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*}