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Explain briefly Cubic Polynomial Paths.
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Solution:

Consider a simple path i.e. from w0 to w1 an interpolated motion between two-knot points w0 to w1.

The equation for the path from w0 to w1 is represented by

a cubic polynomial given by w(t)=at3+bt2+ct+d;0tT;T>0; where w(t) is a cubic polynomial trajectory in Tool Configuration Space R6 and a,b,c,d are the unknown polynomial coefficients which has to be determined.

Once, the equation for the path is determined, the robot moves along the specified path.

Interpolation:

There are four constraints on the four unknown coefficients of the cubic interpolating polynomial.

Consider the equation w(t)=at3+bt2+ct+d

At the start of the path, t=0 in w(t), Putting t=0 in w(t) w(0)=d=w0

At the end of the path, t=T, Putting t=T in w(t) w(T)=aT3+bT2+cT+d=w1

If the tool starts with velocity of v0 at the beginning of the trajectory i.e. t=0 and ends with velocity v1 at the end of trajectory t=T, then the two end-point velocity constraints are

Velocity =ddt[at3+bt2+ct+d]

Therefore, ˙w(t)=3at2+2bt+c

At the start of the path, t=0, Putting t=0 in ˙w(t); we get ˙w(0)=c=v0

At the end of the path t=T, Putting t=T in ˙w(t); we get ˙w(T)=3aT2+2bT+c=v1

To get the values of a and b; solving by trial and error method

For finding the value of a ;

1) Add v1 to v0 and multiply the sum by T

2) Subtract w0 from w1 and multiply the result by 2

We get,

a=T(v1+v0)2(w1w0)T

For finding the value of b;

1) Take v1, add it to twice v0 and multiply the result by T2

2) Subtract w0 from w1 and multiply the result by 3

3) the Result of step 1 and the result of step 2, divide by T2

4) Take the negative of the entire result

We get,

b=T[(v1+2v0)3(w1w0)]T2

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