From Continuous Design to Delay-Aware Discrete Synthesis: Guaranteed High-Bandwidth Joint Control for PMSM Drives

Edmundo Pozo Fortunić; Mehmet C. Yildirim; Sami Haddadin

cs.ROAug 14, 2026

From Continuous Design to Delay-Aware Discrete Synthesis: Guaranteed High-Bandwidth Joint Control for PMSM Drives

Authors: Edmundo Pozo Fortunić, Mehmet C. Yildirim, Sami Haddadin

Organizations: Chair of Robotics and Systems Intelligence and the Munich Institute of Robotics and Machine Intelligence (MIRMI), Technical University of Munich, Germany · Mohamed Bin Zayed University of Artificial Intelligence, Abu Dhabi, United Arab Emirates

Abstract

The increasing dynamic demands of modern robotic joints require current controllers to achieve high bandwidth over wide operating ranges of speed, acceleration, and torque, where communication, computation, and discrete-time effects can no longer be neglected. Conventional PMSM current controllers are typically designed in continuous time and subsequently discretized, leaving the sampling frequency and the impact of implementation delays largely to heuristic selection and iterative validation. This paper introduces a task-aware, delay-extended discrete-time joint model that explicitly accounts for physical communication and computation delays and enables direct synthesis of a discrete PI current controller with prescribed bandwidth and delay guarantees throughout the operating envelope. The framework analytically determines the minimum required sampling frequency, controller gains, and DC-link voltage needed to satisfy the specified motor and joint performance. Simulations across a range of dynamic requirements validate the methodology and demonstrate substantially reduced sampling-frequency and DC-link-voltage requirements compared with conventional continuous-time-based design. Experiments on a newly developed custom robotic joint further validate the proposed framework under real embedded implementation conditions.

Figures & tables

Fig. 1: Design Synthesis for a PMSM embedded discrete controller based on its joint-level specifications.

Fig. 2: Magnitudes (top) and Phases (bottom) of the closed loop current controllers 26 with different $Kp$ parameters. Each $\Omega_{b}w$ is highlighted and marked on the top axis. The bottom axis has the equivalent $f_{bw}/f_{s}$ ratio. Each corresponding phase is also marked.

Fig. 3: Boundaries required to select the correct minimum sampling frequency

f_{s}

that ensure the proper operation of two motors (see Tab. I ) for the required conditions of bandwidth, maximum speed and current ripple.

Parameter	Symbol	Unit	$M_{1}$	$M_{2}$
Phase Resistance	$R$	$\Omega$	12.65e-2	7.55e-2
Phase Inductance	$L$	$\mathrm{H}$	3.21e-5	5.95e-5
Pole Pairs	$n_{p}$	$-$	2	10
Torque Constant Back-EMF Constant	$k_{t}$	$\mathrm{N\cdot m/A}$ $(\mathrm{V\cdot s/rad})$	2.81e-2	3.1e-2
Rated Current	$i_{\mathrm{max}}$	$\mathrm{A}$	6.6	10
Rated Speed	$\omega_{\mathrm{max}}$	$\mathrm{rad/s}$	924	717

TABLE I: Motor parameters of the robotic joint prototype.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

$\begin{gathered}\textbf{Actuator}\\ \textbf{Model}\end{gathered}$	Closed-Loop Transfer Function $\boldsymbol{H^{A,t}_{cl}(s)}$	$\boldsymbol{\omega_{n}^{2}}$	$\boldsymbol{\zeta}$	Additional pole(s)
$\begin{gathered}\text{Ideal}\\[2.84526pt] 1\end{gathered}$	$\dfrac{1}{b_{\theta}}\dfrac{k+ds}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}}$	$\dfrac{k}{J_{\theta}}$	$\dfrac{d_{\theta}}{2\omega_{n}}$	–
$\begin{gathered}\text{Delay}\\[2.84526pt] e^{-sT_{d}}\end{gathered}$	$\dfrac{\frac{T_{d}}{2}s+1}{\frac{1}{p}s+1}\!\left(\dfrac{2}{T_{d}p\,b_{\theta}}\right)\dfrac{(k+ds)e^{-sT_{d}}}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}}$	$\dfrac{2}{T_{d}p}\dfrac{k}{J_{\theta}}$	$\dfrac{2\frac{d}{J}-d_{\theta}+\frac{2}{T_{d}}-p}{2\omega_{n}}$	$p=\dfrac{2}{T_{d}}-2\!\left(d_{\theta}-\dfrac{d}{J}\right)+\mathcal{O}(T_{d})$
$\begin{gathered}\text{Finite}\\[2.84526pt] \dfrac{\omega_{bw}}{s+\omega_{bw}}\end{gathered}$	$\dfrac{\omega_{0}}{s+\omega_{0}}\!\left(\frac{\omega_{bw}}{\omega_{0}b_{\theta}}\right)\dfrac{k+ds}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}}$	$\dfrac{\omega_{bw}}{\omega_{0}}\dfrac{k}{J_{\theta}}$	$\dfrac{\frac{d}{J}+\omega_{bw}-\omega_{0}}{2\omega_{n}}$	$\omega_{0}=\omega_{bw}-\left(d_{\theta}-\dfrac{d}{J}\right)+\mathcal{O}(\omega_{bw}^{-1})$
$\begin{gathered}\text{Finite +}\\ \text{Delay}\\[2.84526pt] \dfrac{\omega_{bw}e^{-sT_{d}}}{s+\omega_{bw}}\end{gathered}$	$\dfrac{\omega_{0}}{s+\omega_{0}}\dfrac{\frac{T_{d}}{2}s+1}{\frac{1}{p}s+1}\!\left(\dfrac{2\omega_{bw}}{T_{d}p\,\omega_{0}b_{\theta}}\right)\dfrac{(k+ds)e^{-sT_{d}}}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}}$	$\dfrac{2\omega_{bw}}{T_{d}p\omega_{0}}\dfrac{k}{J_{\theta}}$	$\!\!\dfrac{\frac{d}{J}\!+\!\frac{2}{T_{d}}\!\!-\!p\!+\!\omega_{bw}\!-\!\omega_{0}\!}{2\omega_{n}}$	$\begin{gathered}p=\dfrac{2}{T_{d}}-\!\ \frac{\omega_{bw}\left(2d_{\theta}-\dfrac{d}{J}\right)}{\frac{2}{T_{d}}-\omega_{bw}}+\mathcal{O}\left(\!T_{d},\omega_{bw}^{-1}\!\right)\!\\[2.84526pt] \!\omega_{0}=\omega_{bw}\!-\!\frac{\frac{2}{T_{d}}\!\left(\!d_{\theta}\!-\!\dfrac{d}{J}\!\right)\!+\!\omega_{bw}d_{\theta}}{\frac{2}{T_{d}}\!-\!\omega_{bw}}\!+\!\mathcal{O}\!\left(\!T_{d},\omega_{bw}^{-1}\!\right)\!\end{gathered}$

Appendix

TABLE II: Equivalent actuator models and resulting closed-loop transfer functions.

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A cable-driven continuum robot arm is an underactuated mechanism and may suffer residual vibration at the end of a rest-to-rest maneuver. In this work, a time-delay filter is applied as an input shaper to the system to eliminate the excitation of vibratory modes. A non-robust and a robust time-delay filter are designed based on a linear system model and demonstrate improved response compared to a velocity-driven pulse input. Experimental results using the continuum robot validate the application of the input shaper, with reduced overshoot and settling time exemplifying the reduction in oscillation at the end of the maneuver. It is also shown that utilizing the robust shaper further improves the response of the arm in comparison to applying the non-robust shaper. These results are significant towards the precise and robust implementation of continuum robots in applications involving arbitrary end-effector trajectories.

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