System Identification for Virtual Sensor-Based Model Predictive Control: Application to a 2-DoF Direct-Drive Robotic Arm

TL;DR

NMPC for nonlinear systems is hampered by the difficulty of identifying accurate dynamics and measuring control-relevant variables. The authors propose Predictive Virtual Sensor Identification (PVSID), which jointly learns a dynamics model and virtual sensors through neural networks that produce near-future multi-step predictions, ready for NMPC. On a 2-DoF direct-drive robotic arm, PVSID enables precise tip tracking using an IMU in operation while bypassing the need for continuous motion capture. This approach offers a practical route to deploying high-performance NMPC in systems where expensive sensing is impractical, with strong potential for scaling to more complex, higher-DoF platforms.

Abstract

Nonlinear Model Predictive Control (NMPC) offers a powerful approach for controlling complex nonlinear systems, yet faces two key challenges. First, accurately modeling nonlinear dynamics remains difficult. Second, variables directly related to control objectives often cannot be directly measured during operation. Although high-cost sensors can acquire these variables during model development, their use in practical deployment is typically infeasible. To overcome these limitations, we propose a Predictive Virtual Sensor Identification (PVSID) framework that leverages temporary high-cost sensors during the modeling phase to create virtual sensors for NMPC implementation. We validate PVSID on a Two-Degree-of-Freedom (2-DoF) direct-drive robotic arm with complex joint interactions, capturing tip position via motion capture during modeling and utilize an Inertial Measurement Unit (IMU) in NMPC. Experimental results show our NMPC with identified virtual sensors achieves precise tip trajectory tracking without requiring the motion capture system during operation. PVSID offers a practical solution for implementing optimal control in nonlinear systems where the measurement of key variables is constrained by cost or operational limitations.

Figures (5)

• Figure 1: Structure of the neural network model for Predictive Virtual Sensor IDentification (PVSID). The sequence of past input-output $\left(u_t^{{\mathrm{p}}}, y_t^{{\mathrm{p}}}\right)$ is reduced to current estimate $\hat{x}_t$ and then reflected in the prediction of the sequence of future output $w_t^{{\mathrm{f}}}$ driven by $u_t^{{\mathrm{f}}}$.
• Figure 2: Overview of the experimental setup. Left: The developed 2 DoF direct-drive robotic arm and the schematic diagram. Middle: Experimental device configuration, including the motion capture system (MoCap) and controlled hardware. Right: Communication architecture between components.
• Figure 3: Mean squared error of $k$-step ahead predictions for test data. The black line shows the model with $\gamma = 1.0$, $n_{\hat{x}} = 8$, and IMU. Other $10$ lines represent models trained with $\gamma = 0.9$ for $n_{\hat{x}} \in \{2, 4, 6, 8, 10\}$, with and without IMU. The solid and dotted lines correspond to the results with and without IMU, respectively.
• Figure 4: Model prediction results at $t=20s$ in the test dataset. The red dotted line indicates the current time. The black solid line shows the past actual output, the black dotted line shows the future actual output, and the blue line shows the predicted future output.
• Figure 5: Time series of tip position coordinates during star-shaped trajectory tracking. The red line represents the reference tip position, which also corresponds to the input of inverse kinematics-based feedforward control (IK-based FF) after conversion to tip coordinates via forward kinematics. The black and blue lines show the measured tip positions under NMPC control and IK-based FF, respectively. The green line depicts the control input generated by NMPC, expressed in tip coordinates using forward kinematics. The tip position MSE under NMPC control was significantly lower at $(1.3mm)^2$, compared to $(11.5mm)^2$ with IK-based FF control.

Theorems & Definitions (8)

• Remark 1
• Definition 1: uniform $k$-observability moraal1995observer
• Theorem 1
• proof
• Remark 2
• Theorem 2
• proof
• Remark 3