Small tracking error correction for moving targets of intelligent electro-optical detection systems

Abstract

Small tracking error correction for electro-optical systems is essential to improve the tracking precision of future mechanical and defense technology. Aerial threats, such as “low, slow, and small (LSS)” moving targets, pose increasing challenges to society. The core goal of this work is to address the issues, such as small tracking error correction and aiming control, of electro-optical detection systems by using mechatronics drive modeling, composite velocity–image stability control, and improved interpolation filter design. A tracking controller delay prediction method for moving targets is proposed based on the Euler transformation model of a two-axis, two-gimbal cantilever beam coaxial configuration. Small tracking error formation is analyzed in detail to reveal the scientific mechanism of composite control between the tracking controller’s feedback and the motor’s velocity–stability loop. An improved segmental interpolation filtering algorithm is established by combining line of sight (LOS) position correction and multivariable typical tracking fault diagnosis. Then, a platform with 2 degrees of freedom is used to test the system. An LSS moving target shooting object with a tracking distance of S = 100 m, target board area of A = 1 m², and target linear velocity of v = 5 m/s is simulated. Results show that the optimal method’s distribution probability of the tracking error in a circle with a radius of 1 mrad is 66.7%, and that of the traditional method is 41.6%. Compared with the LOS shooting accuracy of the traditional method, the LOS shooting accuracy of the optimized method is improved by 37.6%.

Key words： electro-optical detection system; small tracking error; moving target; visual servo; aiming control

Nomenclature

Abbreviations
BLDC	Brushless direct-current motor
DOF	Degree of freedom
EODS	Electro-optical detection system
FOV	Field of view
LOS	Line of sight
LSS	Low, slow, and small
MD	Miss distance
PID	Proportional–integral–differential
Variables
A	Target board area
B	Viscous friction coefficient of the motor
E	Counter-electromotive force
Ea	Material elastic modulus
e	Counter-electromotive force
ea, eb, ec	Counter-electromotive force of each phase winding
F	Constant force
f	Sinusoidal response signal frequency
G (t)	Intermediate function
G_{\mathrm{q}}^{\mathrm{s}}\left(s\right)	Current-loop delay
G_{\mathrm{f}}^{\mathrm{e}}\left(s\right)	Delay compensator
G_{\mathrm{k}}^{\mathrm{a}}\left(s\right)	Position-loop delay
GLag (s)	Improved Lagrange interpolation compensator
Go (z)	Compensate function
Goc (s), Gcc (s)	Transfer functions of the open and closed loops
Gop (s)	Open-loop transfer function of the position-loop
G_{\text{op}}^{\mathrm{e}}\left(s\right)	Open-loop transfer function of the current-loop
G_{\text{PI}}^{\mathrm{a}}\left(s\right)	Position-loop controller
Gov (s)	Open-loop transfer function of the velocity-loop
G_{\text{PI}}^{\mathrm{b}}\left(s\right)	Velocity-loop controller
G_{\text{PI}}^{\mathrm{e}}\left(s\right)	Current-loop controller
Gpr (s)	Closed-loop transfer function of the position-loop
Gs (s)	Transfer function
G_{\mathrm{s}}^{\mathrm{b}}\left(s\right)	Velocity-loop delay
G_{\mathrm{s}}^{\mathrm{e}}\left(s\right)	Current-loop controlled object
GTrack	Image tracker
GT (s)	Electromagnetic torque
G_{\text{ZOH}}^{\mathrm{a}}\left(s\right)	Zero-order hold
Gθ (s)	Controller of the position loop
Gω (s), GFF (s), GI (s)	Controllers of the velocity loop
I	Material cross-sectional moment of inertia
I (s)	Current function
ia, ib, ic	Current of each phase winding
id, iq	Current of the d–q axis
if	Current
J	Moment of inertia
K	Free quantity
Ke	Coefficient of counter-electromotive force
Kk	Target manipulator deceleration ratio
Ko, Kδ, Kw	Conversion coefficient
KN	Coefficient of pulse conversion
KV, KI, Kω, Kθ	Coefficients of visual servo
kP, kI, kD	Coefficients of the velocity-loop controller
kp, ki, kd	Coefficients of the current-loop controller
k_{\mathrm{p}}^{\mathrm{a}},k_{\mathrm{i}}^{\mathrm{a}},k_{\mathrm{q}}^{\mathrm{a}}	Coefficients of the position-loop controller
L	Stator inductance
L (x)	Interpolation function
Ld, Lq	Inductance of the d–q axis
Ln (x)	Lagrange interpolation polynomial
Lo	Target manipulator rod length
l	Length of the beam
li	Length of the ith structure
Lk (x)	Interpolation basis function
M	Coil mutual inductance of each phase winding
M (x)	The function of bending moment
Ma	Bending moment
M × N	Resolution
{mn, mn+1, ..., mn+4}	Dataset of miss distance
N	Interpolation step size
n	Stepper motor speed
n0	Speed output of the decelerator
OA	Line of sight line
OB	Fire line
Oa, Ob	Reference axis of the calibration tool
P	The stress on the beam
Pn	Number of motor poles
q	Uniformly distributed load
R	Stator resistance
s	Differential module
S	Tracking distance
ds	Differential arc of a cantilever beam
T	Sampling time
T1	Unit step signal duration
T2	Sinusoidal response signal duration
Td, Tf, Tk, Tb	Constant of delay time
Te, TL	Electromagnetic torque and load torque
Ts	Sampling period of the encoder
t	Unit step signal starting time
U (s)	Voltage function
u	Voltage
ua, ub, uc	Voltage of each phase winding
ud, uq	Voltage of the d–q axis
v	Linear velocity
w	Deflection curve
wmax	Maximum deflection
X (t)	Sampling value
x	Displacement
x0, x1, ..., xn	Independent variables
Y	Feedback coordinate
y0, y1, ..., yn	Function values
ya1	Section deflection
yα1, yα2, yα3, yα4	Deflection distance
yβ1, yβ2, yβ3, yβ4	Total deflection distance
z	A variable after the difference transformation
α	Adjustment coefficient
β1, β2	Orthogonal decomposition distances
ψ	Total flux of each phase winding
ψa, ψb, ψc	Flux linkage of each phase winding
ψf	Magnetic linkage of the permanent magnet
ψm	Rotor permanent magnet flux
{\psi }_{\mathrm{s}}{\mathrm{e}}^{\mathrm{j}{\theta }_{\mathrm{s}}}	Flux components of each phase winding
θ	Offset angle
θ*	Output control instruction
θα1(l − x)	Offset angle of section A
θe	Relative angle
θmax	Maximum offset angle
ρ	Curvature radius
δ1, δ3	Calibration deviation
δ2	Vertical distance
δmax	Calibration error
|δ|, δx, δy	Shooting accuracy judgment threshold
|δ|	Tracking error
ε	Stability coefficient of the closed loop
τI	Integral time constant of the velocity loop
τi	Integral time constant of the current loop
ω	Target manipulator angular velocity
ωe	Electrical angular velocity
ωr	Angular velocity
(x, y)	Target pixel coordinate
(Δx, Δy)	Pixel distance
(θM, θN)	Angle of the lens
(θx, θy)	Miss distance
(α, β)	Output pulse of the motor
\left(w/2,h/2\right)	Center of the lens

Acknowledgements

The present work was funded by the National Natural Science Foundation of China (Grant No. U19A2072), the Provincial Department of Education Postgraduate Scientific Research Innovation Project of Hunan Province of China (Grant No. QL20210007), and the Ministerial Level Postgraduate Funding Project of China (Grant No. JY2021A007).

Conflict of Interest

The authors declare that they have no conflict of interest.

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