According to Shneydor (1998), guidance is defined as: The process for guiding the path of an object towards a given point, which in general may be moving. Also, the father of inertial navigation, Charles Stark Draper, states in (Draper 1971) that: Guidance depends upon fundamental principles and involves devices that are similar for vehicles moving on land, on water, under water, in air, beyond the atmosphere within the gravitational field of earth and in space outside this field, see Fig. 2. Thus, guidance represents a basic methodology concerned with the transient motion behavior associated with the achievement of motion control objectives. The most rich and mature literature on guidance is probably found within the guided missile community. In one of the earliest texts on the subject (Locke 1955), a guided missile is defined as: A space-traversing unmanned vehicle which carries within itself the means for controlling its flight path. Today, most people would probably think about unmanned aerial vehicles (UAVs) when hearing this definition. However, guided missiles have been operational since World War II (Spearman 1978), and thus organized research on guidance theory has been conducted almost as long as organized research on control theory. The continuous progress in missile hardware and software technology has made increasingly advanced guidance concepts feasible for implementation. Today, missile guidance theory encompass a broad spectrum of guidance laws, namely: classical guidance laws; optimal guidance laws; guidance laws based on fuzzy logic and neural network theory; differential-geometric guidance laws; and guidance laws based on differential game theory.

Fig. 2. Fundamental guidance principles apply from subsea to space

As already mentioned, a classical text on missile guidance concepts is (Locke 1955), while more recent work include (Lin 1991), (Shneydor 1998), (Zarchan 2002), (Siouris 2004), and (Yanushevsky 2008). Relevant survey papers include (Pastrick et al. 1981), (Cloutier et al. 1989), (Lin & Su 2000), and (White & Tsourdos 2001). Also, very interesting personal accounts of the guided missile development during and after World War II can be found in (Haeussermann 1981), (Battin 1982), and (Fossier 1984), while MacKenzie (1990) and Westrum (1999) put the development of guided missile technology into a larger perspective. The fundamental nature and diverse applicability of guidance principles can be further illustrated through a couple of examples. In nature, some predators are able to conceal their pursuit of prey by resorting to so-called motion camouflage techniques (Mizutani et al. 2003). They adjust their movement according to their prey so that the prey perceive them as stationary objects in the environment. These predators take advantage of the fact that some creatures detect the lateral motion component relative to the predator-prey line of sight far better than the longitudinal component. Hence, approaching predators can appear stationary to such prey by minimizing the relative lateral motion, only changing in size when closing in for the kill. Interestingly, this behavior can be directly related to the classical guidance laws from the missile literature (Justh & Krishnaprasad 2006). Also, such guidance laws have been successfully applied since the early 1990s to avoid computationally-demanding optimization methods associated with motion planning for robot manipulators operating in dynamic environments (Piccardo & Honderd 1991).

2. Motion control fundamentals

This section reviews some basic motion control concepts, including operating spaces, vehicle actuation properties, motion control scenarios, as well as the motion control hierarchy. It concludes with some preliminaries.

2.1 Operating spaces

It is useful to distinguish between different types of operating spaces when considering vehicle motion control, especially since such characterizations enable purposeful definitions of various motion control scenarios. The two most fundamental operating spaces to consider are the work space and the configuration space.

The work space, also known as the operational space (Sciavicco & Siciliano 2002), represents the physical space (environment) in which a vehicle moves. For a car, the work space is 2-dimensional (planar position), while it is 3-dimensional (spatial position) for an aircraft. Thus, the work space is a position space which is common for all vehicles of the same type. The configuration space, also known as the joint space (Sciavicco & Siciliano 2002), is constituted by the set of variables sufficient to specify all points of a (rigid-body) vehicle in the work space (LaValle 2006). Thus, the configuration of a car is given by its planar position and orientation, while the configuration of an aircraft is given by its spatial position and attitude.

2.2 Vehicle actuation properties

Every variable associated with the configuration of a vehicle is called a degree of freedom (DOF). Hence, a car has 3 degrees of freedom, while an aircraft has 6 degrees of freedom. The type, amount, and distribution of vehicle thrust devices and control surfaces, hereafter commonly referred to as actuators, determine the actuation property of a vehicle. We mainly distinguish between two qualitatively different actuation properties, namely full actuation and underactuation. A fully actuated vehicle is able to independently control all its DOFs simultaneously, while an underactuated vehicle is not. Thus, an underactuated vehicle is generally unable to achieve arbitrary tasks in its configuration space. However, it will be able to achieve tasks in the work space as long as it can freely project its main thrust in this space, e.g., through a combination of thrust and attitude control. In fact, this principle is the mode by which most vehicles that move through a fluid operate, from missiles to ships. Even if these vehicles had the ability to roam the work space with an arbitrary attitude, this option would represent the least energy-efficient alternative.

2.3 Motion control scenarios

In the traditional control literature, motion control scenarios are typically divided into the following categories: point stabilization, trajectory tracking, and path following. More recently, the concept of maneuvering has been added to the fold as a means to bridge the gap between trajectory tracking and path following (Skjetne et al. 2004). These scenarios are often defined by motion control objectives that are given as configuration-space tasks, which are best suited for fully actuated vehicles. Also, the scenarios typically involve desired motion that has been defined apriori in some sense. Little seems to be reported about tracking of target points for which only instantaneous motion information is available.

However, in this work, both apriori and non-apriori scenarios are considered, and all the motion control objectives are given as work-space tasks. Thus, the scenarios cover more broadly, and are also suited for underactuated vehicles. The considered scenarios are defined in the following.

The control objective of a target-tracking scenario is to track the motion of a target that is either stationary (analogous to point stabilization) or that moves such that only its instantaneous motion is known, i.e., such that no information about the future target motion is available. Thus, in this case it is impossible to separate the spatio-temporal constraint associated with the target into two separate constraints.

In contrast, the control objective of a path-following scenario is to follow a predefined path, which only involves a spatial constraint. No restrictions are placed on the temporal propagation along the path.

However, the control objective of a path-tracking scenario is to track a target that moves along a predefined path (analogous to trajectory tracking). Consequently, it is possible to separate the target-related spatio-temporal constraint into two separate constraints. Still, this scenario can be viewed as a target-tracking scenario and handled with target-tracking methods, thus disregarding any apriori path information that is available.

Finally, the control objective of a path-maneuvering scenario is to employ knowledge about vehicle maneuverability to feasibly negotiate (or somehow optimize the negotiation of) a predefined path. As such, path maneuvering represents a subset of path following, but is less constrained than path tracking since spatial constraints always take precedence over temporal constraints. Path-maneuvering methods can also be used to handle path-tracking scenarios.

2.4 Motion control hierarchy

A vehicle motion control system can be conceptualized to involve at least three levels of control in a hierarchical structure, see Fig. 3. This figure illustrates the typical components of a marine motion control system, encompassing strategic, tactical, and execution levels of control (Valavanis et al. 1997). All the involved building blocks represent autonomyenabling technology, but more instrumentation and additional control levels are required to attain fully autonomous operation. An example involves collision avoidance functionality, which demands additional sense and avoid components.

This work is mainly concerned with the highest (strategic) control level of Fig. 3. Termed the kinematic control level, it is responsible for prescribing vehicle velocity commands needed to achieve motion control objectives in the work space. Thus, in this work, kinematic control is equivalent to work-space control, and kinematic controllers are referred to as guidance laws. This level purely considers the geometrical aspects of motion, without reference to the forces and moments that generate such motion.

Next, the intermediate (tactical) level encompass kinetic controllers, which do consider how forces and moments generate vehicle motion. These controllers are typically designed by model-based methods, and must handle both parametric uncertainties and environmental disturbances. For underactuated vehicles, they must actively employ the vehicle attitude as a means to adhere to the velocities ordered by the guidance module. The intermediate control level also contains a control allocation block which distributes the kinetic control commands among the various vehicle actuators.

Finally, the lowest (execution) level is constituted by the individual actuator controllers, which ensure that the actuators behave as requested by the intermediate control module, and ultimately that the vehicle moves as prescribed by the guidance laws.

High Level Control

Intermediate Level Control

Low Level Control

Kinematic Control f Human"

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Control Allocation

Actuator Control

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