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1
Modern Technologies forTracking the Baseball
Alan NathanUniversity of Illinois and Complete Game Consulting
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Here’s what I’ll talk about:
• Brief review of baseball aerodynamics
• The new technologies– Camera-based systems:
• PITCHf/x and HITf/x:
– Doppler radar-based systems:• TrackMan
• Using these technologies for analysis– Lots of examples
3
Review of Baseball Aerodynamics Forces on a Spinning Baseball in Flight
D2
D C1
ˆF = - ρAv v2
2LM
1ˆ ˆF = ρAv (ωC v)
2
v
ω
mgFD
FM
• Drag slows ball down
• Magnus + mg deflects ball from straight line
See Michael Richmond’s talk
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Example: Bonds’ record home run
5
Familiar (and not so familiar) Effects:
• Drag– Fly balls don’t travel as far (factor of ~2!)– Pitched balls lose ~10%
• Magnus– Movement on pitches (many examples later)– Batted balls
• Backspin longer fly balls; tricky popups• Topspin nosedive on line drives; tricky
grounders• Sidespin balls curve toward foul pole
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PITCHf/x and HITf/xTwo video cameras @60 fps• “high home” and “high first”• tracks every pitch in every MLB ballpark
– data publicly available
• tracks initial trajectory of batted ball– data not publicly available
Image, courtesy of Sportvision
Marv White, Physics,UIUC, 1969
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PITCHf/x and HITf/x• Used for TV broadcasts, MLB Gameday, analysis,…• See http://www.sportvision.com/baseball.html
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Camera Registration• T(x,y,z) screen coordinates (u,v)• 7 parameters needed for T
– Camera location (xC,yC,zC)– Camera orientation (pan, tilt, roll)– Magnification (focal length of zoom lens)
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Details of Tracking Process
• Each camera image determines LOP
• If cameras were synchronized– LOP intersection (x,y,z)
• Cameras not synchronized– Need a clever idea
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Sportvision’s Clever Idea
• Physics trajectory is smooth• Parametrize smooth trajectory mathematically
– e.g., constant acceleration (9 parameters)
• Adjust parameters to fit pixel data– We then have full trajectory
1
z
y
x
T
1
v
u
k
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Possible Parametrizations
• Constant acceleration– x(t) = x0 +vx0t + ½axt2 (etc. for y,z)– Solve simultaneous linear equations for 9P– This is scheme used in PITCHf/x
• Constant “jerk”– x(t) = x0 +vx0t + ½ax0t2 +1/6jxt3
– Solve simultaneous linear equations for 12P
• “exact”– Non-linear least-squares fit to get 9P*
x0,y0,z0,vx0,vy0,vz0,Cd,Cl,
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9P vs. Exact Trajectory
0.5
1.0
1.5
2.0
2.5
3.0
-0.1
-0.05
0
0.05
0.1
0.15
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
t
-135
-130
-125
-120
-115
-0.4
-0.2
0
0.2
0.4
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
vy (ft/s) v
t (sec)
x(t) vy(t)
Many studies like this show that 9P works extremely well
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z vertically up
All useful parameters derived from 9P• Release point NOT measured
• x0,z0 are locations at y0=50 ft• easily extrapolated to 55 ft
• Derived parameters• v0, vf = speed at y=50,HP• px, pz: = location at y=HP• pfxx, pfxz = movement y=40-HP• spin axis = related to direction of movement• Cd, Cl related to vf/v0 , pfx
• Spin rpm is NOT measured• but approximate value inferred from pfx values
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PITCHf/x Precison:A Monte Carlo Simulation
• Start with exact trajectories
• Use cameras to get pixels
• Add random “noise” (1 pixel rms)
• Get 9P and derived quantities
• Compare with the exact quantities
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• Central values close to exact 9P works well
• 1 pixel rms rms on following quantities:v0 : 0.23 mph ; x0, z0 : 0.4” ; px, pz: 0.7” ; pfx_x, pfx_z: 1.6”
v0
x0px
pfx_xexact-inferred
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Some Comments on Registration • In-game monitors
– “blue-field” vs. actual field– LOP error
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Registration Studies in Progress
• Could accuracy be improved with additional “pole” calibrations?
• Can the data themselves be used to recalibrate the cameras?– An example follows
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Drag Coefficient: Anaheim, 2009
Camera registrations changed between days 187,188
188187
187188*
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Some Remarks on Hitf/x
• Pixel data fit to constant velocity (6P)– Not enough of trajectory to do any better
• Impact location inferred from intersection of pitched and batted ball trajectories
• BBS and VLA are systematically low due to drag and gravity– Not a big effect– One could correct for it fairly easily
• Balls hitting ground in field of view are somewhat problematic
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Phased Array Doppler Radar: TrackMan
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TRANSMITTER
RECEIVER
FTX
FTX
FRX
V
VR
fd
fd = FTX-FRX = 2*VR*FTX/c
Measurement principle IDoppler Frequency
fD = Doppler Shift = FTX - FRX = 2FTX(VR/c)
Example: FTX = 10.5 GHz; c=0.67 Gmph; VR=90 mphfD = 2.82 kHz
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Doppler shift Radial velocity
Time
Pitched ball
Batted ball
Frequency/Velocity vs. Time
Bat
Bounce
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Measurement principle IIPhase Shift
TRANSMITTER
RECEIVER 1
FTX
FTX
FRX1fd1
(fd1 ) - (fd2) = 2**sin()*D*FTX/cRECEIVER 2FRX2
fd2
FTX
D
Phase shift = 2DFTXsin()/c
Measurement principle IIPhase Shift
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1
2
3
1-2: Vertical angle
1-3: Horizontal angle
Measurement principle IIPhase Shift
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Spin Measurement principle
w*r
w*r
VradRadar
Strong reflection
Weaker reflection
Weaker reflection
rw
Vtot = Vrad - w*r
Vtot = Vrad + w*r
Vtot = Vrad
Conclusion:A Doppler radar does not only see one velocity, but avelocity spectrum.
The velocity spectrum turns out to have discretefrequency components spaced with the spin frequency w.
Doppler frequency modulated by rotation frequency sidebands
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• Doppler radar measures radial velocity– VR R(t) = distance of ball from radar
– …provided initial R is known
• 3-detector array to measure phase– two angles (t), (t) location on sphere
• R(t), (t), (t) gives full 3D trajectory• Spin modulates to give sidebands
– spin frequency
Summary of Technique
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Additional Details
• Need location and orientation of TM device (just like PFX)
• Need R(0)
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TrackMan Capabilities I
• Full pitched ball trajectory– Everything PITCHf/x gives plus….
• Actual release point perceived velocity• Total spin (including “gyro” component)• Many more points on the trajectory
But given smooth trajectory, additional points are not necessarily useful
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Comment about Spin
• Tracking (either TM or PFX) only determines component of spin in the x-z plane– No deflection due to y (gyro) component
• Many pitches have a gyro component– Especially slider
• Combining TrackMan total spin with the indirect determination of x-z component gives 3D spin axis– …a potentially useful analysis tool
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TrackMan Capabilities II
• Full batted ball trajectory, including…
• Batted ball speed, launch & spray angles– Equivalent to HITf/x– Landing point coordinates at ground level and
hang time• Equivalent to Hittracker
– Initial spin– and more, if you want it
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TrackMan Data Quality I
• Comparisons with Pitchf/x– Pitch-by-pitch comparisons from May 2010 in
StL and Bos look excellent– Comparable in precision and accuracy to PFX– Our Red Sox friends could tell us more, if we
ask them really nicely!
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TrackMan Data Quality II
• My Safeco Field experiment, October 2008– Project fly balls with pitching machine– Track with TrackMan– Measure initial velocity and spin with high-
speed video camera– Measure landing point with a very long tape
measure (200-300 ft)
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Landing: TrackMan vs. Actual
200
220
240
260
280
300
320
340
200 220 240 260 280 300 320 340
actual distance (ft)
Tra
ckM
an
dis
tan
ce (
ft)
Landing Point Comparison
TrackMan high by about 2.5 ft.: Could be R0 issue
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spin: TrackMan vs. video
0
500
1000
1500
2000
2500
3000
3500
4000
0 1000 2000 3000 4000
video spin (rpm)
Tra
ckM
an s
pin
(rpm
)Spin Comparison
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Summary of Safeco Results
• Initial velocity vector excellent
• Initial spin mostly excellent– But sometimes off by an integer factor (?)
• Landing point correlates well– But systematic difference ~2.5 ft
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One final point about batted balls
• We need a convenient way to tabulate batted ball trajectories
• Current TM scheme:– Initial velocity vector– Landing point and hang time, both
extrapolated to field level
• Constant jerk (12P) might work
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Some Examples of Analysis
• Pitched ball analysis– Dan Brooks will do much more
• Batted ball analysis
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"Every time that I come here to San Diego, it's always good. Everything moves different. The breaking ball is really nasty, and my fastball moves a lot. So I love it here."
Ex 2 Ubaldo Jimenez Pitching at High Altitude
Denver
vf/v0
DenverDenverDenver
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Ex 2 Ubaldo Jimenez Pitching at High Altitude
Denver
vf/v0
DenverDenverDenver
San Diego
San Diego
Denver
Denver
vf/v0
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Ex 3: Effect of batted ball speed and launch angle on fly balls: TrackMan from StL, 2009
R vs. v0 R vs. 0
USEFUL BENCHMARK400 ft @ 103 mph
~5 ft per mph
peaks @ 25o-35o
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Ex 4: What Constitutes a Well-Hit Ball? Hitf/x from April 2009
w/o home runs
HR
BABIP V0>90 Basis for outcome-independent batting metrics
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Combining HITf/x with Hittracker
• HITf/x (v0,,)
• Hittracker (xf,yf,zf,T)
• Together full trajectory– HFX+HTT determine unique Cd, b, s
– Full trajectory numerically computed
• T b
• horizontal distance and T Cd
• sideways deflection s
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How well does this work?Test experimentally (Safeco expt)
010203040506070
0 50 100 150 200 250 300 350distance (ft)
My Safeco Experiment w/TrackMan
It works amazingly well!
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Some examples of HFX+HTT Analysis
• Windy Yankee Stadium?
• Quantifying the Coors Field effect
• Home runs and batted ball speed
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HITf/x + hittracker Analysis: The “carry” of a fly ball
• Motivation: does the ball carry especially well in the new Yankee Stadium? • “carry” ≡ (actual distance)/(vacuum distance)
for same initial conditions
(379,20,5.2)
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HITf/x + Hittracker Analysis:4354 HR from 2009
Denver
Cleveland Yankee Stadium
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Average Relative Air Density
Denver
Phoenix
SF
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The Coors Effect
~26 ft
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Phoenix vs. SF
Phoenix +5.5 ft
SF -5.5 ft
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Home Runs and BBS
• 4% reduction in BBS– 20 ft reduction in fly ball distance (~5%)– 50% reduction in home runs– NOTE: typical of NCAA reduction with new bats
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Now that you (think you) understand everything…
Slo-mo video here
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My Final Slide
• Lots of new information from tracking data
• We have only just begun to harvest it
• These new data will keep us all very busy!
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