Vehicle Body 3DOF

3DOF rigid vehicle body to calculate longitudinal, lateral, and yaw motion

  • Library:
  • Vehicle Dynamics Blockset / Vehicle Body

  • Vehicle Body 3DOF block

Description

The Vehicle Body 3DOF block implements a rigid two-axle vehicle body model to calculate longitudinal, lateral, and yaw motion. The block accounts for body mass and aerodynamic drag between the axles due to acceleration and steering.

Use this block in vehicle dynamics and automated driving studies to model nonholonomic vehicle motion when vehicle pitch, roll, and vertical motion are not significant.

In the Vehicle Dynamics Blockset™ library, there are two types of Vehicle Body 3DOF blocks that model longitudinal, lateral, and yaw motion.

BlockVehicle Track SettingImplementation

Vehicle Body 3DOF Single TrackVehicle Body 3DOF block single track

Single (bicycle)

  • Forces act along the center line at the front and rear axles.

  • No lateral load transfer.

Vehicle Body 3DOF Dual TrackVehicle Body 3DOF block

Dual

Forces act at the four vehicle corners or hard points.

Use the Axle forces parameter to specify the type of force.

Axle Forces SettingImplementation

External longitudinal velocity

  • The block assumes that the external longitudinal velocity is in a quasi-steady state, so the longitudinal acceleration is approximately zero.

  • Because the motion is quasi-steady, the block calculates lateral forces using the tire slip angles and linear cornering stiffness.

  • Consider this setting when you want to:

    • Generate virtual sensor signal data.

    • Conduct high-level software studies that are not impacted by driveline or nonlinear tire responses.

External longitudinal forces

  • The block uses the external longitudinal force to accelerate or brake the vehicle.

  • The block calculates lateral forces using the tire slip angles and linear cornering stiffness.

  • Consider this setting when you want to:

    • Account for changes in the longitudinal velocity on the lateral and yaw motion.

    • Specify the external longitudinal motion through a force instead of an external longitudinal velocity.

    • Connect the block to tractive actuators, wheels, brakes, and hitches.

External forces

  • The block uses the external lateral and longitudinal forces to steer, accelerate, or brake the vehicle.

  • The block does not use the steering input to calculate vehicle motion.

  • Consider this setting when you need tire models with more accurate nonlinear combined lateral and longitudinal slip.

You can use these block parameters to create additional input ports. This table summarizes the settings.

Input Signals Pane Parameter

Input PortDescription

Front wheel steering

WhlAngF

Front wheel angle, δF

External wind

WindXYZ

Wind speed, WX, WY, WZ, in the inertial reference frame

External forcesFExt

External force on vehicle center of gravity (CG), Fx, Fy, Fz, in the vehicle-fixed frame

Rear wheel steeringWhlAngR

Rear wheel angle, δR

External frictionMu

Friction coefficient

External moments

MExt

External moment about vehicle CG, Mx, My, Mz, in vehicle-fixed frame

Hitch forcesFh

Hitch force applied to the body at the hitch location, Fhx, Fhy, and Fhz, in the vehicle-fixed frame

Hitch momentsMh

Hitch moment at the hitch location, Mhx, Mhy, and Mhz, about the vehicle-fixed frame

Initial longitudinal position

X_o

Initial vehicle CG displacement along the earth-fixed X-axis, in m

Initial lateral position

Y_o

Initial vehicle CG displacement along the earth-fixed Y-axis, in m

Initial longitudinal velocity

xdot_o

Initial vehicle CG velocity along the vehicle-fixed x-axis, in m/s

Initial lateral velocity

ydot_o

Initial vehicle CG velocity along the vehicle-fixed y-axis, in m/s

Initial yaw angle

psi_o

Initial rotation of the vehicle-fixed frame about the earth-fixed Z-axis (yaw), in rad

Initial yaw rate

r_o

Initial vehicle angular velocity about the vehicle-fixed z-axis (yaw rate), in rad/s

Air temperature

AirTemp

Ambient air temperature. Considering this option if you want to vary the temperature during run-time.

Theory

The Vehicle Body 3DOF block implements a rigid two-axle vehicle body model to calculate longitudinal, lateral, and yaw motion. The block accounts for body mass, aerodynamic drag, and weight distribution between the axles due to acceleration and steering. To determine the vehicle motion, the block implements these equations for the single track, dual track, and drag calculations.

Single Track

CalculationDescription

Dynamics

The block uses these equations to calculate the rigid body planar dynamics.

y¨=x˙r+Fyf+Fyr+Fyextmr˙=aFyfbFyr+MzextIzzr=ψ˙

If you set Axle forces to either External longitudinal forces or External forces, the block uses this equation for the longitudinal acceleration.

x¨=y˙r+Fxf+Fxr+Fxextm

If you set Axle forces to External longitudinal velocity, the block assumes a quasi-steady state for the longitudinal acceleration.

x¨=0

External forces

External forces include both drag and external force inputs. The forces act on the vehicle CG.

Fx,y,z ext=Fd x,y,z+Fx,y,z inputMx,y,z ext=Md x,y,z+Mx,y,z input

If you set Axle forces to External longitudinal forces, the block uses these equations.

Fxft=FxfinputFyft=CyfαfμfFzfFznomFxrt=FxrinputFyrt=CyrαrμrFzrFznom 

If you set Axle forces to External longitudinal velocity, the block uses these equations.

Fxft=0Fyft=CyfαfμfFzfFznomFxrt=0Fyrt=CyrαrμrFzrFznom

The block divides the normal forces by the nominal normal load to vary the effective friction parameters during weight and load transfer. The block uses these equations to maintain pitch and roll equilibrium.

Fzf=bmg(x¨y˙r)mh+hFxext+bFzextMyexta+bFzr=amg+(x¨y˙r)mhhFxext+aFzext+Myexta+b

Tire forces

The block uses the ratio of the local and longitudinal and lateral velocities to determine the slip angles.

αf=atan(y˙+arx˙)δfαr=atan(y˙brx˙)δr

To determine the tire forces, the block uses the slip angles.

Fxf=Fxftcos(δf)Fyftsin(δf)Fyf=Fxftsin(δf)+Fyftcos(δf)Fxr=Fxrtcos(δr)Fyrtsin(δr)Fyr=Fxrtsin(δr)+Fyrtcos(δr)

If you set Axle forces to External forces, the block sets the tire forces equal to the external input force.

Fxf=Fxft=FxfinputFyf=Fyft=FyfinputFxr=Fxrt=FxrinputFyr=Fyrt=Fyrinput

Dual Track

Isometric view of vertical, longitudinal, and lateral forces acting at suspension locations

CalculationDescription

Dynamics

The block uses these equations to calculate the rigid body planar dynamics.

x¨=y˙r+Fxfl+Fxfr+Fxrl+Fxrr+Fxextmy¨=x˙r+Fyfl+Fyfr+Fyrl+Fyrr+Fyextmr˙=a(Fyfl+Fyfr)b(Fyrl+Fyrr)+wf(FxflFxfr)2+wr(FxrlFxrr)2+MzextIzzr=ψ˙

If you set Axle forces to External longitudinal velocity, the block assumes a quasi-steady state for the longitudinal acceleration.

x¨=0

External forces

External forces include both drag and external force inputs. The forces act on the vehicle CG.

Fx,y,z ext=Fd x,y,z+Fx,y,z inputMx,y,z ext=Md x,y,z+Mx,y,z input

If you set Axle forces to External longitudinal forces, the block uses these equations.

Fxflt=FxflinputFyflt=CyflαflμflFzfl2FznomFxfrt=FxlrinputFyfrt=CyfrαfrμfrFzfr2FznomFxrlt=FxrlinputFyrlt=CyrlαrlμrlFzrl2FznomFxrrt=FxrrinputFyrrt=CyrrαrrμrrFzrr2Fznom

If you set Axle forces to External longitudinal velocity, the block uses these equations.

Fxflt=0Fyflt=CyflαflμflFzfl2FznomFxfrt=0Fyfrt=CyfrαfrμfrFzfr2FznomFxrlt=0Fyrlt=CyrlαrlμrlFzrl2FznomFxrrt=0Fyrrt=CyrrαrrμrrFzrr2Fznom

The block divides the normal forces by the nominal normal load to vary the effective friction parameters during weight and load transfer. The block uses these equations to maintain pitch and roll equilibrium.

Fzf=bmg(x¨y˙r)mh+hFxext+bFzextMyexta+bFzr=amg+(x¨y˙r)mhhFxext+aFzext+Myext(a+b)Fzfl=Fzf+(mh(y¨+x˙r)hFyextMxext)2wfFzfr=Fzf+(mh(y¨+x˙r)+hFyext+Mxext)2wfFzrl=Fzr+(mh(y¨+x˙r)hFyextMxext)2wrFzrr=Fzr+(mh(y¨+x˙r)+hFyext+Mxext)2wr

Tire forces

The block uses the ratio of the local and longitudinal and lateral velocities to determine the slip angles.

αfl=atan(y˙+arx˙+rwf2)δflαfr=atan(y˙+arx˙rwf2)δfrαrl=atan(y˙arx˙+rwr2)δrlαrr=atan(y˙arx˙rwr2)δrr

The block uses the steering angles to transform the tire forces to the vehicle-fixed frame.

Fxf=Fxftcos(δf)Fyftsin(δf)Fyf=Fxftsin(δf)+Fyftcos(δf)Fxr=Fxrtcos(δr)Fyrtsin(δr)Fyr=Fxrtsin(δr)+Fyrtcos(δr)

If you set Axle forces to External forces, the block uses these equations. The blocks assumes that the externally provided forces are in the vehicle-fixed frame at the axle-wheel location.

Fxf=Fxft=FxfinputFyf=Fyft=FyfinputFxr=Fxrt=FxrinputFyr=Fyrt=Fyrinput

Drag

CalculationDescription

Coordinate transformation

The block transforms the wind speeds from the inertial frame to the vehicle-fixed frame.

wx=Wxcos(ψ)+Wysin(ψ)wy=Wycos(ψ)Wxsin(ψ)wz=Wz

Drag forces

To determine a relative airspeed, the block subtracts the wind speed from the CG vehicle velocity. Using the relative airspeed, the block determines the drag forces.

w¯=(x˙bwx)2+(x˙ywx)2+(wz)2Fdx=12TRCdAfPabs(w¯)2Fdy=12TRCsAfPabs(w¯)2Fdz=12TRClAfPabs(w¯)2

Drag moments

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

Mdr=12TRCrmAfPabs(w¯)2(a+b)Mdp=12TRCpmAfPabs(w¯)2(a+b)Mdy=12TRCymAfPabs(w¯)2(a+b)

Lateral Corner Stiffness and Relaxation Dynamics

DescriptionImplementation

Constant values.

The block uses constant stiffness values for Cyf and Cyr.

Lookup tables as a function of corner stiffness data and slip angles.

The block uses lookup tables that are functions of the corner stiffness data and slip angles.

Cyf=f(αf,Cyfdata)Cyr=f(αr,Cyrdata)

Lookup tables as a function of corner stiffness data and slip angles.

Slip angles include the relaxation length dynamic settings.

The block uses lookup tables that are functions of the corner stiffness data and slip angles. The slip angles include the relaxation length dynamic settings. The relaxation length approximates an effective corner stiffness force that is a function of wheel travel.

Cyf=f(αfσ,Cyfdata)Cyr=f(αrσ,Cyrdata)αfσ=1s[(αfαfσ)vwfαf]αrσ=1s[(αrαrσ)vwrαr]

The equations use these variables.

x,x˙,x¨

Vehicle CG displacement, velocity, and acceleration, along the vehicle-fixed x-axis

y,y˙,y¨

Vehicle CG displacement, velocity, and acceleration, along the vehicle-fixed y-axis

ψ

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

r,Ψ˙

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

Fxf, Fxr

Longitudinal forces applied to front and rear wheels, along the vehicle-fixed x-axis

Fyf, Fyr

Lateral forces applied to front and rear wheels, along vehicle-fixed y-axis

Fxext, Fyext, Fzext

External forces applied to vehicle CG, along the vehicle-fixed x-, y-, and z-axes

Fdx, Fdy, Fdz

Drag forces applied to vehicle CG, along the vehicle-fixed x-, y-, and z-axes

Fxinput, Fyinput, Fzinput

Input forces applied to vehicle CG, along the vehicle-fixed x-, y-, and z-axes

Mxext, Myext, Mzext

External moment about vehicle CG, about the vehicle-fixed x-, y-, and z-axes

Mdx, Mdy, Mdz

Drag moment about vehicle CG, about the vehicle-fixed x-, y-, and z-axes

Mxinput, Myinput, Mzinput

Input moment about vehicle CG, about the vehicle-fixed x-, y-, and z-axes

Izz

Vehicle body moment of inertia about the vehicle-fixed z-axis

Fxft, Fxrt

Longitudinal tire force applied to front and rear wheels, along the vehicle-fixed x-axis

Fyft, Fyft

Lateral tire force applied to front and rear wheels, along vehicle-fixed y-axis

Fxfl, Fxfr

Longitudinal force applied to front left and front right wheels, along the vehicle-fixed x-axis

Fyfl, Fyfr

Lateral force applied to front left and front right wheels, along the vehicle-fixed y-axis

Fxrl, Fxrr

Longitudinal force applied to rear left and rear right wheels, along the vehicle-fixed x-axis

Fyrl, Fyrr

Lateral force applied to rear left and rear right wheels, along the vehicle-fixed y-axis

Fxflt, Fxfrt

Longitudinal tire force applied to front left and front right wheels, along the vehicle-fixed x-axis

Fyflt, Fyfrt

Lateral force tire applied to front left and front right wheels, along the vehicle-fixed y-axis

Fxrlt, Fxrrt

Longitudinal tire force applied to rear left and rear right wheels, along the vehicle-fixed x-axis

Fyrlt, Fyrrt

Lateral force applied to rear left and rear right wheels, along the vehicle-fixed y-axis

Fzf,Fzr

Normal force applied to front and rear wheels, along vehicle-fixed z-axis

Fznom

Nominal normal force applied to axles, along the vehicle-fixed z-axis

Fzfl,Fzfr

Normal force applied to front left and right wheels, along vehicle-fixed z-axis

Fzrl,Fzrr

Normal force applied to rear left and right wheels, along vehicle-fixed z-axis

m

Vehicle body mass

a, b

Distance of front and rear wheels, respectively, from the normal projection point of vehicle CG onto the common axle plane

h

Height of vehicle CG above the axle plane

d

Lateral distance from the geometric centerline to the center of mass along the vehicle-fixed y-axis

hh

Height of the hitch above the axle plane along the vehicle-fixed z-axis

dh

Longitudinal distance of the hitch from the normal projection point of tractor CG onto the common axle plane

hl

Lateral distance from center of mass to hitch along the vehicle-fixed y-axis.

αf, αr

Front and rear wheel slip angles

αfl, αfr

Front left and right wheel slip angles

αrl, αrr

Rear left and right wheel slip angles

δf, δr

Front and rear wheel steering angles

δrl, δrr

Rear left and right wheel steering angles

δfl, δfr

Front left and right wheel steering angles

wf, wr

Front and rear track widths

Cyf, Cyr

Front and rear wheel cornering stiffness

Cyfdata, Cyrdata

Front and rear wheel cornering stiffness data

σf, σr

Front and rear wheel relaxation length

α, αFront and rear wheel slip angles that include relaxation length
vwf, vwrMagnitude of front and rear wheel hardpoint velocity
μf, μr

Front and rear wheel friction coefficient

μfl, μfr

Front left and right wheel friction coefficient

μrl, μrr

Rear left and right wheel friction coefficient

Cd

Air drag coefficient acting along vehicle-fixed x-axis

Cs

Air drag coefficient acting along vehicle-fixed y-axis

ClAir drag coefficient acting along vehicle-fixed z-axis
Crm

Air drag roll moment acting about the vehicle-fixed x-axis

Cpm

Air drag pitch moment acting about the vehicle-fixed y-axis

Cym

Air drag yaw moment acting about the vehicle-fixed z-axis

Af

Frontal area

RAtmospheric specific gas constant
TEnvironmental air temperature
PabsEnvironmental 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

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Front wheel steering angles, δF, in rad.

Vehicle Track Setting

Variable

Signal Dimension

Single (bicycle)δF

Scalar – 1

Dual

δF=[δflδfr]  or  [δflδfr]

Array – [1x2] or [2x1]

Dependencies

To enable this port, on the Input signals pane, select Front wheel steering.

Rear wheel steering angles, δR, in rad.

Vehicle Track Setting

Variable

Signal Dimension

Single (bicycle)δR

Scalar – 1

Dual

δR=[δrlδrr]  or  [δrlδrr]

Array – [1x2] or [2x1]

Dependencies

To enable this port, on the Input signals pane, select Rear wheel steering.

Vehicle CG velocity along the vehicle-fixed x-axis, in m/s.

Dependencies

To enable this port, set Axle forces to External longitudinal velocity.

Force on the front wheels, FwF, along the vehicle-fixed axis, in N.

Vehicle Track Setting

Axle Forces Setting

Description

Variable

Signal Dimension

Single (bicycle)External longitudinal forces

Longitudinal force on the front wheel

FwF=Fxf

Scalar – 1

External forces

Longitudinal and lateral forces on the front wheel

FwF=[FxfFyf]  or [FxfFyf]

Array – [1x2] or [2x1]

DualExternal longitudinal forces

Longitudinal force on the front wheels

FwF=[FxflFxfr] or [FxflFxfr]

Array – [1x2] or [2x1]

External forces

Longitudinal and lateral forces on the front wheels

FwF=[FxflFxfrFyflFyfr]

Array – [2x2]

Dependencies

To enable this port, set Axle forces to one of these options:

  • External longitudinal forces

  • External forces

Force on the rear wheels, FwR, along the vehicle-fixed axis, in N.

Vehicle Track Setting

Axle Forces Setting

Description

Variable

Signal Dimension

Single (bicycle)External longitudinal forces

Longitudinal force on the rear wheel

FwR=Fxr

Scalar – 1

External forces

Longitudinal and lateral forces on the rear wheel

FwR=[FxrFyr]  or [FxrFyr]

Array – [1x2] or [2x1]

DualExternal longitudinal forces

Longitudinal force on the rear wheels

FwR=[FxrlFxrr] or [FxrlFxrr]

Array – [1x2] or [2x1]

External forces

Longitudinal and lateral forces on the rear wheels

FwR=[FxrlFxrrFyrlFyrr]

Array – [2x2]

Dependencies

To enable this port, set Axle forces to one of these options:

  • External longitudinal forces

  • External forces

External forces applied to vehicle CG, Fxext, Fyext, Fzext, in vehicle-fixed frame, in N. Signal vector dimensions are [1x3] or [3x1].

Dependencies

To enable this port, on the Input signals pane, select External forces.

External moment about vehicle CG, Mx, My, Mz, in the vehicle-fixed frame, in N·m. Signal vector dimensions are [1x3] or [3x1].

Dependencies

To enable this port, on the Input signals pane, select External moments.

Hitch force applied to the body at the hitch location, Fhx, Fhy, Fhz, in the vehicle-fixed frame, in N, specified as a 1-by-3 or 3-by-1 array.

Dependencies

To enable this port, under Input signals, select Hitch forces.

Hitch moment at the hitch location, Mhx, Mhy, Mhz, about the vehicle-fixed frame, in N·m, specified as a 1-by-3 or 3-by-1 array.

Dependencies

To enable this port, under Input signals, select Hitch moments.

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

Dependencies

To enable this port, on the Input signals pane, select External wind.

Tire friction coefficient, μ. The value is dimensionless.

Vehicle Track Setting

Description

Variable

Signal Dimension

Single (bicycle)

Longitudinal force on the front wheel

Mu=[μfμr] or [μfμr]

Array – [1x2] or [2x1]

Dual

Longitudinal force on the front wheels

Mu=[μflμfrμrlμrr]

Array – [2x2]

Dependencies

To enable this port, on the Input signals pane, select External friction.

Ambient air temperature, in K.

Dependencies

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

Initial vehicle CG displacement along the earth-fixed X-axis, in m.

Dependencies

To enable this port, on the Input signals pane, select Initial longitudinal position.

Initial vehicle CG displacement along the earth-fixed Y-axis, in m.

Dependencies

To enable this port, on the Input signals pane, select Initial lateral position.

Initial vehicle CG velocity along the vehicle-fixed x-axis, in m/s.

Dependencies

To enable this port:

  1. Set Axle forces to one of these options:

    • External longitudinal forces

    • External forces

  2. On the Input signals pane, select Initial longitudinal velocity

Initial vehicle CG velocity along the vehicle-fixed y-axis, in m/s.

Dependencies

To enable this port, on the Input signals pane, select Initial lateral velocity.

Rotation of the vehicle-fixed frame about the earth-fixed Z-axis (yaw), in rad.

Dependencies

To enable this port, on the Input signals pane, select Initial yaw angle.

Vehicle angular velocity about the vehicle-fixed z-axis (yaw rate), in rad/s.

Dependencies

To enable this port, on the Input signals pane, select Initial yaw rate.

Output

expand all

Bus signal containing these block values.

SignalDescriptionValueUnits
InertFrmCgDispXVehicle CG displacement along the earth-fixed X-axis

Computed

m
YVehicle CG displacement along the earth-fixed Y-axis

Computed

m

ZVehicle CG displacement along the earth-fixed Z-axis0m
VelXdotVehicle CG velocity along the earth-fixed X-axis

Computed

m/s

YdotVehicle CG velocity along the earth-fixed Y-axis

Computed

m/s
ZdotVehicle CG velocity along the earth-fixed Z-axis0m/s
AngphiRotation of the vehicle-fixed frame about the earth-fixed X-axis (roll)0rad
thetaRotation of the vehicle-fixed frame about the earth-fixed Y-axis (pitch)0rad
psiRotation of the vehicle-fixed frame about the earth-fixed Z-axis (yaw)

Computed

rad
FrntAxlLftDispXFront left wheel displacement along the earth-fixed X-axis

Computed

m
YFront left wheel displacement along the earth-fixed Y-axis

Computed

m
ZFront left wheel displacement along the earth-fixed Z-axis0m
VelXdotFront left wheel velocity along the earth-fixed X-axis

Computed

m/s
YdotFront left wheel velocity along the earth-fixed Y-axis

Computed

m/s
ZdotFront left wheel velocity along the earth-fixed Z-axis0m/s
RghtDispXFront right wheel displacement along the earth-fixed X-axis

Computed

m
YFront right wheel displacement along the earth-fixed Y-axis

Computed

m
ZFront right wheel displacement along the earth-fixed Z-axis0m
VelXdotFront right wheel velocity along the earth-fixed X-axis

Computed

m/s
YdotFront right wheel velocity along the earth-fixed Y-axis

Computed

m/s
ZdotFront right wheel velocity along the earth-fixed Z-axis0m/s
RearAxlLftDispXRear left wheel displacement along the earth-fixed X-axis

Computed

m
YRear left wheel displacement along the earth-fixed Y-axis

Computed

m
ZRear left wheel displacement along the earth-fixed Z-axis0m
VelXdotRear left wheel velocity along the earth-fixed X-axis

Computed

m/s
YdotRear left wheel velocity along the earth-fixed Y-axis

Computed

m/s
ZdotRear left wheel velocity along the earth-fixed Z-axis0m/s
RghtDispXRear right wheel displacement along the earth-fixed X-axis

Computed

m
YRear right wheel displacement along the earth-fixed Y-axis

Computed

m
ZRear right wheel displacement along the earth-fixed Z-axis0m
VelXdotRear right wheel velocity along the earth-fixed X-axis

Computed

m/s
YdotRear right wheel velocity along the earth-fixed Y-axis

Computed

m/s
ZdotRear right wheel velocity along the earth-fixed Z-axis0m/s
HitchDispXHitch offset from axle plane along the earth-fixed X-axis

Computed

m
YHitch offset from center plane along the earth-fixed Y-axis

Computed

m
ZHitch offset from axle plane along the earth-fixed Z-axis

Computed

m
VelXdotHitch offset velocity from axle plane along the earth-fixed X-axis

Computed

m
YdotHitch offset velocity from center plane along the earth-fixed Y-axis

Computed

m
ZdotHitch offset velocity from axle plane along the earth-fixed Z-axis

Computed

m
GeomDispXVehicle chassis offset from axle plane along the earth-fixed X-axis

Computed

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

Computed

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

Computed

m
VelXdotVehicle chassis offset velocity along the earth-fixed X-axis

Computed

m/s
YdotVehicle chassis offset velocity along the earth-fixed Y-axis

Computed

m/s
ZdotVehicle chassis offset velocity along the earth-fixed Z-axis

Computed

m/s
BdyFrmCgVelxdotVehicle CG velocity along the vehicle-fixed x-axis

Computed

m/s
ydotVehicle CG velocity along the vehicle-fixed y-axis

Computed

m/s
zdotVehicle CG velocity along the vehicle-fixed z-axis0m/s
AngBeta

Body slip angle, β

β=VyVx

Computed

rad
AngVelpVehicle angular velocity about the vehicle-fixed x-axis (roll rate)0rad/s
qVehicle angular velocity about the vehicle-fixed y-axis (pitch rate)0rad/s
rVehicle angular velocity about the vehicle-fixed z-axis (yaw rate)

Computed

rad/s
AccaxVehicle CG acceleration along the vehicle-fixed x-axis

Computed

gn
ayVehicle CG acceleration along the vehicle-fixed y-axis

Computed

gn
azVehicle CG acceleration along the vehicle-fixed z-axis0gn
xddotVehicle CG acceleration along the vehicle-fixed x-axis

Computed

m/s^2
yddotVehicle CG acceleration along the vehicle-fixed y-axis

Computed

m/s^2
zddotVehicle CG acceleration along the vehicle-fixed z-axis0m/s^2
AngAccpdotVehicle angular acceleration about the vehicle-fixed x-axis0rad/s
qdotVehicle angular acceleration about the vehicle-fixed y-axis0rad/s
rdotVehicle angular acceleration about the vehicle-fixed z-axis

Computed

rad/s
DCM

Direction cosine matrix

Computed

rad
ForcesBodyFxNet force on vehicle CG along the vehicle-fixed x-axis

Computed

N
FyNet force on vehicle CG along the vehicle-fixed y-axis

Computed

N
FzNet force on vehicle CG along the vehicle-fixed z-axis0N
ExtFxExternal force on vehicle CG along the vehicle-fixed x-axis

Computed

N
FyExternal force on vehicle CG along the vehicle-fixed y-axis

Computed

N
FzExternal force on vehicle CG along the vehicle-fixed z-axis0N
HitchFx

Hitch force applied to body at the hitch location along the vehicle-fixed x-axis

Input

N
Fy

Hitch force applied to body at the hitch location along the vehicle-fixed y-axis

Input

N
Fz

Hitch force applied to body at the hitch location along the vehicle-fixed z-axis

Input

N
FrntAxlLftFx

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

Computed

N
Fy

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

Computed

N
Fz

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

Computed

N
RghtFx

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

Computed

N
Fy

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

Computed

N
Fz

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

ComputedN
RearAxlLftFx

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

Computed

N
Fy

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

Computed

N
Fz

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

ComputedN
RghtFx

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

Computed

N
Fy

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

Computed

N
Fz

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

ComputedN
TiresFrntTiresLftFx

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

ComputedN
Fy

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

ComputedN
Fz

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

ComputedN
RghtFx

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

ComputedN
Fy

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

ComputedN
Fz

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

ComputedN
RearTiresLftFx

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

ComputedN
Fy

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

ComputedN
Fz

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

ComputedN
RghtFx

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

ComputedN
Fy

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

ComputedN
Fz

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

Computed 
DragFxDrag force on vehicle CG along the vehicle-fixed x-axis

Computed

N
FyDrag force on vehicle CG along the vehicle-fixed y-axis

Computed

N
FzDrag force on vehicle CG along the vehicle-fixed z-axis

Computed

N
GrvtyFxGravity force on vehicle CG along the vehicle-fixed x-axis

Computed

N
FyGravity force on vehicle CG along the vehicle-fixed y-axis

Computed

N
FzGravity force on vehicle CG along the vehicle-fixed z-axis

Computed

N
MomentsBodyMxBody moment on vehicle CG about the vehicle-fixed x-axis0N·m
MyBody moment on vehicle CG about the vehicle-fixed y-axis

Computed

N·m
MzBody moment on vehicle CG about the vehicle-fixed z-axis0N·m
DragMxDrag moment on vehicle CG about the vehicle-fixed x-axis0N·m
MyDrag moment on vehicle CG about the vehicle-fixed y-axis

Computed

N·m
MzDrag moment on vehicle CG about the vehicle-fixed z-axis0N·m
ExtMxExternal moment on vehicle CG about the vehicle-fixed x-axis0N·m
MyExternal moment on vehicle CG about the vehicle-fixed y-axis

Computed

N·m
MzExternal moment on vehicle CG about the vehicle-fixed z-axis0N·m
HitchMxHitch moment at the hitch location about vehicle-fixed x-axis0N·m
MyHitch moment at the hitch location about vehicle-fixed y-axis

Computed

N·m
MzHitch moment at the hitch location about vehicle-fixed z-axis0N·m
FrntAxlLftDispxFront left wheel displacement along the vehicle-fixed x-axis

Computed

m
yFront left wheel displacement along the vehicle-fixed y-axisComputedm
zFront left wheel displacement along the vehicle-fixed z-axis

Computed

m
VelxdotFront left wheel velocity along the vehicle-fixed x-axis

Computed

m/s
ydotFront left wheel velocity along the vehicle-fixed y-axis

Computed

m/s
zdotFront left wheel velocity along the vehicle-fixed z-axis0m/s
RghtDispxFront right wheel displacement along the vehicle-fixed x-axis

Computed

m
yFront right wheel displacement along the vehicle-fixed y-axisComputedm
zFront right wheel displacement along the vehicle-fixed z-axis

Computed

m
VelxdotFront right wheel velocity along the vehicle-fixed x-axis

Computed

m/s
ydotFront right wheel velocity along the vehicle-fixed y-axis

Computed

m/s
zdotFront right wheel velocity along the vehicle-fixed z-axis0m/s
SteerWhlAngFL

Front left wheel steering angle

Computed

rad
WhlAngFR

Front right wheel steering angle

Computed

rad
RearAxlLftDispxRear left wheel displacement along the vehicle-fixed x-axis

Computed

m
yRear left wheel displacement along the vehicle-fixed y-axisComputedm
zRear left wheel displacement along the vehicle-fixed z-axis

Computed

m
VelxdotRear left wheel velocity along the vehicle-fixed x-axis

Computed

m/s
ydotRear left wheel velocity along the vehicle-fixed y-axis

Computed

m/s
zdotRear left wheel velocity along the vehicle-fixed z-axis0m/s
RghtDispxRear right wheel displacement along the vehicle-fixed x-axis

Computed

m
yRear right wheel displacement along the vehicle-fixed y-axisComputedm
zRear right wheel displacement along the vehicle-fixed z-axis

Computed

m
VelxdotRear right wheel velocity along the vehicle-fixed x-axis

Computed

m/s
ydotRear right wheel velocity along the vehicle-fixed y-axis

Computed

m/s
zdotRear right wheel velocity along the vehicle-fixed z-axis0m/s
SteerWhlAngRL

Rear left wheel steering angle

Computed

rad
WhlAngRR

Rear right wheel steering angle

Computed

rad
HitchDispxHitch offset from axle plane along the vehicle-fixed x-axis

Input

m
yHitch offset from center plane along the vehicle-fixed y-axis

Input

m
zHitch offset from axle plane along the earth-fixed z-axis

Input

m
VelxdotHitch offset velocity along the vehicle-fixed x-axis

Computed

m/s
ydotHitch offset velocity along the vehicle-fixed y-axis

Computed

m/s
zdotHitch offset velocity along the vehicle-fixed z-axis

Computed

m/s
PwrExtApplied external power

Computed

W
HitchPower loss due to hitch

Computed

W
DragPower loss due to drag

Computed

W
GeomDispxVehicle chassis offset from axle plane along the vehicle-fixed x-axis

Input

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

Input

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

Input

m
VelxdotVehicle chassis offset velocity along the vehicle-fixed x-axis

Computed

m/s
ydotVehicle chassis offset velocity along the vehicle-fixed y-axis

Computed

m/s
zdotVehicle chassis offset velocity along the vehicle-fixed z-axis0m/s
BetaBeta

Body slip angle, β

β=VyVx

Computed

rad

SignalDescriptionValueUnits
PwrInfoPwrTrnsfrdPwrFxExtExternally applied longitudinal force power

Computed

W
PwrFyExtExternally applied lateral force power

Computed

W
PwrMzExtExternally applied roll moment power

Computed

W
PwrFwFLxLongitudinal force applied at the front left axle power

Computed

W
PwrFwFLyLateral force applied at the front left axle power

Computed

W
PwrFwFRxLongitudinal force applied at the front right axle power

Computed

W
PwrFwFRyLateral force applied at the front right axle power

Computed

W
PwrFwRLxLongitudinal force applied at the rear left axle power

Computed

W
PwrFwRLyLateral force applied at the rear left axle power

Computed

W
PwrFwRRxLongitudinal force applied at the rear right axle power

Computed

W
PwrFwRRyLateral force applied at the rear right axle power

Computed

W
PwrNotTrnsfrdPwrFxDragLongitudinal drag force power

Computed

W
PwrFyDragLateral drag force power

Computed

W
PwrMzDragDrag pitch moment power

Computed

W
PwrStoredPwrStoredGrvtyRate change in gravitational potential energy

Computed

W
PwrStoredxdotRate of change of longitudinal kinetic energy

Computed

W
PwrStoredydotRate of change of lateral kinetic energy

Computed

W
PwrStoredrRate of change of rotational yaw kinetic energy

Computed

W

Vehicle CG velocity along the vehicle-fixed x-axis, in m/s.

Vehicle CG velocity along the vehicle-fixed y-axis, in m/s.

Rotation of the vehicle-fixed frame about the earth-fixed Z-axis (yaw), in rad.

Vehicle angular velocity, r, about the vehicle-fixed z-axis (yaw rate), in rad/s.

Normal force on front wheels, FzF, along the vehicle-fixed z-axis, in N.

Vehicle Track Setting

Description

Variable

Signal Dimension

Single (bicycle)

Normal force on front axle

FzF=Fzf

Scalar – 1

Dual

Normal force on the front wheels

FzF=[FzflFzfr]

Array – [1x2]

Normal force on rear wheels, FzR, along the vehicle-fixed z-axis, in N.

Vehicle Track Setting

Description

Variable

Signal Dimension

Single (bicycle)

Normal force on rear wheel

FzR=Fzr

Scalar – 1

Dual

Normal force on the rear wheels

FzR=[FzrlFzrr]

Array – [1x2]

Parameters

expand all

Options

In the Vehicle Dynamics Blockset library, there are two types of Vehicle Body 3DOF blocks that model longitudinal, lateral, and yaw motion.

BlockVehicle Track SettingImplementation

Vehicle Body 3DOF Single TrackVehicle Body 3DOF block single track

Single (bicycle)

  • Forces act along the center line at the front and rear axles.

  • No lateral load transfer.

Vehicle Body 3DOF Dual TrackVehicle Body 3DOF block

Dual

Forces act at the four vehicle corners or hard points.

Use the Axle forces parameter to specify the type of force.

Axle Forces SettingImplementation

External longitudinal velocity

  • The block assumes that the external longitudinal velocity is in a quasi-steady state, so the longitudinal acceleration is approximately zero.

  • Because the motion is quasi-steady, the block calculates lateral forces using the tire slip angles and linear cornering stiffness.

  • Consider this setting when you want to:

    • Generate virtual sensor signal data.

    • Conduct high-level software studies that are not impacted by driveline or nonlinear tire responses.

External longitudinal forces

  • The block uses the external longitudinal force to accelerate or brake the vehicle.

  • The block calculates lateral forces using the tire slip angles and linear cornering stiffness.

  • Consider this setting when you want to:

    • Account for changes in the longitudinal velocity on the lateral and yaw motion.

    • Specify the external longitudinal motion through a force instead of an external longitudinal velocity.

    • Connect the block to tractive actuators, wheels, brakes, and hitches.

External forces

  • The block uses the external lateral and longitudinal forces to steer, accelerate, or brake the vehicle.

  • The block does not use the steering input to calculate vehicle motion.

  • Consider this setting when you need tire models with more accurate nonlinear combined lateral and longitudinal slip.

Input Signals

Specify to create input port WhlAngF.

Specify to create input port WindXYZ.

Specify to create input port FExt.

Specify to create input port MExt.

Specify to create input port WhlAngR.

Specify to create input port Mu.

Select to create input port Fh.

Specify to create input port Mh.

Specify to create input port X_o.

Specify to create input port Y_o.

Specify to create input port xdot_o.

Specify to create input port ydot_o.

Specify to create input port psi_o.

Specify to create input port r_o.

Specify to create input port AirTemp.

Longitudinal

Number of wheels on front axle, NF. The value is dimensionless.

Number of wheels on rear axle, NR. The value is dimensionless.

Vehicle mass, m, in kg.

Horizontal distance a from the vehicle CG to the front wheel axle, in m.

Horizontal distance b from the vehicle CG to the rear wheel axle, in m.

Height of vehicle CG above the axles, h, in m.

Longitudinal distance from center of mass to hitch, dh, in m.

Dependencies

To enable this parameter, on the Input signals pane, select Hitch forces or Hitch moments.

Vertical distance from hitch to axle plane, hh, in m.

Dependencies

To enable this parameter, on the Input signals pane, select Hitch forces or Hitch moments.

Initial vehicle CG displacement along earth-fixed X-axis, in m.

Initial vehicle CG velocity along vehicle-fixed x-axis, in m/s.

Dependencies

For the Vehicle Body 3DOF Single Track or Vehicle Body 3DOF Dual Track blocks, to enable this parameter, set Axle forces to one of these options:

  • External longitudinal forces

  • External forces

Lateral

Front tire corner stiffness, Cyf, in N/rad.

Dependencies

For the Vehicle Body 3DOF Single Track or Vehicle Body 3DOF Dual Track blocks, to enable this parameter:

  1. Set Axle forces to one of these options:

    • External longitudinal velocity

    • External longitudinal forces

  2. Clear Mapped corner stiffness.

Rear tire corner stiffness, Cyr, in N/rad.

Dependencies

For the Vehicle Body 3DOF Single Track or Vehicle Body 3DOF Dual Track blocks, to enable this parameter:

  1. Set Axle forces to one of these options:

    • External longitudinal velocity

    • External longitudinal forces

  2. Clear Mapped corner stiffness.

Initial vehicle CG displacement along earth-fixed Y-axis, in m.

Initial vehicle CG velocity along vehicle-fixed y-axis, in m/s.

Enables mapped corner stiffness calculation.

Dependencies

To enable this parameter, set Axle forces to one of these options:

  • External longitudinal velocity

  • External longitudinal forces

Enables relaxation length dynamics.

Dependencies

To enable this parameter:

  1. Set Axle forces to one of these options:

    • External longitudinal velocity

    • External longitudinal forces

  2. Clear Mapped corner stiffness.

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.

Lateral distance from geometric centerline to the hitch, hl, in m, along the vehicle-fixed y. Positive values indicate that the hitch is to the right of the geometric centerline. Negative values indicate that the hitch is to the left of the geometric centerline.

Dependencies

To enable this parameter, on the Input signals pane, select Hitch forces or Hitch moments.

Track width, w, in m.

Dependencies

To enable this parameter, set Vehicle track to Dual.

Front tire relaxation length, σf, in m.

Dependencies

To enable this parameter:

  1. Set Axle forces to one of these options:

    • External longitudinal velocity

    • External longitudinal forces

  2. Do either of these:

    • Select Mapped corner stiffness.

    • Clear Mapped corner stiffness and select Include relaxation length dynamics.

Rear tire relaxation length, σr, in m.

Dependencies

To enable this parameter:

  1. Set Axle forces to one of these options:

    • External longitudinal velocity

    • External longitudinal forces

  2. Do either of these:

    • Select Mapped corner stiffness.

    • Clear Mapped corner stiffness and select Include relaxation length dynamics.

Front axle slip angle breakpoints, αfbrk, in rad.

Dependencies

To enable this parameter:

  1. Set Axle forces to one of these options:

    • External longitudinal velocity

    • External longitudinal forces

  2. Select Mapped corner stiffness.

Front axle corner data, Cyfdata, in N/rad.

Dependencies

To enable this parameter:

  1. Set Axle forces to one of these options:

    • External longitudinal velocity

    • External longitudinal forces

  2. Select Mapped corner stiffness.

Rear axle slip angle breakpoints, αrbrk, in rad.

Dependencies

To enable this parameter:

  1. Set Axle forces to one of these options:

    • External longitudinal velocity

    • External longitudinal forces

  2. Select Mapped corner stiffness.

Rear axle corner data, Cyrdata, in N/rad.

Dependencies

To enable this parameter:

  1. Set Axle forces to one of these options:

    • External longitudinal velocity

    • External longitudinal forces

  2. Select Mapped corner stiffness.

Yaw

Yaw polar inertia, in kg*m^2.

Rotation of the vehicle-fixed frame about earth-fixed Z-axis (yaw), in rad.

Vehicle angular velocity about the vehicle-fixed z-axis (yaw rate), in rad/s.

Aerodynamic

Effective vehicle cross-sectional area, Af, to calculate the aerodynamic drag force on the vehicle, in m2.

Air drag coefficient, Cd. The value is dimensionless.

Air lift coefficient, Cl. The value is dimensionless.

Longitudinal drag pitch moment coefficient, Cpm. The value is dimensionless.

Relative wind angle vector, βw, in rad.

Side force coefficient vector coefficient, Cs. The value is dimensionless.

Yaw moment coefficient vector coefficient, Cym. The value is dimensionless.

Environment

Environmental absolute pressure, Pabs, in Pa.

Environmental absolute temperature, T, in K.

Dependencies

To enable this parameter, clear Air temperature.

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

Nominal friction scale factor, μ. The value is dimensionless.

Dependencies

For the Vehicle Body 3DOF Single Track or Vehicle Body 3DOF Dual Track blocks, to enable this parameter:

  1. Set Axle forces to one of these options:

    • External longitudinal velocity

    • External longitudinal forces

  2. Clear External Friction.

Simulation

Longitudinal velocity tolerance, in m/s.

Nominal normal force, in N.

Dependencies

For the Vehicle Body 3DOF Single Track or Vehicle Body 3DOF Dual Track blocks, to enable this parameter, set Axle forces to one of these options:

  • External longitudinal velocity

  • External longitudinal forces

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.

References

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

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Introduced in R2018a