Small tracking error correction for moving targets of intelligent electrooptical detection systems
Cheng SHEN, Zhijie WEN, Wenliang ZHU, Dapeng FAN, Mingyuan LING
Small tracking error correction for moving targets of intelligent electrooptical detection systems
Small tracking error correction for electrooptical 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 electrooptical 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 twoaxis, twogimbal 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 m^{2}, 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%.
electrooptical detection system / small tracking error / moving target / visual servo / aiming control
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Abbreviations  
BLDC  Brushless directcurrent motor 
DOF  Degree of freedom 
EODS  Electrooptical 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  Counterelectromotive force 
E_{a}  Material elastic modulus 
e  Counterelectromotive force 
e_{a}, e_{b}, e_{c}  Counterelectromotive force of each phase winding 
F  Constant force 
f  Sinusoidal response signal frequency 
G (t)  Intermediate function 
${G}_{\mathrm{q}}^{\mathrm{s}}\left(s\right)$  Currentloop delay 
${G}_{\mathrm{f}}^{\mathrm{e}}\left(s\right)$  Delay compensator 
${G}_{\mathrm{k}}^{\mathrm{a}}\left(s\right)$  Positionloop delay 
G_{Lag} (s)  Improved Lagrange interpolation compensator 
G_{o} (z)  Compensate function 
G_{oc} (s), G_{cc} (s)  Transfer functions of the open and closed loops 
G_{op} (s)  Openloop transfer function of the positionloop 
${G}_{\text{op}}^{\mathrm{e}}\left(s\right)$  Openloop transfer function of the currentloop 
${G}_{\text{PI}}^{\mathrm{a}}\left(s\right)$  Positionloop controller 
G_{ov} (s)  Openloop transfer function of the velocityloop 
${G}_{\text{PI}}^{\mathrm{b}}\left(s\right)$  Velocityloop controller 
${G}_{\text{PI}}^{\mathrm{e}}\left(s\right)$  Currentloop controller 
G_{pr} (s)  Closedloop transfer function of the positionloop 
G_{s} (s)  Transfer function 
${G}_{\mathrm{s}}^{\mathrm{b}}\left(s\right)$  Velocityloop delay 
${G}_{\mathrm{s}}^{\mathrm{e}}\left(s\right)$  Currentloop controlled object 
G_{Track}  Image tracker 
G_{T} (s)  Electromagnetic torque 
${G}_{\text{ZOH}}^{\mathrm{a}}\left(s\right)$  Zeroorder hold 
G_{θ} (s)  Controller of the position loop 
G_{ω} (s), G_{FF} (s), G_{I} (s)  Controllers of the velocity loop 
I  Material crosssectional moment of inertia 
I (s)  Current function 
i_{a}, i_{b}, i_{c}  Current of each phase winding 
i_{d}, i_{q}  Current of the d–q axis 
i_{f}  Current 
J  Moment of inertia 
K  Free quantity 
K_{e}  Coefficient of counterelectromotive force 
K_{k}  Target manipulator deceleration ratio 
K_{o}, K_{δ}, K_{w}  Conversion coefficient 
K_{N}  Coefficient of pulse conversion 
K_{V}, K_{I}, K_{ω}, K_{θ}  Coefficients of visual servo 
k_{P}, k_{I}, k_{D}  Coefficients of the velocityloop controller 
k_{p}, k_{i}, k_{d}  Coefficients of the currentloop controller 
${k}_{\mathrm{p}}^{\mathrm{a}},{k}_{\mathrm{i}}^{\mathrm{a}},{k}_{\mathrm{q}}^{\mathrm{a}}$  Coefficients of the positionloop controller 
L  Stator inductance 
L (x)  Interpolation function 
L_{d}, L_{q}  Inductance of the d–q axis 
L_{n} (x)  Lagrange interpolation polynomial 
L_{o}  Target manipulator rod length 
l  Length of the beam 
l_{i}  Length of the ith structure 
L_{k} (x)  Interpolation basis function 
M  Coil mutual inductance of each phase winding 
M (x)  The function of bending moment 
M_{a}  Bending moment 
M × N  Resolution 
{m_{n}, m_{n+1}, ..., m_{n+4}}  Dataset of miss distance 
N  Interpolation step size 
n  Stepper motor speed 
n_{0}  Speed output of the decelerator 
O_{A}  Line of sight line 
O_{B}  Fire line 
O_{a}, O_{b}  Reference axis of the calibration tool 
P  The stress on the beam 
P_{n}  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 
T_{1}  Unit step signal duration 
T_{2}  Sinusoidal response signal duration 
T_{d}, T_{f}, T_{k}, T_{b}  Constant of delay time 
T_{e}, T_{L}  Electromagnetic torque and load torque 
T_{s}  Sampling period of the encoder 
t  Unit step signal starting time 
U (s)  Voltage function 
u  Voltage 
u_{a}, u_{b}, u_{c}  Voltage of each phase winding 
u_{d}, u_{q}  Voltage of the d–q axis 
v  Linear velocity 
w  Deflection curve 
w_{max}  Maximum deflection 
X (t)  Sampling value 
x  Displacement 
x_{0}, x_{1}, ..., x_{n}  Independent variables 
Y  Feedback coordinate 
y_{0}, y_{1}, ..., y_{n}  Function values 
y_{a1}  Section deflection 
y_{α1}, y_{α2}, y_{α3}, y_{α4}  Deflection distance 
y_{β1}, y_{β2}, y_{β3}, y_{β4}  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 
ψ_{a}, ψ_{b}, ψ_{c}  Flux linkage of each phase winding 
ψ_{f}  Magnetic linkage of the permanent magnet 
ψ_{m}  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 
θ_{α1}(l − x)  Offset angle of section A 
θ_{e}  Relative angle 
θ_{max}  Maximum offset angle 
ρ  Curvature radius 
δ1, δ3  Calibration deviation 
δ2  Vertical distance 
δ_{max}  Calibration error 
δ, δ_{x}, δ_{y}  Shooting accuracy judgment threshold 
δ  Tracking error 
ε  Stability coefficient of the closed loop 
τ_{I}  Integral time constant of the velocity loop 
τ_{i}  Integral time constant of the current loop 
ω  Target manipulator angular velocity 
ω_{e}  Electrical angular velocity 
ω_{r}  Angular velocity 
(x, y)  Target pixel coordinate 
(Δx, Δy)  Pixel distance 
(θ_{M}, θ_{N})  Angle of the lens 
(θ_{x}, θ_{y})  Miss distance 
(α, β)  Output pulse of the motor 
$\left(w/2,h/2\right)$  Center of the lens 
/
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