CEAI, Vol.17, No. 3, pp. 30-40, 2015 Printed in Romania

On the Velocity and AccelerationEstimation from Discrete Time-Position

Sensors ?

L.J. Puglisi, R.J. Saltaren, C.E. Garcia Cena

Centro de Automatica y Robotica, UPM-CSIC,Jose Gutierrez de Abascal, Madrid, 28006, Spain

(e-mail: lisandro.puglisi@ alumnos.upm.es, rsaltaren@etsii.upm.es,cgarcia@upm.es).

Abstract: In this work is addressed the topic of estimation of velocity and acceleration fromdigital position data. It is presented a review of several classic methods and implemented withreal position data from a low cost digital sensor of a hydraulic linear actuator. The results areanalyzed and compared. It is shown that static methods have a limited bandwidth application,and that the performance of some methods may be enhanced by adapting its parametersaccording to the current state.

Keywords: digital encoder, velocity estimation, acceleration estimation, Kalman Filter

1. INTRODUCTION

Many work has been done in the field of estimation ofvelocity and acceleration from digital encoder data (Brownet al. (1992), Merry et al. (2010), Liu (2002), Ovaska andValiviita (1998), Baran et al. (2010), Song et al. (2009)).However, most of them are focussed on angular sensorsand scarce information regarding their implementation inlinear position encoder can be found.

In typical applications, angular sensors are not directlycoupled to the measured variable instead there’s a gear be-tween the encoder and the angular variable. Consequently,the constraints imposed by the resolution of the encoderis somehow diminished, since the relative variation of theangular variable is mapped into the encoder amplified bythe multiplication factor of the gear.

A linear position sensor is coupled directly to the measuredvariable and it is not possible to make use of this treach-erous enhancement. In general, the resolution of mediumcost sensors is allocated in the 0.1 − 0.5mm range. Eventhough this resolution could be considered appropriate forgeneral applications, the implementation of these sensorsin low velocity tasks brings several complications, espe-cially when the control loop depends on the estimation ofvelocity and its time derivatives (Han et al. (2007), Choiand Jung (2011), Emaru et al. (2009), Ohmae et al. (1982),Tsuji et al. (2005)).

Therefore, in this work is presented a review of the classicmethods used for the estimation of velocity and accelera-tion and they are implemented on a digital linear positionsensor. Several concepts are introduced and combined pro-viding more than twelve alternatives for the estimation of

? The authors would like to thank the financial support of theSpanish Government through the CICYT Project Ref. DPI2009-08778 and also to Comunidad de Madrid who supports the projectROBOCITY2030-II Ref. P2009/DPI-1559.

velocity and acceleration. This survey could be though asa small tutorial for those readers that are beginning toexplore the field of velocity and acceleration estimationremarking the pros and cons of each method.

This work is organized as follows. Firstly, a brief introduc-tion to the problem associated to digital signals and thecalculation of its time derivatives is presented. Followingthis, some of the most common methods for velocity andacceleration estimation are introduced and tested with realdata. An objective performance comparison is presentedby introducing the definition of some Key PerformanceIndexes and finally, the results are discussed.

2. ENCODER SIGNAL ANALYSIS

In order to understand the problem associated with theestimation of velocity and acceleration from a digital posi-tion signal it is better to clarify some of its characteristics.

First of all, when dealing with the information providedby a digital sensor, there are two main issues that must beremarked:

(1) The resolution of the encoder is finite. Its outputcan only take some defined states, therefore the truevalue of the measured variable resides between twoadjacent states. The output of the encoder should notbe interpreted as the true value.

(2) The information corresponds only to a finite sampleof time. There’s no true notion of what’s happeningbetween samples.

These characteristics are clearly depicted in Fig.1(a). Thereal data is presented in black line and the output ofthe encoder as red dots. As it can be observed, thesignal obtained from the encoder clearly shows its discretenature caused by the finite resolution and consequentlythe uncertainty of the measurement (represented in grey).

Control Engineering and Applied Informatics 31

The problem with the finite resolution of the encoder isthat for slow variations it delivers the same informationfor several consecutive samples, until the variation exceedsits resolution, presenting a laddered type signal (observethe red dots).

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Fig. 1. Digital position signal analysis

Based on the afore mentioned, the output of the encoderyk for every sample-time k can be expressed as (Shaoweiand Shanming (2012)):

yk =

⌊y

er

⌋er, (1)

where b•c is the rounding to floor operator, and the errorof the measurement is given by yerr = yk − y, withyerrε (−er, 0] and er is the resolution of the encoder.

Let us consider the output of the encoder presented asred dots in Fig.1(a), and let us apply the backward finitedifference method (FD) to obtain its time derivative.The result is presented in Fig.1(b) in red dots, and itcan be observed that two consecutive identical positiondata result in a zero flat line, while a change in theposition turns into a spike. This estimation of the firsttime derivative does not resemble at all the real velocity(see black curve).

On the other hand, since the data has an intrinsic errorgiven by the finite resolution of the encoder it can be easilyfound, by error propagation theory, that the estimatedvelocity will have an error bounded to yerrε(−er/T, er/T )(represented in grey in Fig.1(b)) , where T is the sampletime.

From the bounded error it can be observed that the errorof the estimated time derivative is related proportionallywith the resolution of the encoder, therefore to reduce itthe resolution should improve. It can also be extractedthat by reducing the sampling period T , the bounded errorincreases.

3. EXPERIMENTAL SETUP

The experimental setup used in this work is a linearhydraulic actuator with a low cost digital linear position

sensor LX-EP40 with a resolution of 2.45 counts/mm. AdSPACE DS1103 board is implemented as the controlhardware, it captures data from the position sensor (yk),and it also provides the control signal (xv) to the servo-valve (see Fig. 2).

In order to objectively analyze the performance of eachmethod, the following Key Performance Indexes (KPI) aredefined:

(1) Average error:

KPI1 =1

n

n∑k=1

(θk − θk). (2)

(2) Absolute maximum relative error:

KPI2 = maxkε[1,n]

∣∣∣∣∣θk − θkθk

∣∣∣∣∣. (3)

(3) Average relative error:

KPI3 =1

n

n∑k=1

∣∣∣∣∣θk − θkθk

∣∣∣∣∣. (4)

Where θk is the real data and θk is the estimated variableat the kth sampled time. These KPI are evaluated for halfa period (discarding the first and last 25 samples in orderto avoid θk ≈ 0 and thus KPIi ≈ ∞), of the referencetrajectory: y = 100 + 25 sin(2πf), with f = 0.1Hz,f = 0.5Hz and f = 1Hz (y is in mm), implementing aproportional position controller.

linear encoder

servo-valve

piston's rod

external load

linear guide

cylinder

Test-bed

pressure lines

Hydraulic station

DS1103 board

PC

xv

y

Fig. 2. Experimental Setup.

4. VELOCITY ESTIMATION

In this section, it is introduced some of the techniquesimplemented for the estimation of velocity. Hereafter, ykis the output of the sensor and •k is the estimation for •at the kth sampled time.

32 Control Engineering and Applied Informatics

4.1 Finite Difference and Filtering

The most simple practice for the estimation of the velocityand acceleration is to implement finite difference (FD) onthe discrete time position data and then filter the resultwith a low pass filter (LPF).

The first order FD applied to the position data (in itsbackward formulation), leads to the following expression:

yk =yk − yk−1

T. (5)

The analysis presented in (Belanger (1992)) suggests thatexists an optimal sampling time to estimate the velocityand acceleration with FD, though in this work the sampletime period is settled to T = 1ms for all the methods.

The result of the implementation of the FD over the rawdata (yk) is presented in Fig.5(a) - Fig.5(c) in red. Asit was mentioned before, the estimation depends on thechange between two consecutive samples. Therefore, inthe low-speed range the estimation is rather poor andit improves while increasing the real velocity (i.e. moreposition changes between two consecutive samples). Infact, if the real velocity is y > er

T , the estimation withFD will be very accurate with a time-lag T .

For the improvement of the estimation on the low-speedrange a LPF is implemented. The design of the LPF iscrucial for the performance of the estimated variable. Acommon practice is to implement Butterworth filters sincethey provide maximally flat magnitude for the wantedfrequencies. The cut-off frequency of the filter must beselected depending on the frequency range of the applica-tion.

In Fig.5(a) - Fig.5(c) are presented the estimation ofvelocity implementing a 4th order Butterworh LPF withcut-off frequencies: fc = 10Hz (blue curve), fc = 1Hz (cyancurve) and fc = 0.1Hz (green curve).

As it can be seen, the filtering with fc = 10 Hz, does noteliminate the noise introduced by the derivative actionof the FD in the low velocity range (see blue curve inFig.5(a)). On the other hand, for greater velocity thefiltering seems to work properly (i.e. it eliminates thenoise), but it introduces a time-lag in the estimatedvelocity (see the blue curve and compare it with the realvelocity in Fig.5(b) and Fig.5(c)).

On the pursuit of the suppression of the noise in the low-speed range, it is implemented a filter with fc = 0.1 Hz.The action of this filter eliminates the noise in the lowvelocity range (see green curve in Fig.5(a)), but it alsoaffects the amplitude and time-lag of the estimation forhigher velocities (see green curve in Fig.5(b) and Fig.5(c)).

However, if the frequency range of work of yk is restrictedto a narrowed bandwidth, an acceptable performance canbe achieved by selecting the cut-off frequency at the midof this bandwidth. Even though, it must be remarked thatthe estimation will have some noise in the low-velocityrange and time-lag for the high-velocity range.

4.2 Finite Difference from encoder events

The problem that arises from the finite resolution of theencoder is that for slow motions the signal provided by

the encoder delivers the same data for several consecutivesamples. In order to overcome this issue two classic meth-ods based on two different sampling approaches (Goodwinet al. (2013), Astrom and Bernhardsson (2002)) are com-monly implemented (Tsuji et al. (2005)):

• M method (fixed-time method): it counts the numberof pulses from the optical encoder during a fixedinterval of time.

• T method (fixed-position method): it measures thetime that it takes to count a predefined number ofpulses.

The latter one provides better results in the low-speedrange, while the first one deteriorates it.

Based on the T-method approach and defining only onepulse to count, it is detected the minimum change ofthe encoder. This minimum change and the time of itsoccurrence is defined as an encoder event (EE) (Merryet al. (2010)) and it is given by the pair (tee, yee), i.e.the time when the event occurs and the current measure,respectively. An analysis of event sampling for nonlinearfiltering is presented in (Cea and Goodwin (2012)).

It must be recalled that in this work the sampling timeis constant, therefore tee = nT , where n is the number ofconsecutive samples until an event is detected.

By definition every EE represents a change on the outputof the encoder, thus there’s no identical consecutive EE(see the blue dots in Fig.3(a)). Hence, when the FDmethod is implemented over the EE, the estimation resultsin a spike-free curve as shown in Fig.3(b) in blue (compareit with the red dots in Fig.1(b)). On the other hand, it canalso be observed that the estimation is much more likethe real velocity (black curve in Fig.3(b)), even though itpresents a laddered wave form.

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Fig. 3. Definition of encoder events. 3(a): position data(real position in black line, encoder’s signal in red,and EE in blue dots). 3(b): Velocity estimation im-plementing FD over EE. (real velocity in black andvelocity estimation in blue dots.

The concept of EE with FD is implemented on the positiondata and the results are depicted in Fig.5(d) - Fig.5(f) in

Control Engineering and Applied Informatics 33

red. As it is observed, there still exists some spurious spikesnear the region of low accelerations (maximum velocity).In the low-speed region it is appreciated that the signalis distorted. This is caused by the fact that there’s a longperiod of time without detecting any new event (see Fig.3).

The spikes on the low-acceleration region can be elim-inated by smoothing the estimation by skipping someevents in the FD, providing a filtering effect (Merry et al.(2010)). This approach is presented in Fig.5(d) - Fig.5(f),by skipping one (blue curve), five (green curve) and ten(cyan curve) consecutive events.

As it can be seen, the skipping option provides filteringto the estimation (compare them with the red curve). Thequestion now is how many events must be skipped in orderto provide a good estimation. In these figures, it can beobserved that skipping one event is not enough (observethe blue curves), and the smoothness of the estimationimproves when more events are skipped. However, skip-ping events leads to time-lags in the estimated velocity,thus there is always a compromise between smoothnessand time delays. An alternative could be to skip eventsaccording to the last estimated velocity as proposed in(Liu (2002)), providing adaptability to the method.

4.3 Polynomial Fitting

The lack of information between the states of the encodercould be surpassed by estimating the real value of themeasured variable by fitting yk to a polynomial of order n(Merry et al. (2010), Brown et al. (1992)), as follows:

yk = cntnk + cn−1t

n−1k−1 + · · ·+ c1tk−n−1 + c0, (6)

where yk is the estimation of yk at sample k, and tk =k T, ∀k = 1, 2, · · · , n is the time sample. The coefficientscn, · · · , c0 are chosen to minimize ‖yk− yk‖2. Based on thepolynomial fitting, the estimated velocity can be found as:

ˆyk = ncntn−1k + (n− 1)cn−1t

n−2k−1 + · · ·+ c1. (7)

The polynomial fitting can be extended to the m mostrecent data points:

y =Tc (8)ykyk−1

...yk−m

=

tnk tn−1k−1 · · · tk−n−1k−n−1 1

tnk−1 tn−1k−2 · · · tk−n−1k−n−2 1...

... · · ·...

...tnk−m tn−1k−m−1 · · · t

k−n−1k−m−n−1 1

cncn−1

...c0

.The coefficient vector c that minimizes ‖yk− yk‖2 is givenby:

c = (T TT )−1T Ty. (9)

The challenge of the polynomial fitting is to find the ap-propriate order of the polynomial and number of previouspoints to use. The higher the order and number of pointsthe better is the accuracy of the fitting, however the lag-time and the computation time will increase.

In this work is implemented a third order polynomial overfour data points and the quality of the estimation is ratherpoor (see the red curves in Fig.5(g) - Fig.5(i)). This is dueto the fact that the fitting is performed over data with thesame information.

This problem leads once again to the concept of EE. In thesame charts, it is presented the estimation using polyno-mial fitting over the EE considering three and five skippedEE (blue and cyan curves, respectively) providing betterresults. The estimation will improve with the quantity ofskipped events. However if they are compared with theestimation obtained with FD with EE, (see Fig.5(d) -Fig.5(f)), it can be observed that they have worse noiseto signal ratio.

The poor efficiency of the estimation based on the poly-nomial fitting, is that it requires of the analysis of thecontinuity and trend of the higher order derivatives whenconsecutive data points do not add new information, lead-ing to the use of spline or nurbs approaches which requiremuch computation efforts.

4.4 Estimation with Kalman Filter

The Kalman Filter (KF) is perhaps the most used tech-nique for the estimation of state variables of a system. Thisrecursive filter, which encloses the least square estimatorproposed long ago by F. Gauss (Sorenson (1970)) was firstpublished in 1960 (Kalman (1960)). Since then, it has beenimplemented in many applications and several formula-tions were derived from the former one (e.g. Extended KF(Einicke and White (1999), Ligorio and Sabatini (2013),Perala and Piche (2007), Quine (2006)) , Unscended KF(Julier and Uhlmann (2004),LaViola (2003)), among oth-ers), but keeping the core concept of a two step predictor-corrector type estimator that minimizes the estimatederror covariance.

The KF tries to estimate the state x ∈ Rn of a discrete-time controlled process that is governed by a linearstochastic difference equation with a measurement z ∈Rm, given by the following general expressions:

xk =Axk−1 + Buk + wk−1, (10)

zk =Hxk + vk, (11)

where uk is the control input and k is the time sample. Therandom variables wk−1 and vk represent the process andmeasurement noise respectively, and it is assumed thatthey are independent with normal probability distribu-tions, (i.e. p(wk) ∼ N(0,Qk) and p(vk) ∼ N(0,Rk), whereQk and Rk are the corresponding covariance matrices).

The estimation problem is solved recursively by imple-menting the following sequential relations at each sampletime (Welch and Bishop (2001)):

x−k =Axk−1 + Buk, (12)

P−k =AP k−1AT + Q, (13)

Kk =P−kH

T

HP−kHT + R

, (14)

xk =x−k + Kk(zk −Hx−k ), (15)

P k =(1−KkH)P−k , (16)

For the particular case of velocity and acceleration es-timation from position data, the process model can beeasily derived by considering the Taylor series expansionof each state variable and assuming that the plant noise isthe discrete representation of the remainder of the Taylorseries (Zhou et al. (2008), Sabatini (2003), Shaowei and

34 Control Engineering and Applied Informatics

Shanming (2012), Belanger (1992)). Therefore, the processand measurement models are given by:ykˆyk

ˆyk

=

1 T T 2/20 1 T0 0 1

[ykykyk

]+

T 3/6T 2/2T

, (17)

yk+1 =yk + vk, (18)

and the covariance matrices are:

Q =q

T 5/20 T 4/8 T 3/6T 4/8 T 3/6 T 2/2T 3/6 T 2/2 T

, (19)

R =e2r/4 + 2r

3, (20)

where r = E[εi] and εi is the error associated to themeasurement. A comprehensive study of the stochasticcomponents of the model is presented in (Belanger (1992)).

The position data is filtered with the KF and is presentedin red in Fig.5(j) − Fig.5(`).

As it can be observed, the estimation can’t keep itsperformance along all the range of work. In order toenhance the performance of the KF many authors proposeto modify the noise covariance matrices considering thelast estimation (Shaowei and Shanming (2012), Zhou et al.(2008)), or rescaling the gain of the filter (Duan et al.(2003)) turning it into an adaptive filter. In this work, itis proposed the following adaptive law:

Rak =10R

1 + ˆy2k−1. (21)

This adaptation law allows to preserve the performanceof the estimation along the full range of work providingbetter result as it can be observed on the blue curves inFig.5(j) - Fig.5(`).

Even though this formulation gives a suitable estimationit may not be appropriate for embedded real time system,since the third order nature of the expressions requiresof time consuming computation. Therefore, some authorspropose to use a reduced order system (Shaowei and Shan-ming (2012), Sabatini (2003)). More precisely, in (Shaoweiand Shanming (2012)) is implemented the following firstorder model, which leads to a Single Dimensional KalmanFilter (SDKF):

θk =θk−1 + wk−1, (22)

θk+1 =θk + vk. (23)

The implementation of the SDFK leads to two new ap-proaches, one that implements the KF to estimate theposition and the other to estimate the velocity.

Kalman Filter for position estimation In this approach,the SDKF is used to estimate the real position, thusθk = yk. The position estimation with the SDKF providesa smoother signal, as it can be observed in Fig. 4. Fromthis estimated position the velocity can be estimated byany of the previous techniques mentioned.

In particular, the invariant sampled time result from theSDKF makes it suitable to implement the Savitzky-Golay(SG) method. This method provides a polynomial fittingwith constant coefficients obtained from specific tablesfor digital signals with invariant sample time (Madden(1978), Gorry (1990), Schafer (2011)). Therefore, there is

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Fig. 4. Position estimation using SDKF. The real positionis in black, the measurements from the sensor is inred, and the position estimation implementing SDKFis in blue.

no need to obtain the coefficients for each new incomingdata reducing the computation time. Some properties ofthe SG method as digital differentiator are presented in(Luo et al. (2005))

The SG method is implemented with a fourth orderpolynomial fitting with ten points and the estimation ispresented in magenta in Fig.5(j) - Fig.5(`).

As it can be seen, the velocity estimation has improvedcomparing it with the results obtained with the FDmethod with the raw data (see Fig.5(d)) and the poly-nomial fitting with the EE (see Fig.5(g)).

Kalman Filter for velocity estimation In this approach,θk = ˜yk, where ˜yk is obtained from the FD method overEE. For this particular example it is implemented the FDmethod skipping two events.

The estimated velocity with the SDKF provides a smootherestimation than the FD method with practical no time-lag,as it can be appreciated in cyan in Fig.5(j) - Fig.5(`).

4.5 Sliding Mode Differentiation

The main problem with differentiation is to combine ac-curacy and robustness in face of possible measurementerrors and input noises. In (Levant (1998)) is proposeda robust first-order differentiator based on the slidingmode technique which features finite-time convergence androbustness against the measurement noise. This techniqueis implemented in (Damiano et al. (2004)) for the esti-mation of velocity and acceleration for a DC-drive motorcontroller.

The discrete formulation of the differentiator is given by:

hk =− λ|zk − xk|1/2sign(zk − xk) + wk, (24)

wk =wk−1 − Tαsign(zk−1 − xk−1), (25)

zk =zk−1 + Thk−1, (26)

where xk is the measurable signal, w0 = z0 = 0, and αand λ are set according to the following inequalities:

α > Xdd, (27)

λ ≥ 2

√Xdd

α+Xdd

α−Xdd, (28)

and Xdd > 0 is the upperbound for the magnitude of thesecond derivative.

The Sliding Mode Differentiator (SMD) is implementedwith real data and the velocity estimation is presented inred in Fig.5(m). The parameter λ and α are selected toprovide the best result possible for low-speed range (see

Control Engineering and Applied Informatics 35

Fig.5(m) - Fig.5(o)). However, the estimation deterioratesfor higher velocities (see Fig.5(n) and Fig.5(o)).

In order to enhance the performance of the estimation, itis considered a variable acceleration upperbound:

Xak = Xdd

∣∣∣ ˆyk − ˆyk−1T

∣∣∣. (29)

This small modification provides of adaptability to themethod and the result improves along all the range ofwork as it can be observed in blue in Fig.5(o), howeverthe estimation for the low speed range becomes more noisy(compare it with the red curve).

5. ACCELERATION ESTIMATION

In order to estimate the acceleration there are two possibleapproaches regarding the model implemented, i.e. theacceleration can be thought as the second time-derivativeof the position or it can also be though as the first time-derivative of the velocity. The latter one, leads to thesuccessive implementation of any of the first order methodspresented in the previous section.

5.1 Finite Difference and Filtering

The most common method implemented to estimate theacceleration is also the finite difference method. Its formu-lation for the second order is given by (henceforth FD2):

yk =yk − 2yk−1 + yk−2

T 2. (30)

As it was shown in the velocity estimation section, workingwith FD over the raw data, doesn’t lead to suitable results.Therefore, for the acceleration estimation it is consideredonly EE.

In Fig.8(a)-Fig.8(c) are presented the estimation usingonly FD2 over the EE considering two (red curve) andfive (blue curve) skipped events. This formulation leads tovery poor estimation. Therefore, the implementation of aLPF is mandatory for the acceleration estimation. As itcan be seen in the green and cyan curves (filtered signalof the FD2 over EE with two and five skipped events,respectively), the filter improves the estimation but still itis not accurate enough for any application.

5.2 Kalman Filter

In this section is implemented several combinations of theKF to estimate the acceleration.

The raw position data is filtered implementing a SDKF aspresented in Fig.4, and then it is submitted to a FD2 oper-ation. The result is presented in red in Fig.8(d) - Fig.8(f).As it can be seen, the estimation is poor, especially inthe low-speed range, where the estimated signal is farfrom resemblance the real acceleration signal. Therefore, asecond SDKF is implemented over the estimated signal,whose result is presented in blue in the same figures.Even though, the result improves considerably for highvelocities, the estimation for the low-speed range still isunacceptable for any purpose.

As it was already presented, the filtering improves byconsidering only those states of the encoder that represent

a change. Therefore, the FD2 estimation is implementedon the EE and then the signal is filtered using a SDKF.The result shows that the estimation improves (see cyancurve in Fig.8(d) - Fig.8(f)), though the signal still isfar to be used in a control loop. Thus, another SDKF isimplemented over this first estimated data, and the newestimation is presented in green in the same charts. Theestimation improves drastically its performance in the low-speed range and in the high-speed range, though time-lagappears in the estimated data.

The SDKF as it was mentioned is a simplified formulationthat saves computation efforts, by implementing a firstorder model. This simplification of the model reducesthe performance of the estimation. This can be seen bysimple comparison between the results obtained with theSDKF and the KF with the third order model presented inmagenta in Fig.8(d) - Fig.8(f). This last result shows muchbetter performance for all the range of velocity, despite thefact that for high velocity there is a considerably time-lag. However, if the adaptation rule (21) for the noisecovariance matrix is implemented, the time-lag reducesconsiderably and the overall performance of the estimationenhances, as it can be observed in the yellow curve.

5.3 Sliding Mode Differentiation

Based on the estimator proposed in (Damiano et al.(2004)), the acceleration is estimated as a successivederivation using a first order SMD. The result is presentedin red in Fig.8(g)-Fig.8(i). It can be observed that theresult is not as promising as with the velocity estimation.The best estimation for the low-speed range has an unsat-isfactory noise to signal ratio estimation, and furthermorefor higher velocities, the acceleration estimation presentssevere deformation and time-lag. Considering the adaptivebehavior on the upper-bound parameter Xdd of the filter,it can be seen that the estimation does not improve (seeblue curve), and it remains practically the same.

5.4 Integration within a Control Loop

In general, the second order time derivative of the positionis the most implemented relation for the estimation of theacceleration. Even though, the implementation of differ-entiators inherently carries with them noise amplificationin the high frequency range, therefore its performance willbe always limited.

It must be recalled that the differentiation is not theonly relation that links these two variables, there is alsothe integration. Therefore, some authors have developedapproaches based on the implementation of integratorswithin a closed-loop. One example can be found in (Tilliand Montanari (2001)), where feedforward with a inte-grator feedback is proposed as velocity and accelerationestimator.

In this work two different approaches are analyzed.

The first approach considers the estimation problem as aclassical position control of a ordinary double integratortype system, as presented in Fig.6. This approach has asmany alternatives as possible controllers. A comprehensivestudy and comparison between several control strategies

36 Control Engineering and Applied Informatics

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(n)

0 0.5 1 1.5−200

−100

0

100

200

y

t

(o)

Fig. 5. Velocity estimation with different methods for different velocity ranges: first column f = 0.1Hz, second columnf = 0.5Hz. and third column f = 1Hz.

1

s

ˆy yy 1

s

ˆy

Double integrator

Controller

Fig. 6. Acceleration estimation based on a controlleddouble integrator type system.

for the classical double integrator type system is presentedin (Rao and Bernstein (2001)), where it is shown that theclassical PD controller exhibits good robustness.

In particular, in this survey is considered the algorithmpresented in (Lee and Song (2001)), which implements thefollowing PD controller:

ˆy = Kp(y − y)−Kddy

dt(31)

that tries to force to the estimated position y to followthe actual displacement y. Based on the later relation, thefollowing transfer function can be obtained:

Control Engineering and Applied Informatics 37

y

y=

Kp

s2 +Kds+Kp, (32)

where Kp defines the bandwidth of the acceleration esti-mation.

The result found implementing this formulation presents asuitable estimation for the low-range acceleration, howeverit also posses a lag-time for higher accelerations, as it canbe observed in blue in Fig.8(j) - Fig.8(`).

In (Shaowei and Shanming (2012)) it is presented an al-gorithm based on the Phase Locked Loop structure (PLL)in order to obtain the estimation of the acceleration byintegral calculation using the previously estimated veloc-ity. The overall concept is presented in Fig.7, where ˆy isthe SDKF filtered velocity and y′ is the velocity estimatedthrough integral computation of the estimated accelerationˆy through the PLL method. The parameters τ , Kp andKi are chosen to satisfy the tracking performance of thevelocity loop.

˜yKp +

Ki

s

1

τs+ 1

1

s

Velocity

Estimation

y ˆy

y′

ˆy

PLL

Fig. 7. Acceleration estimation based on a PLL structure.

The estimated acceleration implementing this approachpresents noise in the low acceleration region, even thoughthe time-lag for high acceleration is improved in compari-son with the double-integrator type controller (see the redcurve in red in Fig.8(j) - Fig.8(`)).

6. KPI ANALYSIS

In Table.1 is summarized the results from the performanceanalysis of all the methods presented in this work using theKPI defined previously. The table is divided into two parts.The first part (the first fifteen rows) shows the resultsof the KPI obtained for the velocity estimation, and thesecond part ( the last fourteen rows) shows the results ofthe KPI obtained for the acceleration estimation.

For the velocity estimation, the KPI analysis confirmsthat the methods based on the FD over EE with theskipping option and the methods based on KF has betterperformance over all the frequency range analyzed.

In particular, the KPI analysis shows that the KF im-plementing a third order model with adaptive covariancematrix (KF 3rdadap) has the best performance of all themethods implemented.

For the acceleration estimation, the KPI analysis alsoshows that the KF 3rdadap formulation presents betterperformance for all the frequency range. However, the KPIanalysis also demonstrates that the second order integratorwith a PD controller method (2ndInt+PD), presents thebest performance of all the methods analyzed for the low-speed range (f = 0.1Hz).

7. CONCLUSION AND DISCUSSION

In this work the problem of the velocity and accelerationestimation from position data measured with a generaldigital position sensor from a linear actuator was assessed.

The main issue regarding a digital encoder is its finiteresolution. This gap between two consecutive data fromthe digital sensor derives in the impossibility of detectingsmall changes of the measured variable. Therefore, theestimation of velocity and acceleration will be mostlyaffected in the low-speed range.

Several methods were introduced and their results werecompared.

It was shown that independently of the method imple-mented for the estimation it is clearly settled that a staticmethod provides an appropriate solution for reduced band-width applications, even though with small variations itis easy to convert the method into an adaptable solutionenhancing the overall performance and increasing its band-width.

The estimation of the acceleration can be addressed asa first order time-derivative relation of the velocity or aa second order time-derivative relation of the position.The first one implies that a successive implementationof a first order derivation or integration must be carriedon which may lead to modeling simplification, such asthe SDKF approach. This simplification in the model hasthe drawback that the performance will also be limited.If a complete model is implemented the performance isenhanced with the counterpart that the computation effortrises and therefore this approach could not be embeddedin simple RT hardware.

From all the methods implemented it was found that theKF with a third order model with an adaptable covariancematrix provides better result for all the velocity rangeanalyzed.

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40 Control Engineering and Applied Informatics

Tab

le.1.

KP

Icom

pariso

nM

ethod

f=

0.1H

zf

=0.5

Hz

f=

1.0Hz

Figu

reKPI1

KPI2

KPI3

KPI1

KPI2

KPI3

KPI1

KPI2

KPI3

FD

+L

PFfc

=10H

z14

.96

21.0

70.5

321.2

93.9

0.1

5376.31

1.440.24

blu

e,5(a)-5(c)

FD

+L

PFfc

=1H

z0.45

8.4

60.1

7205.0

64.1

0.4

63267.59

4.20.72

cyan,

5(a)-5(c)F

D+

LP

Ffc

=0.1H

z22

.54

33.4

41.1

310313.8

211.0

72.0

117724.96

2.31.14

green,

5(a)-5(c)F

Dw

ithE

E(sk

ip=

1)

1.1110.6

30.2

755.5

92.5

10.2

1480.63

2.040.26

red,

5(d)-5(f)

FD

with

EE

(skip

=5)

1.2612.7

0.3

251.6

53

0.2

4448.86

2.240.26

blu

e,5(d

)-5(f)F

Dw

ithE

E(sk

ip=

10)

2.0515.4

0.4

174.4

93.5

90.2

9593.4

2.630.31

cyan,

5(d)-5(f)

Pol.

with

rawd

ata2800.9

198.8

93.2

26886.1

716.6

21.5

75592.49

6.520.66

red,

5(g)-5(i)P

ol.

with

EE

(skip

=3)

1.795.1

90.1

7286.7

91.8

10.2

51077.8

1.130.28

blu

e,5(g)-5(i)

Pol.

with

EE

(skip

=5)

0.84.8

10.1

548.9

81.8

20.1

6396.52

0.940.2

cyan,

5(g)-5(i)S

DK

F(y

)+

SG

13.9

60.1

782.6

32.9

60.2

91342.29

2.370.45

magen

ta,5(j)-5(`)

FD

(EE

)+

SD

KF

0.663.4

30.1

664.9

82.7

80.2

61113.3

2.160.42

cyan,

5(j)-5(`)K

F3rd

0.14.4

60.0

741.5

31.2

20.1

72189.63

2.750.54

red,

5(j)-5(`)K

F3rdadap.

0.184.4

10.0

811.0

31.0

60.1

1229.6

0.510.16

blu

e,5(j)-5(`)

SM

D0.71

4.4

20.1

52346.4

28.8

11.1

514212.64

3.281.04

red,

5(m)-5(o)

SM

Dad

ap

.6.3

9.4

10.3

329.3

1.9

90.1

5548.75

1.810.25

blu

e,5(m

)-5(o)F

D2(E

E)

1274

.75

456.1

19.4

43647951.5

578.7

112.1

345893518.83

95.4611.25

red,

8(a)-8(c)F

D2(E

E)

129.53

114.9

63.2

4267806.5

1107.1

14.1

62088631.79

22.831.83

blu

e,8(a)-8(c)

FD

2(E

E)

+L

PF

137.29

39.4

42.4

2116178.2

413.9

62.6

1391160.92

3.91.11

green,

8(a)-8(c)F

D2(E

E)

+L

PF

35.8

743.6

91.7

718608.5

7.8

61.2

2252869.78

4.110.91

cyan,

8(a)-8(c)S

DK

F(y

)+F

D2

89501.13

5804.0

673.8

1202674.6

545.3

33.7

5388375.48

8.770.91

red,

8(d)-8(f)

SD

KF

(y)+

FD

2+

SD

KF

881.03

274

7.9

75360.5

99.4

70.6

8242745.68

5.050.88

blu

e,8(d

)-8(f)S

DK

F(E

E)

+F

D2

129.53

114.9

63.2

4267806.5

1107.1

14.1

62088631.79

22.831.83

cyan,

8(d)-8(f)

SD

KF

(EE

)+

FD

2+

SD

KF

29.5

922.5

41.3

414992.3

716.6

21.1

4122605.79

3.380.62

green,

8(d)-8(f)

KF

3rd

3.3514.9

60.4

66319.6

77.7

70.8

421449.176.73

1.2m

agenta,

8(d)-8(f)

KF

3rdadap.

14.7

616.3

91

1444.5

53.7

80.3

682299.66

3.170.52

yellow,

8(d)-8(f)

SM

D21

.26

36.8

50.8

644491.7

74.8

71.2

614764.041.42

1.09red

,8(g)-8(i)

SM

Dad

ap

.697.97

289.6

98.5

617087.0

310.2

71.0

7650052.41

8.971.44

blu

e,8(g)-8(i)

2nd

Int

+P

D1.7

12.1

20.4

517953.6

713

1.3

1959689.94

7.241.52

blu

e,8(j)-8(`)

ˆy+P

LL

20.3

325.0

80.8

26542.5

97.3

10.7

8287468.49

5.471

red,

8(j)-8(`)