# To Study Trajectory Of Shuttlecock In Badminton

• Vivek Bhagat
In summary, the team was looking for equations of trajectory to predict the final position of the shuttlecock, but they were not able to find any. They used a method of integrating differential equations to calculate time of flight. If they had more data, they could use numerical solutions.f

#### Vivek Bhagat

Okay, so I am a team member of a college robotics team, and this year we have quite challenging theme of playing badminton with robots, so for this reason we decided to track the position of shuttlecock in air.
Now our approach was to predict the final position where it will land on ground from the points we get from our camera,
So for that we searched for the equations of trajectory considering air drag, sadly we didnt find any, and the on we found had terms initial velocity and initial angle, now our camera gives us readings in around 60 millisecond, so the time between two frames is not that small.
So we want some equations that will take position and time as input,
So can anyone tell me any approach to this problem, we have found equations of trajectory by integrating some differential equations we got after doing free body diagram of shuttlecock, while going up and falling down, I can post them, so you guys might check if my maths is correct, and if I have applied correct mechanics.

I would be surprised by analytical solutions for the trajectory. If you have enough computing power, you can use numerical solutions based on the observed points. Do you have a second viewing angle, a distance measurement or some other method to track the position in 3D?

My suggestion is to try dimensional analysis rather than a detailed modelling of shuttlecock dynamics. You will probably run into additional things that affect the path of the shuttlecock, like the angle at which it is struck, whether the striker hits the nose of the shuttlecock or the side of the shuttlecock, etc. Dimensional analysis might reveal the general form of an equation that could be fit to empirical data.

If you have enough computing power, you can use numerical solutions based on the observed points.

Well our computational work will be done by laptop, so it will be pretty good, Can you please explain how can I use numerical solutions, And for a given trajectory, our camera gives us roughly 10-15 points of shuttlecock, will they be enough to predict??

Do you have a second viewing angle, a distance measurement or some other method to track the position in 3D?

Yes sir, we have two cameras placed right angle to each other, one just near to net, and other one facing opponent's zone, but however both cameras can give us 3D co-ordinates of positions of shuttlecock, out of them, we require firstly the vertical distance to calculate time required by shuttle to fall, then by using this time we can calculate how much distance it will travel in remaining axes

I would be surprised by analytical solutions for the trajectory.

I will first state the assumptions we took:-
1. We assumed that there are only two forces acting on shuttlecock, first is drag force, which we assumed to be proportional to square of velocity, and other is gravitational force.
2. We assumed that drag coefficient to be constant throughout the motion
3. We split velocity of shuttlecock in 3 axes viz, x,y and z. y-axis being vertical. We assumed that drag coefficient is same in all three axes. This was done to calculate the time of flight from equations of vertical motion only.
4. We assumed Buoyancy effect to be negligible as it was only 0.002N.
So we first draw FBD for shuttle going upwards, solved the differential equation, and got y=f(t);
then we draw FBD for shuttlecock falling downwards, and found y=f(y);
so now we can get time of flight, so we draw FBD of shuttlecock for its motion in x and z direction.
If you want I can post those equations here.

And If you have any other approach or have any suggestion in assumptions or equations Please do tell me. Thank You Sir.

My suggestion is to try dimensional analysis rather than a detailed modelling of shuttlecock dynamics. You will probably run into additional things that affect the path of the shuttlecock, like the angle at which it is struck, whether the striker hits the nose of the shuttlecock or the side of the shuttlecock, etc. Dimensional analysis might reveal the general form of an equation that could be fit to empirical data.

Thank You Sir, I really liked this approach, Can you suggest while forming a general equation, which factors should I consider in time of flight, So that by doing dimensional analysis I can get a function for time of flight, and also do you have any reference for how to use dimensional analysis??
Actually I am not in touch with classical mechanics for 3 years, So I might need to revise my concepts.

This is a very challenging project. You probably need the literal 'large research team and 10 years' to get this from the ground, but this is probably also how robot football started, and they've come a long way. Here's some links for you:

detailed simulations of a shuttlecock from Sheffield-Hallam university:

http://www.ansys.com/staticassets/ANSYS/Hall%20of%20Fame/staticassets/2012-sheffieldhallam-bg.JPG [Broken]
http://www.ansys.com/Hall+of+Fame/Archive/2012 [Broken]

a conference proceeding giving some measurements for drag:

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detailed simulations of a shuttlecock from Sheffield-Hallam university:

http://www.ansys.com/staticassets/ANSYS/Hall%20of%20Fame/staticassets/2012-sheffieldhallam-bg.JPG [Broken]
http://www.ansys.com/Hall+of+Fame/Archive/2012 [Broken]

a conference proceeding giving some measurements for drag:
Thank You very much for the pdf and links, Can you tell me how can I use Ansys Fluent software, I have no idea of this software, so I don't know where to start from...

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Can you suggest while forming a general equation, which factors should I consider in time of flight, So that by doing dimensional analysis I can get a function for time of flight
Is finding time-of-flight your objective or do you want to know the entire trajectory of the shuttlecock?

and also do you have any reference for how to use dimensional analysis??
Online, there is: http://web.mit.edu/2.25/www/pdf/DA_unified.pdf

Can you please explain how can I use numerical solutions, And for a given trajectory, our camera gives us roughly 10-15 points of shuttlecock, will they be enough to predict??
Based on position measurements you can estimate position and velocity and feed that into your differential equations to calculate future motion step by step. Adjust as necessary as more datapoints are coming in.
Yes sir, we have two cameras placed right angle to each other, one just near to net, and other one facing opponent's zone, but however both cameras can give us 3D co-ordinates of positions of shuttlecock, out of them, we require firstly the vertical distance to calculate time required by shuttle to fall, then by using this time we can calculate how much distance it will travel in remaining axes
The two coordinates are coupled, you cannot analyze them independently. That violates assumption 3, and it is a significant violation.

@Stephen Tashi: I don't think dimensional analysis will help because "no drag" would be perfectly fine in such an analysis, but gives completely wrong predictions. On the other hand, the details how the shuttlecock got hit should not matter beyond the initial flight phase.

The two coordinates are coupled, you cannot analyze them independently. That violates assumption 3, and it is a significant violation.

What do you mean by coupled?? In our projectile motion we split the velocities in two perpendicular directions, so why can't we do the same here??
Is it because of drag force or something else?

If the air resistance is function of the square of the magnitude of the velocity vector, the x-component of the air resistance isn't a function of the x-velocity alone. The magnitude of the velocity vector depends on the y and z components of velocity also.

The magnitude of the velocity vector depends on the y and z components of velocity also.

But look at this case then, if I calculate all three air resistance components in x y and z direction, as per your saying they all should be proportional to net velocity and not to their respective components, so the magnitude of this net resistive force will be square root of addition of squares of each component.
Now I can also calculate air resistance force by considering only the net velocity,
So here the two magnitudes of drag forces are coming different...

Here is an analysis and approximate solutions: http://users.df.uba.ar/sgil/physics_paper_doc/papers_phys/mechan/air0.pdf [Broken]

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But look at this case then, if I calculate all three air resistance components in x y and z direction, as per your saying they all should be proportional to net velocity and not to their respective components, so the magnitude of this net resistive force will be square root of addition of squares of each component.
Now I can also calculate air resistance force by considering only the net velocity,
So here the two magnitudes of drag forces are coming different...
They are not proportional to the net velocity, they are proportional to the squared velocity - but they point in the direction of motion (well, against it). With drag proportional to velocity you could separate the two directions, with velocity squared you cannot.

A simple example: Let's ignore units, assume have drag directly given by the velocity squared. With a velocity vector of (1,1), drag is ##\sqrt{2}##(-1,-1). With a velocity vector of (1,2) where just the y component changed, drag is ##\sqrt{5}## (-1,-2) where both components changed.