Modified Dual Hierarchical Terminal Sliding Mode Control Design for Two-Wheeled Self-Balancing Robot
Abstract
1. Introduction
- A hierarchical sliding mode control (HSMC) scheme is combined with terminal sliding mode control (TSMC) to address the underactuated control problem of the two-wheeled self-balancing robot (TWSBR), resulting in a dual-terminal sliding mode control (DTSMC) law. Furthermore, a modified dual hierarchical terminal sliding mode control (MDHTSMC) scheme is designed based on the duality concept, with stability rigorously verified through Lyapunov theory.
- A modified dual hierarchical terminal sliding mode control (MDHTSMC) scheme is proposed for the TWSBR nonlinear system, accounting for disturbances and uncertainties. Using Lyapunov theory, finite-time convergence on the newly defined MDHTSMC surface is proven, and the arrival and sliding times are explicitly calculated.
- The Jellyfish Search Optimization (JSO) algorithm is employed to minimize the integral of the time-weighted absolute error (ITAE) by adjusting the x and parameters of the TWSBR, achieving optimal control parameters. These parameters are subsequently fine-tuned for enhanced optimization.
2. Preliminary
2.1. Dynamic Models
2.2. Actuated and Underactuated Subsystems
2.2.1. -Subsystem
2.2.2. -Subsystem
3. Modified Dual Hierarchical Terminal SMC
3.1. Dynamic Model
3.2. Hierarchical Terminal Sliding Mode Control (HTSMC)
3.3. Dual Hierarchical Terminal SMC (DHTSMC)
3.4. MDHTSM Controller Design for TWSBR
4. Simulation Result
4.1. Optimization Algorithm
4.2. Parameter Tuning
- The performance of each sliding mode controller under ideal, disturbance-free conditions;
- The performance of each sliding mode controller in the presence of disturbances .
4.2.1. Case 1: Ideal State
4.2.2. Case 2: With Disturbance
4.3. Indicator Evaluation
5. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
TWSBR | Two-Wheeled Self-Balancing Robot |
SMC | Sliding Mode Control |
TSMC | Terminal Sliding Mode Control |
HSMC | Hierarchical Sliding Mode Control |
HTSMC | Hierarchical Terminal Sliding Mode Control |
DHTSMC | Dual Hierarchical Terminal Sliding Mode Control |
MDHTSMC | Modified Dual Hierarchical Terminal Sliding Mode Control |
JSO | Jellyfish Search Optimization |
ITAE | Integral of Time-weighted Absolute Error |
LQR | Linear Quadratic Regulator |
PD | Proportional-Derivative |
Appendix A
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Notation | Definition |
---|---|
Torques acting on the left and right wheels provided by wheel motors | |
Interacting forces between the left and right wheels and the chassis | |
Friction forces acting on the left and right wheels | |
External forces acting on the left and right wheels | |
Rotational angles of the left and right wheels | |
Displacements of the left and right wheels along the x-axis | |
Tilt angle of the vehicle body | |
Rotational angle of the vehicle | |
x | Displacement of the vehicle along the direction of the longitudinal velocity |
v | Longitudinal velocity of the vehicle |
m | Mass of the inverted pendulum |
M | Mass of the chassis |
Mass of the wheels | |
R | Radius of the wheels |
l | Distance between the body center of gravity and the wheel axis |
D | Distance between the two wheels along the axle center |
Current position of the vehicle on the plane | |
Interacting force between the pendulum and the chassis on the x-axis |
Symbol | Description |
---|---|
State vector | |
Nonlinear dynamic function in the ith subsystem | |
Control gain function in the ith subsystem, with | |
Bounded disturbance in the ith subsystem, satisfying | |
Control input applied to the ith subsystem | |
, | Tracking errors for pitch angle and yaw/position |
, | Sliding surface variables |
, | Sliding mode control gains |
, | Positive constants defining sliding surface slopes |
, | Reaching law coefficients |
, | Terminal sliding exponents in subsystem i |
Signum function used in sliding mode control |
Disturbance | Controller | ITAE () | Settling Time (s) | Overshoot (%) | Steady-State Error () | Chattering Level |
---|---|---|---|---|---|---|
Without | HSMC | 2.3 | 4.2 | 18 | 0.025 | Severe |
HTSMC | 1.6 | 3.0 | 11 | 0.010 | Moderate | |
MDHTSMC | 0.9 | 1.8 | 4 | 0.000 | Minimal | |
With | HSMC | 3.7 | 5.1 | 22 | 0.035 | Severe |
HTSMC | 2.1 | 3.4 | 14 | 0.015 | Moderate | |
MDHTSMC | 1.2 | 2.2 | 6 | 0.000 | Minimal |
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Zhang, H.; Mohamad Nor, N.; Umar, S.N.H. Modified Dual Hierarchical Terminal Sliding Mode Control Design for Two-Wheeled Self-Balancing Robot. Electronics 2025, 14, 2692. https://doi.org/10.3390/electronics14132692
Zhang H, Mohamad Nor N, Umar SNH. Modified Dual Hierarchical Terminal Sliding Mode Control Design for Two-Wheeled Self-Balancing Robot. Electronics. 2025; 14(13):2692. https://doi.org/10.3390/electronics14132692
Chicago/Turabian StyleZhang, Huaqiang, Norzalilah Mohamad Nor, and Siti Nur Hanisah Umar. 2025. "Modified Dual Hierarchical Terminal Sliding Mode Control Design for Two-Wheeled Self-Balancing Robot" Electronics 14, no. 13: 2692. https://doi.org/10.3390/electronics14132692
APA StyleZhang, H., Mohamad Nor, N., & Umar, S. N. H. (2025). Modified Dual Hierarchical Terminal Sliding Mode Control Design for Two-Wheeled Self-Balancing Robot. Electronics, 14(13), 2692. https://doi.org/10.3390/electronics14132692