Evaluation of K⊖ from [η]-M-data of binary and ternary polymer systems

Unperturbed dimensions of flexible linear macromolecules can be obtained from [η]-M-data in any solvent, good or poor, single or mixed. Usually K_θ is estimated by a relationship between [η]/M_W^0.5 and M_w^0.5 first proposed by Burchard and by Stockmayer and Fixman. But, it is well-known that the Burchard-Stockmayer-Fixman-plot shows downward curvature, especially for good solvent systems. Various efforts have been made to achieve relations with better linearity. One of the first was the semi-empirical relation between ([η]/M_w^0.5)^0.5 and M_w/[η] by Berry. Predicting a relationship of the excluded volume parameter z to the viscosity expansion factor by α⁵_η instead of α⁵_η Tanaka obtains that ([η]/M_w^0.5)^5/3 is linear in M_w^0.5. By allowing for the dependence of the viscometric interaction parameter B, which is correlated to the second virial coefficient A₂, on molar mass, Gavara, Campos and Figueruelo predict a linear dependence of [η]/M_w^0.5 against A₂.M_w^0.5. It is not our intention here to discuss the validity of these theories, but to compare them with experimental data.     

Time Optimal Path Planning for Industrial Robots: A Dynamic Programming Approach Considering Torque Derivative and Jerk Constraints

This paper presents a dynamic programming approach for calculating time optimal trajectories for industrial robots, subject to various physical constraints. In addition to path velocity, motor torque, joint velocity and acceleration constraints, the present contribution also shows how to deal with torque derivative and joint jerk limitations. First a Cartesian path for the endeffector is defined by splines using Bernstein polynomials as basis functions and is parameterized via a scalar path parameter. In order to compute the belonging quantities in configuration space, inverse kinematics is solved numerically. Using this and in combination with the dynamical model, joint torques as well as their derivatives can be constrained. For that purpose the equations of motion are calculated with the help of the Projection Equation. As a consequence of the used optimization problem formulation, the dynamical model as well as the restrictions have to be transformed to path parameter space. Due to the additional consideration of jerk and torque derivative constraints, the phase plane is expanded to a phase space. The parameterized restrictions lead to feasible regions in this space, in which the optimal solution is sought. Result of the optimization is the time behavior of the path parameter and subsequently the feed forward torques for the optimal motion on the spatial path defined by previously mentioned splines. Simulation results as well as experimental results for a three axes industrial robot are presented. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)