High-precision gyro-stabilized control of a gear-driven platform with a floating gear tension device

doi:10.1007/s11465-021-0635-5

Front. Mech. Eng.

2021, Vol. 16

Issue (3) : 487-503 https://doi.org/10.1007/s11465-021-0635-5

RESEARCH ARTICLE

High-precision gyro-stabilized control of a gear-driven platform with a floating gear tension device

Xianliang JIANG, Dapeng FAN(

), Shixun FAN, Xin XIE, Ning CHEN

College of Intelligence Science and Technology, National University of Defense Technology, Changsha 410073, China

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Abstract

This study presents an improved compound control algorithm that substantially enhances the anti-disturbance performance of a gear-drive gyro-stabilized platform with a floating gear tension device. The tension device can provide a self-adjustable preload to eliminate the gap in the meshing process. However, the weaker gear support stiffness and more complex meshing friction are also induced by the tension device, which deteriorates the control accuracy and the ability to keep the aim point of the optical sensors isolated from the platform motion. The modeling and compensation of the induced complex nonlinearities are technically challenging, especially when base motion exists. The aim of this research is to cope with the unmeasured disturbances as well as the uncertainties caused by the base lateral motion. First, the structural properties of the gear transmission and the friction-generating mechanism are analyzed, which classify the disturbances into two categories: Time-invariant and time-varying parts. Then, a proportional-integral controller is designed to eliminate the steady-state error caused by the time-invariant disturbance. A proportional multiple-integral-based state augmented Kalman filter is proposed to estimate and compensate for the time-varying disturbance that can be approximated as a polynomial function. The effectiveness of the proposed compound algorithm is demonstrated by comparative experiments on a gear-drive pointing system with a floating gear tension device, which shows a maximum 76% improvement in stabilization precision.

Keywords inertially stabilized platform floating gear tension device nonlinear friction disturbance compensation proportional multiple-integral observer

Corresponding Author(s): Dapeng FAN

Just Accepted Date: 30 April 2021 Online First Date: 27 May 2021 Issue Date: 24 September 2021

Cite this article:

Xianliang JIANG,Dapeng FAN,Shixun FAN, et al. High-precision gyro-stabilized control of a gear-driven platform with a floating gear tension device[J]. Front. Mech. Eng., 2021, 16(3): 487-503.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-021-0635-5
https://academic.hep.com.cn/fme/EN/Y2021/V16/I3/487

Fig.1 Two-axis gyro-stabilized platform with anti-backlash gear-drive transmission configuration.

Fig.2 Dynamic model of azimuth transmission chain with anti-backlash mechanism.

Fig.3 Equivalent schematic diagram of azimuth transmission chain with anti-backlash mechanism.

Fig.4 Dynamic system of azimuth transmission chain with anti-backlash mechanism with torque disturbances.

Fig.5 Schematic diagram of vehicle base coupling.

Fig.6 Meshing force analysis diagram of anti-backlash gear drive.

Fig.7 Block diagram of the proposed stabilization loop.

Fig.8 Experimental device for the test of the control performance of the gyro-stabilized platform.

Tab.1 Gear transmission parameters

Fig.9 Stability performance of the azimuth transmission chain with anti-backlash mechanism under (a) Condition 1, (b) Condition 2, (c) Condition 3, and (d) Condition 4.

Base motion	Residual velocity/ $((°) ? s − 1)$		Improvement/%
Base motion	Original	Proposed	Improvement/%
Condition 1	0.745	0.452	39.3
Condition 2	0.948	0.559	41.0
Condition 3	0.583	0.383	34.3
Condition 4	1.005	0.877	12.7

Tab.2 Comparison of residual velocity

Base motion	Residual angle/ $(°)$		Improvement/%
Base motion	Original	Proposed	Improvement/%
Condition 1	0.104	0.036	65.4
Condition 2	0.123	0.045	63.4
Condition 3	0.117	0.027	76.9
Condition 4	0.127	0.050	60.6

Tab.3 Comparison of residual angle

Fig.10 Bode plot of the control system. (a)

G open (s) = ω l (s) / ω err (s)

; (b)

G close (s) = ω l (s) / ω cmd (s)

$a i$	Constant but unknown coefficients
$A$ , $B$ , $B d$ , $C$	Coefficients of Eq. (34)
$A ˜$ , $B ˜$ , $C ˜$ , $K ˜ o$ , $K ˜ o 0$ , $K ˜ o 1$ , $K ˜ o q − 1$	Observer matrixes for the augmented state space
$A ˜ d$	Discrete state matrix of $A ˜$
$B ˜ d$	Discrete state matrix of $B ˜$
$C s$	Damping coefficients of the anti-backlash spring
$C m$	Damping coefficient of the gear mesh
$d i$	Estimates of the (q-i) th derivative of time-varying disturbance
$d^i$	Estimate of the $d i$
$D$	Damping matrix
$e ˜ x$	Estimation error. On behalf of $x ˜^− x ˜$
$e ˜ d i$	On behalf of $d^i − d q − i$
$e$	Transfer function of the tracking err error
$E ˜$	State vector of Eq. (43)
$F b$	Total equivalent force borne by the bearing
$F gf$	Gear force from the driving gear to the driven gear
$F n$	Normal force
$F p$	Pretightening force
$F p0$	Initial value of the pretightening force
$F r$	Radial force
$F t$	Tangential force
$G close$	The closed-loop transfer function
$G gyro$	Gyros model
$G notch$	Notch filter
$G open$	Open loop transfer function
$G VEL$	Stabilization loop controller
$I dg$ , $I dn$	Inertias of the driving and driven gear, respectively
$J m$ , $J l$	Equivalent inertias of the azimuth motor and the load, respectively
$k$	Number of iterations
$K eq$	Equivalent stiffness
$K g$	Gain of a coefficient conversion
$K i$	Integral gain of PI controller
$K I$	Voltage conversion factor of the driver
$K I i$	Integral factors of the observer
$K m$	Gear mesh stiffness coefficient
$K p$	Proportional gain of PI controller
$K s$	Stiffness coefficient of the anti-backlash spring
$K T$	Motor moment coefficient
$K ω$	Observer gain
$K$	Stiffness matrix
$K ˜ o 0 (k)$	Value of $K ˜ o 0$ for iteration $k$
$Q$	Torque input matrix
$m dg$ , $m dn$	Masses of the driving and driven gear, respectively
$M$	Mass matrix
$N$	Transmission ratio
$N b$ , $M i$ , $f i$	Number, amplitude, and frequency of frequency components
$O dg$ , $O dn$	enter of the rotation of the driving and driven gear, respectively
$O ′ dg$	Changed rotation center of the driving gear
$P$ , $G$	Meshing points in driving and driven gear
$P ˜ (k − 1 \| k − 1)$	Covariance of the estimation error vector
$P ˜ (k \| k − 1)$	Prediction of $P ˜ (k − 1 \| k − 1)$ to the next iteration
$R b$	Bearing gyration radius
$R dg$ , $R dn$	Circle radius of the driving and driven gear, respectively
$R d i$ $(i ∈ {1, 2, …, q})$	Variance of each order component of disturbance
$R ˜ M$	Measurement noise
$R ˜ P$	Process noise
$R u$	Variance of the control voltage
$R ω l$	Noise covariance of $ω l$
$t s$	Settling time of the stabilization loop
$T b$	Base coupling disturbance
$T d$	Total disturbance torque
$T dc$	Constant part of $T d$
$T dv$	Variable part of $T d$
$T f$	Equivalent friction torque including the meshing friction and the rotation friction
$T fb$	Frictions exist between and within the gimbal’s rotating bearings
$T fg$	Friction torque in gear transmission
$T l$	Input torque of the load
$T m$	Output torque of the azimuth motor
$T s$	Sampling time
$T tor$	Driving torque to the azimuth driving gear
$U c$	Compensation voltage
$U in$	Input voltage
$U pi$	Output vlotage of the PI controller
$W ˜$	Noise input matrix
$x$	State variable of Eq. (34)
$x^$	Estimate of $x$
$x ˜$	New vectors are constructed for an augmented state space
$x ˜^$	Estimate of the state $x ˜$
$X$	Generalized coordinate vector
$y dg$	Movement in the Y direction
$ω cmd$	Speed command
$ω err$	Residual gyro rate
$ω dg$ , $ω dn$	Equivalent relative rotation speed of the driving and driven gear, respectively
$ω m$ , $ω l$	Angular rate of the motor and the load, respectively
$ω n$	Undamped natural frequencies
$ω re$	Natural resonant frequency
$ω s$	Integral of the velocity residual
$θ b$	Angular disturbance
$θ dg$ , $θ dn$	Rotation angle of the driving and driven gear, respecively
$θ lb$ , $θ mb$	Rotation angle of the load and the motor with respect to the vehicle base, respecively
$θ m$ , $θ l$	Rotate angle of the motor and the load, respecively
$μ f$	Constant meshing friction coefficient
$μ b$	Viscous damping friction coefficient of the bearing
$α$	Operating pressure angle
$ξ$	Damping ratio of the stabilization loop
$ξ d$	Denominator damping factor
$ξ m$	Molecular damping factor
$λ 1, 2$	Poles of the desired stabilization loop
$δ u$	Resolution of DA
$Δ r$	Displacement increment of the rotation center of the driving gear

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[1]	Jianyong YAO. Model-based nonlinear control of hydraulic servo systems: Challenges, developments and perspectives[J]. Front. Mech. Eng., 2018, 13(2): 179-210.

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