Method and Apparatus for Determining the Position of a Retroreflective Element

The present invention relates generally to laser-based position sensing systems useful in construction applications and, more particularly, to a method and system for determining the position of a retroreflective element in a predetermined frame of reference and for providing this position to a position sensing system for use in subsequently determining the position of any point in the predetermined frame of reference.

In the construction industry, the level of automation and robotization continues to remain very low. A significant reason for this is the difficulty of properly positioning machines and tools. In the construction of commercial buildings, for example, various points of reference have to be established, such as placement of floors, ceilings, electrical wiring, plumbing, and heating and cooling ducts. Establishing reference points is time consuming and expensive, particularly as such work is often contracted out to companies which specialize in this work.

Various laser-based position sensing systems are known which continually track the position of a reference point in a predetermined frame of reference. For example, commonly assigned U.S. Pat. No. 5,076,690 discloses a position sensing system which calculates the X-Y coordinates of a point using triangulation and determines the direction in which the point is moving. The triangulation calculation is based on the known coordinates of at least three retroreflective elements spaced apart from each other around the periphery of a two-dimensional coordinate frame, and the measured angles between the lines projected radially outward from the point to each of the retroreflective elements. The accuracy of the measured angles is achieved by using a rotating member supported by dedicated hardware and controlled by software. The member rotates with a beam of light generated by a light transmitting and detecting device positionable at the point. The light transmitting and detecting device receives the beam of light reflected back from the retroreflective elements and generates an output signal in response thereto. A computer processes the output signals for use in calculating the X-Y position of the point and the orientation of the light transmitting and detecting device when it is positioned at the point.

The precision of a position sensing system as just described is highly dependent on the ability of the operator to measure the coordinates of the three retroreflective elements accurately . Moreover, each of the respective coordinates of the retroreflective elements must be measured and entered into the computer of the system. Miscalculation of, or errors in entering, the positions of the retroreflective elements could result in time consuming and expensive removal and repositioning of various building fixtures, such as walls, plumbing and heating and cooling ducts.

Consequently, in view of the shortcomings of the above laser-based position sensing system, there is a need for a method and system for easily and accurately determining the position coordinates of the retroreflective elements while concomitantly reducing the likelihood of operator error.

This need is met by the method and system of the present invention which accurately determines the positions of any number of position retroreflective elements from the known coordinates of two benchmark positions. A light transmitting and detecting means and a reference retroreflective element are positioned at a primary benchmark position located at known first coordinates and a secondary benchmark position located at known second coordinates, respectively. The light transmitting and detecting means takes angular measurements by sensing a reflected light beam from the position and reference retroreflective elements. After taking angular measurements in the above configuration, the light transmitting and detecting means and the reference retroreflective element are transposed and additional angular measurements are taken. From the angular measurements, the position of the position retroreflective element is determined in the coordinate frame of reference.

In accordance with one aspect of the present invention, the method for determining the position of at least one position retroreflective element located in a predetermined coordinate frame of reference having an X axis comprises the steps of: (a) providing a light transmitting and detecting means capable of transmitting a rotating beam of light, detecting reflections of the beam of light, providing an index position along the rotation of the beam of light, and providing a reference position which is parallel to said X axis irrespective of the orientation of said light transmitting and detecting means; (b) providing a reference retroreflective element capable of reflecting the beam of light back toward the light transmitting and detecting means; (c) positioning the light transmitting and detecting means at a primary benchmark position at known first coordinates in the coordinate frame of reference; (d) positioning the reference retroreflective element at a secondary benchmark position at known second coordinates in the coordinate frame of reference; (e) transmitting the beam of light to illuminate the at least one position retroreflective element and the reference retroreflective element during a first rotation of the beam of light; (f) monitoring the first rotation of the beam of light as it travels between the index position, the at least one position retroreflective element and the reference retroreflective element; (g) repositioning the light transmitting and detecting means at the second benchmark position and the reference retroreflective element at the primary benchmark position; (h) transmitting the beam of light to illuminate the at least one position retroreflective element and the reference retroreflective element during a second rotation of the beam of light; (i) monitoring the second rotation of the beam of light as it travels between the index position, the at least one position retroreflective element and the reference retroreflective element; and (j) determining the position of the at least one position retroreflective element in the coordinate system based on the first coordinates, the second coordinates, the reference position and the first and second rotations of the beam of light.

Preferably, the steps of monitoring the first and second rotations of the beam of light further include the steps of: mounting a member on a rotating shaft, the member having a periphery and further having a plurality of apertures located at the periphery which divide a revolution of the member into a plurality of generally equal partial revolutions; and registering a time differential between each adjacent pair of the plurality of apertures as the member rotates. The step of interpolating between the each adjacent pair of the plurality of apertures may be provided to determine exact angles between each of the adjacent pair of the plurality of apertures.

In accordance with another aspect of the present invention, the method for determining the position of at least one position retroreflective element located in a predetermined coordinate frame of reference having an X axis comprises the steps of: (a) providing a light transmitting and detecting means capable of transmitting a rotating beam of light, detecting reflections of the beam of light, providing an index position along the rotation of the beam of light, and providing a reference position parallel to said X axis irrespective of said orientation of said light transmitting and detecting means; (b) providing a reference retroreflective element capable of reflecting the beam of light back toward the light transmitting and detecting means; (c) providing an intermediate benchmark position located at third coordinates; (d) positioning the light transmitting and detecting means at a primary benchmark position at known first coordinates in the coordinate frame of reference; (e) positioning the reference retroreflective element at a secondary benchmark position at known second coordinates in the coordinate frame of reference; (f) transmitting the beam of light to illuminate the at least one position retroreflective element and the reference retroreflective element during a first rotation of the beam of light; (g) monitoring the first rotation of the beam of light as it travels between the index position, the at least one position retroreflective element and the reference retroreflective element; (h) repositioning the reference retroreflective element at an intermediate benchmark position; (i) transmitting the beam of light to illuminate the at least one position retroreflective element and the reference retroreflective element during a second rotation of the beam of light; (j) monitoring the second rotation of the beam of light as it travels between the index position, the at least one position retroreflective element and the reference retroreflective element; (k) repositioning the light transmitting and detecting means at the secondary benchmark position; (1) transmitting the beam of light to illuminate the at least one position retroreflective element and the reference retroreflective element during a third rotation of the beam of light; (m) monitoring the third rotation of the beam of light as it travels between the index position, the at least one position retroreflective element and the reference retroreflective element; (n) repositioning the reference retroreflective element at the primary benchmark position; (o) transmitting the beam of light to illuminate the at least one position retroreflective element and the reference retroreflective element during a fourth rotation of the beam of light; (p) monitoring the fourth rotation of the beam of light as it travels between the index position, the at least one position retroreflective element and the reference retroreflective element; (q) repositioning the light transmitting and detecting means at the intermediate benchmark position; (r) transmitting the beam of light to illuminate the at least one position retroreflective element and the reference retroreflective element during a fifth rotation of the beam of light; (s) monitoring the fifth rotation of the beam of light as it travels between the index position, the at least one position retroreflective element and the reference retroreflective element; and (t) determining the position of the at least one position retroreflective element in the coordinate system based on the reference position, the first and second coordinates and at least two of the first, second, third, fourth, and fifth rotations of the beam of light.

The step of determining the position of the at least one position retroreflective element may preferably include the steps of: calculating at least two X coordinate values and at least two Y coordinate values for the at least one position retroreflective element based on the reference position, the first, second, and third coordinates and the first, second, third, fourth and fifth rotations of the beam of light; comparing the at least two X coordinate values to determine a most accurate X coordinate value; comparing the at least two Y coordinate values to determine a most accurate Y coordinate value; and using the most accurate X coordinate value and the most accurate Y coordinate value as the coordinates of the at least one position retroreflective element.

In another embodiment of the present invention, the step of determining the position of the at least one position retroreflective element includes the steps of: (t) calculating a X coordinate value and a Y coordinate value for the at least one position retroreflective element based on the reference position, the first, second and third coordinates and at least two of the first, second, third, fourth and fifth rotation of the beam of light; (u) setting a maximum X derivative value; (v) setting a maximum Y derivative value; (x) calculating a derivative of the calculated X coordinate value to produce a X derivative value; (y) calculating a derivative of the calculated Y coordinate value to produce a Y derivative value; (z) comparing the X derivative value to the maximum X derivative value; (aa) comparing the Y derivative value to the maximum Y derivative value; (bb) using the calculated X coordinate value as the X coordinate of the at least one position retroreflective element if the X derivative value is less than the maximum X derivative value; (cc) using the calculated Y coordinate value as the Y coordinate of the at least one position retroreflective element if the Y derivative value is less than the maximum Y derivative value; (dd) repeating steps (t) through (cc) using the at least two of the first, second, third, fourth and fifth rotation of the beam of light until a X derivative value and a Y derivative value are less than the maximum X derivative value and the maximum Y derivative value respectively.

Preferably, the position of the position retroreflective element of the present invention is used to initialize a light transmitting and detecting means which typically uses at least three retroreflective elements at known locations to determine the position of any point in a given frame of reference.

In accordance with yet another aspect of the present invention, a system for determining the position of at least one position retroreflective element in a predetermined coordinate frame of reference having an X axis is provided. A light transmitting and detecting means, which is positioned at a first benchmark position of known first coordinates and, subsequently, at a second benchmark position at known second coordinates, provides a rotating beam of light which is reflected from a reference retroreflective element and the at least one position retroreflective element. The light transmitting and detecting means further provides an index position along the rotation of the beam of light and a reference position which is parallel to the X axis irrespective of the orientation of the light transmitting and detecting means such that angular measurements may be referenced thereto. A computer means computes the position of the at least one position retroreflective element in the frame of reference from the rotations of the beam of light at the respective positions of the light transmitting and detecting means.

Preferably, the position retroreflective element includes an identification means thereon for reflecting light which includes information generated by the identification means for uniquely identifying a particular position retroreflective element. Additionally, the light transmitting and detecting means may be comprised of means for generating a beam of light, means for projecting the beam of light at a substantially constant rotating angular velocity toward the at least one position retroreflective element and the reference retroreflective element, means for receiving the beam of light reflected from the at least one position retroreflective element and the reference retroreflective element corresponding to the illumination of each of the retroreflective elements during the rotation of the beam, and means for transmitting a plurality of angle signals indicative of a plurality of angles at which the means for receiving the beam of light is positioned with respect to the index position, the at least one position retroreflective element and the reference retroreflective element.

It is thus an object of the present invention to provide a method and system for determining the position of at least one position retroreflective element in a given frame of reference which eliminates the need for manually measuring the location of each position retroreflective element, provides improved accuracy over manual surveying techniques and reduces the possibility of operator error. Other objects and advantageous of the invention will be apparent from the following description, the accompanying drawings and the appended claims.

In order that the invention may be more readily understood, reference will now be made to the accompanying drawings, in which.

FIGS. 1A and 1B are top plan views of a structured environment defined by four intersecting walls and including primary and secondary benchmark positions at known coordinates, a light transmitting and detecting means, a retroreflective element and a position retroreflective element in which the method and system of the present invention may be advantageously applied;

FIGS. 2A and 2B are top plan views of a structured environment defined by four intersecting walls and including primary, secondary and intermediate benchmark positions, a light transmitting and detecting means, a reference-retroreflective element and three position retroreflective elements in which an alternative embodiment of the method and system of the present invention may be advantageously applied;

FIG. 3 is a side view, with parts broken away, of a light transmitting and detecting means of a laser-based positioning system;

FIG. 4 is an exploded partial top view of a rotating member illustrated in FIG. 3; and

FIG. 5 is a schematic block diagram of a hardware interface, controlled by software, which supports the initializing system of the present invention.

The method and system of the present invention are advantageously utilized in either a structured environment 100, as shown in FIGS. 1A to 2B, or an external environment, to determine the coordinate position of a position retroreflective element 102 in a given coordinate frame of reference. In FIG. 1A, light transmitting and detecting means 104, best illustrated in FIG. 3, and a reference retroreflective element 106 are shown positioned at a primary benchmark position 108 located at known first coordinates and a secondary benchmark position 110 located at known second coordinates, respectively. A light beam 112, generated by the light transmitting and detecting means 104, strikes each of the retroreflective elements 102 and 106 producing reflections of the beam 112 which are detected by the light transmitting and detecting means 104. After taking angular measurements in the above configuration, the light transmitting and detecting means 104 and the reference retroreflective element 106 are transposed, as shown in FIG. 1B, and additional angular measurements are taken. From the angular measurements, the position of the position retroreflective element 102 is determined in the coordinate frame of reference.

Position sensing systems using a light transmitting and detecting means in which the present invention may be advantageously employed are known in the art. For example, such a system is disclosed in commonly assigned U.S. Pat. No. 5,076,690 which is incorporated herein by reference. A brief description of such a light transmitting and detecting means 104 will now be given with reference to FIGS. 3 through 6.

Referring now to FIG. 3, a side diagrammatic view of a housing 300 containing the light transmitting and detecting means 104 is shown. The light transmitting and detecting means 104 includes an electric motor 302 mounted to rotate a shaft 304. A member 306, such as a code wheel, and a light diverting mirror 308 are mounted on the shaft 304. As described more fully below, a member pick up element 309 detects the rotation of the member 306. Member pick up element 309 further detects an index element, shown as 402 in FIG. 4, and produces a single reference pulse indicating an index position, shown as line 114 in FIGS. 1A and 1B, along the rotation of the beam of light 112 in response thereto. As one skilled in the art will appreciate, index element 402 may be of any configuration which produces a reference pulse that is distinguishable from signals produced by pick up elements 400. A light source, such as solid state laser 312, directs the beam of light 112 onto the rotating mirror 308 so that a plane of rotation is created.

When the rotating laser beam 112 strikes the retroreflective elements 102 and 106 during a revolution of the shaft 304, the beam of light 112 is reflected back to the light transmitting and detecting means 104 and is transformed into an analog signal by a suitable detector, such as photodetector 305, and transmitted to a signal processing means 502 of FIG. 5, which outputs two digital signals. The light transmitting and detecting means 104 may include means for diverting and focusing 310 the returning beam to the photodetector 305. In FIG. 3, the rotating mirror 308 diverts the light beam 112 toward a collimating lens 314, which lens 314 focuses the light beam 112 toward the photodetector 305.

As shown in FIG. 4, the member or code wheel 306 has a plurality of angularly positioned elements, preferably apertures 400, spaced around its periphery which divide a revolution of the member 306 into a plurality of generally equal partial revolutions. The index element 402, as previously described, provides an indication of the index position 114. The size and spacing of these apertures are greatly exaggerated in the drawing for clarity of illustration. For example, the code wheel 306 may divide a revolution into one thousand generally equal parts positioned approximately 0.36 degrees apart by spacing one thousand elements or apertures 400 around the periphery of the member 306.

Preferably, angles between pairs of adjacent, such as 400a and 400b, are measured to a great degree of precision. This information is then stored in computer 522. Therefore, even though the spacings between apertures are not exactly equal, it is possible to make accurate angular measurements with the laser-based positioning system. Interpolation is used for angular measurements that fall between apertures too. Since any speed fluctuation of the motor 302 between two adjacent apertures 400, particularly when there are one thousand such apertures spaced around the periphery of the member 306, will be negligible, it is possible to interpolate between an adjacent pair of apertures to determine an exact angle of a point M between the pair of apertures 400a and 400b, according to the equation

where <400a is the measured angle between the index position 114 and the aperture 400a; Tm is the time elapsed between passage of the previous aperture, here aperture 400a, and the moment M in time that the reflecting light strikes the sensor or photodetector 305; and Tcw is the time it takes the code wheel 306 to move between element 400a and element 400b.

The position sensing system which combines the use of the code wheel 306 and the motor 302 is supported by a hardware interface 504, illustrated in FIG. 5. An event occurs every time an aperture 400 on the code wheel 306 passes, or a retroreflective element commences or ends a reflection of the beam of light 112. Due to the high precision time measurements required between each adjacent pair of apertures 400, a reference clock 506 is used in keeping a record of an event. If an event occurred during this time, it is stored in a circuit 508, such as a 16 bit first-in-first-out circuit. The circuit 508 records the movement of the code wheel 306 at register 0. The actual element or aperture 400 which is currently passing is sensed at the member or code wheel pick up element 309 and counted by a member rotation counter 512. Each time the member 306 has completed a full rotation, the member pick up element 309, in response to the index element 402, sends an index position signal to reset the member rotation counter 512. The member pick up element 309 comprises means for detecting movement of each of the elements 400 past a predetermined point as the member 306 rotates. In a preferred embodiment of the present invention, the pick up element 309 comprises a light source paired with a photodetector element.

Signal processing means 502 detects when receiving optics 516, consisting of collimating lens 314 and photodetector 305, is either commencing receipt of the reflection of the light beam 112 or ending receipt of the reflection of the light beam 112 from a retroreflective element to the light transmitting and detecting means 104. Signal processing means 502 can transform the analog signal from photodetector 305 into two digital signals which are received at register 1 of circuit 508. The first digital signal represents a START signal which indicates that the reflection from a retroreflective element is commencing the reflection of beam 112 from the retroreflective element to the light transmitting and detecting means 104, and the second digital signal is an END signal which indicates that the reflection is ending.

Alternatively, the signal processing means 502 digitizes the analog signal from photodetector 305. The digital signal is then analyzed to determine when a reflection of the beam 112 from a retroreflective element is starting and when the reflection is ending. Such analysis may include indicating successive start and end signals as the digital signal exceeds and falls below a certain value.

Referring again to FIG. 5, register 2 receives signals for measuring the time elapsing between the passage of the last aperture 400 and an event, which event may be the time Tm or the time Tcw shown in FIG. 4. A clock pulse counter 518 is reset at every passage of an aperture 400 as detected by the element 309 and, consequently, the counter 518 counts the time elapsing between each pair of adjacent elements 400.

The circuit 508 stores the information received and provides an output signal 520 to the computer 522 which includes a microprocessor having memory means. The computer 522 is responsive to the output signal 520 and computes the coordinates of the position retroreflective element 102 in the coordinate frame of reference 100. After determining the position of at least three retroreflective elements, the computer 522 can compute the position of the light transmitting and detecting means 104 at any point in the frame of reference. It will be apparent to one skilled in the art that the position of any position retroreflective element can be determined either by the method of the present invention or by conventional surveying techniques.

Preferably, an identification means, such as a conventional bar code pattern, is located on the face of the retroreflective elements. The identification means provides information in the light reflected from the retroreflective element such that the computer 522 can uniquely identify each retroreflective element.

The method for determining the position of a position retroreflective element in a predetermined coordinate frame of reference using the primary and secondary benchmark positions 108 and 110 in accordance with the present invention will now be described with reference to FIGS. 1A and 1B. Initially, the coordinate positions of the primary benchmark position 108, hereinafter designated as PX and PY, and the secondary benchmark position 110, hereinafter designated as SX and SY, are determined, for example, by conventional surveying techniques. By using these coordinates, the present invention can determine the position of one or more retroreflective elements in the frame of reference. The coordinates are entered into the computer 522, having a microprocessor (not shown), such as a Motorola 68332. Appropriate software for controlling the above described hardware in accordance with the present invention is stored in memory associated with the microprocessor. An exemplary listing of the software is provided below.

As shown in FIG. 1A, the light transmitting and detecting means 104 is positioned at the primary benchmark position 108 and the reference retroreflective element 106 is positioned at the secondary benchmark position 110. The rotating beam of light 112 illuminates the position retroreflective element 102 and the reference retroreflective element 106. A first angle .gamma..sub.p between the index position 114, generated by the index element 402, and the reference retroreflective element 106, located at the second benchmark position 110, and a second angle .gamma..sub.1p between the index position 114 and the position retroreflective element 102 are measured by detecting the reflected light beam 112 from each of the respective retroreflective elements 102 and 106 during a rotation of the light beam I12.

The computer 522 initially establishes, based on the given coordinates of the primary and secondary benchmark positions 108 and 110, a coordinate frame of reference. From the coordinate frame of reference, the computer 522 continually calculates a reference position, shown as a reference line 115 in FIGS. 1A through 2B, which is always parallel to the X axis of the coordinate frame of reference, irrespective of the position of the light transmitting and detecting means 104. A primary-secondary axis angle .theta..sub.p is then calculated by the computer 522 relative to the reference line 115.

The light transmitting and detecting means 104 is then repositioned at the secondary benchmark position 110 and the reference retroreflective element 106 is repositioned at the primary benchmark position 108, as shown in FIG. 1B. The light beam 112 is again rotated to illuminate the position retroreflective element 102 and the reference retroreflective element 106. A third angle .gamma..sub.s between the index position 114 and the reference retroreflective element 106, located at the first benchmark position 108, and a fourth angle .gamma..sub.1s between the index position 114 and the position retroreflective element 102 are then measured based on the detected reflections of the light beam 112. A secondary-primary axis angle .theta..sub.s is then determined by the computer 522 relative to the reference line 115 which is parallel to the X axis of the coordinate frame of reference. Based on the known first coordinates, PX and PY, and the known second coordinates, SX and SY, of the respective primary and secondary benchmark positions 108 and 110 and the measured angles to the respective retroreflective elements, the position of the position retroreflective element 102 in the predetermined coordinate frame of reference 100 is determined and stored by the computer 522.

Although the above description is directed to determining the position of a single position retroreflective element 102, it is readily apparent to one skilled in the art that the position of more than one retroreflective element can be determined by the present invention. As noted above, each retroreflective element may have an identification means thereon which produces a reflected light beam having characteristics unique to the particular retroreflective element. Consequently, the computer means 522 can distinguish between each retroreflective element and calculate their positions separately in accordance with the method of the present invention.

In a preferred embodiment of the present invention, the X and Y coordinates of the position retroreflective element 102 are measured by determining a first benchmark angle .alpha.1 between the position retroreflective element 102 and the reference line 115, as measured from the primary benchmark position 108. The first benchmark angle .alpha.1 may be calculated according to the following equation:

wherein .gamma..sub.p is the first angle between the index position 114 and the position retroreflective element 102, .gamma..sub.1p is the second angle between the index position 114 and the secondary benchmark position 110, and .theta..sub.p is the primary-secondary axis angle between the reference line 115 and the secondary benchmark 110.

Similarly, a second benchmark angle .beta.1 between the position retroreflective element 102 and the reference line 115 may be calculated according to the following equation:

wherein .gamma..sub.s is the third angle between the index position 114 and the reference retroreflective element 106, located at the primary benchmark position 108, .gamma..sub.1s is the fourth angle between the index position 114 and the position retroreflective element 102 and .theta..sub.s is the second reference angle between the reference line 115 and the secondary benchmark position 108.

The X and Y coordinate values of the position retroreflective element 102 are then calculated from the first benchmark angle .alpha.1, the second benchmark angle .beta.1 and the known coordinates of the respective benchmark positions PX, PY, SX and SY. The X coordinate value of the position retroreflective element 102 may preferably be calculated by solving the following equation: ##EQU1## The Y coordinate value may be determined using .alpha.1 by solving the following equation:

and using .beta.1 by solving the following equation:

In addition to calculating the X and Y coordinates of the position retroreflective element 102, the following derivatives are also calculated. ##EQU2##

The computer 522 then uses the following criteria to determine which set of calculated coordinates for the position retroreflective element 102 provides the best accuracy.

A. If dx/d.sub.TOT <dx, then T.sub.X is stored as the X coordinate for the position retroreflective element 102 and dx/d.sub.TOT is stored as dx.

B. If dy/d.alpha.1<dy, then T.sub.YA is stored as the Y coordinate for the position retroreflective element 102 and .vertline.dy/d.alpha.1.vertline. is stored as dy.

C. If dy/d.beta.1<dy, then T.sub.YB is stored as the Y coordinate for the position retroreflective element 102 and .vertline.dy/d.beta.1.vertline. is stored as dy.

Thus, by using the aforementioned criteria, the present invention assures that the most accurate measured X and Y coordinates of the position retroreflective element 102 are used.

In an alternative embodiment of the present invention, a third benchmark position, illustrated as an intermediate benchmark position 116 of FIG. 2A, is used to improve the accuracy of the coordinate calculation of position retroreflective element 102. The intermediate benchmark position 116 is located at a third set of coordinates.

Initially, the operator of the light transmitting and detecting means 104 indicates to the computer 522, via an input device (not shown), that the three benchmarks are being used. Referring now to FIG. 2A, it should be noted that the reference retroreflective element 106 is shown in phantom for clarification purposes. The reference retroreflective element 106 will occupy one of the individual phantom areas during a particular step of the method of the present invention, as describe more fully below.

In the intermediate benchmark mode, referencing the angles of the scans to the reference line 115, and therefore to the coordinate frame of reference, is of primary concern, however, some of the scans cannot be referenced until the third coordinates of the intermediate benchmark 116 are known. Therefore, the coordinates of the intermediate benchmark 116 must first be determined. Although conventional surveying techniques may be used to determine the position of the intermediate benchmark position 116, the method of the present invention provides better accuracy and less chance for error. The method of the present invention uses the `intersection of lines` principle to determine the coordinates of the intermediate benchmark position 116. First, the reference retroreflective element 106 is positioned at the secondary benchmark position 110 and the primary-secondary axis angle .theta..sub.p is calculated as indicated above in the two benchmark mode. An angle .tau..sub.0 between a position retroreflective element, shown for example in FIG. 2A at 204, and the secondary benchmark position 116 is also determined.

Next, the reference retroreflective element 106 is repositioned at the intermediate benchmark position 116 and an angle .tau..sub.1, located between the element 204 and the intermediate benchmark position 110, is determined. Then a first interior angle .delta..sub.PI can be determined from the following equation:

The light transmitting and detecting means 104 is then repositioned at the secondary benchmark position 110, as shown in FIG. 2B. With the reference retroreflective element 106 positioned at the intermediate benchmark position 116, angle .tau..sub.3 is measured. The reference retroreflective element 106 is repositioned at the primary benchmark position 108 and angles .tau..sub.2 is measured. A second interior angle .delta..sub.SI is then determined by the following equation:

To determine the primary-intermediate axis angle .lambda..sub.PI and the secondary-intermediate axis angle .lambda..sub.SI, which are both relative to the reference line 115, and thus the coordinate system, the following equations are solved:

The coordinates of the intermediate benchmark position 116 is then calculated using equations (3), (4) and (5), as delineated above, substituting angles .lambda..sub.PI and .lambda..sub.SI for angles .alpha.1 and .beta.1, respectively. Since the intermediate benchmark coordinates, designated IX and IY, are now known, the coordinates of the position retroreflecting element 106 can be measured with increased accuracy. The light transmitting and detecting means 104 and the reference retroreflective element 106 are positioned in five different scan combinations to determine the coordinates of all position retroreflective elements struck by light beam 112. The coordinates of the position retroreflective elements 102, 202 and 204 are then calculated from each scan combination using the same method described above with reference to the two benchmark position procedure.

Specifically, the element coordinates will be calculated for each of the following scan combinations. It should be noted that to be calculated the position retroreflective element must have been acquired in both of the scans.

Scan [0] occurs with the light transmitting and detecting means 104 on the primary benchmark position 108 and the reference retroreflective element 106 on the secondary benchmark position 108. Scan [1] occurs with the light transmitting and detecting means 104 on the primary benchmark position 108 and the reference retroreflective element 106 on the intermediate benchmark position 116. Scan [2] occurs with the light transmitting and detecting means 104 on the secondary benchmark position 108 and the reference retroreflective element 106 on the intermediate benchmark position 116. Scan [3] occurs with the light transmitting and detecting means 104 on the secondary benchmark position 108 and the reference retroreflective element 106 on the primary benchmark position 110. Finally, Scan [4] occurs with the light transmitting and detecting means 104 on the intermediate benchmark position 116 and the reference retroreflective element 106 on the primary benchmark position 108. The coordinate calculation derivatives, as determined from equations (6) through (10), will again be used to determine which values are the most accurate.

Alternatively, the derivatives may be calculated utilizing successive scan combinations until the derivative values for both the X coordinate and the Y coordinate are less than predetermined maximum X and Y derivative values. For example, X and Y coordinate values are initially calculated from scan [0] and scan [2]. The derivatives of the just calculated X and Y coordinates are calculated as described above using equations (6) through (10). The resulting X and Y derivative values are then compared to their respective maximum X and Y derivative values. If the X derivative value is less than the maximum X derivative value, then the calculated X coordinate value is used as the X coordinate of the position retroreflective element. Similarly, if the Y derivative value is less than the maximum Y derivative value, the calculated Y coordinate value is used as the Y coordinate of the position retroreflective element. In the event that one or both of the calculated X and Y derivative values are not less than their respective maximums, successive scan combinations, as listed above, are employed until acceptable X and Y coordinate values of the position retroreflective element are determined.

In another embodiment of the present invention, the method of the present invention is used to initialize, or set up, a position sensing system, such as the system briefly discussed above and further disclosed in U.S. Pat. No. 5,076,690, to allow operation of the system in a frame of reference where the position of one or more retroreflective elements is initially unknown. After determining the positions of the retroreflective elements, the position sensing system can then determine the position of a point in a predetermined coordinate frame of reference. The operation of the prior art position sensing system will not be further discussed herein since the present invention only concerns the initial determination of the positions of one or more of the retroreflective elements.

To initialize the position sensing system, the operator enters the coordinates of the primary and secondary benchmark positions 108 and 110 into the computer 522. The light transmitting and detecting device 104 is positioned at the primary benchmark position 108 and the reference retroreflective element 106 is positioned at the secondary benchmark position 110. The appropriate angle measurements are then taken as described above. The light transmitting and detecting means 104 and the reference retroreflective element 106 are transposed and further angular measurements are taken. After the positions of at least three stationary retroreflective elements are known, the position sensing system is setup to continuously determine the position of a point in the coordinate frame of reference 100. In addition, a third benchmark, as described above, may preferably be utilized to improve the accuracy of the initialization procedure.

A representative listing of exemplary software stored in the computer 522 for performing angle measurements for both the two benchmark and three benchmark procedures, determining the coordinates of at least one retroreflective element and initializing a position sensing system is as follows. ##SPC1##

Having thus described the method and system for determining the position of at least one retroreflective element of the present invention in detail and by reference to preferred embodiments thereof, it will be apparent that modifications and variations are possible without departing from the scope of the invention defined in tee appended claims.