- #1

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I have the following angular displacement function for time T0 through T3 and the following acceleration and velocity constraints in Cartesian coordinates. At t = T0, theta = 0, velocity = 0. At t = T3, velocity = 0.

I want to calculate the maximum

**A_T**and

**V_T**in terms of the linear A_max and V_max quantities (A_max and V_max put a limit on A_T and V_T). I want to eventually be able to calculate T1, T2, T3.

[tex]

\forall t \in [T_0, T_1]

\begin{cases}

\frac{d^2\theta}{dt^2}(t) = A_T \\

\frac{d\theta}{dt}(t) = \dot{\theta}_{0} + A_T(t-T_0) \\

\theta(t) = \theta_{0} + \dot{\theta}_{0}(t-T_0)+\frac{1}{2}A_T(t-T_0)^2 \\

\end{cases}

\\

\forall t \in [T_1, T_2]

\begin{cases}

\frac{d^2\theta}{dt^2}(t) = 0 \\

\frac{d\theta}{dt}(t) = V_T \\

\theta(t) = \theta(T_1)+V_T(t-T_1) \\

\end{cases}

\\

\forall t \in [T_2, T_3]

\begin{cases}

\frac{d^2\theta}{dt^2}(t) = -A_T \\

\frac{d\theta}{dt}(t) = \dot{\theta}(T_2) - A_T(t-T_2) \\

\theta(t) = \theta(T_2) + \dot{\theta}(T_2)(t-T_2) - \frac{1}{2}A_T(t-T_2)^2 \\

\end{cases}

\\

\\

\vec{A_{max}} = \begin{pmatrix}

A_{max,x} \\

A_{max,y} \\

\end{pmatrix}

\\

\vec{V_{max}} = \begin{pmatrix}

V_{max,x} \\

V_{max,y} \\

\end{pmatrix}

\\

\vec{Position} =

\begin{pmatrix}

x \\

y \\

\end{pmatrix} =

\begin{pmatrix}

\cos(\theta(t)) \\

\sin(\theta(t)) \\

\end{pmatrix}

[/tex]

At first I figured I could solve

[tex]\vec A =

\begin{pmatrix}

\frac{d^2}{dt}[\cos(\theta(t))] \\

\frac{d^2}{dt}[\sin(\theta(t))]\\

\end{pmatrix}

[/tex]

for A_T at t = T0 with all of the initial conditions applied but I'm afraid this is not correct.

How do I approach this problem?

Thanks