Stiffness and Damping Effect in the Foot Contact Controller Perfomance

[Taherifar]

Hi Dr. Caron,
I have recently reviewed the discussion on the "Stair Climbing Stabilization of the HRP-4 Humanoid Robot using Whole-body Admittance Control" paper. It was mentioned that the damping control is used instead of acceleration-based controller in the LIPM controller but the reason of that is not still clear for me. So, my questions are as listed below:
1- It is mentioned that the KΔθ≫BΔθ` in your application. Generally, Is it a must for the contact to have large stiffness and low damping to have stable damping controller?
2- I have done some simulation based on Kajita et al. (2001), and it showed that in the presence of damping, the controller would be stable and even damping can reduce torque tracking error and improves the performance. Based on my simulation the foot contact should be designed such that the damping be as much as possible in order to increase the performance of the foot contact control. Is this an acceptable result?Are you agree with it?
3- Would you please explain why did you use the damping controller instead of acceleration-based controller?

[stephane-caron]

Why use the damping controller?

Large stiffness and low damping were rather a design choice that came with the robots we worked with. The second-order model with both stiffness K and damping B coefficients has more parameters to tune, but if KΔθ ≫ BΔdot(θ) this more complex model will anyway behave like a first-order model. Let us apply Occam's razor then: using the first-order model, we get similar performance with less parameters to tune and simpler code to maintain 😉

I hope this helps with your questions 1 and 3. Regarding your question 2:

Shall we try moving away from high-stiffness low-damping at the contact interface?

Absolutely: moving away from the "high stiffness low damping" setting makes sense in my opinion. We tried one such move when walking on gravel with soft soles, however we reduced K rather than increase B: the soles had a Young's modulus of 0.32 MPa, versus 5 MPa in previous works, and I don't know how much for the default hard robot soles. (We still walked on hard floor though, so only the robot-side parameters of the contact interface had changed.) I see two observations from this work that may be relevant here:

First, the interaction with the ground through soft soles had more delay (to be precise: the delay between ankle displacement and reaction force response), i.e. lower force control bandwidth. This is not great for balance control where, when the stabilizer outputs a desired contact wrench, we want this wrench to be applied to the environment as fast as possible. That's one thing to look at: what combinations of K and B maximize the force control bandwidth?

If you simulation shows that B should be as high as possible, that's a very interesting answer and I hope you can share the details with the world by e.g. putting out a pre-print on it 🙂

Second, foot impacts at touchdown with soft soles were better mitigated mechanically. Walking controllers typically touch down earlier than expected at every step, which incurs undesired CoM velocity disturbances (foot impacts after they propagate across the kinematic chain). But with the soft soles these impacts were lower. If we imagine a function impact(K, B) at touchdown, it is likely decreasing in B yet increasing in K. This leads us to another question: for a given K (and some model of early-touchdown impact), what values of B minimize impact(K, B)?

Hoping this helps!

[Taherifar]

Hi Dr. Caron,

Thank you for you informative post.
I just wanted to categorize what you said and completed them based on what I have simulated.
It seems that the compliance of the contact has major effect on the following aspects:
a) Impact mitigation
b) Bandwidth of torque control system
c) Tracking error of the torque control system
d) Robot Stability (viability)

The first three items could be evaluated by simulation, but the last one must be determined based on the experimental results.

For impact mitigation, I have considered a mass-spring-damper model, which is excited from the bottom, simulation of the prescribed vibration system shows that by decreasing the stiffness, the natural frequency will reduce and the magnitude of output will merge to zero in lower frequencies, which is desirable. Also, high damping ratio will results in lower peak in the magnitude of the output. So, in order to have lower impact, we should reduce the stiffness and increase the damping.
To analyze the bandwidth and torque tracking error, the transfer function of the close-loop control system based on what presented in section 2 of Kajita et al. (2001) is done. The results revealed that higher stiffness would results in higher bandwidth and lower tracking error. The same result is obtained for damping. To verify the results, a double pendulum model is considered based on section3 of the Kajita et al. (2001). The simulation analysis of this model verified the results of the previous simulation unless the effect of stiffness on tracking error. I am still investigating the reason behind this difference. Actually, I am not still confident about the way the simulation was run in section3.
To sum up, high damping in the contact point would mitigate the impact and increase the performance of torque control in terms of bandwidth and tracking error. On the other hand, high stiffness would increase the negative effects of the impact on the system, but it would increase the performance of the control system. So, based on what I have simulated, to find the optimum stiffness, there is a trade-off between the impact mitigation and controller performance. This could be done by experimenting on the real robot.
I hope it could be helpful for you and other researchers. It would be appreciated if you could share any comment or concern related to this topic.