Modeling PWM-Time-SOC Interaction in a Simulated Robot

1. Problem Statement

Autonomous mobile robots operating in environments with limited or no recharging opportunities face a critical challenge: energy management. Without accurate models to predict battery State of Charge (SOC) depletion, robots risk failing mid-mission.

• The Gap: Traditional approaches rely on simplified linear models (which fail to capture nonlinear battery discharge dynamics) or extensive pre-programmed lookup tables (which lack generalizability across different motion profiles and conditions).
• The Specific Challenge: The relationship between the Pulse Width Modulation (PWM) duty cycle (motor control input), operational time, and battery SOC is inherently nonlinear. This nonlinearity arises from:
  • Motor electrical characteristics (resistance, inductance, back-EMF).
  • Mechanical dynamics (mass, drag, rolling resistance).
  • Battery internal resistance effects (voltage droop under load).
• Goal: Develop a unified, interpretable, and computationally efficient mathematical model that predicts SOC as a function of time ($t$) and PWM duty cycle ($p$) to enable energy-aware path planning.

2. Methodology

The authors propose a physics-informed, data-driven framework that bridges high-fidelity simulation with sparse regression techniques.

A. Simulation Environment

A physics-based simulation of a 4-wheel Arduino robot was constructed to generate training data. The simulation incorporates:

• Electrical Model: DC motor dynamics including resistance ($R$), inductance ($L$), back-EMF ($K_e$), and torque constant ($K_t$).
• Mechanical Model: Newtonian dynamics accounting for mass, aerodynamic drag, rolling resistance, and wheel radius.
• Battery Model: Includes internal resistance ($R_{int}$) and open-circuit voltage ($V_{oc}$) to simulate voltage droop under load.
• Data Generation: The robot was simulated moving forward for 300 seconds at PWM duty cycles ranging from 1% to 100% (in 10% increments). SOC was recorded every 10 seconds, creating a dataset of 310 points across 10 trajectories.

B. Model Discovery: SINDy

The core of the methodology is Sparse Identification of Nonlinear Dynamics (SINDy).

1. Library Construction: A candidate function library ($\Theta$) was built based on physical intuition, including:
   • Polynomial terms in PWM ($p, p^2, p^3$) to capture nonlinear current draw.
   • Logarithmic time terms ($\ln(1+t)$) to model transient startup behavior.
   • Rational interaction terms (e.g., $\frac{t}{p+1}$) to capture coupled effects.
2. Sparse Regression: The system solves a regularized least-squares problem to find a sparse coefficient matrix ($\Xi$) such that $\dot{X} \approx \Theta(X)\Xi$. This identifies the minimal set of terms required to describe the dynamics, enforcing sparsity to prevent overfitting and ensure interpretability.
3. Integration: The identified rate equation ($\frac{dSOC}{dt}$) was integrated to produce the final closed-form SOC model.

3. Key Contributions

The paper makes three primary contributions to energy-aware robotics:

1. Unified Nonlinear SOC Model: Development of a closed-form analytical model, $SOC(t, p)$, that explicitly captures the coupled effects of time and PWM. Unlike lookup tables, this model generalizes across the full range of operating conditions and arbitrary time horizons.
2. Physics-Informed Data-Driven Methodology: A systematic approach that uses physics-based simulation to generate data but relies on SINDy to discover the governing equations. This bridges the gap between purely empirical black-box models and rigid theoretical derivations, achieving high accuracy with interpretable coefficients.
3. Scalable Framework for Deployment: A complete workflow demonstrating how to extend these models to real-world scenarios, including handling varying initial SOC levels and adapting to environment-dependent parameters (e.g., terrain-specific rolling resistance) without full model retraining.

4. Results and Validation

The derived models were validated against held-out simulation data at both high-load (90% PWM) and low-load (40% PWM) conditions.

Derived Models

Three complementary models were identified:

1. General Time-Series Model ($SOC(t, p)$):
   $$SOC(t, p) = a_1 - b_1t - c_1p - d_1p^2 - e_1p^3 - f_1\ln(1+t) - g_1\frac{t}{p+1} + h_1\frac{\ln(1+t)}{1+t}$$
   • Captures the full interaction, including the "hook" (steep initial drop) and steady-state linear depletion.
2. Fixed-Horizon Model ($SOC(p)$): A quadratic function predicting final SOC after a fixed time (e.g., 300s) based solely on PWM.
3. Instantaneous Rate Model ($\frac{\partial SOC}{\partial t}$): Describes the instantaneous depletion rate, highlighting the dominance of cubic PWM terms at high loads.

Performance Metrics

• Average Prediction Error: 0.162%
• Maximum Prediction Error: 0.82% (occurring at $t=0$ during high-load transients).
• Validation:
  • 90% PWM: The model accurately captured the rapid discharge and the initial transient "hook" caused by battery internal resistance.
  • 40% PWM: The model showed excellent agreement with a maximum error of 0.27%, demonstrating robustness in low-load conditions.

5. Significance and Implications

• Energy-Aware Planning: The closed-form nature of the model allows for microsecond-level evaluation, making it suitable for real-time onboard implementation in path planning algorithms (e.g., A*, Model Predictive Control) without the computational overhead of running full physics simulations or accessing large lookup tables.
• Interpretability: The model terms have clear physical meanings (e.g., $p^3$ relates to $I^2R$ losses and drag; $\ln(1+t)$ relates to transient current decay). This provides insights into how specific robot parameters affect energy consumption.
• Generalizability: The framework is not tied to a specific robot platform; by adjusting physical parameters (mass, drag, battery capacity), the model can be adapted to different robots.
• Future Impact: This approach paves the way for adaptive energy management systems that can update coefficients online to account for battery aging, temperature changes, and payload variations, significantly extending the operational autonomy of mobile robots.