# Understanding the kinetic energy expressions for a 1 link vs 2 link robot arm

I'm studying these lecture notes on dynamic models of robot arm links, slides 33-36, where two examples are given for the kinetic energies for a single link and two link robot arm.

In the single link case, the kinetic energy is posed as:

$$K=\frac{1}{2}I\dot{\theta}^2$$

This appears to account for the rotational kinetic energy of the single link around the joint.

In the two link case, the kinetic energy of the first link is posed as:

$$K=\frac{1}{2}I_1\dot{\theta_1}^2 + \frac{1}{2}m_1a_{c1}^2\dot{\theta_1}^2$$

This appears to mean that the total kinetic energy in the first link of the two link chain contains a contribution from rotation around the first joint ($$m_1a_{c1}^2$$ term) and from rotation around the center of mass ($$I_1$$ term).

Comparing the two link and single link cases, shouldn't $$I$$ from the single link case be decomposed similarly into energies from joint rotation and center of mass rotation as well? Or is the single link case fundamentally different in that one only needs to consider energy from rotation around the joint?

It's my intuition that if the parameters of the single link case are equal to the parameters of the first link (of the two-link case), then we could conceivably write $$I=I_1+m_1a_{c1}^2$$. Is this correct? If not, why not?