# Vehicle Body 6DOF

6DOF rigid vehicle body to calculate translational and rotational motion

• Library:
• Vehicle Dynamics Blockset / Vehicle Body

• ## Description

The Vehicle Body 6DOF block implements a six degrees-of-freedom (DOF) rigid two-axle vehicle body model to calculate longitudinal, lateral, vertical, pitch, roll, and yaw motion. The block accounts for body mass, inertia, aerodynamic drag, road incline, and weight distribution between the axles due to suspension and external forces and moments. Use the Inertial Loads parameters to analyze the vehicle dynamics under different loading conditions.

You can connect the block to virtual sensors, suspension system, or external systems like body control actuators. Use the Vehicle Body 6DOF block in ride and handling studies to model the effects of drag forces, passenger loading, and suspension hardpoint locations.

• Front powertrain

• Front and rear row passengers

• Rear cargo

For each of the loads, you can specify the mass, location, and inertia.

The dots in this illustration indicate example load locations. The table provides the corresponding location parameter sign settings. This table summarizes the parameter settings that specify the load locations indicated by the dots. For the location, the block uses this distance vector:

• Front suspension hardpoint to load, along the vehicle-fixed x-axis

• Vehicle centerline to load, along the vehicle-fixed y-axis

• Front suspension hardpoint to load, along the vehicle-fixed z-axis

Parameter

Example Location

Front

Distance vector from front axle, z1R

• `z1R(1,1)<0` — Forward of the front axle

• `z1R(1,2)>0` — Right of the vehicle centerline

• `z1R(1,3)>0` — Above the front axle suspension hardpoint

Distance vector from front axle, z2R

• `z2R(1,1)>0` — Rear of the front axle

• `z2R(1,2)<0` — Left of the vehicle centerline

• `z2R(1,3)>0` — Above the front axle suspension hardpoint

Row 1, left side

Distance vector from front axle, z3R

• `z3R(1,1)>0` — Rear of the front axle

• `z3R(1,2)<0` — Left of the vehicle centerline

• `z3R(1,3)>0` — Above the front axle suspension hardpoint

Row 1, right side

Distance vector from front axle, z4R

• `z4R(1,1)>0` — Rear of the front axle

• `z4R(1,2)>0` — Right of the vehicle centerline

• `z4R(1,3)>0` — Above the front axle suspension hardpoint

Row 2, left side

Distance vector from front axle, z5R

• `z5R(1,1)>0` — Rear of the front axle

• `z5R(1,2)<0` — Left of the vehicle centerline

• `z5R(1,3)>0` — Above the front axle suspension hardpoint

Row 2, right side

Distance vector from front axle, z6R

• `z6R(1,1)>0` — Rear of the front axle

• `z6R(1,2)>0` — Right of the vehicle centerline

• `z6R(1,3)>0` — Above the front axle suspension hardpoint

Rear

Distance vector from front axle, z7R

• `z7R(1,1)>0` — Rear of the front axle

• `z7R(1,2)>0` — Right of the vehicle centerline

• `z7R(1,3)>0` — Above the front axle suspension hardpoint

### Equations of Motion

To determine the vehicle motion, the block implements calculations for the rigid body vehicle dynamics, wind drag, inertial loads, and coordinate transformations. The body-fixed and the vehicle-fixed are the same coordinate systems.

The Vehicle Body 6DOF block considers the rotation of a body-fixed coordinate frame about a flat earth-fixed inertial reference frame. The origin of the body-fixed coordinate frame is the vehicle center of gravity of the body. The block uses this equation to calculate the translational motion of the body-fixed coordinate frame, where the applied forces [Fx Fy Fz]T are in the body-fixed frame, and the mass of the body, m, is assumed constant.

`$\begin{array}{l}{\overline{F}}_{b}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m\left({\stackrel{˙}{\overline{V}}}_{b}+\overline{\omega }×{\overline{V}}_{b}\right)\\ \\ {\overline{M}}_{b}=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=I\stackrel{˙}{\overline{\omega }}+\overline{\omega }×\left(I\overline{\omega }\right)\\ \\ I=\left[\begin{array}{ccc}{I}_{xx}& -{I}_{xy}& -{I}_{xz}\\ -{I}_{yx}& {I}_{yy}& -{I}_{yz}\\ -{I}_{zx}& -{I}_{zy}& {I}_{zz}\end{array}\right]\end{array}$`

To determine the relationship between the body-fixed angular velocity vector, [p q r]T, and the rate of change of the Euler angles, $\left[\begin{array}{ccc}\stackrel{˙}{\varphi }\text{ }\text{\hspace{0.17em}}& \stackrel{˙}{\theta }\text{\hspace{0.17em}}\text{ }\text{ }& \stackrel{˙}{\psi }\end{array}{\right]}^{T}$, the block resolves the Euler rates into the body-fixed frame.

`$\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ 0\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& -\mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right]\left[\begin{array}{c}0\\ \stackrel{˙}{\theta }\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& -\mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right]\left[\begin{array}{ccc}\mathrm{cos}\theta & 0& -\mathrm{sin}\theta \\ 0& 1& 0\\ \mathrm{sin}\theta & 0& \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{c}0\\ 0\\ \stackrel{˙}{\psi }\end{array}\right]\equiv {J}^{-1}\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ \stackrel{˙}{\theta }\\ \stackrel{˙}{\psi }\end{array}\right]$`

Inverting J gives the required relationship to determine the Euler rate vector.

`$\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ \stackrel{˙}{\theta }\\ \stackrel{˙}{\psi }\end{array}\right]=J\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\text{\hspace{0.17em}}=\left[\begin{array}{ccc}1& \left(\mathrm{sin}\varphi \mathrm{tan}\theta \right)& \left(\mathrm{cos}\varphi \mathrm{tan}\theta \right)\\ 0& \mathrm{cos}\varphi & -\mathrm{sin}\varphi \\ 0& \frac{\mathrm{sin}\varphi }{\mathrm{cos}\theta }& \frac{\mathrm{cos}\varphi }{\mathrm{cos}\theta }\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]$`

The applied forces and moments are the sum of the drag, gravitational, external, and suspension forces.

`$\begin{array}{l}{\overline{F}}_{b}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=\left[\begin{array}{c}{F}_{d}{}_{{}_{x}}\\ {F}_{d}{}_{{}_{y}}\\ {F}_{d}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{g}{}_{{}_{x}}\\ {F}_{g}{}_{{}_{y}}\\ {F}_{g}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{ext}{}_{{}_{x}}\\ {F}_{ext}{}_{{}_{y}}\\ {F}_{ext}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{FL}{}_{{}_{x}}\\ {F}_{FL}{}_{{}_{y}}\\ {F}_{FL}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{FR}{}_{{}_{x}}\\ {F}_{FR}{}_{{}_{y}}\\ {F}_{FR}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{RL}{}_{{}_{x}}\\ {F}_{RL}{}_{{}_{y}}\\ {F}_{RL}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{RR}{}_{{}_{x}}\\ {F}_{RR}{}_{{}_{y}}\\ {F}_{RR}{}_{{}_{z}}\end{array}\right]\\ \\ {\overline{M}}_{b}=\left[\begin{array}{c}{M}_{x}\\ {M}_{y}\\ {M}_{z}\end{array}\right]=\left[\begin{array}{c}{M}_{d}{}_{{}_{x}}\\ {M}_{d}{}_{{}_{y}}\\ {M}_{d}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{ext}{}_{{}_{x}}\\ {M}_{ext}{}_{{}_{y}}\\ {M}_{ext}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{FL}{}_{{}_{x}}\\ {M}_{FL}{}_{{}_{y}}\\ {M}_{FL}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{FR}{}_{{}_{x}}\\ {M}_{FR}{}_{{}_{y}}\\ {M}_{FR}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{RL}{}_{{}_{x}}\\ {M}_{RL}{}_{{}_{y}}\\ {M}_{RL}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{RR}{}_{{}_{x}}\\ {M}_{RR}{}_{{}_{y}}\\ {M}_{RR}{}_{{}_{z}}\end{array}\right]+{\overline{M}}_{F}\end{array}$`

CalculationImplementation

Block uses parallel axis theorem to resolve the individual load masses and inertias with the vehicle mass and inertia.

`${J}_{ij}={I}_{ij}+m\left({|R|}^{2}{\delta }_{ij}-{R}_{i}{R}_{j}\right)$`

Gravitational forces, Fg

Block uses direction cosine matrix (DCM) to transform the gravitational vector in the inertial-fixed frame to the body-fixed frame.

Drag forces, Fd, and moments, Md

To determine a relative airspeed, the block subtracts the wind speed from the vehicle center of mass (CM) velocity. Using the relative airspeed, the block determines the drag forces.

`$\begin{array}{l}\overline{w}=\sqrt{{\left({\stackrel{˙}{x}}_{b}-{w}_{x}\right)}^{2}+{\left({\stackrel{˙}{x}}_{y}-{w}_{x}\right)}^{2}+{\left({w}_{z}\right)}^{2}}\\ \\ {F}_{dx}=-\frac{1}{2TR}{C}_{d}{A}_{f}{P}_{abs}{\left(}^{\overline{w}}\\ {F}_{dy}=-\frac{1}{2TR}{C}_{s}{A}_{f}{P}_{abs}{\left(}^{\overline{w}}\\ {F}_{dz}=-\frac{1}{2TR}{C}_{l}{A}_{f}{P}_{abs}{\left(}^{\overline{w}}\end{array}$`

Using the relative airspeed, the block determines the drag moments.

`$\begin{array}{l}{M}_{dr}=-\frac{1}{2TR}{C}_{rm}{A}_{f}{P}_{abs}{\left(}^{\overline{w}}\left(a+b\right)\\ {M}_{dp}=-\frac{1}{2TR}{C}_{pm}{A}_{f}{P}_{abs}{\left(}^{\overline{w}}\left(a+b\right)\\ {M}_{dy}=-\frac{1}{2TR}{C}_{ym}{A}_{f}{P}_{abs}{\left(}^{\overline{w}}\left(a+b\right)\end{array}$`

External forces, Fin, and moments, Min

External forces and moments are input via ports `FExt` and `MExt`.

Suspension forces and moments

Block assumes that the suspension forces and moments act on these hardpoint locations:

• FFL, MFL — Front left

• FFR, MFR — Front right

• FRL, MRL — Rear left

• FRR, MRR — Rear right

The equations use these variables.

 $x,\stackrel{˙}{x},\stackrel{¨}{x}$ Vehicle CM displacement, velocity, and acceleration along the vehicle-fixed x-axis $y,\stackrel{˙}{y},\stackrel{¨}{y}$ Vehicle CM displacement, velocity, and acceleration along the vehicle-fixed y-axis $z,\stackrel{˙}{z},\stackrel{¨}{z}$ Vehicle CM displacement, velocity, and acceleration along the vehicle-fixed z-axis φ Rotation of the vehicle-fixed frame about the earth-fixed X-axis (roll) θ Rotation of the vehicle-fixed frame about the earth-fixed Y-axis (pitch) ψ Rotation of the vehicle-fixed frame about the earth-fixed Z-axis (yaw) FFLx, FFLy, FFLz Suspension forces applied to front left hardpoint along the vehicle-fixed x-, y-, and z-axes FFRx, FFRy, FFRz Suspension forces applied to front right hardpoint along the vehicle-fixed x-, y-, and z-axes FRLx, FRLy, FRLz Suspension forces applied to rear left hardpoint along the vehicle-fixed x-, y-, and z-axes FRRx, FRRy, FRRz Suspension forces applied to rear right hardpoint along the vehicle-fixed x-, y-, and z-axes MFx, FFy, FFz Suspension moments applied to vehicle CM about the vehicle-fixed x-, y-, and z-axes Fextx, Fexty, Fextz External forces applied to vehicle CM along the vehicle-fixed x-, y-, and z-axes Fdx, Fdy, Fdz Drag forces applied to vehicle CM along the vehicle-fixed x-, y-, and z-axes Mextx, Mexty, Mextz External moment about vehicle CM about the vehicle-fixed x-, y-, and z-axes Mdx, Mdy, Mdz Drag moment about vehicle CM about the vehicle-fixed x-, y-, and z-axes I Vehicle body moments of inertia a, b Distance of front and rear wheels, respectively, from the normal projection point of vehicle CM onto the common axle plane h Height of vehicle CM above the axle plane wF, wR Front and rear track widths γ Road grade angle Cd Air drag coefficient acting along vehicle-fixed x-axis Cs Air drag coefficient acting along vehicle-fixed y-axis Cl Air drag coefficient acting along vehicle-fixed z-axis Crm Air drag roll moment acting about vehicle-fixed x-axis Cpm Air drag pitch moment acting about the vehicle-fixed y-axis Cym Air drag yaw moment acting about vehicle-fixed z-axis Af Frontal area R Atmospheric specific gas constant T Environmental air temperature Pabs Environmental absolute pressure wx, wy, wz Wind speed along the vehicle-fixed x-, y-, and z-axes Wx, Wy, Wz Wind speed along inertial X-, Y-, and Z-axes

## Ports

### Input

expand all

Suspension longitudinal, lateral, and vertical suspension forces applied to the vehicle at the hardpoint location, in N. Signal dimensions are `[3x4]`.

`$FSusp=\left[\begin{array}{cccc}{F}_{{x}_{FL}}& {F}_{{x}_{FR}}& {F}_{{x}_{RL}}& {F}_{{x}_{RR}}\\ {F}_{{y}_{FL}}& {F}_{{y}_{FR}}& {F}_{{y}_{RL}}& {F}_{{y}_{RR}}\\ {F}_{{z}_{FL}}& {F}_{{z}_{FR}}& {F}_{{z}_{RL}}& {F}_{{z}_{RR}}\end{array}\right]$`

Array ElementAxleTrackForce Axis
`FSusp(1,1)`FrontLeftVehicle-fixed x-axis (longitudinal)
`FSusp(1,2)`FrontRight
`FSusp(1,3)`RearLeft
`FSusp(1,4)`RearRight
`FSusp(2,1)`FrontLeftVehicle-fixed y-axis (lateral)
`FSusp(2,2)`FrontRight
`FSusp(2,3)`RearLeft
`FSusp(2,4)`RearRight
`FSusp(3,1)`FrontLeftVehicle-fixed z-axis (vertical)
`FSusp(3,2)`FrontRight
`FSusp(3,3)`RearLeft
`FSusp(3,4)`RearRight

Suspension longitudinal, lateral, and vertical suspension moments applied about the vehicle at the hardpoint location, in N. Signal dimensions are `[3x4]`.

`$MSusp=\left[\begin{array}{cccc}{M}_{{x}_{FL}}& {M}_{{x}_{FR}}& {M}_{{x}_{RL}}& {M}_{{x}_{RR}}\\ {M}_{{y}_{FL}}& {M}_{{y}_{FR}}& {M}_{{y}_{RL}}& {M}_{{y}_{RR}}\\ {M}_{{z}_{FL}}& {M}_{{z}_{FR}}& {M}_{{z}_{RL}}& {M}_{{z}_{RR}}\end{array}\right]$`

Array ElementAxleTrackMoment Axis
`MSusp(1,1)`FrontLeftVehicle-fixed x-axis (longitudinal)
`MSusp(1,2)`FrontRight
`MSusp(1,3)`RearLeft
`MSusp(1,4)`RearRight
`MSusp(2,1)`FrontLeftVehicle-fixed y-axis (lateral)
`MSusp(2,2)`FrontRight
`MSusp(2,3)`RearLeft
`MSusp(2,4)`RearRight
`MSusp(3,1)`FrontLeftVehicle-fixed z-axis (vertical)
`MSusp(3,2)`FrontRight
`MSusp(3,3)`RearLeft
`MSusp(3,4)`RearRight

External forces on vehicle, in N. Signal vector dimensions are `[1x3]` or `[3x1]`.

`$\text{FExt}={F}_{ext}=\left[\begin{array}{ccc}{F}_{ex{t}_{x}}& {F}_{ex{t}_{y}}& {F}_{ex{t}_{z}}\end{array}\right]or\left[\begin{array}{c}{F}_{ex{t}_{x}}\\ {F}_{ex{t}_{y}}\\ {F}_{ex{t}_{z}}\end{array}\right]$`

Array ElementForce Axis

`FExt(1,1)`

Vehicle-fixed x-axis (longitudinal)

`FExt(1,2)` or `FExt(2,1)`

Vehicle-fixed y-axis (lateral)

`FExt(1,3)` or `FExt(3,1)`

Vehicle-fixed z-axis (vertical)

External moments acting on vehicle, in N·m. Signal vector dimensions are `[1x3]` or `[3x1]`.

`$\text{MExt}={M}_{ext}=\left[\begin{array}{ccc}{M}_{ex{t}_{x}}& {M}_{ex{t}_{y}}& {M}_{ex{t}_{z}}\end{array}\right]or\left[\begin{array}{c}{M}_{ex{t}_{x}}\\ {M}_{ex{t}_{y}}\\ {M}_{ex{t}_{z}}\end{array}\right]$`

Array ElementForce Axis
`MExt(1,1)`Vehicle-fixed x-axis (longitudinal)

`MExt(1,2)` or `MExt(2,1)`

Vehicle-fixed y-axis (lateral)

`MExt(1,3)` or `MExt(3,1)`

Vehicle-fixed z-axis (vertical)

Wind speed, Wx, Wy, Wz along inertial X-, Y-, and Z-axes, in m/s. Signal vector dimensions are `[1x3]` or `[3x1]`.

Ambient air temperature, Tair, in K.

#### Dependencies

To enable this port, on the Environment pane, select Air temperature.

### Output Arguments

expand all

Bus signal containing these block values.

SignalDescriptionValueUnits
`InertFrm``Cg``Disp``X`Vehicle CM displacement along the earth-fixed X-axis

Computed

m
`Y`Vehicle CM displacement along the earth-fixed Y-axis

Computed

m

`Z`Vehicle CM displacement along the earth-fixed Z-axis

Computed

m
`Vel``Xdot`Vehicle CM velocity along the earth-fixed X-axis

Computed

m/s

`Ydot`Vehicle CM velocity along the earth-fixed Y-axis

Computed

m/s
`Zdot`Vehicle CM velocity along the earth-fixed Z-axis

Computed

m/s
`Ang``phi`Rotation of the vehicle-fixed frame about the earth-fixed X-axis (roll)

Computed

`theta`Rotation of the vehicle-fixed frame about the earth-fixed Y-axis (pitch)

Computed

`psi`Rotation of the vehicle-fixed frame about the earth-fixed Z-axis (yaw)

Computed

`FrntAxl``Lft``Disp``X`Front left axle displacement along the earth-fixed X-axis

Computed

m
`Y`Front left axle displacement along the earth-fixed Y-axis

Computed

m
`Z`Front left axle displacement along the earth-fixed Z-axis

Computed

m
`Vel``Xdot`Front left axle velocity along the earth-fixed X-axis

Computed

m/s
`Ydot`Front left axle velocity along the earth-fixed Y-axis

Computed

m/s
`Zdot`Front left axle velocity along the earth-fixed Z-axis

Computed

m/s
`Rght``Disp``X`Front right axle displacement along the earth-fixed X-axis

Computed

m
`Y`Front right axle displacement along the earth-fixed Y-axis

Computed

m
`Z`Front right axle displacement along the earth-fixed Z-axis

Computed

m
`Vel``Xdot`Front right axle velocity along the earth-fixed X-axis

Computed

m/s
`Ydot`Front right axle velocity along the earth-fixed Y-axis

Computed

m/s
`Zdot`Front right axle velocity along the earth-fixed Z-axis

Computed

m/s
`RearAxl``Lft``Disp``X`Rear left axle displacement along the earth-fixed X-axis

Computed

m
`Y`Rear left axle displacement along the earth-fixed Y-axis

Computed

m
`Z`Rear left axle displacement along the earth-fixed Z-axis

Computed

m
`Vel``Xdot`Rear left axle velocity along the earth-fixed X-axis

Computed

m/s
`Ydot`Rear left axle velocity along the earth-fixed Y-axis

Computed

m/s
`Zdot`Rear left axle velocity along the earth-fixed Z-axis

Computed

m/s
`Rght``Disp``X`Rear right axle displacement along the earth-fixed X-axis

Computed

m
`Y`Rear right axle displacement along the earth-fixed Y-axis

Computed

m
`Z`Rear right axle displacement along the earth-fixed Z-axis

Computed

m
`Vel``Xdot`Rear right axle velocity along the earth-fixed X-axis

Computed

m/s
`Ydot`Rear right axle velocity along the earth-fixed Y-axis

Computed

m/s
`Zdot`Rear right axle velocity along the earth-fixed Z-axis

Computed

m/s
`Geom``Disp``X`Vehicle chassis offset from axle plane along the earth-fixed X-axis

Computed

m
`Y`Vehicle chassis offset from center plane along the earth-fixed Y-axis

Computed

m
`Z`Vehicle chassis offset from axle plane along the earth-fixed Z-axis

Computed

m
`Vel``Xdot`Vehicle chassis offset velocity along the earth-fixed X-axis

Computed

m/s
`Ydot`Vehicle chassis offset velocity along the earth-fixed Y-axis

Computed

m/s
`Zdot`Vehicle chassis offset velocity along the earth-fixed Z-axis

Computed

m/s
`BdyFrm``Cg``Vel``xdot`Vehicle CM velocity along the vehicle-fixed x-axis

Computed

m/s
`ydot`Vehicle CM velocity along the vehicle-fixed y-axis

Computed

m/s
`zdot`Vehicle CM velocity along the vehicle-fixed z-axisComputedm/s
`AngVel``p`Vehicle angular velocity about the vehicle-fixed x-axis (roll rate)

Computed

`q`Vehicle angular velocity about the vehicle-fixed y-axis (pitch rate)

Computed

`r`Vehicle angular velocity about the vehicle-fixed z-axis (yaw rate)

Computed

`Acc``ax`Vehicle CM acceleration along the vehicle-fixed x-axis

Computed

gn
`ay`Vehicle CM acceleration along the vehicle-fixed y-axis

Computed

gn
`az`Vehicle CM acceleration along the vehicle-fixed z-axis

Computed

gn
`xddot`Vehicle CM acceleration along the vehicle-fixed x-axis

Computed

m/s^2
`yddot`Vehicle CM acceleration along the vehicle-fixed y-axis

Computed

m/s^2
`zddot`Vehicle CM acceleration along the vehicle-fixed z-axis

Computed

m/s^2
`DCM`

Direction cosine matrix

Computed

`Forces``Body``Fx`Net force on vehicle CM along the vehicle-fixed x-axis

Computed

N
`Fy`Net force on vehicle CM along the vehicle-fixed y-axis

Computed

N
`Fz`Net force on vehicle CM along the vehicle-fixed z-axis

Computed

N
`FrntAxl``Lft``Fx`

Longitudinal force on front left axle along the vehicle-fixed x-axis

Computed

N
`Fy`

Lateral force on front axle left along the vehicle-fixed y-axis

Computed

N
`Fz`

Normal force on front axle left along the vehicle-fixed z-axis

ComputedN
`Rght``Fx`

Longitudinal force on front right axle along the vehicle-fixed x-axis

Computed

N
`Fy`

Lateral force on front axle right along the vehicle-fixed y-axis

Computed

N
`Fz`

Normal force on front axle right along the vehicle-fixed z-axis

Computed

N
`RearAxl``Lft``Fx`

Longitudinal force on rear left axle along the vehicle-fixed x-axis

Computed

N
`Fy`

Lateral force on rear left axle along the vehicle-fixed y-axis

Computed

N
`Fz`

Normal force on rear left axle along the vehicle-fixed z-axis

Computed

N
`Rght``Fx`

Longitudinal force on rear right axle along the vehicle-fixed x-axis

Computed

N
`Fy`

Lateral force on rear right axle along the vehicle-fixed y-axis

Computed

N
`Fz`

Normal force on rear right axle along the vehicle-fixed z-axis

Computed

N
`Tires``FrntTires``Lft``Fx`

Front left tire force along the vehicle-fixed x-axis

Computed

N
`Fy`

Front left tire force along the vehicle-fixed y-axis

Computed

N
`Fz`

Front left tire force along the vehicle-fixed z-axis

Computed

N
`Rght``Fx`

Front right tire force along the vehicle-fixed x-axis

Computed

N
`Fy`

Front right tire force along the vehicle-fixed y-axis

Computed

N
`Fz`

Front right tire force along the vehicle-fixed z-axis

Computed

N
`RearTires``Lft``Fx`

Rear left tire force along the vehicle-fixed x-axis

Computed

N
`Fy`

Rear left tire force along the vehicle-fixed y-axis

Computed

N
`Fz`

Rear left tire force along the vehicle-fixed z-axis

Computed

N
`Rght``Fx`

Rear right tire force along the vehicle-fixed x-axis

Computed

N
`Fy`

Rear right tire force along the vehicle-fixed y-axis

Computed

N
`Fz`

Rear right tire force along the vehicle-fixed z-axis

Computed

N
`Drag``Fx`Drag force on vehicle CM along the vehicle-fixed x-axis

Computed

N
`Fy`Drag force on vehicle CM along the vehicle-fixed y-axis

Computed

N
`Fz`Drag force on vehicle CM along the vehicle-fixed z-axis

Computed

N
`Grvty``Fx`Gravity force on vehicle CM along the vehicle-fixed x-axis

Computed

N
`Fy`Gravity force on vehicle CM along the vehicle-fixed y-axis

Computed

N
`Fz`Gravity force on vehicle CM along the vehicle-fixed z-axis

Computed

N
`Moments``Body``Mx`Body moment on vehicle CM about the vehicle-fixed x-axis

Computed

N·m
`My`Body moment on vehicle CM about the vehicle-fixed y-axis

Computed

N·m
`Mz`Body moment on vehicle CM about the vehicle-fixed z-axis

Computed

N·m
`Drag``Mx`Drag moment on vehicle CM about the vehicle-fixed x-axis

Computed

N·m
`My`Drag moment on vehicle CM about the vehicle-fixed y-axis

Computed

N·m
`Mz`Drag moment on vehicle CM about the vehicle-fixed z-axis

Computed

N·m
`FrntAxl``Lft``Disp``x`Front left axle displacement along the vehicle-fixed x-axis

Computed

m
`y`Front left axle displacement along the vehicle-fixed y-axis

Computed

m
`z`Front left axle displacement along the vehicle-fixed z-axis

Computed

m
`Vel``xdot`Front left axle velocity along the vehicle-fixed x-axis

Computed

m/s
`ydot`Front left axle velocity along the vehicle-fixed y-axis

Computed

m/s
`zdot`Front left axle velocity along the vehicle-fixed z-axis

Computed

m/s
`Rght``Disp``x`Front right axle displacement along the vehicle-fixed x-axis

Computed

m
`y`Front right axle displacement along the vehicle-fixed y-axis

Computed

m
`z`Front right axle displacement along the vehicle-fixed z-axis

Computed

m
`Vel``xdot`Front right axle velocity along the vehicle-fixed x-axis

Computed

m/s
`ydot`Front right axle velocity along the vehicle-fixed y-axis

Computed

m/s
`zdot`Front right axle velocity along the vehicle-fixed z-axis

Computed

m/s
`RearAxl``Lft``Disp``x`Rear left axle displacement along the vehicle-fixed x-axis

Computed

m
`y`Rear left axle displacement along the vehicle-fixed y-axis

Computed

m
`z`Rear left axle displacement along the vehicle-fixed z-axis

Computed

m
`Vel``xdot`Rear left axle velocity along the vehicle-fixed x-axis

Computed

m/s
`ydot`Rear left axle velocity along the vehicle-fixed y-axis

Computed

m/s
`zdot`Rear left axle velocity along the vehicle-fixed z-axis

Computed

m/s
`Rght``Disp``x`Rear right axle displacement along the vehicle-fixed x-axis

Computed

m
`y`Rear right axle displacement along the vehicle-fixed y-axis

Computed

m
`z`Rear right axle displacement along the vehicle-fixed z-axis

Computed

m
`Vel``xdot`Rear right axle velocity along the vehicle-fixed x-axis

Computed

m/s
`ydot`Rear right axle velocity along the vehicle-fixed y-axis

Computed

m/s
`zdot`Rear right axle velocity along the vehicle-fixed z-axis

Computed

m/s
`Pwr``PwrExt`Applied external power

Computed

W
`Drag`Power loss due to drag

Computed

W
`Geom``Disp``x`Vehicle chassis offset from axle plane along the vehicle-fixed x-axis

Input

m
`y`Vehicle chassis offset from center plane along the vehicle-fixed y-axis

Input

m
`z`Vehicle chassis offset from axle plane along the earth-fixed z-axis

Input

m
`Vel``xdot`Vehicle chassis offset velocity along the vehicle-fixed x-axis

Computed

m/s
`ydot`Vehicle chassis offset velocity along the vehicle-fixed y-axis

Computed

m/s
`zdot`Vehicle chassis offset velocity along the vehicle-fixed z-axis

Computed

m/s
`Ang``Beta`

Body slip angle, β

`$\beta =\frac{{V}_{y}}{{V}_{x}}$`

Computed

Vehicle CM velocity along the vehicle-fixed x-, y-, z- axes, respectively, in m/s.

Vehicle CM angular velocity about the vehicle-fixed x(roll rate)-, y(pitch rate)-, z(yaw rate)- axes, respectively, in rad/s.

Euler angles, φ, θ, and ψ, respectively, in rad.

Vehicle CM position along inertial-fixed X-, Y-, Z- axes, respectively, in m.

Vehicle CM velocity along inertial-fixed X-, Y-, Z- axes, respectively, in m/s.

## Parameters

expand all

### Chassis

Vehicle mass, m, in kg.

Distance from vehicle CM to front axle, a, in m. Distance from vehicle CM to front axle, b, in m. Lateral distance from geometric centerline to center of mass, d, in m, along the vehicle-fixed y. Positive values indicate that the vehicle CM is to the right of the geometric centerline. Negative values indicate that the vehicle CM is to the left of the geometric centerline. Vertical distance from vehicle CM to axle plane, h, in m. Initial position of vehicle in the inertial frame, Xeo, in m.

Initial vehicle CM velocity along the vehicle-fixed x, y-, and z- axes, respectively, in m/s.

Initial Euler rotation of the vehicle-fixed frame about the earth-fixed X(roll)-, Y(pitch)-, Z(yaw)- axes, respectively, in rad.

Initial vehicle CM angular velocity about the vehicle-fixed x(roll rate)-, y(pitch rate)-, z(yaw rate)- axes, respectively, in rad/s.

Vehicle inertia tensor, Iveh, in kg*m^2. Dimensions are `[3-by-3]`.

Front and rear track width, in m. Dimensions are `[1-by-2]`.

#### Front

Mass, z1m, in kg.

Distance vector from front axle to load, z1R, in m. Dimensions are `[1-by-3]`.

Array ElementDescription
`z1R(1,1)`

Front suspension hardpoint to load, along vehicle-fixed x-axis

`z1R(1,2)`

Vehicle centerline to load, along vehicle-fixed y-axis

`z1R(1,3)`

Front suspension hardpoint to load, along vehicle-fixed z-axis For example, this table summarizes the parameter settings that specify the load location indicated by the dots.

Example Location

Sign

• Forward of the front axle

• Right of the vehicle centerline

• Above the front axle suspension hardpoint

• `z1R(1,1)` < 0

• `z1R(1,2)` > 0

• `z1R(1,3)` > 0

Inertia tensor, z1I, in kg·m^2. Dimensions are `[3-by-3]`.

`$z1I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]$`

The tensor uses a coordinate system with an origin at the load CM.

• x-axis along the vehicle-fixed x-axis

• y-axis along the vehicle-fixed y-axis

• z-axis along the vehicle-fixed z-axis Mass, z2m, in kg.

Distance vector from front axle to load, z2R, in m. Dimensions are `[1-by-3]`.

Array ElementDescription
`z2R(1,1)`

Front suspension hardpoint to load, along vehicle-fixed x-axis

`z2R(1,2)`

Vehicle centerline to load, along vehicle-fixed y-axis

`z2R(1,3)`

Front suspension hardpoint to load, along vehicle-fixed z-axis For example, this table summarizes the parameter settings that specify the load location indicated by the dot.

Example Location

Sign

• Rear of the front axle

• Left of the vehicle centerline

• Above the front axle suspension hardpoint

• `z2R(1,1)` > 0

• `z2R(1,2)` < 0

• `z2R(1,3)` > 0

Inertia tensor, z2I, in kg·m^2. Dimensions are `[3-by-3]`.

`$z2I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]$`

The tensor uses a coordinate system with an origin at the load CM.

• x-axis along the vehicle-fixed x-axis

• y-axis along the vehicle-fixed y-axis

• z-axis along the vehicle-fixed z-axis #### Row 1, left side

Mass, z3m, in kg.

Distance vector from front axle to load, z3R, in m. Dimensions are `[1-by-3]`.

Array ElementDescription
`z3R(1,1)`

Front suspension hardpoint to load, along vehicle-fixed x-axis

`z3R(1,2)`

Vehicle centerline to load, along vehicle-fixed y-axis

`z3R(1,3)`

Front suspension hardpoint to load, along vehicle-fixed z-axis For example, this table summarizes the parameter settings that specify the load location indicated by the dot.

Example Location

Sign

• Rear of the front axle

• Left of the vehicle centerline

• Above the front axle suspension hardpoint

• `z3R(1,1)` > 0

• `z3R(1,2)` < 0

• `z3R(1,3)` > 0

Inertia tensor, z3I, in kg·m^2. Dimensions are `[3-by-3]`.

`$z3I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]$`

The tensor uses a coordinate system with an origin at the load CM.

• x-axis along the vehicle-fixed x-axis

• y-axis along the vehicle-fixed y-axis

• z-axis along the vehicle-fixed z-axis #### Row 1, right side

Mass, z4m, in kg.

Distance vector from front axle to load, z4R, in m. Dimensions are `[1-by-3]`.

Array ElementDescription
`z4R(1,1)`

Front suspension hardpoint to load, along vehicle-fixed x-axis

`z4R(1,2)`

Vehicle centerline to load, along vehicle-fixed y-axis

`z4R(1,3)`

Front suspension hardpoint to load, along vehicle-fixed z-axis For example, this table summarizes the parameter settings that specify the load location indicated by the dot.

Example Location

Sign

• Rear of the front axle

• Right of the vehicle centerline

• Above the front axle suspension hardpoint

• `z4R(1,1)` > 0

• `z4R(1,2)` > 0

• `z4R(1,3)` > 0

Inertia tensor, z4I, in kg·m^2. Dimensions are `[3-by-3]`.

`$z4I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]$`

The tensor uses a coordinate system with an origin at the load CM.

• x-axis along the vehicle-fixed x-axis

• y-axis along the vehicle-fixed y-axis

• z-axis along the vehicle-fixed z-axis #### Row 2, left side

Mass, z5m, in kg.

Distance vector from front axle to load, z5R, in m. Dimensions are `[1-by-3]`.

Array ElementDescription
`z5R(1,1)`

Front suspension hardpoint to load, along vehicle-fixed x-axis

`z5R(1,2)`

Vehicle centerline to load, along vehicle-fixed y-axis

`z5R(1,3)`

Front suspension hardpoint to load, along vehicle-fixed z-axis For example, this table summarizes the parameter settings that specify the load location indicated by the dot.

Example Location

Sign

• Rear of the front axle

• Left of the vehicle centerline

• Above the front axle suspension hardpoint

• `z5R(1,1)` > 0

• `z5R(1,2)` < 0

• `z5R(1,3)` > 0

Inertia tensor, z5I, in kg·m^2. Dimensions are `[3-by-3]`.

`$z5I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]$`

The tensor uses a coordinate system with an origin at the load CM.

• x-axis along the vehicle-fixed x-axis

• y-axis along the vehicle-fixed y-axis

• z-axis along the vehicle-fixed z-axis #### Row 2, right side

Mass, z6m, in kg.

Distance vector from front axle to load, z6R, in m. Dimensions are `[1-by-3]`.

Array ElementDescription
`z6R(1,1)`

Front suspension hardpoint to load, along vehicle-fixed x-axis

`z6R(1,2)`

Vehicle centerline to load, along vehicle-fixed y-axis

`z6R(1,3)`

Front suspension hardpoint to load, along vehicle-fixed z-axis For example, this table summarizes the parameter settings that specify the load location indicated by the dot.

Example Location

Sign

• Rear of the front axle

• Right of the vehicle centerline

• Above the front axle suspension hardpoint

• `z6R(1,1)` > 0

• `z6R(1,2)` > 0

• `z6R(1,3)` > 0

Inertia tensor, z6I, in kg·m^2. Dimensions are `[3-by-3]`.

`$z6I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]$`

The tensor uses a coordinate system with an origin at the load CM.

• x-axis along the vehicle-fixed x-axis

• y-axis along the vehicle-fixed y-axis

• z-axis along the vehicle-fixed z-axis #### Rear

Mass, z7m, in kg.

Distance vector from front axle to load, z7R, in m. Dimensions are `[1-by-3]`.

Array ElementDescription
`z7R(1,1)`

Front suspension hardpoint to load, along vehicle-fixed x-axis

`z7R(1,2)`

Vehicle centerline to load, along vehicle-fixed y-axis

`z7R(1,3)`

Front suspension hardpoint to load, along vehicle-fixed z-axis For example, this table summarizes the parameter settings that specify the load location indicated by the dot.

Example Location

Sign

• Rear of the front axle

• Right of the vehicle centerline

• Above the front axle suspension hardpoint

• `z7R(1,1)` > 0

• `z7R(1,2)` > 0

• `z7R(1,3)` > 0

Inertia tensor, z7I, in kg·m^2. Dimensions are `[3-by-3]`.

`$z7I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]$`

The tensor uses a coordinate system with an origin at the load CM.

• x-axis along the vehicle-fixed x-axis

• y-axis along the vehicle-fixed y-axis

• z-axis along the vehicle-fixed z-axis ### Aerodynamic

Effective vehicle cross-sectional area, Af to calculate the aerodynamic drag force on the vehicle, in m^2.

Air drag coefficient, Cd, dimensionless.

Air lift coefficient, Cl, dimensionless.

Longitudinal drag pitch moment coefficient, Cpm, dimensionless.

Relative wind angle vector, βw, in rad.

Side force coefficient vector coefficient, Cs, dimensionless.

Yaw moment coefficient vector coefficient, Cym, dimensionless.

### Environment

Environmental air absolute pressure, Pabs, in Pa.

Ambient air temperature, Tair, in K.

#### Dependencies

To enable this parameter, clear Air temperature.

Gravitational acceleration, g, in m/s^2.

### Simulation

Longitudinal velocity tolerance, xdottol, in m/s.

The block uses this parameter to avoid a division by zero when it calculates the body slip angle, β.

Vehicle chassis offset from axle plane along body-fixed x-axis, in m. When you use the 3D visualization engine, consider using the offset to locate the chassis independent of the vehicle CG.

Vehicle chassis offset from center plane along body-fixed y-axis, in m. When you use the 3D visualization engine, consider using the offset to locate the chassis independent of the vehicle CG.

Vehicle chassis offset from axle plane along body-fixed z-axis, in m. When you use the 3D visualization engine, consider using the offset to locate the chassis independent of the vehicle CG.

Wrap the Euler angles to the interval `[-pi, pi]`. For vehicle maneuvers that might undergo vehicle yaw rotations that are outside of the interval, consider deselecting the parameter if you want to:

• Track the total vehicle yaw rotation.

• Avoid discontinuities in the vehicle state estimators.

 Gillespie, Thomas. Fundamentals of Vehicle Dynamics. Warrendale, PA: Society of Automotive Engineers (SAE), 1992.

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