**1. Introduction**

In practice, many control problems involve the "underactuated" behavior of mechanical systems. In underactuated systems, the number of equipped actuators is less than that of the controlled variables. That is, actuators do not directly control several degrees of freedom. For example, we consider a tracking control problem for a marine vessel (**Figure 1**). In many cases, ships are equipped with either two independent aft thrusters or one main aft thruster and one rudder, without any bow or side thruster. Therefore, no sway control force acting on the ship is assumed. From the aforementioned condition, Lefeber et al. [1] investigated tracking control for underactuated ships in which three state variables, namely, surge, sway, and yaw, are

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

driven by only two inputs: surge force and yaw torque. We can find many underactuated systems in engineering, such as mobile robots, aircraft, and gantry cranes, among others.

**Figure 1.** Tracking control of an underactuated ship [1].

According to the study of Tedrake [2], a mechanical system that can be described mathematically by

$$\mathbf{M(q)}\ddot{\mathbf{q}} + \mathbf{C(q, \dot{q})}\dot{\mathbf{q}} + \mathbf{G(q)} = \mathbf{B(q)}\mathbf{u} \tag{1}$$

is regarded an underactuated system if the rank of matrix **B**(**q**) is less than the dimension of vector **q**, that is,

$$\text{rank}\left(\mathbf{B}\left(\mathbf{q}\right)\right) < \text{dim}\left(\mathbf{q}\right). \tag{2}$$

Otherwise, system (1) has a "fully actuated" property in configuration , ˙ , if it can control instantaneous acceleration in an arbitrary direction in **q**.

$$\text{rank}\left(\mathbf{B}\left(\mathbf{q}\right)\right) = \text{dim}\left(\mathbf{q}\right) \tag{3}$$

Unlike modern control techniques, such as fuzzy logic and neural networks, traditional control methods require knowing the physical properties of a system, which are generally governed by its mathematical model. For dynamical systems, a mathematical model is constructed based on mechanics principles, such as Newton's law, Lagrange equation, Lagrange multiplier method, Euler-Lagrange methodology, and so on. In mechanical systems with multiple degrees of freedom, system dynamics will comprise a set of second-order differential equations (1) in terms of displacements **q**, velocities ˙ , and time *t*. From this point of view, dynamical systems can be classified according to the type of mathematical model. *Partial differential equations* are used to describe *distributed systems* mathematically, whereas *ordinary differential equations* govern the motions of *discrete systems*.

**Figure 2.** Cart-pole system [3].

(1)

(2)

(3)

driven by only two inputs: surge force and yaw torque. We can find many underactuated systems in engineering, such as mobile robots, aircraft, and gantry cranes, among others.

According to the study of Tedrake [2], a mechanical system that can be described mathemat-

is regarded an underactuated system if the rank of matrix **B**(**q**) is less than the dimension of

Otherwise, system (1) has a "fully actuated" property in configuration , ˙ , if it can control

Unlike modern control techniques, such as fuzzy logic and neural networks, traditional control methods require knowing the physical properties of a system, which are generally governed by its mathematical model. For dynamical systems, a mathematical model is constructed based on mechanics principles, such as Newton's law, Lagrange equation, Lagrange multiplier method, Euler-Lagrange methodology, and so on. In mechanical systems with multiple degrees of freedom, system dynamics will comprise a set of second-order differential equations (1) in terms of displacements **q**, velocities ˙ , and time *t*. From this point of view, dynamical systems can be classified according to the type of mathematical model.

**Figure 1.** Tracking control of an underactuated ship [1].

244 Nonlinear Systems - Design, Analysis, Estimation and Control

instantaneous acceleration in an arbitrary direction in **q**.

ically by

vector **q**, that is,

Most realistic systems exhibit nonlinear behavior. A nonlinear system is generally described by nonlinear differential equations. Nonlinearities appear in a mathematical model because of its nonlinear components or geometric relationship. For example, a system that consists of an inverted pendulum mounted on a cart, as shown in **Figure 2**, has the following equations of motion:

$$(m\_c + m\_p)\ddot{\mathbf{x}} + m\_p l \cos\theta \ddot{\theta} - m\_c l \dot{\theta}^2 \sin\theta = \mathbf{u} \,\,\,\,\tag{4}$$

(5)

The nonlinearities of the aforementioned dynamics originate from geometric constraint.

$$f\left(\mathbf{x}, l, \sin\theta, \cos\theta\right) = 0\tag{6}$$

The other example is a spring-damper system, which is illustrated in **Figure 3**. The force of nonlinear spring

$$F = k\_1 \mathbf{x} + k\_2 \mathbf{x}^3 \tag{7}$$

leads to the nonlinear modeling of the system, as follows:

$$
\dot{\mathbf{x}}\ \dot{\mathbf{x}}\dot{\mathbf{x}} + b\dot{\mathbf{x}} + k\_1 \mathbf{x} + k\_2 \mathbf{x}^3 = \mathbf{u} \ . \tag{8}
$$

The nonlinear feedback technique, also called feedback linearization, is a representative method for controlling nonlinear systems. The main concept of feedback linearization is to transfer the original nonlinear system algebraically into the linear system by inserting equivalent inputs to suppress the nonlinearities of the former. The feedback linearization control of fully actuated systems has been discussed in several well-known textbooks [4, 5] in which this theory has been completely developed. Previous studies have pointed out that fully actuated systems are feedback linearizable through nonlinear feedback [6, 7]. In this chapter, we introduce the feedback linearization control for a class of multiple-input and multipleoutput (MIMO) underactuated systems. The analysis process is conducted using an algebra foundation in which the mathematical model is simplified through matrix equations.

**Figure 3.** Mechanical system with a viscous damper and a nonlinear spring [3].

First, the mathematical model of underactuated mechanical systems is separated into two subsystems: actuated states and unactuated states. Then, we design a controller in which nonlinear feedback is partly applied to both actuated and unactuated dynamics. Subsequently, actuated submodel is "linearized" using a nonlinear feedback method; thus, the unactuated dynamics is regarded as internal model. Seeing actuated states as system outputs, a nonlinear control law is designed to drive state trajectories to the references. However, this controller does not promise the stability of unactuated states. Therefore, its structure should be adjusted to guarantee the stability of both actuated and unactuated states based on the nonlinear feedback of all system states. The control scheme now exhibits the linear combination of two components that are distinctly acquired from the nonlinear feedback of both the actuated and unactuated submodels.

In comparison with traditional controllers, such as the proportional-integral-derivative (PID) controller, partial feedback linearization (PFL) exhibits several advantages. In the PID controller design, most of the nonlinear factors of a system are not mentioned. By contrast, in the design of PFL, all the nonlinearities of a system considered in the system dynamics are entirely vanished by the PFL controller. However, the PFL approach requires a precise model to achieve good control action. Additionally, the approach is not convenient in systems with uncertain parameters.

As an enhancement of Tuan et al.'s [8] paper, where PFL was applied for three-dimensional (3D) overhead crane, we introduce the PFL theory in the generalized form for a class of nonlinear underactuated mechanical systems. The outline of this chapter is as follows. Section 1 introduces the chapter. Section 2 presents the general form of the mathematical modeling of an underactuated mechanical system. Section 3 constructs a nonlinear controller based on the partial nonlinear feedback technique. Section 4 discusses system stability. Section 5 provides an example to illustrate the proposed theory. Finally, Section 6 provides the conclusion of the chapter.
