Exercise on Units and Quaternions - Percro

Elements of MATLAB and Simulink
Scuola Superiore S.Anna
Lecture 5 <strong>Exercise</strong>s, 05/05/2009
Lecturer: Emanuele Rualdi
1 Subsystems
1.1 Wiring and Triggering
Create a sequence of Subsystems that are activated only once at the start of the
Simulation and are executed in the correct order of dependencies
1.2 Math and Libraries
1.2.1 Quaternions
A unit quaternion (normalized) is a four dimensional vector that can be used for
expressing rotation in Three dimensional space. Create a library of blocks that can
be used for manipulating quaternions:
Quaternion: (w, (x, y, z)) also for a quaternion p expressed as (p r , p v ) where p v
is a vector
From Axis Angle (a, (x, y, z)) becomes (cos a/2, (x sin a/2, y sin a/2, z sin a/2))
A real value (scalar) is a quaternion with q v = 0 while a vector is a quaternion
with no scalar part q r = 0
1. Conjugate and Inverse (q r , −q v )
2. Conversion From Axis Angle representation
3. Multiplication (p r q r − p v q v , p r q v + q r p v + p v × q v )
4. Rotation of Vector by Quaternion
5. Decomposition of Quaternion in Axis Angle
6. Exponentiation q t = (cos t · q r , q v sin t · q r )
7. Slerp (Linear Interpolation): Slerp(q0, q1, t) = q 0 (q −1
0 q 1 ) t
8. Finally package the Blocks in a Library
References: some functionalities are already in the Aerospace Blockset— and in an
online QLib Toolbox (http://www.mathworks.com/matlabcentral/leexchange/259)
Elements of MATLAB and Simulink-1

1.3 Masking
1.3.1 Unit Conversion
Create a simple unit conversion block that takes some input dimension. Examples
of units:
ˆ Mass
ˆ Length
ˆ Currency (input currency rate as a matrix or as input)
1. Example of parameter specied conversion
2. Example of Fixed Unit (e.g. kg) to user specied Unit
3. Example of double unit
1.3.2 Unit Check with Bus
Create a set of blocks that allows to manage values with their associated unit. Each
dimensioned value is now represented by a Bus object with two signals the value
and the unit (value, unit). For the moment the unit can be chosen from a limited
list (e.g. 1 for kg, 2 for ton, 3 for pund, 4 for ounces).
Required Blocks:
1. From Adimensional value to Dimensional (selecting Dimension by popup)
2. Unit check between 2 inputs (Stop Simulation in case of mismatch)
3. Unit conversion from input bus to selected Dimension
4. From Dimensional to Value (just a Bus Selector)
Note: ideally this unit check and propagation should be done only at Editing time
before the start of the Simulation
Optional variant of this approach: instead of unit scalar component use a vector
component in which the second value is a scaling factor. In this way it is possible
to represent the unit as: (coreunit, scale) for example kg are (1, 1) and ton are
(1, 1000) .
Optional variant II: a more sophisticated approach uses instead a vector unit plus
scale in which each element of the vector is the power of an SI unit. For example
(length, mass, time, current, temp) and for example Energy is (2, 1, −2).
Elements of MATLAB and Simulink-2