetrator in Terraflight and its rotational properties of angular acceleration α (Alpha), angular velocity ω (Omega), and angular path Θ (Theta).
DETAILED DESCRIPTION OF DRAWINGS
[0031] FIG. 1 depicts the Euler and Poncelet ‘g’ loading mathematical formulation for Terraflight of a Penetrator thru high density Terramedia and shows graphical solutions of the Euler Poncelet formulations. The geometric and Terramedia variables and constants of the equation are defined.
[0032] FIG. 2 depicts the common geometry features of a Penetrator body, that is, defines the nose, wall, shank, tail and body length features and its dynamic flight events of Ejecta and Cavitation, and contrasts acquired data of a Penetration event to the Euler Solution for an axial only Terraflight event thru a high density Terramedia.
[0033] FIG. 3 shows actual ‘g’ loading data from a 3-body length Terraflight of a Penetrator thru rock Terramedia with both actual axial (with Euler’s projected solution overlayed) and lateral ‘g’ loading on the left-hand side of the figure and on the right-hand side contrasts the actual data to US National Laboratory researcher’s empirical coded solutions for the lateral and axial ‘g’ loading of the same event.
[0034] FIG. 4 are ‘x’, ‘y’, ‘z’ Penetrator cartesian co-ordinate ‘g’ loading results from a gun launched Penetration event with initial conditions of 1000′/sec into a constructed Terramedia target and compares the results to the Euler “A” solution thereby identifying undulations around the Euler solution and demonstrating that lateral ‘g’ loading is vectorially transferred from the ‘y’ and ‘z’ axis to the ‘x’ axis via the Euler Coriolis acceleration and revealing and determining the existence of an additional non-inertial frame of reference, FIG. 8, that mathematically predicts the transfer.
[0035] FIG. 5 is the algorithmic method, 100, of solving Euler’s rotational rigid body vector equation of motion with one analog piezoresistive accelerometer measurement a .sub.x 140. Euler’s vector equation, 120, is separated into 3 scalar equations, 112, in a cartesian non-inertial frame ‘x’, ‘y’ and ‘z’ and with the ‘x’ role term set to zero, 114, determining the Euler solution “A”. Simultaneously Euler’s vector equation in a polar non-inertial frame is separated returning the ‘y’ and ‘z’ spatial acceleration Rω.sup.2, 124, and that the 2ωv.sub.r Euler Coriolis acceleration is transferred to a .sub.x via the vector cross product in the polar non-inertial frame determining the final solution 130. The singular axial measurement 140, in running fashion, point by measured point, is digitized and AC coupled 142 to remove the DC terms, which are the Euler “A” solution and any ω.sup.2 accelerations, which are DC rectified terms. This operation yields the Euler Coriolis acceleration, that is, Euler’s 2ωv.sub.r , 144, term which is angular acceleration α.sub.x (Alpha.sub.x). 1.sup.st and 2.sup.nd running integrals, 146 and 148, are taken determining point by measured point the Penetration event’s lateral angular velocity ω (Omega) and traversed path θ (Theta) in units of feet per second and feet. Dividing by the distance from a .sub.x to the point of rotation, which is the torque impulse location at the center of pressure, FIG. 2, on the nose, yields Omega in units of radians/sec and Theta in units of radians.
[0036] FIG. 6 separates Euler’s rotational rigid body vector equation of motion into 3 scaler penetrator body difference equations in a non-inertial cartesian co-ordinate frame, 112, and reduces the axial equation to 114 for the case of a fixed ‘x’ co-ordinate uni-axial (no ‘y’ or ‘z’ co-ordinate) piezoresistive accelerometer sensor positioned as a constant mass m″ on a Penetrator body “B” and also reduces the ‘z’ and ‘y’ separated scaler equations for fixed ‘xz’ and ‘xy’ co-ordinates, that is a singular ‘y’ facing sensor, FIG. 6; Eq. 2, with no ‘z’ and no ω.sub.x roll co-ordinate and a singular ‘z’ facing sensor, Eq. 3, with no ‘y’ and no ω.sub.x roll co-ordinate. This unveils the Euler Coriolis acceleration content in the signal allowing the scaler Euler Coriolis acceleration signal to be isolated from the total signal with AC Coupling.
[0037] FIG. 7 are the difference solutions for the non-inertial body “B” Penetrator frame in a Cartesian co-ordinate system and defines the dynamic and geometry features of the Penetrator and location of sensors a.sub.x , a.sub.y, a.sub.z and their centers of rotation for a Penetration event. The cartesian non-inertial frame solutions, FIG. 6; Eqs. 1, 2 and 3, for Euler’s rotational rigid body vector equation of motion are re-arranged in a difference form unlocking the rotational terms within the signal. For a singular sensor a.sub.x the axial ‘x’ longitudinal axis the solution -x (w.sub.y.sup.2 + w.sub.z.sup.2), FIG. 6; 114, only remains as there are no rotation around the ‘y’ and ‘z’ co-ordinates and in the same manner a.sub.y and a.sub.z and no ‘z’ or ‘y’ co-ordinate and no ω.sub.x (Omega.sub.x) roll yield
[00014]
respectively.
[0038] FIG. 8 unmasks the missing terms, 130, of FIG. 6; Eqs. 1, 2 and 3 solutions to Eq. 120 in a cartesian co-ordinate system and details the polar non-inertial frame and the solution in this frame to Euler’s rotational rigid body vector equation of motion, 120, and forecasts the scaler axial 2ωv.sub.r term, the Euler Coriolis acceleration, and further the Rω.sup.2 via the triple product of FIG. 8; 120, that is, the causal event that produces the lateral ‘y’ and ‘z’ pulses seen in FIG. 4. The Orbital ‘R’ is the spatial, not a body, radius to the center of rotation and starts at infinity for a perfectly straight Penetrator. This large spatial ‘R’, which relegates the Penetrator body “B” to a particle, m’, within the polar non-inertial frame, produces substantially an order of magnitude greater acceleration then a body small radii, ω.sup.2 terms, which are the fixed cartesian co-ordinate vales of ‘x’, ‘y’, and ‘z’. Body radius to the center of rotation are substantially inches while spatial large radii average many tens of feet over the sector Theta travel and for normal manufacturing tolerances start at 100’s of feet for the manufacturing bow. This further relegates the
[00015]
terms, Eqs. 2 and 3 on FIG. 6, a small percentage acceleration but does not completely annihilate their contribution to the ‘y’ and ‘z’ accelerations with respect to Rω.sup.2 but does relegate the terms to integral contaminates and must be removed to uncover the Euler Coriolis acceleration content in the signal. Further, while the Rω.sup.2 term overwhelms the lateral signals forcing them to appear as pulses, their removal with AC Coupling effectively unmasks the Euler Coriolis scaler accelerations.
[0039] FIG. 9 details a single axial piezoresistive accelerometer, a.sub.x , sensor solution that with an applied AC Coupling algorithm, 142, tracks the resultant vector, 148, of the Pitch (17°) and Yaw (22°), that is, the 28° (which is the square root of the sum of the squares of the Pitch and Yaw) vector total Θ (Theta) sector travel in radians and point by point. The sensor is positioned 22 inches behind the torque impulse. The undulations are shown around a superimposed totally axial Euler solution of -1193 ‘gs’. The AC coupling algorithm removes the DC components of the axial acceleration, that is, the Euler solution “A” and the -x (w.sub.y.sup.2 + w.sub.z.sup.2), FIG. 6; 114, contaminate solution for the Penetrator body in a cartesian non-inertial frame of FIG. 6, producing the AC coupled axial acceleration, 142, point by point, which is the 2ω X v.sub.r Euler Coriolis acceleration. The algorithm takes a running 1.sup.st integral and simultaneously a running 2.sup.nd integral producing ω angular velocity, 146, and θ (Theta) distance, 148, travelled in units of feet/sec and feet respectively. Division of each integral point by point by the sensor placement of 22″ behind the torque impulse produces the rotational velocity and rotational travel in units of radians/sec and radians respectively.
[0040] FIG. 10 is the algorithmic Penetrator tracking solution for Penetrator Pitch and Yaw using three axial facing planer accelerometer sensors with no ‘y’ or ‘z’ co-ordinates. Three piezoresistive accelerometers 151, termed a.sub.x1, a.sub.x2 and a.sub.x3, are placed in a plane substantially near the back end of the Penetrator. A running digitization and difference 152 are taken with respect to a.sub.x2; a.sub.x1 being the Pitch sensor and a.sub.x3 the Yaw sensor. Operation 152 yields point by measured point the Pitch acceleration and the Yaw acceleration in units of ‘gs’. Running 1.sup.st and 2.sup.nd integrals 154 and 156 respectively are taken and point by point determine Pitch and Yaw angular velocity ω (Omega), and angular path θ (Theta) in units of radians/sec and radians respectively, when each point is divided by the distances, r1 (Pitch) and r2 (Yaw) FIG. 11; 150, respectively.
[0041] FIG. 11 is the geometry of the difference algorithms and accelerometer placement of FIG. 10 to parse Pitch and Yaw from a Penetration event. FIG. 11 shows the placement of 3 axial facing sensors, x1, x2 and x3 placed substantially at the rear of the Penetrator and defines the points of rotation r.sub.1 (Pitch) and r.sub.2 (Yaw) distances with respect to the common terminal accelerometer x2. The event shown is constructed concrete Terramedia. The 3 axial accelerometer difference method parses the Pitch and Yaw resultant from Euler’s rotational rigid body vector equation of motion, FIGS. 6 and 8; 120, into its ‘z’ and ‘y’ components respectively. In each case the difference produces a scaler Alpha acceleration term 2ωv.sub.r. Differencing is a close relative of AC coupling in that it removes the common modes from the two signals which in the case of Penetration is the Euler solution “A” of FIG. 1 and the -x (w.sub.y.sup.2 + w.sub.z.sup.2) 114 of FIG. 6. A signal that is squared on itself is an AC rectifier and produces a DC pulse and commonly used in electrical circuits, in both analog and digital electronics and power electronics to change AC to DC. The three accelerometer signals are placed in a plane 150 and in an “L” configuration. The “L’s” horizontal and vertical intersection is the Common terminal, and must lie on the primary longitudinal axis, while the top of the “L” is the Pitch sensor and the sensor to the right of the Common terminal is the Yaw sensor. The dimensional spacing of the sensors is labeled r.sub.1 (Pitch) and r.sub.2 (Yaw), FIG. 11. These dimensions are used on the 1.sup.st 154 and 2.sup.nd 156 integrals, FIG. 10, of the acceleration difference to change feet/sec to radians/sec and feet to radians respectively.
[0042] FIG. 12 is the verification of the algorithmic method of extracting the Pitch and Yaw components from an in-flight Penetrator and the rotational properties of angular acceleration α (Alpha), angular velocity ω (Omega), and angular path Θ (Theta). The event is Terraflight thru a granite rock target that penetrated 3.45 body lengths with a striking velocity of 1000′/sec. It did not Pitch but Yawed 19°. The x3 sensor is subtracted from the common x2 sensor as shown on FIG. 11. The result is the undulations, 152, around the Euler solution are captured just as AC Coupling captured these features from a single accelerometer in FIG. 4. The 1.sup.st ,154, and 2.sup.nd 156, integrals are taken, and the units of radians/sec and radians recovered by dividing by the Yaw spacing which is r.sub.2. The spacing in this test was only 0.82 inches. This demonstrates the strength of the Euler Coriolis 2ωv.sub.r and the discovery’s usefulness in extracting from an axial signal of an in-flight Penetrator its rotational properties of angular acceleration α (Alpha), angular velocity ω (Omega), and angular path Θ (Theta) enabling the solution to Euler’s rotational rigid body vector equation of motion, heretofore considered insoluble in closed form and implementation of a precision tracking algorithm.
[0043] It is noted that the terms “substantially” and “about” may be utilized herein to represent the inherent degree of uncertainty that may be attributed to any quantitative comparison, value, measurement, or other representation. These terms are also utilized herein to represent the degree by which a quantitative representation may vary from a stated reference without resulting in a change in the basic function of the subject matter at issue. Furthermore, these terms are also utilized herein to represent the degree by which a quantitative representation may vary from a stated reference due to manufacturing tolerances or fabrication tolerances. While particular embodiments have been illustrated and described herein, it should be understood that various other changes and modifications may be made without departing from the spirit and scope of the claimed subject matter. Moreover, although various aspects of the claimed subject matter have been described herein, such aspects need not be utilized in combination. It is therefore intended that the appended claims cover all such changes and modifications that are within the scope of the claimed subject matter.
