Mont Hubbard

Higher order linear approximations to nonlinear control systems

One traditional approach in the analysis and design of nonlinear control systems is a first order approximation by a linear system. A new approach is to use nonlinear change of coordinates and feedback to construct linear approximations that are accurate to second and higher orders. However, the algebraic calculations required to obtain these aproximations are somewhat lengthy. In this paper, the theoretical framework for finding such change of coordinates for a nonlinear system are described. A software package that symbolically solves these transformations is currently being prepared.

Experimental determination of baseball spin and lift

The aim of this study was to develop a new method for the determination of lift on spinning baseballs. Inertial trajectories of (a) ball surface markers during the first metre of flight and (b) the centre of mass trajectory near home-plate were measured in a pitch using high-speed video. A theoretical model was developed, incorporating aerodynamic Magnus-Robins lift, drag and cross forces, which predicts the centre of mass and marker trajectories. Parameters including initial conditions and aerodynamic coefficients were estimated iteratively by minimizing the error between predicted and measured trajectories. We compare the resulting lift coefficients and spin parameter values with those of previous studies. Lift on four-seam pitches can be as much as three times that of two-seam pitches, although this disparity is reduced for spin parameters greater than 0.4.

How to hit home runs: Optimum baseball bat swing parameters for maximum range trajectories

~Received 12 June 2002; accepted 3 July 2003! Improved models for the pitch, batting, and post-impact flight phases of a baseball are used in an optimal control context to find bat swing parameters that produce maximum range. The improved batted flight model incorporates experimental lift and drag profiles ~including the drag crisis! .A n improved model for bat‐ball impact includes the dependence of the coefficient of restitution on the approach relative velocity and the dependence of the incoming pitched ball angle on speed. The undercut distance and bat swing angle are chosen to maximize the range of the batted ball. The sensitivity of the maximum range is calculated for all model parameters including bat and ball speed, bat and ball spin, and wind speed. Post-impact conditions are found to be independent of the ball‐bat coefficient of friction. The lift is enhanced by backspin produced by undercutting the ball during batting. An optimally hit curve ball will travel farther than an optimally hit fastball or knuckleball due to increased lift during flight.

Hoof accelerations and ground reaction forces of Thoroughbred racehorses measured on dirt, synthetic, and turf track surfaces

OBJECTIVE To compare hoof acceleration and ground reaction force (GRF) data among dirt, synthetic, and turf surfaces in Thoroughbred racehorses. ANIMALS 3 healthy Thoroughbred racehorses. PROCEDURES Forelimb hoof accelerations and GRFs were measured with an accelerometer and a dynamometric horseshoe during trot and canter on dirt, synthetic, and turf track surfaces at a racecourse. Maxima, minima, temporal components, and a measure of vibration were extracted from the data. Acceleration and GRF variables were compared statistically among surfaces. RESULTS The synthetic surface often had the lowest peak accelerations, mean vibration, and peak GRFs. Peak acceleration during hoof landing was significantly smaller for the synthetic surface (mean + or - SE, 28.5g + or - 2.9g) than for the turf surface (42.9g + or - 3.8g). Hoof vibrations during hoof landing for the synthetic surface were < 70% of those for the dirt and turf surfaces. Peak GRF for the synthetic surface (11.5 + or - 0.4 N/kg) was 83% and 71% of those for the dirt (13.8 + or - 0.3 N/kg) and turf surfaces (16.1 + or - 0.7 N/kg), respectively. CONCLUSIONS AND CLINICAL RELEVANCE The relatively low hoof accelerations, vibrations, and peak GRFs associated with the synthetic surface evaluated in the present study indicated that synthetic surfaces have potential for injury reduction in Thoroughbred racehorses. However, because of the unique material properties and different nature of individual dirt, synthetic, and turf racetrack surfaces, extending the results of this study to encompass all track surfaces should be done with caution.

Dependence of release variables in the shot put

When the shot is released above a horizontal plane, range from this point depends on release height, speed and angle. Measured distance is the sum of this range and horizontal distance of the release point from the throwing circle edge. Optimal release conditions can be calculated only if the dependence of release velocity on other variables, due to thrower limitations, is known. Experiments on two shot-putters investigated the hypothesis that there are constraint relationships among these four release parameters. A variable scaling scheme, using measurement of impact point and the known magnitude of g, corrected 2D data from one camera for out-of-plane motion and yielded accurate estimates of release parameters. Multivariate regression analyses determined approximate constraint surfaces limiting performance. Achievable release speed decreases with increasing release angle at about 1.7(m/s)/rad and decreases with increasing release height at about 0.8(m/s)/m, with only small differences in sensitivities between the throwers. Horizontal release distance also decreases with increasing release angle at about 1.7m/rad and increases with increasing release height at about 1.3m/m, again with only small differences between the two throwers. Optimal release conditions producing maximum range for a particular athlete can be determined using similar constraints for that athlete.

Open channel transient flow control by discrete time LQR methods

Abstract A real-time compensation scheme for multipool canals is developed using linear quadratic methods. A special response of the simple wave equation is used as a basis for developing the performance index which is minimized for a linear method of characteristics flow model (discrete time). State estimation, using only depths adjacent to underflow gates, is shown always to be possible. Fixed compensation for a large flow transition is demonstrated in an example where the controlled object is a realistic, nonlinear numerical flow model. A reference input which results in deadbeat control of the wave equation and allows approximate prediction of surge heights is used. Development and implementation of the compensation algorithm involves only moderate computational effort, but substantial simplification seems possible.

Dynamics of the pole vault

Abstract The nonlinear differential equations of motion of the pole vault are derived using bond graph modeling methods. The model for the pole builds on large deflection theory of a slender rod but incorporates the possibility of a vaulter-applied torque at the top. A pinned rigid body model of the human vaulter is used with internally generated control torques applied at the hip, shoulder and wrist joints. Time simulations of vaults are presented. The complete system model is used to study the effects of different initial conditions and the effect of time variations in the control torques during the vault.

Dynamics of the basketball shot with application to the free throw

Abstract A completely general three-dimensional dynamic model is presented for the motion of basketball shots that may contact the rim, the backboard, the bridge between the rim and board, and possibly the board and the bridge simultaneously. Non-linear ordinary differential equations with six degrees of freedom describe the ball angular velocity and ball centre position. The model includes radial ball compliance and damping and contains five sub-models: purely gravitational flight, and ball – rim, ball – bridge, ball – board, and ball – bridge – board contact. Each contact sub-model has both slipping and non-slipping motions. Switching between the sub-models depends on the reaction force at, and velocity of, the contact point. Although the model can be used to study shots from any point on the court, we here use it to study the sets of free throw release angle, velocity, angular velocity, and lateral deviation angle that result in success (capture), as well as underhand free throws and those using an under-inflated ball. Free throw shots with larger backspin, lower inflation pressures, and underhand release conditions are shown to result in larger capture percentages.

Rapid and accurate estimation of release conditions in the javelin throw

We have developed a system to measure initial conditions in the javelin throw rapidly enough to be used by the thrower for feedback in performance improvement. The system consists of three subsystems whose main tasks are: (A) acquisition of automatically digitized high speed (200 Hz) video x, y position data for the first 0.1-0.2 s of the javelin flight after release (B) estimation of five javelin release conditions from the x, y position data and (C) graphical presentation to the thrower of these release conditions and a simulation of the subsequent flight together with optimal conditions and flight for the sam release velocity. The estimation scheme relies on a simulation model and is at least an order of magnitude more accurate than previously reported measurements of javelin release conditions. The system provides, for the first time ever in any throwing event, the ability to critique nearly instantly in a precise, quantitative manner the crucial factors in the throw which determine the range. This should be expected to much greater control and consistency of throwing variables by athletes who use system and could even lead to an evolution of new throwing techniques.

Optimal javelin trajectories

A companion paper has treated computer simulation of javelin flight using measured lift, drag and pitching moments. In the present paper we present, categorize and discuss the relative significance of various initial conditions in such a simulation. Since the differential equations describing flight are autonomous, the eventual javelin range and entry angle are unique functions of the initial conditions. A series of successively less constrained optimum solutions is defined, the last of which is the global optimum javelin trajectory. Sensitivities of these trajectories to perturbations from the optima and their implications for throwers are discussed. Finally, we investigate the effects of some design and environmental parameters on optimal initial conditions and trajectories.