Math Is Fun Forum

Emad

Guest

convert sum into equation.

I have a summation function and I want to convert it into a fucntion of K alone.

summation cos(theta_i), from i=0 to k.

how can I get the above summation into a function of k alone.

Best.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: convert sum into equation.

Hi;

What is theta_i?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Emad

Member

Registered: 2015-01-31

Posts: 3

Re: convert sum into equation.

theta_i is angles in degree. from 1 up to k.. k is finite number.

I want to minimize this term ... j(K) = summation from i=1 to k ( cos( theta_i)+ Sin(theta_i))

I want to find Theta1,Theta2.... Theta k that minmize j(k)..

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: convert sum into equation.

hi Emad,

Arhhh!  That's made it clearer.  Each term can be simplified like this:

The minimum value of the cosine is -1, so the minimum will be -root(2)

The final answer will be independent of the values that the thetas have.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Emad

Member

Registered: 2015-01-31

Posts: 3

Re: convert sum into equation.

thank you Bob !

I want K to appear in the final expression. K is number of steps or turns, and its an integer value. everytime an incriment of 1 to k, there is a new angle theta.

the problem arises from turns of a robot that walk. so I want to know the minmim steps of the robot to the goal point. each step is 1 meter.

How can I choose Theta_i in order to get min steps for man robot ( K)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: convert sum into equation.

hi Emad,

You have a summation with different values of theta.  That's what was puzzling bobbym and me.  How can you minimise when you don't know any of the thetas ?

But each term in the summation has a trig expression cos(A) + sin(A) and that can always be written as cos(something) which will always have a minimum value of -1.

So if you have k terms like this the minimum value of the sum is -1 + (-1) + (-1) + ... = -k

From what you say about a robot it doesn't sound as if that helps much, sorry  .

How about going back a stage and telling us how the expression arises; why you want a minimum; and where you want to go with this next ?

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Emad

Member

Registered: 2015-01-31

Posts: 3

Re: convert sum into equation.

Theta is ranging from 0-90 degree. 

the Idea behind this is, if we have 2 points ( lets say (0,0 ) as initial point and (10,10) as final point ) and we want to move from initial point to final point in steps ( each step is 1 meter ). if we encounter obstackles , then we have some turns, how can we can minimize the turns in order to minimize the steps, and thats only by finding theta_i that results in minmime turns and   therefore steps to final point.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: convert sum into equation.

So, let's say there is an obstacle at (5,5)  . 

Have I got this right : You want to steer around the obstacle by turning a little bit on each step.  We need to know what is an acceptable clearance distance between the moving object and the obstacle. 

And then, if there's another obstacle,  this will completely throw all the calculations.

It sounds like all the thetas will be the same for a given setup.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob