Integrating control systems defined on the frame bundles of the space forms

This paper considers left-invariant control systems defined on the orthonormal frame bundles of simply connected manifolds of constant sectional curvature, namely the space forms Euclidean space E-3, the sphere S-3 and Hyperboloid H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. In the Euclidean case the elements of the Lie algebra se(3) are often referred to as twists. For constant twist motions, the corresponding curves g(t) is an element of SE(3) are known as screw motions, given in closed form by using the well known Rodrigues' formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

Planning rigid body motions using elastic curves

This paper tackles the problem of computing smooth, optimal trajectories on the Euclidean group of motions SE(3). The problem is formulated as an optimal control problem where the cost function to be minimized is equal to the integral of the classical curvature squared. This problem is analogous to the elastic problem from differential geometry and thus the resulting rigid body motions will trace elastic curves. An application of the Maximum Principle to this optimal control problem shifts the emphasis to the language of symplectic geometry and to the associated Hamiltonian formalism. This results in a system of first order differential equations that yield coordinate free necessary conditions for optimality for these curves. From these necessary conditions we identify an integrable case and these particular set of curves are solved analytically. These analytic solutions provide interpolating curves between an initial given position and orientation and a desired position and orientation that would be useful in motion planning for systems such as robotic manipulators and autonomous-oriented vehicles.

Switching curve control of functional electrical stimulation assisted rowing exercise in paraplegia

An indoor rowing machine has been modified for functional electrical stimulation (FES) assisted rowing exercise in paraplegia. To perform the rowing manoeuvre successfully, however, the voluntarily controlled upper body movements must be co-ordinated with the movements of the electrically stimulated paralysed legs. To achieve such co-ordination, an automatic FES controller was developed that employs two levels of hierarchy. At the upper level, a finite state controller identifies the state or phase of the rowing cycle and activates the appropriate lower-level controller, in which electrical stimulation to the paralysed leg muscles is applied with reference to switching curves representing the desired seat velocity as a function of the seat position. In a pilot study, the hierarchical control of FES rowing was shown to be intuitive, reliable and easy to use. Compared with open-loop control of stimulation, all three variants of the closed-loop switching curve controllers used less muscle stimulation per rowing cycle (73% of the open-loop control on average). Further, the closed-loop controller that used switching curves derived from normal rowing kinematics used the lowest muscle stimulation (65% of the open-loop control) and was the most convenient to use for the client.

Finding the point on Bezier Curves with the normal vector passing an external point

Two algorithms for finding the point on non-rational/rational Bezier curves of which the normal vector passes through a given external point are presented. The algorithms are based on Bezier curves generation algorithms of de Casteljau's algorithm for non-rational Bezier curve or Farin's recursion for rational Bezier curve, respectively. Orthogonal projections from the external point are used to guide the directional search used in the proposed iterative algorithms. Using Lyapunov's method, it is shown that each algorithm is able to converge to a local minimum for each case of non-rational/rational Bezier curves. It is also shown that on convergence the distance between the point on curves to the external point reaches a local minimum for both approaches. Illustrative examples are included to demonstrate the effectiveness of the proposed approaches.

Calibration of closed loop controllers for setting impedances in force-reflecting systems

This paper discusses a new method of impedance control that has been successfully implemented on the master robot of a teleoperation system. The method involves calibrating the robot to quantify the effect of adjustable controller parameters on the impedances along its different axes. The empirical equations relating end-effector impedance to the controller's feedback gains are obtained by performing system identification tests along individual axes of the robot. With these equations, online control of end-effector stiffness and damping is possible without having to monitor joint torques or solving complex algorithms. Hard contact conditions and compliant interfaces have been effectively demonstrated on a telemanipulation test-bed using appropriate combinations of stiffness and damping settings obtained by this method.

Closed form approximations for spread options

This article expresses the price of a spread option as the sum of the prices of two compound options. One compound option is to exchange vanilla call options on the two underlying assets and the other is to exchange the corresponding put options. This way we derive a new closed form approximation for the price of a European spread option and a corresponding approximation for each of its price, volatility and correlation hedge ratios. Our approach has many advantages over existing analytical approximations, which have limited validity and an indeterminacy that renders them of little practical use. The compound exchange option approximation for European spread options is then extended to American spread options on assets that pay dividends or incur costs. Simulations quantify the accuracy of our approach; we also present an empirical application to the American crack spread options that are traded on NYMEX. For illustration, we compare our results with those obtained using the approximation attributed to Kirk (1996, Correlation in energy markets. In: V. Kaminski (Ed.), Managing Energy Price Risk, pp. 71–78 (London: Risk Publications)), which is commonly used by traders.

The role of closed ecological systems in carbon cycle modelling

Acquiring a mechanistic understanding of the role of the biotic feedbacks on the links between atmospheric CO2 concentrations and temperature is essential for trustworthy climate predictions. Currently, computer based simulations are the only available tool to estimate the global impact of the biotic feedbacks on future atmospheric CO2 and temperatures. Here we propose an alternative and complementary approaches by using materially closed and energetically open analogue/physical models of the carbon cycle. We argue that there is potential in using a materially closed approach to improve our understanding of the magnitude and sign of many biotic feedbacks, and that recent technological advance make this feasible. We also suggest how such systems could be designed and discuss the advantages and limitations of establishing physical models of the global carbon cycle.

Spatial and temporal effects of free-air CO2 enrichment (POPFACE) on leaf growth, cell expansion and cell production in a closed canopy of poplar

Leaf expansion in the fast-growing tree,Populus × euramericana was stimulated by elevated [CO2] in a closed-canopy forest plantation, exposed using a free air CO2 enrichment technique enabling long-term experimentation in field conditions. The effects of elevated [CO2] over time were characterized and related to the leaf plastochron index (LPI), and showed that leaf expansion was stimulated at very early (LPI, 0–3) and late (LPI, 6–8) stages in development. Early and late effects of elevated [CO2] were largely the result of increased cell expansion and increased cell production, respectively. Spatial effects of elevated [CO2] were also marked and increased final leaf size resulted from an effect on leaf area, but not leaf length, demonstrating changed leaf shape in response to [CO2]. Leaves exhibited a basipetal gradient of leaf development, investigated by defining seven interveinal areas, with growth ceasing first at the leaf tip. Interestingly, and in contrast to other reports, no spatial differences in epidermal cell size were apparent across the lamina, whereas a clear basipetal gradient in cell production rate was found. These data suggest that the rate and timing of cell production was more important in determining leaf shape, given the constant cell size across the leaf lamina. The effect of elevated [CO2] imposed on this developmental gradient suggested that leaf cell production continued longer in elevated [CO2] and that basal increases in cell production rate were also more important than altered cell expansion for increased final leaf size and altered leaf shape in elevated [CO2].

The elasticity of demand is about occupancy, not prices: building European demand curves based on occupancy elasticity

The peak congestion of the European grid may create significant impacts on system costs because of the need for higher marginal cost generation, higher cost system balancing and increasing grid reinforcement investment. The use of time of use rates, incentives, real time pricing and other programmes, usually defined as Demand Side Management (DSM), could bring about significant reductions in prices, limit carbon emissions from dirty power plants, and improve the integration of renewable sources of energy. Unlike previous studies on elasticity of residential electricity demand under flat tariffs, the aim of this study is not to investigate the known relatively inelastic relationship between demand and prices. Rather, the aim is to assess how occupancy levels vary in different European countries. This reflects the reality of demand loads, which are predominantly determined by the timing of human activities (e.g. travelling to work, taking children to school) rather than prices. To this end, two types of occupancy elasticity are estimated: baseline occupancy elasticity and peak occupancy elasticity. These represent the intrinsic elasticity associated with human activities of single residential end-users in 15 European countries. This study makes use of occupancy time-series data from the Harmonised European Time Use Survey database to build European occupancy curves; identify peak occupancy periods; draw time use demand curves for video and TV watching activity; and estimate national occupancy elasticity levels of single-occupant households. Findings on occupancy elasticities provide an indication of possible DSM strategies based on occupancy levels and not prices.

A closed paramagnetic electron-precise cluster

A pentametallic cluster is formed by addition of two [SnR2][R = CH(SiMe3)2] groups to [Os3(CO)8(µ-H)(C6H4PhPCH2PPh2)], in a reaction reversing the ortho-hydrogen abstraction and giving the first closed electron-precise paramagnetic cluster.