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Feedback controls have applications in various fields including engineering, mechanics, biomathematics, and mathematical economics; see (Ogata, 1970), (de Queiroz, et al. 2000), (Murray, 2002), and (Seierstad & Sydsaeter, 1987) for more details. Lyapunov based control of mechanical system is a well-known technique. This includes Lyapunov direct/indirect methods. Such techniques can be employed to control the whole state variables or a part of the state variables. Sometimes there are some uncertainties or some reference trajectories which requires adaptive control. Back-stepping is a yet powerful approach to design the required controller. However, this approach leads to a complicated controller, especially when the chain of integrators is long. Back-stepping can also be used when the aim of control is the stability with respect to a part of the variables. These three concepts emerge in a mechanical system like a robot. Adaptive control can be carried out through two different approaches: indirect and direct adaptive control. Nevertheless there are some drawbacks in such control systems which are a matter of concern. For example, when there is the possibility of fault or it is considered to turn off the adaptation for saving energy, when the system seems to be relaxed at its equilibrium situation, the outcome can be dramatically destructive. Adaptively controlled systems with unknown parameters exhibit partial stability phenomenon when the persistence of excitation is not assumed to be satisfied by the designed controllers. Partial stability technique is most useful when a fully stabilized system losses some control engine or some phase variables are not actively controlled. Such situation is most applicable for automatic systems which need to work remotely without a proper access to maintenance; e.g., satellite, robots to work on other planets or under hard conditions which are required to continue their mission even if some fault happens, or when a minimum of controller is required. It is also applicable to biped robots when one of the engines is turned off, or weakened, for lack of energy or fault or when the robot is passively designed. It is worth noting that another useful aspect of partial stability and control is the possibility of controlling the required part of the phase variables without spending energy to control the part of the variables which is not relevant to the mission of the designed system. These concepts will be explained through some examples. The results will be illustrated by numerical computations. This chapter is organized as follows. In section 2 the

Closed-Loop Feedback Systems in Automation and Robotics, Adaptive and Partial Stabilization