Load-independent Metrics for Benchmarking Force Controllers

TL;DR

The paper addresses load-dependent biases in benchmarking torque/force controllers by introducing a load-independent modeling framework that splits the actuator impedance into $Z_b$ (blocked-load tracking) and $Z_t$ (transparency), yielding the transfer function $T_y(s)=\dfrac{F_l(s)}{F_r(s)}=\dfrac{Z_b(s)}{1-Z_t(s)Y(s)}$. It defines four quantitative metrics—$\text{LCS}$, $\text{TR}$, $\text{PII}$, and $\text{LRT}$—that quantify how load dynamics affect closed-loop performance, stability, and robustness, including a novel Passivity Index Interval that blends passivity with small-gain theory. The metrics are validated experimentally on two DOB-based controller tunings for a linear motor, revealing differences in transparency and load-robustness that are not easily discerned from standard frequency responses. This work aims to standardize load-independent benchmarking for force/torque controllers, enabling more robust and transparent actuator design across electromechanical and hydraulic systems and motivating extensions to nonlinear regimes.

Abstract

Torque-controlled actuators are critical components in mechatronic systems that closely interact with their environment, such as legged robots, collaborative manipulators, and exoskeletons. The performance and stability of these actuators depend not only on controller design and system dynamics but also significantly on load characteristics, which may include interactions with humans or unstructured environments. This load dependence highlights the need for frameworks that properly assess and compare torque controllers independent of specific loading conditions. In this short paper, we concisely present a modeling approach that captures the impact of load on the closed-loop dynamics of torque-controlled systems. Based on this model, we propose new methods and quantitative metrics, including the Passivity Index Interval, which blends passivity and small-gain theory to offer a less conservative measure of coupled stability than passivity alone. These metrics can be used alongside traditional control performance indicators, such as settling time and bandwidth, to provide a more comprehensive characterization of torque-controlled systems. We demonstrate the application of the proposed metrics through experimental comparisons of linear actuator force controllers.

Figures (4)

• Figure 1: Block diagram for a closed-loop force controller acting on a load through an actuator. Dashed lines represent signals, while continuous lines represent physical connections. The controller and the actuator dynamics can be grouped into an transparency block $Z$, while the load dynamics forms an admittance block $Y$. The feedback of the load velocity $v_l$ from the load into the actuator is intrinsic to physics and always present in force dynamics, while its feedback in the controller is optional and dependent on the controller design. The closed-loop force controller feedbacks the load force $f_l$ and compares it with a reference force $f_r$.
• Figure 2: The closed-loop force-controlled actuator, represented by $Z$, can be divided into two subsystems: blocked ($Z_b$) and transparency ($Z_t$). These two functions are dependent on both the controller and the actuator dynamics.
• Figure 3: Illustration of the Passivity Index Interval. The thick red segment highlights the frequency range $[\omega_1, \omega_2]$ where the gain $R_g$ is less than $1 - \epsilon$. The $M$ represents the maximum value of $|Z_t|$ outside this interval (thin blue line). According to mixed-sector small gain arguments, if the load is passive and satisfies $|Z_t||Y| < 1$ in the blue line, then the interconnected system remains stable. A smaller $|Z_t|$ allows for larger values of $|Y|$ while still guaranteeing stability.
• Figure 4: Frequency response of the PMLSM system with two DOB controllers. The plot on the left shows the response of $Z_b$, and the one on the right shows $Z_t$. Controller DOB-2 was tuned for a faster rise time. Transparency initially appears low, increases as the controller attempts to track the load at mid-frequencies, and decreases again at higher frequencies due to spring dynamics.