Note: this answer is no longer valid as it has been established that robots are not affectionate and, therefore, cannot hug or hold hands.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

$$25 = v_{x1}*t_1$$
$$v_{y1}=a_g*t_1/2$$
$$10=v_{y1}*t_1/2-a_g*t_1^2/8=a_g*t_1^2/8$$
$$t_1=4\sqrt 5/a_g$$

Second Robot:

$$15 = v_{x2}*t_2$$
$$v_{y2}=a_g*t_2/2$$
$$20=v_{y2}*t_2/2-a_g*t_2^2/8=a_g*t_2^2/8$$
$$t_2=4\sqrt {10}/a_g$$

Robots Hugging tightly and jumping at same time:

$$v_{xc}=\frac{m_1*v_{x1}+m_2*v_{x2}}{m_1+m_2}$$
$$v_{yc}=\frac{m_1*v_{y1}+m_2*v_{y2}}{m_1+m_2}$$

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

$$0=v_{yc}*t_c-a_g*t_c^2/2=2(\sqrt {10}+\sqrt 5)t_c-a_g*t_c^2$$
$$t_c=2(\sqrt {10}+\sqrt 5)/a_g$$
$$v_{xc}t_c=(\frac{25}{8\sqrt 5/a_g}+\frac{15}{8\sqrt{10}/a_g})*2(\sqrt {10}+\sqrt 5)/a_g$$
$$v_{xc}t_c=21.49>20$$