Research On Task-space Control Based On Quaternion Algebra And A Lie-group Framework

Task-space is a typical configuration space of mechanical systems, whose geometrical correspondence is the Lie-group SE(3). The control problem on task space covers a variety of cases under research. However, due to the complexity of the space structure, theoretical research on task-space control advances slowly while being away from engineering applications. It is the defects of current research and the demands from engineering applications that initiates the research in this thesis, which focuses on two key issues of task-space control: the computation burden brought by matrix description of Lie-groups, and the no-literal-drift motion control.The first half of this thesis aims at releasing the computation burden within control schemes, in seeking solution for whole task space control. Research on attitude control and posture control were carried out via the usage of quaternion algebra as representing tools, aiming at control schemes with both computational efficiency and utilization of task- space's geometric structures. The main achievements lay in the following three aspects.1. The Lie-group structures of unit quaternions and normalized dual quaternions were formulated. Thereafter, generalized proportional-derivative(PD) control laws were developed for attitude control and posture control respectively, aiding by the Lie-algebras of the unit-quaternion Lie-group (Q_u) and the dual-quaternion Lie-group (DQ_u). With proper handling of the double-equilibrium problem brought by quaternion-like representations, both regulation and trajectory tracking control were realized for fully-actuated models.2. Research on optimal attitude control and rotational trajectory design were carried out using unit quaternion. In the case study carried on the reorientation problem of free-floating space robot(FFSR), orientation control that can optimizing energy was achieved utilizing the Lie-algebra of Q_u and DQ_u. In discussing the two-point rotating interpolation problem with minimum acceleration, approximate analytical solutions were proposed using polynomial splines.3. Research on control of multi-body mechanisms was carried out based on the notion of using dual-quaternion in posture control. The case study of multi-freedom manipulator shows how to combine a specific system model with the generalized PD law within the structure of the dual-quaternion Lie-Group DQ_u. Regulation law in whole task space was deduced, inducing the notation double task space control which leads to a control law for FFSR with minimum disturbance for the basement.On the other hand, observing driftless motion which has aircrafts as the typical example, the model's characteristics makes it difficult to perform posture control, leading to the research on two sub-problems: trajectory planning of particles, and model analysis of driftless motion. Also, as a simplified version of driftless motion, control of ground mobile robots calls for a control scheme depending merely on angle detection. The latter part of the thesis has been focused on related problems of driftless motion control, arriving at achievements in three aspects.4. Achievements in real-time trajectory planning: Focusing on the typical motion of moving particles with constant speed, combining with the notion of aircraft guidance, using the information brought by the variation of line-of-sight(LOS)'s orientation, the literal accelaration was calculated and real-time trajectory planning was accomplished. Description of the LOS's orientation can be performed by Lie-group or twist, yielding two algorithms for real-time trajectory planning. The two algorithms can also be used as guidance laws, which can fix the defects in traditional method where decoupling of channels were demanded, and partly meet the demands in multi-constraints guidance.5. Achievements in model analysis: The analysis of driftless motion points out that, the degree-of-freedom(DOF) of motion affects motion particularity. With the known actuators, 2-dimensional(2D) driftless motion is naturally differential flat due to its minor DOFs. For 3-dimensional(3D) driftless motion, the kinematic model is differential flat only when the longitudinal velocity is constant, and the key for dynamic modeling and control is the relative rotation between the body axis and the velocity vector.6. Achievements in control law design: A design on the flat output of 2D driftless motion results in a unified control law for both posture stabilization and trajectory tacking. When performing path following, the system was simplified to be 1-dimensional; With proper design of flat output, control law can be developed via exact linearizing. When only angle measurement of LOS was available, revising existing 2D guidance law, adding a control on the longitudinal velocity, posture stabilization with certain optimization was developed. For 3D driftless motion, design on the flat output, a new algorithm for trajectory generation is achieved for particle motion with constant speed.The idea of combining algebraic method with geometric method, together with the skill of space decomposition, was utilized in the thesis, which deepens the research on control and trajectory planning of various kind of motion in task-space, and pave the way for a final solution to both the control of manipulators in whole task space and integrated guidance-control of aircrafts.