The design and implementation of an accelerometer-assisted velocity observer

ISA Transactions 59 (2015) 418–423

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Research Article

The design and implementation of an accelerometer-assisted velocity observer Yu-Sheng Lu a,n, Sheng-Hao Liu b a b

Department of Mechatronic Engineering, National Taiwan Normal University, 162, He-ping East Rd., Sec. 1, Taipei 106, Taiwan Department of Mechanical Engineering, National Yunlin University of Science & Technology, Yunlin, Taiwan

art ic l e i nf o

a b s t r a c t

Article history: Received 4 May 2015 Received in revised form 30 July 2015 Accepted 6 September 2015 Available online 28 September 2015

This paper presents a dynamically compensated velocity observer (DCVO), in which acceleration measurement is employed to estimate velocity. Its sensitivity to the noise that is associated with the acceleration measurement is formulated and compared with that of a conventional state-space velocity observer (SSVO). Unlike the SSVO, the DCVO is completely insensitive to an accelerometer offset. The DCVO, the SSVO and a common ITM-based estimator are all realized in a linear motion stage, to determine their effectiveness. A sliding-mode controller is also implemented with different velocity observers, and it is shown that the DCVO enables high-frequency switching of the sliding-mode controller and also practically improves the positioning accuracy. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Velocity observer Acceleration Dynamic compensation Inverse time method Control system

1. Introduction Precise linear motions are required in a variety of industrial applications, including semiconductor wafer inspection, lasercutting machines and many numerically controlled machine tools. In these applications, motion control systems are employed, and their performance inevitably depends on the accuracy and sensitivity of the sensors for the feedback systems [1]. Position sensors, such as optical incremental encoders, are widely used to provide positional information. However, velocity sensors may not be available or suitable for certain motion control systems. For example, the velocity signals provided by tachometers are generally contaminated with noise. System cost and size are also increased when velocity sensors are incorporated into motion control systems. As a consequence, velocities are usually estimated from the positional signal using a filtering algorithm. The quality of the velocity estimation can be improved by a more advanced filtering algorithm. Real-time velocity estimation approaches can be divided into two categories: model-based and non-model-based methods [2]. Model-based methods, such as sliding-mode observers [3,4], Generalized Proportional Integral-based observers [5], neural networks [6], dual-sampling rate observers [7], Kalman filters [8] and extended Kalman filters [9], use models of the dynamic n

Corresponding author: Tel.: +886 2 7734 3512; fax: +886 2 2358 3074. E-mail address: [email protected] (Y.-S. Lu).

http://dx.doi.org/10.1016/j.isatra.2015.09.001 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

systems for which the velocities are to be estimated. In general, these model-based approaches require a model of a mechanical load, but the mechanical properties of the load are uncertain or difficult to ascertain. Therefore, this paper focuses on non-modelbased methods, which usually use signal processing techniques. Two of the most common non-model-based methods for velocity estimation that use positional signals from incremental encoders are the finite difference method (FDM) and the inversetime method (ITM) [10]. The FDM is also referred to as the fixedtime (clock-driven, pulse-counting, or lines-per-period) method and evaluates the backward difference for encoder counts at fixed time intervals. This method is easily implemented in a real-time control system, since the control algorithm is also normally calculated at fixed time intervals. However, this method fails in the low-speed region, where the positional increments are relatively small for fixed time intervals, which significantly increases the effect of quantization noise and results in inferior velocity estimation. In other words, resolution by this method becomes very coarse at faster sampling rates, when the pulse counting period and thus the positional increment are small. The ITM is also referred to as the fixed-position (encoder-driven, pulse-width measuring, or reciprocal-time) method and estimates the velocity by using the time that elapses between two consecutive encoder pulses. This method gives improved resolution in the lowspeed region, as the interval between two consecutive pulses is relatively long. However, this method has limitations in the highspeed regime. To detect velocities using the discrete pulse train

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from an incremental encoder, Ref. [11] provides an overview of existing techniques such as Taylor series expansions, backward difference expansions and least-squares fits. Ref. [12] uses the time stamping concept to improve a least-squares fit velocity estimator. Non-linear filtering techniques, including tracking differentiators [13–15], enhanced differentiators [16], high-order nonlinear differentiator [17] and differentiators using the so-called supertwisting algorithm [18], have also been developed for velocity estimation. However, they either require bounds for the acceleration or are complex in design and implementation. Because of the advances in electromechanical microsystem (MEMS) technology, the cost of MEMS accelerometers is extremely low, when compared with that of positional sensors, such as incremental encoders. Therefore, the addition of these ubiquitous, low-cost accelerometers to positional sensors is an attractive option for high-quality velocity estimation. Ref. [19] proposes a position- and acceleration-based velocity estimation law, in which an observation time period is defined as the interval between certain past time, t i , and the current time, t, and is chosen to be a constant multiple of the sampling period in discrete-time implementation. This law involves a double integral of acceleration over the observation time period, ½t i ; t, and can be written as: Rt Rt xm ðtÞ xm ðt i Þ t τ am ðτ a Þdτ a dτ þ i ; ð1Þ v^ ðtÞ ¼ t t i t  ti in which τ and τa are integration variables, v^ ðtÞ is the velocity estimate and xm and am respectively denote the measured position and acceleration. In practice, the measured acceleration signal usually contains an uncertain offset, which leads to a drift in the velocity estimate when the velocity estimation law (1) is used. Using the state-space approach, Refs. [1,20] present a Luenberger observer-like velocity estimator, which is referred to as state-space velocity observer (SSVO) in this paper. However, a SSVO can also be susceptible to accelerometer offset. In [21], the so-called twochannel approach is proposed for the fusion of positional and acceleration measurements. This approach uses two frequencyweighted second-order filters: one for the acceleration signal and the other for the positional signal. The velocity estimate then comprises a positional component and an acceleration component. The complementary nature of the two filters' transfer functions yields an exact all-pass characteristic, so the velocity estimate gives the true velocity in this transfer function analysis. In order to take account of the accelerometer offset, this approach uses a high-pass filter, whose pole is also one pole of the velocity observer. When this pole is close to the origin of the complex plane, the observer dynamics are limited, which produces slow transient responses. When this pole moves away from the origin, the observer dynamics are improved, but the sensitivity to highfrequency position noise, such as the position quantization error of an incremental encoder, increases. This paper presents an accelerometer-assisted velocity observer that is completely insensitive to the accelerometer offset. Compared with the two-channel approach, the proposed approach neither limits pole locations for the observer dynamics, nor is it vulnerable to high-frequency positional noise. The two-channel approach also requires the use of two second-order filters, but the proposed approach only needs to realize a third-order model, which allows simpler implementation and a lower computational cost. In this paper, various velocity estimation methods are used practically to control the position of a screw-based linear motion stage using an accelerometer and an incremental optical linear encoder. More specifically, the so-called Integral VariableStructure Control (IVSC), a switching control law, is used to determine the effectiveness of various velocity estimation methods.

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2. A revisit to state-space velocity observer (SSVO) The state-space velocity observer (SSVO) can be described by:   x_^ ðtÞ ¼ v^ ðtÞ þ l1 xm ðtÞ  x^ ðtÞ

ð2Þ

  v_^ ðtÞ ¼ am ðtÞ þ l2 xm ðtÞ  x^ ðtÞ ;

ð3Þ

where x^ ðtÞ denotes the positional estimate and l1 and l2 are constant observer gains. If xðtÞ;vðtÞ and aðtÞ denote the true position, velocity and acceleration, respectively, the positional and acceleration noises are defined as nx ðtÞ ¼ xm ðtÞ  xðtÞ and na ðtÞ ¼ am ðtÞ  aðtÞ, respectively, and the positional and velocity estimation errors are defined as e1 ðtÞ ¼ xðtÞ  x^ ðtÞ and e2 ðtÞ ¼ vðtÞ  v^ ðtÞ, respectively. To analyze the observer dynamics, differentiate the positional estimation error with respect to time, which gives e_ 1 ðtÞ ¼ x_ ðtÞ  x_^ ðtÞ ¼ vðtÞ  x_^ ðtÞ, and substitute (2) into the equation, which yields:   ð4Þ e_ 1 ðtÞ ¼ e2 ðtÞ  l1 xm ðtÞ  x^ ðtÞ ¼ e2 ðtÞ  l1 e1 ðtÞ  l1 nx ðtÞ: Similarly, substituting (3) into the time derivative of the velocity estimation error gives: e_ 2 ðtÞ ¼ aðtÞ  v_^ ðtÞ ¼  na ðtÞ  l2 e1 ðtÞ  l2 nx ðtÞ:

ð5Þ

If f ðsÞ is defined as the Laplace transform of a time-domain signal, f ðtÞ, that is to say, the s-domain representative of f ðtÞ, then neglecting initial conditions, the s-domain representation of (4) and (5) is: se1 ðsÞ ¼ e2 ðsÞ  l1 e1 ðsÞ  l1 nx ðsÞ

ð6Þ

se2 ðsÞ ¼  na ðsÞ  l2 e1 ðsÞ  l2 nx ðsÞ:

ð7Þ

Solving (6) and (7) for e2 ðsÞ gives: e2 ðsÞ ¼ 

s þ l1 l2 s na ðsÞ  2 nx ðsÞ: s þ l1 s þ l2 s2 þ l1 s þ l2

ð8Þ

Since e2 ðsÞ ¼ vðsÞ  v^ ðsÞ, then: v^ ðsÞ ¼ vðsÞ þ

s þl1 l2 s na ðsÞ þ 2 nx ðsÞ: s þ l1 s þl2 s2 þ l1 s þ l2

ð9Þ

Therefore, the characteristic equation for the SSVO is s2 þ l1 s þ l2 ¼ 0; from which the observer gains, l1 and l2 , can be determined. The transfer function from na ðsÞ to v^ ðsÞ is ðs þ l1 Þ=ðs2 þ l1 s þ l2 Þ, which demonstrates that when there is an accelerometer offset, the velocity estimate drifts. That is, the SSVO is unable to eliminate any adverse effects of an accelerometer offset.

3. A dynamically compensated velocity observer (DCVO) This paper presents a dynamically compensated velocity observer (DCVO). The structure of the DCVO is presented in Fig. 1, where CðsÞ is a dynamic compensator that shapes the DCVO's dynamics. The discrepancy between the measured and the estimated position signals is defined as e ¼ xm  x^ and the output of the compensator is defined as μðsÞ ¼ CðsÞeðsÞ. For the structure of am

xm

Fig. 1. Dynamically compensated velocity observer (DCVO).

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the DCVO shown in Fig. 1,

Ball screw fixed to the frame

  sv^ ðsÞ ¼ am ðsÞ þ μðsÞ ¼ am ðsÞ þ CðsÞeðsÞ ¼ am ðsÞ þ CðsÞ xm ðsÞ  x^ ðsÞ :

Brushless permanentmagnet motor

ð10Þ Using the relationship, x^ ðsÞ ¼ v^ ðsÞ=s, (10) is rewritten as:   CðsÞ v^ ðsÞ ¼ aðsÞ þ na ðsÞ þ CðsÞðxðsÞ þ nx ðsÞÞ: sþ s

ð11Þ

Payload Scale of the optical linear encoder

Substituting the relationship, aðsÞ ¼ s2 xðsÞ, into (11) gives:   s2 þ CðsÞ v^ ðsÞ ¼ s2 þ CðsÞ xðsÞ þ na ðsÞ þ CðsÞnx ðsÞ: s

ð12Þ Slider of the optical linear encoder

Finally, because vðsÞ ¼ sxðsÞ, (12) is rearranged as: v^ ðsÞ ¼ vðsÞ þ

s sCðsÞ na ðsÞ þ 2 nx ðsÞ; s2 þ CðsÞ s þ CðsÞ

ð13Þ

which describes the velocity estimate as a function of the true velocity and the acceleration and positional measurement noises. In principle, there is a wide choice of dynamic compensators, CðsÞ. In this paper, the dynamic compensator has the form: CðsÞ ¼

c2 s þc1 ; s þ c3

ð14Þ

Accelerometer circuitry Fig. 2. Photo of the experimental linear motion stage.

s3 þ c

s2 þ c 3 s c2 s2 þ c1 s na ðsÞ þ 3 nx ðsÞ: 2 s þ c3 s2 þc2 s þ c1 3 s þ c2 s þ c1

Linear motion stage

Regulated current converter

ð15Þ

This equation shows that the characteristic equation of the DCVO is s3 þ c3 s2 þ c2 s þc1 ¼ 0: Therefore, the compensator parameters, c1 , c2 and c3 , can be decided using, for example, the poleassignment technique. From (15), it is also    found that the transfer  function from nx ðsÞ to v^ ðsÞ is c2 s2 þ c1 s = s3 þ c3 s2 þ c2 s þ c1 ; and its magnitude response approaches zero as the frequency approaches infinity. In the so-called two-channel approach [21], the corresponding magnitude response approximates to a constant for large frequencies, and this constant equals the sum of the observer gains. A small observer gain results in low sensitivity to high-frequency positional noises, but slow observer dynamics. Increasing the observer gains increases the observer bandwidth, but leads to an increased sensitivity to high-frequency positional noises. In contrast to the two-channel approach, the proposed DCVO is insensitive to positional noises at large frequencies and this complete insensitivity at infinite frequency is a robust property because there is no dependence on the compensator parameters. The transfer function na ðsÞ to v^ ðsÞ can also be    from obtained from (15), as s2 þ c3 s = s3 þ c3 s2 þ c2 s þ c1 . When there is an acceleration offset, e.g. na ðtÞ ¼ 1; it gives na ðsÞ ¼ 1=s and the corresponding deviation of the velocity estimate from the true   velocity is ðs þ c3 Þ= s3 þc3 s2 þ c2 s þ c1 . According to the final value theorem, the of the velocity estimate is  steady-state deviation  limsðs þ c3 Þ= s3 þ c3 s2 þ c2 s þ c1 , demonstrating that the deviation

Payload

Motor

where c1 , c2 and c3 are constant compensator parameters. Substituting (14) into (13) yields: v^ ðsÞ ¼ vðsÞ þ

Linear encoder

A, B-phase signals

Accelerometer Personal computer

u ADC/DAC

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of the velocity estimate converges to zero when an accelerometer offset appears. In contrast to a SSVO, the DCVO allows no velocity drift when there is an accelerometer offset.

4. Experimental study 4.1. Experimental system Figs. 2 and 3 respectively show a photo and a schematic of the experimental system, in which a rotational, brushless permanentmagnet synchronous motor is accompanied by a regulated current converter that operates in torque mode. The motor drives a screwbased linear stage that, in turn, produces a linear motion for a payload. An accelerometer and an incremental optical linear encoder are

Fig. 4. Schematic of the FPGA interconnection and functionality.

both attached to the payload, to measure its acceleration and position, respectively. The accelerometer is a model ADXL320 from Analog Device, Inc., with a full-scale range of 75 g. The optical linear encoder is a model WTB5-0600MM from Carmar Technology Co., Ltd., with a grating pitch of 20 μm. As shown in Fig. 4, a fieldprogrammable gate array (FPGA) of model XCV50PQ240-C6 from Xilinx, Inc., receives A- and B-phase signals from the encoder and implements a quadrature decoder, which multiplies the resolution of digitally filtered A- and B-phase signals by a factor of four and gives a positional resolution of 5 μm. The quadrature decoder produces pulse and direction signals to an integral position counter and an interval counter. With a clock of 100 MHz, the interval counter

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position (mm)

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Fig. 5. Flowchart of an ISR with the DSP.

measures the time between two successive pulses for velocity estimation using the ITM. The FPGA also reads the accelerometer signal through a 16-bit analog-to-digital converter (ADC) and is able to send a control effort to the regulated current converter through a 12-bit digital-to-analog converter (DAC) and analog amplifier circuits. The controller/observer core is a TMS320C6713 digital signal processor (DSP), and an external memory interface (EMIF) connects the DSP with the FPGA. A flowchart of the interrupt service routine (ISR) with the DSP is provided in Fig. 5, in which ADOB denotes an accelerationbased disturbance observer [22]. In the ISR with a sampling rate of 12.2 kHz, the DSP reads the positional and acceleration information from the FPGA, calculates the controller/observer algorithms and sends the control effort through the FPGA. In the experimental system, a personal computer is used to develop the control program, to download the code to DSP and to acquire experimental data. The control input to the plant is actually a torque command to the regulated current converter, which is denoted as u in the form of voltage. Using the swept-sine technique, a dynamic signal analyzer measures the frequency response of the plant. Curvefitting the frequency-response data produces the transfer function for the plant, and the plant is modeled as: ð16Þ

where a ¼ 10:89; b ¼ 13449:58;xðtÞ denotes the payload position in mm, uðtÞ is the torque command in V, and dðtÞ represents an unknown input disturbance to the plant. 4.2. Observer design and open-loop experimental results The proposed DCVO is compared with a common ITM estimator and a SSVO. To realize the ITM, the FPGA implements a quadrature decoder that increases the resolution of the A- and Bphase signals by a factor of four, and measures time interval for two consecutive pulses of the quadrature decoder's output using a 100 MHz clock. The DSP reads the time-interval information from

velocity (mm/s)

End of the ISR

x€ ðtÞ þ ax_ ðtÞ ¼ bðuðtÞ þ dðtÞÞ;

8

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t (s) Fig. 7. Open-loop experimental results for the SSVO and the DCVO.

the FPGA and performs the inverse-time operation to give an ITMbased velocity estimate. For the SSVO, the observer gains, l1 ¼ 2  300 and l2 ¼ 3002 , are chosen so that its eigenvalues are  300 with a multiplicity of two. For the proposed DCVO, the characteristic polynomial, s3 þ c3 s2 þ c2 s þ c1 , is specified to be ðs þ 6ωn Þðs2 þ 2ωn s þ ω2n Þ, where both dominant poles are located at  ωn and the third pole is six times further away from the imaginary axis. The parameter, ωn ¼ 83:2, is determined so that both the SSVO and the DCVO have the same high-frequency sensitivity to measurement noise. HðtÞ denotes the Heaviside step function. An input voltage, uðt Þ ¼ 0:2HðtÞ V, was applied to the plant that was initially at rest. Fig. 6 shows the open-loop experimental results for the ITM and the DCVO. Because of the accelerometer, the DCVO produces a different velocity estimate even when the linear encoder detects no position variation. Before the second encoder pulse appears, the ITM yields a velocity estimate of zero but the DCVO reacts to this low-velocity motion. The velocity estimate for the DCVO leads and is also smoother than that for the ITM, even when the speed increases. To determine the effect of an accelerometer offset on velocity estimation, an accelerometer offset of  0:2 V was added to the acceleration signal and the same open-loop experiment was used again. Fig. 7 shows the estimation results for the SSVO and the DCVO, both with and without the accelerometer offset. It is seen that the accelerometer offset causes a drift in the velocity estimation for the SSVO but the proposed DCVO asymptotically rejects the effect of this accelerometer offset.

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Fig. 10. ADOB-augmented experimental results for the DCVO.

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n o ^ ¼ Q b  1 x€ þ ax_   u , in which Q f U g disturbance estimate, dðtÞ denotes a low-pass filter. A second-order low-pass Butterworth filter with a cutoff frequency of 100π rad/s is used in the ADOB. The ADOB-augmented control is the IVSC (17) subtracted by the ^ disturbance estimate, dðtÞ. Fig. 10 shows the dynamic responses for the DCVO and the ADOB. It is seen that the acceleration measurement can be used for both velocity estimation and disturbance compensation.

5. Conclusions

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4.3. Controller design and closed-loop experimental results A switching-type control, the so-called Integral Variable Structure Control (IVSC) [23], is used to investigate the effect of different velocity estimation schemes on control performance. The tracking error is defined as e ¼ x  r, in which the positional reference r ¼ 10 mm. A switching function is defined as s ¼ v^ þ k1 e þ k0 z, where k1 and k0 are positive design parameters Rt and the integral variable, z ¼ 0 edt þ z0 . The integral variable's   1 initial value, z0 , is assigned as z0 ¼  k0 v^ ð0Þ þ k1 eð0Þ , so that the switching function is initially zero. The IVSC law is designed as:      ð17Þ u ¼ αv^  β^ k1 v^ þ k0 eþ k2 s  D þ Δβ k1 v^ þ k0 e sgnðsÞ   where sgnð UÞ denotes the sign function, β^ ¼ β max þ βmin =2;  Δβ ¼ βmax  βmin =2;βmax ¼ 1:4=b; βmin ¼ 0:6=b; α ¼ a=b; D ¼ 0:1, k1 ¼ 2ωn ; k0 ¼ ω2n ; k2 ¼ ωn ; and ωn ¼ 30. In these experiments, the plant is initially at rest; i.e. xð0Þ ¼ 0 and x_ ð0Þ ¼ 0:Figs. 8 and 9 show respectively the dynamic responses for the ITM-based observer and the DCVO. It is seen that the DCVO yields a more precise position than the ITM-based observer. In view of the control effort in the steady state, the control effort for the DCVO switches also faster than that for the ITM, so the DCVO reacts more quickly to velocity changes and enables the IVSC to correct the output error almost immediately. In addition to estimating the velocity, the acceleration measurement can be used to compensate for disturbance. For the experimental system (16), the ADOB gives a

The DCVO combines positional and acceleration measurements to estimate velocity and contains a dynamic compensator that shapes the observer dynamics. In this paper, a conventional SSVO is compared with the DCVO, both analytically and experimentally. The results show that in contrast to the SSVO, the DCVO is completely insensitive to an acceleration offset, which regularly happens during acceleration measurement. The DCVO also practically outperforms a common ITM-based estimator in velocity estimation, in terms of smoothness and phase lead. Since the DCVO is more sensitive to low-speed motion than the ITM-based estimator, the DCVO allows high-frequency switching control and gives more precise positioning, in practice. The experimental results also show that the ADOB can further augment the control system and improve transient response during the positioning process.

Acknowledgments This work was supported by the Ministry of Science and Technology of ROC under Grant number MOST 102-2221-E-003020-MY2.

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