ImpedanceController weight gains interpretation?

[sageshoyu]

Hello!

I'm interested in having the RBY-1 manipulate a set of drawers using the provided ImpedanceController, but am having a hard time finding an interpretation of the 'weight' gains.

Would it be possible provide any details on how position and weights are converted into a control signal? Having the equations used (like the ones provided for the OptimalController in the RBY-1 documentation) would be very helpful.

Thanks!

[messy-snail]

First, apologies for the delayed response. Providing specific equations is currently under consideration.

In the meantime, you can indirectly understand the behavior by reviewing the 07_impedance_control.py example. If you examine the example_impedance_control_command_1 section, specifically the right_arm_command, you’ll notice that the weight values [3000, 3000, 0] are applied to the XYZ positions. This means the Z direction becomes significantly smoother, as a weight of 0 effectively disables position control in that direction, making it feel “free” or “smooth.”

You might already be aware of this general behavior, and we understand you are likely asking for detailed equations. The team in charge is currently evaluating whether to release the equations or provide additional materials to facilitate understanding. Thank you for your patience and understanding while we work on this.

@keunjun-choi please provide your feedback on this.

[sageshoyu]

Thanks for the response! Yes, the behavior that @messy-snail describes is what we observe from the controller.

We're interested in the interpretation of the weights when they are nonzero. Detailed equations would be the most helpful, if it is possible to provide them.

Thank you!

[sageshoyu]

Just following up -- would it be possible to given any high-level details about the impedance controller implementation?

We've continued to experiment with the controller on the RBY-1 and found that there can be a large amount of steady-state error (even with stiff gains, e.g. 1500 for position). More details would be greatly appreciated.

Thanks!

[keunjun-choi]

I sincerely apologize for the extremely delayed response—I realize it's been nearly one month. Thank you for your patience.

The impedance controller is designed as follows:

$$ \tau = J^T F $$

where the force $F$ is defined as:

$$ F = -K x - D \dot{x} $$

with

$$ x = \text{log} (T^{-1}T_d) $$

where $T_d$ represents the desired position and orientation in $SE(3)$.

The stiffness and damping matrices are defined as:

$$ K = \text{diag}(k_x, k_y, k_z, k_{\theta_x}, k_{\theta_y}, k_{\theta_z}) $$

$$ D = \sqrt{M} \sqrt{K} + \sqrt{K} \sqrt{M} $$

translation-weight is defined as

$$ \begin{bmatrix} k_x & k_y & k_z \end{bmatrix}^T $$

and

rotation-weight is defined as

$$ \begin{bmatrix} k_{\theta_x}, k_{\theta_y}, k_{\theta_z} \end{bmatrix}^T $$

---

Regarding the steady-state error you observed, one possible explanation is that while the current model includes gravity compensation, it does not incorporate friction compensation.

Without explicit friction compensation, the controller must rely solely on the impedance gains to overcome static friction, which can lead to residual steady-state error.

Additionally, there is a fundamental limitation in the current implementation: velocity computation is performed at a higher level rather than directly at the motor drivers. As a result, the estimated velocity has both delay and deviation from the actual velocity. This introduces latency in the damping term, making it challenging to increase the stiffness $K$ beyond a certain point while maintaining system stability.

We are actively working on improving this by refining our velocity estimation and implementing friction compensation. However, we do not have a concrete timeline for its completion yet.

[sageshoyu]

Thank you the detailed answer! We look forward to an update where friction compensation has been included, and we'll come up with a workaround for the steady-state error for now.

[siddancha]

We're also wondering how the impedance controller performs EE force control when there is no force/torque sensing at the joints?

[messy-snail]

As mentioned above, @keunjun-choi will address the question regarding impedance control. Since I’m not an internal controller developer, I don’t know the exact implementation details of the impedance control.

Our robot uses a force/torque (FT) sensor attached to the End Effector. There are no FT sensors on the joints, and we rely on a current-based torque estimation model. This value is an approximation and not precise. I’m not sure if this estimated torque is used in impedance control or if this answers your question. Hopefully, @keunjun-choi will provide clarity soon.

[messy-snail]

Currently, @keunjun-choi is very busy with a business trip and other tasks, so the response may be slightly delayed. We appreciate your understanding and hope a more precise answer will be provided soon.

[keunjun-choi]

Sorry for the late reply.

As @messy-snail mentioned, we are using a current-based torque controller, meaning that we estimate torque from the measured current. Using this method, we can implement an impedance controller.