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RADIANCE TECHNOLOGIES, INC. PROPRIETARY INFORMATION: The information contained herein may not be used in whole or in part except for the limited purpose for which it was furnished. Do not distribute, duplicate, or reproduce in whole or in part without the prior written consent of an authorized official of Radiance. Exempt from disclosure to third parties under 5 USC 552(b)(4). 18 USC 1905 may apply. RT MDA Phase I SBIR Final Report Advanced Guidance, Navigation, and Control (GNC) Algorithm Development to Enhance Lethality of Interceptors Against Maneuvering Targets Radiance Technologies, Inc. (256) January 2005

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RT SBIR PHASE I Interceptor Guidance / Estimation Algorithm Development The team members for this SBIR are University of Alabama at Huntsville (UAH), Princeton University, and Radiance Technologies, where Radiance Technologies is the prime contractor. UAH provides support in the area of Sliding Mode Guidance and Control. Princeton University provides support in the area of Particle Filtering. Radiance Technologies works in the area of Multiple Model Linear Kalman Filtering and Six Degree-of-Freedom (6-DOF) simulation of the various engagement scenarios and algorithms developed.

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RT SBIR PHASE I Interceptor Guidance / Estimation Algorithm Development Team Members – Tasks UAH– Sliding Mode Guidance/Estimation/Autopilot Princeton– Target State Estimation/Guidance via Two-Step Filtering and/or Particle (UKF) Filters Radiance– Target State Estimation/Guidance via Single Linear Multiple Target Model Kalman Filters

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RADIANCE TECHNOLOGIES, INC. PROPRIETARY INFORMATION: The information contained herein may not be used in whole or in part except for the limited purpose for which it was furnished. Do not distribute, duplicate, or reproduce in whole or in part without the prior written consent of an authorized official of Radiance. Exempt from disclosure to third parties under 5 USC 552(b)(4). 18 USC 1905 may apply. RT Preliminary Simulation Analyses of Estimation and Guidance Algorithms

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RT Analysis Outline The Following Sections will provide an overview of the 6-DOF simulation used in the Radiance Analysis, Linear Multiple Model Kalman Filtering (LMMKF) coupled with Optimal Guidance, Sliding Mode Guidance and Control methodologies, and non-linear filtering methodologies including the two step extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF).

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RT Analysis Outline Simulation Overview Single Linear, Multiple Model Kalman Filter Methodologies Sliding Mode Guidance/Estimation/Autopilot Two-Step Filters and UKF

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RT Functional Block Diagram of Interceptor 6-DOF Simulation and Engagement Models The 6-DOF Simulation functional block diagram illustrates the general flow of the simulation, the various interceptor component modeling options, and some of the guidance, navigation, and control algorithm options. For example, there are options for both gimbaled and strapdown seekers. There are several guidance filter options such as the three decoupled 3-state Kalman filter, the linear multiple model 8- state Kalman filter, and the super twist sliding mode filter. Currently there is only a fin control system model, however the structure is in place to upgrade the simulation to include Thrust Vector Control (TVC), single-shot (squib) thrusters, and pulse width modulated thrusters models.

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RT Functional Block Diagram of Interceptor 6-DOF Simulation and Engagement Models

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RT Interceptor Model Assumptions The following assumptions were made in order to perform a credible 6-DOF simulation analysis of the various guidance and control algorithms: The seeker measures range, azimuth and elevation angles at a sample rate of 100Hz. The gimbaled seeker has a track loop bandwidth of approximately Hz, the stabilization loop bandwidth is 30Hz. The interceptor is tail control via 2nd order 30Hz actuator with a damping ratio of 0.7. The autopilot is a two loop pole placement acceleration autopilot with a natural frequency of 18 rad/s (or 2.9Hz) with a damping ratio of 0.7. If a full flyout trajectory is performed either pursuit or Optimal Trajectory Shaping Guidance is used, depending on range. Shorter range engagements use pursuit guidance during boost/midcourse to quickly turn the velocity vector toward the target. The airframe static margin varies from slightly stable to slightly unstable or near neutral stability.

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RT Interceptor Model Assumptions Gimbaled Seeker – Measures Angles and Range – Track Loop Bandwidth From 10-20r/s – Stabilization Loop Bandwidth Approximately 30Hz – Seeker Sample Rate 100Hz Tail Fin Controlled Airframe – Assumes 30Hz 2nd Order Actuator Two Loop Acceleration Autopilot – n =18r/s, =0.7 For Region of Interest – 20g Acceleration Command Limit Pursuit Guidance or Optimal Trajectory Shaping Guidance – Boost/Midcourse Airframe Static Margin Varies From Slightly Unstable to Stable

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RT Autopilot Used in 6-DOF Simulation Guidance / Estimation Studies Both pitch and yaw channels use the same autopilot topology. The autopilot consists of three gains and feedback signals of acceleration and body angular rate. The gains are computed using on-board estimates of the airframe stability derivatives. Where the gain calculations assume that the gyro, accelerometer, and actuator bandwidths are high (relative to the autopilot and airframe). The gains are specified to achieve a desired autopilot natural frequency, damping ratio, and a DC gain of 1.0.

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RT Autopilot Used in 6-DOF Simulation Guidance / Estimation Studies

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RADIANCE TECHNOLOGIES, INC. PROPRIETARY INFORMATION: The information contained herein may not be used in whole or in part except for the limited purpose for which it was furnished. Do not distribute, duplicate, or reproduce in whole or in part without the prior written consent of an authorized official of Radiance. Exempt from disclosure to third parties under 5 USC 552(b)(4). 18 USC 1905 may apply. RT Interceptor Guidance/Estimation Algorithms Single Linear, Multiple Model Kalman Filter Methodology

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RT Single Linear, Multiple Model Kalman Filter Methodologies The basis for applying the single Linear, Multiple Model Methodologies is that the target will provide via its signature near DC, a single frequency, multiple frequencies, or frequencies that vary with time. For example a step maneuver would provide primarily low frequency or near DC kinematic frequency content. A spiraling target could produce single, multiple, or frequencies that vary as a function of time. A tumbling target would probably present multiple frequencies varying with time. The initial filter design modeled discrete frequencies (or the sinusoidal frequencies were constant over time). The analysis results presented in this briefing only consider the discrete frequency filter. However a filter that models a chirp in frequencies has been designed and has been modeled in the Matlab environment. The current maximum number of states in the weave filter is 8, where the states are indicated on the chart. In the future more frequencies will be added to the filter as well as implementing the chirp form of the filter in the 6-DOF simulation.

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RT Single Linear, Multiple Model Kalman Filter Methodologies Target Could Present “DC”, Single, or Multiple Frequencies – Step Maneuver – Weaving Target – Tumbling Target? Filter Initially Designed to Model Discrete Frequencies Evolution to Chirp Frequency Design Allows Filter to Capture a Larger Range of Frequencies More Accurately Current Filter is Two Decoupled Eight (8) State Filters Per Channel, (Pitch and Yaw) for a total or Sixteen (16) States Filtered States – Target to Missile Relative Position – Target to Missile Relative Velocity – Target Acceleration at Three Frequencies – Target “Jerk” at Three Frequencies

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RT Discrete Frequency Kalman Filter State Equations* The slide below defines the state equations for a 6-state discrete frequency Linear Multiple Model Kalman Filter (LMMKF). The states in the filter are the relative position, relative velocity, target acceleration at frequency 1, target jerk at frequency 1, target acceleration at frequency 2, and target jerk at frequency 2. This filter is an example of applying the internal model principal. The internal model of the filter contain difference equations that closely approximate the differential equations that describe sinusoidal motion.

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RT Discrete Frequency Kalman Filter State Equations* *Note: Only Six States Shown Here

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RT Optimal Weave Guidance Law* In order to effectively apply the filtered states from the LMMKF, it must be integrated with a complementary guidance law. The Optimal Weave Guidance Law is a form of zero effort miss guidance (or Augmented Proportional Guidance) that accounts for the sinusoidal motion of the target. Using the estimates provided by the LMMKF, this closed form predictive guidance law computes the zero effort miss at closest approach. The estimated miss distance is then scaled by the guidance gain and the inverse of the estimated time-to-go squared to form the acceleration command that is processed by the missile autopilot.

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RT Optimal Weave Guidance Law* *Note: Only Six Terms Shown Here Guidance Law Must be Consistent with Filter Mechanization to Gain Improved Miss and Reduced Maneuver Benefits Guidance Law Must be Consistent with Filter Mechanization to Gain Improved Miss and Reduced Maneuver Benefits

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RT Filter Frequency Chirp Impacts Zero Effort Miss Guidance Law In order to gain full benefit from the chirp Kalman filter form, the guidance law must be modified from the previous form. Because the frequencies of the target are now varying with time, the integration (or prediction) in the guidance algorithm must account for this as well. We now numerically propagate the relative states, target acceleration, target jerk until closest approach, as the closed form weave solution no longer applies. At closest approach the zero effort miss distance is determined. Once the zero effort miss is determined, the guidance law is the same as before. The acceleration command is just the estimated miss distance scaled by the guidance gain and the inverse of time-to- go squared, as shown in the chart.

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RT Filter Frequency Chirp Impacts Zero Effort Miss Guidance Law Guidance Uses Filter Model Frequencies and Current State Estimates Provided by Kalman Filter Guidance Algorithm is Numerically Propagated Form of Optimal Weave Guidance (due to Frequency Chirping) Target and Interceptor Propagated to Target to Provide Predicted or Estimated Miss Distance at Closest Approach The Acceleration Command Becomes the Familiar:

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RT Linear Single Frequency Kalman Filter Tolerates 10% Weave Frequency Mismatch The following chart illustrates the interceptor miss performance when the filter model assumes a target weave frequency of 0.5Hz, while the target frequency is varied from 0.4Hz to 0.6Hz. The target frequency is held constant for a particular set of runs. From this set of sensitivity runs, the plot indicates that the filter can tolerate up to about a +/- 10% frequency error before miss performance is severely affected. Actually the filter models two frequency with the second frequency near zero to capture near DC behavior such as target drag, etc. So in this instance a six state filter was used. The acceleration command plots show the affect of filter and target frequency mismatch on the guidance commands. As the filter frequencies become more mismatched the acceleration commands contain more of the target frequency. When the target and filter frequency are well matched the interceptor guides directly to the predicted intercept point.

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RT Linear Single Frequency Kalman Filter Tolerates +10% Weave Frequency Mismatch Filter Pre-Selected Frequency 0.5Hz

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RT Second Frequency in Filter Model Increases Performance Window The following set of plots show the effect of adding a second filter weave frequency on the frequency performance window. In this set of cases runs where the target frequency was varied from 0.35Hz to 0.6Hz, using the same methodology for varying the target frequency as before. Including a second “non-near DC” frequency increased the window of frequency performance from 0.45Hz to 0.55Hz to 0.375Hz to 0.575Hz or another 10% on the low end of the frequency window.

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RT Second Frequency in Filter Model Increases Performance Window Target Acceleration Amplitude 8g’s Target =5000kg/m 2 Filter Model Frequencies ~ DC, 0.4, and 0.5 Hz Target Weave Frequency Varied from Hz Pole Placement Autopilot w/DC Gain Adjustment n =18r/s, =0.7 Guidance Gain Target Weave Frequency (Hz) Guidance Gain Target Weave Frequency (Hz) CEP R95 R99 CEP R95 R99 Filter Frequencies Filter Frequencies Guidance Gain Target Weave Frequency (Hz) CEP R95 R99 Filter Frequencies

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RT PN And APN Guidance Performance Sensitivity to Target Weave Frequency Both Proportional Navigation (PN) and Augmented Proportional Navigation are sensitive to target weave frequency. It is important to note the reasons for the severe sensitivity are the target maneuver level of 8 gees combined with an interceptor maneuver limit of 20 gees. The standard rule of thumb for maneuver advantage for PN and APN is 3:1 for “reasonable” guidance gains. In practice it is difficult to maintain guidance gains greater than 4 during homing due to seeker noise. Although game theory indicates a high gain PN solution as optimal, it is difficult to apply in practice due to real world error sources such as seeker noise and boresight slope error. (Although we do have compensation algorithms for the latter).

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RT PN And APN Guidance Performance Sensitivity to Target Weave Frequency Weave Amplitude 8g’s, Target = 5000kg/m 2 PN Guidance, K G = Target Weave Frequency (Hz) CEP R95 R Target Weave Frequency (Hz) CEP R95 R99 APN Guidance, K G = 4

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RT Section Summary The Linear, Multiple Model Kalman Filter (LMMKF) is an example of the internal model principle, i.e., the filter model contains the weave terms expected in the target threat set. Given that there is some region around the filter frequency, the idea is to allow the filter to trap several frequencies between the filter frequencies. The filter model is fairly simple, i.e., the state transition matrix is sparse and there is no real cross coupling between the acceleration and jerk states. The acceleration terms do not couple until they are linearly added and integrated to determine the relative velocity state. So the usual concern about more states implying more lag is not as big of a concern here. Although this approach did not yield a “silver bullet” it still provided improvements over other conventional methods.

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RT Section Summary Linear, Multiple Model Kalman Filter (LMMKF) is an example of the “internal model” principle Idea is to “trap” the true target weave motion between filter models in order to obtain better performance Additional models can be easily incorporated No “silver bullet” but results show definite improvement over conventional approaches

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RT Sliding Mode Methodologies The following section describes the sliding mode control methodology applied to the guidance, autopilot, and control problem.

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RADIANCE TECHNOLOGIES, INC. PROPRIETARY INFORMATION: The information contained herein may not be used in whole or in part except for the limited purpose for which it was furnished. Do not distribute, duplicate, or reproduce in whole or in part without the prior written consent of an authorized official of Radiance. Exempt from disclosure to third parties under 5 USC 552(b)(4). 18 USC 1905 may apply. RT Interceptor Guidance / Estimation / Autopilot Algorithms Sliding Mode Methodologies

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GNC Algorithms for Advanced Homing Interceptor In this chart the tasks to be addressed while studying hit-to-kill interceptor GN&C systems using second order sliding mode control are listed.

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GNC Algorithms for Advanced Homing Interceptor Second-Order Sliding (2-Sliding) Mode Approach: 2-Sliding LOS Rate Filter (a part of analog-to-digital converter of seeker output) 2-Sliding Guidance Law (extension of Proportional Navigation) 2-Sliding Missile Autopilot (normal acceleration tracking system)

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GNC Algorithms for Advanced Homing Interceptor The benefits of an application of second order sliding mode control to the GN&C of hit-to-kill missile interceptors are discussed. The main advantages are: 1) robustness to measurement noise helps designing a navigation system 2) robustness to target maneuvers helps designing a smooth guidance law 3) robustness to external disturbances helps in designing an autopilot

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GNC Algorithms for Advanced Homing Interceptor Why Second-Order Sliding Mode? Navigation: robust to noise high accuracy signal differentiation/filtration Guidance: smooth, robust to target acceleration Guidance Laws Autopilot: robust to disturbances/uncertainties continuous (if necessary) Control Laws

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Objective The main R&D and performance objectives for the accomplished Phase I research effort are listed and discussed in this chart.

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Objective Main R&D objective determine combinations of measurements/estimations and accuracies required to achieve performance objectives with the varying approaches to guidance law development Performance objective achieve hit-to-kill intercept and minimize acceleration advantage under given intercept geometry

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Given Intercept Geometry The chart illustrates a given collision course versus aspect angle in a given intercept geometry for the missile-target engagement scenario, upon which the developed SMC-based GN&C system must be tested. The chart shows initial aspect angle envelope for the missile homing mode: it is requested to be between –400 and +400 while initial range changing from 5 km until 50 km

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Given Intercept Geometry Aspect Angle Envelope Collision Course vs. Aspect Angle

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Given Target Maneuver The chart illustrates a given target maneuver envelope for the missile-target engagement scenario, upon which the developed GN&C must be tested. The chart shows the step-constant maneuver envelope during the homing mode intercept while target acceleration normal to the LOS is equal to –7 g and +7 g. Also, target spiral maneuver is described for the designed GN&C system testing.

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Given Target Maneuver Step-Constant Maneuver Envelope Spiral Maneuver Envelope

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Meeting the Objectives The chart discusses the innovations in designing the navigation system for hit-to-kill interceptors proposed in a Phase I effort. The second order sliding mode filter for filtering the measured with noise line of sight rate is proposed (see the formula). The implementation of the proposed filter as an analog-to-digital converter is discussed (see Figure).

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Meeting the Objectives 2-Sliding Filter as a Part of Analog-to-Digital Converter Measurement Input : Seeker LOS Rate Signal ADC Sampling Rate: 10[kHz]-1[GHz] 2-Sliding Filter is a Sliding-Mode-Controlled Signal Observer The continuous-time model: Super-Twisting 2-Sliding Observer Block- Scheme

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Meeting the Objectives The chart discusses the innovations in designing the guidance law for hit-to-kill interceptors proposed in Phase I effort. A second order smooth sliding mode guidance law is proposed (see the formula). The second order sliding mode-based guidance system configuration/digital implementation is proposed in order to have a fare comparison with APN via computer simulation. During the compute simulations the normal acceleration commands generated by a sliding mode guidance system and AGL are 100 Hz step-wise signals.

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Meeting the Objectives 2-Sliding Guidance Law Missile Acceleration Command Transversal to LOS A Combination of PN-Guidance and “Zero LOS Rate” Strategy 2-Sliding Guidance System Configuration A Simulation Setup with Reference Guidance System – Augmented Proportional Navigation with 3-state Kalman Filter

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Meeting the Objectives The parameters of THAAD-like interceptor are given in the chart. During the Phase I effort the mathematical pitch plane model of THAAD-like interceptor is developed based on the given parameter values (see the chart). A sliding mode autopilot is designed to control divert and attitude thrusters in order to follow the normal acceleration commands. Use of the attitude thrusters allows to increase divert capabilities of the missile up to 100%.

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Meeting the Objectives 2-Sliding Flight Control System THAAD-Like Interceptor Control challenge: Uncertainties created by the interactions between the airflow and the thruster’s jets KEKV Parameters: Moment of inertia (pitch plane): 4 kg m 2 ; Mass: 100 kg, Reference diameter: 0.3 m Divert force: 10,000 N Attitude normal force: 1,000 N

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Meeting the Objectives A structure of the autopilot (flight control system) for THAAD-like missile developed during Phase I effort is presented at this chart. A control challenge ( uncertainties created by the interaction between airflow and jet-thrusters) that is addressed by the designed autopilot is mentioned in the chart.

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Meeting the Objectives 2-Sliding Flight Control System THAAD-Like Interceptor Control challenge: Uncertainties created by the interactions between the airflow and the thruster’s jets

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2-Sliding Autopilot The second order sliding mode autopilot designed for the THAAD-like missile during the Phase I effort is discussed in this chart. Block diagrams of an Attitude thruster control system based on nonlinear dynamic sliding manifold and a Divert thruster control system based on smooth second order sliding mode control are presented.

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2-Sliding Autopilot Attitude Thruster Control Divert Thruster Control Smooth 2-Sliding Control Nonlinear Dynamic Sliding Manifold

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2-Sliding Flight Control System A THAAD-like missile was simulated with the designed second order sliding mode control-based guidance system and the autopilot during the Phase I effort. The plots presented in the chart correspond to a direct hit and characterize an excellent, high accuracy, robust to target maneuver autopilot performance.

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2-Sliding Flight Control System Flight Path Angle Tracking Angle-of-Attack Tracking Pitch Rate Tracking Attitude and Divert Thruster Commands Missile and Target Accelerations Transversal to LOS

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Sensor Information Requirements The requirements to the seeker characteristics are formulated in the chart for the GN&C tests accomplished during Phase I effort. Information requirements and requirements to a compensated missile dynamics are also formulated for the simulation tests.

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Sensor Information Requirements Seeker: LOS noise StdDev = 1[mrad], Time Lag = 100[msec] Active Seeker: Range, Range Rate Required accuracies not specified, and not investigated in this simulation study SMC Guidance Information Requirements: LOS, LOS Rate, Range, Range Rate Flight Control System Requirements: Time Lag T=0.1[sec] and max(Am)=20G

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Performance Analysis The designed sliding mode control-based GN&C system has been fairly compared with the APN guidance supported by KF via computer simulations accomplished during Phase I effort. The sliding mode filter and KF are tuned up to have comparable dynamics in order to provide a fair comparison of the GN&C systems’ performances via computer simulation. In this chart the performance characteristics of the correspondingly tuned up sliding mode filter and KF are demonstrated.

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Performance Analysis LOS rate estimation 2-Sliding Filter and Kalman Filter Tune-Up Objective is to make response times to target maneuver equal for fare competition

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Performance Analysis During the Phase I effort a given missile engagement scenario was simulated in a perfect information condition to identify the engagement configurations, upon which the intercept could not happen. The presented table indicates that at aspect angle equal to 40 0 the missile misses the target almost in each considered case. This case is not include in the Monte-Carlo simulations accomplished during the Phase I effort.. Evaluation of Intercept Envelope under full information guidance

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Performance Analysis Evaluation of Intercept Envelope under full information guidance Miss Occurs when Target Having Higher Speed Overruns Missile

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Performance Analysis Mean Miss Distance vs. Time-to-Go, Step-Constant Maneuver The Monte-Carlo simulations of the miss distance versus time-to-go for the missile with given compensated frame dynamics and with the sliding mode guidance versus APN guidance were accomplished during Phase I effort. The results of the simulations are shown in the chart. Aspect angle was equal to 100, and the target performs 7g step maneuver at various times-to-go. The sliding mode guidance demonstrated less sensitivity to the step target target maneuvers at various times-to-go.

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Performance Analysis Mean Miss Distance vs. Time-to-Go, Step-Constant Maneuver Average of 25 trials for each 7g-step maneuver 0.25m is a direct hit Aspect Angle 10deg

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Performance Analysis Acceleration Ratio vs. Time-to-Go, Step-Constant Maneuver The Monte-Carlo simulations of the acceleration ratio versus time-to-go for the SMC guidance versus the APN guidance and a step-constant 7g target maneuver were accomplished during the Phase I effort. The results of the simulations are presented at the chart. Better acceleration advantage (10%-20% percent decrease) was demonstrated at large time-to-go by the SMC guidance.

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Performance Analysis Acceleration Ratio vs. Time-to-Go, Step-Constant Maneuver Aspect Angle 10deg 10%-20% Decrease in AR at large times-to-go

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Performance Analysis Missile Acceleration Profiles, Spiral Target Maneuver Acceleration advantage was studied for the target spiral maneuver during the Phase I effort. The plot presented at the chart demonstrated much better performance of the SMC guidance against the APN guidance. The SMC guidance provided a very close following the target acceleration profile while no knowledge about this profile is used in the guidance law. Acceleration advantage was 30% better for the SMC guidance versus the APN guidance.

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Performance Analysis Missile Acceleration Profiles, Spiral Target Maneuver SMC acceleration is 30% less than APN+KF acceleration (not including end-game phase) Aspect Angle 10[deg] Weave Frequency 0.1[Hz] Direct Hit achieved For all times-to-go

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Overall Simulation Results This chart contains the conclusions based on Monte-Carlo simulations of the given engagement scenario using the developed during the Phase I effort sliding mode-based GN&C.

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Overall Simulation Results SMC-Guidance System Conditions: Given Seeker noise sigma 1[mrad], Seeker first order lag with T=0.1[sec] and flight control system lag with T=0.1[sec] and max(Am)=20G. Results: Direct hit (miss<0.25m) at initial aspect angles 10,20,30deg is achieved: For 7-g step-constant target maneuver not sooner than 1[sec] time-to-go or larger than 5[km] range-to-target; For 7-g weave with frequencies up to 0.2[Hz]; Requires Acceleration Advantage 2-3 times over the target 10-30% less than APN Requirements

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SMC Methodology Strengths over APN The chart illustrates the advantages of the SMC-based guidance law developed during the Phase I effort over the APN guidance.

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SMC Methodology Strengths over APN Performance Gain RobustComplementary Simple Less Demanding to Navigation (No Kalman Filter in Guidance Computer, SMC Filtering is a Part of ADC) Advantage at Short Times-to-Go (Robust to Target Sharp Turn at the End-Game Phase) Less Control Energy Hitting Weaving Target (Actuated Acceleration Needs 2-3 Times Advantage over Target only at End-Game)

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Key SMC Methodology Benefits The chart describes the key benefits of using SMC-base GN&C methodology developed during the Phase I effort for the GN&C design of a hit-to-kill missile interceptor.

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Key SMC Methodology Benefits Variety of Design Techniques for Guidance Law Development (Conversion of Intercept Strategy into Robust Control Design) SMC Guidance Term Appears as a Complementary Addition (May Augment PN, APN, or Any Other Guidance Laws) Synthesis of Filtering/Estimation and Analog-to-Digital Conv. A Unified Integrated Approach to Missile Interceptor GNC Simple and Robust Autopilot Design (Continuous or Pulse-Modulated Control Signal of Equal Performance) –Robustness to plant uncertainties, in particular robust to interaction between jet thrust and air flow –Conserving divert thrust propellant due to special control allocation between divert and attitude thrusters

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Next Step Mature and extend the developed sliding mode GNC technology to higher fidelity 6 DOF interceptor models, including dual thruster and thruster-fins autopilot configurations

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RT Sliding Mode Estimation Equations The chart below shows the sliding mode algorithms employed to estimate the Target to Interceptor Line-of-Sight (LOS), LOS Rate, Range, and Rate Rate. The LOS and LOS rate estimators are in the form of the Super Twist Algorithm. The Range/Range Rate estimation are performed using a Non-Linear Dynamic Sliding Mode (NDSM) differentiator. The outputs of these estimators were used by the sliding mode guidance algorithms. Note, the sliding mode algorithms can also be driven by outputs of a LOS/LOS rate and Range/Range rate Kalman filter set.

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RT Sliding Mode Estimation Equations LOS, LOS Rate Estimation LOS, LOS Rate Estimation Range, Range Rate Estimation Equations and Block Diagram Range, Range Rate Estimation Equations and Block Diagram

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RT Sliding Mode Guidance Equations The following set of equations illustrate the “most general” of the three forms of sliding mode guidance presented in the report. The guidance command is constructed by summing a proportional navigation (or PN) term, a term inversely proportional to the square root of range to go to the target, and a robust finite time convergence term. The finite convergence term is a function of the sliding mode surface, . Where is defined as: And the variables are defined on the chart. Sliding mode guidance by its nature will produce a high gain solution. Note the target acceleration is not explicitly required by the guidance law, i.e., no estimate of target acceleration is required.

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RT Sliding Mode Guidance Equations

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RT Representative Altitude and Cross Range vs. Downrange Trajectories, SMC Guidance The following plots illustrate representative altitude and cross range vs. downrange for an initial near head-on trajectory until the target pulls a 8gee step yaw maneuver. The target maneuver time constant, , is 0.4 seconds. The maneuver resulted in an out of plane distance of approximately 1750m which the interceptor successfully pursued by the sliding mode guidance algorithm. In this instance a end-to-end interceptor trajectory was flown, which includes boost and midcourse guidance. In this case Optimal Trajectory Shaping Guidance (OTSG) was used to set up the endgame conditions. The target step maneuver began approximately 6.5 seconds prior to intercept.

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RT Representative Altitude and Cross Range vs. Downrange Trajectories, SMC Guidance Altitude (m) Downrange (m) Target Interceptor Cross Range (m) Downrange (m) Target Step Maneuver 8g’s, Yaw Plane, =0.4sec Target Interceptor

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RT Sliding Mode Acceleration Commands and Interceptor Responses The chart below illustrates the acceleration commands generated by the sliding mode guidance algorithm and the airframe responses. The seeker angle noise is assumed to be 0.5 mr, 1 , the range noise is 5.0m, 1 , and the seeker sample rate is 100Hz. The target maneuver is in the yaw plane. The target is commanded to a maneuver of 8gees, where the target response time constant, , is 0.4 seconds. The light chatter on the acceleration commands is due to the nature of sliding mode guidance and of course seeker noise. It appears that the interceptor to target maneuver ratio is approximately 1.5 : 1.0 for this case.

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RT Sliding Mode Acceleration Commands and Interceptor Responses Pitch Acceleration Command (m/s 2 ) Pitch Acceleration Response (m/s 2 ) Yaw Acceleration Command (m/s 2 ) Yaw Acceleration Response (m/s 2 ) Target Step Maneuver 8g’s, Yaw Plane, =0.4sec Seeker Noise 0.5mr (1 ), 5m (1 ),Seeker/Guidance Update Rate 100Hz

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RT First Variation of Sliding Mode Guidance Algorithm The first variation of the sliding mode guidance algorithm simplifies the sliding surface from: To the following: This first variation results in a more “aggressive” guidance law as it is not offset by the term proportional to the square root of the range. Basically the guidance law simplifies to PN plus the sliding mode augmentation term. Once again the target acceleration term is not explicitly estimated.

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RT First Variation of Sliding Mode Guidance Algorithm Simple Form of Sliding Mode Guidance Consists of: – PN Term + Sliding Mode Augmentation Term – C o Term in Generic Sliding Mode Guidance is set to Zero – Note: Target Acceleration Not Explicitly Estimated Guidance Equations

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RT Sliding Mode Guidance Variation 1 The following plots illustrate representative altitude and cross range vs. downrange for an initial near head-on trajectory until the target pulls a 8gee step yaw maneuver. The target maneuver time constant, , is 0.4 seconds. The maneuver resulted in an out of plane distance of approximately 1750m which the interceptor successfully pursued by the sliding mode guidance algorithm. In this instance a end-to-end interceptor trajectory was flown, which includes boost and midcourse guidance. In this case Optimal Trajectory Shaping Guidance (OTSG) was used to set up the endgame conditions. The target step maneuver began approximately 6.5 seconds prior to intercept.

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RT Sliding Mode Guidance Variation 1 Altitude (m) Downrange (m) Target Interceptor Cross Range (m) Downrange (m) Target Interceptor Target Step Maneuver 8g’s, Yaw Plane, =0.4sec

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RT Variation 1 Sliding Mode Guidance Acceleration Commands and Interceptor Responses The chart below illustrates the acceleration commands generated by the sliding mode guidance algorithm and the airframe responses. The seeker angle noise is assumed to be 0.5mr, 1 , the range noise is 5.0m, 1 , and the seeker sample rate is 100Hz. The target maneuver is in the yaw plane. The target is commanded to a maneuver of 8gees, where the target response time constant, , is 0.4 seconds. The increased chatter on the acceleration commands is due to more aggressive form of the sliding mode guidance algorithm. It appears that the interceptor to target maneuver ratio is approximately 1.5 : 1.0 for this case.

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RT Variation 1 Sliding Mode Guidance Acceleration Commands and Interceptor Responses Target Step Maneuver 8g’s, Yaw Plane, =0.4sec Seeker Noise 0.5mr (1 ), 5m (1 ), Seeker/Guidance Update Rate 100Hz Pitch Acceleration Command (m/s 2 ) Pitch Acceleration Response (m/s 2 ) Yaw Acceleration Command (m/s 2 ) Yaw Acceleration Response (m/s 2 )

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RT Linearized Sliding Mode Guidance Law The block diagram illustrates a linearized form of the sliding mode guidance law. It is basically PN augmented by a term that represents the velocity normal to the LOS. This augmented term is processed by a proportional plus integral or PI compensation. The augmented term has the effect of causing the interceptor to respond sooner the target maneuver, however this term has the potential to be sensitive to seeker noise and radome aberration particularly at long ranges.

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RT Linearized Sliding Mode Guidance Law KISKIS K P T GO + 1 T GO Ks R EST V CEST KGKG A CMD Estimated LOS Rate Velocity Normal To LOS Estimated Range Estimated Closing Velocity PN Gain Integral Gain Proportional Gain s Gain ACAC A CPN Guidance Command PN Guidance Augmented by Velocity Normal to LOS Using Constant Integral Gain and Proportional Gain as Function of Time-to-go

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RT Performance Sensitivity to Target Maneuver Weave Frequency The chart below indicates that all three forms of sliding mode guidance have a sensitivity to target weave frequency for an 8gee maneuvering target. However, variation 1 and the linearized form of sliding mode guidance appear to have the best potential in terms of miss distance. It is important to note that the filters used to generated the estimates were not “exhaustively” optimized, and the seeker noise terms were larger than those discussed with the government evaluators toward the end of the briefing. So there is still room for further performance improvement.

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RT Performance Sensitivity to Target Maneuver Weave Frequency Target Maneuver Amplitude 8g’s, Target =5000kg/m2 Sliding Mode Guidance (Variant 1) Degrades Linearly (Gracefully) With Target Frequency Linearized Guidance Derived From First Variant of Sliding Mode Guidance Sliding Mode Guidance Most Sensitive to Increase in Target Weave Frequency Sigma Guidance Decent Performance to 0.5 Hz Linearized Sliding Mode Guidance Target Weave Frequency (Hz) Sliding Mode Guidance Target Weave Frequency (Hz) Sliding Mode Guidance Variant Target Weave Frequency (Hz) CEP R95 R99 CEP R95 R99 CEP R95 R99

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RT PN And APN Guidance Performance Sensitivity to Target Weave Frequency This chart is included again here in order to more easily compare PN, APN and sliding mode guidance performance. Both Proportional Navigation (PN) and Augmented Proportional Navigation are sensitive to target weave frequency. It is important to note the reasons for the severe sensitivity are the target maneuver level of 8 gees combined with an interceptor maneuver limit of 20 gees. The standard rule of thumb for maneuver advantage for PN and APN is 3:1 for “reasonable” guidance gains. In practice it is difficult to maintain guidance gains greater than 4 during homing due to seeker noise. Although game theory indicates a high gain PN solution as optimal, it is difficult to apply in practice due to real world error sources such as seeker noise and boresight slope error. (Although we do have compensation algorithms for the latter).

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RT PN And APN Guidance Performance Sensitivity to Target Weave Frequency Weave Amplitude 8g’s, Target = 5000kg/m 2 PN Guidance, K G = Target Weave Frequency (Hz) CEP R95 R Target Weave Frequency (Hz) CEP R95 R99 APN Guidance, K G = 4

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RT Section Summary In this section we presented our initial analyses for three forms of sliding mode guidance. The sliding mode guidance methodologies performed better than PN and APN against weaving targets, with a better required interceptor to target maneuver ratio. Again this approach did not yield the “silver bullet”, however results warrant further study.

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RT Section Summary Explored Three Variations of Sliding Mode Control Approach Sliding Mode Approaches Performed Better Than PN and APN Against Weaving Targets Sliding Mode Guidance Algorithms Provide Better Interceptor to Target Maneuver Ratio Characteristics Than PN or APN Again, no “Silver Bullet”, Further Exploration Warranted

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RADIANCE TECHNOLOGIES, INC. PROPRIETARY INFORMATION: The information contained herein may not be used in whole or in part except for the limited purpose for which it was furnished. Do not distribute, duplicate, or reproduce in whole or in part without the prior written consent of an authorized official of Radiance. Exempt from disclosure to third parties under 5 USC 552(b)(4). 18 USC 1905 may apply. RT Interceptor Guidance/Estimation Algorithms Two-Step Filters and Unscented Kalman Filters (UKF) Methodologies

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RADIANCE TECHNOLOGIES, INC. PROPRIETARY INFORMATION: The information contained herein may not be used in whole or in part except for the limited purpose for which it was furnished. Do not distribute, duplicate, or reproduce in whole or in part without the prior written consent of an authorized official of Radiance. Exempt from disclosure to third parties under 5 USC 552(b)(4). 18 USC 1905 may apply. RT Interceptor Guidance/Estimation Algorithms Future Weaponization Tasks

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RT Weaponization Tasks At a minimum, the following items will need to be addressed in any future study: Airframe/Autopilot designs/implementations, Radome and/or seeker slope type compensation methods (software and hardware) and rolling versus non-rolling airframe configurations. Radiance has at its disposal several methodologies which have been developed and tested which can be utilized in this weaponization task.

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RT Weaponization Tasks Non-Linear Autopilot Implementation Adaptive Radome Aberration Compensation Non-Rolling/Rolling Airframe/Autopilot Implementations As Required All Of Above Previously Developed And Tested

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RT Non-Linear Autopilot The Non-Linear Airframe configuration results in the generation of highly non-linear (near cubic in nature) body loads (lift) as a function of total angle of attack (A-O-A) for relatively moderate values of A-O-A. Additionally, there exists significant center of pressure (C.P.) travel as a function of total A-O-A as well as a non-linear control moment effect. All of these phenomena result in an airframe which is subject to non- linear aerodynamic effects which will cause significant coupling between the normally uncoupled autopilot axes. Compounding matters are the high performance levels demanded of the airframe and autopilot which tend to negate the more conventional approaches to autopilot design. Indeed, current control methodologies rely on local linearizations and linear design techniques which, in general, will not be valid over the global non-linear flight regime. Specifically, this can include the current “buzz” word methodologies of LQG/LTR, H- Infinity and Mu-Synthesis. For example, any designs utilizing conventional "local linearization" techniques can result in either too sluggish a system when linearized about a "high" A-O-A or an unstable system (at "high" A-O-A) when linearized about a "low" A-O-A.

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RT Non-Linear Autopilot Non-linear Aerodynamics - Missile lift and moment (center of pressure location) characterizations are a non-linear function of total angle of attack (pitch/yaw axes). Induced moment coupling into the roll channel (vane shading, airframe asymmetries, etc.). NLAP design methodology allows for the easy creation of an arbitrary autopilot topology Arbitrary and decoupled autopilot configurations/channels.

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RT Non-Linear Autopilot There exists a "state of the art" technique known as "Feedback Linearization (Dynamic Inversion)" which can be applied to an appropriately constructed non-linear model of the airframe. This results in a unified autopilot design methodology which solves the non-linear problem while still preserving a large portion of the usefulness of linear "point stability" design methods. Implicit in this approach is the development of a suitable non-linear, non-rolling coordinate frame characterization of the non-linear aerodynamic airframe control model. This design method is inherently a non-linear synthesis technique which directly allows the construction of a non-linear control (autopilot) law which utilizes all of the information present in the non-linear airframe database. The use of the “Feedback Linearization” design method results in a user selectable, decoupled, linear time-invariant relationship (map) between the commanded accelerations and the obtained accelerations. Furthermore, the autopilot “gain scheduling” is implemented so as to be table driven by the aerodynamic and mass properties characterization models. This allows for a simple and highly maintainable (and flexible) autopilot structure despite changes in the airframe characteristics. Additionally, the same autopilot topology can be used throughout the entire flight regime. Specifically, the potential blending of aerodynamic and thruster controls during terminal guidance is easily handled within the non-linear design framework. Finally, the technique, as applied, allows the designer to be much more attuned to the pertinent modeling details and controller topology than is usually the case. Indeed the coupling mechanisms and decoupling rules become particularly evident.

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RT Non-Linear Autopilot NLAP design implementation allows for a quick and effective way to design a table-driven autopilot structure which can be updated in a trivial manner. NLAP design allows for ease of blending aero and thruster type controls in order to minimize thruster motor usage. Precise guidance and autopilot stabilization and performance requirements can be met even with a highly non-linear aerodynamic missile

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RT Block Diagram for Objective Non-Linear Autopilot A Functional Block diagram of the Non-linear Autopilot configuration (NLAP). Note that the pitch/yaw channels are coupled together in order to obtain the desired decoupled response characteristics.

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RT Block Diagram for Objective Non- Linear Autopilot

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RT Adaptive RF Radome Aberration Compensation Radome Aberrations (General Concepts) Electrical (phase front) distortions of the RF plane waves as they pass through the dielectric material of the radome, interaction of the plane waves with the internal components of the missile front end, and receiver processing characteristics give rise to non-linear and time varying boresight aberration errors. In general, these “radome” errors are characterized (in each of two orthogonal guidance axes), by aberration maps as a function of target look angle, received RF polarization and RF frequency. Polarization of the incoming RF wavefront becomes an issue of concern when semi-active or passive systems are employed since any factory compensation must be performed at an assumed polarization unless the seeker front end employs a polarimetric estimation function (an expensive proposition). Active seekers essentially see only the polarization which they transmit and are thus relatively immune to this effect (cross polarization transmissions). Rolling Active Seeker / Airframe Model The co-polarization aberration map in each of two orthogonal rolling antennae axes is characterized by an assumed Fourier expansion consisting of a fundamental and up to six additional harmonics of the roll (or de-roll) angle. This assumed model is chosen since the airframe roll rate (several HZ.) makes the aberrations look “near periodic” while the missile is undergoing “slower” non-rolling pitch/yaw motion. These aberrations, along with monopulse boresight angles, are then de-rolled to a non-rolling guidance frame.

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RT Adaptive RF Radome Aberration Compensation Active, Two Axis RF Seekers in a Rolling Airframe Appear to Offer an Advantage with Respect to Slope Type Errors (Primarily Radome). This is Partially a Result of Replacing Linear Slope Errors with Periodic Spatial Noise-Like Errors as a Consequence of Rolling The Obtained Performance is Greatly Enhanced by the Use of Adaptive Filtering Techniques Which Can “Learn” a Significant Portion Of The Spatial Contamination. Additionally, Seeker Null Biases and Gyro Misalignments Can Also Be “Learned” During the Course of the Engagement. This Adaptation Process Essentially Constructs a Fourier Expansion of the Induced Aberrations

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RT Adaptive RF Radome Aberration Compensation The Filtering Concepts Presented Here Can Be Easily Adapted to Other RF Missiles Which Satisfy the Underlying Operational Mode (Two Axis Gimbal or Strapdown, Rolling Airframe)

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RT Functional Filter Block Diagram (Non-Rolling) Coordinate Frame A functional Block Diagram of the Adaptive, Rolling Airframe Aberration Filter. The filter is implemented in a non-rolling guidance frame.

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RT (Elevation) Residual K K acc. KwKw + ++ +. - + 1/R K acc.. - + 1/R K Residual KwKw + + + + Z 2 LOS + Corruption + noise (Azimuth) - + Z 1 LOS + Corruption + noise. Independent Filters Aberration Fourier Expansion Coefficients Stacked Filters Coupled Measurement Fourier Expansion Coefficients OR Functional Filter Block Diagram (Non-Rolling) Coordinate Frame

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RT Conclusions / Recommendations While the results of this study have shown potential design approaches which may yield an acceptable solution to the “weaving” target problem, no overall, definitive solution has been found. Indeed, it appears that the most fruitful approach will be to combine a properly designed airframe/autopilot configuration which will allow for the fastest possible response time with an appropriate guidance law. This guidance law may still prove to be “predictive” in nature, or may simply be a highly responsive adaptation of pro-nav without explicit estimation of the weaving target motion. Finally, airframe enhancements such as dual control (tail, canard), axial control and reactive warheads should be considered.

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RT Conclusions/Recommendations No “Silver Bullet” Has Been Found to Date Some Methods Show Enough Promise to Warrant Their Inclusion in an Overall “Strategy” Overall Strategy Should Incorporate Airframe / Autopilot Enhancements (Dual Control, Axial Control, etc.) and Enhanced (Reactive) Warhead to Ensure Kill Even Without HTK Performance

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Mechanical & Aerospace Engineering, Princeton University Princeton, NJ. Sep 28 th, 2004 Filtering for an Interceptor Mission with a Weaving Target by Egemen Kolemen & N. Jeremy Kasdin SBIR 2004 Presentation

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Models We are Using The objective of the study is to examine the potential effectiveness of various nonlinear filters, particularly the UKF and other particle filtering approaches, on the estimation problem of determining the states of a weaving target. In all missile estimation work, one of the critical choices is the particularly reference frame and coordinate system within which to model the problem. In this study we simplify by only looking at the 2-D engagement, but we model in both a Cartesian system and in LOS coordinates to begin comparing the advantages and disadvantages of each. This is only the a preliminary look at the various filters to help inform future work. Any estimation analysis must include some control law to stabilize the engagement. Here, we use augmented proportional navigation with and without weaving but we include delay compensation.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Models We are Using 2. 2D LOS Coord. EKF and UKF (Working-Checked) PN, APN with and without weaving + delay compensation Coordinate SystemFilter 1. 2D Cartesian Coord. Control Model APN with and without weaving + delay compensation UKF (Working-Not Checked for different initial conditions) /EKF can be added easily

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Problem Statement This figure establishes the problem we will be studying. The engagement is 2-D, both missile and target have a constant velocity, and the command accelerations for both are always perpendicular to the velocity vectors. Also shown is the Cartesian coordinate system used for the modeling.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Problem Statement Accelerations are perpendicular to velocities. is the LOS angle. is the target velocity angle. L is the Miss Angle. HE is the Heading error. 2D target-missile model X inertial Y inertial vmvm amam atat vtvt y L+HE

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Problem Statement This slide states the specific problem we were asked to examine. We are to assume a target with constant velocity but a possible weave with frequencies anywhere from 0.1 to 2 Hz and accelerations up to 17 g. The interceptor also has a constant velocity but a maneuvering acceleration capability of up to 20 g. We also assume a time lag in the acceleration command model. We are not considering specific autopilots in this study. That is, both the target and interceptor are point masses.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Problem Statement TARGET Constant V T =3 km/s Target weaving with frequency 0.1 to 2 Hz. A T =A T 0 sin(wt+w 0 ) where w=2 f and A T 0 varying between 7g and 17g. INTERCEPTOR Constant V I =2 km/s A I m ax =20g Dynamic model: Time Lag of 0.1 sec. GEOMETRY Range=5-50km LOS angle=-30 to +30 degrees Heading Error=-10 to +10 degrees

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Non-Linear Dynamical Equations This chart sets up the non-linear equations of motion for the missile and target in both Cartesian and line-of-sight coordinates. We have not yet included the target weave dynamics; these are just the natural dynamical equations of motion. It is useful to observe that the LOS coordinates are highly nonlinear. This is typical of missile estimation problems. In LOS coordinates, the dynamics is very nonlinear but the measurement equations are quite simple and linear. The converse is true in Cartesian coordinates. It is a fact that different filtering approaches tend to be more or less robust depending upon whether the dominant nonlinearity is in the dynamics or in the measurement equation.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Non-Linear Dynamical Equations 2D Non-Linear Cartesian Coordinate Equations where 2D Non-Linear LOS Coordinate Equations where

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Optimal Guidance Law This chart shows the various guidance laws that we apply to the problem, from a simple proportional navigation to a Lag compensated augmented proportional navigation. In all of our simulations we have implemented lag compensated augmented proportional navigation.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Optimal Guidance Law Proportional Navigation: where N’ is chosen between ~3-5. Augmented Proportional Navigation: If we have some knowledge of the target acceleration Lag Compensated Augmented Proportional Navigation: If there is a known lag in the missile dynamics, T, where N’=N’(x) is an increasing function of time. i.e. large as t go 0 and

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Optimal Guidance Law against a Weaving Target This slide shows the optimal inclusion of target weave into the proportional navigation guidance law with lag. Assuming the weave frequency is known, this will result in a guaranteed hit. We have not yet investigated the effect of modeling errors, that is, if the target performs a non-sinusoidal weave different from the model.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Optimal Guidance Law against a Weaving Target Augmented Proportional Navigation for a Weaving Target If we have some knowledge of the target acceleration where N’ is chosen between ~3-5. Lag Compensated Augmented Proportional Navigation for a Weaving Target If there is a known lag in the missile dynamics, T. where N’=N’(x) is an increasing function of time. i.e. large as t go 0 and and x=t go /T.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Linearized Dynamical Equations for a Weaving Target It is useful, and common, to attempt a linearized model. This has the advantage of allowing a simple, linear Kalman filter (assuming all parameters are known) and greatly simplifies the dynamics. It is a good starting point and basis for comparison. Here, we assume small line of sight angle and small target velocity angle. In other words, the engagement is almost head on. Under that scenario, it is straightforward to linearize the Cartesian equations of motion. Here, we have also included the dynamics of the target sinusoidal weave. If the frequency of the weave is known, then the model is entirely linear as in the top set of equations. If the frequency is unknown and needs to be estimated, then we have a parameter estimation problem. This is typically handled by augmenting the state with the unknown parameter model. This is done in the lower set of equations above. The resulting equations are still linear in the state, but nonlinear in the parameter; what we call a pseudo- linear model.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Linearized Dynamical Equations for a Weaving Target 2D Linear Cartesian Equations for known frequency. (Assume small 2D Pseudo-Linear Cartesian Equations for unknown frequency 2D target-missile model X inertial Y inertial λ atat vtvt y amam vmvm

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Linearized Dynamical Equations for a Weaving Target This chart shows the same pseudo-linear equations in LOS coordinates. In order to obtain linear state equations, we assumed that the range and range rate was perfectly known (or measured) and thus could be removed from the dynamics. Even in this idealized, head on scenario this is a somewhat suspect assumption, but it does allow for a baseline case. Note the big advantage of the pseudo-linear LOS case; the line of sight measurement is perfectly linear (again, we do not include airframe or sensor models here).

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Linearized Dynamical Equations for a Weaving Target 2D Linearized LOS Coordinate Equations for unknown frequency (Assume small and r, r_dot known Where the measurement is: 2D target-missile model X inertial Y inertial λ atat vtvt y amam vmvm

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Sub-Optimal Estimation Philosophies This chart begins a brief summary of the various filtering techniques in the literature that might be brought to bare on this problem. Recall that the problem is to estimate the missile and target states (mainly the accelerations) and the target weave frequency using the range and line of sight measurements. These are then used in the optimal guidance laws we showed earlier. For all of these studies we assume active seekers that provide range information (or that range information may be available from ground sites via a Comm link). Passive systems have observability questions and are much more complicated; they were not addressed in this study. There are essentially three approaches to estimation: Fisher, Bayesian, and Least Squares. Fisher methods essentially assume nothing about the state or model and will not be discussed further. Bayesian methods use prior or assumed knowledge of the distribution of the states (and noises) to find the most probable estimates of the states. Least Squares approaches use no prior assumptions on the distributions to find the estimates; they simply minimize a quadratic cost function of the states and measurements. It is true that for Gaussian noises and linear models, the Bayesian and Least Squares approaches converge.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Sub-Optimal Estimation Philosophies Fisher: –Assume x to be completely unknown Bayesian –Assume x to be a distribution Least Square –Find x according to a Least Square Function defined by the engineer. No assumptions.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Estimation Techniques Within the context of both Bayesian and Least Squares methods comes the problem of how to handle the nonlinearities in the model and measurements. For linear problems, all techniques converge to the Kalman filter, but nonlinearities greatly complicate. The most common approach to handling nonlinearities is to simply expand the nonlinear functions in a Taylor series about the previous estimates. As long as the estimation error is small, this can be quite effective. The Extended Kalman Filter is the most familiar example of this approach. Unfortunately, since errors are usually rather large initially, this approach can result in large biases and inconsistent estimates. In stochastic nonlinear dynamic analysis, a simple, efficient and commonly-employed method of solution is to solve a statistically-equivalent linearized set of governing equations. By minimizing the error between the original and the equivalent system in the mean square sense, reasonably good mean square responses of the system compared with simulation results can be obtained. Another approach is to use many points and the law of large numbers to attempt to reproduce the probability density function of the estimate. Particle filters and Monte Carlo techniques are general classes of this type of approach. One could also take a brute force approach and simply numerically minimize some nonlinear cost. Iterated batch smoothers are examples of this. The two-step estimator is a method that uses numerical minimization recursively to handle non-linear measurements.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Estimation Techniques Taylor Series Expansion Statistical Linearization Employ Many Points (imitation of the PDF) Numerical Minimization of a Cost Function

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Least-Squares Type Filters This chart shows the most common, nonlinear least squares type filters. The pseudo- linear, extended, and iterated extended Kalman filter all use Taylor series expansions to handle the nonlinearities. The iterated EKF improves the response slightly by iterating the measurement update with repeated expansions. The two-step filter is a newer method that does not Taylor expand the measurement but uses a numerical minimization for the optimal measurement update (though an expansion is necessary for the covariance estimate). All of these filters handle nonlinearities in the dynamics via numerical integration and Taylor expansion for the covariance.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Least-Squares Type Filters Pseudo-Linear Kalman Filter Extended Kalman Filter Iterated Extended Kalman Filter Two-Step Filter (+ numerical minimization)

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Types of Particle Based Filters Particle filters avoid expansions by tracking the propagation of many points through the filter. The law of large numbers is then used to reproduce the probability distribution functions of the estimates, from which mean and covariance estimates can be obtained. A Monte-Carlo type estimator is the simplest, though most impractical, of these methods. While these methods offer great potential improvements to nonlinear estimation problems, this comes at a potentially large computational cost. The Gaussian Particle Filter (GPF) approximates the posterior mean and covariance of the unknown state variables using importance sampling. It is essentially a PF optimized for additive Gaussian covariances (noise). Particle filters compute the posterior distribution through discrete approximations to the exact one. Particles are points chosen from a posterior distribution in each time step (Nt particles). The key idea is to represent the required predictive or filtering distribution by a set of random samples (possibly with weights) and to compute estimates based on these samples and on these weights. The Sigma-Point Particle Filter is a hybrid particle filter that uses a sigma-point Kalman filter (SRUKF or SRCDKF) for proposal distribution generation and is an extension of the original "Unscented Particle Filter" of Merwe et al. In this paper, we present a novel algorithm to recursively update the posterior density called the Gaussian Mixture Sigma- Point Particle Filter (GMSPPF). This filter has equal or better estimation performance when compared to standard particle filters and the SPPF, at a largely reduced computational cost. The GMSPPF combines an importance sampling (IS) based measurement update step with a SPKF based Gaussian sum filter for the time update and proposal density generation. The GMSPPF uses a finite Gaussian mixture model (GMM) representation of the posterior filtering density, which is recovered from the weighted posterior particle set of the IS based measurement update stage, by means of a weighted Expectation-Maximization (WEM) algorithm. The WEM stage also replaces the resampling step of the standard particle filter and mitigates the “sample depletion” problem.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Types of Particle Based Filters UKF, SRUKF (Square-Rook UKF) PF -Generic Particle Filter Different Types of Partical Filters and UKF/PF Hybrids –GPF - Gaussian Particle Filter –SPPF- Sigma-Point Particle Filter –GMSPPF- Sigma-Point Bayesian Filter –… Two-Step/UKF Hybrid

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin EKF / IEKF The EKF is by the far the most commonly used and simplest to implement of the nonlinear filters. The equations come from simply expanding the nonlinearity about the previous estimates in the cost function and developing linearized versions of the Kalman filter equations. This chart is here simply for reference.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin EKF / IEKF Procedure: –Keep the 1 st order terms in Taylor Expansion for both the time and measurement update. –Assume linear measurement update.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Two Step Estimator The two-step filter attempts to improve on the EKF by mapping the nonlinear measurement to a new variable y that is linear. A Kalman filter measurement update is then performed on the new variable. The original states are then estimated via a numerical minimization of a second quadratic cost. For static problems, this has been shown to be the optimal state estimator. The two-step filter offers improvement only in the event of nonlinear measurements. The time update is the same as the EKF. For the simplified problems examined in this study we have assumed linear measurements in every case, so we have not yet examined the two-step estimator.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Measurement Update: –Define a second stage variable y s.t. y=f(x) z=Hy+n –So the Linear Kalman Filter measurement update can be utilized i.e. y k(+) =y k(-) +K*(z – H y ) P y k(+)=(P y k(-)-1+H’*R*H) -1 l –Find x by minimizing (using Gaussian Iteration) the most likelihood cost function J=[y-f(x)]G[y-f(x)]’ Time Update: Until now, first or second order TSE. We want to include a Unscented time update Two Step Estimator

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Examples of Sub-Optimal Filters In this figure, different types of sub-optimal filters are compared graphically. In an EKF, the value of y is estimated simply by a point estimation i.e. y_{est} = h(x_{est}) while the covariance is obtained by linearizing at the state estimate x_{est}. In a PF, the distribution of x is approximated by utilizing a set of random sample points (possibly with weights). In the figure we showed the most basic form, where there is no weighting on the random points. Each of these points are then propagated though the equation and the new mean and covariance are found by averaging the results. The UKF is a particle based filter where the sample points are chosen using whats is known as a “sigma point approach” rather than randomly. The Two-step filter goes backwards through the nonlinear equation. First, an estimate for y is found then the “best” x_{est} is solved by iteration. The problem with an EKF is that it linearizes the function at the estimation of the state x_{est} and does not make use of the changing shape of the covariance. It also assume, incorrectly, that the mean of the estimation is h(x_{est}).

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin h(x) y H(x)| x 0 x PF UKF EKF y est x est Two-Step Examples of Sub-Optimal Filters

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin UKF The unscented Kalman filter (UKF) is a derivative free approach to Kalman filtering. Essentially, 2L + 1 sigma points (where L is the state dimension), are chosen based on a square-root decomposition of the prior state covariance. These sigma points are propagated through the true nonlinearity, without approximation, and then a weighted sum is taken to approximate the mean and covariance. In the figure, the height of the probability density function at the sigma points is used to give a sense of how the weights are used. (i.e., highest weights near the mean while decreasing symmetrically away from the center)

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin UKF h(x) y H| x 0 x Instead of linearization for Covariance Update –choose sigma points with a weight proportional to the height in the probability density function. –propagate in the nonlinear function –find the weighted mean and covariance Weight

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Particle Filters While there are many different types of particle filters, the basic idea is to represent the required predictive or filtering distribution by a set of random sample points (possibly with weights) and to compute estimates based on these samples and on the weights. As the number of samples in the Monte-Carlo sample becomes very large, the empirical measure associated with this sample becomes an approximation of the predictive or the filtering distribution... provided that the Monte-Carlo samples are drawn in a consistent way. In the figure we show an example of a PF without any weighting on the sample points.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Particle Filters Idea: Idea: Imitate Probability Density Function with a collection of points. a.Choose many points at random with the given covariance. b.Propagate these points in the nonlinear function. c.Find the un-weighted mean and covariance h(x) y x y est x est

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin EKF Filter Equations This slide shows the linearized, Cartesian dynamical equations with unknown weave frequency used in an extended Kalman filter. The weaving guidance law is also shown. This filter has been implemented in simulation.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin EKF Filter Equations EKF with and without Lag and Wave Compensation using the Linearized Cartesian Coordinate Dynamic Equations. Running-Checked. Lag Compensated Augmented Proportional Navigation for a Weaving Target If there is a known lag in the missile dynamics, T. where N’=N’(x) is an increasing function of time. i.e. high when as t go 0 and and x=t go /T.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin EKF Filter Equations Any EKF needs to solve the linearized dynamic equations for the state transition matrix and the discrete noise covariance matrix in order to perform the time update. For reference, this slide provides those matrices for the previous model.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin EKF Filter Equations State Transition Matrix Used only to propagate the error covariance Process Noise Matrix Used only to propagate the error covariance

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin UKF Filter Equations An unscented Kalman filter (UKF) was also designed and implemented. The Cartesian system is straightforward and shown earlier. Here is the model developed in LOS coordinates. This slide provides the linearized dynamic model and the weaving guidance law. Once again, the range (and range rate) are assumed known.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin UKF Filter Equations UKF with and without Lag and Wave Compensation using the Linearized LOS Coordinate Dynamic Equations. Running-Not Checked. Lag Compensated Augmented Proportional Navigation for a Weaving Target If there is a known lag in the missile dynamics, T. where N’=N’(x) is an increasing function of time. i.e. high when as t go 0 and and x=t go /T.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin A Comparison of EKF and UKF for Linear Case This slide provides a side by side comparison of an EKF and UKF for the simplest possible head on engagement using the pseudo-linear Cartesian models. The target has a constant weave frequency. In the figure, the green lines are the estimated states while blue are the real value of the state. This is the simplest possible head on engagement. The aim of using this simple scenario is to compare the effectiveness of the EKF and UKF in find the weave frequency with minimum interference of any other effects. There are 4 graphs plotted with time on the x-axis and the real-estimated acceleration, real-estimated jerk, real-estimated frequency, error in acceleration estimate and error in jerk estimation respectively. The EKF and UKF approximations are presented on the left and right respectively. From this analysis we can see that UKF does a much better job of estimating the frequency and the state of the target.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin A Comparison of EKF and UKF for Linear Case EKF UKF

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Non-linear Equations w/ Unknown Frequency Our simulations showed that the pseudo-linear model was not a very effective approach. It only took a few degrees off of direct for the simulation to break down. Also, it was necessary to have at least 3 to 4 full weave periods of the target to get a lock on the frequency. In any scenario, it is likely that at the very least the transient will involve errors larger than 10 deg. Therefore, it is inevitable and necessary that we develop full nonlinear filtering approaches. Nevertheless, the pseudo-linear approach was quite useful for developing the code and gaining insight into the trade-offs. As shown earlier, we can develop non-linear models in both Cartesian and LOS coordinates. However, it is our belief at the moment that LOS offers an advantage because, at least in this simplified approach with no autopilot, the measurements remain linear and we only need to develop complex algorithms for the time update. In the future, we expect to examine the trade-offs between the two approaches. This chart shows the nonlinear model in LOS coordinates. Now, it is necessary to include the range and range rate as states for the best behavior. However, we assume perfect noise free measurements.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Non-linear Equations w/ Unknown Frequency LOS Coordinate Equations with unknown frequency where With measurements

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Preliminary Results for Nonlinear Case We were unable in the time we had to do a thorough study of the various filtering methods on the full problem. This is proposed for future projects. However, we were able to demonstrate that we had functioning code on a sample problem. Using the non-linear LOS equations, we obtained the following trajectories for different types of particle type filters. Here blue is the path of the interceptor and red is the target. On the left top we see that the UKF and SRUKF. They give the same results since SRUKF is simply a faster version of the UKF. Other results we see are very different in nature since the control is dependent on the frequency. In the beginning of the simulation the target moves upwards and depending on the frequency, the filter interprets this as either an upward motion with a small period or a horizontal motion with a large period. The control output changes dramatically–either to accelerate upwards like the PF or to continue on moving on the horizontal direction without much acceleration like the UKF. These figures demonstrate that the type of filter selected can make a significance difference in the engagement. They also show that control is sensitive to the calculated frequency of the target. While this a single scenario, it signifies the importance of the estimator type on motion.

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Preliminary Results for Nonlinear Case UKF,SRUKF PF UKF, SRUKF PF GPF GMSPPF Interceptor Target

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Conclusions This Page Intentionally Left Blank

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SBIR Presentation Egemen Kolemen& N. Jeremy Kasdin Built Simulations for linear and non-linear models Demonstrated UKF in linear model, showed improvement over EKF Built a variety of Particle filters, including UKF, for nonlinear model Demonstrated on simple problem Very limited range of effectiveness so far Still to do: –Tune filters, debug, and run monte-carlo cases –Implement more complicated weaving scenarios –Incorporate realistic missile model and measurements Two Step EstimatorConclusions

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