FABRICATING A TENNIS ROBOT BASED ON THE COUNTER ROTATING WHEELS R. Yousefi Moghaddam, M. Alitavoli, A. Chaibakhsh, A. Rezaiefar and A. Bagheri Department of Mechanical Engineering Guilan University Rasht POBox 3756, Iran Email:
[email protected]
Abstract The finalization and optimization of the mechanical systems whose fabricating technologies are obtained, is one of the main aims of the engineering processes. Tennis Robot is a kind of these systems, which its design and manufacturing technology is defined in this article.It is tried to explain the effective design parameters in short and show the application of them in fabricating the machine under a critical situation. Then according to the experimental results that are taken from the former version of the machine some fundamental replacements are done in the most of its accessories in order to provide some of the engineering viewpoints such as reliability, effectiveness and productivity. Based on the experimental results, it is tried to find a correction factor to predict the initial velocity of the thrown ball and the mean throwing angle. Also, some discussions are made according to find an agreement between the experimental and theoretical results.
Keywords Tennis Robot – Counter Rotating Wheels (C. R. W.) – Optimizing – Spin – Mechanism
1 Introduction The application of tennis robots in training amateur and professional players is a very common practice nowadays in the world of sports. The main purpose is to simulate hits by a machine more similar to reality. There are three main mechanisms for throwing the ball: springs, pneumatics and counter rotating wheels (C. R. W.). Although tennis machines are not new accessories, but using counter-rotating wheels in these machines is a new method in simulating the hits and there are a lot to do to obtain the practical parameters. The authors applied the basic mechanism of CRW for throwing ball[1],[2] to obtain a very similar simulation to the real hits, which produces spinal ball [4] that is one of the unique parameters that can be considered in simulation only by this exact type of mechanism. It works by the idea of pressing ball between two counter-rotating wheels which their axes are parallel. In this process spinal velocity of wheels transforms to linear velocity of the ball. Different velocities of upper and lower wheels will result top or backspin. According to the increasingly need of helping accessories for this world - favor sport, the authors of this paper decided to design and manufacture one prototype machine respecting all aspects of hits simulation [4]. Then by testing the machine in the tennis court, they took some results to optimize and improve the efficiency of essential parameters such as angle of throw, range of throw, optimal ball pressure and etc. They tried to apply and extend this information in second generation of machine which is produced and presented in this paper. Four sub-mechanisms of machine are Automatic Ball Feeding, Ball Directing, Ball throwing, and Axial Rotation systems.
2 Designs Four following mechanisms and their sub-mechanisms have been designed. 2.1 Automatic Ball Feeding Different rate of ball feeding is very necessary for such machines, because of different level of players’ skills. The mechanism provides a continuous range of ball feeding between 3 to 15 balls per minute (bpm). There’s a small gadget for entering, the ball into the ball directing throat (Fig. 1). 2.2 Ball Directing An upgrading in second generation of this machine, simplified the difficult process of ball directing. The main idea was inserting ball between two rotating wheels, by an arm. In new version a cycloid profile devised under the upper wheel. When the ball enters the cycloid, it is under pressure: on top; by the upper wheel and, on bottom; by the cycloid. Finally when the ball reaches the lower wheel, it will be under a very tact pressure which produces a firm and powerful shot , also it can provide top or back spin (Fig. 2). 2.3 Ball Throwing Mechanism Final velocity, maximum range and minimum height are the parameters of a problem which solving it, will conclude into a relation between appropriate input force and wheels diameter. Maximum kinetic energy can be considered as additional boundary condition. There made frictional surface on the external layer of wheels for the best rate of energy transferring to the ball. Also an electrical inverter is employed for controlling the rotational velocity of the wheels. The main idea of using this implement is to create spin in the thrown ball (Fig. 3). 2.4 Axial Rotation The more sweeping of the court, shows the efficiency of the machine. Thus, axial rotation is applied which provides covering the court from the base line at the back, up to the net in front of the court. The role of axial rotation is considerable, when the machine shoots the ball on the right half or left half of the court or shoots it directly. Rotation achieved by the mean of a three-rod mechanism, which produces a motion, similar to Scotch-yoke mechanism. Its duty is to convert a 360 degree rotation of input, into a reciprocating movement, and finally, change it into a 30 degree rotational movement in the out put. The boundary condition is, throwing the ball inside the court’s lines.
3 Calculations There are some parameters which need to be considered. Some of them are problem’s input and some of them such as “shaft’s diameter, input power, rotational velocity of wheels, distance between wheels, angle, height and range of shot.... need to be calculated. Here we bring some of calculus for one of shafts. For the upper and lower shafts we have the same conditions of dynamic load and speed, but according to static loads, we understand that, lower shaft is in a more critical state. So we calculate the equations to find the diameter for critical shaft, and then we can use it for upper shaft as well. 3.1 Power Transmission For the best efficiency, belts are used in power transmission system between input power supply and wheels’ shafts. Input power considers as a known parameter of problem, and then the forces are the answers of it. Maximum Load exerts at the minimum spinal velocity. H = 0.5 (hp) = 370 (W) ? = 1100 (rpm) = 115.13 (rad/s)
(1)
V = r.? = 6 (m/s)
(2)
H = ( F1 − F2 ).V
(3)
370 F1 − F2 = = 61.67 (N) 6 F1 / F2 = e µθ
(4)
θ = π + 2 sin −1 (
D−d ) = 3.14 2L
F1 = 60.01 (N) F2 = 6.9 (N)
(5) (6)
3.2 Calculating the Lower Shaft under Maximum Ball Compression For the critical state, a new ball is considered, under the maximum internal pressure, which exerts a Maximum force of 392.8 (N) [5], [6]. F = 392.8 N (7) According to boundary conditions, angle of shot can be found around 15 degree. Main boundary conditions that can help us are the length of court and height of the throw. Wp = 4
(8)
W = 392.4 + 28.4 = 420.8 Fb = 68.14
(9) (10)
∑M ∑M
DY
=0
Dx
=0
⇒
C x C y D x D y
(11) (12)
= 104.86 = 233.79
(13)
= 26.9 = 208.49
In profile A: M = 208.49 × 0.09 = 18.76 1 M 2 = 26.99 × 0.09 = 2.43 2 2 M = M 1 + M 2 = 18.92
(14)
In profile B : M = 6.99 × 0.09 = 2.43 1 M 2 = 208.49 × 0.09 = 18.76 2 2 M = M 1 + M 2 = 18.9
(15)
And finally we have: µ = 0.7 Ν = 392.4 f = 392.4 × 0.7 = 274.68
(16)
Τ f = 274.68 × 0.13 = 35.71 Τtotal = Τ + Τ f = 38.35
Then for the lower shaft we have. 32 n ds = (M 2 +T π S y
1
2
3 ) 0.5
n = 4 , S =160 Mpa
y → d s = 22.62
And by searching in standard tables d= 0.2 m is appropriate diameter for shafts.
(17) (18)
4 Results and Discussion The distance between two wheels and the angle of the throwing ball can be adjusted together. As it is shown in (Fig. 3), to obtain the best efficiency, some tests are needed. Optimizing the angle and the distance of the wheels, affect the ball pressure. In a hard court ground, some results are taken, that are used to optimize the output. Here are some of the results according to three parameters for two sessions: shot angle ( α ), shot velocity (V0) and the range of the thrown ball (R). RT =
V02T sin ( 2α ) g
(19)
y=−
1 g .x 2 + x. tan(α T ) 2 2 V 0T cos 2 (α T )
(20)
V 0T =
x2 2 cos 2 (α T ) ( x. tan(α T ) − y) g
×
(21)
4.1 Experimental Results Some parameters as well as the mean range and the angle of throw are obtainable by precise measurements. These measures are considered as the experimental results of the problem. The linear velocities of the upper and lower wheels could be computed directly from the input velocity. Adjustment of the input velocity will done applying the inverter interface system. The inverter has some control keys to adjust the rotational velocity of the electro motors as an output of the system. The final out put of such a system are V1E (the experimental velocity of the upper wheel) and V2E (the experimental velocity of the lower wheel). The throwing angle α E is computable by the positions of the wheels. By measuring the falling point of the ball the experimental range (RE) will be found. 4.2 Theoretical Results The velocity that is calculated (Equation. 19) using the RE and α E would be the theoretical velocity (V0T) (Table. 1). As the system’s governing equations are very complicated, it is very difficult to find the output velocity of the throwing ball. Thus to simplify the equations, the summation of the upper and the lower velocities are assumed to be (VBE), the experimental velocity of the ball. VBE = V1E + V2 E ( 22) V1E = r.ω1 , V2 E = r.ω2 If it is assumed that RE= RT and α E = α T , so it is possible to find a relation between the Theoretical velocity (V0T) and the ball experimental velocity (VBE) by exerting a correction factor (CF) as below. V0T = CF × VBE (23 ) The mean value that is obtained is CF=0.83 (Fig. 11). Using these simplifications makes it possible to estimate the experimental range of the throwing ball applying VBE. CF 19 ) V1E , V2 E → VBE → V0T ( → RE
Two coming discussions are also noticeable First: The test based on different degrees of throw. All test trials were made on the 350 (N) load on the ball. It is clear that by increasing the angle there’s a pick in the range and velocity but after that it will decrease again. Depend on the location of the machine (on the base line or the service line), there were limitations for ball velocity. The variation of velocity range must be in a way that the ball could pass over the net but not exceed form the court dimensions. The default position of this machine is behind the base line (Fig. 10) on the centre mark. Base on the mathematical calculations, the minimum velocity of the wheels should be 5.65 to 6.30 (m/s) in order to pass the ball over the net ( V1 is the upper wheel (UW) and V2 is the lower wheel (LW) velocities).
[ Boundary conditions : y = 1.05 ( m) ,
x = R E = 11.885 ( m) ]
if α = 30 : V0T = 12.6076 ⇒ V1T = V2T = 6.30 ( m / s ) if α = 45 : V0T = 11.3089 ⇒ V1T = V 2T = 5.65 ( m / s)
(24)
Obtaining different linear velocities of the wheels is accessible by varying the rotational velocity. Variations of the wheels’ rotational velocity are in the range of 0 - 2000 (rpm). For these three parameters of α , UW velocity and LW velocity, different groups of answers are obtained and shown in (Fig. 13) The summation of two velocities must always become more than 1100 (rpm). By taking all above matters to the consideration, programming of speed control system’s microchip (AT89C51), must be satisfied below conditions.
if v1(i) + v 2 (i) > 2000 if v1(i) > v 2 (i) v1(i) = 1800,v 2 (i) = 0, else v 2(i) = 1800,v1(i) = 0,
( 25)
end end if v1(i) + v 2 (i) < 1100 v1(i) = 560,v 2 (i) = 560, end Second: The test based on different rates of pressure. By decreasing the distance, the pressure on ball will increase and it has some effects on the range and shot velocity. It’s clear that by increasing the pressure, the range and velocity will increase. Pressure can be shown, by decreasing the distance, so it will be shown indirectly in degrees. Axial rotation will cause some variations in the range. These variations have a direct influence on the rotation of the higher and the lower wheels. If the angle of axial rotation assumed as ß this will manage the thrown ball in the direction of x / (fig. 14) for a fixed place in the middle of the base line. And the range will verify as below.
x′ = x 2 + z 2 ; β = tg −1 ( z / x) ′ = Rmin / Cosβ x′ = x / Cosβ ⇒ Rmin
(26)
According to these parameters, the diagrams of the minimum range and the minimum velocity can be plotted. For the considered length of the court (x=11.885m) and the width (z=5.485), ß will obtain as well. β = tg −1 ( x / z ) ⇒ β = tg −1 (5.485/11.885) = 24.78 o
(27)
⇒ - 24 o < β < 24ο
And by exerting the limitation of the sidelines on that, we would have − 24o < β < −13o &
V0 =
13o < β < 24o
5.485 2 Sinβ g × 2 Sin 2 β .Cos 2 α .(5.485 − 1.05Sinβ )
(28) (29)
By varying the angle of a from 0 to 45 by the step of 3 degrees, and assuming the angle of ß between the mentioned ranges, the diagrams are plotted (fig. 15)
5 Graphs, Tables and Photograph
Figure 1 Automatic Ball Feeding Mechanism.
Figure 3 Ball Throwing Mechanism.
Figure 5 Free Body Diagram of Lower Wheel.
Figure 7 General Overview.
Figure 2 Ball Directing Mechanism.
Figure 4 Free Body Diagram of Belt and Wheel.
Figure 6 Angle of Shot.
Figure 8 Complete Machine.
Figure 9 Three Different Views of Machine.
Figure 10 Dimension of Tennis Court.
Table 1- the effect of angle on range and velocity under fixed load of 350 (N).
step 1 5 10 15 20
α(rad) 0.52 0.56 0.62 0.67 0.72
R (m) 15.5 17.9 19.8 23.2 24.6
VE (m/s) 15.6 16.3 16.8 17.7 18.0
V0T (m/s) 12.6 13.2 13.9 14.64 15.4
Table 2- The Effect of Pressure on Range and Velocity.
step 1 5 10 15 20
α(rad) 0.52 0.56 0.62 0.67 0.72
R (m) 15.5 17.9 19.8 23.2 24.6
P (N) 300 320 345 370 395
VE (m/s) 14.7 16.9 19.2 22.2 24.8
Figure 11 Experimental & Theoretical Velocities and Correction Factor.
Figure 12 Effect of increasing Ball pressing Force (N)
Figure 13 Variation of
α , UW
and
LW
as input and R as output.
Figure 14 The relation between angle ß and the direction x /
Figure 15 Variation of ß on the minimum range and the minimum velocity
6 Conclusions The appropriate pressure and the angle of throw are two main factors in this machine .For the best efficiency it is very necessary to find the most optimum rate of these two factors. As it is realized in this study, based on the critical pressure of ball, the study focused on the optimal angle and range of throw. By using the practical results a correction factor has been found to estimate the initial velocity of the thrown ball directly from the rotational velocity of the motors. This fact will lead us to find the approximatly range of the thrown ball. For having a sucsessful throw, the place of throwing and the maximum and the minimum velocity are determind. The entire graph for different amount of the α and UW and LW is plotted and as it is shown in the graph the output is the range of throw. Increasing the compression of the ball, will increase the frictional energy transfering and will effect on some unrecognized parameters.it will effect on the speed and the range of the thrown ball. However all ranges and ball speeds are needed for different players, also, a good safety factor for shafts and a powerful electrical engine can satisfy all common needs of such a machine.
References [1] Alitavoli M., Zerafatangiz F., Chaibakhsh A., Yousefi Moghadam R, Design and Modeling a Tennis Robot Based on Contour Rotating Wheels, Proc. 5th Iranian Manufacturing Engineering Conference, Tehran, Amirkabir University of technology , 2003 [2] Yousefi Moghadam R., Alitavoli M., Chaibakhsh A., Rezaeifar A., Optimization and Manufacturing a Tennis Robot Based on Contour Rotating Wheels system. Proc. 6th Iranian Manufacturing Engineering Conference, Tehran, Amirkabir University of technology, 2004 [3] J. Coursey, D. Cunningham, Development of product Requirements, September 27th, Creative solutions. 1988 [4] A. Armenti, The Physics of Sports. (American Institute of Physics, New York, 1992) [5] B. Howard. , Tennis Science for Tennis Players, (Philadelphia, University of Pennsylvania press, 1987) [6] B. Howard. , How would a physicist design a tennis racket?, (Physics today, vol. 8), 1995 [7] Manual Handbook of U.S. Tennis Court and Track Builders Association. Copyright, 1998 [8] W. Beitz, K. H. Kuttner, Dubbel Handbook of Mechanical Engineering , (Springer-Verlag, 2001)
