we provide some background on the main task at hand and relevant motivation , before the control methodology is elaborated .
this procedure is successfully performed in human oocytes as well as other species such as mouse and bovine .
first , a holding pipette is used to immobilize an individual oocyte with a slight suction .
then an injection pipette ( with outer diameter of about 8  m ) , which contains the sperm head to be injected , is forced into the cell .
the piercing through the zona layer and the membrane needs to be achieved with minimal biological damage to facilitate rapid healing .
a significant amount of research effort has been devoted towards developing microscopic instruments for icsi from this perspective alone .
however , its piercing performance is successful only by using a small mercury droplet in the pipette tip [ 2 , 3 ] . without this addition ,
undesirable lateral oscillations occur at the tip and severely hamper the performance of piercing . due to high toxicity of mercury ,
on the other hand , the piezo - assisted icsi procedure is forbidden in many biological laboratories . in recent years , an improved remedial technology , called rotationally oscillating drill ( ros - drill ) ,
this technique shows comparable results to those obtained by the piezo - assisted icsi process , with one major difference , that ros - drill does not have the mercury problem .  
the injection pipette is connected to a small - precision micromotor which is controlled to track a desired sinusoidal trajectory : 
 ( 1)d = adsin(2fdt ) , 
 where ad is the oscillation amplitude ( typically around 0.2  deg ) , and fd is the frequency ( in the range of 500700  hz ) .
these selections are based on a simple bandwidth analysis ; at this frequency range of external stimuli the cell membrane is not expected to follow the pipette tip motion .
the ensuing relative rotational motion between the pipette and the cell membrane creates a clean piercing action , which facilitates the rapid healing of membrane after piercing .  
considerable amount of experimental effort has been invested to demonstrate the validity of ros - drill technology .
it is shown that the success rate in icsi by ros - drill is comparable to that of piezo - assisted technology , provided that the pipette oscillations are maintained as close as possible to the desired trajectory in ( 1 ) .
many healthy mouse offspring are produced using the ros - drill methodology , as shown in figure 1(a ) .
these biological tests are conducted by a group of experimentalists from the university of connecticut and the university of california , davis , usa . two outstanding requirements are noted in these experiments towards an acceptable ros - drill performance : ( i ) rotational oscillation to track the harmonic trajectory very closely ; ( ii ) the flexible pipette to be concentric to the rotational axis of ros - drill .
the later condition is shown to satisfy because of the extreme bending compliance and whirling effects [ 2 , 3 ] .
the smooth variation of the rotational motion in accordance with a harmonic function is the most natural desired trajectory .
the contribution of this paper is a proper control law which can serve the objective under severe sensing constraints .
overarching restriction in this study comes from the pricing aspect . in order to make the ros - drill accepted by a wide range of ivf clinics , the cost of this automated device must remain within $ 1500 per copy ; otherwise , it would be difficult to compete against the commonly - used but inefficient method adopted by trained icsi specialists .
let us take a closer look at the physics of the ros - drill microinjection procedure .
the rotational stiffness of the pipette holder and the pipette including the extremely fine tip are assumed to be high .
consequently , the angular displacement ,  , of the injection pipette is transmitted from the shaft of the micromotor to the pipette tip without a loss .
the lateral ( bending ) vibrations are the major concern at the tip , and the present operating scheme is intended to suppress them to ignorable levels especially when compared with those oscillations caused by the piezo - assisted icsi [ 2 , 3 ] . with these assumptions , the major objective of this study is directed to insure desirably precise harmonic motion tracking capabilities at the pipette tip [ 68 ] despite the very coarse sensory capabilities . for position servoing , in general ,
the sensors are expected to have high resolution vis -  - vis the range of the intended motion , which can yield a desirable tracking capability .
for instance , in representing a harmonic trajectory , one expects to have a minimum of 10 discrete data points per cycle . however , resolution of digital encoders is limited by the number of slots on a rotating disk , through which the encoder 's light beam travels .
although advances in encoder technology have wonderfully progressed to increase the number of slots in order to improve the resolution of the encoders , the trade - off between resolution and cost is unavoidable . in such applications as ros - drill where the cost limitations are very stringent , sensor resolution is often compromised .  
recent model - based speed observers   make the velocity estimate robust using an interesting disturbance observer . in another effort , kwon et al .  
bautista - quintero and pont   propose an h - infinity control algorithm for sensor - constrained mechatronic systems using the position sensors with relatively low resolutions .
they demonstrate how this procedure allows a faithful reproduction of observed motion starting from a limited sensing ability using relatively common ( and inexpensive ) microcontrollers .
furthermore , the methods which deal with control applications with low - resolution sensors usually have high computational demands to compensate the sensory shortfall .
this aspect quickly makes the method prohibitive in a cost - sensitive design such as ros - drill .
we wish to familiarize the reader with the components of the first - generation ros - drill setup and the current improvements .
the first - generation design having a 512 lines / revolution - encoder with quadrature signature is the finest selection we could find within the cost and spatial confines .
it allows a maximum of 2-step reading over the desired peak - to - peak stroke ( note that the desired amplitude is 0.2  deg ) [ 4 , 12 ] .
this sensor makes our best observation of a ros - drill harmonic cycle with a 2-step representation , which is a colossal handicap to perform the control .
the spirit of the proposed control scheme and the focus of this paper are primarily on this crucial feature . moving on to another component ,
the first trial generation of ros - drill employs a plc ( programmable logic controller ) as its digital controller , which has a maximum sampling speed of 1  khz .
this constraint clearly limits the maximum frequency of the controlled trajectory to 500  hz .
most recently reported icsi tests use injection pipette oscillations up to 0.3  deg amplitude and maximum frequency of 500  hz .
these oscillations last a certain length of time which we name the duration of oscillation , d. typically , d varies within 250500  msec .
the preliminary reports claim that ros - drill - assisted icsi results in embryo survival , embryo development , birth , and weaning rates comparable to those of piezo - assisted icsi using mercury .
although these biological results are very promising , the trajectory tracking performance of the existing prototype is not satisfactory mainly because of the low resolution of position sensor and low control sampling rate of plc .
this work represents an effort to further improve this performance .   in the second - generation ( current )
ros - drill microinjector prototypes , the low - sampling - frequency matter is considerably improved by replacing the plc with a microcontroller ( which brings the sampling speed from 1000  hz to 10000  hz ) and the encoder resolution issue by selecting one - level higher - capacity sensor ( 1000 lines and 0.09 degree resolution after quadrature ) . however , despite this upgrade , the position sensor with the resolution of 0.09 degrees still presents the biggest hurdle in this control system design because the required harmonic motion of 0.2 degree amplitude displays only 4-step peak - to - peak encoder recording , which is still low . in , a look - up - table - based adaptively
the most appropriate pid ( proportional , integral , and derivative ) control gains are a priori selected corresponding to the given operating frequency using dynamic simulations . for a given desired trajectory , d , with ad and fd attributes , an ad hoc 3d search routine is performed using the representative simulink simulation program .
this yields a set of feedback gains , which are then utilized through a look - up adaptation structure .
they are regarded as nominal control gains as they are tuned based on the nominal model of ros - drill , and the control design is considered as the first adaptation stage towards the second - generation ros - drill .
an important point to discuss for this first adaptation stage is that the look - up - table - based control law can handle the potential parameter uncertainties , disturbances , and the unmodeled mechanical properties of the system only via the supervisory interference from the user .
for instance , the external load originating from the membrane resistance torque may vary with the type of species and it may affect the control performance of the system .
still worse is the extremely stringent requirement of concentricity on the shaft of motor and pipette holder via a coupling and ensuing resistance to the servosystem . in order to achieve better updates of the control gains ,
furthermore , the low - resolution encoder is a range indicator ( as discussed later ) rather than a measurement device .
the text is organized as follows . in the first part of the paper , a dedicated novel adaptive feedback control method
, the second - stage adaptive law is developed to update the control gains in situ , which is validated by experiments . in the second part of the paper ,
a stochastic perspective is presented which correlates the encoder readings with the actual angular motion .  
the highlighted feature in this study is the unusually low - resolution sensor ( encoder ) .  
figure 2 gives a description of the operating principle of an optical encoder which is our main sensor .
for simplicity , we overlook the quadrature effect and depict an encoder disk as the combination of transparent and dark lines .
the small circle indicates the position of light beam . the angular motion  denotes the rotational motion of the encoder . at the start , the encoder is declared at its  zero position  and  = 0 .  
we now present a critical argument on the position detection ability of the optical encoder .
it is about an encoder offset angle , which we will denote by a , 0 < a <
. it is the angular motion needed for the first encoder pulse to register under a counterclockwise rotation ( see figure 2 ) . 
a  can be taken as a dead zone during which the encoder does not respond .
notice that this offset angle is not really an  offset position  from a baseline configuration . on the contrary
this value presents no importance at all , when the monitored motion is a few orders of magnitude larger than the resolution , . however , for the particular application here the complete range of motion is composed of only a few  's ( e.g. , 4 ) .
thus , the offset angle a plays a very critical role and we present a set of novel observations on this issue , later in the paper . clearly , a is an unknown quantity which is random and uniformly distributed within 0 < a < . that is , from the starting position  = 0 the encoder does not register any reading until a degree of rotation ( in counterclockwise sense ) or a +  degree ( in clockwise sense ) is completed ( see figure 2 ) . in figure 3 , we further depict the sensing ability of the encoder on a hypothetical oscillation  ( shown as the red line ) .
the encoder can only register when  reaches angular displacements of a +  , a + 2 ,  , a + m clockwise and a , a   , 
therefore , the responses to the same angular motion may vary ; that is , the encoder may sense the same  differently , depending on encoder offset angle , a. of course , if the resolution of encoder is high compared to the total stroke , such an offset angle would not cause a noticeable difference in detecting the motion .
we represent this behavior using a quantizer block in simulink if the resolution is high . however , for low - resolution encoders ( i.e. , relative to the stroke ) , such a quantizer block can not reveal the true response of the encoder , as depicted in figure 3 . for such cases ,
some definitions are provided next , in figure 3 ; a1 and a2 are the upper and lower amplitudes of  , respectively . without loss of generality ,
we assume ( a1 > a2 > 0 ) and define the complete actual peak - to - peak stroke as a12 = a1 + a2 .
the blue line represents the rotational motion being sensed by the encoder and the encoder reading is denoted as enc .
the peak - to - peak angular stroke which is recorded by the encoder is named aenc , which is an integer multiple of  ( e.g. , 4 in figure 3 ) .
we denote the average of a12 and aenc over a certain number of oscillations by a12 and aenc , respectively .
the bias of actual angular motion is defined as the distance of center of a12 from 
zero  and it is expressed as 
 ( 2)a12-b = a1a22 . 
 for a certain frequency , the hypothetical actual oscillations of the pipette can be described by the bias and the amplitude of actual angular motion .
let us express  as 
 ( 3)=a12-b+asint , 
 where  = 2f and a and f are amplitude and frequency of harmonic wave , respectively , and we have a12 = 2a . to summarize these definitions ,
we list them again as follows:(i)resolution of position sensor = ;(ii)encoder offset angle = a ( deg);(iii)upper stroke of actual angular position from rest = a1 > 0 ( deg);(iv)lower stroke of actual angular position from rest = a2 > 0 ( deg);(v)actual peak - to - peak angular stroke = a12 = a1 + a2 ( deg);(vi)bias of actual angular stroke = a12-b;(vii)peak - to - peak angular stroke detected by the encoder = aenc ( deg);(viii)average actual angular stroke over a certain number of oscillations = a12 ( deg);(ix)average encoder stroke over a certain number of oscillations = aenc ( deg ) .
resolution of position sensor =  ; encoder offset angle = a ( deg ) ; upper stroke of actual angular position from rest = a1 > 0 ( deg ) ; lower stroke of actual angular position from rest = a2 > 0 ( deg ) ; actual peak - to - peak angular stroke = a12 = a1 + a2 ( deg ) ; bias of actual angular stroke = a12-b ; peak - to - peak angular stroke detected by the encoder = aenc ( deg ) ; average actual angular stroke over a certain number of oscillations = a12 ( deg ) ; average encoder stroke over a certain number of oscillations = aenc ( deg ) .
this section presents a brief review of the earlier work , which establishes the departure points of the present effort . in ros - drill application ,
the desired angular trajectory requires high control sampling frequency , fs , in the microcontroller ( such as 10  khz ) in order to perform a meaningful tracking .
this imposes some further constraints on the limited computational capabilities of selected microcontroller . keeping these restrictions in mind ,
the ros - drill microinjector can be considered as a simple rotational mass , which is attached to a torque - generating dc servomotor ( figure 1 ) .
this yields a transfer function of ros - drill system and the corresponding frequency response creates the magnification factor as a function of  , kp , ki , and kd : 
 ( 4)m=|g(s = j)|=m(fd , kp , ki , kd ) , 
 where  = 2  fd and kp , ki , and kd are the control gains . the objective of the control is to achieve a flat response over a given range of operating frequencies .
we wish to emphasize that the phase angle between the desired harmonic input and the resulting output is not considered as a part of the performance description .
another important point to stress is that since we have limited access to the actual rotationally oscillating motion via the coarse measurements of the encoder , this flat response characteristic can be achieved , at best , by enforcing the peak - to - peak strokes of the encoder readings to be equal to aenc = floor  ( 2ad/ )   deg .
it is shown in   that m is much more sensitive to the kd variations rather than kp and ki .
we compare the effects of variation of these three control gains around their nominal values ( k / k ) .
this results in some percentage variation of magnification factor ( m / m ) . among the three gains ( kd , kp , and ki ) , the most effective one is kd , from this perspective .
thus , the tuning of the magnification factor can be achieved more directly by utilizing only kd variations .
this tuning methodology needs to be further adapted to the actual sample data system when using a low - resolution encoders .
the logic steps of this tuning methodology , or the adaptive update laws of kd , form the main contributions of this study , and they will be detailed in the following sections . in this section , we present a realistic dynamic model of the system using matlab - simulink platform .  
the encoder block is created to reflect the limitations of the position sensor as closely as possible to the reality .
two pieces of information are used from the quad signature encoder outputs : one is the angular step counter , and the other is the velocity of the angular motion .
these data divide the angular movement space into four sections : 
 ( 5)(i )  >0 ,  >0(ii )  >0 ,  <0(iii )  <0 ,  <0(iv )  <0 ,  >0 . 

notice that only one of these four cases will be active at any sampling period ; thus , the summation of the four outputs will actually declare the one - step angular accumulation .
the angular measurements , ad , in zero - order - hold ( z - o - h ) mode is created with sampling time ts just to simulate the sample data procedures of the microcontroller , see figures 5(a ) and 5(b ) . the integration step size for the simulation routine is chosen to be one - tenth of the sampling period of microcontroller ( 10  s ) , which is sufficient to represent the transient behavior within a sampling period with a desirable fidelity . the amplifier gain , k , in figure 4 is chosen by the user based on experimental knowledge such that the saturation in the d / a converter is prevented .
the section enclosed by dotted lines in figure 4 is the dynamic model for ros - drill microinjector itself .
using this simulink tool , we first determine some starting values of the control gains kp , ki , and kd using a pole - placement - based method as described in . then , we systematically vary kd .
the objective in this tuning procedure is to obtain m = 1 as close as possible for the particular interval of frequencies .
for a desired motion of d = adsin(2  fd  t ) , m = 1 implies a peak - to - peak stroke of 2ad .
this is the continuous rotational angle and the corresponding peak - to - peak encoder recording should be aenc = floor  ( 2ad/ ) 
.   we continue the systematic variations of kd until the frequency response amplitude condition , m = 1 , is achieved for the given operating frequency .
we then repeat the same operation for other fd values on a list of potential ( i.e. , biologically required ) operating frequencies .
this set of kd gains will form a look - up table which can be used by the real - time control program on the microcontroller .
once the user identifies a preferred harmonic frequency , the program selects a kd feedback gain from a given list and adaptively sets the new control logic .
this completes the first adaptation scheme , which follows the steps below:(a)declare the desired amplitude and frequency via gui ( ad , fd ) ; ( b)select the corresponding control gain ( kd ) using the look - up table;(c)observe the encoder registrations of peak - to - peak strokes , aenc;(d)evaluate the average of aenc over a predetermined duration d , aenc;(e)signal the operator the direction of the error e = floor  ( 2ad/)-aenc when it is outside a tolerance range , |e| > tolerance . notice that ad is typically not an integer multiple of . therefore , we deploy 2ad/ to create a comparable basis with aenc;(f)manually adjust the gain , kd , as supervisory fine tuning and repeat the steps ( c)(f ) until no violation of tolerance in ( e ) remains .
declare the desired amplitude and frequency via gui ( ad , fd ) ;   select the corresponding control gain ( kd ) using the look - up table ; observe the encoder registrations of peak - to - peak strokes , aenc ; evaluate the average of aenc over a predetermined duration d , aenc ; signal the operator the direction of the error e = floor  ( 2ad/)-aenc when it is outside a tolerance range , |e| > tolerance .
notice that ad is typically not an integer multiple of . therefore , we deploy 2ad/ to create a comparable basis with aenc ; manually adjust the gain , kd , as supervisory fine tuning and repeat the steps ( c)(f ) until no violation of tolerance in ( e ) remains . this procedure is shown to perform the task so that the peak - to - peak strokes of ros - drill are guaranteed to stay desirably close to aenc = floor  ( 2ad/ )   .  
figure 6 shows that experimental setup , the mechanical device along with the controller box . a faulhaber
an optical encoder with 1000 lines is utilized as position sensor , enabling the angular resolution at 0.09  deg ( with quad - signature characteristics ) .
it is important to clarify that this is the best option for an encoder in terms of size , resolution , and costs demanded by the application .
a control box contains the components that handle the logical operations between the sensor ( encoder ) and the actuator ( micromotor ) .
its main unit ( the cpu and i / o device ) is a silicon lab 's c8051f121  controller .
this box also contains the necessary peripheral circuits such as the converter of encoder 's quadrature - signatures into the rotational pulse counter and direction determination of the rotation , optoisolators , and power output chip which feeds directly into the dc micromotor .  
we wish to give an idea to the reader about the microcontroller 's time management structure . for the aimed ros - drill application ,
10  khz siso ( single input - single output ) control sampling rate is selected , and the c8051f121 micro - controller can easily accommodate this speed . out of this 100  s total loop time , 71.8  s is used for sensing , control logic evaluation , and d / a conversion ; the remaining 28.9  s is the idle time which can be devoted to other applications ( such as filtering ) to be added later .   in order to crossvalidate the capabilities of the tracking control , we use an independent angular motion monitoring tool .
the motion validations provided in the entire text are from this sensing channel instead of the unit which handled the same task in the loop .
the look - up - table - based control law offered in , as we summarized above , can handle the potential parameter uncertainties , disturbances , and the unmodeled mechanical properties of the system only via the supervisory interference from the user , see step ( f ) above . for a typical application such as ros - drill ,
these uncertainties originate mainly from friction - related sources , which are impossible to model accurately .
we include them in our simulink model ( figure 4 ) as some arbitrary combinations of viscous and coulomb friction effects , just to get a feel .
the control objective is again to achieve a sustained peak - to - peak stroke of aenc = floor  ( 2ad/ )   as the encoder registers it ( as an integer multiple of  ) .
the average deviation of the aenc ( over a predetermined number of cycles ) from this desired aenc = floor  ( 2ad/ )   is used as the error for tuning of kd .
one must pay attention , however , to another angle of these encoder readings . during the experiments , it is commonly observed that aenc fluctuates typically between two successive steps ( say 3 and 4 ) .
this feature occurs due to the random offset angle a as mentioned above , but also stems from the transient regimes where the oscillations are not yet set into periodically repeated format .
to discriminate the two causes of the same effect is an impossible task . throughout this study
, we consider the motion to have reached a steady regime for simplicity , therefore disqualifying the latter cause .
the effects of the transient behavior are left for a future document .   in this paper , we confine the execution of supervisory gain adjustments : ( a ) through a windowing and averaging procedure ; ( b ) with sufficient update - and - wait period . in short ,
therefore , kd can change only after a certain number of oscillations . in the icsi experiment , typically entire duration of oscillations ,
the tuning of kd is expected to be completed within a small fraction of d and in the very early stages of the period d. after extensive tests using simulink with additional disturbances imposed , it is observed that the angular position of the servo system (  ) reaches the steady state no later than 5 oscillations for all the operating frequencies ( from 400 to 1000  hz ) .
therefore , we use the average of peak - to - peak strokes within 15 cycles of oscillations , which we denote by  aenc , to assess the performance of current control gains and to update them . obviously , if  aenc  is equal to aenc = floor  ( 2ad/ )   that is considered to be satisfactory for the objectives of the control .   in summary
, this adaptation scheme follows the steps listed below:(a , b , c ) are identical to the first adaptation scheme above;(d ) evaluate the average of aenc over 15 cycles , aenc ; ( e ) utilize an update law for kd adjustments if the error , e , is outside a tolerance range , that is , |e| > tolerance .
if the error is within the tolerance , no kd update is needed.(f ) repeat step ( d ) .  
are identical to the first adaptation scheme above ;   evaluate the average of aenc over 15 cycles , aenc ;    utilize an update law for kd adjustments if the error , e , is outside a tolerance range , that is , |e| > tolerance .
these steps will again guarantee the execution of the desired aenc using an adaptive control gain update law .
for the sake of simplicity , from this point onwards we will take ad = 0.2 degrees and the corresponding aenc = 4 = 0.36 degrees . let us define kd(n ) as the nth update of the derivative gain and e(n ) as ensuing amplitude error which is obtained over 15 oscillatory cycles ( an informed selection based on observation of experimental data ) , after kd(n ) is applied .
that is , 
 ( 6)e(n)=4aenc. 
 for the next 15-cycle period , we use a new feedback gain with the following update law:(7a)kd(n)=kd(n1)+kd(n ) , 
 where 
 ( 7b)kd(n)=ce(n1),and c is an update constant .
this update process is performed in the following sequences:(i ) start with kd(0 ) which is taken from a look - up table as explained in section 3 .
( ii ) determine the resulting e(0 ) at the end of the following 15 cycles .
( iii ) for n = 1 , use kd(1 ) = c  e(0 ) and evaluate kd(1 ) from ( 7a).(iv ) again after 15 cycles of using this control gain , determine e(1 ) .
( v)(a ) if |e(1)|  |e(0)| , evaluate kd(2 ) = c  e(1 ) , kd(2 ) = kd(1 ) + kd(2 ) , assign kd - ref = kd(1 ) and eref = e(1 ) , and go to ( vi).(b ) if |e(1)| > |e(0)| , evaluate kd(2 ) = 0.5  c  e(0 ) , kd(2 ) = kd(0 ) + kd(2 ) and determine e(2 ) . if |e(2)|  |e(0)| , assign kd - ref = kd(2 ) and eref = e(2 ) , go to ( vi ) .
if |e(2)| > |e(0)| , repeat ( vb ) but this time with kd(3 ) = 0.5  0.5c  e(0 ) and continue until the absolute error falls below e(0 ) . assign the current kd(n ) value to kd - ref , current e(n ) value to eref  and move to ( vi).(vi ) after 15 cycles , determine the new e(n ) and go to ( va ) .
start with kd(0 ) which is taken from a look - up table as explained in section 3 .   
determine the resulting e(0 ) at the end of the following 15 cycles .    for n = 1 , use kd(1 )
= c  e(0 ) and evaluate kd(1 ) from ( 7a ) .   again after 15 cycles of using this control gain , determine e(1 ) .  
( a ) if |e(1)|  |e(0)| , evaluate kd(2 ) = c  e(1 ) , kd(2 ) = kd(1 ) + kd(2 ) , assign kd - ref = kd(1 ) and eref = e(1 ) , and go to ( vi).(b ) if |e(1)| > |e(0)| , evaluate kd(2 ) = 0.5  c  e(0 ) , kd(2 ) = kd(0 ) + kd(2 ) and determine e(2 ) . if |e(2)|  |e(0)| , assign kd - ref = kd(2 ) and eref = e(2 ) , go to ( vi ) .
if |e(2)| > |e(0)| , repeat ( vb ) but this time with kd(3 ) = 0.5  0.5c  e(0 ) and continue until the absolute error falls below e(0 ) . assign the current kd(n ) value to kd - ref , current e(n ) value to eref  and move to ( vi ) .   if |e(1)|  |e(0)| , evaluate kd(2 ) = c  e(1 ) , kd(2 ) = kd(1 ) + kd(2 ) , assign kd - ref = kd(1 ) and eref = e(1 ) , and go to ( vi ) .  
if |e(1)| > |e(0)| , evaluate kd(2 ) = 0.5  c  e(0 ) , kd(2 ) = kd(0 ) + kd(2 ) and determine e(2 ) . if
|e(2)|  |e(0)| , assign kd - ref = kd(2 ) and eref = e(2 ) , go to ( vi ) . if |e(2)| > |e(0)| , repeat ( vb ) but this time with kd(3 ) = 0.5  0.5c  e(0 ) and continue until the absolute error falls below e(0 ) .
assign the current kd(n ) value to kd - ref , current e(n ) value to eref  and move to ( vi ) .  
after 15 cycles , determine the new e(n ) and go to ( va ) .
some nuances on this gain adaptation process are discussed later in this section over an example data set along with the role of kd - ref and eref .
the following portion is devoted to the selection of the update constant c. uncertain friction term is modeled as some combination ( in simulink model of figure 4 ) of viscous and coulomb frictions : 
 ( 8)tf(t)=tcsgn()+b , 
 where tc is the coulomb friction torque and b is the viscous friction coefficient .
note that different friction scenarios refer to the model in figure 4 with the same system parameters but with combination of different tc and b. scenarios with various combinations of frictions are artificially created by varying tc and b components in ( 8) .
the selection of c is performed offline via the following steps:(a)estimate the bounds of tc and b based on experimental studies .
( b)deploy kd(0 ) in simulink and find the corresponding e(0 ) over a certain number of cycles ( say 15 ) .
e(0 ) , using systematically increasing c values ( starting from zero ) , so that the resulting error e(1 ) becomes desirably small ( e.g. , 75% as we used in our tests ) compared with e(0 ) under most adverse friction conditions . at the same time , we note that kd should be upper - bounded as it multiplies the encoder - based angular speed and tends to saturate the actuator input ( u in figure 4 ) .
note that the determination of c becomes critical from the concern of minimizing the number of gain updates before the error converges to zero . using the proposed method , determination of constant c
deploy kd(0 ) in simulink and find the corresponding e(0 ) over a certain number of cycles ( say 15 ) .  
determine kd(1 ) = c  e(0 ) , using systematically increasing c values ( starting from zero ) , so that the resulting error e(1 ) becomes desirably small ( e.g. , 75% as we used in our tests ) compared with e(0 ) under most adverse friction conditions . at the same time
, we note that kd should be upper - bounded as it multiplies the encoder - based angular speed and tends to saturate the actuator input ( u in figure 4 ) .
note that the determination of c becomes critical from the concern of minimizing the number of gain updates before the error converges to zero . using the proposed method , determination of constant c
the following portion is devoted to the convergence of e to zero and the determination of kd over some experimental data .
reference kd , kd - ref  is the updated derivative gain which leads to smaller amplitude error |eref| than that which evolves under the previous kd - ref . and the corresponding new error is denoted by eref . to ensure that e converges to zero
, the update law of kd always moves the gain in the direction of smaller |eref| .
the update of kd - ref is based on eref ( i.e. , kd = c  eref ) . if the resultant |e| < |eref| , this update results in a new kd - ref and eref .
otherwise , we can continue reducing this kd by 50% until a new kd - ref as described in the updating sequence earlier .  
this update process of kd is shown over an example experimental case study . in this case
, experiment is done on our ros - drill setup for the desired amplitude of 0.2 degrees and frequency of 500  hz . the update of kd is performed every 15 oscillations and throughout d ( i.e. , 500  ms ) .  
the microcontroller is programmed to monitor the average peak - to - peak value of enc over 15 oscillations ( i.e. , aenc ) .  
table 1 shows the update process of kd for this experiment . in the first round of tuning ( marked as k = 0 in table 1 and
figure 7 ) , kd(0 ) = 0.0036 is selected from the look - up table for operating frequency of 500  hz
. however , e(0 ) = 1.9 is large ( see the inset in figure 7 ) , meaning that the peak - to - peak strokes are much smaller than 4. in the second round of tuning ( k = 1 in table 1 and figure 7 ) , c  e(0 ) is used to update kd and kd(1 ) = kd - ref + c  e(0 ) = 0.008 . because kd(1 ) results in |e(1)| > |eref| , kd(1 ) is bisected for the third round of tuning ( k = 2 in table 1 and figure 7 ) , kd(2 ) = kd - ref + kd(1)/2 .
because |e(2)| < |eref| , kd(2 ) and e(2 ) are taken as new kd - ref and eref , respectively .
the controlled oscillation of pipette lasts for a predetermined duration and then the pipette is returned to starting  zero position  in order to prevent the wrap - around effect ( on the tubing attachments ) . in figure 7 , at 0.98  sec , oscillation of pipette ends and begins to return to  zero position .
 figure 8 shows the flowchart of this monitoring and on - line tuning mechanism .
the first part of the paper , which is related to the new , second - stage adaptation scheme , ends here . in what follows
, we will demonstrate the influence mechanism for the actual angular motion , if the encoder reading of the stroke is aenc = 4 just to provide a better insight to the reader .
the above control scheme is designed to perform an adaptively updating logic to assure a given aenc peak - to - peak stroke .
let us take aenc = 4 for the sake of simplified arguments , without loss of generality .
we query which operational conditions are satisfied if aenc = 4 reading is guaranteed , next .
note that the derivations pertaining to aenc = 4 in what follows can be generalized with ease to cases when desired control performance is aenc = 2m , for m = 1,2,  . to clarify some definitions about encoder reading ,
we list them as follows:(i)a duration of oscillations = d;(ii)absolute deviation of a12 from its nearest odd integer multiple of  = . a duration of oscillations = d ; absolute deviation of a12 from its nearest odd integer multiple of  = . a very important observation is stated next .
one can see from figure 3 that for the encoder to register 4-step stroke ( i.e. , aenc = 4 ) , the following are the necessary and sufficient conditions:(9a)a+2<a1<a+3,(9b)a+<a2<a+2.by summing ( 9a ) and ( 9b ) , we obtain 
 ( 10)3<a12<5. 
 here , we define  as the absolute deviation of a12 from the nearest odd integer multiple of  and 0 <  < . for 3 < a12 < 5 , we have(11a )  either  3<a12<4 ,  =a123(11b )  or  4a12<5 ,  =5a12.for the simplicity of expressions , let us denote 
 ( 12)a12-b=2a2 . 

note that the offset angle is an uniformly distributed random variable between zero and . hence , a12-b is also an uniformly distributed random variable but between /2 and /2 . 

proposition 1 . for any harmonic motion with ( 2 m  1) < a12 < ( 2 m + 1) , peak - to - peak encoder reading will always be aenc = 2m if and only if the bias of stroke ( a12-b ) satisfies 
 ( 13)2<a12-ba12-b<2 . 
 furthermore if a12-b  =  a12-b , for such a harmonic motion the encoder reading will always be 2m. 
  for any harmonic motion with ( 2 m  1) < a12
< ( 2 m + 1) , peak - to - peak encoder reading will always be aenc = 2m if and only if the bias of stroke ( a12-b ) satisfies 
 ( 13)2<a12-ba12-b<2 . 
 furthermore if a12-b  =  a12-b , for such a harmonic motion the encoder reading will always be 2m. 
 proofwithout loss of generality , let us take m = 2 and prove the sufficiency of ( 13 ) for aenc = 4. by substituting ( 2 ) and ( 12 ) into ( 13 ) , we obtain(14a)2a<a1a2<2a+. 
 ( a ) first consider the interval 3 < a12
< 4. note from ( 11a ) : 
 ( 14b)a12=a1+a2=3+.using ( 14a ) and ( 14b ) once for a1 and again for a2 , we obtain(15a)a+2<a1<a+2+,(15b)a+<a2<a++.keeping in mind that 0 <  <  , one can see that satisfying ( 15a ) and ( 15b ) automatically satisfies ( 9a ) and ( 9b ) , which are the necessary and sufficient conditions for aenc = 4 measurement ( as mentioned at the beginning of section 5).(b ) one should next consider the interval 4 < a12 < 5. note from ( 11b ) : 
 ( 16)a12=a1+a2=5. 
 using ( 14a ) and ( 16 ) once for a1 and again for a2 , we obtain(17a)a+3<a1<a+3,(17b)a+2<a2<a+2.with the condition 0 <  <  , ( 17a ) and ( 17b ) guarantee the fulfillment of ( 9a ) and ( 9b ) , respectively . again
this completes the proof of sufficiency of proposition 1 . 
  without loss of generality , let us take m = 2 and prove the sufficiency of ( 13 ) for aenc = 4. by substituting ( 2 ) and ( 12 ) into ( 13 ) , we obtain(14a)2a<a1a2<2a+. 
 ( a ) first consider the interval 3 < a12 < 4. note from ( 11a ) : 
 ( 14b)a12=a1+a2=3+.using ( 14a ) and ( 14b ) once for a1 and again for a2 , we obtain(15a)a+2<a1<a+2+,(15b)a+<a2<a++.keeping in mind that 0 <  <  , one can see that satisfying ( 15a ) and ( 15b ) automatically satisfies ( 9a ) and ( 9b ) , which are the necessary and sufficient conditions for aenc = 4 measurement ( as mentioned at the beginning of section 5 ) . ( b ) one should next consider the interval 4 < a12 < 5. note from ( 11b ) : 
 ( 16)a12=a1+a2=5. 
 using ( 14a ) and ( 16 ) once for a1 and again for a2 , we obtain(17a)a+3<a1<a+3,(17b)a+2<a2<a+2.with the condition 0 <  <  , ( 17a ) and ( 17b ) guarantee the fulfillment of ( 9a ) and ( 9b ) , respectively . again
we take into account that aenc = 4 and , therefore , ( ( 9a ) and ( 9b ) ) conditions hold .
4 first . under this assumption , the right inequalities of ( ( 9a ) and ( 9b ) ) are satisfied automatically .
the left inequality of ( 9a ) is a + 2 < a1 and it can be expressed as(18a)(a1+a23)<a1a2(2a ) . 
 similarly , the left inequality of ( 9b ) is a +  < a2 and it can be rewritten as 
 ( 18b)a1a2(2a)<a1+a23.using the definitions in ( 2 ) and ( 12 ) and the constraint ( 14b ) , the combined inequalities of ( ( 18a ) and ( 18b ) ) render exactly the conditions given in ( 13 ) .
< 5. we also claim that this condition , this time , together with the right inequalities of ( ( 9a ) and ( 9b ) ) , will satisfy the left inequalities of ( ( 9a ) and ( 9b ) ) . following the similar procedure above , with the right inequality of ( 9a ) , a1
< a + 3 , we arrive at(19a)a1a2(2a)<5a1a2 . 
 the right inequality of ( 9b )
, a2 < a + 2 , creates 
 ( 19b)(5a1a2)<a1a2(2a).once again , using the definitions in ( 2 ) and ( 12 ) and the constraint ( 16 ) , the combined inequalities of ( ( 19a ) and ( 19b ) ) render exactly the conditions given in ( 13 ) .
furthermore , if a12-b = a12-b , ( 13 ) automatically holds for any a12 within 3 <
a12 < 5 ; therefore , the causality follows . this proof for m = 2
can be extended for cases when aenc = 2m , for m = 1,2,  .
the important implications of proposition 1 can be summarized in the following logical sequences:(i)a12-b is a random number ( due to the term  a  ) . however , 
a  remains unchanged during the oscillatory period d ; thus , a12-b remains constant in the same period.(ii)bounds expressed by ( 13 ) suggest that the bias , a12-b , of each cycle during d stays within a bounded distance from a12-b  ( i.e. , /2 ) if and only if we maintain the encoder reading of aenc = 2m.(iii)in essence , if a control mechanism which is described earlier assures the aenc = 2m , this also brings a guarantee for the boundedness of |a12-b - a12-b|</2 , where  is defined as ( 11a ) and ( 11b ) .
this implies that the bias a12-b has an attractive bonding to a12-b. it can fluctuate around this value but can not run away from it .
a  remains unchanged during the oscillatory period d ; thus , a12-b remains constant in the same period .
bounds expressed by ( 13 ) suggest that the bias , a12-b , of each cycle during d stays within a bounded distance from a12-b  ( i.e. , /2 ) if and only if we maintain the encoder reading of aenc = 2m. in essence , if a control mechanism which is described earlier assures the aenc = 2m , this also brings a guarantee for the boundedness of |a12-b - a12-b|</2 , where  is defined as ( 11a ) and ( 11b ) .
this implies that the bias a12-b has an attractive bonding to a12-b. it can fluctuate around this value but can not run away from it .
a novel control system with low - resolution encoder for the desired harmonic trajectory is studied on a cellular microinjection technology called the ros - drill ( rotationally oscillating drill ) . in the first part of paper
, a novel adaptive control logic is developed to facilitate the tracking of the harmonic rotational motion under uncertainties , especially frictions .
we demonstrate via dynamics simulations first , followed by experiments that the tracking of the harmonic rotational motion is achieved with desirable fidelity . in the second part , a stochastic analysis connecting the actual motions and their low - resolution sensory recordings is presented .
it is observed that when the control structure guarantees a fixed peak - to - peak stroke , the bias of actual angular motion is bounded in a range and it is attracted to a certain predefined value .
 a novel control methodology which uses a low - resolution encoder is presented for a cellular microinjection technology called the ros - drill ( rotationally oscillating drill ) . 
 it is developed primarily for icsi ( intracytoplasmic sperm injection ) operations , with the objective of generating a desired oscillatory motion at the tip of a micro glass pipette . 
 it is an inexpensive setup , which creates high - frequency ( higher than 500  hz ) and small - amplitude ( around 0.2 deg ) rotational oscillations at the tip of an injection pipette . 
 these rotational oscillations enable the pipette to drill into cell membranes with minimum biological damage . 
 such a motion control procedure presents no particular difficulty when it uses sufficiently precise motion sensors . 
 however , size , costs , and accessibility of technology to the hardware components severely constrain the sensory capabilities . 
 consequently , the control mission and the trajectory tracking are adversely affected . 
 this paper presents two contributions : ( a ) a dedicated novel adaptive feedback control method to achieve a satisfactory trajectory tracking capability . 
 we demonstrate via experiments that the tracking of the harmonic rotational motion is achieved with desirable fidelity ; ( b ) some important analytical features and related observations associated with the controlled harmonic motion which is created by the low - resolution feedback control structure . 
1. Introduction and Motivation 
2. Some Descriptors of the Encoder Signal
3. Review of Earlier Work: The First-Stage Adaptation Scheme
4. New Procedure: Second-Stage Adaptation Scheme
5. Comprehension of Actual Angular Motion from Coarse Encoder Readings
6. Conclusions