# Difference between revisions of "RotorcraftMixing"

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+ | This page describe how to compute "mixing" for an arbitrary multirotors configuration. | ||

+ | |||

[[Image:hexa_3D.png|800px]] | [[Image:hexa_3D.png|800px]] | ||

− | + | == Introduction == | |

+ | |||

+ | "Mixing" consist in converting rotational accelerations commands computed by the autopilot into throttle commands for each of the motor controllers. | ||

Let us consider a vehicle comprising a set of <math>N</math> identical fixed pitch rotors <math>R_i, i \in[1:N]</math> located at coordinates <math>(X_i,Y_i, 0), i\in[1:N]</math> and spinning in the same plane in the direction <math>D_i, i\in[1:N], D_i\in[-1;1]</math> at a rotational speed <math>\omega_i, i\in[1:N]</math>. | Let us consider a vehicle comprising a set of <math>N</math> identical fixed pitch rotors <math>R_i, i \in[1:N]</math> located at coordinates <math>(X_i,Y_i, 0), i\in[1:N]</math> and spinning in the same plane in the direction <math>D_i, i\in[1:N], D_i\in[-1;1]</math> at a rotational speed <math>\omega_i, i\in[1:N]</math>. | ||

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</math> | </math> | ||

− | <math>A</math> is a <math>3*N</math> matrix describing the geometric positions of our rotors. | + | <math> |

+ | \overrightarrow{M}^{B} = A \overrightarrow{U} | ||

+ | </math> | ||

+ | |||

+ | <math>A</math> is a <math>3*N</math> matrix describing the geometric positions of our rotors and <math>U</math> is the vector of throttle commands for our set of motor controllers. | ||

In order to express the command applied to each power train as a function of the momentum we want to apply to our vehicle, we need to find <math>B</math>, a <math>N*3</math> matrix such as | In order to express the command applied to each power train as a function of the momentum we want to apply to our vehicle, we need to find <math>B</math>, a <math>N*3</math> matrix such as | ||

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We know that one solution of this equation is the Moore-Penrose pseudoinverse of <math>A</math>. | We know that one solution of this equation is the Moore-Penrose pseudoinverse of <math>A</math>. | ||

Furthermore, this solution is the one leading to the power train command vector having the smallest euclidian norm, which is interesting as it optimizes the repartion of our control effort across our power trains. | Furthermore, this solution is the one leading to the power train command vector having the smallest euclidian norm, which is interesting as it optimizes the repartion of our control effort across our power trains. | ||

+ | |||

+ | == Example == | ||

+ | |||

+ | Let's consider the following H hexarotors configuration. | ||

+ | |||

+ | [[Image:hexa_h.png|200px]] | ||

+ | |||

+ | <math> | ||

+ | A = | ||

+ | \begin{pmatrix} | ||

+ | -0.35 &-0.35 &0 & 0 & 0.35 & 0.35 \\ | ||

+ | 0.17 &-0.17 &0.25&-0.25& 0.33 &-0.33 \\ | ||

+ | -0.1 & 0.1 &0.1 &-0.1 &-0.1 & 0.1 | ||

+ | \end{pmatrix} | ||

+ | </math> | ||

+ | |||

+ | <math> | ||

+ | B = | ||

+ | \begin{pmatrix} | ||

+ | - 0.714 & 0.241 & - 1.465 \\ | ||

+ | - 0.714 & - 0.241 & 1.465 \\ | ||

+ | 0 & 0.929 & 2.441 \\ | ||

+ | 0 & - 0.929 & - 2.441 \\ | ||

+ | 0.714 & 0.687 & - 1.094 \\ | ||

+ | 0.714 & - 0.687 & 1.094 | ||

+ | \end{pmatrix} | ||

+ | </math> |

## Revision as of 19:24, 27 January 2011

This page describe how to compute "mixing" for an arbitrary multirotors configuration.

## Introduction

"Mixing" consist in converting rotational accelerations commands computed by the autopilot into throttle commands for each of the motor controllers.

Let us consider a vehicle comprising a set of identical fixed pitch rotors located at coordinates and spinning in the same plane in the direction at a rotational speed .

Assuming a quasi hovering regime, the force produced by each rotor can be considered normal to the rotor plane and proportional to the square of its rotational speed. Expressed in body frame ( front, right, down ), this leads to

Under the same assumption, the torque produced by each rotor can also be assumed to be in the same direction and proportional to the square of the rotational speed.

The moment produced by each rotor around the CG, expressed in body frame can then be writen

Where is a "thrust" coefficient and is a "torque" coefficient. It has been measured experimentally that on a mikrokopter.

As a first approximation we can consider that the rotational speed of the propeller is proportional to the square root of the applied command

This allows us to express the momentum produced by the set of rotors as

which can be rewriten as a matrix vector product

is a matrix describing the geometric positions of our rotors and is the vector of throttle commands for our set of motor controllers.

In order to express the command applied to each power train as a function of the momentum we want to apply to our vehicle, we need to find , a matrix such as

If has rank 3, we know that such a matrix exists (yeah, we can't do 2 rotors or have all rotors aligned), and in this case, we have the relationship

We know that one solution of this equation is the Moore-Penrose pseudoinverse of . Furthermore, this solution is the one leading to the power train command vector having the smallest euclidian norm, which is interesting as it optimizes the repartion of our control effort across our power trains.

## Example

Let's consider the following H hexarotors configuration.