Leslie Jennings

A three-dimensional kinematic method for determining the effectiveness of arm segment rotations in producing racquet-head speed

The contribution that a segments anatomical rotations make to racquet-head speed depends on both the segments angular velocity and the instantaneous position of the head of the racquet with respect to the segments axes of rotation. Any analysis of racquet swing technique that does not consider both of these factors simultaneously is, at best, incomplete. With this in mind, a three-dimensional kinematic method was developed to determine the effectiveness of the anatomical rotations of the upper arm, forearm, and hand in producing racquet-head speed. The method entailed developing a system of vector equations for three-dimensional upper limb rotations that used displacement histories of 10 selected landmarks as input. The required three-dimensional displacement histories were obtained using three cine cameras and the DLT approach. To test the diagnostic capabilities of the method, a tennis serve was selected for analysis. For the player and serve analyzed, the greatest contribution to racquet-head speed at impact was produced by internal rotation of the upper arm (8 m s-1). Forearm pronation, although exhibiting the fastest rotation at impact (24 rad s-1), ranked only fourth in terms of its contribution (4 m s-1) to racquet-head speed. To test the performance of the method, a comparison was made between the racquet-head speed measured directly from film and the racquet-head speed computed by summing all of the individual segment contributions to speed commencing at the start of forward swing and ending at ball contact. The results indicate that the method can successfully determine the individual contributions that the different anatomical rotational velocities of the arm segments make to the measured instantaneous racquet-head speed.

THE CONTROL PARAMETERIZATION ENHANCING TRANSFORM FOR CONSTRAINED OPTIMAL CONTROL PROBLEMS

Consider a general class of constrained optimal control problems in canonical form. Using the classical control parameterization technique, the time (planning) horizon is partitioned into several subintervals. The control functions are approximated by piecewise constant or piecewise linear functions with pre-fixed switching times. However, if the optimal control functions to be obtained are piecewise continuous, the accuracy of this approximation process greatly depends on how fine the partition is. On the other hand, the performance of any optimization algorithm used is limited by the number of decision variables of the problem. Thus, the time horizon cannot be partitioned into arbitrarily many subintervals to reach the desired accuracy. To overcome this difficulty, the switching points should also be taken as decision variables. This is the main motivation of the paper. A novel transform, to be referred to as the control parameterization enhancing transform, is introduced to convert approximate optimal control problems with variable switching times into equivalent standard optimal control problems involving piecewise constant or piecewise linear control functions with pre-fixed switching times. The transformed problems are essentially optimal parameter selection problems and hence are solvable by various existing algorithms. For illustration, two non-trivial numerical examples are solved using the proposed method.

Nonlinear optimal control problems with continuous state inequality constraints

In this paper, we consider a class of optimal control problems involving continuous inequality state constraints. This class of optimal control problems can be solved using the technique developed in a paper by Goh and Teo, where a simple constraint transcription is used to convert continuous inequality state constraints into equivalent equality terminal state constraints. However, that constraint transcription has the disadvantage that the equality terminal state constraints so obtained do not satisfy the usual constraint qualification. Thus, convergence is not guaranteed and some oscillation may exist in numerical computation. The aim of this paper is to use a new constraint transcription together with the concept of control parametrization to devise a new computational algorithm for solving this general class of constrained optimal control problems. This new algorithm is much more stable numerically, as we have successfully overcome the above-mentioned disadvantages.

On a class of optimal control problems with state jumps

In this paper, we consider a class of optimal control problems in which the dynamical system involves a finite number of switching times together with a state jump at each of these switching times. The locations of these switching times and a parameter vector representing the state jumps are taken as decision variables. We show that this class of optimal control problems is equivalent to a special class of optimal parameter selection problems. Gradient formulas for the cost functional and the constraint functional are derived. On this basis, a computational algorithm is proposed. For illustration, a numerical example is included.

MISER3: solving optimal control problems—an update

Abstract Dynamic optimization problems occur often in all fields of engineering and management science. The two previous versions of the MISER software for solving constrained optimal control problems, while proving successful on many problems, did have some drawbacks. These are outlined in this paper and the numerical analysis for correcting these problems is also outlined. The approach is one of control parameterization, expressing each control as a linear combination of simple basis functions which makes for efficiencies in the computation.

A Survey of the Control Parametrization and Control Parametrization Enhancing Methods for Constrained Optimal Control Problems

There are many efficient computational methods for solving optimal control problems subject to state and control constraints. In particular, it has been demonstrated through extensive numerical studies that methods based on the control parametrization technique and the control parametrization enhancing technique are effective and reliable. This paper contains a survey of these computational methods for a general class of optional control problems in a canonical form. It is shown that many practical problems can be easily cast in the canonical form and then solved by these methods.

Numerical solution of an optimal control problem with variable time points in the objective function

In this paper, we consider the numerical solution of a class of optimal control problems involving variable time points in their cost functions. The control enhancing transform is first used to convert the optimal control problem with variable time points into an equivalent optimal control problem with fixed multiple characteristic time (MCT). Using the control parametrization technique, the time horizon is partitioned into several subintervals. Let the partition points also be taken as decision variables. The control functions are approximated by piecewise constant or piecewise linear functions in accordance with these variable partition points. We thus obtain a finite dimensional optimization problem. The control parametrization enhancing control transform (CPET) is again used to convert approximate optimal control problems with variable partition points into equivalent standard optimal control problems with MCT, where the control functions are piecewise constant or piecewise linear functions with pre-fixed partition points. The transformed problems are essentially optimal parameter selection problems with MCT. The gradient formulae for the objective function as well as the constraint functions with respect to relevant decision variables are obtained. Numerical examples are solved using the proposed method.

Numerical Computation of Differential-Algebraic Equations for Non-Linear Dynamics of Multibody Systems Involving Contact Forces

The well known Euler-Lagrange equations of motion for constrained variational problems are derived using the principle of virtual work. These equations are used in the modelling of multibody systems and result in differential-algebraic equations of high index. Here they concern an N-link pendulum, a heavy aircraft towing truck and a heavy off-highway track vehicle. The differential-algebraic equation is cast as an ordinary differential equation through differentiation of the constraint equations. The resulting system is computed using the integration routine LSODAR, the Euler and fourth order Runge-Kutta methods. The difficulty to integrate this system is revealed to be the result of many highly oscillatory forces of large magnitude acting on many bodies simultaneously. Constraint compliance is analyzed for the three different integration methods and the drift of the constraint equations for the three different systems is shown to be influenced by nonlinear contact forces.

A computational procedure for suboptimal robust controls

We consider a class of optimal control problems that depend on a set of scalar parameters which could have some uncertainty as to their exact values. We show how to compute the control functions given that we wish to balance two objectives. The first is the original objective while the second is the variation of the original objective with respect to the scalar parameters. That is we wish to move the controls to a position where there is less variation with respect to uncertainty in the scalar parameters, perhaps at the expense of the original objective. The gradient of the combined objective is derived and the method demonstrated using two examples.

Discrete-time optimal control problems with general constraints

This paper presents a computational procedure for solving combined discrete-time optimal control and optimal parameter selection problems subject to general constraints. The approach adopted is to convert the problem into a nonlinear programming problem which can be solved using standard optimization software. The main features of the procedure are the way the controls are parametrized and the conversion of all constraints into a standard form suitable for computation. The software is available commercially as a FORTRAN program DMISER3 together with a companion program MISER3 for solving continuous-time problems.