Multirotor

Multipropeller dynamics

Since R2023a

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Libraries:
Aerospace Blockset / Propulsion

Description

The Multirotor block computes the aerodynamic forces and moments generated by multiple rotating propellers or rotors, such as quadcopters, in all three dimensions. You can also include simplified flap, or steady state, aerodynamics.

Limitations

• The block follows a simplified approach with the option to include steady state flap effects. It does not model dynamic flap, lag, or feathering motion of blade.

• The block does not directly consider the variation in pitch angle input, although you can perform approximate analysis by varying the blade twist.

• The block uses Twist distribution to model only linear or ideal twist distributions. The block assumes that the blade chord and lift curve slope are constant.

• The block does not include interference effects (between rotors) in the model.

Ports

Input

Ω (rad/sec) — Rotor speed

1-by-N vector

Rotor speed, specified as a 1-by-N vector in rad/sec in body frame.

Data Types: double

ρ — Air density

positive scalar

Air density, specified as a positive scalar in specified units.

Data Types: double

V_b (m/s) — Body velocity of rotor

3-by-1 vector | 1-by-3 vector

Body velocity of rotor, specified as a 3-by-1 or 1-by-3 vector, in body frame. To perform a multisystem analysis, consider connecting output from a State-Space or Integrator block to this port.

Data Types: double

ω_b — Angular velocity

3-by-1 vector | 1-by-3 vector

Angular velocity of entire vehicle, specified as a 3-by-1 or 1-by-3 vector in rad/s in body frame.

Data Types: double

Output

F_xyz(N) — Total force

three-element vector

Total force, returned as a three-element vector in body frame. Units depend on the Units parameter.

Data Types: double

M_xyz(Nm) — Net moment

three-element vector

Net moment in the x-y-z direction, returned as a three-element vector in body frame. Units depend on the Units parameter.

C_Q — Computed torque coefficient

scalar positive

Computed torque coefficient, returned as a scalar positive.

Dependencies

To enable this output port, select these check boxes:

• Compute CT and CQ

• Output computed CT and CQ

C_T — Computed thrust coefficient

scalar positive

Computed thrust coefficient, returned as a scalar positive.

Dependencies

To enable this output port, select these check boxes:

• Compute CT and CQ

• Output computed CT and CQ

Parameters

Units — Input and output units

Metric (MKS) (default) | English (Velocity in ft/s) | English (Velocity in kts)

Units	Density	Force	Mass	Velocity	Radius	Chord	Hinge offset
Metric (MKS)	kg/m3	Newtons	Newton-meter	Meters per second	Meters	Meters	Meters
English (Velocity in ft/s)	slug/ft3	Pound force	Pound force-feet	Feet per second	Feet	Feet	Feet
English (Velocity in kts)	slug/ft3	Pound force	Pound force-feet	Feet per second	Feet	Feet	Feet

Programmatic Use

Block Parameter: units

Type: character vector

Values: 'Metric (MKS)' | 'English (Velocity in ft/s)' | 'English (Velocity in kts)'

Default: 'Metric (MKS)'

Modeling — Rotor thrust calculation method

Without flap effects (default) | With flap effects

Rotor thrust calculation method, specified as:

• Without flap effects — Model rotor thrust using force and moment calculations. For more information, see Force and Moment.

• With flap effects — Effect of tilt in rotor disc due to flap motion, while in forward flight, is included. The steady state lateral and longitudinal flap angles are calculated using the equations from [1] and [2].

Programmatic Use

Block Parameter: modelMode

Type: character vector

Values: 'Without flap effects' | 'With flap effects'

Default: 'Without flap effects'

Vehicle

Configuration — Vehicle configuration

Quadcopter X (default) | Quadcopter +

Quadcopter configuration, specified as:

• Quadcopter X — Propellers configured as an X shape, considered as more stable than the Quadcopter + configuration.

• Quadcopter + — Propellers configured as a + shape.

For more information, see Arm Computations.

Programmatic Use

Block Parameter: config

Type: character vector

Values: 'Quadcopter X' | 'Quadcopter +'

Default: 'Quadcopter X'

Arm length (m) — Vehicle arm length

0.0624 (default) | nonzero positive scalar

Vehicle arm length, specified as a nonzero positive scalar. This length is the rotor displacement from the center of mass of the vehicle. For more information on arm lengths, see Algorithms.

Programmatic Use

Block Parameter: l

Type: character vector

Values: '0.0624' | nonzero positive scalar

Default: '0.0624'

Height (m) — Vertical offset from vehicle center of gravity

-0.0159 (default) | nonzero scalar

Vertical offset from vehicle center of gravity, specified as a nonzero scalar.

Programmatic Use

Block Parameter: h

Type: character vector

Values: '-0.0159' | nonzero positive scalar

Default: '-0.0159'

Rotor

Number of blades — Number of blades per rotor

2 (default) | nonzero positive scalar

Number of blades per rotor, specified as a nonzero positive scalar.

Dependencies

To enable this parameter, select the Compute CT and CQ check box and set Modeling to With flap effects.

Programmatic Use

Block Parameter: Nb

Type: character vector

Values: '2' | nonzero positive scalar

Default: '2'

Compute CT and CQ — Option to calculate thrust coefficient and torque coefficient

on (default) | off

Select this check box to let the block calculate the thrust coefficient and torque coefficient. For more information on these calculations, see Thrust Coefficient and Torque Coefficient Calculations. The block assumes the aerodynamic and structural parameters to be constant.

Clear this check box to specify thrust coefficient and torque coefficient values.

Programmatic Use

Block Parameter: CTcheck

Type: character vector

Values: 'on' | 'off'

Default: 'off'

Output computed CT and CQ — Output computed thrust coefficient and torque coefficient

on (default) | off

Select this check box to output the calculated thrust coefficient and torque coefficient to C_T and C_Q output ports. For more information on these calculations, see Thrust Coefficient and Torque Coefficient Calculations. The block assumes the aerodynamic and structural parameters to be constant.

Otherwise, clear this check box.

Programmatic Use

Block Parameter: CTout

Type: character vector

Values: 'on' | 'off'

Default: 'off'

Thrust coefficient (CT) — Thrust coefficient

0.0107 (default) | nonzero positive scalar

Thrust coefficient, specified as a nonzero positive scalar.

Dependencies

To enable this parameter, clear the Compute CT and CQ check box.

Programmatic Use

Block Parameter: CT

Type: character vector

Values: '0.0107' | nonzero positive scalar

Default: '0.0107'

Torque coefficient (CQ) — Torque coefficient

7.8263e-4 (default) | nonzero positive scalar

Torque coefficient, specified as a nonzero positive scalar.

Dependencies

To enable this parameter, clear the Compute CT and CQ check box.

Programmatic Use

Block Parameter: CQ

Type: character vector

Values: '7.8263e-4' | nonzero positive scalar

Default: '7.8263e-4'

Radius (m) — Rotor radius

0.0330 (default) | nonzero positive scalar

Rotor radius, specified as a nonzero positive scalar.

Programmatic Use

Block Parameter: R

Type: character vector

Values: '0.0330' | nonzero positive scalar

Default: '0.0330'

Chord (m) — Blade chord

0.0080 (default) | nonzero positive scalar

Blade chord, specified as a nonzero positive scalar.

Programmatic Use

Block Parameter: c

Type: character vector

Values: '0.0080' | nonzero positive scalar

Default: '0.0080'

Hinge offset (m) — Hinge offset

0 (default) | positive scalar

Hinge offset, specified as a positive scalar. This value is typically 0 for propellers.

Dependencies

To enable this parameter, select the Compute CT and CQ check box and set Modeling to With flap effects.

Programmatic Use

Block Parameter: m

Type: character vector

Values: '0' | positive scalar

Default: '0'

Lift curve slope (per rad) — Lift curve slope

5.5 (default) | nonzero positive scalar

Lift curve slope, specified as a nonzero positive scalar. The block assumes the aerodynamic and structural parameters to be constant. The block does not consider variation with respect to angle of attack.

Dependencies

To enable this parameter, select the Compute CT and CQ check box and set Modeling to With flap effects.

Programmatic Use

Block Parameter: clalpha

Type: character vector

Values: '5.5' | nonzero positive scalar

Default: '5.5'

Lock number — Lock number

0.6051 (default) | nonzero positive scalar

Lock number, which is the ratio of aerodynamics forces to inertial forces, specified as a nonzero positive scalar.

Dependencies

To enable this parameter, set Modeling to With flap effects.

Programmatic Use

Block Parameter: gamma

Type: character vector

Values: '0.6051' | nonzero positive scalar

Default: '0.6051'

Twist distribution — Rotor blade twist distribution

Linear (default) | Ideal

Rotor blade twist distribution, specified as:

• Linear — Close approximation of blade twist distribution.

• Ideal — Optimal approximation of blade twist distribution.

Dependencies

To enable this parameter, select the Compute CT and CQ check box and set Modeling to With flap effects.

Programmatic Use

Block Parameter: twistType

Type: character vector

Values: 'Linear' | 'Ideal'

Default: 'Linear'

Blade root angle (rad) — Blade root pitch angle

0.2548 (default) | nonzero positive scalar

Blade root pitch angle θ₀, specified as a nonzero positive scalar.

Dependencies

To enable this parameter:

• Set Modeling to With flap effects.

• Set Twist distribution to Linear.

Programmatic Use

Block Parameter: theta0

Type: character vector

Values: '0.2548' | nonzero positive scalar

Default: '0.2548'

Blade twist angle (rad) — Blade twist angle

-0.1361 (default) | positive scalar

Blade twist angle θ₁, specified as a positive scalar.

Dependencies

To enable this parameter:

• Set Modeling to With flap effects.

• Set Twist distribution to Linear.

Programmatic Use

Block Parameter: theta1

Type: character vector

Values: '-0.1361' | positive scalar

Default: '-0.1361'

Blade tip angle (rad) — Blade tip pitch angle

0.06 (default) | nonzero positive scalar

Blade tip pitch angle θ_tip for ideal twist distribution, specified as a nonzero positive scalar.

Dependencies

To enable this parameter:

• Set Modeling to With flap effects.

• Set Twist distribution to Ideal.

Programmatic Use

Block Parameter: thetaTip

Type: character vector

Values: '0.06' | nonzero positive scalar

Default: '0.06'

Algorithms

Arm Computations

Quadcopter x

p1=(d,-d,-h)

p2=(d,d,-h)

p3=(-d,d,-h)

p4=(-d,-d,-h)

where:

• l is the arm length.

• h is the offset from the vehicle center of gravity.

In this graphic:

• The x-, y-, z-axes are perpendicular to each other.

• The four arms of the vehicle are perpendicular and equal in length (l).

• All propellers are at same height (h).

Quadcopter +

p1=(l,0,h)

p2=(0,l,h)

p3=(-l,0,h)

p4=(0,-l,h)

where:

• l is the arm length.

• h is the offset from the vehicle center of gravity.

In this graphic:

• The x-, y-, z-axes are perpendicular to each other.

• The four arms of the vehicle are perpendicular and equal in length (l).

• All propellers are at same height (h).

Force and Moment

Force and moment calculated using these equations:

$$\begin{array}{l}F\_xyz=\left[\begin{array}{ccc}0;& 0;& -b{\omega }^2\end{array}\right]\\ M\_xyz=\left[\begin{array}{ccc}0;& 0;& k{\omega }^2\end{array}\right]\end{array}$$

where:

• ω is the rotor in revolutions per minute (RPM).

• b and k are experimental values:

  $$\begin{array}{l}b=C_T\rho (\pi R^2)R^2\\ k=-C_Q\rho (\pi R^2)R^3\end{array}$$

  where:

  • C_T is the thrust coefficient

  • C_Q is the torque coefficient

Thrust Coefficient and Torque Coefficient Calculations

When the Compute CT and CQ check box is selected, the block calculates the thrust coefficient and torque coefficient using these equations.

With the inclusion of Prandtl’s tip loss function, the incremental thrust coefficient using blade element momentum theory is:

$$d{C}_T=4F{\lambda }^{2}rdr$$

where F is the correction factor:

$$\begin{array}{l}F=\frac{2}{\pi }{\cos }^{-1}exp-f\\ f=\frac{1}{2}\frac{Nb(1-r)}{r\phi }\\ \phi =\frac{\lambda }{r}\end{array}$$

Using blade element theory, the incremental thrust coefficient is:

$$d{C}_T=\frac{1}{2}\sigma c_{l\alpha }(\theta (r)r-\lambda )rdr$$

with ideal twist,

$$\theta (r)=\frac{{\theta }_t}{r}$$

Resulting in:

$$d{C}_T=.5\sigma c_{l\alpha }({\theta }_{t-\lambda r})rdr$$

Equating the two equations for dC_T, we have:

$$4F{\lambda }^2=.5\sigma c_{l\alpha }({\theta }_t-\lambda )$$

The equation can be solved to obtain λ using the fsolve from the Optimization Toolbox™. To remove the dependency on the toolbox, the Newton-Raphson method is used:

$$fun=@(\lambda )4F(r,\lambda ){\lambda }^2-.5\sigma c_{l\alpha }({\theta }_t-\lambda )$$

The differential of fun as diff(fun,λ):

$$\begin{array}{l}dfun=\frac{c_{l\alpha }\sigma }{2}+\frac{\text{5734161139222659*}\lambda {\text{*cos}}^{-1}(\exp \left(\frac{Nb*(r-1)}{2*\lambda }\right)}{\text{1125899906842624}}\\ \text{ +}\frac{{\text{5734161139222659*N}}_b*\exp \left(\frac{Nb*(r-1)}{2*\lambda }\right)*(r-1)}{(\text{4503599627370496*(1-exp}\left(\frac{Nb*(r-1)}{\lambda }\right){)}^{\left(\frac{1}{2}\right)})}\end{array}$$

To avoid numerical issues, the ratios are approximated as:

$$\begin{array}{l}dfun=\frac{c_{l\alpha }}{2}+5.0930*\lambda *{\cos }^{-1}\left(\exp \left(\frac{Nb*(r-1)}{2*\lambda }\right)\right)\\ \text{ +}\frac{1.2732*Nb*\exp \left(\frac{Nb*(r-1)}{2*\lambda }*(r-1)\right)}{(1-\exp {\left(\frac{Nb*(r-1)}{\lambda }\right)}^{\left(\frac{1}{2}\right)})}\end{array}$$

After λ is calculated, C_T, C_Q are calculated by summing up dC_T=4λ²r dr and dC_Q =λdC_T across the radius.

References

[1] Pounds, P. E. I. (2007). Design, construction and control of a large quadrotor micro air vehicle (Doctoral dissertation, Australian National University).

[2] Riether, F. (2016). Agile quadrotor maneuvering using tensor-decomposition-based globally optimal control and onboard visual-inertial estimation (Doctoral dissertation, Massachusetts Institute of Technology).

Version History

Introduced in R2023a