Title: MDA Phase I SBIR Final Report Advanced Guidance, Navigation, and Control (GNC) Algorithm Development to Enhance Lethality of Interceptors Against Maneuvering Targets
1MDA Phase I SBIR Final ReportAdvanced Guidance,
Navigation, and Control (GNC) Algorithm
Development to Enhance Lethality of Interceptors
Against Maneuvering Targets
- Radiance Technologies, Inc.
- (256) 704-3400
- 5 January 2005
2SBIR PHASE IInterceptor 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.
3SBIR PHASE IInterceptor 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
4This Page Intentionally Left Blank
5Preliminary Simulation Analyses of Estimation and
Guidance Algorithms
6Analysis 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).
7Analysis Outline
- Simulation Overview
- Single Linear, Multiple Model Kalman Filter
Methodologies - Sliding Mode Guidance/Estimation/Autopilot
- Two-Step Filters and UKF
8Functional Block Diagram of Interceptor6-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.
9Functional Block Diagram of Interceptor6-DOF
Simulation and Engagement Models
10Interceptor 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 1.6-3.1Hz, 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.
11Interceptor 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
- ??n18r/s, z0.7 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
12Autopilot 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.
13Autopilot Used in 6-DOF Simulation Guidance /
Estimation Studies
14This Page Intentionally Left Blank
15Interceptor Guidance/Estimation Algorithms
- Single Linear, Multiple Model Kalman Filter
Methodology
16Single 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.
17Single 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
18Discrete 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.
19Discrete Frequency Kalman Filter State Equations
Note Only Six States Shown Here
20Optimal 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.
21Optimal Weave Guidance Law
Guidance Law Must be Consistent with Filter
Mechanization to Gain Improved Miss and Reduced
Maneuver Benefits
Note Only Six Terms Shown Here
22Filter 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.
23Filter 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
24Linear 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.
25Linear Single Frequency Kalman Filter Tolerates
10 Weave Frequency Mismatch
Filter Pre-Selected Frequency 0.5Hz
26Second 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.
27Second Frequency in Filter Model Increases
Performance Window
- Target Acceleration Amplitude 8gs
- Target b5000kg/m2
- Filter Model Frequencies DC, 0.4, and 0.5 Hz
- Target Weave Frequency Varied from 0.35-0.6Hz
- Pole Placement Autopilot w/DC Gain Adjustment
- wn18r/s, z0.7
28PN 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 31 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).
29PN And APN Guidance Performance Sensitivity to
Target Weave Frequency
Weave Amplitude 8gs, Target b 5000kg/m2
PN Guidance, KG 4
APN Guidance, KG 4
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0.0
0.1
0.2
0.3
0.4
0.0
0.1
0.2
0.3
0.4
Target Weave Frequency (Hz)
Target Weave Frequency (Hz)
CEP
CEP
R95
R95
R99
R99
30Section 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.
31Section 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
32Sliding Mode Methodologies
- The following section describes the sliding mode
control methodology applied to the guidance,
autopilot, and control problem.
33Interceptor Guidance / Estimation / Autopilot
Algorithms
- Sliding Mode Methodologies
34GNC Algorithms for Advanced Homing Interceptor
In this chart the tasks to be addressed while
studying hit-to-kill interceptor GNC systems
using second order sliding mode control are
listed.
35GNC 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)
36GNC Algorithms for Advanced Homing Interceptor
- The benefits of an application of second order
sliding mode control to - the GNC of hit-to-kill missile interceptors are
discussed. - The main advantages are
- robustness to measurement noise helps designing
a navigation system - robustness to target maneuvers helps designing a
smooth guidance law - robustness to external disturbances helps in
designing an autopilot
37GNC 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
38Objective
The main RD and performance objectives for the
accomplished Phase I research effort are listed
and discussed in this chart.
39Objective
- Main RD objective
- determine combinations of measurements/estimation
s 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
40Given 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 GNC 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
41Given Intercept Geometry
Collision Course vs. Aspect Angle
42Given Target Maneuver
The chart illustrates a given target maneuver
envelope for the missile-target engagement
scenario, upon which the developed GNC 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
GNC system testing.
43Given Target Maneuver
- Step-Constant Maneuver Envelope
- Spiral Maneuver Envelope
44Meeting 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).
45Meeting the Objectives
- 2-Sliding Filter as a Part of Analog-to-Digital
Converter - Measurement Input Seeker LOS Rate Signal
- ADC Sampling Rate 10kHz-1GHz
2-Sliding Filter is a Sliding-Mode-Controlled
Signal Observer
The continuous-time model Super-Twisting
2-Sliding Observer
Block- Scheme
46Meeting 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.
47Meeting the Objectives
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
48Meeting 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.
49Meeting the Objectives
THAAD-Like Interceptor
- 2-Sliding Flight Control System
KEKV Parameters
Moment of inertia (pitch plane) 4 kg m2 Mass
100 kg, Reference diameter 0.3 m Divert force
10,000 N Attitude normal force 1,000 N
Control challenge Uncertainties created by the
interactions between the airflow and the
thrusters jets
50Meeting 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.
51Meeting the Objectives
THAAD-Like Interceptor
- 2-Sliding Flight Control System
Control challenge Uncertainties created by the
interactions between the airflow and the
thrusters jets
52The 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.
53Attitude Thruster Control
Divert Thruster Control
Smooth 2-Sliding Control
Nonlinear Dynamic Sliding Manifold
54- 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.
55- 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
56Sensor Information Requirements
The requirements to the seeker characteristics
are formulated in the chart for the GNC tests
accomplished during Phase I effort. Information
requirements and requirements to a compensated
missile dynamics are also formulated for the
simulation tests.
57 Sensor Information Requirements
- Seeker LOS noise StdDev 1mrad, Time Lag
100msec
- 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
T0.1sec and max(Am)20G
58Performance Analysis
The designed sliding mode control-based GNC
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 GNC systems performances via
computer simulation. In this chart the
performance characteristics of the
correspondingly tuned up sliding mode filter and
KF are demonstrated.
59Performance 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
60Performance Analysis
- Evaluation of Intercept Envelope under full
information guidance
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 400 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..
61Performance Analysis
- Evaluation of Intercept Envelope under full
information guidance
Miss Occurs when Target Having Higher Speed
Overruns Missile
62Performance 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.
63Performance Analysis
- Mean Miss Distance vs. Time-to-Go, Step-Constant
Maneuver
Aspect Angle 10deg
Average of 25 trials for each 7g-step
maneuver 0.25m is a direct hit
64Performance 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.
65Performance Analysis
- Acceleration Ratio vs. Time-to-Go, Step-Constant
Maneuver
Aspect Angle 10deg
10-20 Decrease in AR at large times-to-go
66Performance 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.
67Performance Analysis
- Missile Acceleration Profiles, Spiral Target
Maneuver
Aspect Angle 10deg Weave Frequency
0.1Hz Direct Hit achieved For all times-to-go
SMC acceleration is 30 less than APNKF
acceleration (not including end-game phase)
68Overall 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 GNC.
69Overall Simulation Results
- Conditions Given Seeker noise sigma 1mrad,
Seeker first order lag with T0.1sec and flight
control system lag with T0.1sec and
max(Am)20G. - Results Direct hit (misslt0.25m) at initial
aspect angles 10,20,30deg is achieved - For 7-g step-constant target maneuver not
sooner than 1sec time-to-go or larger than
5km range-to-target - For 7-g weave with frequencies up to 0.2Hz
- Requires Acceleration Advantage 2-3 times over
the target 10-30 less than APN Requirements
70SMC 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.
71SMC Methodology Strengths over APN
- 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)
72Key SMC Methodology Benefits
The chart describes the key benefits of using
SMC-base GNC methodology developed during the
Phase I effort for the GNC design of a
hit-to-kill missile interceptor.
73Key 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
74This Page Intentionally Left Blank
75Next 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
76This Page Intentionally Left Blank
77Sliding 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.
78Sliding Mode Estimation Equations
LOS, LOS Rate Estimation
Range, Range Rate Estimation Equations and Block
Diagram
79Sliding 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.
80Sliding Mode Guidance Equations
81Representative 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.
82Representative Altitude and Cross Range vs.
Downrange Trajectories, SMC Guidance
Target Step Maneuver 8gs, Yaw Plane, t0.4sec
Interceptor
Target
Target
Altitude (m)
Cross Range (m)
Interceptor
Downrange (m)
Downrange (m)
83Sliding 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.
84Sliding Mode Acceleration Commands and
Interceptor Responses
Target Step Maneuver 8gs, Yaw Plane, t0.4sec
Seeker Noise 0.5mr (1s), 5m (1s),Seeker/Guidance
Update Rate 100Hz
Pitch Acceleration Command (m/s2)
Pitch Acceleration Response (m/s2)
Yaw Acceleration Command (m/s2)
Yaw Acceleration Response (m/s2)
85First 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.
86First Variation of Sliding Mode Guidance Algorithm
- Simple Form of Sliding Mode Guidance
- Consists of
- PN Term Sliding Mode Augmentation Term
- Co Term in Generic Sliding Mode Guidance is set
to Zero - Note Target Acceleration Not Explicitly Estimated
Guidance Equations
87Sliding 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.
88Sliding Mode Guidance Variation 1
Target Step Maneuver 8gs, Yaw Plane, t0.4sec
Interceptor
Target
Target
Altitude (m)
Cross Range (m)
Interceptor
Downrange (m)
Downrange (m)
89Variation 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.
90Variation 1 Sliding Mode Guidance Acceleration
Commands and Interceptor Responses
Target Step Maneuver 8gs, Yaw Plane, t0.4sec
Seeker Noise 0.5mr (1s), 5m (1s), Seeker/Guidance
Update Rate 100Hz
Pitch Acceleration Command (m/s2)
Pitch Acceleration Response (m/s2)
Yaw Acceleration Command (m/s2)
Yaw Acceleration Response (m/s2)
91Linearized 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.
92Linearized Sliding Mode Guidance Law
93Performance 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.
94Performance Sensitivity to Target Maneuver Weave
Frequency
Target Maneuver Amplitude 8gs, Target b5000kg/m2
Linearized Sliding Mode Guidance
Sliding Mode Guidance Variant 1
10
10
R99
R99
R95
R95
CEP
CEP
5
5
0
0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
Target Weave Frequency (Hz)
Target Weave Frequency (Hz)
Sliding Mode Guidance
R99
10
R95
- 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
CEP
5
0
0.2
0.4
0.6
0.8
1.0
Target Weave Frequency (Hz)
95PN 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 31 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).
96PN And APN Guidance Performance Sensitivity to
Target Weave Frequency
Weave Amplitude 8gs, Target b 5000kg/m2
PN Guidance, KG 4
APN Guidance, KG 4
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0.0
0.1
0.2
0.3
0.4
0.0
0.1
0.2
0.3
0.4
Target Weave Frequency (Hz)
Target Weave Frequency (Hz)
CEP
CEP
R95
R95
R99
R99
97Section 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.
98Section 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
99This Page Intentionally Left Blank
100Interceptor Guidance/Estimation Algorithms
- Two-Step Filters and Unscented Kalman Filters
(UKF) Methodologies
101This Page Intentionally Left Blank
102Interceptor Guidance/Estimation Algorithms
- Future Weaponization Tasks
103Weaponization 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.
104Weaponization Tasks
- Non-Linear Autopilot Implementation
- Adaptive Radome Aberration Compensation
- Non-Rolling/Rolling Airframe/Autopilot
Implementations As Required - All Of Above Previously Developed And Tested
105Non-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.
106Non-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.
107Non-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.
108Non-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
109Block 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.
110Block Diagram for Objective Non-Linear Autopilot
111Adaptive 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.
112Adaptive 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
113This Page Intentionally Left Blank
114Adaptive 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)
115Functional 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.
116Functional Filter Block Diagram
(Non-Rolling) Coordinate Frame
Kw
?
-
?
Residual
?
K?
Z1? LOS Corruption noise
-
?
?
Kacc.
.
1/R
(Elevation)
.
Independent Filters Aberration Fourier Expansion
Coefficients
Stacked Filters Coupled Measurement Fourier
Expansion Coefficients
OR
-
?
Kacc.
.
1/R
?
?
?
Residual
K?
Z2? LOS Corruption
-
noise
Kw
?
(Azimuth)
117Conclusions / 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.
118Conclusions/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
119This Page Intentionally Left Blank
120by Egemen Kolemen N. Jeremy Kasdin SBIR 2004
Presentation
121Models 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.
122Models We are Using
Coordinate System
Filter
Control Model
EKF and UKF (Working-Checked)
PN, APN with and without weaving delay
compensation
1. 2D Cartesian Coord.
2. 2D LOS Coord.
APN with and without weaving delay compensation
UKF (Working-Not Checked for different initial
conditions) /EKF can be added easily
123Problem 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.
124Problem Statement
- Accelerations are perpendicular to velocities.
- l is the LOS angle.
- b is the target velocity angle.
- L is the Miss Angle.
- HE is the Heading error.
-
2D target-missile model
125Problem 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.
126Problem Statement
- TARGET
- Constant VT3 km/s
- Target weaving with frequency 0.1 to 2 Hz.
- ATAT0sin(wtw0)
- where w2?f and AT0 varying between 7g and 17g.
- INTERCEPTOR
- Constant VI2 km/s
- AImax20g
- Dynamic model Time Lag of 0.1 sec.
- GEOMETRY
- Range5-50km
- LOS angle-30 to 30 degrees
- Heading Error-10 to 10 degrees
127Non-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.
128Non-Linear Dynamical Equations
2D Non-Linear Cartesian Coordinate
Equations where 2D
Non-Linear LOS Coordinate Equations where
129Optimal 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.
130Optimal 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 NN(x) is an increasing function of
time. i.e. large as tgo ? 0 and
131Optimal 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.
132Optimal 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 NN(x) is an increasing function of
time. i.e. large as tgo ? 0 and - and xtgo/T.
-
133Linearized 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.
134Linearized Dynamical Equations for a Weaving
Target
- 2D Linear Cartesian Equations for known
frequency. (Assume small ?, b) -
-
- 2D Pseudo-Linear Cartesian Equations for unknown
frequency -
2D target-missile model
135Linearized 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).
136Linearized Dynamical Equations for a Weaving
Target
- 2D Linearized LOS Coordinate Equations for
unknown frequency (Assume small ?, b) and r,
r_dot known -