JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS
`Vol. 38, No. 2, February 2015
`
`Flight Control of a Small-Diameter Spin-Stabilized Projectile
`Using Imager Feedback
`
`Frank Fresconi∗
`U.S. Army Research Laboratory, Aberdeen Proving Ground, Maryland 21005
`and
`Jonathan Rogers†
`Georgia Institute of Technology, Atlanta, Georgia 30332
`
`DOI: 10.2514/1.G000815
`
`Small-diameter gun-launched projectiles pose a challenging platform on which to implement closed-loop guidance
`and control. This paper presents a novel imager-based guidance and control algorithm for small-diameter spin-
`stabilized projectiles. The control law is specifically formulated to rely on feedback only from a strapdown detector
`and roll angle sensors. Following introduction of the projectile nonlinear dynamic model, an integrated guidance and
`control algorithm is presented in which control commands are computed directly from detector feedback using a gain-
`scheduled proportional control. Time-varying controller gains are derived through a surrogate modeling approach,
`and controller performance is further enhanced through use of an observer that filters unwanted angular motion
`components from detector feedback. Example closed-loop flight simulations demonstrate performance of the
`proposed control system, and Monte Carlo analysis shows a factor of 2 accuracy improvement for the closed-loop
`system over ballistic flight. Results indicate that delivery system improvements are achievable in small, gyroscopically
`stabilized projectiles containing low-cost guidance elements using the proposed integrated guidance and control
`approach.
`
`Cm0 ,
`
`Clp
`Cm,
`Cmα , Cmα3
`Cmq
`CN,
`CNα , CNα3
`Cnpα
`CX, CX0 , CX α2
`
`CN0 ,
`
`CMX , CMZ ,
`
`
`
`CMl , CM
`m
`CYpα
`D
`fL
`I
`Ke, K _e
`
`Kε, K _ε
`
`L, M, N
`
`M
`m
`q
`
`qM
`
`rc:g:→c:p:i
`
`=
`=
`
`=
`=
`
`Nomenclature
`roll damping moment coefficient
`total,
`trim,
`linear, and nonlinear pitching
`moment coefficient
`pitch damping moment coefficient
`total, trim, linear, and nonlinear normal force
`coefficient
`= Magnus moment coefficient
`=
`total, zero-yaw, and yaw-dependent axial force
`coefficient
`= wing actuator aerodynamic coefficients
`
`=
`
`=
`
`= Magnus force coefficient
`=
`reference diameter, m
`=
`lens focal length, m
`= moment of inertia matrix, kg · m2
`=
`controller proportional and derivative gain
`values
`surrogate model proportional and derivative
`gain values
`aerodynamic moments acting on projectile,
`Nm
`= Mach Number
`= mass, kg
`=
`dynamic pressure at projectile mass center,
`N∕m2
`dynamic pressure at wing actuator center of
`pressure, N∕m2
`vector from center of gravity to ith movable
`aerodynamic surface center of pressure, m
`
`=
`
`=
`
`Received 7 June 2014; revision received 3 September 2014; accepted for
`publication 4 September 2014; published online 2 January 2015. This material
`is declared a work of the U.S. Government and is not subject to copyright
`protection in the United States. Copies of this paper may be made for personal
`or internal use, on condition that the copier pay the $10.00 per-copy fee to the
`Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923;
`include the code 1533-3884/15 and $10.00 in correspondence with the CCC.
`*Guided Flight Dynamics Lead, Weapons and Materials Research
`Directorate.
`†Assistant Professor, School of Mechanical Engineering.
`
`181
`
`
`
`rIOT, rI
`
`OP
`
`PT, rV
`rB
`
`PT
`
`~ry, ~rz
`
`S
`TBE
`
`TBM
`
`TBMi
`
`TVE
`
`_u, _v, _w
`
`V, Vc:g:∕I
`
`X, Y, Z
`XG, YG, ZG
`x, y, z
`_x, _y, _z
`xB, yB, zB
`xc:p:, rc:p:
`xE, yE, zE
`α
`α
`β
`δ
`ε
`
`
`
`ϵy, ϵ
`
`z
`
`~ϵ
`y, ~ϵ
`
`z
`
`
`
`~ϵ y , ~ϵ
`
`z
`
`=
`
`=
`
`=
`
`=
`=
`
`=
`
`=
`
`=
`
`=
`
`=
`
`=
`=
`=
`=
`=
`=
`=
`=
`=
`=
`=
`=
`
`=
`
`=
`
`=
`
`vector from origin to target and from origin
`to
`projectile
`expressed
`in Earth-fixed
`frame, m
`vector from projectile to target, expressed in
`body frame and velocity vector frame, m
`guidance command reference signals in fixed-
`plane coordinates, m
`reference area, m2
`transformation matrix from body- to Earth-
`fixed coordinates
`transformation matrix from body-fixed to
`wing surface coordinates
`transformation matrix from body-fixed to
`movable aerodynamic surface coordinates
`transformation matrix from velocity vector
`coordinates to Earth-fixed coordinates
`time rate of change of body-fixed coordinate
`system translational velocity, m∕s2
`total velocity of projectile with respect to
`wind, velocity of center of gravity with respect
`to inertial frame in body-fixed coordinates,
`m∕s
`aerodynamic forces acting on projectile, N
`gravity forces acting on projectile, N
`inertial position, m
`inertial translational velocity, m∕s
`body-fixed coordinate system
`axial and radial center of pressure, m
`Earth-fixed coordinate system
`pitch angle of attack, rad
`total angle of attack, rad
`yaw angle of attack, rad
`normalized control magnitude
`state variable for surrogate model representing
`y or z component of rV
`PT
`body-fixed coordinates of lateral components
`of vector from projectile to target in detector
`plane, m
`fixed-plane coordinates of lateral components
`of vector from projectile to target in detector
`plane, m
`filtered ~ϵy, ~ϵz signals from Kalman filter
`
`Downloaded by Michael Daniels on August 3, 2016 | http://arc.aiaa.org | DOI: 10.2514/1.G000815
`
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`

`
`182
`
`FRESCONI AND ROGERS
`
`
`
`ηy, η
`
`z
`
`ϕA
`ξ, ω
`
`n, fμ, μ
`
`C, ϕ
`
`B,
`
`ρ
`ϕ, θ, ψ
`ϕ
`CMD, ϕ
`Φ, ^ϕ
`ω
`B∕I
`
`=
`
`=
`=
`
`=
`=
`=
`
`=
`
`as
`
`zero-mean
`
`noise, modeled
`detector
`Gaussian white noise, m
`aerodynamic roll angle, rad
`damping, natural frequency, control scaling,
`and control input of surrogate model
`atmospheric density, kg∕m3
`roll, pitch, and yaw Euler angles, rad
`Earth-fixed commanded, Earth-fixed control,
`body-fixed control, relative, and stowed roll
`angles, rad
`body-fixed
`coordinate
`velocity, rad∕s
`
`system rotational
`
`I.
`
`Introduction
`
`D EVELOPMENT of viable guidance and control technologies
`
`for gun-launched projectiles is a unique engineering challenge.
`Over the past several decades, numerous solutions have been
`proposed, primarily for large artillery projectiles [1,2] or for slowly
`rolling airframes [3–5]. The general methodology employed by most
`of these studies has been to use GPS [1,6], inertial measurements [7],
`and seekers [8] to provide feedback to a flight control system. Course
`correction maneuvers are then initiated through aerodynamic mech-
`anisms [2,4,9] or jet control [10]. The numerous engineering
`challenges associated with fielding viable smart weapons systems
`have been repeatedly noted in the literature, including lack of payload
`volume for electronics, harsh accelerations at gun launch, highly
`nonlinear dynamics, and low-cost requirements.
`At the same time, guidance and control technologies developed to
`date largely do not scale to smaller man-portable munitions. Assault
`and carbine rifles, grenade launchers, and machine guns have been
`the backbone of the infantry for many years. Currently there is a
`desire to increase lethality of the infantry squad through precision
`guidance technologies designed for small-diameter projectiles.
`Successful application of miniaturized low-cost sensors and actu-
`ators to projectiles in this domain has the potential to revolutionize
`squad-level
`lethal capability. However, small-caliber projectiles
`impose severe constraints on control system development. Volume is
`a critical concern because the diameter of these projectiles ranges
`from only 5.56 to 40 mm. Additionally, these projectiles are launched
`from rifled guns that produce spin rates in the hundreds to thousands
`of rotations per second. The nonlinear dynamics caused by these high
`spin rates complicates sensor stimuli and stresses the actuator
`bandwidth. In addition, high-bandwidth and efficient control inputs
`are necessary because time of flight is often only a few seconds.
`Finally, because small-diameter projectiles are meant for squad-level
`weapons, it is essential that the integrated projectiles can be fielded at
`low cost. Low-cost requirements are manifested in terms of loose
`manufacturing tolerances, limitations in the type of feedback that can
`be obtained, and limitations in the degree of sensor calibration that
`can be performed. As a result, applicable guidance laws must be
`robust to large measurement uncertainty and require minimal state
`feedback.
`This paper outlines a novel guidance and control algorithm for
`small-diameter, gyroscopically stabilized projectiles equipped with a
`strapdown imager. The integrated guidance and control loop is
`designed to transform detector feedback directly into control surface
`commands. The proposed guidance law is formulated through
`control response modeling and development of a surrogate model for
`the input–output response. Once the surrogate model is developed, a
`linear control system is derived and stability bounds are determined.
`Because strapdown sensor feedback is severely perturbed by body
`angular motion during controlled flight, an extended Kalman filter is
`added to the feedback loop to remove body angular rate components
`from the feedback signals. This is feasible because the characteristic
`fast and slow-mode frequencies of the projectile are approximately
`known. It is assumed that the imager used in this study may be
`modeled as a pinhole camera [11]. Thus, details concerning whether
`the strapdown (or “stiff-neck”) detector signals are derived from a
`laser-designated spot or
`through image-based navigation are
`irrelevant for this study, because both seekers and standard cameras
`
`approximately satisfy the pinhole camera modeling assumptions
`[12]. The control system presented here is applied to an example
`40 mm grenade projectile equipped with a rotating wing maneuver
`mechanism. This grenade projectile exhibits flight characteristics
`typical of small-diameter spin-stabilized rounds, including high-
`amplitude angular motion, which highlight controller performance in
`a challenging flight environment.
`The paper proceeds as follows. First, the nonlinear projectile
`dynamic model is defined, and some representative flight behaviors
`are highlighted that complicate the guidance problem and preclude
`traditional solutions. An integrated guidance and control approach is
`presented, including surrogate modeling of the open-loop control
`response, followed by design of a linear controller that computes
`control surface deflections directly from detector outputs. Stability
`bounds on surrogate model control gains are analyzed. Example
`ballistic and controlled flight simulation results are presented, along
`with Monte Carlo simulations, which analyze performance with and
`without the angular rate estimator. Overall, resulting closed-loop
`performance of the proposed guidance system is shown to be favor-
`able, leading to significant reductions in impact point dispersion.
`
`II. Projectile Flight Dynamic Model
`This section describes the nonlinear flight dynamic model used in
`this study, including simulation of dynamics and aerodynamics,
`control actuation, and detector feedback. An example projectile con-
`cept is described, followed by a description of the mathematical
`model for each relevant component. A sample ballistic trajectory is
`provided to highlight key flight behaviors of the system.
`
`A. Projectile Concept
`The guidance and control formulation proposed in this study
`applies to the gamut of small-diameter spin-stabilized projectiles. For
`these small airframes, high control authority is difficult to obtain due
`to gun-launched packaging constraints and dynamic flight insta-
`bilities that often arise [13,14]. In this paper, a 40-mm-diam actively
`controlled grenade projectile is selected as an illustrative example.
`This system features a subsonic launch and a projectile spin rate of
`approximately 60 Hz. A rendering of the example projectile, in-
`cluding maneuver and navigation technologies, is shown in Fig. 1.
`An imaging device is packaged in a strapdown manner in the nose to
`reduce cost and ensure survivability during gun launch. Fuzing and
`warhead are contained in the central portion of the projectile body.
`The maneuver mechanism is placed at the aft end of the spin-
`stabilized projectile to maximize control authority [15,16]. This
`mechanism consists of a rotary motor (shown in red in Fig. 1) linked
`to a deflected wing (light blue in Fig. 1). As the projectile body spins
`over one revolution, the motor rotates the wing in the opposite
`direction at the same rate. This deploys the wing into the airstream for
`a portion of the roll cycle to create an aerodynamic asymmetry.
`
`Fig. 1 Example spin-stabilized 40 mm projectile with navigation and
`maneuver components.
`
`Downloaded by Michael Daniels on August 3, 2016 | http://arc.aiaa.org | DOI: 10.2514/1.G000815
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`

`
`FRESCONI AND ROGERS
`
`MR ˆ qS
`
`
`LR ˆ qSD
`Clp…M† pD
`
`Cmα…M†α ‡ Cmα3…M†α3 ‡ Cmα…M† qD
`
`‡ Cnpα…M; α†β pD
`
`NR ˆ qS
`−Cmα…M†β − Cmα3…M†β3 ‡ Cmq…M† rD
`
`‡ Cnpα…M; α†α pD
`
`
`
`2V
`
`2V
`
`2V
`
`183
`
`(4)
`
`2V
`
`2V
`
`Various means may be used to modulate the control effort because the
`wing has a fixed deflection angle. Control commands can “fishtail”
`the flight, a clutch can decouple the motor and wing, the wing can
`rotate at a frequency off the projectile spin rate, or multiple motor–
`wing assemblies can mix the relative phasing of the roll angles to
`deploy the wings into the airstream. For the purposes of this study,
`two motor–wing assemblies are considered to be packaged near the
`projectile base oriented 180 deg apart in roll angle. Because the focus
`of this study is on the imager-based guidance and control algorithm,
`additional details regarding this particular maneuver mechanism are
`omitted here for brevity but may be found in [17–20].
`
`B. Flight Dynamic Model
`The projectile body-fixed coordinate system is illustrated in Fig. 2,
`including the definition of aerodynamic angles of attack used to
`describe forces and moments. A standard body frame B is defined, as
`is a velocity vector frame V, where the V frame is derived by first
`rotating by β about the body z axis, and then by α about the resulting y
`axis. For this airframe, aerodynamic forces and moments are
`composed of components from the projectile rigid body (denoted by
`superscript R) and movable wing surfaces (denoted by super-
`script M).
`Total forces and moments on the projectile are given by
`
`X ˆ XR ‡ XM
`Y ˆ YR ‡ YM
`Z ˆ ZR ‡ ZM
`
`L ˆ LR ‡ LM
`M ˆ MR ‡ MM
`N ˆ NR ‡ NM
`
`(1)
`
`(2)
`
`Projectile body forces include contributions from axial force, normal
`force, and side (Magnus) force and can be described by the following
`aerodynamic expansion:
`
`
`…M† ‡ CXα2…M† α2Š
`XR ˆ −qS‰CX0
`YR ˆ −qS
`CNα…M†β ‡ CNα3…M†β3 − CYpα…M; α†α pD
`
`CNα…M†α ‡ CNα3…M†α3 − CYpα…M; α†β pD
`
`
`
`
`2V
`
`2V
`
`ZR ˆ −qS
`
`To model the wing forces and moments, first define the relative roll
`angle of wing actuator i with respect to the inertial frame as
`
`Φi ˆ ϕCi
`
`− ϕ
`
`(5)
`
`As the projectile rolls, the wing area exposed to the airstream varies
`such that it is maximum when the projectile roll angle coincides with
`the desired maneuver plane. Thus, define the amount of wing area
`exposed to the airstream through the stowed roll angle as
`
`^ϕ
`
`i ˆ Φ
`
`i
`
`− ϕ
`
`Bi
`
`(6)
`
`The total angle of attack of wing actuator i αMi and the yaw angle of
`attack βMi are given by the following expressions:
`
`q
`q
`
`2 ‡ wMi
`vMi
`2 ‡ vMi
`2 ‡ wMi
`uMi
`
`(7)
`
`1A
`
`2
`
`2
`
`0@
`
`Mi ˆ a sin
`
`α
`
`(8)
`
`1A
`
`q
`
`vMi
`2 ‡ vMi
`2 ‡ wMi
`uMi
`
`2
`
`0@
`
`βMi ˆ a sin
`
`where the local velocity at wing actuator i in local wing coordinates is
`obtained according to
`
`(3)
`
`‰ uMi
`
`vMi wMi ŠT ˆ TBMi…Vc:g:∕I ‡ ωB∕I × rc:g:→c:p:i†
`
`(9)
`
`Likewise, projectile body moments include contributions from roll
`damping, static pitching, pitch damping, and Magnus moments and
`can be described by the following expansion:
`
`Given the stowed roll angle and angle-of-attack relationships, the
`aerodynamic coefficients for each wing are calculated using aero-
`dynamic expansions dependent on the local Mach number and the
`stowed roll angle (which describes the wing area exposed to the
`freestream flow). These coefficients are given by
`
`…M; ^ϕ
`
`i†β
`
`Mi ‡ CM
`
`Xα2
`
`M
`
`M
`
`
`
`…M; ^ϕi†β2
`
`Mi
`
`Xα
`
`Mi
`
`Fig. 2 Body-fixed coordinate system.
`
`Finally, Eq. (11) provides the aerodynamic forces and moments of
`wing actuator i in projectile body coordinates:
`
`lα
`
`M
`
`mα
`
`Mi
`
`Mi ‡ CM
`
`mα3
`
`
`
`…M; ^ϕi†β3
`
`Mi
`
`M
`
`(10)
`
`M
`
`i† ‡ CM
`…M; ^ϕ
`X ˆ CM
`CMi
`‡ CM
`…M; ^ϕi†β3
`…M; ^ϕi† ‡ CM
`Z ˆ CM
`CMi
`l ˆ CM
`…M; ^ϕ
`i† ‡ CM
`CMi
`m ˆ CM
`…M; ^ϕ
`i† ‡ CM
`CMi
`
`Zα
`
`M
`
`…M; ^ϕi†β3
`
`Mi
`
`M
`
`Zα3
`
`…M; ^ϕi†βMi ‡ CM
`…M; ^ϕ
`i† α
`…M; ^ϕ
`i†β
`
`Xα3
`
`M
`
`X0
`
`Z0
`
`l0
`
`m0
`
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`
`184
`
`FRESCONI AND ROGERS
`
`Fig. 3 Earth- and body-fixed coordinate systems and euler angles.
`
`(11)
`
`3775
`
`3775
`
`CMi
`X
`CMi
`
`Z 0
`2664
`
`control system and ϕ
`C be the angle actually achieved by the wing
`actuator. To isolate controlled flight performance, we assume for the
`remainder of this study that the actuator responds at least 10 times
`faster than the projectile roll rate (∼60 Hz) so that actuator dynamics
`CMD ˆ ϕC. Experimental studies with
`may be neglected, and thus ϕ
`this actuator described in [20] have shown that an actuator bandwidth
`of approximately 600 Hz, or 10 times the projectile roll rate, is likely
`achievable using a rotary actuator design.
`
`C. Detector Modeling
`is
`To minimize cost, a simple strapdown feedback concept
`proposed. A strapdown detector provides the location of the target in
`the image plane and is modeled as a pinhole camera. The pinhole
`camera is used to project a three-dimensional scene onto a two-
`dimensional image plane, neglecting distortion effects from the lens
`or finite aperture size [11]. As mentioned previously, a pinhole
`camera model may be used to represent both seekers and standard
`camera devices [12]. Thus, details concerning whether detector data
`arrive from a laser-designated spot detector or through image-based
`navigation are irrelevant for this study and it is assumed that the target
`is identifiable in the detector plane. To formulate the pinhole model
`for the detector, start by defining the relative position between the
`projectile and target according to
`
`where Eq. (17) is written in body coordinates. Note that the presence
`of the roll angle in the transformation matrix in Eq. (17) yields a
`rapidly changing target location as seen by the strapdown detector for
`a spin-stabilized projectile.
`In practice, a passive detector can sense only the lateral relative
`positions, and the actual detector displacements are scaled based on
`the lens properties according to
`
`
`
`
`
`ϵ
`y
`ϵz
`
`ˆ
`
`fL
`
`
`
`PT…2†
`rB
`PT…3†
`rB
`
`
`
`
`
`
`
`‡
`
`η
`y
`ηz
`
`(18)
`
`2664
`
`ˆ −qMi ST
`
`−1
`BMi
`
`3775
`
`3775
`
`2664
`
`XMi
`
`YMi
`
`ZMi
`
`LMi
`
`MMi
`
`NMi
`
`2664
`
`CMi
`
`l 0
`
`ˆ −qMi SDT
`
`−1
`BMi
`
`CMi
`m
`
`
`rB
`
`PT ˆ TTBE…rI
`
`OT
`
`OP†
`− rI
`
`(17)
`
`Given this force and moment model, the equations of motion for the
`projectile are formulated using a Newton–Euler approach. The
`inertial, flat-Earth coordinate system (denoted by frame E) and the
`body-fixed coordinate system B are related by Euler roll ϕ, pitch θ,
`and yaw ψ angles, as shown in Fig. 3. The translational and rotational
`kinematic equations are given by
`
`(12)
`
`35
`
`u v w
`
`24
`35
`
`cθcψ
`cθsψ
`−sθ
`
`sϕsθcψ − cϕsψ
`sϕsθsψ − cϕcψ
`sϕcθ
`
`cϕsθcψ ‡ sϕsψ
`cϕsθcψ ‡ sϕsψ
`cϕcθ
`
`24
`
`ˆ
`
`35
`
`_x
`_y
`_z
`
`24
`
`fL
`PT…1†
`− rB
`
`where the left-hand side of Eq. (18) represents the raw displacements
`of the target in the image plane and ηy, ηz are zero-mean Gaussian
`white noise representing detector quantization error, distortion
`effects, and other sources of detector noise [12]. A diagram of the
`projective transformation for ϵ
`z is shown in Fig. 4, and a similar
`diagram can be constructed for ϵ
`y. Note, however, that the raw
`detector measurements in Eq. (18) are not sufficient to compute a
`desired maneuver plane. Roll estimation is necessary to resolve the
`target error components in the fixed-plane (i.e., nonrolling) co-
`ordinate system. Numerous authors have studied roll angle esti-
`mation for smart weapons based on magnetometer [21–23] or
`infrared sensor measurements [24,25]. For this study, it is assumed
`that some source of roll feedback is available, and thus the target error
`components can be transformed into the fixed-plane frame according
`to
`
`Lf
`
`z
`
`α
`
`Bx
`c.g. IV
`/
`
`PTr
`
`Target
`
`Bz
`
`Ex
`
`Ez
`Fig. 4 Definition of detector displacement ϵz. Image plane is shown in
`gray. (This diagram not to scale.)
`
`(13)
`
`35
`
`p q r
`
`24
`35
`
`sϕtθ
`1
`cϕ
`0
`0 sϕ∕cθ
`
`cϕtθ
`−sϕ
`cϕ∕cθ
`
`24
`
`ˆ
`
`35
`
`_ϕ
`_θ
`_ψ
`
`24
`
`where cα ≡ cos α, sα ≡ sin α, and tα ≡ tan α. Application of
`Newton–Euler kinetics to a projectile in free flight yields the
`following translational and rotational dynamic equations:
`
`24
`
`−
`
`35
`
`X ‡ XG
`Y ‡ YG
`Z ‡ ZG
`
`24
`
`ˆ 1
`m
`
`35
`
`35
`
`L M N
`
`24
`
`ˆ I
`
`−1
`
`35
`
`24
`
`_u
`_v
`_w
`
`_p
`_q
`_r
`
`24
`
`(14)
`
`(15)
`
`35
`
`u v w
`
`p q r
`
`35
`
`24
`35
`
`24
`
`35
`
`I
`
`−r
`q
`0
`0 −p
`r
`−q p
`0
`
`24
`
`− I
`
`−1
`
`−r
`q
`0
`0 −p
`r
`−q p
`0
`
`Note that aerodynamic forces and moments are computed as
`shown earlier, and the gravity force is included according to
`ŠT
`
`‰ XG YG ZG ŠT ˆ TT
`BE‰ 0 0g
`
`(16)
`
`The equations of motion given by Eqs. (12–15) are integrated
`forward in time using a fixed time step Runge–Kutta integration
`routine to obtain a single flight trajectory. Note that control action is
`input to this system through specification of the roll angle ϕC, which
`describes the maneuver plane in which the wing actuator is fully
`extended. Let ϕ
`CMD be the commanded roll angle output from the
`
`Downloaded by Michael Daniels on August 3, 2016 | http://arc.aiaa.org | DOI: 10.2514/1.G000815
`
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`
`

`
`FRESCONI AND ROGERS
`
`185
`
`Fig. 5 is the bias in the yaw angle of attack of about −1 deg, a result of
`the so-called yaw of repose. Yaw of repose is due to an interaction
`between the spin rate and overturning motion [27] and yields a cross-
`range drift of about 1 m over the 200 m range trajectory.
`For projectiles equipped with a maneuver mechanism, previous
`work has proven that flight instabilities occur for spin-stabilized
`projectiles maneuvering perpendicular to the gravity field when the
`control effectiveness is sufficiently high [13,14]. The potential for
`such instability exists in the current application. To demonstrate this,
`the aerodynamic force and moment coefficients for the wing were
`multiplied by a factor of 3 and a flight simulation was conducted with
`a constant maneuver command of ϕ
`to the right). Figure 6 demonstrates the resulting large-amplitude
`angular motion (counterclockwise when viewed from behind the
`projectile) which is a typical feature of this flight instability for spin-
`stabilized rounds. The implications of this instability are that,
`although it is difficult to maneuver small-diameter spin-stabilized
`projectiles, there is an upper bound on possible control authority,
`which is not particularly high, to maintain stability during maneuver.
`This leads to the conclusion that efficient control design and
`disturbance rejection are at least as important, if not more important,
`than generation of large control authority for accuracy improvement
`in these types of rounds.
`
`CMD ˆ 90 deg (full deflection
`
`III. Guidance and Flight Control Design
`Classical missile autopilots use an inner–outer loop guidance and
`control approach where proportional navigation (PN) guidance is
`based on line-of-sight and closing velocity estimates. In many
`implementations of these algorithms, accelerometers are used in an
`inner-loop control to track desired center of mass acceleration
`commands from PN guidance, and gyroscopes may be used to damp
`unwanted angular motion. Although PN-based inner–outer loop
`control may be suitable for some types of guided munition concepts,
`it is difficult to apply here for a variety of reasons. In the strapdown
`detector plane, the target rotates at the spin rate, and motion due to the
`trajectory arc and coning are superimposed. This action, along with a
`time of flight on the order of seconds, precludes high-accuracy
`estimation of line-of-sight rate from a low-cost detector. In addition,
`low-cost measurement devices such as accelerometers are severely
`stressed in the spin-stabilized environment. Manufacturing con-
`straints typically mean that these devices are subject to some error in
`misalignment and misposition, which can yield feedback that is
`largely unusable. To demonstrate this, Fig. 7 shows a component of
`lateral acceleration for a typical ballistic flight of the 40 mm grenade,
`both the truth value and as measured by a notional accelerometer. A
`random misalignment and misposition error were selected from a
`Gaussian distribution with standard deviations of 0.5 deg and 5 mm,
`respectively (small errors on the order of manufacturing tolerance).
`As shown in Fig. 7, high acceleration from the fast-spinning axis
`bleeds over into the lateral channels and largely overwhelms the true
`
`Fig. 5 Angular motion for ballistic flight of the 40 mm grenade concept.
`
`
`
`
`
`
`
`
`
`
`
`ˆ
`
`~ϵy
`~ϵ
`
`z
`
`cos ϕ − sin ϕ
`sin ϕ
`cos ϕ
`
`ϵy
`ϵ
`z
`
`(19)
`
`The target error components in the fixed-plane frame in Eq. (19)
`form the feedback signal used by the controller. Thus, a strapdown
`detector and roll angle sensor(s) are the only measurement devices
`assumed in this study. A detector update rate of 100 Hz, focal length
`
`of fL ˆ 0.1 m, and standard deviation of 0.001 m for ηy and ηz are
`
`used in Sec. IV of this paper for simulation studies.
`
`D. Example Flight Trajectories
`Several example simulations are provided here before control
`design to demonstrate key aspects of the flight behavior of small-
`diameter actively controlled projectiles. The example 40 mm system
`under investigation is launched with conditions necessary to fly to a
`ballistic range of 200 m, and no control inputs are given. Mass
`properties and aerodynamics for the example projectile obtained
`from solid modeling, semi-empirical aeroprediction, wind-tunnel
`tests, and computational fluid dynamics are provided in [18]. The
`angular motion over the entire ballistic flight (with no initial angular
`body rates) is shown in Fig. 5. A counterclockwise coning motion
`with amplitude of approximately 1 deg is visible due to Magnus
`moments on the projectile [26]. Magnus moments are an aero-
`dynamic effect resulting from the high projectile spin rate (which is
`necessary for gyroscopic stabilization) and often result in a limit cycle
`oscillation manifest as a coning motion. Aeromechanics theory
`dictates that this coning occurs at a natural frequency of about 2 Hz
`for this projectile [27]. Note that the Magnus moment is highly
`nonlinear with angle of attack, and thus maneuvering flight may
`significantly alter the response. Another important effect evident in
`
`Fig. 6 Instability of 40 mm grenade projectile with high control
`effectiveness.
`
`Fig. 7 Example true and measured lateral accelerometer signal for
`ballistic flight.
`
`Downloaded by Michael Daniels on August 3, 2016 | http://arc.aiaa.org | DOI: 10.2514/1.G000815
`
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`
`

`
`186
`
`FRESCONI AND ROGERS
`
`lateral acceleration. There is also significant high-frequency content
`in the signal at the spin rate (approximately 60 Hz) as shown in the
`oscillations for the zoomed-in portion of the measured acceleration.
`Clearly, use of this feedback in an inner-loop controller would not
`yield suitable performance. Similar arguments apply to gyroscope
`measurements in this environment.
`To address these issues with sensor measurements, this study
`proposes to use a detector and roll angle sensors as the sole feedback
`measurements in an integrated inner–outer loop control. This inte-
`grated control design is similar to that proposed for missile end-game
`guidance as described in [28–30]. The idea behind integrated
`guidance is that the controller is designed to compute control inputs
`directly from feedback measurements without any intermediate
`loops. This design allows for minimal feedback but requires some
`knowledge of the input–output dynamics of the system.
`Control design proceeds as follows. First, a second-order surrogate
`model is developed to describe the relationship between control
`inputs and lateral vehicle response. Once this model is identified, a
`proportional control gain is derived that provides suitable closed-
`loop response characteristics for the surrogate model. A model
`equivalency procedure is then performed between the surrogate
`model and the full nonlinear model, and a proportional controller is
`designed for the closed-loop nonlinear system. The gain for this
`controller is designed based on the surrogate model feedback gain
`and detector dynamics. This results in a time-varying control gain
`with a linear dependence on time of flight. Finally, an observer is
`introduced that filters body angular motion from detector feedback.
`
`A. Control System Overview
`The basic structure for the reduced-state control system proposed
`here is shown in Fig. 8. The nonlinear system dynamics formulated in
`the preceding flight models is contained in the function H. The
`z ŠT. This detector
`measurements are the raw detector data y ˆ ‰ ~ϵ
`~ϵ
`y
`data are filtered using the observer labeled L, which attempts to filter
`out components due to body angular rates and produces feedback
`signals y ˆ ‰ ~ϵ
`z ŠT. The error signal is formed by subtracting a
`~ϵ
`y
`reference from the feedback. The reference in the downrange
`direction is nonzero to account for trajectory arc. Likewise, the
`reference in the cross-range direction is nonzero to account for the
`difference between the center of the coning motion and the velocity
`vector caused by yaw of repose (as shown in Fig. 5). The control
`vector consists of the magnitude and direction of maneuver given by
`
`Fig. 8 Control system block diagram.
`
`Fig. 9 Representative downrange reference signal.
`
`
`
`q
`CMD ŠT
`u ˆ ‰ δ ϕ
`ˆ
`…
`− ~ry†2 ‡ …~ϵ
`− ~rz†2
`~ϵ
`
`Ke
`
`y
`
`
`
`T
`
`(20)
`
`−1 −…~ϵ z − ~rz†
`
`−…~ϵ
`y − ~ry†
`tan
`
`z
`
`where δ represents the desired normalized control magnitude and
`varies between zero and one. The proportional gain value Ke is a
`critical parameter in control system performance because it provides
`a direct relationship between error magnitude in the detector plane
`and control magnitude. It was noted during numerous simulation
`trials with this projectile that a constant gain value yielded either
`closed-loop instability or, if the gain was lowered, insufficient control
`response. It was determined that the nonlinear dynamics of the
`problem, and the limitations on practical feedback, prohibits the
`classic control design method of finding a constant linear relationship
`between control input and feedback output. Thus, a surrogate model
`approach was used to determine a time-varying gain that yields
`reasonable performance.
`Note that, if a zero reference value ~rz were used in Eq. (20), the
`projectile would unnecessarily spend valuable control effort trying to
`maneuver toward the ground due to the trajectory arc. To overcome
`this, a downrange reference signal representing the curvature of the
`trajectory can be estimated preflight using the approximate launch
`angle derived from a weapon fire control system. Because the muzzle
`velocity, mass properties, and aerodynamics are relatively fixed
`quantities, a downrange reference can be formed for a variety of
`launch angles and stored in a lookup table. These data may be indexed
`as a function of time of flight to effectively zero the target location on
`the detector in the downrange direction. An example of a reference
`signal for a launch angle of about 15 deg and a focal length of
`
`fL ˆ 0.1 m is presented in Fig. 9.
`
`Likewise, the cross-range reference signal is nonzero due to yaw of
`repose, which causes the projectile to cone about a vector that is at
`some angular offset to the velocity vector. Let this angular offset be
`given by β, and assume it is approximately constant throughout the
`direct fire trajectory. For instance, in Fig. 5, β is observed to be
`approximately 1 deg for the example projectile. Using the pinhole
`camera model in Eq. (18), the cross-range reference signal is given
`
`by ~ry ˆ fL tan β. For the example projectile, if ⠈ 1 deg and
`fL ˆ 0.1 m, ~ry is approximately 0.0017 m.
`
`B. Surrogate Model Development
`To develop a surrogate model for control response, nonlinear
`simulations were conducted wherein open-loop maneuvers were
`commanded at different magnitudes and directions (i.e., to cause the
`projectile to travel left, right, up, or down). The resulting response
`data due to these maneuvers were collected. More specifically, the
`relative position (21) and velocity (22) of the projectile with respect to
`the target in velocity vector coordinates was tabulated according to
`
`
`rV
`
`PT ˆ TTVE…rI
`
`OT
`
`OP†
`− rI
`
`PT ˆ _rI
`_rV
`
`PT
`
`− rI
`
`OP