Abstract

Traffic problems such as pollution and congestion are becoming ever more serious in urban areas. A potential solution to these problems may be the development of narrow vehicles, which occupy less space and produce fewer gas emissions. There has been increasing interest in these types of underactuated mechanical systems, such as Mobile Wheeled Inverted Pendulum (MWIP) models, which are widely used in the field of autonomous robotics and intelligent narrow vehicles. A novel structure based on an MWIP and a movable seat above called the UW-Car is investigated in this study. Dynamic models of this underactuated vehicle running on flat ground and in rough terrain are derived using Lagrange’s motion equation. Based on the models and the Terminal Sliding Mode Control (TSMC) method, two terminal sliding mode controllers were designed for the velocity and braking control of a UW-Car. The first one is for heading speed to a set-point while keeping the body upright and the seat in some fixed position. The second one is a switching sliding mode controller made up of three terminal sliding mode controllers. By using the proposed controller, a UW-Car can move at a desired velocity while keeping the seat always upright. To solve the problem of obtaining quick acceleration performance in a UW-Car, a control method combining trajectory generation and dynamics canceling inputs is proposed. Using this method, the UW-Car can achieve high acceleration while keeping its body upright at all times. All the proposed theoretical results are finally demonstrated through numerical simulations.

Keywords

Mobile-wheeled inverted pendulum; narrow vehicle; robust control; sliding mode control; stability; underactuated system; trajectory generation

1.1 Introduction

1.2 Modeling of the UW-Car

1.2.1 Dynamic Model on Flat Ground

1.2.2 Analysis of Equilibria in Set-Point Velocity Control

1.2.3 Dynamic Model in Rough Terrain

1.3 Velocity Control Using a Sliding Mode Approach

1.3.1 Velocity Control of a UW-Car System on Flat Ground

1.3.2 Optimal Braking Controller Designed Using Sliding Mode Control

1.3.3 Velocity Control of a UW-Car System in Rough Terrain

1.4 Stabilization of an Inverted Pendulum Cart by Consistent Trajectories in Acceleration Behavior

1.4.1 Motivation

1.4.2 Feedback Control System

1.4.3 Desired Trajectory Generation

1.4.4 Control Method Based on Desired Trajectory of Acceleration

1.5 Simulation Study

1.5.1 Set-Point Velocity Control Simulation on Flat Ground

1.5.2 Optimal Braking Control Simulation on Flat Ground

1.5.3 Set-Point Velocity Control Simulation in Rough Terrain

1.5.4 Consistent Trajectories in Acceleration Behavior

1.5.5 Dynamics Canceling Inputs

1.6 Conclusion

Appendix

References

1.1 Introduction

Parking, pollution, and congestion problems caused by cars in urban areas have made life uncomfortable and inconvenient. To improve living conditions, development of an intelligent, self-balanced, and less polluting narrow vehicle might offer a good opportunity. In terms of this idea, many autonomous robots and intelligent vehicles have been designed based on Mobile Wheeled Inverted Pendulum (MWIP) models [1–7], such as PMP [1], iBot [2], Segway [3], and so on.

The MWIP models have attracted much attention in the field of control theory because of the nonlinear and underactuated with inherent unstable dynamics. Many previous attempts used linear [8,9] or feedback linearization methods [10–13] for modeling and control. These rely on a rather precise description of nonlinear functions and show a lack of robustness to model errors and external disturbances.

There are also some other control methods implemented with MWIP models. Lin et al. [6] adopted adaptive control for self-balancing and yaw motion control of two-wheeled mobile vehicles. The nested saturation control design technique is applied to derive a control law for two-wheeled vehicles [7]. Adaptive robust dynamic balance and motion control are utilized to handle the parametric and functional uncertainties [14]. Jung and Kim [15] presented a method for online learning and control of an MWIP by using neural networks.

Sliding Mode Control (SMC) might be a comparatively appropriate approach to deal with uncertain MWIP systems because SMC is less sensitive to the parameter variations and noise disturbances. It has been proved that SMC algorithms can robustly stabilize a class of underactuated mechanical systems such as a mobile robot [16]. Park et al. [17] proposed an adaptive neural SMC method for trajectory tracking of non-holonomic wheeled mobile robots with model uncertainties and external disturbances. We proposed a velocity control method for the MWIP based on sliding mode and a novel sliding surface [18]. Ashrafiuon and Erwin [19] proposed an SMC approach for underactuated multibody systems. Tsai et al. [20] proposed an adaptive sliding mode controller to a hierarchical tracking control in triwheeled mobile robots. Terminal Sliding Mode Control (TSMC) of finite time mechanisms is an example of a variable structure control idea whose formation and development is based on introducing a nonlinear function into sliding hyperplanes. Compared to a linear sliding mode surface, terminal sliding mode has no switching control term and the chattering can be effectively alleviated [21]. On the other hand, TSMC can improve the transient performance substantially. TSMC has already been used successfully in control applications [22–26]. Compared to conventional SMC, TSMC provides faster, finite time convergence, and higher control precision. Nonlinear terminal sliding mode surface functions such as cubic polynomials [26] can also be applied.

While the MWIP system has been successfully applied in many fields, there is still much room for improvement. For instance, drivers can only stand on the Segway vehicles during driving, which is not comfortable for a long-term operation. Another deficiency of Segway is that the body will not always be upright during its operation. To overcome these shortcomings, a new narrow vehicle called the UW-Car is introduced in this study. The novel structure includes an MWIP base and a movable seat driven by a linear motor along the straight direction of motion. The adjustable seat can guarantee the vehicle body will always be upright during driving, which is discussed further in the subsequent sections. The mechanism of a UW-Car is shown in Figure 1.1.

It is well known that the brake system is one of the most important parts relating to the safety of vehicles, the main purpose of which is to reduce the speed or stop driving, or keep the stopped vehicle at rest. In the driving procedure, in order to keep a safe distance between vehicles, precise control of the braking procedure is particularly important. Therefore, studying the braking of mobile robots based on the MWIP structure is of great significance from both the practical and theoretical points of view. Kidane et al. [27] proposed a tilt brake algorithm of a narrow commuter vehicle that was verified for different low-speed maneuvers. We propose an optimal braking strategy for the UW-Car, which can guarantee that an optimal braking period is obtained and the vehicle’s body is always upright during the braking process.

For the UW-Car, achieving a high acceleration performance is difficult due to its non-holonomic characteristic. Conventional methods may realize quick acceleration while bringing unsuitable vibrations at the start and end of acceleration [28]. To solve this problem, we proposed a control method using the desired trajectory of acceleration and dynamics canceling inputs so as to provide the high acceleration performance highlighted in this chapter.

1.2 Modeling of the UW-Car