This paper presents a comparative study about the attitude control methods based on four commonly used error indicators, namely the triad of 3-2-1 deviational Euler angles, the error quaternion, the deviational angle–axis pair and the orientation error matrix. These error indicators are used here with the same backstepping control law to have a common basis of comparison. This control law makes the controller track a restoring angular velocity generated here specifically for each error indicator. This comparative study shows that all these error indicators can be used satisfactorily even in the critical orientations associated with them by taking special measures. For the deviational Euler angle triad, a critical orientation is a singularity, at which the angles become indefinite. Unless the vehicle is stationary, this indefiniteness is resolved here by applying L’Hopital’s Rule on the angular velocity information. The Euler angle triad has also a multiplicity problem. It is solved here by using the novel criterion of minimal deviation angles. For the other error indicators, a critical orientation is an antipodal orientation, which is opposite to the desired one. In an antipodal orientation, the error quaternion and the deviational angle–axis pair cannot be determined through the customary formulas. They are determined here by using the novel specially introduced formulas. Besides, they may suffer from the unwinding phenomenon in an ordinary orientation. This phenomenon is prevented here by keeping the scalar part of the error quaternion non-negative and the deviation angle between 0° and 180°. For the orientation error matrix, a stationary and undisturbed antipodal orientation is an unstable equilibrium, in which the ordinary backstepping control law becomes ineffective for driving the system into action. This problem is solved here by adding an extra term to the ordinary backstepping control law.

This study covers the sliding mode control design with adaptive sliding surfaces for a class of affine non-linear systems, which can be described by ẋ = A(x)x + B(x)u + f(x) + d(x,t). The main streamline of the study is the sliding surface design for such systems. The sliding surfaces are designed to be moving with varying slopes and offsets. The varying sliding surface parameters are determined by solving the state-dependent Riccati equations online during the control process. Thus, the sliding surfaces are updated in an adaptive manner to cope with the non-linearities of the system. The adaptive sliding mode control method developed in this study is used to design a longitudinal controller based on the mathematical model of a generic hypersonic aircraft. Then, the successful performance of this controller is demonstrated by means of a simulated manoeuvre of the aircraft.

Two methods are presented to obtain optimal inverse kinematic solutions for redundant manipulators, according to two different performance criteria stipulated in the position and velocity levels. Both methods are analytical throughout except their final stages, which involve the numerical solution of a simplified minimization problem in a position-level case and the numerical integration of a set of differential equations derived optimally in a velocity-level case. Owing to the analytical nature of the methods, the multiple and singular configurations of the manipulator of concern can be identified readily and studied in detail. The methods are applicable for both serial and parallel redundant manipulators. However, they are demonstrated here for a humanoid manipulator with seven revolute joints. In the demonstrations, the first performance criterion is stipulated in the position level as the minimization of the potential energy. In that case, the optimal inverse kinematic solution is first obtained in the position level for a specified position of the hand. Then, it is compatibly extended to the velocity level for a specified motion of the hand. In the main analytical part of the solution, six of the joint variables are expressed in terms of the selected seventh one. Then, the optimal value of the selected joint variable is determined numerically by a simple one dimensional scanning. The second performance criterion is stipulated in the velocity level as the minimization of the kinetic energy. In that case, the optimal inverse kinematic solution is first obtained in the velocity level and then extended to the position level by integration. The main analytical part of the solution provides an optimally determined set of nonlinear differential equations. These differential equations are then integrated numerically in order to obtain the corresponding solution in the position level. However, the corrections needed to eliminate the numerical integration errors are still obtained analytically. The distinct optimal behaviors of the manipulator according to the mentioned criteria are also illustrated and compared for a duration, in which the hand moves in the same specified way.