Course AE4-T40

Lecture 2:

2D ModelsOf Kite and Cable

Overview

• 2D system model

• 3D system model

• Kite steering

• 3D kite models

2D System Model

Tether: lumped masses with rigid links. External Forces applied at the lumped massesKite is controlled via angle of attack, attitude and flexible dynamics are ignored

Generalized coordinates are the angular rotation of each link θj, Φj, with j= 1…n, and n the number of masses

Length of the links lj(t) are a function of time and thus no generalized coordinatesWhen the line length is changed, only line length of segment ln is changed

2D System Model

For 2D case assume out of plane angle Φj = 0 for all cable elements

For illustration purposes now consider a system model with n = 3

With unit vectors i, j, k in x, y and z (using only I and j) we now define the inertialpositions of the three point masses with respect to the reference axes in terms of the generalized coordinates

Position Vectors of line lump masses

Corresponding velocities and accelerations are determined by differentiation of the position vectors

Velocity vectors of line lump masses

In general we have:

Acceleration vectors of line lump masses

Kanes equations of motionLagrange’s equations are second order differential equations in the generalized coordinates qi (i = 1,…,n). These may be converted to first-order differential equations or into state-space form in the standard way, by defining an additional set of variables, called motion variables.

To convert Lagrange’s equations, one defines the motion variables simply as configuration variable derivatives, sometimes called generalized velocities. Then the state vector is made up of the configuration and motion variables:the generalized coordinates and generalized velocities.

In Kane’s method, generalized coordinates are also used as configuration variables. However, the motion variables in Kane’s equations are defined as functions that are linear in the configuration variable derivatives and in general nonlinear in the configuration variables. The use of such functions can lead to significantly more compact equations. The name given to these new motionvariables is generalized speeds and the symbol commonly used is u.

Kanes Equations

Kanes equations

Generalized Inertia Forces for n = 3

Generalized Inertia Forces for n = 3

Generalized Inertia Forces for n = 3

Generalized Inertia Forces for n = n

Generalized External Forces

Kane’s equations of motion

Where for our system the external forces are composed of:

-Tether drag-Kite Lift and Drag-Gravity

Tether Drag

Kite Lift and Drag

Gravity

Model results

Control results

Control Results

Control Results