An Effective Method to Improve the Accuracy of a Vernier-type Absolute Magnetic Encoder
Ton Hoang Nguyen, Ha Xuan Nguyen,Thuong Ngoc-Cong Tran, Jae Wan Park, Kien Minh Le, Vinh Quang Nguyen and Jae Wook Jeon,Senior Member, IEEE
Abstract—This paper proposes a method to improve the accuracy of a vernier absolute magnetic encoder. The encoder consists of a master and a nonius multipolar mag- netic track. Sinusoidal signals from the master and nonius tracks are used to infer the absolute information. Unfortu- nately, these signals are contaminated by non-ideal factors such as different amplitudes, dc-offsets, phase shifts and random noise. Moreover, harmonics existing in the encoder signals distort the vernier principle and significantly affect the accuracy of the encoder. To address these problems, the present paper proposes an efficient method with three main parts. The first is an observer phase-locked loop (OPLL), which is used to estimate the phase and eliminate the non-ideal factors. The second is non-linear phase com- pensation (NLPC), which is used to correct the vernier prin- ciple that deviated due to the existing harmonics. Finally, a pole pitch compensation (PPC) method is introduced to modulate the master phase angle from the OPLL to eliminate the harmonic distortion. The proposed method can eliminate the non-ideal factors, harmonic distortion and improve the accuracy of the encoder. All the proposed methods were implemented on an ARM STM32F407ZG. The experimental results confirm the validity of the proposed method for practical applications.
Index Terms—Vernier absolute magnetic encoders, auto- calibration, gradient descent, harmonic, observed phase- locked loop, interpolation search, pole pitch compensation.
I. INTRODUCTION
A
N absolute encoder is a device for measuring the rotation angle of a rotational motor in industrial applications.The two typical types of absolute encoders are optical and magnetic. Among these, the absolute magnetic encoder is widely used in applications because of its benefits, which include low cost and superior performance in industrial en- vironments. It can achieve the same resolution and accuracy as an optical encoder. In recent years, extensive studies have
This work was supported in part by the Ministry of Science and ICT (MSIP), South Korea, through the G-ITRC Support Program supervised by the Institute for Information and Communications Technology Promo- tion (IITP) under Grant IITP-2018- 20150-00742.
T. H. Nguyen, H. X. Nguyen, T. N.-C. Tran, J. W. Park, V. Q.
Nguyen and J. W. Jeon are with the Department of Electrical and Computer Engineering, Sungkyunkwan University, Suwon 440-746, South Korea (e-mail: [email protected]; [email protected];
[email protected]; [email protected]; [email protected]; jw- [email protected]).
K. M. Le is with the Faculty of Control Engineering, Le Quy Don Tech- nical University, Hanoi, Vietnam (e-mail: [email protected]).
been conducted on vernier-type absolute encoders [1]–[3]. In this paper, we investigate a vernier absolute magnetic encoder (VAME), which is a multipolar dual track magnetic encoder.
It has master and nonius tracks. The master track was located outside and split toN pole pairs, while the nonius was located inside and split to(N−1) pole pairs. Each pole pair of each track will generate a pair of sine and cosine signals and each period of the sine/cosine signal corresponds to a phase of a saw tooth wave in the range of[0,2π]rad. Two phase values are extracted from the master and nonius tracks. The absolute information of any point on the two tracks can be obtained by the vernier principle [2], [3]. In this structure, the accuracy of the phase obtained from the master and nonius in each pole pair is important for improving the accuracy of the VAME. In practice, the sine/cosine signals from the master and nonius tracks are contaminated by non-ideal factors, such as different amplitudes, dc-offsets, phase shifts and random noise [4], [5]. Moreover, the harmonics existing in the encoder signals significantly affect the accuracy of the multipolar magnetic encoder [6], [7]. Therefore, a method to overcome the error factors is needed to obtain the absolute information.
There are many methods that have been presented to re- duce error factors that affect the sinusoidal signals. A d-axis component was introduced in [8], [9] to reduce the non-ideal sinusoidal signals. The authors in [10] presented a method to determine the error parameters offline before using them for the real operation. The authors in [11] demonstrated a closed-loop self-detection method to determine and eliminate the error factors in the encoder signals. Additionally, using a gradient based on the residual is a good online method to calibrate the different amplitudes, dc-offsets and phase shifts [4], [12], [13]. Similar to the gradient method, a neural network was applied in [14] using an adaptive linear neural network (ADALINE) to reduce the effects of non-ideal factors.
A ratiometric linearization conversion method [15], [16] based on basic mathematical manipulation was introduced to convert the sinusoidal encoder signals to nearly perfectly linear output.
However, the harmonic factors were not mentioned in these studies.
Reference [17]–[21] presented harmonic elimination schemes. In [17], [18], the authors introduced an effective method to reduce the selected high-order harmonics in the sinusoidal signals using an adaptive notch filter (ANF). An improvement was made to the ANF to reduce high-order harmonics by Xinda Song [19] by employing synchronous frequency extract filters (SFFs). The ADALINE method was
also applied to reduce the harmonic components in sinusoidal signals [20], [21]. All these studies focused on reducing the high-order harmonic components in the signals, but they failed to discuss the effects of the non-ideal components.
Moreover, the harmonics existing in the VAME signals are high and low-order harmonics located very close to the fundamental frequency of the signals. Hence, these harmonics are difficult to suppress in the sine/cosine signals.
The paper [6] introduced a method to overcome the non- ideal factors and harmonic disturbance using ADALINE based on the PLL method. However, this method needs to adapt many parameters to achieve good performance. Moreover, the magnitude of the harmonics in the VAME are significantly larger (by a factor of three) than the absolute magnetic encoder (AME) that was mentioned in [6]. Therefore, this method cannot work well for the VAME. Other researchers also reduced the harmonics for sine/cosine signals using a gradient descent method [7]. However, this method cannot reduce the harmonics, which are very close to the fundamental frequencies of the sinusoidal signals. Hence, this method cannot be applied for the VAME. Besides, the phase-shift error is ignored in this method.
In practice, the pole pitch error is the main cause of the har- monics in the VAMEs. It creates a non-linear phase difference between the master and nonius tracks. This error significantly affects the accuracy of the encoder. These problems will be analyzed and overcome in this paper.
This paper proposes an effective approach to reduce the mentioned error components, including the non-ideal factors (e.g., different amplitudes, dc-offsets, phase shifts) and the harmonic disturbance. The main contributions of this paper are described below:
1) Two observer phase-locked loops (OPLLs), which are based on an observer phase detector (OPD) scheme, are employed to estimate the phase of the master and nonius tracks and eliminate the different amplitudes, dc-offsets and phase shifts existing in encoder signals.
2) A non-linear phase compensation (NLPC) method is pro- posed to compensate for the non-linear coarse absolute phase, which is distorted by the pole pitch error, to obtain the linear coarse absolute phase.
3) A pole pitch compensation (PPC) method is proposed to address the harmonics existing in the encoder signals.
The rest of this paper is organized as follows. Related works and an overview of the proposed method are introduced in Section II. The phase estimation based on the observer phase detector scheme is explained in Section III. Section IV addresses the non-ideal phase coarse problem using the non-linear phase compensation method. Section V explains the pole pitch compensation method used to solve the harmonic problem. Section VI describes the experimental results, while Section VII summarizes the conclusions of this paper.
Pre-Amplifier & ADC
Observer-PLL Observer-PLL un,s
un,c um,c um,s
φ
ṋφ
m̭φ
v̭φ
m̭cφ
abs̭Signal Processing Part VAME
Nonius Circuit Master Circuit
Absolute Angle Calculation Pole Pitch Compensation
Vernier & LUTs
Fig. 1. Vernier absolute magnetic encoder.
II. RELATED WORKS AND OVERVIEW OF THE PROPOSED METHOD
A. Vernier absolute magnetic encoder
Fig. 1 illustrates the overall structure of a VAME. The system includes: a magnetic disk with master and nonius tracks, two magnetic sensors and a signal processing circuit.
During each rotation of the motor shaft, the master and nonius sensors outputN and(N−1)cycles of quadrature sinusoidal signals, respectively. Next, the sensed signals will go into the processing circuit, where a microcontroller will process these signals to produce the ϕbm andϕbn phase of the master and nonius tracks. These are then combined to generate the absolute phaseϕbabs as follows:
ϕbabs= 2π
Nf loor(Nϕbd
2π ) +ϕbm
N (1)
where ϕbd is the phase difference between ϕbm and ϕbn. The values of the phases ϕbm, ϕbn, ϕbd and ϕbabs should be in the range of[0,2π]for each sinusoidal cycle.
ϕbd = (
ϕbm−ϕbn ϕbm≥ϕbn
ϕbm−ϕbn+ 2π ϕbm<ϕbn . (2) B. Analysis of nonidealities
The sine/cosine signals from the magnetic sensor are con- taminated by many distortions. In practice, the magnetic pole pairs in the multipolar magnetic tracks are not consistent. Fig.
2(a) illustrates an ideal multipolar magnetic track. Assume the track has N pole pairs and each pole pair has the same dimension. Each pole pair will generate a pair of sine/cosine signals. If we consider the sine (or cosine) signals in this case,
T
0T
1T
2T
3T
NT
a) b)
T
0T
0T
0Fig. 2. a) Ideal pole length b) non-ideal pole length.
the signal oscillates with the period T0 and the fundamental frequencyf0= 1/T0.
Fig. 2(b) illustrates a non-ideal multipolar magnetic track.
The track also has N pole pairs and the dimensions of each pole pair are dissimilar. Consequently, each pole pair generates a pair of quadrature sinusoidal signals whose periods are different fromT0. The time interval for each sinusoidal signal can be smaller or larger than the ideal time T0. However, all these signals can be considered as a combined signal x(t): here, the new periodic signal oscillates with the period T =N T0and the frequency f =f0/N.
This signal can be modeled as shown in equation (3) by applying the Fourier series formula [22] to the periodic signal x(t), which consists ofN sinusoidal signals created by each pole pair. The main cause of the harmonic distortion in the encoder system is explained by this equation.
x(t) =
∞
X
n=−∞
Xk.ej2π(f0/N).k.t (3)
whereXk is the amplitude of thekth harmonic:
Xk= 1 N.T0
Z T
0
x(t).e−j2π(f0/N).k.tdt. (4) To explain this problem in greater detail, Fig. 3 shows the fast Fourier transform (FFT) of the real sine encoder signals from the master and nonius tracks. There are a lot of harmonics around the fundamental frequencies of the master fm and noniusfn. The dominant harmonics in the spectrum are 3132fm
and3332fm, corresponding toN = 32andk= 31,33for master signals. These values are 3031fn and 3231fn with N = 31and k= 30,32for nonius signals. In theory, the phase difference ϕbd is a linear angle. However, the existing harmonics distort the phaseϕbd to a non-linear phase, which significantly affects the accuracy of the encoder.
The raw sine/cosine signals from the master and nonius tracks are also distorted by nonideal factors, such as different amplitudes, dc-offsets, phase shifts and random noise [4], [5].
Equation (5) shows the general mathematical models of the encoder signals for the master and nonius tracks in each pole pair. Where As and Ac are the amplitude of the sine and cosine signals, respectively; Bs and Bc are the dc-offsets; δ
0 100 200 300 400 500
0.8 0.6 0.4 0.2 0
Fundamental frequency
Frequency (Hz) Frequency (Hz)
AmplitudeAmplitude
0.15 0.1 0.05 0
0 100 200 300 400 500
0 100 200 300 400 500
0.8 0.6 0.4 0.2 0
Fundamental frequency
X: 320 Y: 0.7296
X: 310 Y: 0.7053
0 100 200 300 400 500
0.15 0.1 0.05 0
Disturbance harmonic
Disturbance harmonic
master nonius
Fig. 3. Harmonics existing in the master and nonius signals.
is the phase shift between sine and cosine; andηsandηc are random noise.
uc=Accos(ϕ) +Bc+ηc
us=Assin(ϕ+δ) +Bs+ηs (5) C. The proposed method
The signal processing part in Fig. 1 illustrates a block diagram of the proposed method used to obtain the high- resolution absolute angle. The inputs are the sine/cosine sig- nals of the master (um,s,um,c) and nonius (un,s,un,c), which are distorted by the non-ideal factors and the harmonics. To obtain the phases of the master and nonius tracks, as well as address issues related to the non-ideal factors, two OPLLs are employed for the master and nonius signals. The proposed PLL has an OPD, which is used to estimate and eliminate the non-ideal factors from the phase detector.
However, the periods of each estimated phase of the master and nonius are non-homogeneous. This causes the phase differenceϕbdto be non-linear. A NLPC is proposed to correct the non-linear phaseϕbdby changing it to the linear phaseϕbv. This is done by using two look-up tables. To eliminate the harmonics caused by the pole pitch error, the corrected phase ϕbv and the phase of the masterϕbm continue to enter the pole pitch compensation block. This block will modulate each non- homogeneous phase of the master into a homogeneous phase ϕbcm. Finally, the absolute angle can be generated by equation (6). Sections III, IV and V will describe more details about the proposed method.
ϕbabs= 2π
Nf loor(Nϕbv
2π ) +ϕbcm
N (6)
III. PHASE ESTIMATION
To estimate the phase from a pair of sinusoidal signals, the quadrature phase-locked loop is an effective method [23].
The PLL works effectively when the input sine and cosine signals are ideal. However, real encoder signals are non-ideal, as shown in equation (5). To overcome these problems an OPD based on [7] was employed with some small improvements in formula. In this paper, the phase-shift error is considered and addressed. There are three components of the OPLL: the observer phase detector (OPD), loop filter and VCO.
u
su
cLoop filter
φ ̭
Adjusting
cos φ ̭ sin φ ̭ cos 2φ ̭ sin 2φ ̭ 1
+ +
#
e + -
pdPD e
e
ṋ
VCO
sin cosφ ̭ φ ̭
,
i
θ1
θ2
θ3
θ4
k
ik
p+
s 1
s
OPD
Fig. 4. Observer phase-locked loop structure.
A. Observer phase detector
1) Effects of non-ideal factors on the phase detector.
With the non-ideal input signals, the phase detector error epd of the PLL is calculated as:
epd=uscos(ϕ)b −ucsin(ϕ)b (7) whereϕandϕbare the input and output phases.
epd= sin(ϕ−ϕ) +b h
(Ascos(δ)−1) sin(ϕ) cos(ϕ)b + (1−Ac) cos(ϕ) sin(ϕ) +b Assin(δ) cos(ϕ) cos(ϕ)b +Bscos(ϕ)b −Bcsin(ϕ)b i
.
(8)
At the in-lock state, ϕ ≈ ϕ. After some trigonometricb manipulations, (8) can be rewritten as:
epd= sin(ϕ−ϕ) +b h
θ1cos(ϕ) +b θ2sin(ϕ)b +θ3(cos(2ϕ) + 1) +b θ4sin(2ϕ)b i
=ei+en
(9) whereei= sin(ϕ−ϕ)b ≈ϕ−ϕbis the difference between the input and output phases, which is used to estimate the desired phase outputϕ; the value ofb en is the non-ideal error, which is the addition component in the brackets of epd. It contains the fundamental frequency and second-order harmonic error terms, which cause the undesired output phase. The vector form of the en can be described as:
en=θTM (10)
where θ1 = Bs, θ2 = −Bc, θ3 = 0.5Assin(δ) and θ4 = 0.5(Ascos(δ) − Ac). θ =
θ1 θ2 θ3 θ4T represents the adjustable compensation vector and M = cos(ϕ)b sin(ϕ)b 1 + cos(2ϕ)b sin(2ϕ)bT
represents the reference input vector. The en should be rejected from the epd before enter the loop filter and VCO.
Fig. 4 shows the structure of the OPD used to suppress the en. It is based on a simple neural network model whose input includes trigonometric functions of the feedback output phase
to estimate the unknown amplitudes of the harmonic compo- nents in the phase error en. The “adjusting” block is used to automatically tune the weight parametersθby employing the gradient descent approach. Therefore,en can be observed and rejected from the phase detector error:ei=epd−ben. 2) Online adjusting algorithms of the weight vector.
The vector of the weight parameter θ is regulated to min- imize the multi-variable cost function J(θ) = 12kepd−benk2. The rule for adjusting θis based on the gradient descent.
θk =θk−1−λ.∇J(θk−1) (11) whereλis the learning rate that decides the convergence speed of the observer and∇J is the first partial derivative of the cost function with respect toθ:
∇J = (epd−ben)(∂epd
∂θ −∂ben
∂θ )≈ei∇ei (12) The gradient ofepd:
∂epd
∂θ =−(ussin(ϕ) +b uccos(ϕ))b ∂ϕb
∂θ (13) and the gradient ofben:
∂ben
∂θ =h∂
ebn
∂θ1
∂ben
∂θ2
∂ben
∂θ3
∂ben
∂θ4
iT
=M+θTU∂ϕb
∂θ
(14)
where U =
−sin(ϕ)b cos(ϕ)b −2 sin(2ϕ)b 2 cos(2ϕ)bT . Therefore, the derivative of the(epd−ben)correspond toθis:
∇ei=−M −(θTU+ussin(ϕ) +b uccos(ϕ))∇b ϕb
=−M −W.∇ϕb (15)
whereW =−Assin(δ) sin(2ϕ) + (Ab scos(δ)−Ac) cos(2ϕ) +b Assin(ϕ) sin(ϕ+δ)+Ab ccos(ϕ) cos(ϕ). Since, the phase shiftb δ <15oandϕ≈ϕbat the lock-in state of the PLL,W can be approximated as unity. Therefore∇ei is rewritten:
∇ei≈ −M− ∇ϕ.b (16) In the PLL, the phase error ei enters to the loop filter and VCO to obtain the phase output ϕ. From (23), the gradientb of the phase error can be calculated from the feedback of the phase output, as described by the following equation:
∇ei(z)≈ − D(z)
D(z) +N(z)M(z). (17) From (11), (12) and (17), the weighting update rule is:
θk=θk−1−λ.ei∇ei (18) B. Loop filter and VCO
In the PLL scheme, the loop filter is used to reduce random noise in the phase error, while VCO is a pure integrator to extract the estimated phase. In this paper, the proportional- integral (PI) controller is employed as a loop filter with the transfer function F(s) = kp+ ksi. The closed-loop transfer function of the proposed PLL is:
H(s) = kps+ki
s2+kps+ki
= 2ζωns+ω2n
s2+ 2ζωns+ω2n (19)
3.5 3.55 3.6 3.65
φ̭m φ̭n φ̭d φ̭v
Time(s)
rad
0 1 2 3 4 5 6
Fig. 5. Non-linear phase difference between the master and nonius.
φm̭
φṋ Vernier
LUT1 LUT2
φḓ Position pos
Determination Interpolation φv̭
Fig. 6. Non-linear phase compensation algorithm.
where kp and ki are the proportional and integral gains.
ζ = kp
2√
ki is the dimensionless damping coefficient and ωn = √
ki is the natural frequency of the system. These parameters determine the response time, damping and noise suppression capability of the PLL. If s = jω is substituted into equation (19), the PLL bandwidth [24] can be obtained:
BW =ωn
q
1 + 2ζ2+p
1 + (1 + 2ζ2)2. (20) For selecting the damping factor ζ, many studies recom- mend ζ = 0.707to achieve a good dynamic response. When ζis assigned, the PLL bandwidth depends solely on the natural frequencyBW ≈2.058ωn. To achieve good noise immunity, a narrow bandwidth is required; thereforeωnshould be small, and vice versa. However, ωn is inversely proportional to the acquisition time of the PLL [10], and a trade-off exists between the noise immunity and the acquisition time. To address this problem, a software algorithm is used to adjust the bandwidth [25]:
BW(k) =k1BW(k−1) +k2|ei| (21) where BW ∈[BW min;BW max]; k1 andk2 are constants that are selected experimentally by the real system.
To implement the proposed PLL in a microcontroller, the open-loop transfer function of the system is recognized as:
Ho(s) = ϕ(s)b
Ei(s) =kps+ki
s2 . (22)
Then, the bilinear approximation s ≈ T2
s
1−z−1
1+z−1 is employed to convert the open loop transfer function to discrete domain:
Ho(z) = ϕ(z)b
Ei(z)= N(z)
D(z) = N0+N1z−1+N2z−2 D0+D1z−1+D2z−2 (23) where N0 = 4ζωnTs + (ωnTs)2, N1 = 2(ωnTs)2, N2 = (ωnT s)2−4ζωnTs,D0 = 4, D1 =−8,D2 = 4. And Ts is the sampling time.
Algorithm 1: Quick search the closest position
Input :ϕbd: The instant phase difference.
Output:pos: The closest position ofϕbdin LUT1.
Data :V: stored the LUT1 indexed from 0 toS−1.
1 begin
2 low= 0;high=S−1;
3 while(V[low]6=V[high])&&(V[low]≤ϕbd&&
ϕbd≤V[high])do
4 slope= (high−low)/(V[high]−V[low]);
5 j=low+f loor(slope∗(ϕbd−V[low]));
6 ifV[j] ==ϕbdthen
7 pos=−j;
8 returnpos; // ϕbd = LU T1[pos]
9 if(V[j]<ϕbd) && (ϕbd< V[j+ 1])then
10 pos=j;returnpos;
11 if(V[j−1]<ϕbd) && (ϕbd< V[j])then
12 pos=j−1;returnpos;
13 if(V[j]<ϕbd)then
14 low=j+ 1;
15 if(V[j]>ϕbd)then
16 high=j−1;
Algorithm 2: Interpolate the linear phase
Input :pos: The closest position ofϕbdinLU T1.
Output:ϕbv: The compensated phase.
Data :LU T1,LU T2indexed from 0 toS−1
1 begin
2 ifpos <0then
3 ϕbv=LU T2[abs(pos)]; // ϕbd = LU T1[pos]
4 else
5 slope=(LU T2[pos+ 1]−LU T2[pos])/(LU T1[pos+ 1]−LU T1[pos]);
6 ϕbv=slope∗(ϕbd−LU T1[pos]);
IV. NONLINEAR PHASE COMPENSATION
Fig. 5 depicts the phasesϕbm,ϕbn andϕbd after employing the OPLL for a real system. As discussed above, the phase ϕbd is not a linear phase due to the pole pitch error, which causes undesired absolute phase. It should be noted that the length of each pole is fixed. Therefore, the effect of this error is the same for all motor speed ranges. This means the phase ϕbd is distorted in the same way in the system. Based on this property, the look-up table method can be employed to correct the non-linear phase ϕbd and convert it to the linear phase ϕbv. Observe in Fig. 5, the curveϕbd increases in the range of [0,2π]. Therefore, the phase difference in one cycle of[0,2π]
can be saved in aLU T1. A linear phase in the range of[0,2π]
can be generated with the same number of elements of the LU T1. The linear phase is stored in a LU T2:
LU T2(i) = 2π
S−1i (24)
where i = 0...(S−1) is the ith element in the LU T2 and S is the number of elements in the LU T2. From this, each value of the non-linear phase difference can be converted to a linear phase.
To improve the accuracy and reduce the average time it takes to consult the look-up table, an algorithm that can quickly look up data in the table is proposed. Fig. 6 illustrates the structure of the proposed algorithm. When the system operates, the instant value of ϕbd is used as the input for the LUT1 to determine the closest position such that its value is the
T
1T
2T
3T
NT
Fig. 7. Nonhomogeneous phase estimated from each pole pair.
B C A
O B
AC
a) b)
φ ̭
m
φ ̭
mc 2π
αid αid
O β β
Fig. 8. The non-homogeneous and homogeneous phase. (a) The shorter pole pair. (b) The longer pole pair.
closest to the input value. Algorithm 1 describes the scheme for extracting the position using the quick search method based on an interpolation search algorithm [26].
The determined position is used to infer the linear phase by interpolating the LUT2 by following Algorithm 2. The average number of times the look-up table is accessed in the proposed algorithm can be approximated as log2(log2(S)). Therefore, the proposed method can quickly convert the non-linear phase ϕbd to the linear phaseϕbv.
V. POLE PITCH COMPENSATION
A. Principle of the proposed method
As mentioned in the discussion of Fig. 2(b), the dimensions of each pole pair in the multipolar magnetic track are different.
Therefore, the period for each cycle of ϕbm is modulated to correspond to each pole pair, as shown in Fig. 7. Moreover, the absolute angle for each rotation is calculated by combining phases ϕbv andϕbm. Consequently, the absolute angle is not correct. The paper proposes a pole-pitch compensation method to compensate the incorrect phase ϕbm to the homogeneous phaseϕbcm. From the above analyses, the period of each phase ϕbmis proportional to the width of each pole pair. This means that the period of ϕbm is shortened because the pole pair is shorter than the nominal length; the opposite is also true.
The black line in Fig. 8 depicts the phase of the master in the case of a non-homogeneous pole pair. Additionally, the orange line illustrates the master phase in the case of a homogeneous pole pair. Considering the triangles OAB and OCB, the relationship between the non-homogeneous angle ϕbm and the homogeneous angleϕbcmis:
ϕbcm(t) = tan(β)
tan(αid)ϕbm(t) (25) where αid and β represent the AOBd and COBd angles, respectively. In practice,αidwill be larger or smaller than the ideal angle β due to the pole pitch error. It should be noted
2π
T
1T
2T
3T
NT
0T
0T
0T T
0c
o o
c
o
c c
o
c o
a)
b)
φ
m̭φ
̭mcFig. 9. Pole pitch compensation.
that problems related to the pole length differences are due to mechanical errors, which means that this error has the same effect for all motor speed ranges. Therefore, the mechanical ratio tan(αtan(β)
id) for each pole pair is constant for all motor speeds. Assuming the ratios of tan(αtan(β)
id) are determined, the non-homogeneous angleϕbm generated in each pole pair can be modulated to the homogeneous angleϕbcmas follows:
ϕbcm(t) =
( tan(β)
tan(αid)ϕbm(t) id= 1
tan(β)
tan(αid)ϕbm(t) +ϕbadj id >1 (26) with the adjacency angle ϕbadj can be described as:
ϕbadj = 2πhid−1X
j=1
tan(β)
tan(αj)−f loorid−1X
j=1
tan(β) tan(αj)
i (27) where id represents the index that determines the position of each pole pair in each rotation of the master track (id ∈ [1, N]). Note that the modulo operation needs to be employed to keep the value of theϕbcmwithin the range of[0,2π]:
ϕbcm(t) =mod(ϕbcm(t),2π). (28) Fig. 9 illustrates an example to clearly explain the PPC method. The black lines of the saw tooth waves represent the phase ϕbm from the master. The colored lines represent the compensated phase. Fig. 9(a) shows the results after applying equation (25) to each phase ϕbm in each pole pair, corresponding to the value ofid. It should be noted that the indexid=jduring the periodTj. Equation (25) is a rotational operation used to convert each non-homogeneous angleαid to the homogeneous angle β. Fig. 9(b) illustrates the corrected phaseϕbcmafter employing equation (26). The additional term ϕbadj in equation (26) when id >1is the phaseϕbcm, which is compensated for when the indexidchanges from the previous pole pair to the current pole pair. It connects the point “o” to
“c” in Fig. 9(a), resulting in the homogeneous phase angle in Fig. 9(b).
To actualize the proposed method, there are two important parameters that need to be determined: the ratio tan(αtan(β)
id) and id. These will be described in detail in the next sections.
B. Identification of compensation parameters
Algorithm 3 presents the method used to exactly determine which pole pair in the master track that the phaseϕbmbelongs
Algorithm 3: Index determination
Input :ϕbm,ϕbv
Output:id: The index of each pole pair from1toN.
1 begin
2 id=f loor(Nϕbv/2π);
3 tmp= (ϕbm+ 2π∗id)/N−ϕbv;
4 iftmp <−(π/N)thenid=id+ 1;
5 iftmp >(π/N)thenid=id−1;
6 ifid <0thenid=N−1;
7 ifid > N−1thenid= 0;
8 id=id+ 1;returnid;
α
1α
2α
3α
NΔT ΔT ΔT ΔT
2 3 N
id=1
Δ2 Δ3 ΔN
Δ1
φ̭
m Δφ̭
=
Δid id Fig. 10. Mechanical ratio determination.to [27]. Fig. 10 illustrates the scheme used to determine the mechanical ratios. In each section of the angle ϕbmthe phase difference is determined by the timer ∆T:
∆ϕbid =ϕbm(t+ ∆T)−ϕbm(t) (29) and the mechanical ratio in each pole pair under the timer∆T is:
tan(β)
tan(αid) = ∆β
∆ϕbid
(30) where∆β is the phase difference of the master corresponding to ∆T when the pole pairs are homogeneous. After deter- mining the ∆ϕbid in each pole pair, ∆β can be calculated as follows:
N
X
id=1
∆β
∆ϕbid =N. (31)
From equation (31),∆βis determined. Therefore, the mechan- ical ratio of each pole pair can be identified. As mentioned above, these ratios affect the encoder in the same way for all ranges of the motor speed. Therefore, there are onlyN ratios, and these can be stored in a small dimension array for con- venient implementation in the real system. It should be noted that the mechanical ratio identification is done only one time for each encoder system to determine the mechanical ratio.
Then, the N-element array of the mechanical ratio is used to convert the non-homogeneous angle into a homogeneous one.
VI. EXPERIMENTAL RESULTS
This section presents some experimental results to verify the effectiveness of the proposed method. Fig. 11 illustrates the experimental setup used to test the proposed method. In this setup, a magnetic disk (MU2S 30-32N) is mounted to the main shaft of the rotary motor. The disk includes two tracks:
the master track and nonius track, which have 32 and 31 pole pairs, respectively. Two evaluation kits from iC-MU EVAL MU1C of iCHaus are used as two sensors to obtain the raw
Heidenhain Ratory Motor
VAME
Sensor Circuits Processing Circuit
MCU
FPGA PC
φHe
̭φabs
AMP
Fig. 11. Experimental setup.
-0.1 0 0.1
0 0.5 1 1.5 2 2.5
-0.1 0 0.1
1.7 1.701 1.702 1.703 1.704 1.705 1.706 1.707 1.708 1.709 1.71
-0.05 0 0.05
0 0.5 1 1.5 2 2.5
Without OPLL With OPLL
Phase eror (rad)Weighting
a)
b)
c) Time(s)
Convergence Time θ3 θ1
θ4 θ2
epd en
̭
ei
Fig. 12. Phase detector error and weight updating.
sine/cosine signals of the encoder. Note that each circuit can provide optional outputs. Specifically, they can provide the absolute angle or a pair of sine/cosine signals for the master or nonius track. In this paper, the circuits are used to pick the magnetic fields of the master and nonius tracks, and the output is the sinusoidal signals of the master and nonius tracks. These signals are used as the input in the proposed method. The sensed signals are input into the amplifier circuits to adjust the magnitudes of the sensed signals to full the range of(0; 3.3V) of the ADC module in the microcontroller (MCU) circuit. The MCU circuit will process the encoder signals to generate the absolute angleϕbabs. Fig. 11 also depicts the method used to evaluate the accuracy of the absolute angleϕbabs estimated by the proposed method. The motor also mounts the Heidenhain (Hei) absolute encoder with 19-bit resolution to provide the reference absolute angleϕHe. A field programmable gate array (FPGA) is used to read the absolute angles of the VAME and Hei encoder at the same time. Then,ϕbabs andϕHe are sent to a computer (PC). The phase difference betweenϕbabsandϕHe is used to evaluate the performance of the proposed method.
Currently, the proposed method is implemented in the MCU (ARM STM32F407ZG). This allows fast computation where the sampling timeTs is about 50 µs. The parameters in the experiments are selected as follows:
- For the observer PLL: The ζ = 0.707,BW min = 600 rad,BW max= 30000rad and the learning rate λ= 0.02.
- For the linear compensation algorithm: The motor is
Time(s) 0
5
Phase(rad)
0 0.02 0.04 0.06 0.08 0.1
0.04 0.042 0.044 0.046 0.048 0.05
0 5
φ̭
m φ̭
mc φ̭
abs
Phase(rad)
Fig. 13. Pole pitch compensation for the master phase.
0 0.005 0.01 0.015 0.02 -1
0 1
a) Time (s)
0 0.005 0.01 0.015 0.02 b) Time (s)
0 200 400 600 800
d)
0 200 400 600 800
e) Frequency (Hz)
0 200 400 600 800
f) Frequency (Hz)
0 200 400 600 800
c)
-1 0 1
0 0.05 0.1
0 0.05 0.1
0 0.05 0.1
0 0.05 0.1 Fundamental Frequency
Fundamental Frequency
Fig. 14. (a) Input sin/cosine. (b) Output sin/cosine. (c) and (d) The FFT results of In/Out sine. (e) and (f) The FFT results of In/Out cosine.
operated at 5 Hz to build up the LUT. Therefore, the size of each LUT isS= 4000.
- For the pole pitch compensation: The motor is operated at 1 Hz to determine the mechanical ratio tan(αtan(β)
id) for 32 pole pairs of the master track.
Fig. 12(a) and Fig. 12(b) illustrate the effectiveness of the observer phase detector to automatically correct for the non- ideal factors. As shown in the above analyses, theepdcontains the fundamental frequency and second-order harmonic error terms. The OPD can track the non-ideal factors to estimate the phase errorebn and then suppress them to obtain the phase error between the input and output ei. Fig. 12(c) illustrates the updating weight parameters. In addition, to illustrate the convergence time of the proposed method, the experiment is switched from the conventional PI-PLL to the OPLL. As shown in the figure, the erroreiwill be reduced and the weight parameters will converge within 0.1 s, which corresponds to 2000 steps, to reconstruct the real value of non-ideal factors.
Fig. 5 illustrates the result after applying the NLPC algo- rithms to convert the non-linear phaseϕbd into the linear phase ϕbv. The average time for accessing the LUT is approximately equal to log2(log2(4000)) ≈4 for compensating each value of the non-linear phase ϕbd toϕbv.
Fig. 13 illustrates the results after employing the pole pitch compensation method for the master phase. The black line is the non-homogeneous phase of the master track and the
0 200 400 600 800 0 200 400 600 800
a) Frequency b)Frequency
0 0.05 0.1
0 0.05
PI-PLL 0.1 ANL-PLL
Fig. 15. The FFT results of PI-PLL and ANL-PLL method.
0 0.5 1 1.5 2 2.5 3 3.5 4
-0.01 0 0.01
Without OPLL-PPC With OPLL-PPC
Time(s)
Error(rad)
Fig. 16. Phase error between the VAME and the Heidenhain encoder with/without the OPLL-PPC.
orange line is the modulated phase that is compensated by equation (26). The blue line is the high-resolution absolute angle of the VAME. Fig. 14 shows the spectra of the input and output signals to demonstrate the effectiveness of the proposed method in reducing the harmonics. It is visually observed that the harmonics existing in the input signals can be reduced after applying the OPLL-PPC method. The reduction is at least 90%. In addition, the PI-PLL and the ALN-PLL [6] methods are also implemented in this system; these results are shown in Fig. 15. It is observed that these methods cannot eliminate the harmonics in this system. In the PI-PLL method, the phase detector contains the non-ideal factor errors. Hence, the output contains the second-order harmonics in its spectrum, as shown in Fig. 15 (a). As demonstrated by these results, the harmonics caused by the pole pitch error are eliminated by the proposed method.
Fig. 16 illustrates the phase errors (eOP LL−P P C =ϕHe− ϕbabs) between the Heidenhain encoder and the absolute angle of the VAME with and without the proposed method when the motor operates at 20 Hz. Initially, the OPLL-PPC was disabled. After 1 s, the proposed method starts to operate.
As shown in the figure, the phase error is reduced from
±0.012 rad to ±0.002 rad in this experiment. Currently, the proposed method can work in the wide frequency range (0 to 1600 Hz) of the multipolar magnetic track, corresponding to variations in the motor speed from 0 to 50 Hz. In order to show the improvement of the proposed method in the wide range of motor operation, the phase error between the Heidenhain encoder and the VAME with and without the proposed method are compared in Fig. 17. The accuracy of the proposed method is 0.006 rad for the whole frequency range (0-50 Hz) of the main encoder shaft, and the accuracy when not using the proposed method is 0.016 rad. Therefore, the accuracy improved 62% when using the proposed method.
As mentioned above, the sensor circuits that are used in this experiment are from the evaluation kit iC-MU EVAL MU1C of iCHaus. This circuit can also output the absolute angle ϕiC. To further prove the effectiveness of the proposed method, the Heidenhain encoder and the evaluation kit will measure the absolute angle of the motor and then send the angle
0 0.5 1 1.5 2
Error (rad) -0.01
0 0.01
0 0.1 0.2 0.3 0.4 0.5
Error (rad) -0.01
0 0.01
0 0.1 0.2 0.3 0.4 0.5
Error (rad) -0.01
0 0.01
Time (s)
0 0.1 0.2 0.3 0.4 0.5
Error (rad) -0.01
0 0.01
Without OPLL-PPC With OPLL-PPC
a) 1hzb) 10hzc) 30hzd) 50hz
Fig. 17. Phase error between the VAME and the Heidenhain encoder with/without the OPLL-PPC. (a) motor run at 1 Hz. (b) motor run at 10 Hz. (c) motor run at 30 hz. (d) motor run at 50 Hz.
0 0.5 1 1.5 2
-0.01 0 0.01
0 0.1 0.2 0.3 0.4 0.5
-0.01 0 0.01
0 0.1 0.2 0.3 0.4 0.5
-0.01 0 0.01
Time(s)
0 0.1 0.2 0.3 0.4 0.5
-0.01 0 0.01
Error (rad)Error (rad)Error (rad)Error (rad)
a) 1hzb) 10hzc) 30hzd) 50hz
eiC e
OPLL-PPC
Fig. 18. Phase error comparison with a commercial product. (a) motor run at 1 Hz. (b) motor run at 10 Hz. (c) motor run at 30 Hz. (d) motor run at 50 Hz.
data to a computer. The phase difference eiC = ϕHe−ϕiC
is compared with the error eOP LL−P P C that is achieved using the proposed method. Fig. 18 illustrates the results when comparing the proposed method with the commercial evaluation kit. Using the same frequency of the running motor, the error derived by the iCHaus circuit is ±0.011 rad while the error derived by the proposed method is ±0.006 rad for the whole frequency range (0-50 Hz) of the motor speed;
Therefore, the accuracy is improved 45% by the proposed
method. These experimental results prove the effectiveness of the proposed method compared to existing methods in terms of improving the accuracy of the dual track absolute encoder.
VII. CONCLUSION
In this paper, an effective method is proposed to improve the accuracy of vernier absolute magnetic encoders. This method is based on an observer phase-locked loop (OPLL), non-linear phase compensation (NLPC), and pole pitch compensation (PPC). Three main goals have been achieved: 1) compensating for non-ideal factors (e.g., the different amplitudes, dc-offsets and phase shifts) using the OPLL, 2) compensating for the non-linear phase difference between the master and nonius phases, which are used to compute the absolute information, and 3) correcting the pole pitch error, which caused harmonics in the encoder signals, using the PPC method. All the proposed methods were described in detail. The experimental results illustrate the effectiveness of the proposed method in reducing the factors that cause errors in the encoder. Therefore, the proposed method can be applied to improve the accuracy of VAMEs.
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Ton Hoang Nguyenreceived the B.S. degree in Mechatronics engineering from the Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam, in 2016. He is currently pursuing the Ph.D. degree in electrical and computer engi- neering with the School of Information and Com- munication Engineering, Sungkyunkwan Univer- sity, Suwon, South Korea.
His research interests include signal process- ing, motion control, robotics, and embedded sys- tems.
Ha Xuan Nguyenreceived the B.S. degree in Mechatronics engineering from the Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam, in 2015. He is currently pursuing the Ph.D. degree in electrical and computer engi- neering with the School of Information and Com- munication Engineering, Sungkyunkwan Univer- sity, Suwon, South Korea.
His research interests include signal process- ing,motion control, robotics, and embedded sys- tems.
Thuong Ngoc-Cong Tran received the B.S.
degree in Mechatronics engineering from the Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam, in 2015. He is cur- rently pursuing the Ph.D. degree in electrical and computer engineering with the School of Information and Communication Engineering, Sungkyunkwan University, Suwon, South Ko- rea.
His research interests include signal process- ing, motion control, robot vision, and machine- learning.
Jae Wan Park received the B.S. degree in Mechatronics engineering from Korea Polytech- nic University, Siheung, Korea, in 2016. He is currently working toward the Ph.D. degree in electrical and computer engineering with the School of Information and Communication En- gineering, Sungkyunkwan University, Suwon, South Korea.
His research interests include embedded sys- tems, automation system, signal processing, and real-time applications.
Kien Minh Lereceived the B.S. degree in elec- trical engineering from Hanoi University of Sci- ence and Technology, Hanoi, Vietnam, in 2006, the M.S. degree in automation from Le Quy Don Technical University, Hanoi, Vietnam, in 2011, and the Ph.D. degree in electronic and electri- cal engineering from Sungkyunkwan University, Suwon, South Korea, in 2017.
From 2018 to 2019, he was a Postdoctoral Researcher with the ICT HRD Institute for Future Value Creation, Sungkyunkwan University, Su- won, South Korea. He is currently a Lecturer with the Faculty of Control Engineering, Le Quy Don Technical University, Hanoi, Vietnam. His research interests include motion control, automation, and embedded systems.
