Finding coefficients of polynomials with inequality constrains

I have a 7th order polynomial equation: x(t) = a*t^7 + b*t^6 + c*t^5 + d*t^4 + e*t^3 + f*t^2 + g*t + h

i know that:

x(0) = 0

x'(0) = 0

x''(0) = 0

hence, f,h,g =0.

however i am trying to solve for the rest of the coefficients and i have the following constrains:

x''' (0) <= Jmax

x(Tm) = D

x'(Tm) = Ve

x''(Tm) = 0

x'''(Tm) = 0

x'''(Tamax1) = 0

x''(Tamax1) = Amax

x''(Tamax2) <= Amax

x'''(Tamax2) = 0

x'''(Tamax3) = 0

x''(Tamax3) <= Amax

x'(Tvmax) <= Vmax

x''(Tvmax) = 0

where Jmax, D, Ve, Amax, Vmax are known inputs.

Hence, this brings us the problem of needing to find out what is a,b,c,d,e, Tm, Tamax1, Tamax2, Tamax3 and Tvmax.

The main issue i am facing is having more equations (from the constrains, 13) than unknowns (10) and the fact that there are inequality constrains.

Any idea how do i find the coefficients and the various T?

thank you!

Answer 1

[Asvin Kumar]

Have a look at the following example to see if it can help you:

https://www.mathworks.com/help/symbolic/solve.html#buql15h-5