Root Trajectory Research Articles

LEAP EFFECTS AMONG ROBUST ROOT TRAJECTORIES

The set of closed-loop eigenvalues in the presence of the worst uncertainty is termed robust root trajectory. The open-loop gain or some other system parameter is changed monotonically. The admissible uncertainties are considered along their envelope. The worst uncertainty is denoted as that one where a specified function of the eigenvalues leads to the maximum disadvantageous effect on the closed-loop stability performance. Leaps in the dynamic behavior occur even for infinitesimally small changes in the gain parameter when system uncertainties are considered that are located especially at their bounds.

FOOTPRINT SAMPLING-BASED MOTION EDITING

In this paper, we present a motion editing algorithm for the human biped locomotion captured by a motion capturing device. Our algorithm adopts footprints to describe the space-time constraints which should be satisfied during biped locomotion. The footprints are also used as an interface to enable the user to control the space-time constraints directly. A real-time Inverse Kinematics (IK) solver is adapted to compute the configuration of the human body and motion displacement mapping is then constructed using hierarchical B-spline. In order to facilitate the IK solver, we propose a sampling-based scheme to generate root trajectory. Hermit interpolation is then employed to generate the whole root trajectory. This scheme provides a speedup to root trajectory generation. The performance of our algorithm is further enhanced by the real-time IK solver, which directly computes the displacement angles as solution.

Dominant pole placement for multi-loop control systems

This paper proposes a method for multi-loop PI controller design which can achieve dominant pole placement for two input two output processes. It is an extension of the original dominant pole design (PID Controllers: Theory, Design, and Tuning, Instrument Society of America, Research Triangle park, NC, 1995.) for SISO systems. Unlike its SISO counterpart, where the controller parameters can be obtained analytically, the multi-loop version amounts to solving some coupled nonlinear equation with complex coefficients, for which closed-form formulae are not possible. A novel approach is developed to solve the equation using a “root trajectory” method, in which the solution to our pole placement problem is found from intersection points between the “root trajectories” and the positive real axis. The design procedure is given and simulation examples are provided to show the effectiveness of the proposed method and comparisons are made with the BLT method.

Plagiogravitropism of maize roots

The direction of root growth can be studied by analyzing the trajectories of roots growing in soil. Both the primary seminal root and nodal roots of maize attain a preferred, or liminal, angle of growth that deviates from the vertical. These roots are said to be plagiogravitropic. Experiments using plants grown in soil-filled boxes revealed that the primary seminal root is truly plagiogravitropic. It shows both positive and negative gravitropism in response to gravity stimuli and tends to maintain its direction even after growing around obstacles. These are experimental results suggesting that plagiogravitropic growth is controlled by internal factors. The orientation of the grain affects the establishment of the liminal angle of the primary seminal root, and both the position of their node of origin and the root diameter are closely related to the plagiogravitropic behaviour of nodal roots. Several external factors are also known to influence plagiogravitropism. Low soil water content causes a decrease in the angle of growth and soil mechanical resistance suppresses the gravitropic curvature. Plagiogravitropic behaviour of both seminal and nodal roots plays a significant role in shaping the root system.

Influence of soil temperature during root appearance on the trajectory of nodal roots of field grown maize

Trajectories of maize nodal roots were studied to test the hypothesis that roots which appear on a common internode have similar geometrical characteristics, and to assess the effect of soil temperature on root trajectory. Treatments consisted of three sowing dates, a comparison between mulched and non-mulched soil, a replication of one sowing date for two years in two locations, and a comparison between two cultivars at one sowing date. All these sources of variation, except the cultivar, had an appreciable effect on the trajectories of roots which appeared on the first four internodes. The horizontal component of the trajectory differed significantly between treatments, ranging from 93 to 700 mm in roots which appeared on the second internode, and from 71 to 569 for those on the third internode. The original hypothesis had, therefore, to be rejected. Mean soil temperature during the 100°C.days after root appearance accounted for the differences in trajectory between location, year, sowing date and mulch treatments, and for the differences between internodes within each location. The critical period during which temperature affected root trajectory probably began at root appearance, and ended between 50 and 100°C days after root appearance, i.e. when the root was less than 100 mm long.

Trajectory of the nodal roots of maize in fields with low mechanical constraints

The trajectories of seventy three nodal roots of maize were studied in two fields with loose soil structure. Their projections on horizontal and vertical planes were traced. These roots tended to remain in a vertical plane. Trajectories were related to each other by an affine transformation. Thus, all the observed trajectories could be obtained by transformation of a common root archetype. The horizontal component of the trajectories was mainly in the first 0.4 m depth of soil, in the layer where soil structure was disturbed by ploughing. This horizontal component decreased with later appearance of roots (upper internodes), but differed between the two sites. The average soil temperature during the week following root appearance accounted for differences between internodes and sites. Lungley's algorithm, which is commonly used in modelling root trajectories, was tested. A general pattern could be simulated, but the model failed to fit the trajectories in the first 100 to 200 mm of soil. As a consequence, the initial angle between the stem and the root, which is a sensitive parameter in Lungley's model, did not account for differences between root trajectories.

Surface waves on a lossy planar ferrite slab

The parallel-polarized surface waves which propagate on a planar ferrite slab are considered. The ferrite material is isotropic and homogeneous, the slab rests on a perfectly conducting ground plane, and the ambient medium is free space. Only one mode can propagate on a thin lossless slab, but more and more of the higher order modes begin to propagate when one increases the thickness or the loss tangent of the slab. As the thickness and the loss are increased, the lowest order mode takes on the properties of the Zenneck wave. At the same time, the higher order modes take on the properties of a homogeneous plane wave propagating in the grazing direction in the slab. Among the most interesting properties of a surface wave are the attenuation constant and the phase velocity. Contour maps are included to show how these quantities depend on the thickness, permittivity, permeability and loss tangents of the ferrite slab. Additional insight is provided by graphs showing the root trajectories, cutoff boundaries and field distributions of the four lowest order surface-wave modes.

On the evaluation of the classical phase for reactive molecular collisions in semiclassical collision theory

A new formula for the generator of a cartesian to action-angle change of variables for a Morse oscillator is presented. It can be used to calculate the classical phase of semiclassical collision theory for both inelastic and reactive collisions. In the reactive case, the calculation of the phase is equivalent to using action-angle variables for the initial (final) oscillator during the first (second) part of the collision. As an example, application is made to the collinear reactions H + Cl2(n 1→HCl(n 2=2) + Cl and H + F2(n 1=0)→HF(n 2=6) + F for an energy range where there is complete transmission and two contributing real root trajectories.

On the Zeros of Polynomials and the Degree of Stability of Linear Systems

A method is presented for finding the zeros of any nth degree real or complex polynomial as its coefficients (or their parameters) are varied. Several new theorems are presented. The method is based on new theorems, some concerning determinants, and on known theorems. It is applied to characteristic equations of linear physical systems with or without feedback. Root trajectories in the complex plane can be visualized without plotting them from ``root trajectory diagrams'' which show curve families for real or complex roots of a polynomial equation in a plane having two coefficients of the polynomial equation as coordinates. The diagrams can be used in the converse manner to determine the coefficient values corresponding to a desired dominant complex pair or real root.