Marek Boryga 
Trajectory parametrisation of manipulators with the use of the higher-degree polynomials 
zeszyt 384, ss. 183
Planning manipulator motion trajectories is a significant issue from the user’s standpoint. During the realization of any task performed by a robot, it is necessary to specify the method of changing the location of all the manipulator’s joints.
The task of trajectory planning consists of two stages. The first stage concerns the spatial planning of the path defined in the generalised coordinates, while the second one involves time-domain parametrisation. The parametrisation is sometimes performed through adopting specific changes in the position time and its derivatives. The majority of the methods of parametrisation have drawbacks, such as discontinuity of acceleration or jerk profile, the necessity to divide the trajectory into intervals, and specifying the correlations describing the kinematic quantities separately for each interval (excluding multipoint movement planning in which in complex operations it is required to use different values of kinematic parameters in different periods of time), and no possibility of including initial and final values, etc. Moreover, a lot of these methods are only of theoretical importance because of a high computational complexity. Computational complexity is one of a few drawbacks of the parametrisation method involving polynomials of higher orders. 
The aim of the thesis is to develop an algorithm of planning motion trajectories with the use of polynomials of higher degrees, simultaneously simplifying computational complexity. The most often used polynomial description is not included in the thesis. In order to build the form of a polynomial specifying the acceleration profile, multiplicities of the polynomial’s zero points are used. The effect of such an approach is the necessity to determine only one factor of a polynomial, regardless of its degree. This factor can be maintained for speed, location and jerk profiles. 
The author of the thesis has presented a review of relevant literature, the most frequently used ways of describing profiles of kinematic quantities, polynomial functions of position, speed, acceleration and jerk profiles, as well as motion factors and times with the limitation of surge, acceleration and speed. Additionally, the method of planning manipulator trajectories with the use of polynomials of higher orders has been presented by six examples.
The first three examples illustrate the method of planning single trajectories. The first example concerns the trajectory of an anthropomorphic manipulator in the external space, where the path of the gripper’s motion is a straight line segment. The second one describes planning the Cartesian manipulator trajectory in the internal space, where the assumed path of the gripper’s motion is a circle. The third example refers to planning the SCARA manipulator trajectory in the joint space. 
The method of planning complex trajectories is also illustrated by three examples. Each of them relates to trajectories in the Cartesian space. The first two examples describe planning the anthropomorphic manipulator trajectory, where the motion path is the line segment. In the first example this segment consists of two parts (a start-up phase and a braking phase), whereas in the second one, the path of the gripper’s motion is divided into three consecutive parts, composing the start-up, steady motion and braking phases. The third example refers to planning the SCARA manipulator trajectory, where the motion path consists of two line segments joined by an arc. On the first line segment, the gripper moves with an accelerated motion (a start-up phase), on the arc the gripper moves with a constant speed value, and on the second line segment, the gripper moves with a deccelerated motion (a braking phase). 
The results of the simulation indicate the potential use of polynomials of higher degrees for planning manipulator motion trajectories. The presented correlations for position, speed, acceleration and jerk profiles, together with the derived correlations for coefficient and the total motion time, can be used as ready pattern of the profiles of kinematic quantities during the design of the manipulators and machine tools. In addition, the examples prove the significant computational effectiveness of the presented method.