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Discussion and Practical Aspects on Control

Allocation for a Multi-rotor Helicopter

Guillaume Ducard Minh Duc Hua

I3S UNS-CNRS, France

UAV-G 2011, ETH Zurich, 16-09-2011

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Outline

1 Problem Statement

2 Limitations of Classical Pseudo-Inverse Matrix Method

3

New Approach: Weighted Pseudo-Inverse Matrix Method

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Problem Statement

Limitations of Classical Method

New Approach

Hexacopter

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Problem Statement


New Approach

What is Control Allocation?

Multi-rotor Helicopter Control Allocation1 given a virtual control input vectorvcmd = [T , L, M , N ]

cmd from the flight

controller,

2 find the set of propeller speedsΩ := [ω2

1; · · · ; ω2

n ], where the numberof propellers is n,

3 such that vcmd = AΩ, with the

constraints

ω2

min ≤ ω2

i ≤ ω2

max, ∀i, i =1

. . .n

.

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Problem Statement


New Approach

What is Control Allocation?

Multi-rotor Helicopter Control Allocation1 given a virtual control input vectorvcmd = [T , L, M , N ]

cmd from the flight

controller,

2 find the set of propeller speedsΩ := [ω2

1; · · · ; ω2

n ], where the numberof propellers is n,

3 such that vcmd = AΩ, with the

constraints

ω2

min ≤ ω2

i ≤ ω2

max, ∀i, i =1

. . .n

.

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S

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Problem Statement


New Approach

Quadricopter vs. Hexacopter

Quadricopter Case

T

L

M

N

v

=

µ µ µ µ0 −µl 0 µlµl 0 −µl 0

−κ κ

−κ κ

A

ω12

ω22

ω32

ω42

Ω

Hexacopter Case

T

L M

N

v

=µ µ µ µ µ µ

0 −√

3lµ

2 −√

3lµ

2 0

√ 3lµ

2

√ 3lµ

2

lµ lµ

2− lµ

2−lµ − lµ

2

lµ

2−κ κ −κ κ −κ κ

A

ω1

2

ω2

2

...ω6

2

Ω

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Problem Statement


New Approach

Classical Control Allocation Methods

1 computing the (pseudo-) inverse of the matrix A,

2 saturating the computed propeller speeds in between the min and the

max of the propeller speeds possible.

Quadricopter Case

Ωc = A−1vc.

Hexacopter Case

Ωc = A+vc, with A

+ = A(AA)−1 =

1

6µl

l 0 2

−µlκ−1

l −√3 1 µlκ−1

l −√3 −1 −µlκ−1

l 0 −2 µlκ−1

l√

3 −1 −µlκ−1

l√

3 1 µlκ−1

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Problem Statement

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Problem Statement


New Approach

Key contributions

1 to show that a control allocation strategy based on theclassical approach of pseudo-inverse only exploits a

limited range of the vehicle capabilities to generate thrust

and moments,

2

a novel approach: based on a weighted pseudo-inversemethod capable of exploiting a much larger domain

achievable in v = [T , L, M , N ]cmd

,

3 and finally, the control allocation algorithm is formulated in

terms of explicit laws for fast operation and low

computational load, suitable for a microcontroller withlimited computation capability.

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Problem Statement

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Problem Statement


New Approach

Key contributions



and moments,

2



,




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Problem Statement

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Problem Statement


New Approach

Classical Method for Control Allocation

Limitations of Classical Pseudo-Inverse Matrix Method

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Problem Statement




New Approach

Classical Method for Control Allocation

T minl ≤ Tl ± 2 M µlκ−1

N ≤ T maxl

T minl ≤ Tl √3 L ± M ± µlκ−1

N ≤ T maxl. (2)

In particular, when N = 0, one verifies that

2| M | ≤ min (T − T min)l, (T max − T )l√

3| L| + | M | ≤ min (T − T min)l, (T max − T )l .

Define T = min

(T

−T min), (T max

−T )

∈[0; T min+T max

2],

2| M | ≤ T l√3| L| + | M | ≤ T l

.

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Problem Statement

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New Approach

Classical Method

2| M | ≤ Tl√3| L| + | M | ≤ Tl

The admissible area interms of moments L, M for a given total thrust T isin red.

This zone can be enlargedwith the method presentedhereafter.

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Problem Statement

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New Approach

New Approach

New Approach: Weighted Pseudo-Inverse Matrix Method

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Problem Statement

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New Approach

General Formulation

Let us introduce a diagonal weighting matrixW := diag([a; b; c; a; b; c]), where a, b, c are non-negative and

satisfy the condition a + b + c = 1.

A+W

=WA(AWA)−1 =1

6µl

3al 0 2

−µlκ−1

3bl −√3 1 µlκ−1

3cl −√3 −1 −µlκ−1

3al 0 −2 µlκ−1

3bl√

3 −1 −µlκ−1

3cl√

3 1

µl

κ

−1

. (3)

The propellers speed are obtained by Ω = A+Wv

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Problem Statement

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New Approach

General Formulation (2)

ω2

min ≤ ω2

i ≤ ω2

max, ∀i, i = 1 . . . 6 (4)

The above constraints are satisfied if and only if

T minl ≤ 3a Tl ± 2 M µlκ−1 N ≤ T maxl

T minl ≤ 3b Tl √3 L ± M ± µlκ−1

N ≤ T maxl

T minl ≤ 3c Tl √3 L M µlκ−1

N ≤ T maxl

. (5)

In the case N = 0 (no yaw torque control), it simplifies to

2| M | ≤ min(3aT −T min) l, (T max−3aT ) l|√3 L− M | ≤ min(3bT −T min) l, (T max−3bT ) l|√3 L+ M | ≤ min(3cT −T min) l, (T max−3cT ) l

(6)

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Problem Statement





New Approach

L, M area attainable in Classical vs. New Method,

(N=0).

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Problem Statement


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New Approach

New method: L, M aera as a function of thrust, (N=0)

In practice: → designhexacopters such that the total

thrust magnitude T always

remains in the interval

[(T max +2T min)/3, (2T max +T min)/3].

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Problem Statement


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New Approach

Computing the Coefficients a, b, c

When the set-point (¯ L, ¯ M ) is

inside or on the borderlines of theclassical admissible hexagon , we

set a = b = c = 1/3.

When the set-point (¯ L, ¯ M ) stays

outside the classical admissible

hexagon but inside the weighted admissible hexagon ,

a = 1

3+aw −

1

3

δ

b = 1

3+

bw −

1

3

δ

c = 1− a− b

(7)

with

δ :=

¯ L−¯ Lc¯ Lw−

¯ Lcif ¯ M w = ¯ M c

¯ M −¯ M c¯ M w−

¯ M cotherwise .

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Problem StatementLimitations of Classical Method

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New Approach




set a = b = c = 1/3.




a = 1

3+aw −

1

3

δ

b = 1

3+

bw −

1

3

δ

c = 1− a− b

(7)

with

δ :=

¯ L−¯ Lc¯ Lw−


¯ M −¯ M c¯ M w−

¯ M cotherwise .

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New Approach

Properties

Smoothness

The proposed interpolation method ensures that the

variations of the values of a, b, c are continuous if the

reference set-point (¯ L, ¯ M ) varies smoothly over time.

This is important in practice: since it ensures that thedesired motors’ speeds ωi, (Ω = A

+Wv) vary continuously if

the control input vector v is continuous in time.

Extension to the Case of Non-null Yaw Control, N = 0

Described in the paper.

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New Approach

Conclusions

1 Enlarged domain in torques L, M using the new method

vs. classical,

2 Continuity in the coefficients a, b, c, → smooth motor

control if input vector is smooth,

3 Computationally efficient → no optimization problem tosolve, availability of explicit solutions.

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New Approach

Questions

Thank you for your attention.

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