Adaptive Control of Underactuated Mechanical Systems

The following sections are included:

• Dedication
• Preface
• Contents

Underactuated systems are mechanical systems with fewer actuators than the system degrees of freedom, i.e., some degrees of freedoms are not directly driven by actuators (Spong, 1997; Liu and Yu, 2013). Many real life systems are underactuated. Examples include the road vehicles, mobile robots, underwater vehicles, surface vessels, helicopters, and some robot manipulators. Since there are fewer actuators available to drive the system, controllability of underactuated systems is very important. To have an effective control, this book considers controllable underactuated systems only.

Various background materials useful in this book are reviewed in this chapter. Most of the results are presented without proof since they can be found in most of the related textbooks. In the first section, we review some mathematical background such as the notions of various spaces, the best approximation problems in the Hilbert space, the concept of orthogonal functions and their applications, the definitions of normed function spaces, and representations of functions, vectors and matrices in function approximation applications. The second section reviews the Lyapunov stability theory which includes the definition of stabilities of equilibrium points in the sense of Lyapunov for autonomous and non-autonomous systems, Lyapunov stability theorems, invariant set theorem, and Barbalat's lemma. In section 2.3, some notions from differential geometry are introduced to facilitate the understanding of the Frobenius theorem which is very useful in concluding solvability for a set of partial differential equations. This theorem plays an important role in chapter 3 to determine existence of solution for decoupling of underactuated dynamics. Some preliminaries in control theories are also given in this chapter as the background for the theoretical development introduced in the later chapters. The concept of sliding control is provided in section 2.4 and some designs of adaptive strategies are presented in section 2.5. Robustness of the adaptive loop is reviewed in section 2.6 with traditional schemes such as dead-zone and σ-modification. Section 2.7 introduces the concept of general uncertainties. Due to the limitations in the traditional robust control and conventional adaptive design, it is challenging to control a system with general uncertainties. In section 2.8, we introduce the function approximation based adaptive controller for dealing with systems containing general uncertainties. When the uncertainties enter the system dynamics in a mismatched fashion, few strategies are feasible. The backstepping procedure is well-known to be effective in coping with mismatched uncertainties. We review the design method of the backstepping procedure in section 2.9. A backstepping-like design called multiple-surface sliding control is derived in section 2.10 which is the main method in this book to cover the general uncertainties entering the system dynamics in a mismatched fashion.

A coordinate transformation is introduced in this chapter to decouple the control input so that an underactuated system can be represented into a special cascade form. The dynamic equations for mechanical systems are reviewed in section 3.1, and the underactuated system is formally defined in section 3.2. The collocated partial feedback linearization is used in section 3.3 to linearize the actuated part of the system into a double integrator such that the coordinate transformation can be applied. We firstly introduce the coordinate transformation for a general underactuated system in section 3.4. Since a set of partial differential equations are to be solved for obtaining the diffeomorphism, to find the solution is usually not easy even though they are proved to be completely integrable. Explicit transformations for 2-DOF systems are given in section 3.5 to facilitate the formation of the special cascade form. The decoupling of high-order systems is introduced in section 3.6.

In this chapter, an adaptive multiple-surface sliding controller (AMSSC) based on the function approximation techniques will be derived to stabilize a class of uncertain underactuated systems. In section 4.1, we clarify the control problem by investigating the system models obtained in chapter 3 more closely. The admissible uncertainties are identified in the X-space and carried to the Z-space without disturbing the coordinate transformation. An AMSSC is given in section 4.2 for an n-th order system in the cascade form represented in the Z-space. It is proved that if the approximation error is ignorable, the asymptotic stability of the output error and boundedness of the internal signals can be obtained. When considering the approximation error, the output error is guaranteed to be uniformly ultimately bounded, and the transient response is bounded by an exponential function. A summary is given in section 4.3 to combine the results from chapter 3 to chapter 4.

A cart pole system consists of an inverted pendulum mounted on a motor driven cart as shown in Fig. 5.1. It is underactuated because both the cart position x and the pendulum angle θ are controlled by the control u on the cart. This system has been widely studied with various control strategies. Many undergraduate control textbooks, e.g., Ogata (1997), use this system as a benchmark problem. They linearize the system dynamics around the vertical position of the pendulum firstly, and then verify controllability of the system based on the linearized model. A state feedback controller is designed to stabilize the system under the assumption that the angle of the pendulum is small to ensure validity of the linearization. To have a wider range of operation, a nonlinear control scheme is needed. If the system contains uncertainties, some sophisticated controllers have to be designed to give desired performance…

Overhead crane systems (Fig. 6.1) are widely used in the industrial environment to carry heavy loads. The trolley is required to carry a payload connected with a rope to travel a finite distance as fast as possible without overshoot when arriving at the destination. To meet safety requirements, the movement of the payload should be performed with small oscillations. Sometimes the positioning accuracy of the trolley when reaching the destination is also very important. In many situations the hoisting mechanism must lift and lower payloads to avoid obstacles during movement, which may excite oscillations if not properly controlled. In some applications, it is preferable to perform loading/unloading operations during the movement of the crane which will also induce payload swing if not effectively treated. In addition, in some accidents such as in the case when payloads drop down, we would like the controller to be robust enough to ensure stability of the operation. Therefore, consideration of varying payload is a nontrivial attempt for a practical control design in industry…

The TORA (Translational Oscillator with Rotational Actuator) system is an undamped mechanical oscillator M controlled by the rotation of a pendulum represented by the mass m as shown in Fig. 7.1. It was firstly studied by Bernstein and co-workers (Wan et al., 1994; Bupp et al., 1994) and various control strategies were reported thereafter in the literature. Since the oscillator is undamped, the open-loop trajectory of M in response to a non-zero initial displacement will exhibit a sustained oscillation. Proper control of the pendulum motion may provide effective intervention to the oscillation. Most of the strategies are developed to regulate both the displacement x of the oscillator and the angle θ of the pendulum to the origin; therefore, it is an underactuated system…

The rotary inverted pendulum as shown in Fig. 8.1 was firstly introduced by Furuta (Furuta et al., 1991). The arm with length L is driven by the motor torque u. The rotation of the arm can then provide power to swing the pendulum which is similar to the traditional inverted pendulum. Since there is only one input to control both the arm and the pendulum, the rotary inverted pendulum is an underactuated system. Most of the controls for the rotary inverted pendulum fall into one of the several categories (Carlos et al., 2010). For example, some considered the problem of stabilizing the pendulum around the unstable vertical position (Carlos et al., 2010; Nair, 2002; Acosta, 2010; Olfati-Saber, 1999; Turker et al., 2012). Some swung the pendulum from its hanging position to its upright vertical position (Gordillo et al., 2003; Hera et al., 2009; Fantoni et al., 2002; Astrom et al., 2008; Sukontanakarn et al., 2009). Some others tried to create oscillations around its unstable vertical position (Shiriaev et al., 2007; Aguilar et al., 2009; Freidovich et al., 2009)…

In this chapter, we would like to consider the vibration suppression of a primary system which is modeled as a one-dimensional mechanical system composed of impedance elements as shown in Fig. 9.1. The primary system exhibits a sustained oscillation due to the external excitation f…

A pendubot is a 2-link robot arm (Fig. 10.1) whose first link is driven by a motor while the second link is not actuated. Since there is only one control effort to regulate the motion of the two links, it is underactuated. Most of the control problems are to swing up the two links to the upright position. Spong and Block (1995) described the dynamics and control of a pendubot. Fantoni et al. (1999, 2000) presented an energy based controller design strategy for stabilizing a pendubot. A hybrid design was developed for the pendubot by Zhang and Tarn (2002). Alvarez and Gallegos (2004) proposed a procedure to analyze the complex behavior of the pendubot and developed an unstable periodic orbits stabilizing controller. Li et al. (2004) presented an acrobatic control of pendubot with a fuzzy controller to keep the first link swinging periodically while the second link maintains standing vertically. Orlov et al. (2005) constructed a sliding-mode-like control design for underactuated systems with applications to the stabilization of a pendubot around the upright position to improve system robustness. Qian et al. (2007) suggested a hierarchical sliding mode control for the swing up of a pendubot. Albahkali et al. (2009) used the impulse momentum approach to swing up a pendubot. Shoji et al. (2010) simulated dexterous human throwing motion by the control of a pendubot supported by passive elements and some control via unstable zero dynamics…

The following sections are included:

• Bibliography
• Index