Time Discretization of the SST-generalized Navier-Stokes Equations: Positive and Negative Results

In the theory of the Navier-Stokes equations, the proofs of some basic known results, like for example the uniqueness of solutions to the stationary Navier-Stokes equations under smallness assumptions on the data or the stability of certain time discretization schemes, actually only use a small range of properties and are therefore valid in a more general context. This observation leads us to introduce the concept of SST spaces, a generalization of the functional setting for the Navier-Stokes equations. It allows us to prove (by means of counterexamples) that several uniqueness and stability conjectures that are still open in the case of the Navier-Stokes equations have a negative answer in the larger class of SST spaces, thereby showing that proof strategies used for a number of classical results are not sufficient to affirmatively answer these open questions. More precisely, in the larger class of SST spaces, non-uniqueness phenomena can be observed for the implicit Euler scheme, for two nonlinear versions of the Crank-Nicolson scheme, for the fractional step theta scheme, and for the SST-generalized stationary Navier-Stokes equations. As far as stability is concerned, a linear version of the Euler scheme, a nonlinear version of the Crank-Nicolson scheme, and the fractional step theta scheme turn out to be non-stable in the class of SST spaces. The positive results established in this thesis include the generalization of classical uniqueness and stability results to SST spaces, the uniqueness of solutions (under smallness assumptions) to two nonlinear versions of the Euler scheme, two nonlinear versions of the Crank-Nicolson scheme, and the fractional step theta scheme for general SST spaces, the second order convergence of a version of the Crank-Nicolson scheme, and a new proof of the first order convergence of the implicit Euler scheme for the Navier-Stokes equations. For each convergence result, we provide conditions on the data that guarantee the existence of nonstationary solutions satisfying the regularity assumptions needed for the corresponding convergence theorem. In the case of the Crank-Nicolson scheme, this involves a compatibility condition at the corner of the space-time cylinder, which can be satisfied via a suitable prescription of the initial acceleration.

Trajectory and Force Control of a Direct Drive Arm

Using the MIT Serial Link Direct Drive Arm as the main experimental device, various issues in trajectory and force control of manipulators were studied in this thesis. Since accurate modeling is important for any controller, issues of estimating the dynamic model of a manipulator and its load were addressed first. Practical and effective algorithms were developed fro the Newton-Euler equations to estimate the inertial parameters of manipulator rigid-body loads and links. Load estimation was implemented both on PUMA 600 robot and on the MIT Serial Link Direct Drive Arm. With the link estimation algorithm, the inertial parameters of the direct drive arm were obtained. For both load and link estimation results, the estimated parameters are good models of the actual system for control purposes since torques and forces can be predicted accurately from these estimated parameters. The estimated model of the direct drive arm was them used to evaluate trajectory following performance by feedforward and computed torque control algorithms. The experimental evaluations showed that the dynamic compensation can greatly improve trajectory following accuracy. Various stability issues of force control were studied next. It was determined that there are two types of instability in force control. Dynamic instability, present in all of the previous force control algorithms discussed in this thesis, is caused by the interaction of a manipulator with a stiff environment. Kinematics instability is present only in the hybrid control algorithm of Raibert and Craig, and is caused by the interaction of the inertia matrix with the Jacobian inverse coordinate transformation in the feedback path. Several methods were suggested and demonstrated experimentally to solve these stability problems. The result of the stability analyses were then incorporated in implementing a stable force/position controller on the direct drive arm by the modified resolved acceleration method using both joint torque and wrist force sensor feedbacks.

Course Notes, Numerical Methods (MATH3018/MATH6111).

Course notes for the Numerical Methods course (joint MATH3018 and MATH6111). Originally by Giampaolo d'Alessandro, modified by Ian Hawke. These contain only minimal examples and are distributed as is; examples are given in the lectures.

Exam 1999 pdf

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Exam 2000

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Exam 1999

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