Multirotor

Multirotor 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 outputs have been verified against the algorithms used, for input and parameter values within reasonable limits (as observed from literature).

• 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.

• The effect of collective pitch input and drag coefficient is incorporated in the block through the thrust and torque coefficients alone. Hence, the collective input port and the drag coefficient is active only when CT and CQ Source dropdown is set to Compute using BEMT.

Ports

Input

Ω (rad/sec) — Rotor speed

N-element vector

Rotor speed, specified as an N-element vector in rad/sec in body frame. The ordering of the rotors for the two quadcopter configurations (N = 4) considered in the block are as shown in Arm Computations.

Data Types: double

ρ — Air density

positive scalar

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

Data Types: double

V_b — Body velocity of rotor

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

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.

Dependencies

This enable this input, set Modeling to With flap effects.

The unit of velocity depends on the value of the Units parameter.

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.

Dependencies

This enable this input, set Modeling to With flap effects.

Data Types: double

θ₀ — Collective blade pitch angle

N-by-1 vector | 1-by-N vector

Collective blade pitch angle, specified as an N-element vector.

Dependencies

To enable this input port, select the Include pitch angle inputs check box and set the CT and CQ Source dropdown to Compute using BEMT.

Data Types: double

θ_c — Lateral cyclic pitch angle

N-by-1 vector | 1-by-N vector

Lateral cyclic pitch angle, specified as an N-element vector.

Dependencies

To enable this input port,

• Select the Include pitch angle inputs check box.

• Set Modeling to With flap effects.

Data Types: double

θ_s — Longitudinal cyclic pitch angle

N-by-1 vector | 1-by-N vector

Longitudinal cyclic pitch angle, specified as an N-element vector.

Dependencies

To enable this input port,

• Select the Include pitch angle inputs check box.

• Set Modeling to With flap effects.

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.

Data Types: double

C_T — Computed thrust coefficient

positive vector

Computed thrust coefficient, returned as a positive vector of N elements.

Dependencies

To enable this output port, select the Output computed CT and CQ check box and set the CT and CQ Source dropdown to Compute using BEMT.

C_Q — Computed torque coefficient

positive vector

Computed torque coefficient, returned as a positive vector of N elements.

Dependencies

To enable this output port, select the Output computed CT and CQ check box and set the CT and CQ Source dropdown to Compute using BEMT.

Parameters

Main

Units — Input and output units

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

Units	Density	Force	Moment	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	Knots	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 used in the computation of forces and moments [2]. For more information, see Force and Moment.

Programmatic Use

Block Parameter: modelMode

Type: character vector

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

Default: 'Without flap effects'

Include pitch angle inputs — Enable pitch angle inputs

off (default) | on

Select this check box to enable the pitch angle (swash plate) input ports.

Dependencies

To enable this checkbox

• Set the CT and CQ Source dropdown to Compute using BEMT

and/or

• Set Modeling to 'With flap effects'

Programmatic Use

Block Parameter: controlInput

Type: character vector

Values: 'on' | 'off'

Default: 'off'

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: armLength

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: rotorHeight

Type: character vector

Values: '-0.0159' | nonzero positive scalar

Default: '-0.0159'

Rotor

CT and CQ Source — Dropdown to select the source of thrust and torque coefficient values

'Dialog' (default) | 'Compute using BEMT'

Select the source of the thrust coefficient and torque coefficient values, specified as a character vector of value 'Dialog' or 'Compute using BEMT'. The block assumes the aerodynamic and structural parameters to be constant.

Select 'Dialog' to manually enter the thrust and torque coefficient values.

Select 'Compute using BEMT' to compute the values using Blade Element Momentum Theory (BEMT). For more information on BEMT, see Thrust and Torque coefficient.

Programmatic Use

Block Parameter: CTCQMode

Type: character vector

Values: 'Dialog' | 'Compute using BEMT'

Default: 'Dialog'

Thrust coefficient (CT) — Thrust coefficient

0.0107 (default) | nonzero positive scalar | nonzero N-element vector

Thrust coefficient, specified as a nonzero positive scalar or N-element vector.

If you define Thrust coefficient (CT) as a scalar, all the rotors are assumed to have same values for CT. If vector, the ordering of the rotors is as shown in Arm Computations.

Dependencies

To enable this parameter, set the CT and CQ Source dropdown to 'Dialog'.

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 | nonzero N-element vector

Torque coefficient, specified as a nonzero positive scalar or N-element vector.

If you define Torque coefficient (CQ) as a scalar, all the rotors are assumed to have same values for CQ. If vector, the ordering of the rotors is as shown in Arm Computations.

Dependencies

To enable this parameter, set the CT and CQ Source dropdown to 'Dialog'.

Programmatic Use

Block Parameter: CQ

Type: character vector

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

Default: '7.8263e-4'

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 checkbox,

• Set the CT and CQ Source dropdown to Compute using BEMT

and/or

• Set Modeling to 'With flap effects'

Programmatic Use

Block Parameter: Nb

Type: character vector

Values: '2' | nonzero positive scalar

Default: '2'

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

off (default) | on

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.

Dependencies

To enable this parameter, set the CT and CQ Source dropdown to 'Compute using BEMT'.

Programmatic Use

Block Parameter: CTout

Type: character vector

Values: 'on' | 'off'

Default: 'off'

Blade

Radius (m) — Rotor radius

0.0330 (default) | nonzero positive scalar

Rotor radius, specified as a nonzero positive scalar.

Programmatic Use

Block Parameter: radius

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.

Dependencies

To enable this checkbox,

• Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

• Set Modeling to 'With flap effects'

Programmatic Use

Block Parameter: chord

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 checkbox,

• Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

• Set Modeling to 'With flap effects'

Programmatic Use

Block Parameter: hingeOffset

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 checkbox,

• Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

• Set Modeling to 'With flap effects'

Programmatic Use

Block Parameter: clalpha

Type: character vector

Values: '5.5' | nonzero positive scalar

Default: '5.5'

Mean drag coefficient — Mean drag coefficient

0 (default) | positive scalar

Mean drag coefficient, specified as a positive scalar. The block does not consider variation with respect to angle of attack.

Dependencies

To enable this parameter, set the CT and CQ Source dropdown to 'Compute using BEMT'.

Programmatic Use

Block Parameter: cd0

Type: character vector

Values: '0' | positive scalar

Default: '0'

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.

  $$\theta (r)={\theta }_{root}+{\theta }_{twist}r$$

• Ideal — Optimal approximation of blade twist distribution.

  $$\theta (r)=\frac{{\theta }_{tip}}{r}$$

where r is the nondimensional radial location along the blade.

Dependencies

To enable this checkbox,

• Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

• 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) | real scalar

Blade root pitch angle θ_root, specified as a real scalar.

Dependencies

To enable this parameter:

• Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

• Set Modeling to 'With flap effects'

• Set Twist distribution to Linear.

Programmatic Use

Block Parameter: thetaRoot

Type: real scalar

Values: '0.2548' | nonzero positive scalar

Default: '0.2548'

Blade twist angle (rad) — Blade twist angle

-0.1361 (default) | real scalar

Blade twist angle θ_twist, specified as a real scalar.

Dependencies

To enable this parameter:

• Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

• Set Modeling to 'With flap effects'

• Set Twist distribution to Linear.

Programmatic Use

Block Parameter: thetaTwist

Type: real scalar

Values: '0.1018' | positive scalar

Default: '0.1018'

Blade tip angle (rad) — Blade tip pitch angle

0.1018 (default) | real scalar

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

Dependencies

To enable this parameter:

• Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

• Set Modeling to 'With flap effects'

• Set Twist distribution to Ideal.

Programmatic Use

Block Parameter: thetaTip

Type: real scalar

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.

• $$d=\frac{l}{\sqrt{2}}$$.

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

• Without flap effects

  $$\begin{array}{l}{F}_{xyz}={\displaystyle \sum _{i=1}^N{F}_{xy{z}_i}=}{\displaystyle \sum _{i=1}^N\left(\left[\begin{array}{c}0\\ 0\\ -C_{T_i}\rho \pi R^2{({\Omega }_{i}R)}^2\end{array}\right]\right)}\\ M_{xyz}={\displaystyle \sum _{i=1}^N\left(\left[\begin{array}{c}0\\ 0\\ -C_{Q_i}\rho \pi R^2{\Omega }_i\left|{\Omega }_i\right|R^3\end{array}\right]+p_i\times F_{xy{z}_i}\right)}\end{array}$$

• With flap effects

  $$\begin{array}{l}{F}_{xyz}={\displaystyle \sum _{i=1}^N{F}_{xy{z}_i}}={\displaystyle \sum _{i=1}^N\left(C_{T_i}\rho \pi R^2{({\Omega }_{i}R)}^2\left[\begin{array}{c}-\cos {\beta }_{s_i}\sin {\beta }_{c_i}\\ \sin {\beta }_{{}_{s_i}}\\ -\cos {\beta }_{c_i}\cos {\beta }_{{}_{s_i}}\end{array}\right]\right)}\\ M_{xyz}={\displaystyle \sum _{i=1}^N\left(\left[\begin{array}{c}0\\ 0\\ -C_{Q_i}\rho \pi R^2{\Omega }_i\left|{\Omega }_i\right|R^3\end{array}\right]+p_i\times F_{xy{z}_i}\right)}\end{array}$$

where:

• i is the rotor index and p_i is the arm location corresponding to the ith rotor.

• Ω is the rotor speed in rad/s.

• β_c and β_s are the steady state flap angles computed using the equations in Chapter 7 of [1].

Thrust Coefficient and Torque Coefficient Calculations

When the Compute CT and CQ check box is selected, the Multirotor 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:

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

For the equations in this section:

• r is the nondimensional radial location.

• λ is the inflow through the rotor disc.

• c_lα is the lift curve slope.

• σ is the rotor solidity defined as $$\sigma =\frac{N_{b}c}{\pi R}$$, where c is the chord, R is the radius and N_b is the number of blades.

According to blade element theory, the incremental thrust coefficient is:

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

Here, θ(r) will depend on the blade twist distribution and the collective pitch input θ₀. Equating the two expressions for dC_T returns 4Fλ² = 0.5 σc_ıα (θ(r)r − λ), which can be iteratively solved for λ.

Sum dС_T = 4Fλ²r dr across the blade span, to find the net thrust coefficient C_T.

For the torque coefficient C_Q, the profile drag component is approximately included as $$\frac{\sigma c_{d0}}{8}$$. The net torque coefficient can be calculated by summing dС_Q = λdС_T across the blade span and adding the profile drag component.

References

[1] Prouty, Raymond W. Helicopter Performance, Stability, and Control Krieger Publishing Company, 1995.

[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

R2024b: Multirotor block now includes swashplate as input

• The Multirotor block has been updated to support swashplate inputs. This addition allows for the inclusion of collective and cyclic blade pitch angles in the calculation of Forces and Moments through new optional input ports.

• Thrust coefficient (CT) and Torque coefficient (CQ) parameters now also accept N-element vectors as inputs.

• Starting R2024b, the Multirotor block has been updated to include the dropdown CT and CQ Source with two options: 'Dialog' and 'Compute using BEMT'.

  • Select 'Dialog' to manually enter the thrust and torque coefficient values.

  • Select 'Compute using BEMT' to compute the values using Blade Element Momentum Theory (BEMT). For more information on BEMT, see Thrust and Torque coefficient. This option is equivalent to the Compute CT and CQ checkbox from previous releases.