Ideal Lift Kinematics

List of Symbols:

$$\begin{array}{l} d : \operatorname{journey}\operatorname{distance} (m)\\ j : \operatorname{maximum}\operatorname{jerk} (m s^{- 3})\\ a : \operatorname{maximum}\operatorname{acceleration}/\operatorname{deceleration} (m s^{- 2})\\ v : \operatorname{maximum}\operatorname{velocity} (m s^{- 1})\\ J (t) : \operatorname{jerk}\operatorname{at}\operatorname{time}t (m s^{- 3})\\ A (t) : \operatorname{acceleration}\operatorname{at}\operatorname{time}t (m s^{- 2})\\ V (t) : \operatorname{velocity}\operatorname{at}\operatorname{time}t (m s^{- 1})\\ D (t) : \operatorname{distance}\operatorname{travelled}\operatorname{at}\operatorname{time}t (m) \end{array}$$

Condition A: when $$d \geqslant \frac{a^2 v + v^2 j}{a j}$$ (lift reaches maximum speed)

\begin{eqnarray*} t_1 & = & \frac{a}{j}\\ & & \\ t_2 & = & \frac{v}{a}\\ & & \\ t_3 & = & \frac{a}{j} + \frac{v}{a}\\ & & \\ t_4 & = & \frac{d}{v}\\ & & \\ t_5 & = & \frac{d}{v} + \frac{a}{j}\\ & & \\ t_6 & = & \frac{d}{v} + \frac{v}{a}\\ & & \\ t_7 & = & \frac{d}{v} + \frac{a}{j} + \frac{v}{a} \end{eqnarray*}

When $$0 \leqslant t \leqslant t_1$$:

\begin{eqnarray*} J (t) & = & j\\ & & \\ A (t) & = & j t\\ & & \\ V (t) & = & \frac{j}{2} t^2\\ & & \\ D (t) & = & \frac{j}{6} t^3 \end{eqnarray*}

When $$t_1 \leqslant t \leqslant t_2$$:

\begin{eqnarray*} J (t) & = & 0\\ & & \\ A (t) & = & a\\ & & \\ V (t) & = & - \frac{a^2}{2 j} + a t\\ & & \\ D (t) & = & \frac{a^3}{6 j^2} - \frac{a^2}{2 j} t + \frac{a}{2} t^2 \end{eqnarray*}

When $$t_2 \leqslant t \leqslant t_3$$:

\begin{eqnarray*} J (t) & = & - j\\ & & \\ A (t) & = & a + \frac{v j}{a} - j t\\ & & \\ V (t) & = & - \frac{a^2}{2 j} - \frac{j v^2_{}}{2 a^2} + \left( a + \frac{v j}{a} \right) t - \frac{j}{2} t^2\\ & & \\ D (t) & = & \frac{a^3}{6 j^2} + \frac{j v^3}{6 a^3} - \left( \frac{a^2}{2 j} + \frac{j v^2}{2 a^2} \right) t + \left( \frac{a}{2} + \frac{j v}{2 a^{}} \right) t^2 - \frac{j}{6} t^3 \end{eqnarray*}

When $$t_3 \leqslant t \leqslant t_4$$:

\begin{eqnarray*} J (t) & = & 0\\ & & \\ A (t) & = & 0\\ & & \\ V (t) & = & v\\ & & \\ D (t) & = & - \frac{a v}{2 j} - \frac{v^2}{2 a} + v t \end{eqnarray*}

When $$t_4 \leqslant t \leqslant t_5$$:

\begin{eqnarray*} J (t) & = & - j\\ & & \\ A (t) & = & \frac{j d}{v} - j t\\ & & \\ V (t) & = & v - \frac{j d^2}{2 v^2}_{} + \frac{j d}{v} t - \frac{j}{2} t^2\\ & & \\ D (t) & = & \frac{j d^3}{6 v^3} - \frac{a v}{2 j} - \frac{v^2}{2 a}_{} + \left( v - \frac{j d^2}{2 v^2} \right) t + \frac{j d}{2 v} t^2 - \frac{j}{6} t^3\\ & & \end{eqnarray*}

When $$t_5 \leqslant t \leqslant t_6$$:

\begin{eqnarray*} J (t) & = & 0\\ & & \\ A (t) & = & - a\\ & & \\ V (t) & = & v + \frac{a d}{v} + \frac{a^2}{2 j} - a t\\ & & \\ D (t) & = & - \left( \frac{a v}{2 j} + \frac{v^2}{2 a} + \frac{a d^2}{2 v^2} + \frac{d a^2}{2 j v} + \frac{a^3}{6 j^2} \right) + \left( v + \frac{a d}{v} + \frac{a^2}{2 j} \right) t - \frac{a}{2} t^2\\ & & \\ & & \end{eqnarray*}

When $$t_6 \leqslant t \leqslant t_7$$:

\begin{eqnarray*} J (t) & = & j\\ & & \\ A (t) & = & - \left( a + \frac{j d}{v} + \frac{j v}{a} \right) + j t\\ & & \\ V (t) & = & \left( v + \frac{a d}{v} + \frac{a^2}{2 j} + \frac{j d^2}{2 v^2} + \frac{j d}{a} + \frac{j v^2}{2 a^2} \right) - \left( a + \frac{j d}{v} + \frac{j v}{a} \right) t + \frac{j}{2} t^2\\ & & \\ D (t) & = & - \left( \frac{v^2}{2 a} + \frac{j d v}{2 a^2} + \frac{j d^2}{2 v a} + \frac{a v}{2 j} + \frac{j d^3}{6 v^3} + \frac{a^3}{6 j^2} + \frac{a d^2}{2 v^2} + \frac{d a^2}{2 j v} + \frac{j v^3}{6 a^3} \right)\\ & & + \left( v + \frac{j d^2}{2 v^2} + \frac{a d}{v} + \frac{a^2}{2 j} + \frac{j d}{a} + \frac{j v^2}{2 a^2} \right) t - \left( \frac{j d}{2 v} + \frac{a}{2} + \frac{v j}{2 a} \right) t^2 + \frac{j}{6} t^3\\ & & \end{eqnarray*}

Condition B: when $$\frac{2 a^3}{j^2} \leqslant d < \frac{a^2 v + v^2 j}{j a}$$ (lift reaching maximum acceleration but not maximum speed)

\begin{eqnarray*} t_1 & = & \frac{a}{j}\\ & & \\ t_2 & = & - \frac{a}{2 j} + \frac{\sqrt{a^3 + 4 d j^2}}{2 j \sqrt{a}}\\ & & \\ t_3 & = & \frac{a}{2 j} + \frac{\sqrt{a^3 + 4 d j^2}}{2 j \sqrt{a}}\\ & & \\ t_4 & = & \frac{3 a}{2 j} + \frac{\sqrt{a^3 + 4 d j^2}}{2 j \sqrt{a}}\\ & & \\ t_5 & = & \frac{\sqrt{a^3 + 4 d j^2}}{j \sqrt{a}}\\ & & \\ t_6 & = & \frac{a}{j} + \frac{\sqrt{a^3 + 4 d j^2}}{j \sqrt{a}} \end{eqnarray*}

When $$0 \leqslant t \leqslant t_1$$:

\begin{eqnarray*} J (t) & = & j\\ & & \\ A (t) & = & j t\\ & & \\ V (t) & = & \frac{j}{2} t^2\\ & & \\ D (t) & = & \frac{j}{6} t^3 \end{eqnarray*}

When $$t_1 \leqslant t \leqslant t_2$$:

\begin{eqnarray*} J (t) & = & 0\\ & & \\ A (t) & = & \overset{}{a}\\ & & \\ V (t) & = & - \frac{a^2}{2 j} + a t\\ & & \\ D (t) & = & \frac{a^3}{6 j^2} - \frac{a^2}{2 j} t + \frac{a}{2} t^2 \end{eqnarray*}

When $$t_2 \leqslant t \leqslant t_3$$:

\begin{eqnarray*} J (t) & = & - j\\ & & \\ A (t) & = & \frac{a}{2} + \frac{\sqrt{a^3 + 4 d j^2}}{2 \sqrt{a}} - j t\\ & & \\ V (t) & = & \frac{\sqrt{a} \sqrt{a^3 + 4 d j^2} - 3 a^2}{4 j} - \frac{j d}{2 a} + \left( \frac{a}{2} + \frac{\sqrt{a^3 + 4 d j^2}}{2 \sqrt{a}} \right) t - \frac{j}{2} t^2\\ & & \\ D (t) & = & \frac{a^3 + \sqrt{a^3} \sqrt{a^3 + 4 d j^2}}{12 j^2} - \frac{d}{4} + \frac{d \sqrt{a^3 + 4 d j^2}}{12 \sqrt{a^3}}\\ & & + \left( \frac{\sqrt{a} \sqrt{a^3 + 4 d j^2} - 3 a^2}{4 j} - \frac{j d}{2 a} \right) t + \left( \frac{a}{4} + \frac{\sqrt{a^3 + 4 d j^2}}{4 \sqrt{a}} \right) t^2 - \frac{j}{6} t^3 \end{eqnarray*}

When $$t_3 \leqslant t \leqslant t_4$$:

\begin{eqnarray*} J (t) & = & - j\\ & & \\ A (t) & = & \frac{a}{2} + \frac{\sqrt{a^3 + 4 d j^2}}{2 \sqrt{a}} - j t\\ & & \\ V (t) & = & \frac{\sqrt{a^{}} \sqrt{a^3 + 4 d j^2} - 3 a^2}{4 j^{}} - \frac{d j}{2 a} + \left( \frac{a}{2} + \frac{\sqrt{a^3 + 4 d j^2}}{2 \sqrt{a}} \right) t - \frac{j}{2} t^2\\ & & \\ D (t) & = & \frac{a^3 + \sqrt{a^3} \sqrt{a^3 + 4 d j^2}}{12 j^2} + \frac{d \sqrt{a^3 + 4 d j^2}}{12 \sqrt{a^3}} - \frac{d}{4} + \left( \frac{\sqrt{a} \sqrt{a^3 + 4 d j^2} - 3 a^2}{4 j} - \frac{j d}{2 a} \right) t\\ & & + \left( \frac{a}{4} + \frac{\sqrt{a^3 + 4 d j^2}}{4 \sqrt{a^{}}} \right) t^2 - \frac{j}{6} t^3\\ & & \end{eqnarray*}

When $$t_4 \leqslant t \leqslant t_5$$:

\begin{eqnarray*} J (t) & = & 0\\ & & \\ A (t) & = & - a\\ & & \\ V (t) & = & \frac{a^2}{2 j} + \frac{\sqrt{a} \sqrt{a^3 + 4 d j^2}}{j} - a t\\ & & \\ D (t) & = & - d - \frac{\sqrt{a^3} \sqrt{a^3 + 4 d j^2}}{2 j^2} - \frac{2 a^3}{3 j^2} + \left( \frac{a^2}{2 j} + \frac{\sqrt{a^{}} \sqrt{a^3 + 4 d j^2}}{j^{}} \right) t - \frac{a}{2} t^2\\ & & \end{eqnarray*}

When $$t_5 \leqslant t \leqslant t_6$$:

\begin{eqnarray*} J (t) & = & j\\ & & \\ A (t) & = & - a - \frac{\sqrt{a^3 + 4 d j^2}}{\sqrt{a^{}}} + j t\\ & & \\ V (t) & = & \frac{2 d j}{a} + \frac{a^2 + \sqrt{a^{}} \sqrt{a^3 + 4 d j^2}}{j^{}} - \left( a + \frac{\sqrt{a^3 + 4 d j^2}}{\sqrt{a^{}}} \right) t + \frac{j}{2} t^2\\ & & \\ D (t) & = & \frac{(a^3 + 4 d j^2)^{\frac{3}{2}}}{3 j^2 \sqrt{a^3}} - d - \frac{2 a^3}{3 j^2} - \frac{\sqrt{a^3} \sqrt{a^3 + 4 d j^2}}{j^2} - \frac{2 d \sqrt{a^3 + 4 d j^2}}{\sqrt{a^3}}\\ & & + \left( \frac{a^2 + \sqrt{a} \sqrt{a^3 + 4 d j^2}}{j} + \frac{2 d j}{a} \right) t - \left( \frac{a}{2} + \frac{\sqrt{a^3 + 4 d j^2}}{2 \sqrt{a^{}}} \right) t^2 + \frac{j}{6} t^3 \end{eqnarray*}

Condition C: when $$d < 2 \frac{a^3}{j^2}$$ (lift not reaching maximum speed or maximum acceleration)

\begin{eqnarray*} t_1 & = & \left( \frac{d}{2 j} \right)^{\frac{1}{3}}\\ & & \\ t_2 & = & \left( \frac{4 d}{j} \right)^{\frac{1}{3}}\\ & & \\ t_3 & = & \left( \frac{27 d}{2 j} \right)^{\frac{1}{3}}\\ & & \\ t_4 & = & \left( \frac{32 d}{j} \right)^{\frac{1}{3}} \end{eqnarray*}

When $$0 \leqslant t \leqslant t_1$$:

\begin{eqnarray*} J (t) & = & j\\ & & \\ A (t) & = & j t\\ & & \\ V (t) & = & \frac{j}{2} t^2\\ & & \\ D (t) & = & \frac{j}{6} t^3\\ & & \end{eqnarray*}

When $$t_1 \leqslant t \leqslant t_2$$:

\begin{eqnarray*} J (t) & = & - j\\ & & \\ A (t) & = & (2 j)^{\frac{2}{3}} d^{\frac{1}{3}} - j t\\ & & \\ V (t) & = & - \frac{1}{2} (2 j)^{\frac{1}{3}} d^{\frac{2}{3}} + (2 j)^{\frac{2}{3}} d^{\frac{1}{3}} t - \frac{j}{2} t^2\\ & & \\ D (t) & = & \frac{d}{6} + \frac{1}{2} (2 j)^{\frac{2}{3}} d^{\frac{1}{3}} t^2 - \frac{1}{12} (2 j)^{\frac{1}{3}} d^{\frac{2}{3}} j t^3 \end{eqnarray*}

When $$t_2 \leqslant t \leqslant t_3$$:

\begin{eqnarray*} J (t) & = & - j\\ & & \\ A (t) & = & (2 j)^{\frac{2}{3}} d^{\frac{1}{3}} - j t\\ & & \\ V (t) & = & - \frac{1}{2} (2 j)^{\frac{1}{3}} d^{\frac{2}{3}} + 2^{\frac{2}{3}} d^{\frac{1}{3}} t - \frac{j}{2} t^2\\ & & \\ D (t) & = & \frac{d}{6} - \frac{1}{2} (2 j)^{\frac{1}{3}} d^{\frac{2}{3}} t + \frac{1}{2} (2 j)^{\frac{2}{3}} d^{\frac{1}{3}} t^2 - \frac{j}{6} t^3 \end{eqnarray*}

When $$t_3 \leqslant t \leqslant t_4$$:

\begin{eqnarray*} J (t) & = & j\\ & & \\ A (t) & = & - 2 (2 j)^{\frac{2}{3}} d^{\frac{1}{3}} + j t\\ & & \\ V (t) & = & 4 (2 j)^{\frac{1}{3}} d^{\frac{2}{3}} - 2 (2 j)^{\frac{2}{3}} d^{\frac{1}{3}} t + \frac{j}{2} t^2\\ & & \\ D (t) & = & - \frac{13 d}{3} + 4 (2 j)^{\frac{1}{3}} d^{\frac{2}{3}} t - j^{\frac{2}{3}} t^2 - \frac{j}{6} t^3 \end{eqnarray*}