Why applied joint torque will affect another joint?

I want to ask a very simple question.

For the robot dynamics Newton-Euler algorithm, suppose there is a two-link planar robot, we can write the simple dynamic equation form as $$M(\theta)\ddot{\theta} = \tau - h(\theta,\dot{\theta})$$.

From the equation, we can see if we applied only one joint torque(suppose applied at joint one and ignore all links gravity), it will affect another joint accelerate(change the second joint angles).

It's not intuitive, in my imagination, if we run the joint one motor and ignore the gravity of links, so just change the joint one angle.

who can explain it?

Thank you!

Best,

Ben.

1. When you apply torque to the first joint, it makes the first link rotate. This means that the far end of the first link translates.

2. Because the second joint pins the near end of the second link to the far end of the first link, this means that the near end of the second link must also move.

3. In order for the second link to move, the pin in the second joint must apply a force to the second link.

4. The force is applied to the second link at a point that is not its center of mass, which means that there is a net torque from this force around the center of mass of the second link.

5. The torque/inertia ratio for the second link will in general not be the same as the torque/inertia ratio for the applied torque on the first link plus the reaction torque on that link from the pin.

6. Different torque/inertia ratios on the two links mean that they experience different rotational accelerations, and therefore rotate relative to one another.

7. When the links rotate relative to one another, the joint angle between them by definition changes.

(There are other explanations with deeper mathematics; this covers the phenomenon at a Newton-Euler force-balance level.)