#### Date of Award

3-13-2011

#### Document Type

Thesis

#### Degree Name

Master of Science in Electrical Engineering (MSEE)

#### Department

Electrical Engineering

#### First Advisor

Christopher A. Kitts

#### Abstract

Cluster space control allows for the precise guidance of a formation of robots. The desired formation geometry is achieved by fixing the positions of individual robots relative to a mobile cluster space coordinate system. The formation is controlled by moving the cluster space origin along a trajectory defined in global coordinates. Robot positions are translated between cluster space and global coordinates by a set of nonlinear kinematic equations; robot velocities are translated by the Jacobian matrix derived from the kinematic equations. Calculation of the Jacobian matrix becomes computationally costly as the number of robots and degrees of freedom increase, which may limit the number of robots in a formation or the accuracy with which they can be controlled.

Multirate control is a method of reducing the computation required for a control loop. The key concept of multirate control is that some portions of a control loop may not need to run as fast as other portions. Partitioning the loop so that some portions can run at a slower rate reduces the overall computational load. The ratio of the fast loop rate to the slow rate is the parameter m; higher values of m mean reduced computation of the slow parts of the loop. In cluster space control, the Jacobian matrix often changes very little between successive iterations of the control loop, suggesting that this expensive calculation can be performed at a slower rate. This thesis explores the potential for multirate control to reduce the computational load of a cluster space control system.

A three-robot cluster space simulation with nine degrees of freedom has been modified to implement multirate control. The ratio m of the main control loop update rate to the update rate of the 9-by-9 Jacobian matrix can be varied to see the effect of multirate control on the error, defined as the difference between the desired and actual trajectories. Formation velocity, shape, and m were parameterized and a test program created to repeatedly run the simulation while varying m and any other single parameter. To graphically display the results, m and the variable parameter are used as the horizontal axes of a surface plot whose height is the peak error during the simulation run.

It was found that proximity to a singularity in the Jacobian requires a faster update rate, diminishing the savings allowed by multirate control. An alternate cluster space orientation which eliminates the singularity allows for a slower update rate, with increased computational savings.

It is shown that the expected computational savings from multirate control is highest in systems for which the Jacobians are dense. As Jacobian sparsity increases, the computational savings offered by multirate control is reduced.

#### Recommended Citation

Meserve, Michael Charles, "Multirate Control Applied to a Cluster Space Robot Formation" (2011). *Electrical and Computer Engineering Master's Theses*. 4.

https://scholarcommons.scu.edu/elec_mstr/4