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Dynamic Modeling and Stability Analysis of Balancing in Riderless Electric Scooters

Yun-Hao Lin, Alireza Jafari, Yen-Chen Liu

TL;DR

This work tackles balancing a riderless electric scooter when steering and speed vary concurrently. It develops a nonlinear bicycle-like dynamic model using $M\Ddot{\theta} = \tau_{\theta} + U\sin(\theta+\theta_0)$ and analyzes two controllers: a PD controller and a feedback-linearized PD controller, with Lyapunov-based stability results. The PD controller guarantees ultimate boundedness of the roll dynamics, while the feedback-linearized PD controller reduces bound sizes and can achieve asymptotic stability under perfect estimation, with estimation errors yielding smaller but bounded performance degradation. Simulations on a realistic scooter parameter set show the feedback-linearized PD controller outperforms the PD controller, especially under demanding maneuvers, highlighting its practical appeal for riderless autonomous e-scooting; future work aims at experimental validation and integration with sidewalk-level perception and time-to-collision aware path planning.

Abstract

Today, electric scooter is a trendy personal mobility vehicle. The rising demand and opportunities attract ride-share services. A common problem of such services is abandoned e-scooters. An autonomous e-scooter capable of moving to the charging station is a solution. This paper focuses on maintaining balance for these riderless e-scooters. The paper presents a nonlinear model for an e-scooter moving with simultaneously varying speed and steering. A PD and a feedback-linearized PD controller stabilize the model. The stability analysis shows that the controllers are ultimately bounded even with parameter uncertainties and measurement inaccuracy. Simulations on a realistic e-scooter with a general demanding path to follow verify the ultimate boundedness of the controllers. In addition, the feedback-linearized PD controller outperforms the PD controller because it has narrower ultimate bounds. Future work focuses on experiments using a self-balancing mechanism installed on an e-scooter.

Dynamic Modeling and Stability Analysis of Balancing in Riderless Electric Scooters

TL;DR

This work tackles balancing a riderless electric scooter when steering and speed vary concurrently. It develops a nonlinear bicycle-like dynamic model using and analyzes two controllers: a PD controller and a feedback-linearized PD controller, with Lyapunov-based stability results. The PD controller guarantees ultimate boundedness of the roll dynamics, while the feedback-linearized PD controller reduces bound sizes and can achieve asymptotic stability under perfect estimation, with estimation errors yielding smaller but bounded performance degradation. Simulations on a realistic scooter parameter set show the feedback-linearized PD controller outperforms the PD controller, especially under demanding maneuvers, highlighting its practical appeal for riderless autonomous e-scooting; future work aims at experimental validation and integration with sidewalk-level perception and time-to-collision aware path planning.

Abstract

Today, electric scooter is a trendy personal mobility vehicle. The rising demand and opportunities attract ride-share services. A common problem of such services is abandoned e-scooters. An autonomous e-scooter capable of moving to the charging station is a solution. This paper focuses on maintaining balance for these riderless e-scooters. The paper presents a nonlinear model for an e-scooter moving with simultaneously varying speed and steering. A PD and a feedback-linearized PD controller stabilize the model. The stability analysis shows that the controllers are ultimately bounded even with parameter uncertainties and measurement inaccuracy. Simulations on a realistic e-scooter with a general demanding path to follow verify the ultimate boundedness of the controllers. In addition, the feedback-linearized PD controller outperforms the PD controller because it has narrower ultimate bounds. Future work focuses on experiments using a self-balancing mechanism installed on an e-scooter.
Paper Structure (13 sections, 3 theorems, 25 equations, 4 figures, 1 table)

This paper contains 13 sections, 3 theorems, 25 equations, 4 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Dynamic Model
  3. Preliminaries
  4. Problem statement
  5. Discussion on holonomy
  6. System modeling
  7. Controller Design
  8. PD Controller
  9. Feedback-linearized PD Controller
  10. Simulations and discussions
  11. Simulation Setup
  12. Simulation Results
  13. Conclusion and future works

Key Result

Lemma 1

Consider the e-scooter dynamics eq:DynFinal. The PD controller with positive $K_d$ and $K_p$ ensures that $\dot \theta$ is ultimately bounded by ${|{\dot \theta}|}_{max}={U}/{K_d}$.

Figures (4)

  • Figure 1: E-scooter geometry in 3D space. The steering pole is normal to the e-scooter frame.
  • Figure 2: Control block diagram of the self-balancing e-scooter
  • Figure 3: The simulation input: (a) desired path; (b) the desired velocity and the steering angle created by the desired path and designed velocity.
  • Figure 4: The simulation results: (a) the roll angle $\theta$; (b) the roll angle rate $\dot \theta$; (c) the controller action $\tau_\theta$. The legend and the abbreviations apply to all; PD controller with no uncertainties is labeled as "PD"; PD controller with uncertainties is labeled as "PDU"; PD Feedback Linearized controller with no uncertainties is labeled as "PDFL"; PD Feedback Linearized controller with uncertainties is labeled as "PDFLU"; $\left | \theta \right |_{max}$ represents the upper limit of the system's roll angle for PD; $\left | \theta \right |_{max,u}$ represents the upper limit of the system's roll angle for PDFLU.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Corollary 1
  • Remark 3