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

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.

💡 Research Summary

The paper addresses two persistent challenges in nonlinear model predictive control (NMPC): (1) obtaining accurate models of highly nonlinear dynamics, and (2) measuring variables that are directly tied to the control objective but are impractical to sense in real‑time due to cost or integration constraints. To tackle both issues simultaneously, the authors introduce the Predictive Virtual Sensor Identification (PVSID) framework. During a dedicated modeling phase, expensive high‑fidelity sensors (e.g., a motion‑capture system) are temporarily installed to collect ground‑truth data for the “unmeasurable” output w (e.g., the tip position of a robotic arm). Simultaneously, low‑cost sensors that remain available during operation (joint angles, IMU data) constitute the measurable output y.

PVSID formulates a joint identification problem: a state estimator E_φ maps a short history of inputs u_p and measurable outputs y_p to an internal state estimate \hat{x}_t, while an output predictor P_θ takes \hat{x}_t and a future input sequence u_f to directly predict the multi‑step horizon of w (i.e., w_f). Unlike recursive models, both networks produce the entire prediction horizon in a single forward pass, which dramatically reduces the computational burden of Jacobian evaluation required by NMPC. The loss function incorporates a discount factor γ∈(0,1] that weights near‑future prediction errors more heavily, reflecting the fact that NMPC only uses a limited prediction horizon.

Theoretical guarantees are provided under two key assumptions: (i) uniform h_p‑observability for the measurable output and uniform h_f‑observability for the unmeasurable output, and (ii) the estimated state dimension is at least as large as the true state dimension. Under these conditions, Theorem 1 proves the existence of an estimator‑predictor pair, while Theorem 2 shows that, assuming “perfect training” (i.e., the neural networks can represent the true mappings), the learned pair reproduces the exact system behavior.

The experimental platform is a 2‑DoF direct‑drive robotic arm built from LEGO components, actuated by two Roller‑485 motors in position‑control mode, and equipped with an MPU6886 IMU. During model construction, an OptiTrack motion‑capture system records the tip position w, while joint angles and IMU data constitute y. The collected dataset is segmented into past windows (length h_p) and future windows (length h_f) and used to train the two neural networks.

For control, at each 50 Hz sampling instant the estimator E_φ produces \hat{x}_t, and the predictor P_θ generates a multi‑step forecast of w_f given a candidate future input sequence \hat{u}_f. The NMPC optimization minimizes a quadratic cost L( \hat{u}_f, \hat{w}_f ) = ½‖ℓ( \hat{u}_f )‖², where ℓ encodes trajectory‑tracking objectives. The problem is solved with a Levenberg‑Marquardt algorithm, leveraging automatic differentiation for fast Jacobian computation. The optimizer is warm‑started with the previous solution, ensuring convergence within the real‑time budget.

Experimental results show that the NMPC using the PVSID‑identified model tracks a prescribed tip‑position trajectory with root‑mean‑square error comparable to a baseline NMPC that directly uses motion‑capture measurements (≈ 2 mm RMSE). Control inputs remain smooth, and the system operates without the high‑cost motion‑capture hardware after the identification phase. This demonstrates that a virtual sensor learned from temporary high‑precision data can replace expensive instrumentation while preserving NMPC performance.

Key contributions of the work include: (1) a systematic framework for creating virtual sensors from temporary high‑cost measurements, (2) a multi‑step, non‑recursive neural network architecture tailored for NMPC, (3) a temporally weighted loss that prioritizes near‑future accuracy, and (4) a thorough experimental validation on a challenging direct‑drive robot arm with significant nonlinear coupling. Limitations are acknowledged: the approach requires sufficient excitation data for reliable identification, the “perfect training” assumption is idealized, and the current formulation omits hard constraints that are common in industrial applications. Future research directions suggested are extensions to higher‑DOF manipulators, multi‑robot coordination, and hybrid schemes that combine PVSID with reinforcement learning to further reduce data requirements and improve robustness.