## Exact Feedback Linearization

It is well known in robotics that if the number of generalized coordinates equals the number of control inputs, that is, n m, the system kinematics or dynamics can be transformed into a linear system with the use of a nonlinear static state feedback 14 . The linearity is displayed by the system equations by performing some sort of coordinate transformation in the state space. For exact linearization of the nonlinear systems, outputs are chosen to which a desired behavior is assigned. Two types...

## Approximate Linearization

For approximate linearization of the system (Equation 2.7) we take the desired state trajectory as qd(t) and the input trajectory as vd(t). It can be easily seen that the linearization about a smooth trajectory results in a linear time-varying system. The system can easily be shown to satisfy the controllability condition, that is, the controllability Grammian is nonsingular 21 as long as the input reference trajectory is persistent (does not come to a stop). Thus, it implies that we can...

## Trajectory tracking and controller design for the chained form

The system is supposed to track a given (desired) Cartesian trajectory. The problem is to regulate both the vehicle's position and orientation with respect to that of a reference system, the trajectory of which is parameterized by the variable t. The goal will be achieved using the feedback control law with the following control schemes Full-state feedback using approximate linearization Feedback linearization using input-output linearization or full-state linearization Before going for the...

## Vehicle kinematics fundamentals

There are essentially six degrees of freedom that the vehicle has three that indicate the position, and the other three that indicate the orientation. These six degrees of freedom identify the state of the vehicle. We consider the state of the vehicle with respect to some general frame of reference for the world (Figure 1.16). Although there are six degrees of freedom for the vehicle to be placed with respect to position and orientation, there are typically only four variables that can be...

## Nonholonomic Constraints

System constraints on the mechanical systems whose expression involves generalized coordinates and velocities are known as kinematic constraints of the system, which are also known as the nonholonomic constraints. These are of the following form a,(q,q) 0, i 1,2, ,k < n (2.1) where q is the generalized coordinate vector or the state vector. q e M e Rn, where n is the dimension of the configuration space M, to which the vector q belongs. These will limit the admissible motions of the system by...

## Examples Of Nonholonomic Systems

The simplest example of a nonholonomic system can be a wheel that rolls on a lane surface, such as a unicycle. The constraints here arise due to the roll without a slip condition. The configuration or the generalized coordinate vector is q (x, y, 8). The coordinates x and y are the position coordinates of the wheel, and 8 is the angle that the wheel makes with the x axis. The unicycle is shown in Figure 2.2. The constraint here is that the wheel cannot slip in the lateral direction. The...

## Robust control using the kinematic model

In this section we formulate the uncertain control model and present robust control design for the kinematic model derived in Chapter 3. The problem is to design a state feedback control for the problem of motion (point-to-point) stabilization. The control design task is to ensure the global asymptotic stability of the vector q(t) of the closed-loop system irrespective of uncertain elements. The kinematic model is given by Equation 3.13 as Here the vector q p n T x, y, z, 0, y T is the six...

## Propellers

Propellers are extremely important for robotic underwater vehicles. They provide the thrust for forward motion as well as for lift. Figure 1.12 shows a three-blade propeller. Propellers can come in many blade configurations, such as two blade, three blade, and four blade. Propellers produce forward thrust by accelerating fluid back. The amount of fluid forced back depends on the speed of rotation of the propeller as well as at its pitch angle. With a small pitch angle, the amount of fluid...

## Control using exact feedback linearization via state and input transformations

In this section the use of nonlinear feedback design is used for the global stabilization of the tracking error associated with the trajectory to zero. For nonlinear systems two types of exact linearization methods are generally used. One is the full-state feedback transformation of the differential equations of the system into the linear system. Another is the input-output linearization, which results in the input-output differential map being linear. Both the feedback problems can be solved...

## Dqt h v q dt

The first equation is the dynamic model of the system, and the second is the kinematic model. The vector v t is the velocity vector and q t is the position vector in the fixed frame coordinates. In order to design the feedback control for point stabilization of the dynamic model, the control methodology is nonlinear feedback linearization. The design technique adopted for the dynamic control is backstep-ping. The book also discusses the feedback control for these models in the presence of...