Abstract:
A method of assessing a condition of a wheel on a vehicle involves contactlessly determining distance of a first location on the wheel from a fixed point not on the wheel at a first time while the vehicle is moving and contactlessly determining distance of a second location on the wheel from the fixed point at a second time after the vehicle has moved. The two distances are compared to determine an offset between the first and second locations on the wheel. The offset provides an indication of tire wearing angle of the wheel while the vehicle is moving. The method can be used to assess wheel alignment and wheel suspension. An apparatus and system for effecting the method involves the use of a displacement sensor, especially an optical displacement sensor (e.g. a laser) for making the distance determinations. The system and apparatus is completely contactless and only one stationary displacement sensor is required to make the appropriate distance measurements to the wheel on a moving vehicle.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of priority to U.S. patent application Ser. No. 13/791,404 filed Mar. 8, 2013, the contents of which are incorporated herein in their entirety. 
     FIELD OF THE INVENTION 
     The present invention relates to methods, systems and apparatuses for assessing the condition of a wheel on a vehicle, particularly to methods, systems and apparatuses where off-vehicle equipment is used to make the assessment. 
     BACKGROUND OF THE INVENTION 
     Vehicle wheels are the part of a vehicle in contact with a driving surface, such as a road, and bear the entire weight of the vehicle during its operation. As such, it is important to monitor wheel condition, for example wheel alignment, wheel suspension and tire inflation, to determine whether maintenance needs to be performed to ensure optimal performance and safety of the vehicle. 
     The prior art is replete with systems for performing wheel alignment assessment. Most of these systems require equipment mounted on the wheels to assist in wheel alignment assessment and require the vehicle to be hoisted on to or otherwise mounted on to rollers or other apparatuses. A number of non-contact or contactless systems have been developed that employ optical measuring means, for example United States patents and Published patent applications: U.S. Pat. No. 6,545,750; U.S. Pat. No. 5,532,816; U.S. Pat. No. 4,899,218; U.S. Pat. No. 5,818,574; U.S. Pat. No. 6,400,451; U.S. Pat. No. 4,863,266; U.S. Pat. No. 7,336,350; U.S. Pat. No. 8,107,062; U.S. Pat. No. 7,864,309; U.S. Pat. No. 7,177,740; U.S. Pat. No. 6,657,711; U.S. Pat. No. 5,978,077; U.S. Pat. No. 7,454,841; U.S. Pat. No. 7,774,946; and US 2006/0152711, the entire contents of all of which are herein incorporated by reference. These systems involve laser displacement sensors, laser illumination, cameras or some combination thereof. Most of them require the vehicle to be stationary while the system operates. Some involve rotation of the wheels. Various parts of the wheel, including the tire sidewalls, can be used as targets for the lasers and/or cameras. 
     In one example, U.S. Pat. No. 5,532,816 discloses a contactless system for determining vehicle wheel alignment in which a point on a rotating wheel is tracked by a laser tracking unit to generate a signal directly representative of the rotational plane of the wheel. This signal is compared to a mathematically stored model to determine wheel alignment conditions. Both the vehicle and laser tracking unit are translationally stationary with respect to each other. The actual laser rotates to be able to follow the point on the rotating wheel. 
     In U.S. Pat. No. 5,532,816 the vehicle is mounted on rollers to allow the wheels to turn while the vehicle itself does not move. It would be advantageous to have a system that could make wheel alignment assessments while the vehicle itself is moving, for example while it is being driven into a garage or test station. Only a very few prior art systems are configured to permit wheel alignment assessment while the vehicle itself is moving. 
     U.S. Pat. No. 6,545,750 discloses a system for determining the dynamic orientation of a vehicle wheel plane. The system involves an orientation determining device that is not mounted on the vehicle or vehicle wheels. The orientation determining device remains stationary as a vehicle is driven by it and the device takes measurements on the wheel as the wheel passes by. The wheel is preferably outfitted with a reflective test surface. The orientation determining device comprises three transducers that emit beams of e/m radiation (e.g. lasers). The beams reflect off the test surface (or wheel hub) at three non-collinear points and the distance information from the three points is used to calculate wheel orientation at one specific instance in time. This system uses distance information from three separate laser beams to measure the distance to three different points on the wheel at a single instance in time. However, because the system is making measurements at only a single instance in time, it provides data only on wheel alignment, not on other wheel conditions such as wheel suspension and/or tire inflation. Further, acquiring data simultaneously on three non-collinear points on a wheel hub is difficult, so a reflective test surface is preferably mounted on the wheel, making the system more laborious and less useful for “on the go” wheel alignment assessment. 
     There remains a need for a simple method and apparatus for assessing the condition of a wheel on a vehicle while the vehicle is being driven and without the need to mount any equipment on the vehicle. 
     SUMMARY OF THE INVENTION 
     There is provided a method of assessing condition of a wheel on a vehicle, comprising: contactlessly determining distance of a first location on the wheel from a fixed point not on the wheel at a first time while the vehicle is moving; contactlessly determining distance of a second location on the wheel from the fixed point at a second time after the vehicle has moved; and comparing the distance at the first location to the distance at the second location to determine an offset between the first and second locations on the wheel, the offset providing an indication of the dynamic toe of the wheel. The dynamic toe (which may also be referred to as the tire wearing angle) is a measurement of the difference in the direction of movement of the vehicle and the direction of orientation of the wheel. In other words, the method can determine whether the wheel is straight while the vehicle is moving. 
     There is further provided a method for assessing play in suspension elements that hold a wheel of a vehicle, comprising: driving the vehicle so that the wheel passes over a suspension testing surface comprising at least first and second undulations which slant downwards laterally towards opposing sides from each other; determining an offset for the wheel at a point on the first undulation and at a point on the second undulation using the method described above; and determining whether there is play in the suspension elements based on a difference in the offsets between the two points. 
     There is further provided an apparatus for determining an offset between two locations on a wheel on a vehicle at two different times, the apparatus comprising: a first displacement sensor not on the vehicle and fixed in position during operation of the apparatus for determining distance along a fixed path from the apparatus to the wheel on the moving vehicle, and one or more further displacement sensors fixed in position during operation of the apparatus for confirming that the wheel is passing the first displacement sensor. 
     There is further provided a system for assessing a condition of a wheel on a vehicle, the system comprising: an apparatus for generating output signals indicative of distances to two locations on the wheel at two different times while the vehicle is moving on a surface, the apparatus not moving during operation; and, a control system configured to receive the output signals from the apparatus and to output data based on the distances to the two locations. 
     There is further provided a method of detecting a wheel on a vehicle, comprising: contactlessly (i.e. without physical contact) determining distances from a series of points on the moving vehicle to a fixed point not on the moving vehicle over a series of instants in time to generate distance data at each instant in time; at each instant in time, calculating an average of the distance data for a predetermined number of instants in time before and after said each instant in time; and, calculating variances of the distances at said each instance in time from the calculated averages, wherein a local minimum in the calculated average over consecutive instants in time and a small variance at each of the consecutive instants in time in comparison to the variance at other instants in time indicates passage of the wheel by the fixed point. 
     In the present invention, the offset between two locations on a wheel of a moving vehicle is determined. The offset is related to the difference in the distance from the first location to the fixed point in comparison to the distance from the second location to the fixed point. The offset is an indication of the tire wearing angle for the wheel, which is the angle between the wheel&#39;s orientation and the direction of movement of the vehicle  10 . A tire wearing angle of zero exists for a wheel that is perfectly parallel to the longitudinal centerline of the vehicle. The offset may be determined by measuring the distance from a fixed point spaced from the vehicle to a first location on the wheel along a fixed path at a first instant in time and then measuring the distance from the same fixed point along the same fixed path to a second location on the wheel at a second instant in time after the vehicle has moved and the wheel has rotated. 
     To ensure that the two measurements are made at appropriate separate instants in time so that the two locations are on opposite sides of the axle and at similar locations on the wheel, it is useful to know the wheel dimensions. In practice, distance measurements can be made continuously across the entire width of the wheel and software is used to track the relative distances over time to develop a histogram or profile of the wheel. The histogram can be used to visually locate suitable data points representing locations on the wheel for use in the offset calculation. For example, the presence of the sidewall of the tire becomes very evident when the data is analyzed graphically with a histogram. The data may also be analyzed by a processor to determine the presence of the wheel, the appropriate data points representing the locations on the wheel from which to take the distance measurements, and hence the distances at the first and second locations. 
     Yet further, there is one point at a certain height on the wheel (about ⅓ of the way up from the driving surface) that will be substantially the same point measured twice thereby guaranteeing that the first and second locations are actually the same points on the wheel. This arises from the fact that the wheel is rotating while the vehicle is translating so by matching the rotational distance of the wheel on a concentric circle at a particular radius on the wheel to the translational distance of the vehicle, it is possible to always take the two measurements at the same spot on the wheel. This is one of the advantageous consequences of the taking the distance measurements at different times from a fixed point not on the vehicle while the vehicle is moving. While it is advantageous to measure precisely the same physical point on the wheel when measuring the forward point on the wheel and when measuring the rearward point on the wheel so as to eliminate any errors that can arise from a local deformation on the wheel, it is alternatively possible to achieve some portion of that advantage if the forward point measurement and the rearward point measurement are taken at locations on the wheel that are in a selected level of proximity to each other. For example, some advantage is achieved if the forward point and rearward point are taken at physical locations that are within 5 degrees of each other. Alternatively, they may be within 25 degrees of each other, or within 50 degrees of each other, or 75 degrees of each other, or even 90 degrees of each other. This advantage may be at least partially realized by measuring points that are between about 25% and about 40% of the height of the wheel. 
     By comparing the two distances, a difference in the two distances can be determined, i.e. the offset. The difference can be expressed as a linear measurement (e.g. in units of length such as millimeters or centimeters) or as an angular measurement (e.g. in degrees) where the angle is an angle formed between a reference line and the actual line formed between the two locations on the wheel as measured at the two different instants in time. The reference line is the line that is representative of the wheel in a perfectly aligned state. Preferably, the reference line is perpendicular to the fixed path. The offset provides an indication of whether or not the wheel is straight while the vehicle is moving. An offset of zero means the wheel has a tire wearing angle of zero. A non-zero value of the offset provides the value for the tire wearing angle. If the distance to the first location is less than to the second location, the wheel has a toe-out orientation. If the distance to the second location is less than to the first location, it has a toe-in orientation. The size of the offset that might indicate a wheel condition problem, e.g. an alignment problem, depends on the type of vehicle and size of the wheel. Offsets of less than 1 degree generally indicate that there is no alignment problem. 
     Distance measurements may be taken by any convenient means. Optical displacement sensors based on emission of any form of electromagnetic (e/m) radiation are preferred. Optical displacement sensors include, for example, laser displacement sensors. Visible light lasers are preferred. The sampling frequency of the displacement sensor generally does not matter, but should be high enough to ensure measurement accuracy depending on the speed of the vehicle. When collecting data on fast moving vehicles, higher sampling frequency is preferred. Sampling frequencies may be in a range of 100-750 Hz, for example. Laser displacement sensors typically function by emitting a beam of light and capturing the reflection with an optical sensor (e.g. a camera). The sensor is in a slightly different location in the displacement sensor than the laser emitter, so triangulation calculations are performed by a processor in the displacement sensor to determine the distance to the spot where the reflection occurred. Suitable optical laser displacement sensors include Acuity AR-700 Series, Keyence IL Series (e.g. Keyence IL-600 and Keyence IL-2000) and Micro Epsilon optoNCDT 1402 displacement sensors. 
     Especially when the first and second locations are located near the centerline of the wheel, the fixed path along which the distance measurements are taken is at a height where it may intersect with others parts of the vehicle, for example the chassis or fender. In such a case, the passage of some part of the chassis or fender may be mistakenly taken as the passage of the wheel leading to errors in the distance measurements. To alleviate this problem, additional distance measurements may be taken along a second fixed path at a level closer to the surface on which the vehicle is moving. Since the wheel is always on the ground, and at the lower level there is less likelihood of encountering features that might be mistaken as a wheel, when the additional distances change dramatically it will be known that a wheel is passing by. Thus, the distance measurements collected along the second fixed path can be used to confirm the passage of the wheel. It should be noted that the data from the second fixed path does not need to be used and is preferably not used to make the wheel condition assessment, e.g. alignment assessment, itself. These confirmatory distance measurements are made separately from the measurements at the first and second locations and can be made by any convenient means, for example one or more further optical displacement sensors (e.g. one or more lasers). To further reduce the risk of falsely identifying the passage of something other than the intended wheel, it is preferable to use at least three further distance determining means in a row parallel to the surface to confirm the passage of the wheel. This will not only help determine when a wheel is encountered but will also help determine when the wheel has passed. The further distance determining means can also be used to determine the direction of travel of the vehicle and the number of axles on the vehicle as each axle will have a wheel that passes by. 
     Because the present invention employs measurements while the vehicle is moving, it can also be used to determine whether there is play in wheel suspension. This ability to assess other wheel conditions besides alignment is advantageous. Play in wheel suspension can cause a wheel to be angled in or out depending on whether the vehicle is moving forward or backward past the fixed point. To determine play in wheel suspension, the vehicle is moved forward and the two distance measurements made. Then the vehicle is moved backward and the same two distance measurements are made. When moving backward, the first and second locations on the wheel are the same as the second and first locations when the vehicle is moving forward. If there is no play in the suspension, the sign of the offset between forward and backward motion of the vehicle should change. If a change in the sign of the offset direction is not seen, then there may be a suspension problem in one or both wheels being measured. Since, as discussed previously, wheel tracking problems may be caused by suspension play and the offset is also dependent on wheel tracking, such suspension information can be collected even when the wheels themselves are aligned properly. 
     For extremely large vehicles such as tractor-trailers, backing up the vehicle to help determine suspension problems is not practical. Further, heavy loads and/or extensive driving may cause the suspension of such a large vehicle to settle in. For these reasons, the surface on which the vehicle moves may be modified by introducing twists and raised patterns or bumps. This is conveniently accomplished with wavy patterned plates that can be placed on the surface over which the vehicle can move. The twists and raised patterns or bumps release the suspension from its settled mode and force play in the wheel if there is a suspension problem. If there is no suspension play, the wheel remains upright as it passes over the twists and no offset arises due to suspension play. If there is suspension play, the wheel tilts and offset in the two distance measurement arises. 
     Distance data generated and offset data calculated in the present invention may be processed by a control system, for example computers, and the data displayed in any suitable fashion, for example on a computer monitor, numerically and/or graphically. Means for taking distance measurements may be in communication with the one or more processors, for example electronically. Electronic communication may be through cables or wireless. 
     As provided in the present description, vehicles are generally motorized transportation having one or more wheels driven by a motor. Vehicles include cars, trucks, trailers, tractors, motorcycles, etc. and have a front, back, right side and left side. The front points to a forward direction while the back points to a backward or rearward direction. Vehicles may have all of their wheels in a single plane (e.g. motorcycles) or have multiple planes of wheels. Most common vehicles have two lines of wheels. Where the vehicle has multiple lines of wheels, the right side is a passenger side of the vehicle in North American model vehicles while the left side is a driver side in North American model vehicles. 
     The present invention provides a simple method, apparatus and system for preliminarily assessing one or more of a number of wheel conditions on a vehicle, including not only wheel alignment, but also wheel camber, wheel suspension and tire inflation. The invention can be employed while the vehicle is moving into a shop, garage or other testing facility without the need to mount anything on the vehicle or to hoist or otherwise mount the vehicle on a separate apparatus. If the invention indicates a problem with the condition of the wheel, a more precise intervention can be made to fix the problem. If not, a more laborious assessment is thereby avoided. The invention is equally applicable to small vehicles (e.g. cars) and large vehicles (e.g. transport trailers). 
     Further features of the invention will be described or will become apparent in the course of the following detailed description. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       In order that the invention may be more clearly understood, embodiments thereof will now be described in detail by way of example, with reference to the accompanying drawings, in which: 
         FIG. 1  is a schematic diagram showing a top view of a system of the present invention comprising two electronically connected apparatuses for determining offset of wheels on the left and right sides of a moving vehicle; 
         FIG. 1A  is a schematic diagram showing a vehicle being driven through the system shown in  FIG. 1 , at an angle relative to the system; 
         FIG. 2  is a schematic diagram showing a front view of apparatus A depicted in the system of  FIG. 1 ; 
         FIG. 3  is a schematic diagram showing a back view of apparatus B depicted in the system of  FIG. 1 ; 
         FIG. 4A  is a schematic diagram of a top view of apparatus A depicted in the system of  FIG. 1  showing a beam from a laser displacement sensor illuminating a tire at a first location on the tire sidewall; 
         FIG. 4B  is a schematic diagram of a side view of the tire depicted in  FIG. 4A  showing the first location on the sidewall of the tire; 
         FIG. 4C  is a schematic diagram of a top view of apparatus A depicted in the system of  FIG. 1  showing a beam from a laser displacement sensor illuminating a tire at a second location on the tire sidewall after the vehicle has moved forward; 
         FIG. 4D  is a schematic diagram of a side view of the tire depicted in  FIG. 4C  showing the second location on the sidewall of the tire; 
         FIG. 5A  and  FIG. 5B  are schematic diagrams of wavy bumpy plates to assist in suspension testing of vehicle wheels; 
         FIG. 6A  is a histogram of distance data collected on the front left wheel of an vehicle as the vehicle was driven forward past apparatus A as depicted in  FIG. 1 , 
         FIG. 6B  is a histogram of distance data collected on the front right wheel of an vehicle as the vehicle was driven forward past apparatus B as depicted in  FIG. 1 , 
         FIG. 7A  is a plan view of a vehicle with trauma to the rear left wheel; 
         FIG. 7B  is a plan view illustrating the vehicle of  FIG. 7A  travelling; and 
         FIG. 8  is a perspective view of one of the apparatuses shown in  FIG. 1 , with an additional sensor for use in determining camber of a vehicle wheel. 
     
    
    
     DESCRIPTION OF PREFERRED EMBODIMENTS 
     In this specification and in the claims, the use of the article “a”, “an”, or “the” in reference to an item is not intended to exclude the possibility of including a plurality of the item in some embodiments. It will be apparent to one skilled in the art in at least some instances in this specification and the attached claims that it would be possible to include a plurality of the item in at least some embodiments. 
     It has been found that, while the measurement of static toe (i.e. the measurement of toe when the vehicle is stationary) can be useful, there are several problems with it as a tool to determine whether a particular wheel or tire will incur undue wear during use of the vehicle. In general, when static toe is measured, the corners of a polygon are determined, wherein the corners correspond to the centers of each of the four wheels of the vehicle. The angle of each wheel is then determined relative to that rectangle. Depending on the vehicle&#39;s suspension and other factors, the orientation of the wheels when the vehicle is stationary are not the same as the orientation of the wheels during operation of the vehicle. An example of a static toe measurement is shown in  FIG. 7A . The vehicle is shown at  10 , and has a body  11  that is represented by a rectangle for simplicity. The vehicle  10  has four wheels shown at  21 , and shown more particularly at  21 FL (the front left wheel as viewed from a viewpoint above the vehicle  10 ),  21 FR (the front right wheel),  21 RL (the rear left wheel) and  21 RR (the rear right wheel). As can be seen, there has been trauma to the vehicle&#39;s rear left wheel  21 RL, causing it to be out of alignment with the other three wheels. A static toe measurement would find that the front left, front right and rear right wheels  21 FL,  21 FR and  21 RR all have a toe of zero, and the rear left wheel  21 RL has a toe value of some non-zero value. However, as can be seen in the view shown in  FIG. 7B , when the vehicle  10  is being driven, due to particular dynamics involved, the rear left wheel  21 RL may drive the direction of movement of the vehicle  10  more than the rear right wheel  21 RR. The driver of the vehicle (not shown) may steer the vehicle  10  in an effort to compensate for the frictional forces that cause the right and left rear wheels  21 RR and  21 RL to urge the vehicle  10  in different directions. The resulting direction of travel of the vehicle  10  may be as shown in  FIG. 7B . As can be seen, when the dynamic toe measurements would be taken, the front and rear toe values would be zero, and each of the rear toe values would be about ½ of the static toe value of the rear left wheel  21 RL. As can be seen, both the right and left rear wheels  21 RR and  21 RL have non-zero tire wear angles relative to the direction of travel of the vehicle  10 . Such a measurement would reveal that both the rear right and rear left wheels  21 RR and  21 RL have non-zero tire wearing angles and would thus incur wear. 
     Another issue relating to measurement of static toe is that, depending on how soft the vehicle&#39;s suspension is, and depending on whether there are any problems with suspension components, it may be possible to measure the static toe of the vehicle  10  and to find that all the wheels have a suitable toe value, but to find that the wheels  21  move depending on frictional and other forces that urge the wheels  10  to take on different toe values when the vehicle  10  is moving. 
       FIG. 1  is a schematic diagram showing a top view of a system  200  that is configured to determine the tire wearing angles of a vehicle in accordance with an embodiment of the present invention. The system  200  comprising two electronically connected optical displacement sensing apparatuses A,B of the present invention for determining an offset in wheels on the left and right sides of a vehicle  10  that is moving forward past the apparatuses A,B in the direction of the arrow.  FIG. 2  is a schematic diagram showing a front view of apparatus A.  FIG. 3  is a schematic diagram showing a back view of apparatus B. Apparatuses A,B are identical and element numbering in  FIGS. 1-3  applies equally to both. 
     Each apparatus A,B comprises tower  2  mounted on base  9  having height adjustable feet  1  at each corner of the base. Visible laser displacement sensor  3  is mounted fixedly in the tower and configured to emit a laser beam parallel to the surface on which the apparatus rests at a height that may be, for example, between about 25% and about 40% of the height of the vehicle wheel  21  and is preferably at a height of about one third of the height of the wheel  21 . Displacement sensor  3  is used to determine distance to the vehicle&#39;s wheels  21  during operation of the apparatus. Three further laser displacement sensors  4  are mounted in a single row in the base and configured to emit laser beams parallel to the surface on which the apparatus rests at a height below the chassis of a typical vehicle. Further displacement sensors  4  are only used to confirm that a vehicle wheel  21  is passing the apparatus A,B (as opposed to some part of the vehicle body) and to confirm when the wheel  21  has passed the apparatus. Sensors  4  need not be laser displacement sensors and may operate on any other suitable principle. Sensors  4  may be referred to as wheel detection sensors. The two apparatuses A,B are electronically connected through a cable  5  and one of the apparatuses, in this case apparatus B, is electronically connected to computer  8  through cable  7  from a data port  6 . The computer  8  is loaded with software for interpreting signals from all of the laser displacement sensors on both apparatuses to determine distances from the displacement sensors to the surfaces on which the laser beams impact. The software determines distances from each displacement sensor  3  to the vehicle&#39;s wheel  21 . Only data from displacement sensors  3  are used in wheel condition assessment. 
     The computer  8  includes processor  8   a , a memory  8   b , and an output device  8   c , which may be, for example, a display. The computer  8  is but one example of a control system. The control system may include a single processor and a single memory, or could have multiple processors and multiple memories. In the event of having a plurality of processors and memory, the processors and memory may be in a single housing, or may be distributed between a plurality of housings. 
     The height of the laser displacement sensor in each apparatus may optionally be adjusted by adjusting the height adjustable feet  1 , to be at about one-third the diameter of the wheel  21  off the surface on which the wheel  21  is traveling. The height adjustable feet  1  may also be used to level the apparatus A,B on an uneven surface. The two apparatuses A,B may be positioned roughly across from each other and so that the beams from the laser displacement sensors  3  are roughly perpendicular to the direction of motion of the vehicle  10 . Each apparatus A,B is an independent unit that is in no way attached to or mounted on the vehicle  10 . 
       FIGS. 4A-4D  depict a single apparatus (apparatus A) and illustrate the measurement of the offset for the front left wheel  21  of the vehicle  10 . Referring to  FIGS. 4B and 4D , the wheel  21  includes a rim and a tire, shown at  21   a  and  21   b  respectively. In the event that a hubcap is provided, the hubcap may be considered part of the rim for the purposes of this description. In operation, the apparatus A is stationary while the vehicle  10  moves forward past it in the direction of the arrow. As the vehicle  10  passes the apparatus A, the laser displacement sensor  3  sends signals back to the computer  8  at a selected frequency (e.g.  200  distance measurement signals per second), and the computer  8  calculates the distance that beam  25  travels to reach the vehicle  10 . The computer  8  tracks and displays the distance data. The distance data for an example vehicle is shown in Tables 1 and 2, and is illustrated graphically in the form of histograms in  FIGS. 6A and 6B . 
     The computer  8  determines the distances to two longitudinally spaced locations on the wheel  21 , and determines the difference between the two distances, which is referred to as the offset, and which is indicative of the tire wearing angle of the wheel  21 . Preferably, the two locations are on opposite sides of the centerpoint of the wheel. In other words, preferably, one location is on the leading half of the wheel  21  and on is on the trailing half of the wheel  21 . Preferably, the two locations are on parts of the wheel  21  that have the same lateral distance to the longitudinal centerline of the wheel, shown at CL in  FIGS. 4A and 4B . The locations could be on the tire sidewall (shown at  21   c ) or the rim or the hub of the wheel. For ease of detection, the locations may be at points of maximum lateral bulge (shown at  30  and  31  respectively in  FIGS. 4B and 4D ) for the tire  21   b  at whatever height the displacement sensor is operating, although other locations on the wheel  21  may be used. For example, the center of the tire sidewall  21   c  may also be a suitable location (the maximum lateral bulge on a tire is typically not at the center of the sidewall, but is instead closer to the radially outer edge of the tire  21   b ). 
     In the example shown in  FIGS. 4A and 4B , as the vehicle wheel  21  passes the laser displacement sensor  3 , beam  25  finds a point of maximum bulge  30  at a first instant in time on the leading part of tire  21   b  on the tire&#39;s sidewall about one-third the way up off the surface. At this point, a first distance is established, which is displayed by the computer  8 . Referring to  FIGS. 4C and 4D , as the vehicle  10  continues to move forward, sometime later at a second instant in time, a corresponding maximum bulge point  31  on the trailing part of tire  21   b  on the tire&#39;s sidewall passes by the beam  25  about one-third the way up off the surface. At this point, a second distance is determined, which is displayed by the computer  8 . The computer  8  calculates the difference between the first and second distances, which is referred to as the offset. The offset may be converted to a value for the tire wearing angle for the wheel, expressed as an angle using trigonometric relationships if the longitudinal distance between the first and second locations is known. The longitudinal distance information may be inputted to the computer  8  prior to measuring the vehicle  10  based on the tire information provided on the sidewalls  21   c  of the tire  21 . If the computer  8  determines that the value for the tire wearing angle is greater than a selected value, such as, for example, about 1 degree, the computer  8  may indicate to a user that there may be a wheel alignment problem (e.g. via output device  8   c ). Thus, the control system is configured to a) receive output signals from however many of the apparatuses A,B there are and to b) output data based on a difference between the distances to the two locations  30  and  31  on the wheel  21  that were determined.  FIGS. 4A-4D  may relate to determining the offset and value for the tire wearing angle for a first wheel  21  (e.g. the left, or driver&#39;s side, front wheel). Data from the other apparatus at the other side of the vehicle (e.g. the right, or passenger side, front wheel) is factored into the determination as to whether the difference is due to the vehicle  10  not tracking straight (i.e. perpendicularly to the emitted beams) as the vehicle passed the apparatuses A,B. If a significant offset is still found to exist, a test for a suspension problem may be undertaken by backing the vehicle past the apparatuses as described above. 
     It will be noted that, if the direction of travel of the vehicle  10  shown by arrow  202  in  FIGS. 1 and 1   a , is not perpendicular to the directions of travel of the beams  25  this will affect the offset that is determined for the wheels  21 . In the example shown in  FIG. 1 , the vehicle  10  is traveling perpendicular to the beams  25  and so no compensation needs to be made for the direction of travel of the vehicle  10 . However, in  FIG. 1 a   , the vehicle&#39;s direction of travel  202  is not perpendicular to the beams  25 . As a result, an offset will be measured even if the vehicle&#39;s wheels  21  are all perfectly aligned with the direction of travel  202  of the vehicle  10 . By having the two apparatuses A,B take their measurements independently, but substantially simultaneously (although not necessarily precisely simultaneously), on corresponding first and second front wheels on both sides of the vehicle  10  and first and second rear wheels on both sides of the vehicle  10 , the control system  8  can determine the direction of travel of the vehicle. 
     More specifically, the control system  8  can determine the distance to the center of each wheel (e.g. by taking the average of the measurements at the points  30  and  31  on each wheel  21 ), and can then determine the offset between the centers of the front and rear wheels  21 . For example, using the example shown in  FIG. 1 a   , the control system  8  may determine that the distance to the front right wheel center is 1.0 m, the distance to the front left wheel center is 1.6 m, the distance to the rear right wheel center is 1.1 m, and the distance to the rear left wheel center is 1.5 m. Using this information, along with information regarding the front and rear tracks of the vehicle and information regarding the wheelbase of the vehicle, the control system  8  can determine the direction of travel of the vehicle  10  and can then use the determined direction of travel to compensate for the determined offsets and tire wearing angles for the wheels  21 . For example, if the front and rear tracks of the vehicle  10  are the same and if the vehicle  10  was traveling perpendicularly to the beams  25 , then there would not be any offset in the distances to the front wheels  21  and the rear wheels  21 . However, using the example data above, an offset of 0.1 m is apparent. This offset of 0.1 m, when combined with the wheelbase information can be used to determine the angle of the vehicle relative to the beams  25 . For example, if the wheelbase of the vehicle  10  is 2.8 m, then the tangent of the angle of the direction of travel  202  of the vehicle  10  is 0.1/2.8 which equals 0.0357, which corresponds to an angle of 2.05 degrees relative to a hypothetical reference line that is perpendicular to the beams  25 . This 2.05 degrees can then be subtracted (or added, as appropriate) to the tire wearing angle values determined for the wheels  21  to arrive at the true tire wearing angles for the wheels  21 . 
     The effect of tracking on the second wheel will be the opposite of that on the first wheel so information from the two sides can be compared to determine if there is actually a misalignment problem or whether the effect is all due to wheel tracking. Because the measurements made on the two wheels are independent, there is no need to perfectly align the locations between the two wheels. However, for better consistency of data accumulation, it is preferred that the locations being measured on the two wheels are at least relatively closely aligned. Wheel tracking problems can also arise from differences in suspension or tire inflation between the two wheels. To further improve consistency of data and compensate for tracking issues, distance data from both sides of the vehicle may be averaged, multiple passes of the vehicle past the fixed point may be done to increase the amount of data, and calibration methods may be employed to compensate for uneven driving surfaces. 
     Using two apparatuses A,B also permits a determination to be made of the wheelbase of the vehicle  10  on each side of the vehicle  10 . This in turn permits the control system  8  to determine if the two determinations match each other. If the control system  8  determines that the determinations do not match it means that the wheelbase on one side of the vehicle  10  is not the same as the wheelbase on the other side of the vehicle  10 , which can be an indication that the vehicle  10  incurred trauma. If this is found by the control system  8 , the control system  8  can notify a user using the output device  8   c.    
     Data collected on the front wheels of a 2012 Dodge Caravan vehicle using the system described in  FIG. 1  are shown in Table 1 and  FIGS. 6A and 6B . During operation, the laser displacement sensors are operated continuously, and as the vehicle drives past the lasers data is collected at high frequency. In order to locate which data represent the passage of the wheels rather than the chassis or fender, and then to determine the appropriate data points from which the offset may be calculated, an algorithm was used to average data over 15 samples surrounding each sample point and then to calculate the variance for each sample. Inspection of the average for a local minimum associated with a low variance is an indication of the passage of a wheel. The data is shown on Table 1 for the front wheels. In Table 1, Local Mean is the mean over 15 samples surrounding a sample point and Local Variance is the variance of the sample point from the mean. The Measurement, the Local Mean and the Local Variance for the appropriate data points for each wheel that may be used for offset calculation are shown in bold underline in the table. It is the value of the Measurement at each of these points that is used in the offset calculation. 
     The data were converted into histograms for easy visual inspection.  FIG. 6A  is the histogram for the front left wheel and  FIG. 6B  for the front right wheel. First, it is immediately evident from the histograms that the region between about Points  45  and  416  for the front left (see  FIG. 6A ) represents the passage of the front left wheel and the region between about Points  30  and  404  for the front right (see  FIG. 6B ) represents the passage of the front right wheel. The tire profile can be readily seen in these histograms with a generalized minimum between two spikes in distance. 
     For the front left wheel, with reference to Table 1 and  FIG. 6A , it can be seen from the data and histogram that Point  118  forms a minimum distance at the leading part of the wheel. This is most readily seen by looking at the Local Variance surrounding this point. The Local Variances at Points  114 - 120  around Point  118  are very small when compared to other points in the histogram, with the Local Variance at Point  118  being the smallest. Thus, Point  118  represents the point of maximum bulge on the sidewall of the leading part of the tire on the front left wheel. The value of the Measurement at Point  118  is 360.15 mm. This is the first location for the offset determination. A similar analysis from Table 1 and  FIG. 6A  for the trailing part of the tire reveals that Point  358  is the point of maximum bulge on the sidewall of the trailing part of the tire on the front left wheel. The value of the Measurement at Point  358  is 358.37 mm. Therefore, the offset for the front left wheel is 360.15−358.37=1.78 mm, which represents a slightly toe-in orientation for the wheel. 
     Similarly for the front right wheel, with reference to Table 1 and  FIG. 6B , it can be seen from the data and histogram that Point  104  forms a minimum distance of 379.65 mm at the leading part of the wheel, while Point  345  forms a minimum distance of 379.35 mm at the trailing part of the wheel. This represents an offset of 0.30 mm, which represents a slightly toe-in orientation of the wheel. 
     The small offsets for both the left and right front wheels are an indication that the wheels are properly aligned. 
     
       
         
               
             
               
               
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Front Wheels 2012 Dodge Caravan 
               
             
          
           
               
                   
                 Left 
                 Right 
               
             
          
           
               
                   
                 Measurement 
                 Local Mean 
                 Local Variance 
                 Measurement 
                 Local Mean 
                 Local Variance 
               
               
                 Point 
                 (mm) 
                 (15 points) 
                 (15 Points) 
                 (mm) 
                 (15 points) 
                 (15 Points) 
               
               
                   
               
             
          
           
               
                 1 
                 1599.98 
                   
                   
                 1599.98 
                   
                   
               
               
                 2 
                 1599.98 
                   
                   
                 1599.98 
               
               
                 3 
                 1599.98 
                   
                   
                 1599.98 
               
               
                 4 
                 1599.98 
                   
                   
                 1599.98 
               
               
                 5 
                 1599.98 
                   
                   
                 1599.98 
               
               
                 6 
                 1599.98 
                   
                   
                 1599.98 
               
               
                 7 
                 1599.98 
                   
                   
                 1599.98 
               
               
                 8 
                 1599.98 
                   
                   
                 1599.98 
               
               
                 9 
                 1599.98 
                 1599.98 
                   
                 1599.98 
                 1599.98 
               
               
                 10 
                 1599.98 
                 1599.98 
                   
                 1599.98 
                 1599.98 
               
               
                 11 
                 1599.98 
                 1599.98 
                   
                 1599.98 
                 1599.98 
               
               
                 12 
                 1599.98 
                 1599.98 
                   
                 1599.98 
                 1599.98 
               
               
                 13 
                 1599.98 
                 1599.98 
                   
                 1599.98 
                 1599.98 
               
               
                 14 
                 1599.98 
                 1599.98 
                   
                 1599.98 
                 1599.98 
               
               
                 15 
                 1599.98 
                 1599.98 
                   
                 1599.98 
                 1599.98 
               
               
                 16 
                 1599.98 
                 1599.98 
                   
                 1599.98 
                 1599.98 
               
               
                 17 
                 1599.98 
                 1599.98 
                 0.00 
                 1599.98 
                 1599.98 
                 0.00 
               
               
                 18 
                 1599.98 
                 1599.98 
                 0.00 
                 1599.98 
                 1599.98 
                 0.00 
               
               
                 19 
                 1599.98 
                 1599.98 
                 0.00 
                 1599.98 
                 1599.98 
                 0.00 
               
               
                 20 
                 1599.98 
                 1599.98 
                 0.00 
                 1599.98 
                 1526.31 
                 86837.77 
               
               
                 21 
                 1599.98 
                 1599.98 
                 0.00 
                 1599.98 
                 1452.71 
                 161943.24 
               
               
                 22 
                 1599.98 
                 1599.98 
                 0.00 
                 1599.98 
                 1379.80 
                 224096.67 
               
               
                 23 
                 1599.98 
                 1599.98 
                 0.00 
                 1599.98 
                 1321.33 
                 251319.34 
               
               
                 24 
                 1599.98 
                 1599.98 
                 0.00 
                 1599.98 
                 1263.20 
                 270829.86 
               
               
                 25 
                 1599.98 
                 1599.98 
                 0.00 
                 1599.98 
                 1205.25 
                 282935.89 
               
               
                 26 
                 1599.98 
                 1599.98 
                 0.00 
                 1599.98 
                 1147.68 
                 287485.77 
               
               
                 27 
                 1599.98 
                 1599.98 
                 0.00 
                 421.25 
                 1090.27 
                 284817.39 
               
               
                 28 
                 1599.98 
                 1599.98 
                 0.00 
                 422.37 
                 1033.13 
                 274924.09 
               
               
                 29 
                 1599.98 
                 1599.98 
                 0.00 
                 433.37 
                 976.31 
                 257871.05 
               
               
                 30 
                 1599.98 
                 1599.98 
                 0.00 
                 664.60 
                 919.60 
                 233875.23 
               
               
                 31 
                 1599.98 
                 1599.98 
                 0.00 
                 669.90 
                 863.18 
                 202913.69 
               
               
                 32 
                 1599.98 
                 1524.66 
                 90763.61 
                 672.65 
                 807.01 
                 165106.33 
               
               
                 33 
                 1599.98 
                 1449.35 
                 169414.87 
                 678.85 
                 751.00 
                 120549.83 
               
               
                 34 
                 1599.98 
                 1374.04 
                 235956.02 
                 681.55 
                 695.13 
                 69303.64 
               
               
                 35 
                 1599.98 
                 1298.70 
                 290464.66 
                 685.75 
                 639.55 
                 11441.68 
               
               
                 36 
                 1599.98 
                 1223.36 
                 332852.40 
                 690.82 
                 657.84 
                 8275.59 
               
               
                 37 
                 1599.98 
                 1148.05 
                 363090.12 
                 692.62 
                 676.27 
                 4452.45 
               
               
                 38 
                 1599.98 
                 1088.39 
                 362521.47 
                 697.30 
                 694.18 
                 303.86 
               
               
                 39 
                 394.90 
                 1029.01 
                 354132.08 
                 701.17 
                 697.82 
                 286.24 
               
               
                 40 
                 394.98 
                 969.99 
                 337977.87 
                 703.80 
                 701.34 
                 274.56 
               
               
                 41 
                 395.05 
                 911.23 
                 314248.97 
                 706.17 
                 704.91 
                 259.98 
               
               
                 42 
                 394.50 
                 852.76 
                 283034.14 
                 710.70 
                 708.16 
                 248.14 
               
               
                 43 
                 394.60 
                 794.55 
                 244458.87 
                 713.82 
                 711.39 
                 232.08 
               
               
                 44 
                 395.02 
                 736.61 
                 198615.25 
                 717.30 
                 714.58 
                 220.11 
               
               
                 45 
                 645.40 
                 678.91 
                 145609.21 
                 719.92 
                 717.65 
                 215.49 
               
               
                 46 
                 649.87 
                 621.49 
                 85534.19 
                 722.87 
                 720.84 
                 207.98 
               
               
                 47 
                 655.60 
                 564.20 
                 18459.20 
                 726.15 
                 723.83 
                 200.73 
               
               
                 48 
                 659.82 
                 582.52 
                 17212.90 
                 729.77 
                 726.55 
                 187.76 
               
               
                 49 
                 664.55 
                 601.05 
                 15293.38 
                 730.80 
                 729.39 
                 178.71 
               
               
                 50 
                 668.57 
                 619.74 
                 12668.73 
                 733.35 
                 732.26 
                 168.50 
               
               
                 51 
                 673.02 
                 638.62 
                 9299.65 
                 736.70 
                 735.02 
                 163.19 
               
               
                 52 
                 676.80 
                 657.79 
                 5199.45 
                 740.00 
                 737.68 
                 156.04 
               
               
                 53 
                 681.12 
                 677.08 
                 339.57 
                 743.67 
                 740.18 
                 147.64 
               
               
                 54 
                 683.40 
                 680.83 
                 311.20 
                 745.10 
                 742.69 
                 139.88 
               
               
                 55 
                 688.05 
                 684.50 
                 284.49 
                 744.75 
                 745.23 
                 135.80 
               
               
                 56 
                 691.50 
                 687.95 
                 262.04 
                 749.15 
                 727.96 
                 5612.03 
               
               
                 57 
                 694.10 
                 691.35 
                 243.05 
                 752.15 
                 709.41 
                 11047.94 
               
               
                 58 
                 696.57 
                 694.64 
                 227.84 
                 754.77 
                 690.22 
                 16063.92 
               
               
                 59 
                 701.25 
                 697.73 
                 209.08 
                 756.35 
                 670.84 
                 20288.40 
               
               
                 60 
                 703.65 
                 700.76 
                 196.08 
                 757.42 
                 650.71 
                 23945.87 
               
               
                 61 
                 705.42 
                 703.71 
                 184.39 
                 760.05 
                 630.05 
                 26837.94 
               
               
                 62 
                 708.65 
                 706.52 
                 175.42 
                 763.55 
                 608.93 
                 28855.06 
               
               
                 63 
                 710.75 
                 709.41 
                 166.45 
                 449.80 
                 587.59 
                 29945.02 
               
               
                 64 
                 714.25 
                 712.09 
                 159.43 
                 432.92 
                 566.35 
                 30040.92 
               
               
                 65 
                 717.10 
                 714.71 
                 153.86 
                 423.77 
                 544.57 
                 29135.19 
               
               
                 66 
                 718.12 
                 717.26 
                 145.80 
                 423.32 
                 522.59 
                 27133.72 
               
               
                 67 
                 721.45 
                 719.79 
                 136.56 
                 414.55 
                 500.38 
                 24024.98 
               
               
                 68 
                 723.95 
                 704.49 
                 4486.36 
                 409.50 
                 478.03 
                 19811.52 
               
               
                 69 
                 726.12 
                 687.30 
                 9241.22 
                 405.77 
                 455.57 
                 14497.12 
               
               
                 70 
                 729.62 
                 669.67 
                 13533.61 
                 403.55 
                 432.76 
                 8005.06 
               
               
                 71 
                 731.00 
                 651.05 
                 17536.14 
                 404.95 
                 409.67 
                 240.77 
               
               
                 72 
                 733.30 
                 631.96 
                 20935.83 
                 400.77 
                 406.20 
                 136.43 
               
               
                 73 
                 734.90 
                 612.13 
                 23742.39 
                 400.35 
                 403.56 
                 97.49 
               
               
                 74 
                 737.05 
                 591.95 
                 25739.77 
                 399.42 
                 401.45 
                 77.77 
               
               
                 75 
                 456.47 
                 571.46 
                 26944.83 
                 398.85 
                 399.36 
                 50.23 
               
               
                 76 
                 428.72 
                 550.81 
                 27157.28 
                 397.92 
                 397.72 
                 40.03 
               
               
                 77 
                 423.32 
                 529.61 
                 26519.47 
                 395.12 
                 396.36 
                 35.44 
               
               
                 78 
                 410.62 
                 508.14 
                 24893.24 
                 394.22 
                 395.17 
                 34.28 
               
               
                 79 
                 405.30 
                 486.37 
                 22190.52 
                 394.25 
                 394.08 
                 33.80 
               
               
                 80 
                 397.10 
                 464.47 
                 18434.35 
                 390.67 
                 392.78 
                 30.57 
               
               
                 81 
                 394.20 
                 442.38 
                 13573.40 
                 390.00 
                 391.71 
                 30.77 
               
               
                 82 
                 390.17 
                 420.13 
                 7609.76 
                 389.82 
                 390.63 
                 29.54 
               
               
                 83 
                 391.17 
                 397.67 
                 496.17 
                 388.37 
                 389.57 
                 27.64 
               
               
                 84 
                 384.67 
                 392.63 
                 270.07 
                 387.75 
                 388.52 
                 24.39 
               
               
                 85 
                 382.65 
                 389.27 
                 192.14 
                 386.67 
                 387.49 
                 20.73 
               
               
                 86 
                 381.27 
                 386.20 
                 119.85 
                 386.10 
                 386.64 
                 18.44 
               
               
                 87 
                 380.67 
                 383.78 
                 87.50 
                 384.25 
                 385.80 
                 16.20 
               
               
                 88 
                 379.85 
                 381.61 
                 63.35 
                 383.55 
                 384.94 
                 12.52 
               
               
                 89 
                 378.87 
                 379.97 
                 52.16 
                 383.05 
                 384.28 
                 11.41 
               
               
                 90 
                 377.60 
                 378.43 
                 43.22 
                 382.45 
                 383.63 
                 10.21 
               
               
                 91 
                 375.95 
                 377.00 
                 40.23 
                 382.15 
                 383.01 
                 8.18 
               
               
                 92 
                 374.90 
                 375.54 
                 30.17 
                 381.42 
                 382.48 
                 6.61 
               
               
                 93 
                 374.25 
                 374.30 
                 30.73 
                 381.55 
                 381.99 
                 4.95 
               
               
                 94 
                 371.90 
                 373.18 
                 30.80 
                 380.70 
                 381.57 
                 3.59 
               
               
                 95 
                 370.50 
                 372.14 
                 30.08 
                 380.50 
                 381.16 
                 2.32 
               
               
                 96 
                 370.87 
                 371.09 
                 28.63 
                 380.12 
                 380.87 
                 1.75 
               
               
                 97 
                 369.67 
                 370.04 
                 26.75 
                 379.65 
                 380.62 
                 1.31 
               
               
                 98 
                 367.22 
                 369.03 
                 24.03 
                 379.90 
                 380.41 
                 0.93 
               
               
                 99 
                 367.85 
                 368.07 
                 21.26 
                 379.90 
                 380.23 
                 0.66 
               
               
                 100 
                 364.75 
                 367.19 
                 18.85 
                 379.90 
                 380.07 
                 0.42 
               
               
                 101 
                 364.77 
                 366.33 
                 16.49 
                 379.90 
                 379.95 
                 0.30 
               
               
                 102 
                 364.70 
                 365.51 
                 13.41 
                 379.50 
                 379.83 
                 0.12 
               
               
                 103 
                 363.87 
                 364.80 
                 11.76 
                 379.60 
                 379.76 
                 0.07 
               
               
                 104 
                 362.95 
                 364.18 
                 10.36 
                 379.65 
                 379.74 
                 0.04 
               
               
                 105 
                 362.72 
                 363.53 
                 7.86 
                 379.64 
                 379.79 
                 0.14 
               
               
                 106 
                 362.20 
                 362.96 
                 5.64 
                 379.62 
                 379.92 
                 0.35 
               
               
                 107 
                 361.87 
                 362.53 
                 4.68 
                 379.56 
                 380.08 
                 0.77 
               
               
                 108 
                 361.20 
                 362.05 
                 2.90 
                 379.50 
                 380.30 
                 1.44 
               
               
                 109 
                 361.10 
                 361.76 
                 2.56 
                 379.70 
                 380.56 
                 2.30 
               
               
                 110 
                 360.60 
                 361.48 
                 2.03 
                 379.55 
                 380.88 
                 3.48 
               
               
                 111 
                 360.62 
                 361.19 
                 1.37 
                 380.15 
                 381.30 
                 5.02 
               
               
                 112 
                 360.45 
                 360.97 
                 0.90 
                 381.00 
                 381.77 
                 6.91 
               
               
                 113 
                 360.42 
                 360.81 
                 0.63 
                 381.65 
                 382.29 
                 8.92 
               
               
                 114 
                 360.35 
                 360.65 
                 0.39 
                 382.52 
                 382.89 
                 11.29 
               
               
                 115 
                 360.22 
                 360.53 
                 0.23 
                 383.37 
                 383.57 
                 13.90 
               
               
                 116 
                 360.20 
                 360.42 
                 0.10 
                 384.07 
                 384.33 
                 16.66 
               
               
                 117 
                 360.20 
                 360.36 
                 0.06 
                 385.00 
                 385.16 
                 19.10 
               
               
                 118 
                 360.15 
                 360.33 
                 0.03 
                 386.15 
                 386.07 
                 21.76 
               
               
                 119 
                 360.32 
                 360.38 
                 0.10 
                 387.20 
                 387.09 
                 24.15 
               
               
                 120 
                 360.32 
                 360.47 
                 0.29 
                 388.02 
                 387.76 
                 21.41 
               
               
                 121 
                 360.20 
                 360.63 
                 0.66 
                 389.25 
                 388.35 
                 18.50 
               
               
                 122 
                 360.20 
                 360.84 
                 1.27 
                 390.45 
                 388.90 
                 15.45 
               
               
                 123 
                 360.22 
                 361.10 
                 2.08 
                 391.75 
                 389.41 
                 12.68 
               
               
                 124 
                 360.12 
                 361.44 
                 3.32 
                 392.75 
                 389.88 
                 10.16 
               
               
                 125 
                 360.70 
                 361.85 
                 4.92 
                 394.27 
                 390.36 
                 7.91 
               
               
                 126 
                 361.40 
                 362.31 
                 6.73 
                 395.82 
                 390.83 
                 6.07 
               
               
                 127 
                 362.10 
                 362.83 
                 8.60 
                 390.85 
                 391.28 
                 4.82 
               
               
                 128 
                 362.92 
                 363.40 
                 10.73 
                 390.52 
                 391.71 
                 4.01 
               
               
                 129 
                 363.77 
                 364.04 
                 13.16 
                 390.35 
                 392.17 
                 3.84 
               
               
                 130 
                 364.52 
                 364.82 
                 16.52 
                 390.70 
                 393.07 
                 11.11 
               
               
                 131 
                 365.70 
                 365.67 
                 19.61 
                 390.87 
                 394.29 
                 28.00 
               
               
                 132 
                 366.75 
                 366.61 
                 22.75 
                 391.82 
                 396.52 
                 95.66 
               
               
                 133 
                 367.62 
                 367.53 
                 23.58 
                 392.50 
                 398.91 
                 168.08 
               
               
                 134 
                 368.40 
                 368.13 
                 20.63 
                 393.37 
                 401.07 
                 221.30 
               
               
                 135 
                 369.42 
                 368.70 
                 17.64 
                 394.00 
                 403.09 
                 263.48 
               
               
                 136 
                 370.65 
                 369.24 
                 14.70 
                 395.52 
                 405.41 
                 289.43 
               
               
                 137 
                 372.70 
                 369.74 
                 11.96 
                 403.60 
                 408.02 
                 315.15 
               
               
                 138 
                 373.72 
                 370.17 
                 9.44 
                 409.92 
                 411.10 
                 351.19 
               
               
                 139 
                 375.20 
                 370.56 
                 7.17 
                 427.47 
                 414.26 
                 373.39 
               
               
                 140 
                 374.85 
                 370.84 
                 5.52 
                 431.05 
                 417.64 
                 387.51 
               
               
                 141 
                 370.40 
                 371.06 
                 4.37 
                 428.82 
                 420.41 
                 357.65 
               
               
                 142 
                 370.50 
                 371.22 
                 3.60 
                 428.00 
                 422.78 
                 306.41 
               
               
                 143 
                 370.77 
                 371.32 
                 3.16 
                 428.10 
                 424.89 
                 245.25 
               
               
                 144 
                 370.92 
                 371.36 
                 3.04 
                 432.17 
                 426.82 
                 177.67 
               
               
                 145 
                 370.65 
                 371.26 
                 3.32 
                 439.72 
                 428.73 
                 108.53 
               
               
                 146 
                 370.70 
                 371.07 
                 3.33 
                 441.25 
                 429.98 
                 66.41 
               
               
                 147 
                 370.20 
                 370.79 
                 2.98 
                 444.95 
                 430.70 
                 43.92 
               
               
                 148 
                 370.23 
                 370.41 
                 1.72 
                 436.12 
                 430.18 
                 51.90 
               
               
                 149 
                 370.25 
                 370.10 
                 0.32 
                 430.42 
                 429.35 
                 61.26 
               
               
                 150 
                 370.00 
                 370.13 
                 0.35 
                 427.10 
                 428.61 
                 70.91 
               
               
                 151 
                 369.95 
                 370.15 
                 0.37 
                 424.87 
                 427.89 
                 80.32 
               
               
                 152 
                 369.15 
                 370.15 
                 0.38 
                 426.05 
                 427.15 
                 88.58 
               
               
                 153 
                 369.62 
                 370.10 
                 0.34 
                 423.73 
                 426.14 
                 94.24 
               
               
                 154 
                 369.32 
                 370.15 
                 0.46 
                 421.42 
                 424.63 
                 86.97 
               
               
                 155 
                 369.12 
                 370.14 
                 0.44 
                 419.10 
                 423.02 
                 71.29 
               
               
                 156 
                 369.90 
                 370.07 
                 0.51 
                 417.85 
                 421.16 
                 39.65 
               
               
                 157 
                 370.80 
                 369.99 
                 0.58 
                 416.95 
                 419.82 
                 25.52 
               
               
                 158 
                 370.77 
                 369.91 
                 0.64 
                 416.37 
                 418.84 
                 18.78 
               
               
                 159 
                 370.90 
                 369.84 
                 0.70 
                 416.37 
                 417.98 
                 15.42 
               
               
                 160 
                 370.00 
                 369.81 
                 0.70 
                 415.90 
                 417.11 
                 14.75 
               
               
                 161 
                 371.55 
                 369.84 
                 0.67 
                 415.55 
                 415.55 
                 23.88 
               
               
                 162 
                 370.42 
                 369.84 
                 0.67 
                 415.55 
                 414.03 
                 34.41 
               
               
                 163 
                 369.10 
                 369.89 
                 0.66 
                 415.15 
                 412.60 
                 44.72 
               
               
                 164 
                 368.95 
                 369.94 
                 0.61 
                 414.80 
                 411.22 
                 55.90 
               
               
                 165 
                 368.95 
                 369.97 
                 0.62 
                 414.65 
                 409.85 
                 66.70 
               
               
                 166 
                 368.95 
                 369.94 
                 0.59 
                 413.40 
                 408.47 
                 76.21 
               
               
                 167 
                 369.52 
                 369.92 
                 0.55 
                 410.95 
                 407.09 
                 83.49 
               
               
                 168 
                 369.65 
                 369.88 
                 0.50 
                 401.12 
                 405.67 
                 87.51 
               
               
                 169 
                 369.62 
                 369.92 
                 0.53 
                 399.37 
                 404.24 
                 89.06 
               
               
                 170 
                 370.05 
                 369.97 
                 0.48 
                 398.47 
                 400.50 
                 71.93 
               
               
                 171 
                 369.95 
                 370.13 
                 0.50 
                 397.10 
                 399.01 
                 59.56 
               
               
                 172 
                 370.30 
                 370.28 
                 0.46 
                 395.85 
                 397.53 
                 42.77 
               
               
                 173 
                 370.40 
                 370.43 
                 0.37 
                 394.90 
                 396.17 
                 24.25 
               
               
                 174 
                 370.40 
                 370.57 
                 0.40 
                 394.25 
                 394.97 
                 7.84 
               
               
                 175 
                 370.30 
                 370.73 
                 0.46 
                 393.72 
                 394.42 
                 5.13 
               
               
                 176 
                 370.60 
                 370.91 
                 0.53 
                 393.00 
                 393.99 
                 3.35 
               
               
                 177 
                 370.70 
                 371.07 
                 0.60 
                 393.47 
                 393.64 
                 1.84 
               
               
                 178 
                 371.25 
                 371.22 
                 0.60 
                 392.40 
                 393.37 
                 0.92 
               
               
                 179 
                 371.25 
                 371.35 
                 0.60 
                 392.45 
                 393.19 
                 0.45 
               
               
                 180 
                 371.25 
                 371.50 
                 0.63 
                 392.50 
                 393.08 
                 0.23 
               
               
                 181 
                 371.17 
                 371.66 
                 0.61 
                 393.02 
                 392.98 
                 0.13 
               
               
                 182 
                 371.65 
                 371.81 
                 0.53 
                 393.00 
                 392.96 
                 0.11 
               
               
                 183 
                 372.00 
                 371.96 
                 0.48 
                 392.85 
                 392.97 
                 0.11 
               
               
                 184 
                 372.40 
                 372.11 
                 0.40 
                 392.90 
                 392.95 
                 0.09 
               
               
                 185 
                 372.36 
                 372.24 
                 0.41 
                 393.17 
                 393.00 
                 0.07 
               
               
                 186 
                 372.32 
                 372.37 
                 0.39 
                 393.14 
                 393.05 
                 0.05 
               
               
                 187 
                 372.25 
                 372.48 
                 0.31 
                 393.14 
                 393.08 
                 0.03 
               
               
                 188 
                 372.67 
                 372.60 
                 0.19 
                 393.15 
                 393.11 
                 0.05 
               
               
                 189 
                 372.66 
                 372.69 
                 0.13 
                 392.77 
                 393.13 
                 0.04 
               
               
                 190 
                 372.65 
                 372.77 
                 0.10 
                 393.45 
                 393.14 
                 0.04 
               
               
                 191 
                 372.85 
                 372.83 
                 0.11 
                 393.12 
                 393.14 
                 0.04 
               
               
                 192 
                 372.90 
                 372.90 
                 0.12 
                 393.22 
                 393.43 
                 1.29 
               
               
                 193 
                 373.17 
                 373.00 
                 0.13 
                 393.15 
                 394.02 
                 6.22 
               
               
                 194 
                 373.20 
                 373.06 
                 0.09 
                 393.15 
                 394.86 
                 15.26 
               
               
                 195 
                 372.92 
                 373.11 
                 0.10 
                 392.90 
                 395.89 
                 27.30 
               
               
                 196 
                 373.02 
                 373.19 
                 0.10 
                 393.57 
                 397.10 
                 41.27 
               
               
                 197 
                 373.02 
                 373.26 
                 0.10 
                 393.20 
                 398.43 
                 57.44 
               
               
                 198 
                 373.14 
                 373.33 
                 0.11 
                 393.06 
                 399.97 
                 75.47 
               
               
                 199 
                 373.25 
                 373.37 
                 0.10 
                 392.92 
                 401.62 
                 92.43 
               
               
                 200 
                 373.52 
                 373.47 
                 0.21 
                 397.47 
                 403.20 
                 100.99 
               
               
                 201 
                 373.72 
                 374.54 
                 16.70 
                 402.05 
                 404.75 
                 103.65 
               
               
                 202 
                 373.15 
                 376.76 
                 82.85 
                 405.77 
                 406.29 
                 100.13 
               
               
                 203 
                 373.55 
                 379.17 
                 151.11 
                 408.55 
                 407.75 
                 92.34 
               
               
                 204 
                 373.72 
                 381.63 
                 209.37 
                 410.97 
                 409.23 
                 79.06 
               
               
                 205 
                 373.79 
                 384.13 
                 257.73 
                 413.42 
                 410.69 
                 60.42 
               
               
                 206 
                 373.85 
                 386.54 
                 288.65 
                 416.22 
                 412.14 
                 36.78 
               
               
                 207 
                 373.50 
                 388.64 
                 295.96 
                 417.97 
                 413.22 
                 20.32 
               
               
                 208 
                 374.70 
                 390.68 
                 293.27 
                 416.75 
                 414.02 
                 10.77 
               
               
                 209 
                 389.22 
                 392.85 
                 282.48 
                 416.40 
                 414.71 
                 5.72 
               
               
                 210 
                 406.20 
                 395.35 
                 272.71 
                 416.00 
                 414.62 
                 7.03 
               
               
                 211 
                 409.27 
                 397.74 
                 247.61 
                 415.50 
                 414.72 
                 6.40 
               
               
                 212 
                 409.90 
                 400.11 
                 210.19 
                 415.40 
                 414.63 
                 6.82 
               
               
                 213 
                 410.67 
                 402.32 
                 159.14 
                 414.92 
                 414.33 
                 7.13 
               
               
                 214 
                 409.40 
                 403.93 
                 98.67 
                 414.77 
                 413.91 
                 6.48 
               
               
                 215 
                 404.92 
                 405.32 
                 40.63 
                 413.57 
                 413.55 
                 6.23 
               
               
                 216 
                 404.37 
                 405.70 
                 29.62 
                 414.10 
                 413.19 
                 6.01 
               
               
                 217 
                 405.75 
                 404.90 
                 38.33 
                 416.17 
                 412.86 
                 5.63 
               
               
                 218 
                 411.00 
                 403.90 
                 43.89 
                 407.20 
                 412.54 
                 5.37 
               
               
                 219 
                 409.57 
                 402.82 
                 47.51 
                 412.47 
                 412.23 
                 4.90 
               
               
                 220 
                 409.32 
                 401.66 
                 48.28 
                 411.95 
                 411.94 
                 4.52 
               
               
                 221 
                 407.07 
                 400.57 
                 47.94 
                 411.75 
                 411.64 
                 4.04 
               
               
                 222 
                 397.55 
                 399.78 
                 49.96 
                 411.75 
                 411.37 
                 4.00 
               
               
                 223 
                 395.52 
                 399.01 
                 51.34 
                 411.35 
                 411.07 
                 3.61 
               
               
                 224 
                 394.97 
                 398.08 
                 50.86 
                 410.92 
                 410.73 
                 1.62 
               
               
                 225 
                 394.22 
                 396.71 
                 41.04 
                 411.12 
                 410.97 
                 0.67 
               
               
                 226 
                 394.32 
                 395.49 
                 29.73 
                 410.65 
                 410.89 
                 0.51 
               
               
                 227 
                 393.70 
                 394.26 
                 16.00 
                 410.80 
                 410.85 
                 0.44 
               
               
                 228 
                 393.20 
                 393.17 
                 3.91 
                 410.47 
                 410.83 
                 0.41 
               
               
                 229 
                 393.13 
                 392.70 
                 2.82 
                 410.30 
                 410.81 
                 0.38 
               
               
                 230 
                 393.05 
                 392.35 
                 2.55 
                 409.57 
                 410.83 
                 0.40 
               
               
                 231 
                 392.75 
                 392.05 
                 2.23 
                 409.55 
                 410.87 
                 0.44 
               
               
                 232 
                 391.82 
                 391.79 
                 2.02 
                 411.05 
                 411.11 
                 1.39 
               
               
                 233 
                 390.50 
                 391.52 
                 1.64 
                 410.90 
                 411.29 
                 1.74 
               
               
                 234 
                 391.30 
                 391.28 
                 1.38 
                 411.25 
                 411.46 
                 1.98 
               
               
                 235 
                 390.85 
                 391.08 
                 1.16 
                 411.33 
                 411.70 
                 2.33 
               
               
                 236 
                 390.70 
                 390.92 
                 0.85 
                 411.40 
                 411.98 
                 2.65 
               
               
                 237 
                 390.50 
                 390.76 
                 0.50 
                 411.52 
                 412.33 
                 2.68 
               
               
                 238 
                 390.25 
                 390.60 
                 0.20 
                 411.56 
                 412.73 
                 2.75 
               
               
                 239 
                 390.40 
                 390.53 
                 0.09 
                 411.60 
                 413.20 
                 4.35 
               
               
                 240 
                 390.35 
                 390.53 
                 0.09 
                 414.65 
                 413.91 
                 8.37 
               
               
                 241 
                 390.32 
                 390.52 
                 0.08 
                 413.47 
                 414.83 
                 15.71 
               
               
                 242 
                 390.15 
                 390.53 
                 0.09 
                 413.30 
                 415.66 
                 19.90 
               
               
                 243 
                 390.15 
                 390.62 
                 0.22 
                 414.05 
                 416.44 
                 21.86 
               
               
                 244 
                 390.67 
                 390.76 
                 0.51 
                 414.45 
                 416.00 
                 29.40 
               
               
                 245 
                 390.62 
                 391.02 
                 1.18 
                 414.82 
                 415.49 
                 38.17 
               
               
                 246 
                 390.40 
                 391.20 
                 1.43 
                 415.67 
                 414.96 
                 46.63 
               
               
                 247 
                 390.80 
                 391.41 
                 1.72 
                 418.07 
                 414.23 
                 55.19 
               
               
                 248 
                 390.50 
                 391.63 
                 1.96 
                 421.52 
                 413.56 
                 62.97 
               
               
                 249 
                 391.15 
                 391.90 
                 2.17 
                 424.97 
                 412.92 
                 69.41 
               
               
                 250 
                 391.05 
                 392.19 
                 2.37 
                 423.85 
                 412.27 
                 74.23 
               
               
                 251 
                 391.95 
                 392.50 
                 2.81 
                 423.05 
                 411.63 
                 77.39 
               
               
                 252 
                 392.70 
                 392.85 
                 3.22 
                 404.92 
                 411.07 
                 78.32 
               
               
                 253 
                 394.02 
                 393.22 
                 3.33 
                 403.90 
                 411.61 
                 88.02 
               
               
                 254 
                 393.10 
                 393.62 
                 3.63 
                 403.75 
                 412.04 
                 96.89 
               
               
                 255 
                 393.52 
                 394.15 
                 4.27 
                 403.65 
                 413.66 
                 169.29 
               
               
                 256 
                 393.72 
                 394.37 
                 3.59 
                 403.45 
                 415.03 
                 230.55 
               
               
                 257 
                 394.12 
                 394.48 
                 2.98 
                 403.75 
                 413.63 
                 233.55 
               
               
                 258 
                 394.57 
                 394.35 
                 3.96 
                 404.25 
                 411.59 
                 254.74 
               
               
                 259 
                 395.35 
                 393.97 
                 7.47 
                 404.85 
                 410.77 
                 276.47 
               
               
                 260 
                 395.85 
                 393.25 
                 15.12 
                 406.35 
                 409.98 
                 297.47 
               
               
                 261 
                 395.95 
                 392.33 
                 28.12 
                 423.77 
                 409.20 
                 316.99 
               
               
                 262 
                 396.75 
                 391.11 
                 47.38 
                 424.60 
                 408.43 
                 335.12 
               
               
                 263 
                 398.40 
                 389.70 
                 69.44 
                 445.85 
                 407.67 
                 351.85 
               
               
                 264 
                 394.50 
                 388.26 
                 86.71 
                 445.50 
                 406.90 
                 367.50 
               
               
                 265 
                 392.72 
                 386.79 
                 99.36 
                 402.82 
                 406.08 
                 382.01 
               
               
                 266 
                 389.97 
                 385.29 
                 105.59 
                 392.47 
                 405.23 
                 395.19 
               
               
                 267 
                 387.00 
                 383.75 
                 106.39 
                 392.65 
                 404.28 
                 406.44 
               
               
                 268 
                 383.25 
                 382.19 
                 101.93 
                 392.05 
                 402.12 
                 386.17 
               
               
                 269 
                 379.30 
                 380.56 
                 90.94 
                 392.06 
                 399.94 
                 352.54 
               
               
                 270 
                 375.20 
                 378.82 
                 69.93 
                 392.07 
                 396.32 
                 192.89 
               
               
                 271 
                 372.52 
                 377.34 
                 53.08 
                 392.07 
                 392.71 
                 7.95 
               
               
                 272 
                 372.60 
                 375.97 
                 36.02 
                 392.07 
                 392.01 
                 0.13 
               
               
                 273 
                 372.47 
                 374.79 
                 21.51 
                 392.06 
                 391.99 
                 0.12 
               
               
                 274 
                 372.85 
                 373.84 
                 10.20 
                 392.05 
                 391.98 
                 0.11 
               
               
                 275 
                 372.70 
                 373.13 
                 3.44 
                 392.10 
                 392.06 
                 0.19 
               
               
                 276 
                 372.67 
                 372.87 
                 1.02 
                 391.40 
                 392.13 
                 0.28 
               
               
                 277 
                 372.30 
                 372.45 
                 1.51 
                 391.82 
                 392.23 
                 0.42 
               
               
                 278 
                 372.22 
                 372.30 
                 1.86 
                 391.62 
                 392.36 
                 0.64 
               
               
                 279 
                 372.26 
                 372.23 
                 1.88 
                 391.30 
                 392.52 
                 0.89 
               
               
                 280 
                 372.26 
                 372.13 
                 1.98 
                 392.40 
                 392.68 
                 1.13 
               
               
                 281 
                 372.25 
                 372.01 
                 2.01 
                 392.20 
                 392.86 
                 1.38 
               
               
                 282 
                 372.70 
                 371.88 
                 2.08 
                 392.50 
                 393.06 
                 1.66 
               
               
                 283 
                 372.62 
                 371.77 
                 2.08 
                 393.12 
                 393.31 
                 1.70 
               
               
                 284 
                 375.40 
                 371.69 
                 2.08 
                 393.17 
                 393.57 
                 1.89 
               
               
                 285 
                 369.00 
                 371.58 
                 2.14 
                 393.60 
                 393.89 
                 2.07 
               
               
                 286 
                 370.17 
                 371.45 
                 2.19 
                 394.05 
                 394.23 
                 1.91 
               
               
                 287 
                 371.60 
                 371.32 
                 2.22 
                 394.37 
                 394.52 
                 2.05 
               
               
                 288 
                 370.95 
                 371.19 
                 2.21 
                 394.52 
                 394.84 
                 2.00 
               
               
                 289 
                 371.10 
                 371.03 
                 2.08 
                 394.77 
                 395.15 
                 1.87 
               
               
                 290 
                 370.70 
                 370.87 
                 1.92 
                 395.10 
                 395.44 
                 1.86 
               
               
                 291 
                 371.00 
                 370.49 
                 0.41 
                 395.15 
                 395.73 
                 1.71 
               
               
                 292 
                 371.10 
                 370.54 
                 0.29 
                 395.75 
                 396.00 
                 1.57 
               
               
                 293 
                 370.55 
                 370.51 
                 0.32 
                 396.37 
                 396.24 
                 1.43 
               
               
                 294 
                 370.42 
                 370.36 
                 0.31 
                 396.37 
                 396.46 
                 1.28 
               
               
                 295 
                 370.30 
                 370.26 
                 0.34 
                 396.80 
                 396.66 
                 1.04 
               
               
                 296 
                 370.30 
                 370.10 
                 0.43 
                 397.02 
                 396.84 
                 0.81 
               
               
                 297 
                 370.28 
                 369.96 
                 0.54 
                 397.10 
                 396.98 
                 0.58 
               
               
                 298 
                 370.25 
                 369.80 
                 0.58 
                 397.45 
                 397.10 
                 0.32 
               
               
                 299 
                 369.65 
                 369.65 
                 0.50 
                 397.50 
                 397.14 
                 0.23 
               
               
                 300 
                 369.70 
                 369.60 
                 0.45 
                 397.65 
                 397.13 
                 0.25 
               
               
                 301 
                 369.80 
                 369.55 
                 0.40 
                 397.65 
                 397.06 
                 0.43 
               
               
                 302 
                 369.32 
                 369.62 
                 0.58 
                 397.65 
                 396.91 
                 0.84 
               
               
                 303 
                 369.45 
                 369.86 
                 1.82 
                 397.50 
                 396.69 
                 1.55 
               
               
                 304 
                 368.70 
                 370.17 
                 3.54 
                 397.52 
                 396.43 
                 2.32 
               
               
                 305 
                 368.63 
                 370.55 
                 5.74 
                 397.17 
                 396.12 
                 3.07 
               
               
                 306 
                 368.55 
                 371.16 
                 10.06 
                 396.95 
                 395.75 
                 4.07 
               
               
                 307 
                 368.80 
                 371.98 
                 17.66 
                 396.40 
                 395.29 
                 5.29 
               
               
                 308 
                 369.87 
                 373.02 
                 29.06 
                 396.20 
                 394.71 
                 7.48 
               
               
                 309 
                 369.70 
                 374.33 
                 44.36 
                 395.35 
                 394.06 
                 9.72 
               
               
                 310 
                 371.32 
                 375.85 
                 62.93 
                 394.57 
                 393.55 
                 9.83 
               
               
                 311 
                 373.95 
                 377.69 
                 85.73 
                 393.65 
                 393.04 
                 9.43 
               
               
                 312 
                 374.94 
                 379.59 
                 102.91 
                 393.24 
                 392.57 
                 8.59 
               
               
                 313 
                 375.92 
                 381.37 
                 108.24 
                 392.82 
                 392.12 
                 7.37 
               
               
                 314 
                 378.72 
                 382.94 
                 103.10 
                 391.87 
                 391.72 
                 6.10 
               
               
                 315 
                 382.05 
                 384.29 
                 92.54 
                 390.87 
                 391.36 
                 4.59 
               
               
                 316 
                 385.42 
                 385.45 
                 76.48 
                 388.87 
                 391.06 
                 3.38 
               
               
                 317 
                 388.95 
                 386.34 
                 61.41 
                 387.90 
                 390.79 
                 2.44 
               
               
                 318 
                 392.17 
                 386.93 
                 50.95 
                 389.90 
                 390.57 
                 1.82 
               
               
                 319 
                 396.37 
                 387.34 
                 43.01 
                 389.80 
                 390.74 
                 3.31 
               
               
                 320 
                 397.10 
                 387.50 
                 39.51 
                 390.10 
                 390.83 
                 3.79 
               
               
                 321 
                 395.20 
                 387.32 
                 43.30 
                 390.30 
                 390.85 
                 3.86 
               
               
                 322 
                 392.47 
                 386.80 
                 53.19 
                 390.42 
                 390.85 
                 3.86 
               
               
                 323 
                 390.02 
                 385.96 
                 66.30 
                 390.75 
                 390.89 
                 3.72 
               
               
                 324 
                 387.17 
                 384.83 
                 78.17 
                 390.87 
                 390.89 
                 3.69 
               
               
                 325 
                 384.67 
                 383.41 
                 86.12 
                 390.50 
                 390.68 
                 4.81 
               
               
                 326 
                 382.82 
                 381.66 
                 83.40 
                 390.30 
                 390.38 
                 6.73 
               
               
                 327 
                 381.00 
                 379.82 
                 73.23 
                 395.90 
                 389.99 
                 9.24 
               
               
                 328 
                 378.30 
                 378.10 
                 61.06 
                 394.07 
                 389.52 
                 12.25 
               
               
                 329 
                 376.07 
                 376.53 
                 49.63 
                 392.27 
                 388.97 
                 15.68 
               
               
                 330 
                 374.27 
                 375.19 
                 37.83 
                 390.80 
                 388.34 
                 19.37 
               
               
                 331 
                 372.80 
                 374.34 
                 26.85 
                 389.45 
                 387.64 
                 22.83 
               
               
                 332 
                 372.02 
                 373.53 
                 18.77 
                 387.97 
                 386.90 
                 26.46 
               
               
                 333 
                 370.85 
                 372.70 
                 12.54 
                 386.72 
                 386.19 
                 28.93 
               
               
                 334 
                 370.15 
                 371.93 
                 7.76 
                 385.30 
                 385.10 
                 24.11 
               
               
                 335 
                 369.55 
                 371.26 
                 5.34 
                 384.25 
                 384.13 
                 19.58 
               
               
                 336 
                 369.30 
                 370.60 
                 5.09 
                 383.25 
                 383.28 
                 15.58 
               
               
                 337 
                 368.97 
                 369.99 
                 5.87 
                 382.22 
                 382.51 
                 12.01 
               
               
                 338 
                 369.92 
                 369.44 
                 7.16 
                 381.17 
                 381.84 
                 8.78 
               
               
                 339 
                 374.42 
                 368.86 
                 8.96 
                 380.45 
                 381.26 
                 6.24 
               
               
                 340 
                 372.47 
                 368.29 
                 11.42 
                 379.45 
                 380.76 
                 4.12 
               
               
                 341 
                 370.47 
                 367.70 
                 14.27 
                 379.57 
                 380.37 
                 2.62 
               
               
                 342 
                 369.42 
                 367.07 
                 17.68 
                 379.50 
                 380.06 
                 1.48 
               
               
                 343 
                 368.27 
                 366.42 
                 20.91 
                 379.51 
                 379.81 
                 0.70 
               
               
                 344 
                 366.15 
                 365.72 
                 24.46 
                 379.52 
                 379.66 
                 0.27 
               
               
                 345 
                 365.12 
                 364.95 
                 26.40 
                 379.35 
                 379.60 
                 0.12 
               
               
                 346 
                 364.47 
                 363.88 
                 21.96 
                 379.42 
                 379.61 
                 0.14 
               
               
                 347 
                 363.37 
                 362.93 
                 17.97 
                 379.15 
                 379.73 
                 0.32 
               
               
                 348 
                 362.27 
                 362.12 
                 14.73 
                 379.30 
                 379.89 
                 0.64 
               
               
                 349 
                 361.32 
                 361.38 
                 11.33 
                 379.40 
                 380.07 
                 0.98 
               
               
                 350 
                 360.15 
                 360.72 
                 8.13 
                 379.60 
                 380.32 
                 1.62 
               
               
                 351 
                 359.55 
                 360.20 
                 6.13 
                 379.55 
                 380.61 
                 2.42 
               
               
                 352 
                 358.45 
                 359.75 
                 4.43 
                 379.95 
                 380.99 
                 3.59 
               
               
                 353 
                 358.40 
                 359.35 
                 2.79 
                 380.25 
                 381.45 
                 5.20 
               
               
                 354 
                 358.25 
                 359.03 
                 1.57 
                 380.60 
                 382.02 
                 7.26 
               
               
                 355 
                 358.28 
                 358.78 
                 0.77 
                 381.25 
                 382.66 
                 9.67 
               
               
                 356 
                 358.30 
                 358.60 
                 0.28 
                 381.95 
                 383.34 
                 11.76 
               
               
                 357 
                 358.40 
                 358.54 
                 0.13 
                 382.20 
                 384.03 
                 13.42 
               
               
                 358 
                 358.37 
                 358.50 
                 0.07 
                 383.25 
                 384.98 
                 17.92 
               
               
                 359 
                 358.36 
                 358.58 
                 0.15 
                 383.95 
                 385.91 
                 20.86 
               
               
                 360 
                 358.35 
                 358.73 
                 0.45 
                 385.10 
                 386.97 
                 24.72 
               
               
                 361 
                 358.45 
                 358.88 
                 0.63 
                 386.30 
                 388.03 
                 27.11 
               
               
                 362 
                 358.55 
                 359.12 
                 1.16 
                 387.70 
                 389.14 
                 29.47 
               
               
                 363 
                 358.58 
                 359.42 
                 2.02 
                 388.90 
                 390.37 
                 33.32 
               
               
                 364 
                 358.60 
                 359.76 
                 3.04 
                 389.50 
                 391.65 
                 35.34 
               
               
                 365 
                 359.20 
                 360.15 
                 4.12 
                 390.00 
                 392.94 
                 37.23 
               
               
                 366 
                 358.95 
                 360.59 
                 5.29 
                 393.87 
                 394.35 
                 39.73 
               
               
                 367 
                 359.60 
                 361.18 
                 7.79 
                 393.90 
                 395.95 
                 46.52 
               
               
                 368 
                 360.72 
                 361.84 
                 10.42 
                 396.05 
                 397.81 
                 59.89 
               
               
                 369 
                 360.50 
                 362.54 
                 12.85 
                 396.50 
                 399.93 
                 81.30 
               
               
                 370 
                 361.82 
                 363.44 
                 17.32 
                 397.90 
                 402.58 
                 124.22 
               
               
                 371 
                 362.87 
                 364.37 
                 20.61 
                 400.47 
                 406.20 
                 219.06 
               
               
                 372 
                 363.55 
                 365.36 
                 24.31 
                 401.30 
                 410.49 
                 346.37 
               
               
                 373 
                 364.15 
                 366.44 
                 26.98 
                 402.70 
                 437.21 
                 10106.87 
               
               
                 374 
                 364.90 
                 367.55 
                 29.20 
                 405.00 
                 463.76 
                 18218.73 
               
               
                 375 
                 367.32 
                 368.75 
                 33.32 
                 409.15 
                 489.94 
                 24701.75 
               
               
                 376 
                 368.30 
                 369.98 
                 34.30 
                 414.17 
                 515.95 
                 29640.19 
               
               
                 377 
                 369.07 
                 371.29 
                 37.13 
                 419.47 
                 541.85 
                 33150.41 
               
               
                 378 
                 372.05 
                 372.77 
                 43.20 
                 428.70 
                 567.49 
                 35241.41 
               
               
                 379 
                 372.52 
                 375.11 
                 79.45 
                 443.75 
                 592.92 
                 35885.63 
               
               
                 380 
                 374.02 
                 377.47 
                 107.23 
                 454.37 
                 617.93 
                 35075.40 
               
               
                 381 
                 375.15 
                 379.82 
                 126.78 
                 794.72 
                 642.61 
                 32949.33 
               
               
                 382 
                 376.25 
                 382.72 
                 175.27 
                 792.20 
                 666.88 
                 29644.13 
               
               
                 383 
                 378.77 
                 386.56 
                 277.82 
                 788.77 
                 690.68 
                 25252.83 
               
               
                 384 
                 378.97 
                 401.48 
                 3057.55 
                 786.62 
                 713.93 
                 19848.29 
               
               
                 385 
                 381.47 
                 427.33 
                 11451.81 
                 786.40 
                 736.29 
                 13681.64 
               
               
                 386 
                 385.02 
                 453.09 
                 18383.57 
                 785.00 
                 757.44 
                 7133.20 
               
               
                 387 
                 398.75 
                 478.60 
                 23818.51 
                 782.75 
                 777.68 
                 133.51 
               
               
                 388 
                 399.45 
                 503.84 
                 27780.10 
                 777.92 
                 775.07 
                 140.30 
               
               
                 389 
                 400.15 
                 528.82 
                 30310.50 
                 775.12 
                 772.38 
                 150.25 
               
               
                 390 
                 410.82 
                 553.41 
                 31473.51 
                 773.25 
                 769.59 
                 168.92 
               
               
                 391 
                 425.90 
                 577.87 
                 31306.11 
                 771.20 
                 766.70 
                 188.80 
               
               
                 392 
                 592.86 
                 601.97 
                 29875.20 
                 768.12 
                 763.66 
                 199.18 
               
               
                 392 
                 759.82 
                 625.64 
                 27275.29 
                 764.15 
                 760.38 
                 210.37 
               
               
                 394 
                 759.00 
                 648.23 
                 23946.50 
                 761.05 
                 756.90 
                 225.40 
               
               
                 395 
                 756.57 
                 670.50 
                 19513.99 
                 757.92 
                 753.46 
                 247.62 
               
               
                 396 
                 753.77 
                 692.67 
                 14043.16 
                 755.60 
                 749.84 
                 276.18 
               
               
                 397 
                 750.95 
                 713.81 
                 7979.00 
                 751.80 
                 745.90 
                 311.86 
               
               
                 398 
                 747.60 
                 733.78 
                 1640.50 
                 746.95 
                 732.47 
                 2287.94 
               
               
                 399 
                 745.92 
                 742.35 
                 154.53 
                 743.25 
                 709.07 
                 8710.35 
               
               
                 400 
                 742.95 
                 739.51 
                 169.16 
                 740.77 
                 685.91 
                 14022.79 
               
               
                 401 
                 740.07 
                 736.54 
                 177.35 
                 735.85 
                 662.95 
                 18234.57 
               
               
                 402 
                 737.57 
                 733.37 
                 191.65 
                 730.50 
                 640.20 
                 21370.56 
               
               
                 403 
                 733.52 
                 730.18 
                 205.04 
                 726.40 
                 617.60 
                 23442.45 
               
               
                 404 
                 732.70 
                 726.88 
                 221.51 
                 720.82 
                 595.26 
                 24505.47 
               
               
                 405 
                 727.95 
                 723.40 
                 248.66 
                 714.05 
                 573.20 
                 24632.74 
               
               
                 406 
                 725.47 
                 719.70 
                 275.75 
                 569.80 
                 551.39 
                 23820.49 
               
               
                 407 
                 721.32 
                 715.83 
                 307.33 
                 417.20 
                 529.72 
                 22069.86 
               
               
                 408 
                 717.22 
                 711.77 
                 343.55 
                 416.75 
                 508.38 
                 19475.53 
               
               
                 409 
                 714.47 
                 707.36 
                 391.72 
                 416.62 
                 487.39 
                 16092.93 
               
               
                 410 
                 709.12 
                 702.69 
                 456.88 
                 416.60 
                 466.68 
                 11919.92 
               
               
                 411 
                 705.87 
                 679.85 
                 6814.29 
                 416.63 
                 446.35 
                 7048.48 
               
               
                 412 
                 701.45 
                 657.30 
                 12118.59 
                 416.65 
                 426.48 
                 1572.26 
               
               
                 413 
                 695.40 
                 634.87 
                 16385.59 
                 416.12 
                 416.22 
                 0.23 
               
               
                 414 
                 690.35 
                 612.74 
                 19632.73 
                 416.10 
                 416.13 
                 0.16 
               
               
                 415 
                 684.95 
                 590.89 
                 21901.99 
                 415.75 
                 416.06 
                 0.14 
               
               
                 416 
                 679.20 
                 569.23 
                 23202.66 
                 415.70 
                 416.00 
                 0.13 
               
               
                 417 
                 671.37 
                 547.88 
                 23640.10 
                 415.71 
                 415.95 
                 0.10 
               
               
                 418 
                 663.50 
                 526.74 
                 23185.94 
                 415.72 
                 415.91 
                 0.07 
               
               
                 419 
                 390.07 
                 505.90 
                 21900.34 
                 415.80 
                 415.88 
                 0.03 
               
               
                 420 
                 389.67 
                 485.46 
                 19867.04 
                 416.02 
                 415.86 
                 0.02 
               
               
                 421 
                 389.10 
                 465.36 
                 17102.19 
                 416.00 
                 415.87 
                 0.03 
               
               
                 422 
                 389.35 
                 445.60 
                 13660.88 
                 415.77 
                 415.90 
                 0.03 
               
               
                 423 
                 389.48 
                 426.18 
                 9596.35 
                 415.74 
                 415.92 
                 0.03 
               
               
                 424 
                 389.60 
                 407.31 
                 5023.09 
                 415.70 
                 415.95 
                 0.03 
               
               
                 425 
                 388.87 
                 388.97 
                 0.33 
                 415.87 
                 415.95 
                 0.03 
               
               
                 426 
                 388.81 
                 388.87 
                 0.25 
                 416.05 
                 415.96 
                 0.03 
               
               
                 427 
                 388.75 
                 388.78 
                 0.21 
                 416.12 
                 415.96 
                 0.03 
               
               
                 428 
                 388.80 
                 388.73 
                 0.21 
                 415.92 
                 415.92 
                 0.04 
               
               
                 429 
                 388.85 
                 388.67 
                 0.19 
                 416.25 
                 415.88 
                 0.08 
               
               
                 430 
                 388.55 
                 388.60 
                 0.14 
                 416.12 
                 415.83 
                 0.14 
               
               
                 431 
                 387.97 
                 388.55 
                 0.07 
                 416.05 
                 415.80 
                 0.16 
               
               
                 432 
                 388.35 
                 388.57 
                 0.10 
                 416.05 
                 415.76 
                 0.18 
               
               
                 433 
                 388.30 
                 388.61 
                 0.14 
                 415.82 
                 415.70 
                 0.19 
               
               
                 434 
                 388.57 
                 388.65 
                 0.18 
                 415.85 
                 415.62 
                 0.21 
               
               
                 435 
                 388.37 
                 388.69 
                 0.21 
                 416.05 
                 415.56 
                 0.23 
               
               
                 436 
                 388.37 
                 388.72 
                 0.24 
                 415.47 
                 415.48 
                 0.21 
               
               
                 437 
                 388.42 
                 388.78 
                 0.27 
                 415.15 
                 415.40 
                 0.19 
               
               
                 438 
                 388.40 
                 388.87 
                 0.23 
                 414.97 
                 415.33 
                 0.17 
               
               
                 439 
                 388.80 
                   
                   
                 415.25 
               
               
                 440 
                 389.30 
                   
                   
                 415.24 
               
               
                 441 
                 389.36 
                   
                   
                 415.22 
               
               
                 442 
                 389.37 
                   
                   
                 414.95 
               
               
                 443 
                 389.37 
                   
                   
                 415.00 
               
               
                 444 
                 389.37 
                   
                   
                 415.00 
               
               
                 445 
                 389.37 
                   
                   
                 415.00 
               
               
                 446 
                 389.37 
                   
                   
                 415.00 
               
               
                   
               
             
          
         
       
     
     One test that can be undertaken after carrying out a test to determine the value for the tire wearing angle for the wheels of the vehicle  10  is a test to determine if any play is present in the suspension system of the vehicle  10 . The ability to assess other wheel conditions besides alignment is advantageous. Play in wheel suspension can cause a wheel to be angled in or out depending on whether the vehicle is moving forward or backward. To determine if there is play in the wheel suspension, the vehicle  10  is driven forward and the two distance measurements made. Then the vehicle is driven backward and the two distance measurements are made. Alternatively, the vehicle may be driven backwards first and then forwards. When moving backward, the first and second locations on the wheel are the same as the second and first locations when the vehicle is moving forward. If there is no play in the suspension, the sign of the offset between forward and backward motion of the vehicle should change (i.e. from positive to negative or from negative to positive). For example, in one of the examples above, a value of 379.65 mm was found at the leading part of the wheel, and a value of 379.35 mm was found at the trailing part of the wheel when the vehicle was driven forward, for an offset of 0.30 mm. When driven backwards, if the wheels remain oriented exactly the same way a leading part value of 379.35 mm and a trailing part value of 379.65 mm will be obtained, providing an offset of −0.30 mm. If, however, there was play in the suspension, and the wheel shifted as a result of friction when being driven backwards, the values may be 379.35 (leading) and 379.65 (trailing) due to the shift in the orientation of the wheel, resulting in an offset of 0.30 mm again. Thus, if a change in the sign of the offset direction is not seen (i.e. if the sign of the offset remains the same), then there may be a suspension problem in one or both wheels being measured. Since, as discussed previously, wheel tracking problems may be caused by suspension play and the offset is also dependent on wheel tracking, such suspension information can be collected even when the wheels themselves are aligned properly. However, a more thorough inspection would be needed to determine whether the issue is a suspension issue or some other issue (e.g. relating, for example, to tire inflation). 
     With reference to  FIG. 8 , in some embodiments two displacement sensors may be provided on each apparatus A,B (apparatus A is shown in  FIG. 8 ), wherein the two displacement sensors  3   a  and  3   b  are vertically aligned but spaced apart along the same vertical axis (shown at Av). For example, one at, for example, about one-third of the height of the wheel and another at, for example, about two-thirds of the height of the wheel, which permits the computer to measure wheel camber. More generally, providing two displacement sensors that are vertically aligned but spaced apart along the same vertical axis, and in particular two sensors that are positioned at symmetrical vertical distances above and below the center of the wheel  21 , permits a determination of the camber of the wheel  21  using the offset between the two different distance measurements. 
       FIGS. 5A and 5B  depict two suspension testing plates  38   a , 38   b  to assist in testing for play in the suspension components holding the vehicle wheels. The following description of the testing plates is with reference to  FIG. 5A , but the one depicted in  FIG. 5B  has corresponding features discussed in relation to  FIG. 5A . The suspension testing plates may include working surface  39  that have undulations  41  thereon. The undulations  41  include at least a first undulation  41   a  that slants downward laterally towards one side of the plate  38   a  and a second undulation  41   b  that slants downward laterally towards the other side of the plate  38   a . By providing successive first and second undulations that slant towards opposite sides, any play in the wheel of the vehicle would cause the vehicle wheel to turn in when traveling over one of the undulations  41 , and to turn out when travelling over the other of the undulations  41 . By measuring the alignment of the wheel as it travels over both undulations  41   a  and  41   b , it can be determined whether the alignment of the wheel changes from one undulation to the other, which would be indicative of play in the suspension elements holding the wheel. 
     As a vehicle  10  travels the weight of the vehicle  10  bears upon the suspension elements and through them, the wheels. Over time, even if there is play in the suspension elements, the weight of the vehicle may cause the joints where the play exists to seize to some degree. As a result, the play that exists in the suspension system is hidden in some situations even though it exists. To eliminate any effect from seizure of any joints, the plate  38   a  may further include bumps  40 , which are provided so as to induce small, sharp movements in the wheel as the wheel travels over them. Such bumps  40  may be spaced relatively far apart such that each bump is individually configured to loosen any seized joints. Alternatively, the bumps may be spaced relatively close together so as to induce a vibration in the wheel as the wheel passes over them in an effort to loosen any seized suspension joints. 
     In the embodiment shown, the bumps  40  may be formed along the mating edges of successive generally triangular surfaces  42  that extend out of plane from one another by a selected angle. 
     If there were no suspension play at the vehicle wheel, the wheel would remain upright as it passes over the undulations  41  and so there would be no change in the distances measured to the points on the wheel. In other words, its degree of alignment would remain constant as it passed over the undulations  41 . If however, there is play in the suspension, then the orientation of the wheel will change as the wheel passes over the undulations  41  and is subject to the changing forces from successive undulations that urge the wheel in different directions. As a result, measurements of the wheel&#39;s alignment would change from one undulation to the next. 
     The novel features of the present invention will become apparent to those of skill in the art upon examination of the detailed description of the invention. It should be understood, however, that the scope of the claims should not be limited by the preferred embodiments set forth in the examples, but should be given the broadest interpretation consistent with the specification as a whole.