Available online at www.sciencedirect.com
Procedia Engineering 34 (2012) 730 – 735
9th Conference of the International Sports Engineering Association (ISEA)
Defense for basketball field shots
Hiroki Okuboa*, Mont Hubbardb a
a
Chiba Institute of Technology, 2-17-1 Tsudanuma, Narashino, Chiba 2750016, Japan
b
University of California, Davis, CA, 95616, USA
Accepted 02 March 2012
Abstract
A three dimensional dynamic model is used to calculate basketball motions for field shots with release conditions:
release position, release velocity, backspin angular velocity, release angle, and lateral deviation angle. The model
includes basketball stiffness and damping and calculates the slipping and non-slipping, and spinning and nonspinning motions at the ball-contact point. The simulations, together with probabilistic selection of release conditions,
analyze ball trajectories of field shots and possible rebounding positions for players. The results instruct the best
rebounding position for placement of rebounders. We also investigate the effectiveness of denying the optimal shot
paths for attempted blocked shots.
© 2012 Published by Elsevier Ltd.
Open access under CC BY-NC-ND license.
Keywords: Basketball; rebound; blocked shot; field shots; dynamic model
1. Introduction
Excellent basketball defenders like Ben Wallace and Dwight Howard have the talents of both a
rebounder and a shot blocker. More often than others, they are able to deny optimal shot paths and predict
rebounding positions, and this plays a large role in regaining possession of the ball. Getting good
rebounding position and blocking shots are considered to be the most important skills of good defensive
players.
There have been some dynamic models for general basketball shots that include stiffness and damping
of basketball [1-5]. Okubo and Hubbard have measured basketball parameters and compared the
calculated ball trajectories of shots to actual experiments [6,7]. Free-throw rebound positions have been
*Hiroki Okubo. Tel.: +81-478-47-0186; fax: +81-478-47-0186.
E-mail address: [email protected].
1877-7058 © 2012 Published by Elsevier Ltd. Open access under CC BY-NC-ND license.
doi:10.1016/j.proeng.2012.04.124
731
Hiroki Okubo and Mont Hubbard / Procedia Engineering 34 (2012) 730 – 735
calculated by a dynamic model [8]. But no previous study has analyzed missed field shots using
simulation models.
Basketball three dimensional rebound trajectories with reasonable release conditions are simulated for
missed bank and direct shots. The ball sometimes bounces chaotically off the rim or backboard. We show
that the results instruct the best position for placement of rebounders. We also investigate the
effectiveness of denying the optimal shot paths for blocked shots.
2. Overall model
In a general shot the basketball may contact the rim, the backboard, the bridge between the rim and
board, the board and bridge, and the rim and board simultaneously. Our overall model for basketball field
shots has six distinct sub-models: gravitational flight with air drag, and ball-contact sub-models for ballrim, ball-board, ball-bridge, ball-bridge-board, and ball-rim-board contacts. We switch between the submodels depending on the reaction forces at the contact points. The flight sub-model is adopted if the
normal force at each contact position vanishes. Each contact sub-model has possible slipping and nonslipping, and spinning and non-spinning motions. The equations of motion have been derived for flight
and ball-contact sub-models, the latter of which include ball stiffness and damping.
3. Numerical simulation
3.1. Simulation parameters
A Newtonian frame with origin at the hoop center has its XY plane horizontal, the direction of Y from
the hoop center toward the board, and Z up. Release location is specified by three parameters: horizontal
and vertical distances, l and h, from the hoop center, and the floor angle, E , between the board surface
and the vertical plane including the release point and hoop center. Shots have four more parameters:
release velocity v, backspin angular velocity Z , release angle D , between the release velocity and the
horizontal plane, and lateral deviation angle G , between the initial ball path plane and the vertical plane
including the hoop center and the release point.
We choose field shots with short (l=2 m, h=0.05 m and Z =1 Hz), middle (l=4 m, h=0.15 m and Z =2
Hz) and long (l=7 m, h=0.30 m and Z =3 Hz) ranges. Each shot can occur at seven different floor angles:
0 (near baseline), 15, 30, 45, 60, 75, and 90 (on court center line) deg. All shots occur from the right side
of the court.
9.6
9.4
50
Direst
Swish
Front rim
Back rim
Margin
55
50
45
α (°)
α (°)
60
Direct
Swish
Front rim
Back rim
Margin
v (m/s)
70
9.2
40
40
35
9
30
8.8
25
30
8.2 8.4 8.6 8.8
9 9.2 9.4 9.6 9.8 10
v (m/s)
(a)
-6
-4
-2
δ (°)
(b)
0
2
4
Direct
Swish
Front rim
Margin
-10 -5
0 5
δ (°)
10
(c)
Fig. 1. First contact combinations in (a)release velocity-release angle space with zero lateral deviation angle; (b) lateral deviation
angle-release velocity space with constant release angle of 45.5 deg; (c) lateral deviation angle-release angle space with constant
release velocity of 9.08 m/s
732
Hiroki Okubo and Mont Hubbard / Procedia Engineering 34 (2012) 730 – 735
3.2. First ball-contact boundaries
We use the model to study capture and rebound of field shots, beginning with first ball-contact
boundaries. Figures 1(a)-(c) show the first contact conditions of the ball for capture and rebound of direct
shots in v - D , G - v, and G - D spaces, respectively. The shot has release point 7 m away from and 0.30 m
below the hoop center, and a floor angle of 45 deg. The simulations use as initial conditions the release
velocity, release angle and lateral deviation angle in increments of 0.02 m/s, 0.5 deg, and 0.15 deg,
respectively. Shown and labeled in Fig. 1 as “Swish” and “Direct” are most capture conditions, those
with no and one rim bounce, respectively. The other first contact conditions are most non-capture initial
conditions. The center of the capture region, the intersection of the black 3-D cross, has coordinates
v , D ,0 and the volume of the capture region is proportional to the non-dimensional product of the
margins (the lengths of the three arms of the 3-D cross) of the release velocity, release angle, and lateral
deviation angle, V G M v M D M / G max v D , where G M , vM, and D M are the margins of the lateral
deviation angle, release velocity, and release angle for capture, G max is the maximum margin of the
lateral deviation angle, v and D are the release velocity and the release angle of capture center. The direct
center shown has the maximum product and is shown by the intersection of margin lines in Figs. 1(a)-(c).
3.3. Rebounding positions
For the three parameters lateral deviation angle, release velocity, and release angle, we use shots
chosen from independent Gaussian probability density functions as
f KG
§ K2 ·
exp¨ G2 ¸ , f Kv
¨ 2V ¸
2S V
©
¹
1
§ K2 ·
exp¨ v2 ¸ , f KD
¨ 2V ¸
2S V
©
¹
1
§ K2 ·
exp¨ D2 ¸
¨ 2V ¸
2S V
©
¹
1
(1)
where KG (G G ) / G M , K v (v v ) / v M , and KD (D D ) / D M . This is equivalent to assuming
that the variances in the three variables are proportional to their corresponding margins. The final scaling
parameter is chosen to produce a desired shot percentage depending on the range. More precise ways of
specifying the three variances in future work could be based on experimental data.
Rebound position is defined at the moment when the height of ball center equals the hoop level on the
way down. It is calculated for direct and bank shots with short, middle, and long ranges. We assume the
successful shot percentages of 0.5, 0.45, and 0.4 for the short, middle and long ranges, respectively.
Figures 2-4 show rebounding positions for field shots. The dark colored regions show locations of high
rebound density and probability for rebounds. Each missed shot has one or two dark colored regions,
which players are able to choose as a rebounding location. The best rebounding position is quite different
between the direct and bank shots, even if the shooter has the same release point for all floor angles. In
short-range shots, the best rebounding position is usually close to the hoop. On the other hand, the darkcolor region for the long-range shots is further from the hoop than that for the short-range.
Hiroki Okubo and Mont Hubbard / Procedia Engineering 34 (2012) 730 – 735
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2. Rebound positions for short-range direct shots with floor angles of (a) 30 deg; (b) 45 deg; (c) 60 deg, and bank shots with
floor angles of (d) 30 deg; (e) 45 deg; (f) 60 deg
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3. Rebound positions for middle-range direct shots with floor angles of (a) 30 deg; (b) 45 deg; (c) 60 deg, and bank shots with
floor angles of (d) 30 deg; (e) 45 deg; (f) 60 deg
733
734
Hiroki Okubo and Mont Hubbard / Procedia Engineering 34 (2012) 730 – 735
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 4. Rebound positions for long-range direct shots with floor angles of (a) 30 deg; (b) 45 deg; (c) 60 deg, and bank shots with
floor angles of (d) 30 deg; (e) 45 deg; (f) 60 deg
3.4. Effectiveness of blocked shots
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 5. Estimated successful shot percentages of (a) short-; (b) middle-; (c) long-range direct shots; (d) short-; (e) middle-; (f) longrange bank shots as a function of release angle
Hiroki Okubo and Mont Hubbard / Procedia Engineering 34 (2012) 730 – 735
We next analyze the effectiveness of attempted blocked shots. A shooter has to change his or her shot
path when defensive players try to block shots. In many cases, the shooter chooses a larger release angle
in order to avoid the defender. We apply the similar Gaussian probability density functions to larger
release-angle shots. The optimal release angle is assumed to have a successful shot percentage of 0.5,
0.45, and 0.4 for the short-, middle-, and long-range shots, respectively. The Gaussian probability density
for the larger release angles has the same variance and margins of the optimal capture center, but has a
different capture center D at the larger release angle. The calculated shot percentages for direct and bank
shots with different floor angles are shown in Fig. 5. Larger release angles lead to a decrease in successful
shot percentages for both direct and bank shots for all floor angles. These results show clearly that trying
to block shots is important for defensive players if it can cause a modification of release angle and
therefore decrease capture probability.
4. Conclusions
We have calculated rebound positions for basketball direct and bank shots using a dynamic model and
random Gaussian initial conditions. The best rebounding positions have been estimated for short, middle,
and long-range shots with the different floor angles. Each missed shot has one or two high-probability
rebounding positions. Angled direct and bank shots have different rebounding positions. The position of
rebounds for short-range shots is close to the hoop. The distance from the hoop for longer shot rebounds
is roughly proportional to shot distance. Attempting blocked shots is effective as a way to decrease
successful shot percentages of opponents.
Acknowledgements
This work was supported by JSPS Grant-in-Aid Scientific Research (C) 21500603.
References
[1] Okubo H, Hubbard M. Dynamics of basketball-rim interaction. Sports Engineering 2004;7:15–29.
[2] Okubo H, Hubbard M. Dynamics of the basketball shot with application to the free throw. J Sports Sciences 2006;24:1304–14.
[3] Silverberg L, Tran C, Adcock K. Numerical analysis of the basketball shot. ASME J Dynamic System, Measurement, and
Control 2003;125:531–40.
[4] Tran C, Silverberg L. Optimal release conditions for the free throw in men’s basketball. J Sports Sciences 2008;26:1147–55.
[5] Silverberg L, Tran C, Adams T. Optimal targets for the bank shot in men’s basketball. J Quantitative Analysis in Sports
2011;7:Issue 1, Article 3.
[6] Okubo H, Hubbard M. Differences between leather and synthetic NBA basketballs. In: Estivalet M, Brisson P., editors. The
Engineering of Sport 7: Proceedings of 7th International Conference on the Engineering of Sport, Paris: Springer; 2008;7:70512.
[7] Okubo H, Hubbard M. Identification of basketball parameters for a simulation model. In: Sabo A, Kafka P, Sabo C, editors. The
Engineering of Sport 8: Proceedings of 8th International Conference on the Engineering of Sport, Amsterdam: Elsevier;
2010;8:3281-6.
[8] Okubo H, Hubbard M. Basketball free-throw rebound motions. In: Subic A, Fuss FK, Alam F, Clifton P, editors. The Impact of
Technology on Sport IV: Proceedings of 5th Asia-Pacific Congress on Sports Technology, Melbourne: Elsevier; 2011;5:194-9.
735