Flightscope / Trackman clubface / spin

[ZMax]

After questions were raised concerning the Zelocity Tracker's lack of real measurements, I decided to do some digging and try to understand what Flightscope and Trackman measures as far as club and spin.

Both of these devices use 3D phased array doppler radar tracking. I read somewhere that the owner of Trackman used to work for EDH, the same company that the Flighscope people works for. My guess is that both use similar designs and techniques?

In terms of the club, both will measure club path and angle of attack, but it appears that they don't measure the club face directly. My guess is that the club face is derived from at least the club path and ball azimuth. Trackman points it out on their list of features that the club face is calculated. I would assume that Flightscope does it the same way.

From Trackman:

By measuring these parameters and applying the law of physics this provides the orientation of the clubface at the point of impact on the clubface. This orientation of the clubface is 3 dimensional and are reflected in the data parameters “face angle” and “dynamic loft”. Even though that “face angle” and “dynamic loft” are derived numbers from direct measurements and a collision model, numerous robotic test has proven that this is indeed a very accurate calculation of “face angle” and “dynamic loft”.

As for spin, both will measure total spin but not spin axis or side spin directly. Spin axis(side spin) is derived from tracking the ball for as long as the radar device can or until the ball drops below the radar. The type of trajectory or shot shape of the ball flight is translated into degrees of spin axis and then translated to the amount of side spin.
From Trackman:

Patented Technology
TrackMan is a technological leader in its field. We defend our technology and patent our key inventions. TrackMan’s comprehensive patent portfolio includes granted patents and pending patent applications relating to:
• Comparing a target direction indicated in an image, e.g. by using a camera, with the ball direction determined by radar. (US2009/0295624, EP1763683, CN1981207B, KR7002764/2007 and JP2008/504534)
• Measuring spin rate of sport balls by radar using multiple harmonic
spectrum traces. (US2009/0075744, EP1698380, DE602006009719.0, GB/EP/1698380, CN101384308A, JP2008/538085A and KR10/0947898)
• Measuring spin axis orientation of sport balls from trajectory measurements by radar. (US2009/0075744, EP1853362, DE602006015036.9, GB/EP/1853362, CN101384308A, JP2008/538085A and KR10/0947898)

From Flightscope:

Outdoors the FlightScope Kudu tracks the ball for the entire trajectory in 3D (elevation angles, horizontal angles and velocity ) until it lands, or as far as the ball is in the radar field of view. Accuracy for carry distance is typically within 2-4 yards at 250 yards and within 1-2 yards at 150 yards. Spin is measured directly and shot shape, which relates to the side spin, is measured directly and not calculated like with launch monitors.

For outdoor use, measuring spin axis directly or not doesn't really matter since you're outside and can see what the ball is doing anyway. But for indoor use, unless you're measuring the club directly, you need to measure spin axis. Otherwise, you have no way of estimating side spin so golf simulation will not be realistic. Since 15-20ft indoors is not enough for Trackman or Flightscope to measure the shot shape, I have come to the conclusion that neither one will be good for indoor golf simulation.

A note on indoor total spin detection:
From Flightscope:

FlightScope has a very advanced indoor spin measurement capability and can measure spin directly. For indoor applications, a dimple-size metallic sticker is used, allowing near-perfect spin measurements, unsurpassed by any other golf launch monitor technology available today, with a 14 feet(or more) ball flight distance..

So, Flightscope requires a dimple size metallic sticker to measure total spin indoors. Trackman only states that they can measure total spin in just 10 ft of ball flight, but no mention of needing a metallic sticker.

For those that might be looking to use one of these doppler radar devices for a home golf simulator, I hope you find my post helpful. I'm no expert so please correct me if I've posted information that is not true .

EDIT:So, thanks to new info from patrice, we now have confirmation that Flightscope estimates spin axis or side spin from club path and ball path, when using it indoors. This is similar to how Foregolf/sportscoach are doing it. Which makes indoors simulation not so bad. Not great but definitely playable.

EDIT: Official explanation from Trackman(Andreas) on 1/15/2013

Official TrackMan response

Ok, to clear things up:

1. Ballflight

When you hit a shot with TrackMan/E6, the entire ballflight are calculated by TrackMan ballflight algorithms.
This data is then forwarded to E6, which then draws the trajectory untouched until it collides with their world. Then they take over and renders the last part of the shot. This is when the ball hits something that isn't air (trees, fairway, green, bunker, water etc).

2. Shortgame

TrackMan measures shortgame down to the range of ~6ft. Concept is the same as full shots - entire ballflight is sent to E6.

3. Putting

Putting with only a TrackMan is currently not avaliable. What you can do if you would like to have the full indoor golf experience is to extend the system with a TruTrac Mini (http://trugolf.com/golf_simulator/tracking_system.php) for putting purposes only.

4. Spin axis

When indoors TrackMan determines the spinaxis from the short amount of ballflight combined with club data. This way of determining the spin axis is very good except for off center impacts with woods (note that it is only off center impact in the heel-toe dimension which is an issue). The way TrackMan determines the spin axis indoors does not take into account the gear-effect which affects spin axis.
Example (right hand): Say, you hit it on the toe-side, then the spin axis should be more negative (draw), but isn't indoors.

When outdoors, TrackMan determines the spin axis from the measured aerodynamics which is quite accurate.

Regards,
Andreas Willadsen
Engineer
TrackMan A/S

[bubba22]

Awesome review ZMax. Thanks for reviewing these launch monitors.

[ZMax]

Thanks buba22. I know what you mean but review is not the right word for it. I was just doing some deductive reasoning on doppler radar's inability to measure spin axis directly.

[bubba22]

I think the radar is fantastic outdoors where as you say it can track the ball over a distance and give good launch data for club fitting especially. Indoor it is really challenged to get that data especially spin. I am not sure how the metallic stickers give one spin. Anybody know the science behind that? Real time measurement of spin and spin axis is certainly not easy. I think camera based measurements are a good option. The sensor based option of club face and path measurement and calculation of spin axis is probably quite accurate at the end of the day.

[ZMax]

Agreed bubba22. At the moment, cameras are the best option to measure spin axis. Hopefully, we'll get to see that soon on our sims.

[CPA]

has anyone tested a flightscope x2 for spin data (using the dimple metalic stickers of course) indoors against a top notch camera sim?

[Asics]

Very good post, Zmax. I almost went with the FS, but decided against it over concern of its inability to measure spin axis when hitting indoors. I also didn't have the space required. Had never noticed the statement you highlighted re TM.

[FaultyClubs]

Indoors they do measure club and ball path and from those one can calculate spin axis. The question then becomes how accurate is their spin calculation and how accurate does it need to be? Certainly the lack of side spin measurement is a big red flag...but without more investigation concluding it won't work for a home simulator might not be correct.

For me it's all moot due to the space requirements. I suspect most home environments have the same problem. Ten or fourteen feet from ball to screen is not that easy. I think I'm about eight.

[ZMax]

Didn't say it wouldn't work. I said neither one would be good for indoor golf simulation.

And like I stated above, they measure club speed and path, but not club face directly. Club face is calculated from ball path and who knows what else. Can they use all this to estimate side spin? Sure, but it's not mentioned anywhere on their websites that I could fine. Even if they are using that data, part of that data(club face) is already an estimate or calculation of its own. Then, to calculate side spin or spin axis, you're looking at using a calculation on top of another calculation.

It's a moot point for me too. Not enough space and too rich for my blood. That's not even including the $5-6K extra for the E6 golf.

[FaultyClubs]

Agreed but being calculated may not be a problem even for a "good" indoor simulator so I think your conclusion may be over-running the known facts.

One would have to really understand the algorithms and error sources/sensitivity to know for sure. Or have good empirical evidence showing calculated side spin can't achieve good results.

Personally I'm not convinced the inherent errors in side spin calculation are that significant.

[ZMax]

Oh, I know that calculated side spin can achieve good results. Protee and GGS do it well and even the DD does a decent job. My point was that all of the club data that might get used in calculating sidespin when the Flightscope/Trackman is used indoors are not all measured directly. How well side spin gets calculated depends on the accuracy of all the club data.

Also, I used the world might because I don't know that this is even being done. Do you know for sure? Link? Not sure if you've read about the Zelocity issues but we can not assume anything in this golf simulator business.

[bubba22]

ZMax you make a good point. I am sure the data is ok but who knows. The more "LEVELS" of calculations, the more chance of error.

[jvincent]

I am not sure how the metallic stickers give one spin. Anybody know the science behind that? Real time measurement of spin and spin axis is certainly not easy.

The metallic stickers improve the radar signature of the golf ball so it makes it easier for the detector to get a good reading.

Think of it as making an anti-stealth golf ball.

The metal will reflect the transmitted waves much more clearly than a ball without the metal stickers so the received radar signature will be much cleaner making the calculation required both easier to do and more precise at the same time.

[bubba22]

Thanks jv.

[ZMax]

The metallic stickers improve the radar signature of the golf ball so it makes it easier for the detector to get a good reading.

Think of it as making an anti-stealth golf ball.

Would these metallic stickers also work with the Trackman?

Thanks

[jvincent]

Not sure what it would do on a Trackman system.

The Flightscope S/W may "look" for a specific signature from the metal strips to calculate the spin. Depending on how the Trackman S/W reads the returned signal from the metal strips it will either get a better reading or get confused.

That would be an interesting test.

[bubba22]

Yep/ Any takers to test this out?

[CPA]

perhaps it will fix the zelocity tracker as well?...........

if it doesn't something with a bit more lead may make it work:

hammer-1.jpg

[ZMax]

Sure, but instead of putting the stickers on the balls, you have to put them on the Zelocity pods. The more stickers, the faster the spin numbers get picked out of the lookup tables.





I'm sorry CPA...I just couldn't help myself....

[CPA]

Sad thing is the same question as the trackman does not appy to the Zelocity- it would not do any harm to the readings.

[patator]

Hello

I contacted Flightsocpe to understand better how their great X2 model is working.
I wanted to share the answers with you :

is the microwave radar of the Flightscope prime less accurate/powerfull than the X2 ?
Yes, X2 has a more advanced radar design and a more powerful and sensitive antennae.

Can you confirm than the AOA is now correctly measured/calculated with the X2 ?
Yes

Total Spin and Spin axis values
Total spin is always measured directly and spin axis calculated using 3D movement of the club and tracking curvature of the ball. Wind has little effect.
With no wind and a controlled environment there shouldn't be any tolerance on spin axis, because the horizontal azimuth is so low. (I didn't understand this last point)

For indoor the total spin is also directly measured but the sidespin component is derived from the club impact trajectory and ball launch vector.


Shots measurement with Wedges
The club trajectory is not yet reliably measured for wedges. This is being worked on and will improve in the future.


Minimum Launch angle
The launch angle depends on the tilt of the radar and distance from the tee, at standard tilt of 10 degrees you will be capturing lob wedges pretty easy, but theoretically you can capture much higher shots if you could tilt the radar to 20 degrees as an example, so I could say that yes we can measure 90 deg shots, but that will be impractical.


Tolerances of the X2
LaunchAngle <0.5deg
Sideways LaunchAngle (Azimuth) < 0.2deg (Very important)
Ball speed <0.2mph
Carry <2% (outdoor)
Total Spin <50rpm
Club speed <1mph

Measured values
Angle of Attack,
Club Path,
Vertical Swing Plane,
Horizontal Swing Plane,
Speed Profile,
Accelleration Profile,
Carry,
Lateral,
Club Speed,
Ball Speed,
Vertical Launch Angle,
Horizontal Launch Angle,
Vertical Descent Angle,
Apex Height,
Flight time.

Calculated values
Smash Factor,
Dynamic Loft/Spin Loft,
Face Angle relative to Path and Target,
Spin axis,
Roll/ Totla distance.

Are all the screens available on the PC software, are also available on the IPAD ?
Yes

Can I use the X2 with a bit of rain or it's better to cease a session when it starts raining outdoor ?
Yes with rain cover it can be used, but not advisable to use in the rain without cover.

if my I change my PC, can I re-install the software on a new PC.
Yes
No charge.

After one year, can I still use the software even if I don't want to pay the 995USD fee for free support and upgrade (at least 2 a year)
Yes and if I want to have access to the newlest release, the price is 399USD.

Why is the unit so accurate ?
The reason for the accuracy is that the radar is that it generates 10,5 billion microwaves per second. The radar gets about 100 000 microwaves per second back from the club and ball and so you can see that its got an extremely high sampling rate. It compares to having 3 cameras, back, side and top running at 20 000 frames per second each. Nothing compares to that even high speed devices like Foresight could max out with multiple flashes at 10 000 frames per second. That's why camera launch monitors have a high azimuth tolerance of 1deg +.

Isnt'it the kind of answer and transparency one is expecting when it comes to buying a LM that is worth several thousands dollars ?
One should be able to get the same level of accuracy of answers from all LM manufacturers.
The good reputation of Flightscope is excellent, I now know why.

Patrice

[HRS]

I completely agree it is good to see corporate transparency. Something I don't understand is why the "total spin" is measured but the axis is calculated. It seems to me that the radar should be able to tell in what direction the ball is rotating if it can tell the rate. (I also don't understand how spin can be measured at all with radar unless somehow it locks on to a dimple and tracks the motion over milliseconds. A camera can do that at optical wavelengths of nanometers but are the microwaves of a radar "small" enough to track a dimple?)

[ZMax]

Patrice, thanks for sharing. The headings "Values calculated" and "Values Measured" need to be swapped.

I applaud Flightscope for being straight forward with their information. The biggest lie told in the golf industry is how spin and spin axis is obtained. We already know about Zelocity, the other one is Bogolf.

So, we now have confirmation that Flightscope estimates spin axis or side spin from club path and ball path, when using it indoors. This is similar to how Foregolf/sportscoach are doing it. Which makes indoors simulation not so bad. Not great but definitely playable.

Interesting the the club path is also used in the side spin estimation when outside.

[bubba22]

Yes I agree ZMax. Thanks a lot Patrice for sharing. ZMax is right, the values under "Values calculated" and "Values Measured" need to be swapped. Flightscope is being honest and it will pay off with customers. Spin axis tilt is more difficult to directly measure. Most likely it is estimated by most simulator companies, which is totally fine, but they should behonest in that claim. Are we assuming that Trackman and Flightscope are identical in their measurements?

[ZMax]

Are we assuming that Trackman and Flightscope are identical in their measurements?

I think it's safe to assume that. Perhaps the Trackman uses a more powerful radar than the X2, which would explain the difference in cost? I don't know. But apparently, the 3D doppler radar technology used by both can not directly measure the axis of spin.

For indoor use, I would assume that Trackman is also using club path and ball path to estimate spin axis.

[ZMax]

I completely agree it is good to see corporate transparency. Something I don't understand is why the "total spin" is measured but the axis is calculated. It seems to me that the radar should be able to tell in what direction the ball is rotating if it can tell the rate. (I also don't understand how spin can be measured at all with radar unless somehow it locks on to a dimple and tracks the motion over milliseconds. A camera can do that at optical wavelengths of nanometers but are the microwaves of a radar "small" enough to track a dimple?)

I read somewhere that the radar can see the spine of the golf ball and it uses that to determine the rate of spin.

[bubba22]

Z what is the spine of the golf ball?

[ZMax]

Z what is the spine of the golf ball?

I don't know bubba22. And I can't remember where I got it from. Maybe it was for something else and not golf. But here is some good reading:http://ip.com/patent/EP1698380B1

The present invention makes it possible to have a continuous measurement of the spin frequency and spin axis orientation during the entire flight of the ball.
Spin frequency
Consider a Doppler radar 3 in figure 1. The Doppler radar comprises a transmitter 4 and a receiver 5. The transmitting wave 6 at frequency Ftx is reflected on the ball 1, the reflected wave 7 from the ball 1 has a different frequency Frx. The difference between the reflected frequency and the transmitted frequency, is called the Doppler shift Fdopp. Fdopp is proportional to the relative speed Vrad of the reflecting point A on the ball 1 relative to the radar 3. Fdopp,A=2/λ∗V⁢rad [Image Omitted]
, where λ is the wavelength of the transmitting frequency.
A coordinate system 2 is defined as having origin in the center of the ball and X-axis always pointing directly away from the radar, the Z-axis is in the horizontal plane.
Vrad is the change in range from the Doppler radar 3 relative to time (Vrad = dR/dt). With the coordinate system 2 in figure 1, Vrad equals the X component of the velocity of the ball 1.
The strongest reflection from the ball 1 will always be the point A which is perpendicular to the line-of-sight from the radar. When the ball 1 is spinning, the point A with the strongest reflection will in fact be different physical locations on the ball over time.
The output signal of the Doppler receiver 5 from the reflection of point A on the ball can be written as: xAt=at∗exp−j∗Fdopp,A∗t [Image Omitted]
, where a(t) is the amplitude of the received signal.
Consider now the situation of a spinning ball 1 with an angular velocity of ω of the ball around the Z-axis. The reflection from a fixed point B on the ball 1, with a radius of r, will have a Doppler shift relative to the radar 1 of: Fdopp,B=2/λ∗V⁢rad−r∗ω∗sinω∗t [Image Omitted]
The output signal of the receiver 5 from the reflection of point B on the ball can be written as: xBt=at∗dt∗exp−j∗Fdopp,B∗t [Image Omitted]
, where d(t) is the relative amplitude of the received signal from point B relative to point A on the ball 1.
By substituting [2] and [3] in [4], one gets: xBt=xAt∗dt∗expj∗2/λ∗r∗ω∗sinω∗t∗t [Image Omitted]
It is seen that the output signal from point B consist of the signal from point A modulated by a signal XmodB(t): xmod⁢Bt=dt∗expj∗2/λ∗r∗ω∗sinω∗t∗t [Image Omitted]
The exponential term of the modulating signal, is recognized as a frequency modulation (FM) signal, with a modulation frequency of ω/2π and a frequency deviation of 2/λ*r*ω.
From modulation theory it is well known that the spectrum of a sinusoid frequency modulation gives a spectrum with discrete frequency lines at the modulation frequency ω/2π and harmonics of this, the power of the spectrum lines of the m'th harmonic are equal to Jm(4π*r/λ), where Jm() is the Bessel function of first kind of m'th order.
The amplitude signal d(t) of the modulating signal in [6], will also have a time dependent variation. d(t) will like the exponential term in [6] also be periodic with the period T = 2π/ω. Consequently will the spectrum from d(t) also have discrete spectrum lines equally spaced ω/2π. The relative strength of the individual harmonics of d(t) will depend on the reflection characteristics for the different aspect angles.
In summary, because of reflection from a physical point B on a spinning ball from other positions than when this point is closest to the radar (at point A), the received signal will have equally spaced sidebands symmetrical around the Doppler shift Fdopp,A , caused by the velocity of the ball. The sidebands will have multiple harmonics and will be spaced exactly the spin frequency of the ball ω/2π. Only in the case of a perfect spherical ball, there will be no modulation sidebands.
On a normal sports ball there will be several areas on the ball that is not perfectly spherical. Each of these points will give discrete sidebands spaced the spin frequency. The total spectrum for all the scatters on the ball will then add up to the resulting received signal, that of course also has discrete sidebands spaced the spin frequency.
In the above the spin axis was assumed to be constant during time and parallel with the Z-axis. If the spin axis is rotated α around the Y-axis and then rotated β around the X-axis, it can easily be shown that the x-component of the velocity of point B equals: V⁢x,B=cosα∗r∗ω∗sinω∗t [Image Omitted]
Note that Vx,B is independent of the rotation β around the X-axis. Since Vx,B also is periodic with the period T = 2π/ω, except for the special case of spin axis along the X-axis (α = 90deg), the corresponding Doppler shift from point B with rotated spin axis will also have discrete sidebands spaced exactly the spin frequency of the ball ω/2π. This means as long as the spin axis orientation changes slowly compared to the spin frequency, the spectrum of the received signal will contain discrete frequency sidebands spaced the spin frequency of the ball ω/2π.
In figure 2 the received signal spectrum of a golf ball in flight is shown. In figure 2 It is clearly seen that the spectrum contains a strong frequency line that corresponds to the velocity of the ball, as well as symmetric sidebands around this velocity that are equally spaced with the spin frequency.
First the ball velocity is tracked 8 using standard tracking methods. Then symmetrical frequency peaks around the ball velocity is detected 9. In figure 3 the frequency offset of the symmetrical sidebands are shown relative to the ball velocity. The different harmonics of the spin sidebands are tracked over time using standard tracking methods 10. The different tracks are qualified 11, requiring the different harmonic tracks to be equally spaced in frequency. The different tracks are solved for their corresponding harmonic number 12. After this, the spin frequency can be determined from any of the qualified harmonic tracks 13, provided that the frequency is divided by the respective harmonic number.
The final spin frequency chart over time is shown in figure 5, which contains all of the harmonic tracks.
The step-by-step procedure for measuring the spin frequency is described in figure 7.
Spin axis orientation
The 3 dimensional trajectory of the ball flight is obtained by appropriate instruments. In the preferred embodiment of the present invention, the radar used for measuring the spin frequency is also used to provide a 3 dimensional trajectory of the ball flight, see figure 4.
Assuming that the ball is spherical rotational symmetric to a high degree, their will be three and only three forces acting on the ball. Referring to figure 8, the accelerations will be:

• gravity acceleration, G
• air resistance / drag acceleration, D
• and lift acceleration, L

The total acceleration acting on a flying ball is consequently: A‾=G‾+D‾+L‾ [Image Omitted]
Examples of balls that satisfy the rotational symmetry criteria are: golf balls, tennis balls, base balls, cricket balls, soccer balls etc.
The drag is always 180 deg relative to the airspeed vector Vair. The lift acceleration L is caused by the spinning of the ball and is always in the direction given by ωxVair (x means vector cross product), i.e. 90 deg relative to the spin vector ω and 90 deg relative to the airspeed vector Vair. The spin vector ω describes the orientation of the spin axis, identified with the spin unity vector ωe, and the magnitude of the spin vector ω is the spin frequency ω found through the algorithm described in figure 7.
The airspeed vector is related to the trajectory velocity vector V by: V⁢air‾=V‾−W‾ [Image Omitted]
The procedure for calculating the orientation of the spin vector ω is described in figure 9.
From the measured 3 dimensional trajectory, the trajectory velocity V and acceleration A are calculated by differentiation 14.
The airspeed velocity is calculated 15 using equation [9], using a priori knowledge about the wind speed vector W.
The gravity acceleration G is calculated 16 from a priori knowledge about latitude and attitude.
Since drag and lift acceleration are perpendicular to each other, the magnitude and orientation of the drag acceleration D can be calculated 17 using equation [10]. D‾=A‾−G‾·V⁢air‾/V⁢air‾2∗V⁢air‾ [Image Omitted]
, where • means vector dot product.
Hereafter the magnitude and orientation of the lift acceleration L can be easily found 18 from [11]. L‾=A‾−G‾−D‾ [Image Omitted]
As mentioned earlier, by definition the lift vector L is perpendicular to the spin vector ω meaning that: L‾·ω⁢e‾=0 [Image Omitted]
The spin unity vector ωe is normally assumed to be constant over time for rotational symmetrical objects due to the gyroscopic effect. If the spin unity vector ωe can be assumed to be constant over a time interval [t1;tn], then equation [12] constructs a set of linear equations [13]. L⁢xt⁢1∗ω⁢e⁢x+L⁢yt⁢1∗ω⁢e⁢y+L⁢zt⁢1∗ω⁢e⁢z=0L⁢xt⁢2∗ω⁢e ⁢x+L⁢yt⁢2∗ω⁢e⁢y+L⁢zt⁢2∗ω⁢e⁢z=0|||=|L⁢xt⁢n∗ω⁢e⁢x+L⁢ yt⁢n∗ω⁢e⁢y+L⁢zt⁢n∗ω⁢e⁢z=0 [Image Omitted]
, where L(t) = [Lx(t), Ly(t) , Lz(t)] and ωe = [ωex, ωey, ωez]
The linear equations In [13] can be solved for [ωex, ωey, ωez] by many standard mathematical methods. Hereby the 3 dimensional orientation of the spin axis in the time interval [t1,tn] can be determined. The only assumption is that the spin axis is quasi constant compared to the variation of the direction of the lift vector L.
By combining the spin frequency ω found from the algorithm described in figure 7 with the spin unity vector ωe found from equation [13], the spin vector ω can be found 20 by using equation [14]. ω‾=ω∗ω⁢e‾ [Image Omitted]
Partwise known orientation of spin axis
In many cases it is known a priori that the spin axis lies in a known plane at a certain point in time. Let this plane be characterized by a normal unity vector n. This means: n‾·ω‾=0 [Image Omitted]
An example of such a case is the spin axis orientation right after launch of ball. When a ball is put into movement by means of a collision, like a golf ball struck by a golf club or a soccer ball hit by a foot, the spin vector ω will right after launch to a very high degree be perpendicular to the initial ball velocity vector V. The normal unity vector n in [15] will in this case be given by equation [16]. n‾=V‾/V‾ [Image Omitted]
The procedure for calculating the orientation of the spin vector ω in the point in time t0 where the spin vector lays in a known plane characterized by the normal unity vector n is described in figure 10.
First following the exact same steps 14-18 as described in Figure 9 to obtain the lift acceleration at the time t0.
Now determine 21 a rotation matrix R that converts the coordinates for the normal unity vector n in the base coordinate system to the x-axis unity vector [1,0,0], see equation [17]. The rotation matrix R can be found by standard algebraic methods from n. 100=R∗n‾ [Image Omitted]
The coordinates for the lift acceleration L from equation [11] is now rotated 22 through R represented by the Lm vector, see equation [18]. L⁢m‾=L⁢x⁢m,L⁢y⁢m,L⁢z⁢m=R∗L‾ [Image Omitted]
Similar coordinate transformation for the spin unity vector ωe, see equation [19]. ω⁢e⁢m‾=ω⁢e⁢x⁢m,ω⁢e⁢y⁢m,ω⁢e⁢z⁢m=R∗ω⁢e‾ [Image Omitted]
Since it known from equation [15] that ωexm equals 0, then equation [13] simplifies to equation [20]. L⁢y⁢m∗ω⁢e⁢y⁢m+L⁢z⁢m∗ω⁢e⁢z⁢m=0 [Image Omitted]
By using that the length of ωem equals 1, the spin unity vector ωe can be found 23 from either equation [21] or [22]. ω⁢e‾=R−1∗0,−L⁢z⁢m/L⁢y⁢m,1/0,−L⁢z⁢m/L⁢y⁢m,1, L⁢y⁢m≠0 [Image Omitted] ω⁢e‾=R−1∗0,1,−L⁢y⁢m/L⁢z⁢m/0,1,−L⁢y⁢m/L⁢z⁢m, L⁢z⁢m≠0 [Image Omitted]
By combining the spin frequency ω found from the algorithm described in figure 7 with the spin unity vector ωe found from equation [21]-[22], the spin vector ω can be found 20 by using equation [14].

[bubba22]

Complex!!! Back to physics class!!!!!

[patator]

Hi Guys,

I have corrected the inversion between measured and calculated value.

Initially I raised the same question about how a radar can measure the total spin.
Apparently (and making it simple...as it was explained to me), the radar is able to measure the difference of turbulences between the top and the bottom of the ball when it flies. This difference will give the total spin....
Then, the spin axis is "calculated using 3D movement of the club and tracking curvature of the ball. Wind has little effect. With no wind and a controlled environment there shouldn't be any tolerance on spin axis"

There is one point I didn't understand is the following :
With no wind and a controlled environment there shouldn't be any tolerance on spin axis, because the horizontal azimuth is so low.

Maybe, some of your advanced user could explain. We lately had the honour to welcome a valuable and very helpful new member on the forum ("OptimalFlght", owner of Optimalflight). Maybe, will he be able to give us some explanations (I dropped him a mail)...

pat