# Why is the contact force of the rigid-body robot model ks*xr?

In the paper "On Dynamic Models of Robot Force Control", which is written by Steven D. Eppinger and Warren P. Seering, there is a model 'rigid-body robot model'. The model is like below.

In the paper, the model is described as:

We model the robot as a mass with a damper to ground. The mass $$m$$,. represents the effective moving mass of the arm. The viscous damper $$b_r$$ is chosen to give the appropriate rigid body mode to the unattached robot. The sensor has stiffness $$k_s$$ and damping $$b_s$$. The workpiece is shown as a "ground state". The robot actuator is represented by the input force $$F$$ and the state variable $$x_r$$, measures the position of the robot mass.

In the paper, the contact force, which is the force across the sensor, is $$F_c=k_s x_r$$. But because the damping source from $$b_s$$ is also the part of the sensor, I think the contact force is $$F_c =k_s x_r + b_s \dot{x}_r$$. However because this paper and model are cited by many studies, I think I'm wrong. So, why $$F_c=k_s x_r$$?

PS : I wrote $$f_s = k_s x_r$$ previously. I modified that to $$F_c = k_s x_r$$.

Edited:

You should have put the subscript $$c$$ on variable $$F$$, because they define $$F_c$$ as the contact force the system is trying to maintain. Since achieving and servoing around that desired contact force is the goal, the velocity at the target state will be zero. Therefore $$F_c = k_sx_r$$. You’ll find the damping terms in the dynamic model - they are in the denominator of the transfer function.

Note: I see you added the subscript now. Thanks.

• Yes, but because the damper is also the element of the sensor, I think the sum of the spring force and the damping force is the contact force. So, I don't know why the damping force is not added when calculating the contact force even there is a damper in the sensor. – Kim Jaewoo Nov 4 '18 at 6:20

This paper is a thought experiment, using simplified models and mathematical analysis to try to gain some insight into the observed instabilities in real systems. As such they make simplifications as needed to fit the thought experiment.

I don't see any clear explanation for why they are not including the sensor damping force. Since this is a dynamic analysis I don't think it is an assumption of zero velocity. Rather, I think it reflects the simpleness of the model and the type of real-world sensors they were using in their other work.

The main thing to take away from this paper is to note that in the end they had no new ideas and no interesting results. The best use for this type of paper is to think about the problem they were trying to solve (induced instability during contact) and think about how you would approach it differently. Do you get better insight if you you use a more complex sensor model? Or is pole-zero analysis of a phenomena caused by abbe error, latency, and overly simplistic plant models not a great approach?

• Yes, that expression is in another paper, whose writer is the same. In the paper which I wrote, $F_c = k_s x_r$. These are the same meaning. – Kim Jaewoo Nov 4 '18 at 6:15
• This is not really an answer, it might have been better as a comment... :-) – Greenonline Nov 4 '18 at 8:59
• @KimJaewoo - It would be a good idea to edit your question, and put that additional info into it, in order to avoid confusion. – Greenonline Nov 4 '18 at 9:00