Roll Angle in 6DOF Tracking

Roll angle in 6DOF tracking

Stephen Kyle

Senior Honorary Research Fellow, University College London Honorary Research Fellow, West Midlands Manufacturing Measurement Centre, UK

Abstract In a 3D polar measurement system which tracks a probing device in 6 degrees of freedom (6DOF), the probes roll angle can be identified as a critical parameter in some system design concepts. This paper reviews methods and proposals for roll angle measurement in single laser tracker systems based on existing implementations and published concepts. Introduction An earlier CMSC paper (Kyle, 2006) highlights roll angle as a critical measurement in a 6DOF single laser (polar) tracking system. Roll angle appears, directly and indirectly, in a number of patents, for example recently by Leica Geosystems (2007), which emphasizes its importance in commercial manufacture. There are currently 3 manufacturers of laser tracking systems, i.e. Hexagon (Leica Geosystems), API and Faro, and the first two now offer real-time probing. Since the tracked position is offset from the measured point of interest, e.g. the ruby ball on a touch probe or the laser spot on a non-contact probe or scanner, the probe must be tracked in 6DOF in order to calculate the offset vector from tracked to measured point. Standard single laser tracking measures the 3D location of a target retro-reflector by range and two angles (approximately horizontal and vertical).

Roll

Pitch

Yaw

Figure 1 Aircraft roll, pitch and yaw angles

The 3 angular elements of the target, here identified as roll, pitch and yaw angles (see Figure 1), can be measured as a combined result from single image processing. Two methods are well represented here: 1. Leicas T-System uses a zoom camera

image of targets on the probing device which surround the reflector. The technique is essentially a photogrammetrists space resection in which the range data is discarded, see Loser and Kyle, 2003.

2. Past research at the Technical University of Vienna (no longer active) projected the return beam onto a CCD chip. The angular orientation could be determined from the shadow image of the reflector edges, see Prenninger et al., 1993.

Alternatively, two of the angles, pitch and yaw, can be separately measured to a relatively high accuracy. Consider, for example, a probe based on a hand-held camera sighting a single target on the tracker. An up/down rotation or a side-to-side rotation will correspondingly move the target image up and down or side to side. Its image position in x, say, is therefore a measure of yaw angle and in y is a measure of pitch angle. Given a normal angle of view, for example 50 which is roughly equivalent to the acceptance angle of a reflector, a 2K x 2K imaging chip and 1/20 pixel interpolation on the image, then accuracy for a well calibrated image could approach 5 arc sec, equivalent to 25m at 1m offset. This is a good accuracy for a large volume metrology (LVM) probe. Unfortunately, roll is still missing in this case. This document will therefore review methods and proposals for roll angle determination as a largely separate task from pitch and yaw measurement.

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1 Introduction: the space triangle

Figure 2 Theodolite orientation (6DOF)

A standard method for orienting non-levelled theodolites, i.e. determining their relative 6 degrees of freedom (6DOF), helps in understanding the geometry of roll measurement, see Figure 2. Each theodolite points at the other (details below), thus obtaining vectors R1 and R2. Relative to theodolite 1, theodolite 2 is then fixed in 3D by R1 and the separation D (see below). Vector R2 provides the pitch and yaw of theodolite 2. By rolling theodolite 2 such that its measured vector V2 to an offset target intersects the equivalent offset target vector V1 from theodolite 1, roll is also fixed. Instruments and target form a space triangle. For accurate scale the theodolite separation D must be measured in some way. Theodolites typically derive scale, and hence D, by measuring a second offset target on a scale bar of length s. However, D, or equivalently d1 or d2, can also be measured separately or directly by substituting a Total Station or laser tracker (see below), for either theodolite.

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Figure 3 Generalized procedure The procedure is generalized in Figure 3, which shows two angle measuring devices (yellow circle and triangle) and 3 possible positions of an offset target. If the measuring

instruments represent a stationary tracker and a moving target probe, it will clearly be a convenience to confine all measurements to a region close to the baseline between them, i.e. an offset target at locations (1) or (3). As a further convenience, the target would be attached to its nearby device and be part of it. In this case the attached instrument cannot directly measure the corresponding offset vector, shown in red, and it must be determined by prior calibration. Further discussion will identify the shaded space triangle as a roll plane and the offset target as a roll target. Note also that each device measures two vectors within the roll plane. The location of the offset target highlights an accuracy issue. For the same pointing accuracy, roll is less accurately determined as the offset target moves closer to the baseline and when it is on the baseline the roll angle cannot be determined at all (the roll plane disappears in this case). There are two general configurations for accurate roll determination: Lo-res/Hi-offset

A device with moderate angle resolution requires a large offset angle A, close to 90. In a triangulation scheme involving theodolites, cameras or a mixture of both, aiming for the optimal offset would be a common strategy.

Hi-res/lo-offset A device with high angle resolution can accommodate a small offset angle A. This is an appropriate strategy, if possible, for 6D single laser tracking.

Figure 4 Reciprocal pointing by theodolite

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There is an interesting aspect to the theodolite orientation method. To obtain the pointings between theodolites it is common to sight offset targets on the theodolite telescopes, as indicated in Figure 4. The vector mean of R1a and R1b, for example, generates the required pointing R1. (Some older Wild theodolites had internal targets visible through the telescope lens with an apparent location at the theodolites centre of rotation. Direct sighting along the baseline was therefore possible, but this was not a common solution.) As can be seen, this process generates several roll planes which could then be used for calculating the relative roll angle, assuming the local offset vectors (in red) have been calibrated as indicated earlier. As far as the author is aware this is never actually done with theodolites. This is the lo-res/hi-angle requirement and a more accurately defined roll plane is obtained by sighting a more offset target, typically on a scale bar, this measurement being required in any case. However, once a polar measuring device, such as a laser tracker, is substituted for a theodolite, the situation changes. The relevance of the discussion above is illustrated by Figure 5 which extends an early concept suggested by Kyle, 1991. The original suggestion was to have a motorized theodolite track a moving camera.

Figure 5 Camera tracked by (video) theodolite

The main space triangle is shown. Scale is found by a range measurement, d, to a fixed offset target, possibly by using an add-on EDM or Total Station in place of the

theodolite. Alternatively, scale is found by locating a second offset target on a scale bar of length s. By adding a stylus and probe ball to the camera it becomes a touch probe for 3D object measurement, provided the probe ball location within the cameras coordinate system is determined by prior calibration. Some of this early work included methods of targeting the cameras perspective centre and theodolites rotation centre so that the vectors between them could be directly identified. A very simple way of doing this is possible with modern tracker technology, as follows. 2 Roll plane by offset targeting 2.1 Roll target in tracker space

Figure 6 Pinhole retro-reflector

Figure 7 Multi-imaging pinhole

reflector A type of camera whose perspective centre can be directly sighted or measured is available to laser trackers and shown in Figure 6. This concept appears in patents by both Boeing, 2002 and Leica, 2003. Typically the apex of a prism retro-reflector is removed to create a small hole. Part of the incoming laser beam, most of which is reflected back on itself, passes through the hole and onto a sensor. If this is a CCD, or similar, then the xy chip coordinates of the laser spot are a measure of the direction of the laser beam (or equivalently the pitch and yaw of the prism). This is, in effect, a pinhole camera and retro-reflector combined in one.


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Leicas later 2007 patent utilizes the device for roll measurement, an example being shown in Figure 7. The tracker follows the pinhole camera/reflector which effectively measures the pointing back to the tracker. An offset, luminous target on the tracker is also imaged by the pinhole camera/reflector, creating the roll plane indicated by the shaded space triangle. Comparison with Figure 5 shows that the essential difference is direct range measurement in the 2007 patent compared with indirect range measurement in the tracking theodolite concept. This concept is a lo-res/hi-angle configuration, i.e. the angle subtended at the pinhole camera should be relatively large in order to optimize roll angle accuracy. For the convenient situation where targets are placed on or near the tracker head, this will be the case when the probe is close to the tracker but not when it moves further away. Multiple roll targets distributed away from the tracker offer one way to improve the situation at longer ranges but more attention must then be given to ensuring lines of sight are maintained. In this case, in order to avoid the need for many targets such that one is always visible to the reflector as it is moved around by the operator, robot etc., a motorized pointing could be employed to maintain an optimal view. In fact, the Boeing patent is designed to do exactly this for the more specific condition of maintaining a good pointing back to the tracker only. APIs recently developed Active Target is a commercial device which also does this. Clearly an actively pointed pinhole prism could be controlled to maintain multiple pointings and more optimal roll.


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Figure 8 Metronor concept

The combination of pinhole camera + retro-reflector can also be replaced by a conventional camera with an attached, offset, standard retro-reflector. This is used in a 6DOF tracking concept patented by Metronor, 2006 and shown in Figure 8. A retro-reflecting tracker target, attached to a camera with probing stylus, is located in 3D. The camera images a minimum of two targets, shown green and known in the trackers coordinate system, for example by prior direct measurement by the tracker. Alternatively, by choosing a target arrangement which directly identifies the trackers origin of measurement, for example as illustrated by the 4 blue targets which define lines intersecting at the required position, then only one offset target would be required and the geometry would be closer to the Leica configuration. The full 3D offset of the reflector from the perspective centre in the cameras coordinate system, shown as a red line, must be known. However, this configuration has excess information to make the 6DOF calculation. The trackers range measurement, D, is not mathematically required although it will considerably enhance accuracy in many geometrical configurations1.

Figure 9 Alternative camera tracking

Figure 9 shows the concept with tracking range measurement removed, and it should be 1 In fact, Leicas 2007 concept, if using a 3D calibration of offset target position, also has excess information. However, it would not be practical to take scale from such a short offset. In fact, the offset need not be calibrated in full 3D and only direction calibration data is required.

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compared with Figure 5. The main difference from the earlier concept is that the camera sights two offset targets rather than one offset target and the theodolite. As with the laser tracker concept, the targets must be known with respect to the tracking instrument. One option amongst others is to use a robot Total Station which can only track real-time in angle but can locate the targets with range measurements d1 and d2 in an initialization procedure. Now that direct measurement between device centres has been removed, the geometry is a little more complex than that of a simple space triangle, but not difficult. Using a geomatic engineers standard horizontal resection it is easy to calculate the circle through the location of the camera and two targets by using the known positions of the targets and the subtended angle measured at the camera. In 2D the camera then lies somewhere on this circle. When the circle is rotated about the chord connecting the targets, a doughnut shape is created which is the surface of all possible 3D positions of the camera, each with a particular angular orientation. The pointing from the theodolite intersects this surface, thus fixing the camera in 6DOF. It is a convenience for the theodolite to use an offset target on the camera, i.e. the blue target and pointing rather than the direct pointing. If the local offset vector is known in the cameras coordinate system, then every point on the doughnut surface has a corresponding offset point which, in turn, creates a nearby surface for the possible target positions. Again the intersection of the theodolite pointing with this offset surface locates the camera probe in 6DOF. In practice the calculations would be done differently but the geometry illuminates the concept. With minor modification, the same geometrical description can also be applied to Figure 5. This is again a lo-res/hi-angle configuration which benefits from roll targets with greater offsets and, potentially, a motor driven probe which maintains optimal pointings.

2.2 Roll target on the probe

Figure 10 Roll target on probe

Figure 11 Alternative probe roll As explained earlier, the roll target can be at either end of the system. Figure 10 shows a concept presented by Kyle, 2005 in which the overview camera of the API tracker could potentially measure a full roll angle at the reflector using a single offset target. The shaded space triangle is only an approximation here since the cameras perspective centre is separated from the trackers measurement origin. This 3D offset must be known by prior calibration. Compare this with Figure 12 (below) and note that the single offset target is the minimum condition. However, a more accurate roll angle would be expected with at least one more offset target, as implied by the additional greyed elements in the diagram. Applying a substitution used earlier, the pinhole retro-reflector can be replaced by a camera/reflector combination, as indicated in Figure 11. The camera on the probe will require its own target on the tracker for which again the 3D offset in local tracker space is required. (It is interesting to compare this with either pointing shown in Figure 4.)

Figure 12 Leica patent 2007


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With a relatively wide-angle, overview camera this is the lo-res/hi-angle condition for optimal roll and this will only be achieved at close ranges. However, if an on-board zoom camera is used, such as the T-Cam in the Leica T-System (a vario zoom lens camera), then the situation becomes hi-res/lo-angle and therefore capable of operation at longer ranges for equivalently good roll angles. This is shown in Figure 12 where the patent specifies a minimum of 2 targets and suggests alternative pattern arrangements.

Figure 13 Roll by barcode

Some optimization might be possible by using the type of barcode target common in digital levelling, a suggestion in Kyle, 2006. A circular barcode was suggested by the concept diagram, repeated in Figure 13, although two or more linear sections of barcode would also work. The potential advantage is that pattern matching might locate the barcode in the image to a very small sub-pixel level, as is the case in digital levelling, so enabling an accurate roll angle to be determined. 3 Roll plane by tilt sensing

Figure 14 Levelled theodolite orientation

When levelled theodolites are oriented to one another their local z axes are set parallel. In fact, they can also be tilted and the local direction of the gravity vector found by tilt sensing or other means, see Kyle, 1990.

In non-levelled theodolite orientation, shown in Figure 2, an offset roll target at infinity will generate offset roll vectors which are parallel. Using tilt sensors for roll determination can therefore be viewed as a variant of the space triangle method, discussed earlier. Roll by tilt sensing has advantages No additional target.

No offset roll target is required and so there is no requirement for lines of sight off the baseline.

Constant sensing accuracy to maximum device separation, typically 40m for laser trackers.

If device separation is largely horizontal, the large offset angle improves roll accuracy.

However, there are disadvantages: There is a failure case when the baseline

between the devices is vertical. The roll plane then disappears. When the line is close to vertical the geometry is also poor.

Restricted range Tilt sensors offering an angle accuracy suitable for metrology applications generally have a limited operational range, instead of the ideal 360 of roll.

Dynamic effects. Acceleration distorts gravity sensing and movement causes oscillations, both a source of inaccuracy.

Potential ways to deal with restricted range and dynamic effects are suggested in Kyle, 2005.

Figure 15 Tilt sensor attached to pinhole reflector


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Figure 16 6DOF tracking from API

Verbal discussions with API indicate that tilt sensing is used in their Intelliprobe and SmartTrack sensors, and use of a single tilt sensor would explain their limitation on the roll angle. It is also not clear if there is tilt sensing only at the target. If so, this would seem to require a setup or calibration routine to determine a datum position for the roll angle. 4 Roll plane by transmitted reference

direction A different approach to measuring roll angle is to manufacture a roll plane in some way. Optical methods of transmitting a reference direction between devices are presented here. 4.1 Plane of polarization. A beam of polarized light, whose direction of polarization can be detected by various means, appears in many suggestions for transmitting a roll orientation angle. The fact that polarized laser beams are used in laser trackers makes this technique an obvious candidate to consider.

Figure 17 Roll angle by polarized light

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Figure 17 is taken from Kyle, 1990, and illustrates the concept. Here it was suggested that polarized light, transmitted from one theodolite and analyzed by the other, could determine the roll angle between their reciprocal pointings.

The idea of using polarimetry to detect roll angle is a long-established concept and there are numerous investigations. As an example, King and Raine, 1981, suggest roll angle could be measured by polarimetry over several metres to a precision of 0.2 arc sec, or a few arc sec over hundreds of metres. There are also a number of patents in this area, such as the one from Daimler Benz, 1985, and from Lau (API), 2003. The latter specifically aims to extract the roll angle from a tracking laser beam. However, obtaining an accurate result in real time, and with light and compact optics and electronics, may be a challenge in practice and a possible reason for APIs selecting instead the simpler technology of tilt sensing. 4.2 Projected patterns: Bird system This early 1983 concept is introduced as background to the remaining discussion. Here the motivation was to track the location and pose of a robot within a large volume and to this end a prototype device was presented by Arai, et al., 1983.

Figure 18 Bird system concept

This is essentially a real-time triangulation system in which two angle trackers project cross-shaped laser beams onto CCD ring sensors placed onto the faces of a moving target cube. It is apparent that the roll of the cross is detected by where the arms intersect the sensor ring, but pitch and yaw are also detected to some extent when the cross is incident at an oblique angle to the surface of the cube. For example, if the arms are generated with a 90 intersection angle, they will, in general, be detected on the cube as having some other angle. (Note that pitch and yaw accuracy would not be high. For example, in

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perpendicular viewing they cannot be detected if they represent rotations about the arms of the cross.) Two projected crosses provide sufficient data to determine the cubes angular orientation in space. The centre of the cross determines a particular location on the cube, thereby placing the cube on a particular line in space with respect to one of the angle trackers. The corresponding line from the second tracker locks it in 6DOF. It is not necessary for the lines to intersect. Some features of this early system: A tracking mechanism

It is provisionally assumed that since the cube is located in 6DOF, the current location in object space of the centres of the ring sensors can always be computed and returned to the trackers as a point which they must follow.

3 CCD sensors were recommended in order to measure a wide angle of rotation The paper does not suggest multiple trackers so it is again provisionally assumed that the trackers would automatically switch to tracking a different ring centre when a change in the cubes orientation blocked the line of sight to the currently tracked ring.

Small sensors The prototype ring sensors were small, only about 4mm in diameter

From the viewpoint of modern laser tracking technology this concept could be considerably improved to provide a 3D/6D measurement system. A single polar device eliminates the need

for the second tracker Accurate pitch and yaw measurement can

be made by pinhole reflector or reflector + camera

A single line is then sufficient for roll measurement but a pattern may also be useful

Modern, large area and linear array sensors would improve sensitivity

With this in mind, some more recent proposals can be evaluated.

4.3 Projected lines and patterns

Roll plane generator

Roll reference plane

Roll plane detection

Figure 19 Added roll reference plane Figure 19 illustrates a concept taken from Leicas 2007 patent. An additional laser roll reference plane is generated by a separately mounted device on the tracker. Linear CCD sensors on a roll ring around a pinhole prism reflector detect the beam and hence define the roll angle. The roll reference plane can be constructed in a number of ways, for example by expanding a laser beam through a cylindrical lens or reflecting it off rotating mirrors. When added as a fixed ancillary device, the plane must approximately cover the vertical operating range of the tracker, indicated by the heavy red dashed lines. However, it could project over a more limited angular range, by being mounted in a tilting unit like the T-Cam and, like the T-Cam, directed to maintain a pointing at the target. In this particular example, the reference plane will be detected by one of the linear sensors in any roll orientation, and only one measurement is required. However, it is potentially more accurate to measure the line of the plane at two or more positions.


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Figure 20 Closed linear arrays for roll plane

detection In fact, Arais ring sensor could be simulated in a much larger and alternative form using modern linear CCD arrays. This could then replace the separated arrangement of arrays above. Assuming relative array positions can be determined, the example arrangements in Figure 20 show ways in which the roll plane (green) could be detected in two locations to a potentially much higher accuracy than in the Bird system There could be advantages in using crossed planes to ensure an optimal intersection with the linear arrays and/or an improved roll accuracy through excess measurement.

Figure 21 Area array for roll detection

Figure 22 Back projected roll beam

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Figure 23 Coaxial roll beam, semi-reflecting target

Line or pattern projection onto area array CCDs would clearly also work. Although defining a much smaller sensing area than the closed arrangements of linear arrays, they also enable measurement of the entire pattern, which might compensate for the lower resolution. Of the many issues to be resolved in a working system, two are highlighted. 1. Placement of sensor

These general approaches are developed from Kyle, 2006. Figure 22 shows a moving pan and tilt target device projecting a roll pattern onto an offset area array on the fixed tracker. The target device tracks in a similar way to the Bird system and the geometry is similar to Figure 4. Alternatively, in Figure 23 a roll beam (green), co-axial with the tracker beam (red) is projected from the fixed tracker onto the target device. Here beam splitting takes the tracker beam through a mask whose shadow projection measures pitch and yaw and the orientation of the pattern on the CCD measures roll.

2. Generation of the pattern Although there are many systems currently commercially available which can project laser patterns, detailed investigation is still required to see if they have a sufficient range (focusable or otherwise) and are sufficiently compact and cost effective to be applied.

5 Conclusions Separately and accurately measured roll angle, when combined with an accurate measurement of pitch and yaw, would enable high accuracy 6DOF laser tracking of object probing devices. For the same spatial accuracy, improved rotational accuracy permits longer offsets from the tracked location to the probed point on the object. Users will always want more from a system, and a longer reach outside the line-of-sight could be regarded as a worthwhile improvement. For hand-held probes, further investigations might focus on: Projected patterns from tracker to probe Optimized imaging at the tracker of

targeting around the reflector All-angle tilt sensing with algorithm

compensation of accelerations and oscillations

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Motorized probes, likely to be heavier than hand-held probes and therefore potentially better applied to robot and machine control or as components in CMM arms, might make use of: Polarized light Back-projected roll planes (from probing

device to tracker) The space triangle with large offset roll

targets 6 Acknowledgements The author is grateful to Leica Geosystems for recognizing his contributions to concepts for 6DOF tracking whilst in their employment, some of which have been incorporated into their 2007 patent referenced below. 7 References Aarai, Endoh, Minokoshi, 1983 Position and orientation measurement of a moving object by CCD photo array sensors Proceedings of 4th. Intl. Conference on Assembly Automation, Tokyo, 1983, pp 133 144. Boeing: Greenwood, T., 2002 Steerable retroreflective system and method. United States Patent 6,420,694 Daimler Benz, 1985 Patent number DE3347833 Device for continuous polarimetric measurement of the angle of roll of a movable machine part King, R., J., Raine, K. W., 1981 Polarimetry applied to alignment and angle measurement Optical Engineering, 20 (1), 39 - 43, 1981. Kyle, S., 1990 Orientation with polarimetry and gravity vectors Land & Minerals Surveying, Vol. 8, no. 3, March 1990, pp 122 - 129. Kyle, S., 1991 Reciprocal observations in photogrammetry Photogrammetric Record, 13(77): 729 - 739 (April 1991).

Kyle, S., 2005 Alternatives in 6DOF probing - more flexibility, lower cost, universal? Presented at CMSC, Austin, Texas, July 2005 Kyle, S., 2006 Optically jointed probing systems for large volume coordinate metrology The Journal of the CMSC, vol. 2, no. 1, Spring 2007. Lau, K., (Automated Precision Inc.), 2003 Six dimensional laser tracking system and method US Patent 20030043362 Leica Geosystems: Loser, R., Markendorf, A., 2003 Method and device for determining spatial positions and orientations. United States Patent 6,667,798 (also WO0109642) Leica Geosystems: Zumbrunn, R. et al., 2007 Measurement system for determining 6 degrees of freedom of an object, US patent: US 7,312,862 B2 Loser, R., and Kyle, S., 2003 Concepts and components of a novel 6DOF tracking system for 3D metrology Optical 3D Measurement Techniques VI, (Eds. Grn, Kahmen), vol. 2, pp 55 - 62, 2003. Metronor, 2006 Measurement of spatial coordinates, US patent: US 7,145,647 B2 Prenninger, Vincze and Gander, 1993 Measuring dynamic robot movements in 6DOF and real time Robot Calibration (Eds. Bernhardt, Albright), Chapman and Hall 1993, Ch 8, pp 124 - 153.

1 Introduction: the space triangle2 Roll plane by offset targeting2.1 Roll target in tracker space2.2 Roll target on the probe

3 Roll plane by tilt sensing4 Roll plane by transmitted reference direction4.1 Plane of polarization.4.2 Projected patterns: Bird system4.3 Projected lines and patterns

5 Conclusions6 Acknowledgements7 References

Roll Angle in 6DOF Tracking

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