Asymptotic trajectory tracking of autonomous bicycles via backstepping and optimal control

Zhong-Ping Jiang, professor of electrical and computer engineering (ECE) and member of the C2SMART transportation research center at NYU Tandon, directed this research. Leilei Cui, a Ph.D. student in the ECE Department is lead author. Zhengyou Zhang and Shuai Wang from Tencent are co-authors.

This paper studies the trajectory tracking and balance control problem for an autonomous bicycle — one that is ridden like a normal bicycle before automatically traveling by itself to the next user — that is a non-minimum phase, strongly nonlinear system.

As compared with most existing methods dealing only with approximate trajectory tracking, this paper solves a longstanding open problem in bicycle control: how to develop a constructive design to achieve asymptotic trajectory tracking with balance. The crucial strategy is to view the controlled bicycle dynamics from an interconnected system perspective.

More specifically, the nonlinear dynamics of the autonomous bicycle is decomposed into two interconnected subsystems: a tracking subsystem and a balancing subsystem. For the tracking subsystem, the popular backstepping approach is applied to determine the propulsive force of the bicycle. For the balancing subsystem, optimal control is applied to determine the steering angular velocity of the handlebar in order to balance the bicycle and align the bicycle with the desired yaw angle. In order  to tackle the strong coupling between the tracking and the balancing systems, the small-gain technique is applied for the first time to prove the asymptotic stability of the closed-loop bicycle system. Finally, the efficacy of the proposed exact trajectory tracking control methodology is validated by numerical simulations (see the video).  

"Our contribution to this field is principally at the level of new theoretical development," said Jiang, adding that the key challenge is in the bicycle's inherent instability and more degrees of freedom than the number of controllers. "Although the bicycle looks simple, it is much more difficult to control than driving a car because riding a bike needs to simultaneously track a trajectory and balance the body of the bike. So a new theory is needed for the design of an AI-based, universal controller." He said the work holds great potential for developing control architectures for complex systems beyond bicycles.

The work was done under the aegis of the Control and Network (CAN) Lab led by Jiang, which consists of about 10 people and focuses on the development of fundamental principles and tools for the stability analysis and control of nonlinear dynamical networks, with applications to information, mechanical and biological systems.

The research was funded by the National Science Foundation (Grant number 10.13039/100000001).