Decoupled trajectory planning for a submerged rigid body subject to dissipative and potential forces

This paper studies the practical but challenging problem of motion planning for a deeply submerged rigid body. Here, we formulate the dynamic equations of motion of a submerged rigid body under the architecture of differential geometric mechanics and include external dissipative and potential forces. The mechanical system is represented as a forced affine-connection control system on the configuration space SE(3). Solutions to the motion planning problem are computed by concatenating and reparameterizing the integral curves of decoupling vector fields. We provide an extension to this inverse kinematic method to compensate for external potential forces caused by buoyancy and gravity. We present a mission scenario and implement the theoretically computed control strategy onto a test-bed autonomous underwater vehicle. This scenario emphasizes the use of this motion planning technique in the under-actuated situation; the vehicle loses direct control on one or more degrees of freedom. We include experimental results to illustrate our technique and validate our method.

Free surface flow into a horizontal slot with zero gravity

The two-dimensional free surface flow of a finite-depth fluid into a horizontal slot is considered. For this study, the effects of viscosity and gravity are ignored. A generalised Schwarz-Christoffel mapping is used to formulate the problem in terms of a linear integral equation, which is solved exactly with the use of a Fourier transform. The resulting free surface profile is given explicitly in closed-form.

Processing load and the use of concrete representations and strategies for solving linear equations

This paper is a report of students' responses to instruction which was based on the use of concrete representations to solve linear equations. The sample consisted of 21 Grade 8 students from a middle-class suburban state secondary school with a reputation for high academic standards and innovative mathematics teaching. The students were interviewed before and after instruction. Interviews and classroom interactions were observed and videotaped. A qualitative analysis of the responses revealed that students did not use the materials in solving problems. The increased processing load caused by concrete representations is hypothesised as a reason.

Communicating assessment criteria is not sufficient for influencing students' approaches to assessment tasks – perspectives from a differential equations class

This report presents the findings of an exploratory study into the perceptions held by students regarding the use of criterion-referenced assessment in an undergraduate differential equations class. Students in the class were largely unaware of the concept of criterion referencing and of the various interpretations that this concept has among mathematics educators. Our primary goal was to investigate whether explicitly presenting assessment criteria to students was useful to them and guided them in responding to assessment tasks. Quantitative data and qualitative feedback from students indicates that while students found the criteria easy to understand and useful in informing them as to how they would be graded, the manner in which they actually approached the assessment activity was not altered as a result of the use of explicitly communicated grading criteria.

Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations

Maximum-likelihood estimates of the parameters of stochastic differential equations are consistent and asymptotically efficient, but unfortunately difficult to obtain if a closed-form expression for the transitional probability density function of the process is not available. As a result, a large number of competing estimation procedures have been proposed. This article provides a critical evaluation of the various estimation techniques. Special attention is given to the ease of implementation and comparative performance of the procedures when estimating the parameters of the Cox–Ingersoll–Ross and Ornstein–Uhlenbeck equations respectively.

Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts

Columns are one of the key load bearing elements that are highly susceptible to vehicle impacts. The resulting severe damages to columns may leads to failures of the supporting structure that are catastrophic in nature. However, the columns in existing structures are seldom designed for impact due to inadequacies of design guidelines. The impact behaviour of columns designed for gravity loads and actions other than impact is, therefore, of an interest. A comprehensive investigation is conducted on reinforced concrete column with a particular focus on investigating the vulnerability of the exposed columns and to implement mitigation techniques under low to medium velocity car and truck impacts. The investigation is based on non-linear explicit computer simulations of impacted columns followed by a comprehensive validation process. The impact is simulated using force pulses generated from full scale vehicle impact tests. A material model capable of simulating triaxial loading conditions is used in the analyses. Circular columns adequate in capacity for five to twenty story buildings, designed according to Australian standards are considered in the investigation. The crucial parameters associated with the routine column designs and the different load combinations applied at the serviceability stage on the typical columns are considered in detail. Axially loaded columns are examined at the initial stage and the investigation is extended to analyse the impact behaviour under single axis bending and biaxial bending. The impact capacity reduction under varying axial loads is also investigated. Effects of the various load combinations are quantified and residual capacity of the impacted columns based on the status of the damage and mitigation techniques are also presented. In addition, the contribution of the individual parameter to the failure load is scrutinized and analytical equations are developed to identify the critical impulses in terms of the geometrical and material properties of the impacted column. In particular, an innovative technique was developed and introduced to improve the accuracy of the equations where the other techniques are failed due to the shape of the error distribution. Above all, the equations can be used to quantify the critical impulse for three consecutive points (load combinations) located on the interaction diagram for one particular column. Consequently, linear interpolation can be used to quantify the critical impulse for the loading points that are located in-between on the interaction diagram. Having provided a known force and impulse pair for an average impact duration, this method can be extended to assess the vulnerability of columns for a general vehicle population based on an analytical method that can be used to quantify the critical peak forces under different impact durations. Therefore the contribution of this research is not only limited to produce simplified yet rational design guidelines and equations, but also provides a comprehensive solution to quantify the impact capacity while delivering new insight to the scientific community for dealing with impacts.