Title: Method for correcting position error in navigation system
Abstract: A method for correcting position error in a navigation system enables one to more accurately match a position measurement of a moving object on a digital map. Particularly, the method comprises the steps of: receiving a current position measurement of a moving object from GPS/DR (Dead Reckoning)-based information; correcting the current location measurement using a displacement-corrected value; performing map matching using the corrected current position measurement; calculating variation of correction angle by extracting a current correction angle out of the map matching result; converting and correcting a previous displacement-corrected value to the current correction angle; and compensating the displacement-corrected value by applying a predetermined constant to the converted displacement-corrected value, and storing the compensated displacement-corrected value.
Patent Number: 6,931,322 Issued on 08/16/2005 to Jung, et al.
| Inventors:
|
Jung; Mun Ho (Seoul-si, KR);
Joe; Moon Jeung (Anyang-si, KR)
|
| Assignee:
|
LG Electronics Inc. (Seoul, KR)
|
| Appl. No.:
|
652031 |
| Filed:
|
September 2, 2003 |
Foreign Application Priority Data
| Sep 02, 2002[KR] | 10-2002-0052413 |
| Current U.S. Class: |
701/216; 701/214; 342/357.14 |
| Intern'l Class: |
G06F 019/00 |
| Field of Search: |
701/214,216,217,208
342/357.14,357.13,358
340/995.25,995.28
|
References Cited [Referenced By]
U.S. Patent Documents
Other References
Abstract for Japanese Publication No. 2002-213979, published Jul. 31, 2002.
Abstract for U.S. Patent No. 5,394,333, issued Feb. 28, 1995.
|
Primary Examiner: Zanelli; Michael J.
Attorney, Agent or Firm: Birch, Stewart, Kolasch & Birch, LLP
Claims
1. A method for correcting position error in a navigation system, the method
comprising the steps of:
receiving a current position measurement of a moving object from GPS/DR (Dead
Reckoning)-based information;
correcting the current location measurement using at least one displacement-corrected
value;
performing map matching using the corrected current position measurement;
calculating variation of correction angle by extracting a current correction
angle out of the map matching result;
converting and correcting the displacement-corrected value in view of the current
correction angle; and
compensating the displacement-corrected value by applying a predetermined constant
to the converted displacement-corrected value, and storing the compensated displacement-corrected
value.
2. The method according to claim 1, wherein the variation of correction angle
is a difference between a current correction angle being calculated and a previously
map-matched correction angle, the current correction angle being calculated based
on the map-matched position of the moving object and a link angle.
3. The method according to claim 1, wherein the current correction angle is calculated
from a formula, atan
2 {(MMx-;GPSx), (Mmy-;GPSy)}, wherein GPSx and GPSy
denote X-axis and Y-axis measurements of the current position of the moving object;
and MMx and Mmy are map-matched X-axis and Y-axis position measurements obtained
by matching GPSx and GPSy with the map.
4. The method according to claim 1, wherein the displacement-corrected values
are obtained in X-axis and Y-axis, respectively.
5. The method according to claim 1, wherein the displacement-corrected values
are obtained by Equations below:
wherein, GPSx and GPSy are current location measurements of the moving object
in X-axis and Y-axis; MMx and Mmy are map-matched X-axis and Y-axis position measurements
obtained by matching GPSx and GPSy with the map; and d is an absolute correction
distance, which is a straight distance between the current location measurement
and the map-matched link position.
6. The method according to claim 5, wherein the absolute correction distance
(d) is a straight distance between a GPS/DR-based location measurement and a map-matched
location measurement.
7. The method according to claim 5, wherein the absolute correction distance
(d) is obtained from Equation of d=√{square root over ((x
1,x
2)
2+(y
1,y
2)
2)}{square
root over ((x
1,x
2)
2+(y
1,y
2)
2)},
wherein x
1 denotes a previously map matched x-displacement corrected value;
y
1 denotes a previously map matched y-displacement corrected value; x
2
denotes a converted x-displacement corrected value of previous map matching using
a current correction angle; and y
2 denotes a converted y-displacement corrected
value of previous map matching using a current correction angle.
8. The method according to claim 7, wherein the converted x-displacement corrected
value (x
2) of previous map matching using the current correction angle is
obtained by x
2=d* sin(θ
2), and the converted y-displacement
corrected value (y
2) of previous map matching using the current correction
angle is obtained by y
2=d* cos(θ
2), in which θ
2
indicates an angle between a new GPS/DR position measurement and a map-matched
value plotted on a Cartesian coordinate system using the direction of true north
as a reference line.
9. The method according to claim 1, wherein a compensation coefficient of the
displacement-corrected value indicates a directional compensation value along heading
of the moving object, being +1 or -;1.
10. The method according to claim 1, wherein a sign of a compensation coefficient
of the displacement-corrected value changes if a variation angle between the current
correction angle and the previous correction angle is less than a predetermined
level, and if GPS and DR values change from the left side to the right side on
a link.
11. The method according to claim 1, wherein a sign of a compensation coefficient
of the displacement-corrected value changes if a variation angle between al ink
angle and the current correction angle is in a predetermined range, and the corrected
GPS value is located at the right side of a link.
12. The method according to claim 1, wherein compensated displacement-corrected
values to which a compensation coefficient is respectively added are represented
by Equations below:
13. A method for correcting position error in a navigation system, the method
comprising the steps of:
receiving a current position measurement of a moving object by making use of
a GPS/DR (Dead Reckoning) method;
correcting the current location measurement using a previous displacement-corrected
value;
performing map matching using the corrected current position measurement; and
calculating out of the map matching result a variation of correction angle, and
a compensated displacement-corrected value along heading by extracting a current
correction angle out of the map matching result.
14. The method according to claim 13, further comprising the steps of:
after map matching is performed, calculating a variation angle by extracting
correction angle out of the map matching result;
converting and correcting the previous displacement-corrected value in view of
the current correction angle; and
calculating a displaced-corrected value by compensating the displacement-corrected
value by applying to the converted displacement-corrected value a predetermined
constant opposite to a correction direction, and storing the compensated displacement-corrected
value.
15. A method for correcting position error in a navigation system, the method
comprising the steps of:
receiving a current position measurement of a vehicle obtained from a GPS signal
and sensors installed in the vehicle;
correcting the current position measurement of the vehicle to a previously map-matched
x- and y-displacement corrected values, and performing map matching on the corrected
values;
calculating a current correction angle by extracting a map-matched position of
the vehicle and a link angle;
calculating variation of correction angle between the current correction angle
and a previously map-matched correction angle, and deciding whether the variation
of correction angle is less than a predetermined angle;
if the variation of correction angle is not less than the predetermined angle,
initializing x- and y-displacement corrected value to '0'; and
storing x- and y-displacement corrected values being calculated.
16. A method for correcting position error in a navigation system, the method
comprising the steps of:
receiving a GPS position measurement;
extracting candidate links within a predetermined search range around the GPS
position measurement;
among a plurality of interpolation points on the candidate links, selecting a
spot between interpolation points with highest possibilities; and
among candidate links including the selected spot between interpolation points,
selecting a link with a highest possibility of having a moving object, and performing
map matching.
Description
This nonprovisional application claims priority under 35 U.S.C. §119(a)
on Patent Application No. 10-2002-0052413 filed in KOREA on Sep. 2, 2002, which
is herein incorporated by reference.
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates in general to a navigation system, more particularly,
to a method for correcting position error in a navigation system which utilizes
a map matching (MM) in order to provide an accurate information on a current position
of a moving object by correcting position errors in consideration of position measurements
of a mobile object including vehicles, airplanes and ships, and the shape of a
road (i.e. link) displayed on a digital map.
2. Discussion of the Background Art
An essential element of navigation systems is the determination of a current
position
of a moving object (e.g. vehicles) within an allowable range of error, and providing
such information to a driver of the moving object continuously in real time mode
because the moving object's current position is used for marking a current position
of the mobile object, providing route guidance information, and perceiving the
surrounding road environment.
Providing incorrect current positions of the moving object or providing
position information discontinuously may cause serious damages on the safety of
an operator of the moving object, so the determination of current positions of
the moving object is very important.
Navigation systems do not always provide accurate information on a current
position of the moving object. That is, the error range of sensors such as a gyro
(or gyroscope) or electronic compass built in the moving object could be great,
depending on rotation direction of the moving object. Also, when the moving object
stops, its heading may be deflected in one direction or be changed, making position
errors on heading of the moving object. Moreover, measurement of traveling distance
of the moving object using a wheel sensor can be erroneous also because the wheel
sensor is very sensitive to external factors like air pressure of wheels/tires
and road surface conditions.
Another material for informing the current position of the moving object
is a digital map that is drawn over a central line (i.e. a solid yellow line).
Many times, however, operators of moving objects do not drive along an intersection
or central line on a road, and as a result, the digital map itself have errors
thereon. In addition, errors are also made when inputting data in the digital map,
marking the position of the moving object at an incorrect spot. Of course, these
errors should be absolutely corrected.
As an attempt to solve the above problems, Korean Patent No. 216535 (Applin.
No.
1997-24564) disclosed a method for measuring a current position of a moving object.
According to the above disclosure, a dead reckoning system (DR) using information
collected from a heading and distance sensors attached to a moving object provides
a current position of the moving object, which is designated as a first position,
and by a map matching method using position matching, the first position is corrected
or maneuvered onto a digital map, and this corrected position is designated as
a second position. Further, the current position of the moving object is measured
using a signal provided from GPS, and the current position at this time is designated
as a third position. Similar to before, applying the map matching method using
position matching, the third position is corrected or maneuvered onto the digital
map, and this corrected position is designated as a fourth position. With these
first through fourth positions, short distance driving patterns of an operator
of the moving object are obtained. Afterwards, similarities between each of the
short distance driving patterns at the first through fourth positions and road
patterns on the digital map are compared with each other, and from the comparison,
a position having the highest degree of similarity is finally selected as the current
position of the moving object.
However, the above technique did not really use the result of map matching
to trace the moving object. Instead, combinations of the DR and GPS systems, namely
signals from sensors and GPS, were mainly used to match the current position of
the moving object being measured with the digital map. Therefore, it was rather
difficult to do perfect matching of the position of the moving object with the
digital map.
For example, depending on geographical features in an area where the moving object
is at, the roads are sometimes straight but sometimes they are curved. The position
of the moving object on a straight road can be easily matched with the digital
map, but on a curved road, accurate positioning is very difficult.
SUMMARY OF THE INVENTION
An object of the invention is to solve at least the above problems and/or disadvantages
and to provide at least the advantages described hereinafter.
Accordingly, one object of the present invention is to solve the foregoing
problems by providing a method for correcting position error in a navigation system,
enabling to match a measurement position of a moving object with a digital map.
Another object of the present invention to provide a method for correcting
position error in a navigation system, capable of accurate matching of a current
position of a moving object with a link on a digital map by measuring the current
position of the moving object, adding to a measurement a displacement-corrected
value obtained from a previous map matching so as to correct the measurement, and
performing map matching.
Another object of the invention to provide a method for correcting position
error in a navigation system, in which a current position of a moving object is
measured by using received signals based on a dead reckoning and GPS systems, and
the measurement is added with a measurement a X- and Y-displacement-corrected values
obtained from a previous map matching so as to correct the measurement of the current
position of the moving object, and map matching is performed thereon.
Still another object of the present invention is to provide a method for correcting
position error in a navigation system, in which measurement position of a moving
object is used to find a node between interpolation points of a candidate link
having a highest possibility to be matched on a map.
The foregoing and other objects and advantages are realized by providing a method
for correcting position error in a navigation system, the method including the
steps of: receiving a current position measurement of a moving object from GPS/DR
(Dead Reckoning)-based information; correcting the current location measurement
using a displacement-corrected value; performing map matching using the corrected
current position measurement; calculating variation of correction angle by extracting
a current correction angle out of the map matching result; converting and correcting
a previous displacement-corrected value to the current correction angle; and compensating
the displacement-corrected value by applying a predetermined constant to the converted
displacement-corrected value, and storing the compensated displacement-corrected value.
Another aspect of the invention provides a method for correcting position
error in a navigation system, the method including the steps of: receiving a current
position measurement of a moving object by making use of a GPS/DR (Dead Reckoning)
method; correcting the current location measurement using a previous displacement-corrected
value; performing map matching using the corrected current position measurement;
and calculating out of the map matching result a variation of correction angle,
and a compensated displacement-corrected value along heading by extracting a current
correction angle out of the map matching result.
Another aspect of the invention provides a method for correcting position
error in a navigation system, the method including the steps of: receiving a GPS
position measurement; extracting candidate links within a predetermined search
range around the GPS position measurement; among a plurality of interpolation points
on the candidate links, selecting a spot between interpolation points with highest
possibilities; and among candidate links including the selected spot between interpolation
points, selecting a link with a highest possibility of having a moving object,
and performing map matching.
Additional advantages, objects, and features of the invention will be
set forth in part in the description which follows and in part will become apparent
to those having ordinary skill in the art upon examination of the following or
may be learned from practice of the invention. The objects and advantages of the
invention maybe realized and attained as particularly pointed out in the appended claims.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention will be described in detail with reference to the following drawings
in which like reference numerals refer to like elements wherein:
FIG. 1 is a block diagram of the configuration of a navigation system according
to one embodiment of the present invention;
FIG. 2 is a flow chart describing a method for correcting position error in
a navigation system according to one embodiment of the present invention;
FIG. 3 is a diagram depicting a calculation method for correction displacement
using a changed angle of map matching according to one embodiment of the present invention;
FIG. 4 is a diagram depicting a method for correcting a GPS and dead reckoning
(DR) positions using a previous map matching value according to one embodiment
of the present invention; and
FIG. 5 is a flow chart describing a map matching method according to one embodiment
of the present invention.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
The following detailed description will present a method for correcting position
error in a navigation system according to a preferred embodiment of the invention
in reference to the accompanying drawings.
FIG. 1 is a block diagram of the configuration of a navigation system according
to one embodiment of the present invention.
As shown in FIG. 1, the navigation system includes a GPS receiver
100
for
detecting current position information of a moving object by using a signal transmitted
from a plurality of satellite GPSs, a sensor
110 for detecting driving speed
and rotational direction of the moving object by making use of a gyro and speed
sensor installed in the moving objection, a memory
120 for storing digital
map information, information on the moving object's driving path and an operational
program, a controller
130 for deciding a current position of the moving
object by correcting the moving object's current position measured using output
signals from the GPS receiver
100 and the sensor
110 into a map-matched
value on a digital map, the controller
130 being operated in accordance
with the operational program stored in the memory
120, and a display
140,
under control of the controller
130, for displaying the digital map and
the current position of the moving object on the digital map.
A method for correcting position error in a navigation system with the configuration
depicted in FIG. 1 is now illustrated in FIG. 2
First of all, a controller
130 receives a measurement (GPSx, GPSy of
a current position of a moving object (S
202) where the measurement is made
by using output signals from a GPS receiver
100 and DR-based sensor
110.
Adding the measurement (GPSx, GPSY) of the current position of the moving object
to a displacement-corrected value obtained from previous map matching, a new measurement
of the current position is obtained (S
204). The displacement-corrected value
from a previous map matching is stored in a memory
120. Here, the map matching
displacement-corrected value can be obtained from Equations 2 and 3 which are explained later.
This new corrected measurement of the current position in step
204 is
then matched with a digital map stored in the memory
120 (S
206).
That is, for more precise map matching, the measurement of the current position
of the moving object is matched with a road on the digital map, and the position
measurement is corrected accordingly.
Next, the map matching result in step
206, namely map-matched position
values of the moving object (MMx, MMy), and heading of a link are extracted (S
208).
And, a current correction angle is calculated applying Equation 1 below (S
210).
Current correction angle=
atan
2{(
MMx-;GPSx),
(
MMy-;GPSy)} [Equation 1]
As shown, to obtain the current correction angle, GPSx value is first subtracted
from a map-matched x value (MMx) of the moving object from the true north direction,
and similarly, GPSy value is first subtracted from a map-matched y value (MMy)
of the moving object. Then, the resulting (x, y) is multiplied by a tan
2.
That is, moving direction of the object can be changed in accordance with the current
correction angle.
When the calculation of the correction angle is finished, a variation of the
correction angle is calculated by subtracting the previously map-matched correction
angle from the current correction angle obtained in step
210 (S
212).
In this way, one can detect how much the current position has been displaced.
Next, a constant (K) (refer to Equation 2 below) is substituted with '1' (S
214).
The constant (K) is a compensation coefficient for the decision of direction, and
it compensates x- and y-displacement-corrected values in the opposite direction
to the horizontal motion (right/left) from a link, namely (+1, -;1).
If the variation of correction angle calculated in step
212 is greater
than a predetermined angle (S
216), x- and y-displacement-corrected values
are all initialized to '0' and reset (S
218), restarting the calculation
of correction values of measurement position. For example, if the variation of
correction angle is greater than 60 degrees, it means the moving object made a
left or right turn, so both x- and y-displacement-corrected values are initialized
to '0'. The angle can also be set to be in opposition to the true north direction
by the link connecting nodes on a driving path.
Meanwhile, if the variation of correction angle being calculated is less
than a predetermined angle, the controller
130 changes the current position
measurement to the left/right side of the link on the digital map. That is to say,
it is decided whether a right previous position of the moving object has been shifted
to the right side of the link from the left side or whether the object has been
shifted to the left side of the link from the right side. If the current position
measurement happens to be changed from left to right of the link on the digital
map in step
222, the sign of the constant (K) is changed (S
220).
In addition, it is decided if the variation of correction angle is within a range
of 0-180 degrees. If so, that is, if the corrected position of the moving object
is positioned on the right side of the link on the digital map (S
226), the
sign of the constant (K) is changed (S
224), and using the difference between
the map matched position and the corrected GPS position, x and y values corresponding
to the absolute correction distance and the constant (K), new x- and y-displacement-corrected
values are obtained (S
228). These x- and y-displacement-corrected values
of the measurement position are then stored/returned for next map matching.
In other words, new x- and y-displacement-corrected values obtained from Equation
2 are respectively replaced with ones for the current position of the moving object.
y-displacement corrected value=(
MMy-;GPSy)+
K*d*cos(link angle+90°)
In the equations, 'd' denotes an absolute correction distance, and 'sin(link
angle+90°)'
and the 'cos(link angle+90°)' are current correction angles. To make a link
angle connect nodes from the true north direction, 90 degrees are added to the
link angle.
Described in the following is a calculation method of GPS/DR positions
and correction displacement using map matching.
As depicted in FIG. 3, when an object moves in a certain direction within a search
range
300 on a position measurement map, x- and y-displacement-corrected
values passing an interpolation point
322 on a link
323 between nodes
321 are calculated.
More specifically, previous/current GPS/DR position measurements
301
and
302 and previous/current position values
311 and
312 by
map matching are obtained, and a beeline (d) between the position measurements
by GPS/DR and the position measurement by map matching is designated as an absolute
correction distance. Here, the previously map-matched position value
311
is obtained by using the previous GPS/DR position measurement
301 and the
absolute correction distance, and the currently map-matched position value
312
is obtained by using the current GPS/DR position measurement
302 and the
absolute correction distance.
To be short, as illustrated in FIGS.
3(
a) and
3(
b),
if a first GPS/DR position
301 and a first MM position
311 are plotted
on a Cartesian coordinate system, it corresponds to the previously map-matched
displacement value (x
1, y
1)
331. Similarly, if a newly map-matched
GPS/DR position
302 and a second MM position
312 are plotted on the
Cartesian coordinate system, it corresponds to the displacement-corrected value
(x
2, y
2)
332 that is obtained from the previous map matching
using the current correction angle. Here, the displacement value from the previous
map matching (x
1, y
1) coincides with the point
331, and the
new displacement value (x
2, y
2) coincides with the point
312.
At this time, as shown in FIG.
3(
b), in case the corrected values
331 and
332 are obtained, the angle from the true north direction
to the corrected value
331 is designated as θ
2, and the
angle to the displacement value is designated as θ
1.
Then the sum of a new GPS value with the absolute correction distance (d) obtained
previously is designated as a new GPS/DR. FIG. 4 illustrates GPS/DR positions corrected
by using the previous map matching value within a search range
400 while
an object is moving. Using GPS/DR position measurements
401,
403,
and
405 and map matched value
402,
404 and
406, it
is possible to obtain a new GPS/DR value
407 that is obtained by adding
a new GPS/DR value
403 to a previous absolute correction distance (d). A
spot
408 can be obtained using the same method.
In FIG. 4, at the spots
403 and
407 where the direction is changed
from the left to the right having the link as a reference line, the sign of the
constant (K) is not changed to (+1). However, the sign of the constant (K) is changed
to (-;1) at the spot
408 in order to change the direction from the right
to the left.
The absolute correction distance (d) is obtained from Equation 3.
Here, x
1 denotes a previously map matched x-displacement corrected value;
y
1 denotes a previously map matched y-displacement corrected value; x
2
denotes a converted x-displacement corrected value of previous map matching using
a current correction angle; and y
2 denotes a converted y-displacement corrected
value of previous map matching using a current correction angle, in which x
2=d*
sin(θ
2), and y
2=d* cos(θ
2).
And, (x, y) of a new GPS value can also be obtained from the equation 3 using
the correction angle θ
2 and the absolute correction distance (d) obtained previously.
To obtain such displacement-corrected values, the displacement-corrected values
(x, y) are separately calculated as shown in Equation 4. As manifested in Equation
4, x-displacement corrected value is obtained by subtracting a GPSx value from
a map matched value MMx and adding the new GPS value obtained before. The y-displacement
corrected value can be obtained using the same method.
If a constant (K) is added to those x- and y-displacement corrected values in
Equation 4, respectively, the Equation 2 is obtained. Here, θ
2 is
the sum of a link angle and 90 degrees.
Using the displacement-corrected values obtained from the Equation 4, map matching
is performed, and as for next map matching, the beeline (distance) between the
original GPS value before its input and a link is again designated as the absolute
correction distance (d).
As illustrated in FIGS. 3 and 4, sometimes position errors are made even when
the moving object (i.e. vehicle) is running on the road without any problem. However,
even when positioning errors are made by the position measurement devices in a
navigation system, or when the digital map itself have errors, or when inaccurate
position information is provided because the solid yellow line on the road is used
as a reference regardless of the actual motion of the vehicle, the position error
correction method of the invention enables an accurate map matching on a road link.
In summary, according to the method for correcting position error in a navigation
system of the present invention, the current position measurement of the moving
object is corrected to a previously map matched displacement-corrected value, and
the current position of the moving object is matched with the link on the digital
map. Therefore, accurate map matching can be performed regardless of possible errors
in position measurement on curved roads. Also, even though errors are found in
the digital map itself, and the moving object is marked around the central line
(i.e. solid yellow line) differently from the actual motion of the object, the
present invention method can accurately map match the moving object with the link
on the digital map.
FIG. 5 is a map-matching algorithm of a navigation system according to another
embodiment of the present invention.
Referring to FIG. 5, a position measurement is obtained using information
from GPS and DR (S
502), and using a position measurement device, candidate
links within a certain search range around the position measurement are extracted
(S
504). In this embodiment, interpolation points always exist between links.
Then, a node between interpolation points within each candidate link is extracted
(S
506), and information on a spot with the shortest distance to each interpolation
point on the candidate links is extracted (S
508), and based on this information,
the probabilities of map-matching with the candidate links are calculated (S
510).
Among the spots between interpolation points being searched, one spot between
interpolation points with the highest possibility is selected (S
512), and
the selected spot between interpolation points is added to a candidate list (S
514).
Next, based on the spot between interpolation points having the highest possibility
for each candidate link, one candidate link having the highest possibility to be
map matched is calculated (S
516). Finally, the spot between the interpolation
points on the candidate link having the highest possibility to be map matched is
designated as a map matching value (S
518).
As such, using the position measurement of the moving object, the spot between
interpolation points on the link with the highest possibility to be map matched
is searched. Hence, whether or not the road is curved, map matching can be effectively
performed on the interpolation points on the link with the highest possibility
to be map matched.
While the invention has been shown and described with reference to certain
preferred embodiments thereof, it will be understood by those skilled in the art
that various changes in form and details may be made therein without departing
from the spirit and scope of the invention as defined by the appended claims. For
example, the present invention can also be applied to airplanes, ships, or portable
PDAs and notebook computers having a built-in navigation system.
The foregoing embodiments and advantages are merely exemplary and are not to
be construed as limiting the present invention. The present teaching can be readily
applied to other types of apparatuses. The description of the present invention
is intended to be illustrative, and not to limit the scope of the claims. Many
alternatives, modifications, and variations will be apparent to those skilled in
the art. In the claims, means-plus-function clauses are intended to cover the structures
described herein as performing the recited function and not only structural equivalents
but also equivalent structures.
*
