Parabolic Blend Trajectory

Standard Steps

Given three points $w^0,w^1,w^2$ and segment durations $T_1,T_2$, set $t=0$ at $w^0$.

1. Choose the blend transition time

$$0<\Delta T<\min \{T_1,T_2\}$$

The blend starts at $T_1-\Delta T$ and ends at $T_1+\Delta T$. Its total duration is $2\Delta T$.

2. Calculate segment displacements

$$\Delta w^1=w^1-w^0$$ $$\Delta w^2=w^2-w^1$$

3. Calculate blend acceleration

$$a=\frac{T_1\Delta w^2-T_2\Delta w^1}{2T_1{T}_2\Delta T}$$

4. Write the trajectory before the blend

$$w(t)=w^0+\frac{\Delta w^1}{T_1}t$$ $$0\le t\le T_1-\Delta T$$

5. Write the trajectory during the blend

$$w(t)=\frac{a}{2}(t-T_1+\Delta T{)}^2+\frac{\Delta w^1}{T_1}(t-T_1)+w^1$$ $$T_1-\Delta T<t<T_1+\Delta T$$

6. Write the trajectory after the blend

$$w(t)=w^1+\frac{\Delta w^2}{T_2}(t-T_1)$$ $$T_1+\Delta T\le t\le T_1+T_2$$

7. Combine the three equations

$$\boxed{w(t)=\{\begin{array}{ll}{w}^0+\frac{\Delta w^1}{T_1}t,& 0\le t\le T_1-\Delta T\\ \frac{a}{2}(t-T_1+\Delta T{)}^2+\frac{\Delta w^1}{T_1}(t-T_1)+w^1,& T_1-\Delta T<t<T_1+\Delta T\\ w^1+\frac{\Delta w^2}{T_2}(t-T_1),& T_1+\Delta T\le t\le T_1+T_2\end{array}}$$