Simple cubic trajectory derivation from end-point constraints

General solution

End point definitions

p0 = {0, θ_0, Overscript[θ, .] _0} ; pf = {t_f, θ_f, Overscript[θ, .] _f} ;

System of equations

SimpleCubicEquationsDisplay[p0, pf] // ViewSolution

θ_0a_0
Overscript[θ, .] _0a_1
θ_fa_3 t_f^3 + a_2 t_f^2 + a_1 t_f + a_0
Overscript[θ, .] _f3 a_3 t_f^2 + 2 a_2 t_f + a_1

Solution for coefficients

SimpleCubicCoefficientsDisplay[p0, pf] //Simplify//Sort// ViewSolution

a_0θ_0
a_1Overscript[θ, .] _0
a_2 (-3 θ_0 + 3 θ_f - t_f (2 Overscript[θ, .] _0 + Overscript[θ, .] _f))/t_f^2
a_3 (2 θ_0 - 2 θ_f + t_f (Overscript[θ, .] _0 + Overscript[θ, .] _f))/t_f^3

Zero velocity at end points

End point definitions

p0 = {0, θ_0, 0} ; pf = {t_f, θ_f, 0} ;

System of equations

SimpleCubicEquationsDisplay[p0, pf] // ViewSolution

θ_0a_0
0a_1
θ_fa_3 t_f^3 + a_2 t_f^2 + a_1 t_f + a_0
03 a_3 t_f^2 + 2 a_2 t_f + a_1

Solution for coefficients

SimpleCubicCoefficientsDisplay[p0, pf] //Simplify//Sort// ViewSolution

a_0θ_0
a_10
a_2 -(3 (θ_0 - θ_f))/t_f^2
a_3 (2 (θ_0 - θ_f))/t_f^3

Created by Mathematica  (November 12, 2003)