Numerical control method and apparatus for controlling acceleration and deceleration of a controlled object

To enable the free setting of an acceleration and deceleration pattern free from the amount of movement and the movement time a numerical control method controls the operation of a controlled object by a target function. The method which computes the target function Y(t) by an amount of change .DELTA.Y.sub.t, a normalized target function y(t), and correction values .beta., .delta., defined as: ##EQU1##

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to a numerical control method and a numerical 
control system used for the control of various types of PTP (point to 
point) operations such as robot control, temperature control, camera focus 
control, slide control of a CD player, video disk, and the like, and DC 
point control of electrical circuits. More particularly, it relates to a 
numerical control method and numerical control system which enable free 
setting of a pattern of acceleration and deceleration free from the amount 
of movement or the time of movement. 
2. Description of the Related Art 
In general, in a servo control circuit, there is known a method for 
converting information on the phase and speed of a controlled object to a 
pattern, storing the same in advance in a memory, and later reading out 
the information in accordance with a target point to control the object at 
different points of time. In this case, the acceleration curve and 
deceleration curve are tinkered with so that the controlled object will be 
operated smoothly and within the desired time. 
For example, the most general pattern of acceleration and deceleration is 
the triangular pattern shown in FIG. 1A, but since the acceleration Jumps 
at three locations: the starting point of movement, the peak point, and 
the target point, it suffers form the disadvantage that a shock is easily 
given to the mechanical system. Further, this cannot be said to be optimal 
as the input to a servo control circuit. 
Therefore, as shown in FIG. 1B, a proposal has been made of a pattern of 
acceleration and deceleration which provides a constant speed portion F as 
a stopping measure immediately before the target point. Even with this 
pattern of acceleration and deceleration, the jumps in acceleration at the 
starting point of movement and the peak point in the transition from 
acceleration to deceleration are not eliminated, so it also suffers from 
the disadvantage that the shock to the mechanical system cannot be 
completely resolved. Further, the disadvantage that the input is not the 
optimal one for a servo control system similarly remains. 
On the other hand, to achieve a smooth operation at the time of 
acceleration and deceleration, a method has been proposed of storing in 
advance in a read-only-memory (ROM) the smooth pattern of acceleration and 
deceleration as shown in FIG. 2A. By this technique, problems in 
overshooting and precision of stopping are remarkably eliminated. However, 
the acceleration and deceleration times are made constant, so if the 
amount of movement is increased, the acceleration and deceleration curve 
also grows higher as shown in FIG. 2B and the problems in the precision of 
stopping at the time of a stop and shock to the mechanical system 
reappear. Further, it suffers from the disadvantage that the form of the 
acceleration and deceleration cannot be freely changed. 
To overcome the above disadvantages, as shown in FIG. 3, a method has been 
proposed of classifying the amount of movement into large, medium, and 
small movements and storing in advance in a ROM patterns of acceleration 
and deceleration with acceleration times and deceleration times suitable 
for each of the same. The disadvantage nonetheless remains in that this 
results in a massive required memory capacity and further unnatural 
characteristics of the amount of movement and movement time, so is not 
preferable in terms of control. Further, it was not possible to freely 
change the acceleration and deceleration by this technique. 
SUMMARY OF THE INVENTION 
The present invention was made in consideration of the above disadvantages 
mentioned above, and thus, an object of the present invention is to enable 
free setting of the pattern of acceleration and deceleration free from the 
amount of movement or movement time with the assumption of realization of 
quick, smooth PTP operations. 
To achieve the above-mentioned object, the numerical control method of the 
present invention lies in a numerical control method for controlling the 
operation of a controlled object by a target function, characterized by 
computing the target function (Y(t)) by an amount of change 
(.DELTA.Y.sub.t), a normalized target function (y(t)), and correction 
values (.beta., .delta.). 
More specifically, it is characterized in that the target function is 
obtained by multiplying the amount of change (.DELTA.Y.sub.t), the 
normalized target function (y(t)), and the correction values (.beta., 
.delta.). 
Further, for the correction values, when the normalized target function at 
the time of acceleration is y.sub.a (T), the normalized target function at 
the time of deceleration is y.sub.d (t), the acceleration time is 
T.sub.pa, and the deceleration time is T.sub.pd, the parameters 
(.delta..sub.a, .delta..sub.d) having time dimensions defined by the 
following equations: 
##EQU2## 
are included in the target function (Y.sub.a (t)) at the time of 
acceleration as defined by the following equation: 
##EQU3## 
or in the target function (Y.sub.d (t)) at the time of deceleration as 
defined by the following equation: 
##EQU4## 
More specifically, when the normalized target function at the time of 
acceleration is y.sub.a (T), the normalized target function at the time of 
deceleration is y.sub.d (t), the acceleration time T.sub.pa, and the 
deceleration time is T.sub.pd, the correction value is expressed in the 
case of a target function (Y.sub.a (t)) at the time of acceleration as 
defined by the following equation: 
##EQU5## 
and in the case of a target function (Y.sub.d (t)) at the time of 
deceleration as defined by the following equation: 
##EQU6## 
and in that the parameters (.beta..sub.a, .beta..sub.d) are defined as 
defined by the following equation: 
##EQU7## 
Note that when the normalized target function is a triangular acceleration 
and deceleration curve, the parameters (.beta..sub.a, .beta..sub.d) are 1. 
On the other hand, to achieve the above-mentioned object, the numerical 
control system of the present invention provides a numerical control 
system for controlling the operation of a controlled object by a target 
function, characterized by being provided with an input unit which 
receives as input an amount of change (.DELTA.Y.sub.t) and a normalized 
target function (y(t)), a correction value calculation unit which 
calculates correction values (.beta., .delta.) based on the amount of 
change (.DELTA.Y.sub.t) and normalized target function (y(t)), and a 
target function computation unit which computes the target function (Y(t)) 
based on the information from the input unit and the correction value 
calculation unit. 
In a servo loop for controlling an arbitrary physical amount Y, reference 
values Y.sub.ref of the physical-state representing amount are input. In 
such a servo system, it is strongly demanded to quickly and smoothly move 
from a reference value Y.sub.ref1 to a reference value Y.sub.ref2. 
For example, a PTP (point to point) operation which does not dictate the 
path from the reference point Y.sub.ref1 to the reference value Y.sub.ref2 
is made use of in a wide range of fields such as robot control, 
temperature control, camera focus control, slide control of a CD player, 
video disk, and the like, and DC point control of electrical circuits. 
Therefore, in the present invention, an algorithm has been constructed for 
enabling free and easy generation of new target functions, by introduction 
of a very small number of parameters, along with changes in the factors 
characterizing the target functions, i.e., (1) the pattern of acceleration 
and deceleration, (2) the amount of change, and (3) the changes in the 
acceleration time and deceleration time (including expansion and 
contraction of time axis). 
That is, if the desired target functions Y.sub.a (t) and Y.sub.d (t) are 
normalized as 
EQU Y.sub.a (t)=.DELTA.Y.sub.a -.beta..sub.a -y.sub.a (t) 
0.ltoreq.t.ltoreq.T.sub.pa 
EQU Y.sub.d (t)=.DELTA.Y.sub.d -.beta..sub.d -y.sub.d (t) 
0.ltoreq.t.ltoreq.T.sub.pd 
the parameter .beta. may be found by any normalized target function as 
##EQU8## 
On the other hand, if another parameter .delta. having a time dimension is 
introduced as 
EQU .DELTA.Y.sub.a =.delta..sub.a (T.sub.pa)-Y.sub.a (T.sub.pa) 
EQU .DELTA.Y.sub.d =.delta..sub.d (T.sub.pd)-Y.sub.d (T.sub.pd) 
the parameter .delta. also may be found by any normalized target function 
as 
##EQU9## 
Accordingly, the desired target functions are given as 
##EQU10## 
and further the above equations are given as the following even in the 
case of expansion and contraction of the time axis 
##EQU11## 
so it is possible to easily compute them no matter what the amount of 
change .DELTA.Y.sub.t, normalized target function y(t), and time.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Below, the principle and examples of application of the present invention 
will be explained in the following order with reference to the drawings: 
Principle of Present Invention 
Target Functions and Target Speed Functions 
Problems 
Principle of Generation of Target Functions 
(1) Normalized Target Functions y.sub.a (t) and y.sub.d (t) 
(2) Parameter .beta. 
(3) Parameter .delta. 
(4) Matching of Acceleration Curve and Deceleration Curve 
(5) Maximum Speed 
(6) Expansion and Contraction of Time Axis 
(i) Normalized Target Function y(t) 
(ii) Parameter .beta. 
(iii) Parameter .delta.(T.sub.p) 
Examples of Application 
Triangular Type 
Sin Type 
Exp Type 
Combination 
Principle of Present Invention 
First, clarification will be made of the problems which arise when building 
an algorithm which enables free and easy production of new target 
functions through introduction of an extremely small number of parameters 
along with changes in the characterizing factors of target functions, that 
is, 
1! the parameter of acceleration and deceleration, 
2! the amount of movement (amount of change), and 
3! the acceleration time and deceleration time (including expansion and 
contraction of time axis) 
and then the principle of production of a target function will be 
explained. 
Target Functions and Target Speed Functions 
The symbols and parameters used in the above explanation are as shown in 
FIG. 4. Here, the acceleration time is given as T.sub.pa, the deceleration 
time as T.sub.pd, the total movement time as T.sub.t, the amount of 
acceleration movement as .DELTA.Y.sub.a, the amount of deceleration 
movement as .DELTA.Y.sub.d, the total amount of movement .DELTA.Y.sub.t, 
and the peak speed as Y.sub.p. 
TABLE 
______________________________________ 
Tpa: ACCELERATION TIME 
Tpd: DECELERATION TIME 
Tt: TOTAL MOVEMENT TIME (= Tpa + TPd) 
.DELTA.Ya: 
AMOUNT OF ACCELERATION MOVEMENT 
.DELTA.Yd: 
AMOUNT OF DECELERATION MOVEMENT 
.DELTA.Yt: 
TOTAL AMOUNT OF MOVEMENT (= .DELTA.Ya + .DELTA.Yd) 
Yp: PEAK SPEED 
______________________________________ 
Here, the following stand: 
Total movement time T.sub.t =acceleration time T.sub.pa +deceleration time 
T.sub.pd 
Total amount of movement .DELTA.Y.sub.t =amount of acceleration movement 
.DELTA.Y.sub.a +amount of deceleration movement .DELTA.Y.sub.d 
Further, when the time is t and the target value (position, temperature, 
voltage, etc.) is Y, Y(t) expresses the target function and the time 
differential dY(t)/dt of the target function expresses the target speed 
function. The relationship of the target value Y with respect to the time 
is shown at the top of FIG. 4, while the relationship of the target speed 
to the time is shown at the bottom of FIG. 4. 
Problems 
The main factors characterizing a target function may be said to be (1) the 
pattern of acceleration and deceleration, (2) the amount of movement 
.DELTA.Y.sub.t, and (3) the expansion and contraction of the time axis, 
such as the acceleration time T.sub.pa and the deceleration time T.sub.pd. 
Accordingly, a task given to the present invention is to construct an 
algorithm which enables free and easy production of a new target function 
by introduction of a very small number of parameters to deal with changes 
in such factors. 
For example, when using the triangular acceleration and deceleration curve 
shown in FIG. 4, to ensure that no shock was given to a robot arm at the 
stopping point (t=T.sub.t) of the arm, it was necessary to increase the 
deceleration time T.sub.pd or change the deceleration target function 
Y.sub.d (t)dot to a smoother function. Further, a need arose for changing 
the shape of the target function or the acceleration and deceleration time 
along with changes in the amount of movement .DELTA.Y.sub.t. 
In the past, there were no clear cut guidelines regarding such changes, so 
everything was left to the ability of experienced designers. Sometimes, 
the control data had to be decided on after a process of repeated trial 
and error. 
Accordingly, there were the problems that the control data decided on was 
governed by the ability of the designer and that inefficiency of design 
was caused. 
Principle of Generation of Target Functions 
Next, an explanation will be made of the principle for solving the 
above-mentioned task. 
(1) Normalized Target Functions y.sub.a (t) and y.sub.d (t) 
According to the above-mentioned task, since it is necessary to deal with 
all types of the amount of movement .DELTA.Y.sub.t, first the target 
function is normalized. That is, when the target function at acceleration 
is Y.sub.a (t) and the target function at deceleration is Y.sub.d (t), the 
following is set 
EQU Y.sub.a (t)=.DELTA.Y.sub.a -.beta..sub.a -y.sub.a (t) 
0.ltoreq.t.ltoreq.T.sub.pa (1) 
EQU Y.sub.d (t)=.DELTA.Y.sub.d -.beta..sub.d -y.sub.d (t) 
o.ltoreq.t.ltoreq.T.sub.pd (2) 
Here, y.sub.a (t) and y.sub.d (t) in equation (1) and equation (2) are 
dimension-less smooth continuous functions. These are referred to as 
normalized target functions. 
Note that in the case of a normalized target function, the following 
stands: 
EQU Y.sub.a (0)=0, y.sub.d (0)=0 (3) 
and y.sub.a (t) gives the pattern (curve) at the time of acceleration and 
y.sub.d (t) gives the pattern (curve) at the time of deceleration. That 
is, by changing the normalized target functions y.sub.a (t) and y.sub.d 
(t) in accordance with need, it is possible to change the shape of the 
target function. 
Note that in equation (2), y.sub.d (t) is defined by 
0.ltoreq.t.ltoreq.T.sub.pd, but it is possible to change to deceleration 
by applying a reversal operation of the time axis. More specifically, it 
is possible to substitute T.sub.pd +T.sub.pa -t in place of t in equation 
(2). 
(2) Parameter .beta. 
Further, .beta..sub.a and .beta..sub.d in equation (1) and equation (2) are 
dimension-less parameters and are introduced for the following two 
purposes. 
First, the first purpose is to give a degree of freedom to the selection of 
the normalized target function y(t). That is, when using the triangular 
pattern as shown in FIG. 4, the normalized target function y.sub.a (t) is 
introduced to enable absorption by the parameter .beta. no matter if 
multiplied by t.sup.2 /T.sub.p.sup.2, 10t.sup.2 /T.sub.p.sup.2, or any 
other coefficient. 
Further, the effect of the parameter .beta. is not exhibited that much in 
the case of a simple function such as a triangular pattern, but a much 
more remarkable effect is obtained if a complicated function such as a 
later mentioned exp type pattern is adopted. 
Further, the second purpose of the introduction of .beta. is to establish a 
relationship between the normalized target function y(t) and the servo 
loop characteristic. That is, the normalized target function y(t) is an 
extremely important factor for a certain servo loop. By defining this 
relationship by .beta., use may be made of it when producing a target 
function. 
Further, from equation (1) and equation (2), 
EQU Y.sub.a (T.sub.pa)=.DELTA.Y.sub.a -.beta..sub.a -y.sub.a 
(T.sub.pa)=.DELTA.Y.sub.a (1') 
EQU Y.sub.d (T.sub.pd)=.DELTA.Y.sub.d -.beta..sub.d -y.sub.d 
(T.sub.pd)=.DELTA.Y.sub.d (2') 
so, .beta..sub.a and .beta..sub.d may be expressed by 
##EQU12## 
If a normalized target function is determined by equation (4), it is 
possible to find .beta..sub.a and .beta..sub.d by calculation. 
(3) Parameter .delta. 
As stated in relation to the task of the present invention mentioned 
earlier, the algorithm for producing the target function must be able to 
be freely used even with respect to expansion or contraction in the 
direction of the time axis, so another parameter .delta. (sec) is 
introduced. This parameter .delta. is defined as follows: 
EQU .DELTA.Y.sub.a =.delta..sub.a (T.sub.pa)-Y.sub.a (T.sub.pa) (5) 
EQU .DELTA.Y.sub.d =.delta..sub.d (T.sub.pd)-Y.sub.d (T.sub.pd) (6) 
That is, the parameter .delta. means the length of the horizontal axis 
(time axis) of the rectangular shape of the height Y.sub.a (T.sub.pa) 
(shown by the dotted line in FIG. 7 which gives an area equal to the area 
.DELTA.Y.sub.a surrounded by the acceleration (deceleration) curve as 
shown in FIG. 7. Note that this is expressed as a function of T.sub.p like 
.delta.(T.sub.p) because the value of .delta. is dependent on T.sub.pa and 
T.sub.pd. 
The parameter .delta. may be found in the following way: 
That is, if the two sides of equation (1) and equation (2) are 
differentiated by the time t and t=T.sub.p substituted in them, then the 
result becomes 
##EQU13## 
If the normalized target functions are determined by these equation (9) and 
equation (10), then it is possible to find the parameter .delta. by 
calculation. 
(4) Matching of Acceleration curve and Deceleration Curve 
Up to here, the acceleration and deceleration have been treated 
independently, but the target speed function which is produced in the end 
must be continuously linked around t=T.sub.pa. Accordingly, below, the 
relationship linking acceleration and deceleration is found. 
If the total amount of movement is .DELTA.Y.sub.t and the total movement 
time is T.sub.t, then as mentioned above, the following relationship 
stands: 
EQU .DELTA.Y.sub.t =.DELTA.Y.sub.a +.DELTA.Y.sub.d (11) 
EQU T.sub.t =T.sub.pa +T.sub.pd (12) 
In this case, the total amount of movement .DELTA.Y.sub.t and the total 
movement time T.sub.t are by nature determined by a large system including 
the servo system and the algorithm of the present invention does not have 
anything to do with the determination of the system. In particular, the 
movement time T.sub.t, that is, T.sub.pa and T.sub.pd, is an important 
factor in the system and is determined by the magnitude of the load of the 
servo system and other physical situations, such as the capacities of the 
actuator and power system, etc. For example, as shown in FIG. 8, there are 
cases where it is determined as a function of .DELTA.Y.sub.t : 
Further, if the peak speed Y.sub.a (T.sub.pa) at the time of acceleration 
and the peak speed Y.sub.d (T.sub.pd) at the time of deceleration are not 
equal, then as shown in FIG. 9, the speed jumps around t=T.sub.pa, which 
is not preferable as a target function. Therefore, if the two peak speeds 
are made equal, then 
EQU Y.sub.a (T.sub.pa)=Y.sub.d (T.sub.pd)=Y.sub.p (13) 
and from equation (5), equation (6), and equation (11), 
EQU .DELTA.Y.sub.t ={.delta..sub.a (T.sub.pa)+.delta..sub.a (T.sub.pd)}Y.sub.p 
(14) 
is obtained. 
The acceleration time T.sub.pa and deceleration time T.sub.pd are already 
determined, so .delta..sub.a (T.sub.pa) and .delta..sub.d (T.sub.pd) 
become known from equation (9) and equation (10) and in the end the peak 
speed Y.sub.p is determined by the following equation (15): 
##EQU14## 
Therefore, from equation (5) and equation (6), the amount of movement 
during acceleration .DELTA.Y.sub.a and the amount of movement during 
deceleration .DELTA.Y.sub.d may be found by the following equation (16) 
and equation (17). 
##EQU15## 
Therefore, if equation (16) and equation (17) are substituted in equation 
(1) and equation (2), 
##EQU16## 
The right sides of equation (18) and equation (19) are all determined, so 
it becomes possible to use these two equations to produce the target 
function Y.sub.a (t) of acceleration and the target function Y.sub.d (t) 
of deceleration. 
(5) Maximum Speed 
When a target function with symmetrical right and left sides, that is, a 
target function of Y.sub.a (t)=Y.sub.d (t), is used at the time 
t=T.sub.pa, the speed Y.sub.p of the peak point increases if the amount of 
movement .DELTA.Y.sub.t increases. However, since there is a maximum speed 
Y.sub.max in an actuator, the target speed function grows as shown in FIG. 
10 for example. 
In this case, the area enclosed by the speed curve and the time axis is the 
amount of movement .DELTA.Y.sub.t, and the following stands: 
EQU .DELTA.Y.sub.t1 &lt;.DELTA.Y.sub.t2 &lt;Y.sub.t3 &lt; . . . &lt;.DELTA.Y.sub.t6 (20) 
In particular, with the amount of movement .DELTA.Y.sub.t3 of (3) shown in 
FIG. 10, there is is a match with the maximum speed Y.sub.MAX, so in this 
sense if .DELTA.Y.sub.t3 is written as .DELTA.Y.sub.tMAX, then from 
equation (5) and equation (6), the following stands: 
EQU .DELTA.Y.sub.max =.delta.(T.sub.p)-Y.sub.max (21) 
Using this, when the amount of movement .DELTA.Y.sub.t and the peak time 
T.sub.p are given, it is possible to find the speed Y.sub.p of the peak 
point from 
EQU .DELTA.Y.sub.t =.delta.(T.sub.p)-Y.sub.p (22) 
and judge if this is larger or smaller than the maximum speed Y.sub.MAX of 
the actuator and thereby Judge if the operation is at the maximum speed 
(trapezoidal operation (4) to (6) in FIG. 10. The parameter 
.delta.(T.sub.p) may be used in this way as well. Note that even if the 
patterns of acceleration and deceleration differ, they may be similarly 
applied. 
Therefore, in FIG. 10, when the amount of movement is given as 
.DELTA.T.sub.t5 and the acceleration and deceleration time is given as 
T.sub.p, it is possible to judge that 
EQU .delta.(T.sub.p)-Y.sub.max =.DELTA.Y.sub.max &lt;.DELTA.Y.sub.t5 (23) 
Therefore, the result is 
EQU .DELTA.Y.sub.t5 =.DELTA.Y.sub.max +.DELTA.Y.sub.0 (24) 
and the amount of movement .DELTA.Y.sub.0 of the maximum speed portion can 
be easily calculated. 
In the same way, the time T.sub.0 for .DELTA.Y.sub.0 to be moved at the 
speed of Y.sub.MAX can be found by the following equation (25): 
##EQU17## 
As a result, the total movement time T.sub.t in this case becomes as in the 
following equation (26). This situation is shown in FIG. 11. 
##EQU18## 
Note that in this explanation, the peak time T.sub.P was treated as a 
constant, but in the actual application, the peak time T.sub.p is changed 
in accordance with the amount of movement .DELTA.Y.sub.t as shown in FIG. 
8 and the min point and max point shown in FIG. 8 appear due to the 
presence of the f-characteristic of the servo loop or the motor maximum 
speed. Further, this application is easy even if the form of the 
acceleration and deceleration changes. 
(6) Expansion and Contraction of Time Axis 
The total movement time T.sub.t, the acceleration and deceleration times 
T.sub.pa and T.sub.pd etc. are determined by the situation and laws of the 
control system such as the actuator, load, and power system. Therefore, 
consideration is given to a method for producing a target function in 
accordance with the determined times, that is, producing a target function 
with respect to the expansion and contraction of the time axis. 
The situation when multiplying the time axis by .alpha. when the normalized 
target function y(t) is given is shown in FIG. 12. An object in this case 
is to derive an equation when the equation (18) and equation (19) of the 
target function are subjected to expansion or contraction of the time 
axis. 
Therefore, considering the effects of expansion and contraction of the time 
axis on the various amounts included at the right sides of equation (18) 
and equation (19), the following is concluded: 
(i) Normalized target function y(t) From the graph shown in FIG. 12, 
##EQU19## 
stands and the value at t=T.sub.p and the value at .tau.=.alpha.T.sub.p 
become equal. 
(ii) Parameter .beta. 
From equation (4), equation (5), and equation (28), 
##EQU20## 
so the value of the parameter .beta. does not change depending on the 
expansion or contraction of the time axis. 
(iii) Parameter .delta.(T.sub.p) 
From equation (9) and equation (10), 
##EQU21## 
Here, if the time axis is multiplied by .alpha., then the result is 
##EQU22## 
and the value of .delta. is multiplied by .alpha. equal to the 
multiplication of the time axis by .alpha.. This is understood as well 
from the fact that .delta. has a time (sec) dimension and the physical 
meaning of the parameter .delta. explained in FIG. 7. 
In this way, the normalized target function y(t), the parameter .beta., and 
the parameter .delta. are transformed as follows by the expansion or 
contraction of the time axis (multiplication by .alpha.): 
##EQU23## 
Accordingly, considering the expansion and contraction of the time axis 
(.tau.=.alpha.t) in the target function found by equation (18) and 
equation (19), the result becomes: 
##EQU24## 
and equation (18') and equation (19') are derived equations of the target 
functions Y.sub.a (t) and Y.sub.d (t) enabling free change of the shape of 
the acceleration and deceleration, the amount of movement, and the 
movement time (.alpha.). 
Examples of Application 
Next, the present invention will be explained in further detail by 
illustrating specific normalized target functions set based on the 
principle of the present invention explained above. However, the present 
invention may be applied to any normalized target function. The specific 
examples shown below of course are only examples of the same. 
Triangular Type 
Regarding the triangular acceleration and deceleration pattern shown in 
FIG. 13, the normalized target function y(t) is determined so as to be 
dimensionless, so for example if 
##EQU25## 
the target speed function becomes the following by differentiation of the 
two sides of equation (31) by the time t: 
##EQU26## 
On the other hand, according to equation (4) and equation (5), the 
parameter .beta. is 
##EQU27## 
so from equation (9) and equation (10), the parameter .delta. becomes 
##EQU28## 
In such a triangular acceleration and deceleration pattern, there is the 
advantage that calculation at the time of finding the target function is 
easy, but since the acceleration is discontinuous, there are the 
disadvantages that shock is given to the servo system and the movement 
time cannot be shortened. 
Sine Type 
In the sine type deceleration pattern shown in FIG. 14, the following 
stand: 
##EQU29## 
so like with the above-mentioned triangular type, if use is made the 
normalized target functions the same as at acceleration and deceleration 
(y.sub.a (t)=y.sub.d (t)) and the acceleration and deceleration times are 
equal (T.sub.pa =T.sub.pd), the target function becomes as follows if the 
amount of movement is made .DELTA.Y.sub.t : 
##EQU30## 
Note that during deceleration, the above Y(t/.alpha.) may be returned back 
at t=(.alpha.T.sub.p). 
In such a sine type acceleration and deceleration pattern, calculation at 
the time of finding a target function becomes easy and further the 
function is regular sine wave, so phase control of the input and output 
becomes possible. However, there is the defect of a long time until 
stopping. 
Exp Type 
The exponential type acceleration and deceleration pattern is one of the 
optimal target functions considering the characteristic of a positional 
servo loop (so called "f-characteristic"). Further, this target function 
is a function which can be infinitely differentiated at all points, so it 
has the advantage of not causing unnecessary vibration in the mechanical 
system. Further, calculation becomes easier if the servo loop is simulated 
by software. 
FIG. 15A is a signal flow chart of the servo loop. Assuming that as the 
target function the step input of the amount of movement 
.beta..DELTA.Y.sub.t /2 
##EQU31## 
is input, since there is no input faster than the step as Y.sub.i, the 
output changes most quickly by this input. 
In FIG. 15A, if the transfer function of Y.sub.i .fwdarw.Y.sub.0 is 
calculated, it becomes as shown in the following equation (42): 
##EQU32## 
,and T.sub.1, T.sub.2 and T.sub.3 are the time constants in the third 
servo-loop. 
On the other hand, if this equation (41) is subjected to an inverse-Laplace 
transformation, then 
##EQU33## 
If the two sides are differentiated by the time t to find the speed, then 
the result is 
##EQU34## 
If this movement distance Y.sub.0 (t) and speed Y.sub.0 (t) are shown with 
respect to time, then they can be expressed as shown in FIG. 15B and FIG. 
15C. At this time, it is understood that the distance of movement up to 
the time T.sub.p becomes: 
##EQU35## 
Therefore, if the speed Y.sub.0 (t) between 0.ltoreq.t.ltoreq.T.sub.p is 
reversed with respect to the time axis at the time t=T.sub.p, then the 
target functions shown by the dotted lines in FIGS. 15B and 15C are 
obtained. 
This target function is output to the step input of the servo loop shown in 
FIG. 15A, so it is the maximum speed for the servo loop. Further, the 
function is filtered, so it is considered the most suitable as the input. 
Further, the .beta. defined by equation (41) is the same as the parameter 
.beta. of equation (4) defined by the normalized target function as will 
be understood from a comparison of the forms of equation (44) and equation 
(45) and equation (18') and equation (19'). 
Incidentally, the normalized target function of the exp type acceleration 
and deceleration pattern and the parameters .beta. and .delta. may be 
found as follows: 
First, from the form of equation (44), the normalized target function y(t) 
is 
##EQU36## 
By differentiating the two sides by the time t, the result becomes 
##EQU37## 
Therefore, from equation (4), the parameter .beta. is given by 
##EQU38## 
At this time, T.sub.p is an inherent value for the normalized target 
function y(t) and is the same even if the amount of movement changes. 
Further, if the time axis is multiplied by .alpha., the exp (-t/T.sub.1) 
in equation (49) is transformed to exp (-t/.alpha.T.sub.1), but at the 
same time, T.sub.p .fwdarw..alpha.T.sub.p, so in the end 
##EQU39## 
and the same value is obtained. 
Next, the parameter .delta. becomes 
##EQU40## 
but if the time axis is multiplied by .alpha., then the result becomes 
EQU .delta.(.alpha.T.sub.p)=.alpha..delta.(T.sub.p) (52) 
Note that if the signal flow chart shown in FIG. 15A is digitalized and 
realized by software and steps are input, then it is possible to easily 
find the target function even without complicated calculations as shown in 
the above-mentioned equation (44) and equation (45). 
Combinations 
The specific examples described above were examples of use of the same type 
of functions for the normalized target functions of acceleration and 
deceleration, but in the present invention, it is also possible to use 
patterns where the normalized target functions for acceleration and 
deceleration differ. 
For example, the acceleration and deceleration pattern shown in FIG. 16A is 
an example of use of a triangular type acceleration pattern and an exp 
type deceleration pattern. Further, the acceleration and deceleration 
pattern shown in FIG. 16B is an example of use of the exp type 
acceleration pattern and the sine type deceleration pattern. 
In short, which kind of acceleration and deceleration pattern to use may be 
determined by the characteristics of the servo loop, the characteristics 
of the load of the actuator, etc., for example. 
Note that the embodiments explained above were given to facilitate the 
understanding of the present invention and were not given to limit the 
present invention. Accordingly, the elements disclosed in the above 
embodiments include all design modifications and equivalents falling under 
the technical scope of the present invention. 
For example, when the above-mentioned normalized target functions are 
complicated and the types of the same are complicated, if real time 
processing is required, a load may be placed on the calculation time of 
the CPU. In such a case, when the system is started up and initialized, 
the normalized target functions are calculated in advance Just once and 
the results are stored in the memory. Further, when performing processing 
in real time, it is possible to perform linear interpolation in accordance 
with need while referring to the memory. 
As explained above, it is possible to freely set the acceleration and 
deceleration pattern regardless of the amount of movement or the movement 
time and as a result it is possible to improve the precision of stopping 
at the time of stopping and to shorten the movement time. Further, changes 
in the movement time and movement distance may be flexibly dealt with as 
well, so it becomes possible to freely design characteristics of the 
movement time and movement distance. 
Such an application can be expected to be of broad use in the control of 
various types of PTP (point to point) operations such as robot control, 
temperature control, camera focus control, slide control of a CD player, 
video disk, and the like, DC point control of electrical circuits, and the 
like.