Note: this answer is no longer valid as it has been established that robots are not affectionate and, therefore, cannot hug or hold hands.

First Robot:

$$\begin{align}
25&=v_{x_1}t_1\\
v_{y_1}&=a_g\frac{t_1}2\\
10&=v_{y_1}\frac{t_1}2-a_g\frac{t_1^2}8=a_g\frac{t_1^2}8\\
t_1&=\frac{4\sqrt 5}{a_g}
\end{align}$$

Second Robot:

$$\begin{align}
15&=v_{x_2}t_2\\
v_{y_2}&=a_g\frac{t_2}2\\
20&=v_{y_2}\frac{t_2}2-a_g\frac{t_2^2}8=a_g\frac{t_2^2}8\\
t_2&=\frac{4\sqrt{10}}{a_g}
\end{align}$$

Robots Hugging tightly and jumping at same time:

$$
v_{x_c}=\frac{m_1v_{x_1}+m_2v_{x_2}}{m_1+m_2}\quad
v_{y_c}=\frac{m_1v_{y1}+m_2v_{y2}}{m_1+m_2}
$$

Assume the robots weigh the same (I can do more if you need):

$$\begin{align}
0&=v_{yc}t_c-a_g\frac{t_c^2}2\\&=2(\sqrt{10}+\sqrt5)t_c-a_gt_c^2\\
t_c&=\frac{2(\sqrt {10}+\sqrt 5)}{a_g}\\
v_{x_c}t_c&=\left(\frac{25}{\left(\frac{8\sqrt 5}{a_g}\right)}+\frac{15}{\left(\frac{8\sqrt{10}}{a_g}\right)}\right)\frac{2(\sqrt{10}+\sqrt 5)}{a_g}\\
v_{xc}t_c&=21.49>20
\end{align}$$