Combining Load and Motor Encoders to Compensate ... - MDPI

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Combining Load and Motor Encoders to Compensate Nonlinear Disturbances for High Precision Tracking Control of Gear-Driven Gimbal Tao Tang 1,2, *, Sisi Chen 1,2,3 , Xuanlin Huang 1,2,3 , Tao Yang 1,2 and Bo Qi 1,2 1 2 3

*

Institute of Optics and Electronics, Chinese Academy of Science, Chengdu 610209, China; [email protected] (S.C.); [email protected] (X.H.); [email protected] (T.Y.); [email protected] (B.Q.) Key Laboratory of Optical Engineering, Chinese Academy of Sciences, Chengdu 610209, China Department of Mechanical engineering, University of Chinese Academy of Sciences, Beijing 100039, China Correspondence: [email protected]; Tel.: +86-28-8510-0191

Received: 27 January 2018; Accepted: 26 February 2018; Published: 2 March 2018

Abstract: High-performance position control can be improved by the compensation of disturbances for a gear-driven control system. This paper presents a mode-free disturbance observer (DOB) based on sensor-fusion to reduce some errors related disturbances for a gear-driven gimbal. This DOB uses the rate deviation to detect disturbances for implementation of a high-gain compensator. In comparison with the angular position signal the rate deviation between load and motor can exhibits the disturbances exiting in the gear-driven gimbal quickly. Due to high bandwidth of the motor rate closed loop, the inverse model of the plant is not necessary to implement DOB. Besides, this DOB requires neither complex modeling of plant nor the use of additive sensors. Without rate sensors providing angular rate, the rate deviation is easily detected by encoders mounted on the side of motor and load, respectively. Extensive experiments are provided to demonstrate the benefits of the proposed algorithm. Keywords: sensor-fusion; nonlinear disturbances; rate deviation; DOB; gear-driven

1. Introduction Due to the practical issues, such as size and power, which are associated with the increase of the motor capacity, more and more designers tend to seek for the designs with gear attached motors [1,2]. Thus, low power DC motor coupled with gear-box to obtain high torque is well adopted in modern servo systems. However, it is inevitable that the gear-driven control also introduces some new problems to the systems, such as backlash [3–5], flexibility [6], hysteresis [7], and nonlinear friction [8]. The friction and backlash are the most common non-smooth nonlinearities that may degrade the control performance [9,10]. Backlash occurs due to the gap existing between the inner motor axis and the outer load axis with gear box used to connect. As a result, the motion transmission is affected by the spacing of the gear. Especially, when reciprocating movement happens in the gear-driven control system, the position precision could deteriorates, and even the closed-loop system may fall into limit cycle [10], which could lead to the closed-loop control system unstable. In fact, reciprocating movement always occurs in a tracking control system of gimbal, because target trajectory can move forth and back or down and up. An approach based on dual motors connected in parallel to the load to eliminate backlash without by means of software and feedback control [11]. But, these mechanical techniques require changing or adding some parts on the mechanical systems, leading to high production cost. Therefore, advanced control techniques are necessary to compensate these nonlinear disturbances. Methods of Model-based [12–14] control to eliminate these disturbances are developed by some authors, but these techniques are deeply dependent on an accurate model. Sensor-less backlash

Sensors 2018, 18, 754; doi:10.3390/s18030754

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these disturbances are developed by some authors, but these techniques are deeply dependent on an accurate model. Sensor-less backlash compensation is very interesting [15]. However, at the compensation is very However, atwill the motion reversal, theeffects gap due non-contaction motion reversal, the interesting gap due to[15]. non-contaction cause undesirable ontoposition control. will cause undesirable effects on position control. Adaptive control [16] based on an estimation the Adaptive control [16] based on an estimation of the disturbing torque is developed, but a ofgood disturbing torque iscontrol developed, but a good adaptive control will lead to be a good choice of adaptive parameters will choice lead toof a good result. The parameters DOB techniques could very result. Theto DOB techniques could be very withon disturbances, the DOB is based attractive cope with disturbances, butattractive the DOBtoiscope based the inverse but model of the plant, on the inverse of to theestimate plant, combined Q-filter to[17,18]. estimate the torque disturbance [17,18]. combined withmodel Q-filter the torquewith disturbance We a mode-free DOB DOB based on sensor-fusion control to improve closed-loop Wepresent present a mode-free based on sensor-fusion control to improve performance closed-loop of position control for a gear-driven gimbal. This proposed method uses the rate deviation detect performance of position control for a gear-driven gimbal. This proposed method uses to the rate disturbances implementing this for DOB, which can bethis plugged the originally feedbackinto control deviation to for detect disturbances implementing DOB,into which can be plugged the mode. Thefeedback rate deviation loadrate anddeviation motor can exhibitsload the and disturbances exiting in the originally controlbetween mode. The between motor can exhibits the gear-driven compared with position signal. compared Without rate sensors providing disturbancesgimbal exitingquickly in thewhen gear-driven gimbal quickly when with position signal. angular the gimbal, the rate deviation detected encoders mounting ondetected the side by of Withoutrate rateofsensors providing angular rate isofeasily the gimbal, thebyrate deviation is easily motor andmounting load respectively. Dueoftomotor high bandwidth of the motorDue ratetoclosed loop, the inverse encoders on the side and load respectively. high bandwidth of themodel motor of theclosed plant loop, is not the necessary. this does notnecessary. require either complex a plant or rate inverseBesides, model of theDOB plant is not Besides, this modeling DOB doesofnot require the usecomplex of additive sensors.ofThis paper an example of two-axis to testify either modeling a plant orgives the use of additive sensors. gear-driven This paper gimbal gives an[19] example of our proposed control method. The remainder is organized as follow: Section 2 presents a detailed two-axis gear-driven gimbal [19] to testify our proposed control method. The remainder is introduction controlSection model 2ofpresents gear-driven gimbal,introduction mainly describing control model. Section 3 organized astofollow: a detailed to control model of gear-driven discusses and analyzes system robustness. 4 focuses on theand implementation of therobustness. proposed gimbal, mainly describing control model.Section Section 3 discusses analyzes system DOB. Section 5 setson up the simulations and experiments to testify DOB. the theorems remarks Section 4 focuses implementation of the proposed Sectionabove. 5 sets Concluding up simulations and are presentedto intestify Section 6. theorems above. Concluding remarks are presented in Section 6. experiments the

2. 2. Control Control Model Model of of Gear-Driven Gear-Driven Gimbal Gimbal A A two-axis two-axis of of gear-drive gear-drive system system is is depicted, depicted, as as seen seen in in Figure Figure 1. 1. It It is is made made of of DC DC motor, motor, gear gear box reducer, load, and control unit. In a modern control system, the angular position and velocity box reducer, load, and control unit. In a modern control system, the angular position and velocity can encoder. Due to existing transmission flexibility between the load motor, can be beprovided providedbyby encoder. Due to existing transmission flexibility between theand loadtheand the encoders are mounted in the loadinside the motor provide theto rate deviation, motor, encoders are mounted theand loadinside and inside, the respectively, motor side, to respectively, provide the which can be used to detect disturbances. such as backlash, friction, and unmodelled dynamics. Due to rate deviation, which can be used to detect disturbances. such as backlash, friction, and unmodelled the symmetry, the elevation axis is considered as an example to verify the proposed control method in dynamics. Due to the symmetry, the elevation axis is considered as an example to verify the this paper.control method in this paper. proposed

Load encoder

Motor encoder DC motor

Control unit

Gear box

Load

Gear box

DC motor Motor encoder

Figure Figure 1. 1. The The layout layout of of gear-driven gear-driven system. system.

The block structure cancan be be depicted in Figure 2. The The block diagram diagram ofofgear-driven gear-drivencontrol controlsystem system structure depicted in Figure 2. classical control mode includes the motor rate loop and the load position loop. The classical control mode includes the motor rate loop and the load position loop.

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S PM2 1 

3 of 10 3 of 10

1

PP 1  2 2 3 2  Cm P1 P2 N  P3 P4 N

.

(5)

TL

P2P3 0 N 2  P3 P4 N 2

When considering

4 because of N  1 , we P have

L S PM2 1

1 s

L

1 1  Cm P1 P2

. Cm P1P  1 is

required if the change of motor feature can be neglected. 1 In fact, Cm P1 P is considered as the rate

P

open-loop transfer function of motor and its magnitude N response3 is located above the 0 dB below the cutoff frequency. Based on the above analysis, we have L / Tm  0T, and m

r

Cp

C

1

P

P

m 1 P4  P1P2 P4Cm 2 L N  TL  1  P3 P4 Motor Cmrate P1loop P2 P3 P4  Cm P1P2

(6)

m

Considering Cm P1 P has high gain in the low frequencies and approaches to zero in the high frequencies. The (6) is rewritten into

position loop

ofL gear-drive P4 gimbal. Figure 2. 2. The diagram C p (s)C: position controller; Cm (s): Figure The classical classicalcontrol controlblock block diagram gimbal. : position controller; of gear-drive p (s ) (7) speed loop controller; ω L (s): load angular velocity; : motor angular velocity; θ L (s): load angular T 1ωm (Ps)P 4 m (s) : motor angular velocity; Cm (s) : speed loop controller; L (s) : loadL angular 3velocity; position; θr (s): reference angular position; P1 (s): torque constant of motor; P2 (s): mechanical transfer angular position; : reference angular in position; : torque constant of rate function (swe ) : load As can see from (7), L ratio; , istransfer reasonable lowP frequency. Thus, the motor L /TrL(s of motor; N: reduction P)31(s/)2 : the functionthe of gear; P14((ss)): load transfer function; N : forces motor; mechanical transfer reduction ratio; : the transfer P3side. ( s ) system, P2 ( s) : finitely TLcontributes (s): disturbance forces the load function side; Tm (of s)disturbance. : motor; disturbance in the motor loop toinrejecting the load In gear-drive control one of the function problems of gear; P : load transfer function;disturbances in box the in load important is4 to the nonlinear induced forces by gear theside; load side. TL (s) : disturbance ( s)suppress 3.When Robustness Analysis of the Rate Loop adding the rate closed loop in the load side, the load rate is expressed below, : disturbance forces in the motor side. Tm (s)

With only the rate closed loop in  T in Figure 3,the CLmotor M 1 side shown T transfer function ω L (s) can be 3.represented RobustnessasAnalysis LRate  Loop u  L m Tm  L L TL (8) follows: of the

1  CL M 1

1  CL M 1

1  CL M 1

With only the rate closed0 loop in motor side Cmshown P1 P2 P3 P4in N Figure 3, the transfer function L (s ) can ω = u L 2 2 N + P3 P4 N + Pimprove N 2 +Cm P1 P2 N 2performance for the input u In the same way, the load rate loop can 2 P3 +Cm P1 Pthe 2 P3 P4closed-loop be represented as follows: P P P N 3 4the motor side, but it has a limitation of rejecting and further reduces the effect disturbances2 in + Nof (1) 2 + P P N 2 + P P +C P P P P N 2 +C P P N 2 Tm m 1 2 3 4 2 C 3 2 3P 4N Pm1 P12 P mgain 3 4C M the load disturbances because the open-loop of must be restrained by the mechanical 2 2  L  2 u L 1 P N + P P P4 N Cm + P2 P3 P4 2 2 T N +NP N4 N242+PP22PP313+2 P24PN3 2P+4 N 2+ P43 P CmC P1mPP Cm P1P2C Nm2 P1LP2 N 3P resonances of P4 . 2 1P3 P



P2 P3 P4 N Tm N  P3 P4 N  P2 P3  Cm P1 P2 P3 P4 N 2  Cm P1 P2 N 2



P4 N 2  P1 P2 P4 N 2Cm  PT2LP3 P4 L T L N 2  P3 P4 N 2  P2 P3  Cm P1 P2 P3 P4 N 2  CPm4P1 P2 N 2

2

2

(1)

The sensitive expression will explain that whether the performance of the control system will change or not, when one of system’s characteristics is1 changed. P3 The sensitive expression defined by N Horowitz is shown below:

u

 ( s) /  (s) T( s)   ( s) k ( s)  m 1 Cm k ( sP)1 / k ( s) k ( s) Pk2 ( s)  ( s)

Sk 

(2)

N

Here, it assumes that the characteristic of motor is changed. Namely, P2 is replaced by P2 . We Motor rate loop easily get the below equations m

Cm P1P2 P3 Pthe  L  3. The control block diagram 4N rate loop. M1  Figure Figure diagramwith with themotor motor  3. The 2 control block 2 2 rate loop. 2 ,  u  N  P3 P4 N  P2 P3  Cm P1P2 P3 P4 N  Cm P1P2 N

(3)

Cm P1P2P3 P4 N    M1   L   2 2 2 2 .  u  N  P3 P4 N  P2P3  Cm P1P2P3 P4 N  Cm P1P2N

(4)

Substituting (3) and (4) into (2), yielding

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The sensitive expression will explain that whether the performance of the control system will change or not, when one of system’s characteristics is changed. The sensitive expression defined by Horowitz is shown below: ∆ϕ(s)/ϕ(s) ϕ0 (s) − ϕ(s) k (s) = 0 ∆k(s)/k(s) k (s) − k(s) ϕ(s)

ϕ

Sk =

(2)

Here, it assumes that the characteristic of motor is changed. Namely, P2 is replaced by P0 2 . We easily get the below equations  M1 =

M

0

 1

=

ω0 L u

ω0 L u



=



=

N2

+ P3 P4

N2

Cm P1 P2 P3 P4 N , + P2 P3 + Cm P1 P2 P3 P4 N 2 + Cm P1 P2 N 2

Cm P1 P0 2 P3 P4 N . N 2 + P3 P4 N 2 + P0 2 P3 + Cm P1 P0 2 P3 P4 N 2 + Cm P1 P0 2 N 2

(3)

(4)

Substituting (3) and (4) into (2), yielding 1

M

S P0 1 = 2

1+

P0 2 P3 N 2 + P3 P4 N 2

+ Cm P1 P0 2

.

(5)

P0 2 P3 N 2 + P3 P4 N 2

≈ 0 because of N >> 1, we have SPM0 1 = 1+Cm1P P0 . |Cm P1 P0 | ≥ 1 2 1 2 is required if the change of motor feature can be neglected. In fact, Cm P1 P0 is considered as the rate open-loop transfer function of motor and its magnitude response is located above the 0 dB below the cutoff frequency. Based on the above analysis, we have ω L /Tm ≈ 0, and When considering

ω0 L P4 + P1 P2 P4 Cm ≈ TL 1 + P3 P4 + Cm P1 P2 P3 P4 + Cm P1 P2

(6)

Considering Cm P1 P has high gain in the low frequencies and approaches to zero in the high frequencies. The (6) is rewritten into ω0 L P4 (7) ≈ TL 1 + P3 P4 As we can see from (7), ω 0 L /TL ≈ 1/2, is reasonable in the low frequency. Thus, the motor rate loop contributes finitely to rejecting the load disturbance. In gear-drive control system, one of the important problems is to suppress the nonlinear disturbances induced by gear box in the load side. When adding the rate closed loop in the load side, the load rate is expressed below, ωL =

CL M1 ω 0 L /Tm ω 0 L /TL u+ Tm + TL 1 + CL M1 1 + CL M1 1 + CL M1

(8)

In the same way, the load rate loop can improve the closed-loop performance for the input u and further reduces the effect of disturbances in the motor side, but it has a limitation of rejecting the load disturbances because the open-loop gain of CL M1 must be restrained by the mechanical resonances of P4 . 4. Disturbance Observer of the Gear-Drive Gimbal Note that the disturbances in the load side cannot be well mitigated by the motor rate loop with or without the load rate loop. In this section we introduce a DOB based on the rate deviation to reject disturbances, which is depicted in Figure 4.

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4. 4. Disturbance Disturbance Observer Observer of of the the Gear-Drive Gear-Drive Gimbal Gimbal Note Note that that the the disturbances disturbances in in the the load load side side cannot cannot be be well well mitigated mitigated by by the the motor motor rate rate loop loop with or without the load rate loop. In this section we introduce a DOB based on the rate deviation Sensors 2018, 18, 754 the load rate loop. In this section we introduce a DOB based on the rate deviation 5 of 10 with or without to to reject reject disturbances, disturbances, which which is is depicted depicted in in Figure Figure 4. 4.

TTL L

11 N N

uu

C Cmm

  mm

TTm m

P P11

 LL

P P44

P P33

R R

11 N N

P P22

Motor rate loop Motor rate loop

Q Q Figure 4. Disturbance observer observer (DOB) (DOB) of the gear-drive gimbal. Figure Figure 4. 4. Disturbance (DOB) of of the the gear-drive gear-drive gimbal. gimbal.

The rate deviation between the motor side and the load side is used to implement DOB, The rate rate deviation deviation between between the the motor motor side side and and the the load load side side is is used used to to implement implement the the The the DOB, DOB, resulting in disturbance rejection. However, it is very complicate to model the transfer function resultinginindisturbance disturbancerejection. rejection. However, is very complicate to model the transfer resulting However, it isitvery complicate to model the transfer functionfunction from u from such that DOB is to design meeting the In the motor  u to from such thatisthe the DOB to is difficult difficult to meeting design for for theInintention. intention. In fact, fact,rate the closed motor  LL the u to that to ω L such DOB difficult design for themeeting intention. fact, the motor rate closed loop makes the input nearly equal to the output. Therefore, a proposed control structure rate closed makesnearly the input to the output. aTherefore, proposed controlisstructure loop makesloop the input equalnearly to theequal output. Therefore, proposedacontrol structure given in is is given given5.in in Figure Figure 5. 5. Figure

TTL L

11 N N

uu

C Cmm

  mm

TTm m

P P11

 LL

P P44

P P33

P P22

R R

11 N N

Motor Motor rate rate loop loop

Q Q Figure 5. The proposed Disturbance observer. 5. The observer. Figure 5. proposed Disturbance observer.

The load rate with DOB plugging into the motor rate loop shown in Figure derived below The load load rate rate with with DOB DOB plugging plugging into into the the motor motor rate rate loop loop isis isshown shownin inFigure Figure55 5derived derivedbelow below The

M  T (1  Q)  L( M11 LL T T (11  Q Q)) T  u  0LL Tmm (1  Q) T 0 L/T (1 − m  LL   1 T  TLLQ) MQ ω1L/T ( 1 − Q ) ω m m L L 1  QM1R u  11  1  Q  QM1uR+ 1  Q Q  QM QM11R R Tm + Q Q  QM QM11R R ωL = TL 1 − Q + QM1 R

1 − Q + QM1 R

1 − Q + QM1 R

(9) (9) (9)

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How to design Q and R is the first step. The perfect condition of canceling disturbances is 1 − Q = 0 under the condition of the perfect control stability − Q + QM1 R = 0. Therefore, M1 plays an important role for the parameters design. The (3) is rewritten into below M1 =

1 Cm P1 P2 N 1 + Cm P1 P2 1 +

When considering N >> 1 and

Cm P1 P2 1+Cm P1 P2

P3 P4 Cm P1 P2 1+Cm P1 P2 P3 P4

+

P2 P3 N2

(10)

≈ 1, the (10) is simplified, as follows

M1 =

1 P3 P4 N 1 + P3 P4

(11)

For the DOB not affecting the closed-loop stability, the perfect condition is required R = M1−1 = N (1 +

1 ) P3 P4

(12)

Due to either P3 (s) or P4 (s) all being the second-order low-pass filter, P3 P4 ≈ 1 is reasonable in the low frequency. The Q-filter design is guided by below 1 − Q + QM1 R = 1

(13)

Substituting R = 2N into (13), we have Q

1 − P3 P4 =0 1 + P3 P4

(14)

It is impossible to arrive at − Q + QM1 R = 0. So, the bandwidth of Q needs to be less than that

of

1− P3 P4 1+ P3 P4 .

5. Experimental Setup The two-axis gear-driven gimbal is shown in Figure 6. Two optical encoders are installed in the two-axis gimbals to measure the angular position, and also can provide gimbal velocity through encoder difference. The control units include mainly Analog Digital Converter (A/D), Digital Analog Converter (D/A), computer (PC104), and Field Programmable Gate Array (FPGA). The motor is a high speed brush DC motor. The absolute-type rotary encoder is installed at the motor side. Its linear numbers are 1024 counts/round. The reduction ratio of the gear is 1227. The load encoder is 21-bit, and has a resolution of is 0.61800 . The digital signals of optical encoders are transmitted into the computer through serial port communication. The PC104 uses these sensor signals to implement controller for activating the power driving amplification to control the Gimbal. A Bode Plot based on Fourier Transform Theorem is a useful tool that shows the gain and phase response of a given control system for different frequencies, which describes the control system straightforward. The motor rate closed-loop response is shown by Figure 7. In fact, this response is measured from u. to w L , because the load encoder has enough precision to provide more precise velocity than that with the motor encoder. The closed-loop response can provide design criteria for the position controller in Figure 2 and the DOB controller in Figure 5. A Q31 -filter [20,21] is considered as the robust Q-filter shown below Q=

3as + 1 ( as + 1)3

(15)

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Motor(P2)& Load Motor gear box(P ) Load(P ) Figure 6. Experimental 3configuration of the gear-driven gimbal. 4 encoder encoder A Bode Plot based on Fourier Transform Theorem is a useful tool that shows the gain and phase response of a given control system for different frequencies, which describes the control system straightforward. The motor rate closed-loop response is shown by Figure 7. In fact, this response is measured from to  L , because the load encoder has enough precision to provide

u

more precise velocity than that with the motor encoder. The closed-loop response can provide design criteria for the position controller in Figure 2 and the DOB controller in Figure 5. A Q31 -filter [20,21] is considered as the robust Q-filter shown below

Q

3as  1 as  13

Control unit

.

(15)

Substituting the fitting transfer function in Figure 7 in to (13), an optimal Q31 -filter is chosen, resulting in the rate response with the DOB based on the motor rate loop in Figure 7. In comparison with the motor closed-loop response in Figure 7, the bandwidth shown in Figure Experimental configuration the gear-driven gimbal. of Q -filter. Figure 8 becomes narrow for6.6.assuring the control stability with a low bandwidth Figure Experimental configuration ofofthe gear-driven gimbal. 31

Magnitude(dB)

A Bode Plot based -20 on Fourier Transform Theorem is a useful tool that shows the gain and Measured phase response of a given control system for different frequencies, which describes the control Fitted -40 The motor rate closed-loop response is shown by Figure 7. In fact, this system straightforward. response is measured from to  L , because the load encoder has enough precision to provide

u

-60

more precise velocity than that with the motor encoder. The closed-loop response can provide design criteria for the position controller in Figure 2 and the DOB controller in Figure 5. -80 0 1 2 10 A Q31 -filter [20,21] 10 is considered as the robust10Q-filter shown below Frequency(Hz)

Phase(Degree)

200 100

Q

3as  1 as  13

.

(15)

0 Substituting the fitting transfer function in Figure 7 in to (13), an optimal Q31 -filter is chosen,

resulting in the rate response with the DOB based on the motor rate loop in Figure 7. -100 In comparison with the motor closed-loop response in Figure 7, the bandwidth shown in -200 0 1 2 Figure 8 becomes narrow with a low bandwidth of Q31 -filter. 10 for assuring the control stability 10 10 Frequency(Hz)

Magnitude(dB)

-20

Figure M11. . Figure 7. 7. Bode Bode response response of of M

-40

Measured Fitted

Phase(Degree)

Substituting the fitting transfer function in Figure 7 in to (13), an optimal Q31 -filter is chosen, -60 resulting in the rate response with the DOB based on the motor rate loop in Figure 7. In comparison with the motor closed-loop response in Figure 7, the bandwidth shown in Figure 8 -80 0 the control stability with1 a low bandwidth of Q 2 becomes narrow for assuring -filter. 3110 10 10 Frequency(Hz) For verifying the proposed method to be effective, experiments regarding the trajectory track are demonstrated. The200 resulting errors with the position loop closed are exhibited in Figure 9. When the reference angular position is θ r = 0.01◦ sin0.5t, the improved DOB can keep the steady 100 state error lower than 0.01 degree, when approximated to the encoder precision. Without the DOB, 0 friction and backlash lead to dead zone, which can be seen obviously disturbances regarding the in Figure 9 if the reference angular position is θ r = 5◦ sin0.5t. The DOB based on the sensor-fusion -100 makes the tracking error smaller than that without the DOB both in the low-velocity tracking and the high-velocity tracking. -200100 1 2 10 10 Frequency(Hz)

Figure 7. Bode response of M 1 .

Magnitude(d

-40 -50 -60

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1

10

-30 200

Phase(Degree) Magnitude(dB)

-40 100 -50 0

-60 -100 -70 -200 0 0 10 10

1

1 10 10

Frequency(Hz) 200

Phase(Degree)

Figure 8. Rate Bode response with DOB. 100

For verifying the proposed method to be effective, experiments regarding the trajectory track 0 are demonstrated. The resulting errors with the position loop closed are exhibited in Figure 9. When the reference angular position is θr = 0.01°sin0.5t, the improved DOB can keep the steady -100 state error lower than 0.01 degree, when approximated to the encoder precision. Without the DOB, -200 the friction and backlash lead to dead zone, which can be seen obviously in disturbances regarding 0 1 10 10 Figure 9 if the reference angular position is θr = 5°sin0.5t. The DOB based on the sensor-fusion Frequency(Hz) makes the tracking error smaller than that without the DOB both in the low-velocity tracking and the high-velocity tracking. Figure8.8.Rate RateBode Boderesponse responsewith withDOB. DOB. Figure error Tracking error the trajectory track For verifying theTracking proposed method to be effective, experiments regarding 0.05 w/o w/o 9. are demonstrated. The resulting errors with the position loop closed are exhibited in Figure w w 0.04 When the reference angular position is θr = 0.01°sin0.5t, the improved DOB can keep the steady 0.01 0.03 state error lower than 0.01 degree, when approximated to the encoder precision. Without the DOB, 0.02 disturbances regarding the friction and backlash lead to dead zone, which can be seen obviously in 0.005 Figure 9 if the reference angular position is θr = 5°sin0.5t. The DOB based on the sensor-fusion 0.01 makes 0 the tracking error smaller than that without the0 DOB both in the low-velocity tracking and the high-velocity tracking. -0.01 Madnitude(Degree)

-0.005

-0.02

Tracking error 0.015 w/o w

-0.01 0.01

Magnitude(Degree)

-0.015 0.005

5

10

15 20 25 Sample Time(Second)

30

35

40

Tracking error

0.05 -0.03

w/o w

0.04 -0.04 0.03 -0.05 0 0.02 Madnitude(Degree)

Magnitude(Degree)

0.015

5

10

15 20 25 Sample Time(Second)

30

35

40

0.01

Figure 9. Tracking different reference angular positions. The left reference angular angular position Figure Trackingerror errorunder under different reference angular positions. The left reference 0 0 ◦ is the sinusoidal trajectorytrajectory of θ r = 0.01 and the right theright sinusoidal trajectory oftrajectory θ r = 5◦ sin0.5t. position is the sinusoidal of sin0.5t, θr = 0.01°sin0.5t, andisthe is the sinusoidal of θr -0.01 = 5°sin0.5t.

-0.005

-0.02 6. Conclusions 6. Conclusions -0.03 A mode-free DOB based on sensor fusion for a gear-driven gimbal is proposed to reduce some -0.01 -0.04 mode-free DOB based sensor for a into gear-driven gimbal is proposed to reduce errorArelated disturbances. Theon DOB can fusion be plugged the originally control mode, which can some bring error related The system. DOB can be plugged into originally control mode, which can -0.015 -0.05 the a high gain5 todisturbances. the closed-loop Complex modeling of disturbances and the use of additive 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Sample Time(Second) Sample Time(Second) bring a high gain to the closed-loop system. Complex modeling of disturbances and the use of sensors are not needed. In this paper, we focus on the implementation of the DOB, the optimization of the control parameters, and the analysis of the close-loop stability. Experimental results show the Figure 9. Tracking error under different reference angular positions. The left reference angular control method of the DOB has great performance to eliminate the nonlinear disturbances and reduce position is the sinusoidal trajectory of θr = 0.01°sin0.5t, and the right is the sinusoidal trajectory of θr the turning error. In comparison with the originally two closed loops, the tracking error with the = 5°sin0.5t. proposed control mode has a reduction by a factor of three. In the future work, the application of AI methods (evolutionary algorithms and neural networks) to develop the model for control application 6. Conclusions is promising to further improve closed-loop system [22–24]. A mode-free DOB based on sensor fusion for a gear-driven gimbal is proposed to reduce some error related disturbances. The DOB can be plugged into the originally control mode, which can bring a high gain to the closed-loop system. Complex modeling of disturbances and the use of

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Acknowledgments: We would like to gratefully acknowledge M.L. Zhang for technical support. Author Contributions: Tao Tang is the head of the research group that conducted this study. He contributed to the research through his general guidance and advice. Sisi Chen and Xuanlin Huang analyzed the DOB and set up experiments to test the result. Tao Yang gave some advices and help on experimental setup. Bo Qi designed the gear-box gimbal. All authors contributed to the writing and revision of the manuscript. Conflicts of Interest: The authors declare no conflict of interest.

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