# Vehicle Body 3DOF Three Axles

Three-axle vehicle body with longitudinal, lateral, and yaw motion

Since R2020a

Libraries:
Vehicle Dynamics Blockset / Vehicle Body

## Description

The Vehicle Body 3DOF Three Axles block implements a rigid, three-axle vehicle body model to calculate longitudinal, lateral, and yaw motion. The block accounts for the axle and hitch reaction forces due to the vehicle body mass acceleration, aerodynamic drag, 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.

Use the Vehicle track parameter to specify the number of wheels.

Vehicle Track SettingImplementation

`Single (bicycle)`

• Forces act along the center line of the axles.

`Dual`

Forces act at the axle hard-point locations.

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.

To create additional input ports, under Input signals, select these block parameters.

Input Signals Pane Parameter

Input PortDescription

Front wheel steering

`WhlAngF`

Front wheel angle, δF

Middle wheel steering`WhlAngM`

Middle wheel angle, δM

Rear wheel steering`WhlAngR`

Rear wheel angle, δR

External wind

`WindXYZ`

Wind speed, WX, WY, and WZ, in an inertial reference frame

External friction`Mu`

Friction coefficient

External forces`FExt`

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

External moments

`MExt`

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

Front hitch forces`FhF`

Hitch force applied to the body at the front hitch location, FhFx, FhFy, and FhFz, in the vehicle-fixed frame

Front hitch moments`MhF`

Hitch moment at the front hitch location, MhFx, MhFy, and MhFz, about the vehicle-fixed frame

Rear hitch forces`FhR`

Hitch force applied to the body at the rear hitch location, FhRx, FhRy, and FhRz, in the vehicle-fixed frame

Rear hitch moments`MhR`

Hitch moment at the rear hitch location, MhRx, MhRy, and MhRz, about the vehicle-fixed frame

Initial longitudinal position

`X_o`

Initial vehicle CG displacement along the earth-fixed X-axis

Initial yaw angle

`psi_o`

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

Initial longitudinal velocity

`xdot_o`

Initial vehicle CG velocity along the vehicle-fixed x-axis

Initial yaw rate

`r_o`

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

Initial lateral position

`Y_o`

Initial vehicle CG displacement along the earth-fixed Y-axis

Air temperature

`AirTemp`

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

Initial lateral velocity

`ydot_o`

Initial vehicle CG velocity along the vehicle-fixed y-axis

### Theory

To determine the vehicle motion, the block solves the rigid body planar dynamics equations of motion.

CalculationDescription

Dynamics

The block solves the rigid-body planar dynamics equations to determine the vehicle longitudinal motion. If you set Axle forces to ```External longitudinal velocity```, the block assumes a quasi-steady state for the longitudinal acceleration.

External forces

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

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

Tire forces

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

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

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

Single Track

Dual Track

The illustrations use these variables.

 a, b, c Longitudinal distance of the front, middle, and rear axles, respectively, from the normal projection point of the vehicle CG onto the common axle plane h Height of vehicle CG above the axle plane along the vehicle-fixed z-axis d Lateral distance from geometric centerline to 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 normal projection point of the vehicle CG onto the common axle plane hl Lateral distance from center of mass to hitch along the vehicle-fixed y-axis. wf, wm, wr Front, middle, and rear track width, respectively

Drag

This table summarizes the block implementation for the drag calculation.

CalculationDescription

Coordinate transformation

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

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.

Drag moments

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

Lateral Corner Stiffness and Relaxation Dynamics

To enable the mapped corner stiffness and relaxation length dynamic parameters, set Axle forces to ```External longitudinal force``` or ```External longitudinal velocity```.

Parameter SettingsDescription
Mapped Corner StiffnessInclude Relaxation Length Dynamics

`Off` (default)

`On` (default)

The block uses constant corner stiffness values.

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.

`On`

`On` (default)

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.

`Off` (default)

`Off`

The block uses constant corner stiffness values.

## Ports

### Input

expand all

Front wheel steering angles, δF, in rad.

Vehicle Track Setting

Variable

Signal Dimension

```Single (bicycle)```δF

Scalar – `1`

`Dual`

Array – `[1x2]` or `[2x1]`

#### Dependencies

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

Middle wheel steering angles, δM, in rad.

Vehicle Track Setting

Variable

Signal Dimension

```Single (bicycle)```δM

Scalar – `1`

`Dual`

Array – `[1x2]` or `[2x1]`

#### Dependencies

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

Rear wheel steering angles, δR, in rad.

Vehicle Track Setting

Variable

Signal Dimension

```Single (bicycle)```δR

Scalar – `1`

`Dual`

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

`$FwF=F{x}_{f}$`

Scalar – `1`

```External forces```

Longitudinal and lateral forces on the front wheel

Array – `[1x2]` or `[2x1]`

`Dual````External longitudinal forces```

Longitudinal force on the front wheels

Array – `[1x2]` or `[2x1]`

```External forces```

Longitudinal and lateral forces on the front wheels

`$FwF=\left[\begin{array}{cc}{F}_{xfl}& {F}_{xfr}\\ {F}_{yfl}& {F}_{yfr}\end{array}\right]$`

Array – `[2x2]`

#### Dependencies

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

• `External longitudinal forces`

• `External forces`

Force on the middle wheels, FwM, 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 middle wheel

`$FwM=F{x}_{r}$`

Scalar – `1`

```External forces```

Longitudinal and lateral forces on the middle wheel

Array – `[1x2]` or `[2x1]`

`Dual````External longitudinal forces```

Longitudinal force on the middle wheels

Array – `[1x2]` or `[2x1]`

```External forces```

Longitudinal and lateral forces on the middle wheels

`$FwM=\left[\begin{array}{cc}{F}_{xml}& {F}_{xmr}\\ {F}_{yml}& {F}_{ymr}\end{array}\right]$`

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

`$FwR=F{x}_{r}$`

Scalar – `1`

```External forces```

Longitudinal and lateral forces on the rear wheel

Array – `[1x2]` or `[2x1]`

`Dual````External longitudinal forces```

Longitudinal force on the rear wheels

Array – `[1x2]` or `[2x1]`

```External forces```

Longitudinal and lateral forces on the rear wheels

`$FwR=\left[\begin{array}{cc}{F}_{xrl}& {F}_{xrr}\\ {F}_{yrl}& {F}_{yrr}\end{array}\right]$`

Array – `[2x2]`

#### Dependencies

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

• `External longitudinal forces`

• `External forces`

External forces applied to the vehicle CG, Fxext, Fyext, Fzext, in vehicle-fixed frame, in N. The signal array dimensions are `[1x3]` or `[3x1]`.

#### Dependencies

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

External moment about the vehicle CG, Mx, My, Mz, in the vehicle-fixed frame, in N·m. The signal array 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 the inertial X-, Y-, and Z-axes, in m/s. The signal array dimensions are `[1x3]` or `[3x1]`.

#### Dependencies

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

Tire friction coefficient, μ, dimensionless.

Vehicle Track Setting

Variable

Signal Dimension

`Single (bicycle)`

Array – `[1x3]` or `[3x1]`

`Dual`
`$Mu=\left[\begin{array}{cc}{\mu }_{fl}& {\mu }_{fr}\\ {\mu }_{ml}& {\mu }_{mr}\\ {\mu }_{rl}& {\mu }_{rr}\end{array}\right]$`

Array – `[3x2]`

#### 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.

#### Dependencies

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

### Output

expand all

Vehicle data, returned as a bus signal containing these block values.

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

Computed

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

Computed

m

`Z`Vehicle CG displacement along the earth-fixed Z-axis`0`m
`Vel``Xdot`Vehicle CG velocity along the earth-fixed X-axis

Computed

m/s

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

Computed

m/s
`Zdot`Vehicle CG velocity along the earth-fixed Z-axis`0`m/s
`Ang``phi`Rotation of the vehicle-fixed frame about the earth-fixed X-axis (roll)`0`rad
`theta`Rotation of the vehicle-fixed frame about the earth-fixed Y-axis (pitch)`0`rad
`psi`Rotation of the vehicle-fixed frame about the earth-fixed Z-axis (yaw)

Computed

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

Computed

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

Computed

m
`Z`Front left wheel displacement along the earth-fixed Z-axis`0`m
`Vel``Xdot`Front left wheel velocity along the earth-fixed X-axis

Computed

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

Computed

m/s
`Zdot`Front left wheel velocity along the earth-fixed Z-axis`0`m/s
`Rght``Disp``X`Front right wheel displacement along the earth-fixed X-axis

Computed

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

Computed

m
`Z`Front right wheel displacement along the earth-fixed Z-axis`0`m
`Vel``Xdot`Front right wheel velocity along the earth-fixed X-axis

Computed

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

Computed

m/s
`Zdot`Front right wheel velocity along the earth-fixed Z-axis`0`m/s
`MidlAxl``Lft``Disp``X`Middle left wheel displacement along the earth-fixed X-axis

Computed

m
`Y`Middle left wheel displacement along the earth-fixed Y-axis

Computed

m
`Z`Middle left wheel displacement along the earth-fixed Z-axis`0`m
`Vel``Xdot`Middle left wheel velocity along the earth-fixed X-axis

Computed

m/s
`Ydot`Middle left wheel velocity along the earth-fixed Y-axis

Computed

m/s
`Zdot`Middle left wheel velocity along the earth-fixed Z-axis`0`m/s
`Rght``Disp``X`Middle right wheel displacement along the earth-fixed X-axis

Computed

m
`Y`Middle right wheel displacement along the earth-fixed Y-axis

Computed

m
`Z`Middle right wheel displacement along the earth-fixed Z-axis`0`m
`Vel``Xdot`Middle right wheel velocity along the earth-fixed X-axis

Computed

m/s
`Ydot`Middle right wheel velocity along the earth-fixed Y-axis

Computed

m/s
`Zdot`Middle right wheel velocity along the earth-fixed Z-axis`0`m/s
`RearAxl``Lft``Disp``X`Rear left wheel displacement along the earth-fixed X-axis

Computed

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

Computed

m
`Z`Rear left wheel displacement along the earth-fixed Z-axis`0`m
`Vel``Xdot`Rear left wheel velocity along the earth-fixed X-axis

Computed

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

Computed

m/s
`Zdot`Rear left wheel velocity along the earth-fixed Z-axis`0`m/s
`Rght``Disp``X`Rear right wheel displacement along the earth-fixed X-axis

Computed

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

Computed

m
`Z`Rear right wheel displacement along the earth-fixed Z-axis`0`m
`Vel``Xdot`Rear right wheel velocity along the earth-fixed X-axis

Computed

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

Computed

m/s
`Zdot`Rear right wheel velocity along the earth-fixed Z-axis`0`m/s
`Hitch``Disp``X`Trailer hitch offset from the axle plane along the earth-fixed X-axis

Computed

m
`Y`Trailer hitch offset from the center plane along the earth-fixed Y-axis

Computed

m
`Z`Trailer hitch offset from the axle plane along the earth-fixed Z-axis

Computed

m
`Vel``Xdot`Trailer hitch offset velocity along the earth-fixed X-axis

Computed

m/s
`Ydot`Trailer hitch offset velocity along the earth-fixed Y-axis

Computed

m/s
`Zdot`Trailer hitch offset 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 CG velocity along the vehicle-fixed x-axis

Computed

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

Computed

m/s
`zdot`Vehicle CG velocity along the vehicle-fixed z-axis`0`m/s
`Ang``Beta`

Body slip angle, β

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

Computed

`AngVel``p`Vehicle angular velocity about the vehicle-fixed x-axis (roll rate)`0`rad/s
`q`Vehicle angular velocity about the vehicle-fixed y-axis (pitch rate)`0`rad/s
`r`Vehicle angular velocity about the vehicle-fixed z-axis (yaw rate)

Computed

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

Computed

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

Computed

gn
`az`Vehicle CG acceleration along the vehicle-fixed z-axis`0`gn
`xddot`Vehicle CG acceleration along the vehicle-fixed x-axis

Computed

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

Computed

m/s^2
`zddot`Vehicle CG acceleration along the vehicle-fixed z-axis`0`m/s^2
`AngAcc``pdot`Vehicle angular acceleration about the vehicle-fixed x-axis`0`rad/s
`qdot`Vehicle angular acceleration about the vehicle-fixed y-axis`0`rad/s
`rdot`Vehicle angular acceleration about the vehicle-fixed z-axis

Computed

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

Computed

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

Computed

N
`Fz`Net force on vehicle CG along the vehicle-fixed z-axis`0`N
`Ext``Fx`External force on vehicle CG along the vehicle-fixed x-axis

Computed

N
`Fy`External force on vehicle CG along the vehicle-fixed y-axis

Computed

N
`Fz`External force on vehicle CG along the vehicle-fixed z-axis`0`N
`Hitch``Fx`

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

Computed

N
`Fy`

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

Computed

N
`Fz`

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

Computed

N
`FrntAxl``Lft``Fx`

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
`Rght``Fx`

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
`MidlAxl``Lft``Fx`

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

Computed

N
`Fy`

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

Computed

N
`Fz`

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

Computed

N
`Rght``Fx`

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

Computed

N
`Fy`

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

Computed

N
`Fz`

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

ComputedN
`RearAxl``Lft``Fx`

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
`Rght``Fx`

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

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
`Rght``Fx`

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
`RearTires``Lft``Fx`

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
`Rght``Fx`

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

Computed

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

Computed

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

Computed

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

Computed

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

Computed

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

Computed

N
`Moments``Body``Mx`Body moment on vehicle CG about the vehicle-fixed x-axis`0`N·m
`My`Body moment on vehicle CG about the vehicle-fixed y-axis

Computed

N·m
`Mz`Body moment on vehicle CG about the vehicle-fixed z-axis`0`N·m
`Drag``Mx`Drag moment on vehicle CG about the vehicle-fixed x-axis`0`N·m
`My`Drag moment on vehicle CG about the vehicle-fixed y-axis

Computed

N·m
`Mz`Drag moment on vehicle CG about the vehicle-fixed z-axis`0`N·m
`Ext``Mx`External moment on vehicle CG about the vehicle-fixed x-axis`0`N·m
`My`External moment on vehicle CG about the vehicle-fixed y-axis

Computed

N·m
`Mz`External moment on vehicle CG about the vehicle-fixed z-axis`0`N·m
`Hitch``Mx`Hitch moment at the hitch location about vehicle-fixed x-axis`0`N·m
`My`Hitch moment at the hitch location about vehicle-fixed y-axis

Computed

N·m
`Mz`Hitch moment at the hitch location about vehicle-fixed z-axis`0`N·m
`FrntAxl``Lft``Disp``x`Front left wheel displacement along the vehicle-fixed x-axis

Computed

m
`y`Front left wheel displacement along the vehicle-fixed y-axisComputedm
`z`Front left wheel displacement along the vehicle-fixed z-axis

Computed

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

Computed

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

Computed

m/s
`zdot`Front left wheel velocity along the vehicle-fixed z-axis`0`m/s
`Rght``Disp``x`Front right wheel displacement along the vehicle-fixed x-axis

Computed

m
`y`Front right wheel displacement along the vehicle-fixed y-axisComputedm
`z`Front right wheel displacement along the vehicle-fixed z-axis

Computed

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

Computed

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

Computed

m/s
`zdot`Front right wheel velocity along the vehicle-fixed z-axis`0`m/s
`Steer``WhlAngFL`

Front left wheel steering angle

Computed

`WhlAngFR`

Front right wheel steering angle

Computed

`MidlAxl``Lft``Disp``x`Middle left wheel displacement along the vehicle-fixed x-axis

Computed

m
`y`Middle left wheel displacement along the vehicle-fixed y-axisComputedm
`z`Middle left wheel displacement along the vehicle-fixed z-axis

Computed

m
`Vel``xdot`Middle left wheel velocity along the vehicle-fixed x-axis

Computed

m/s
`ydot`Middle left wheel velocity along the vehicle-fixed y-axis

Computed

m/s
`zdot`Middle left wheel velocity along the vehicle-fixed z-axis`0`m/s
`Rght``Disp``x`Middle right wheel displacement along the vehicle-fixed x-axis

Computed

m
`y`Middle right wheel displacement along the vehicle-fixed y-axisComputedm
`z`Middle right wheel displacement along the vehicle-fixed z-axis

Computed

m
`Vel``xdot`Middle right wheel velocity along the vehicle-fixed x-axis

Computed

m/s
`ydot`Middle right wheel velocity along the vehicle-fixed y-axis

Computed

m/s
`zdot`Middle right wheel velocity along the vehicle-fixed z-axis`0`m/s
`Steer``WhlAngRL`

Middle left wheel steering angle

Computed

`WhlAngRR`

Middle right wheel steering angle

Computed

`RearAxl``Lft``Disp``x`Rear left wheel displacement along the vehicle-fixed x-axis

Computed

m
`y`Rear left wheel displacement along the vehicle-fixed y-axisComputedm
`z`Rear left wheel displacement along the vehicle-fixed z-axis

Computed

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

Computed

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

Computed

m/s
`zdot`Rear left wheel velocity along the vehicle-fixed z-axis`0`m/s
`Rght``Disp``x`Rear right wheel displacement along the vehicle-fixed x-axis

Computed

m
`y`Rear right wheel displacement along the vehicle-fixed y-axisComputedm
`z`Rear right wheel displacement along the vehicle-fixed z-axis

Computed

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

Computed

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

Computed

m/s
`zdot`Rear right wheel velocity along the vehicle-fixed z-axis`0`m/s
`Steer``WhlAngRL`

Rear left wheel steering angle

Computed

`WhlAngRR`

Rear right wheel steering angle

Computed

`Hitch``Disp``x`Hitch offset from axle plane along the vehicle-fixed x-axis

Input

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

Input

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

Input

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

Computed

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

Computed

m/s
`zdot`Hitch offset velocity along the vehicle-fixed z-axis`0`m/s
`Pwr``Ext`Applied external power

Computed

W
`Hitch`Power loss due to hitch

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`0`m/s
`Ang``Beta`

Body slip angle, β

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

Computed

SignalDescriptionValueUnits
`PwrInfo``PwrTrnsfrd``PwrFxExt`Externally applied longitudinal force power

Computed

W
`PwrFyExt`Externally applied lateral force power

Computed

W
`PwrMzExt`Externally applied yaw moment power

Computed

W
`PwrFwFLx`Longitudinal force applied at the front left axle power

Computed

W
`PwrFwFLy`Lateral force applied at the front left axle power

Computed

W
`PwrFwFRx`Longitudinal force applied at the front right axle power

Computed

W
`PwrFwFRy`Lateral force applied at the front right axle power

Computed

W
`PwrFwMLx`Longitudinal force applied at the middle left axle power

Computed

W
`PwrFwMLy`Lateral force applied at the middle left axle power

Computed

W
`PwrFwMRx`Longitudinal force applied at the middle right axle power

Computed

W
`PwrFwMRy`Lateral force applied at the middle right axle power

Computed

W
`PwrFwRLx`Longitudinal force applied at the rear left axle power

Computed

W
`PwrFwRLy`Lateral force applied at the rear left axle power

Computed

W
`PwrFwRRx`Longitudinal force applied at the rear right axle power

Computed

W
`PwrFwRRy`Lateral force applied at the rear right axle power

Computed

W
`PwrNotTrnsfrd``PwrFxDrag`Longitudinal drag force power

Computed

W
`PwrFyDrag`Lateral drag force power

Computed

W
`PwrMzDrag`Drag pitch moment power

Computed

W
`PwrStored``PwrStoredGrvty`Rate change in gravitational potential energy

Computed

W
`PwrStoredxdot`Rate of change of longitudinal kinetic energy

Computed

W
`PwrStoredydot`Rate of change of lateral kinetic energy

Computed

W
`PwrStoredr`Rate 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 the 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

`$FzF=F{z}_{f}$`

Scalar – `1`

`Dual`

Normal force on the right and left front wheels

`$FzF=\left[\begin{array}{cc}F{z}_{fl}& F{z}_{fr}\end{array}\right]$`

Array – `[1x2]`

Normal force on the middle wheels, FzM, along the vehicle-fixed z-axis, in N.

Vehicle Track Setting

Description

Variable

Signal Dimension

`Single (bicycle)`

Normal force on middle axle

`$FzM=F{z}_{m}$`

Scalar – `1`

`Dual`

Normal force on the right and left middle wheels

`$FzM=\left[\begin{array}{cc}F{z}_{ml}& F{z}_{rl}\end{array}\right]$`

Array – `[1x2]`

Normal force on the 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

`$FzR=F{z}_{r}$`

Scalar – `1`

`Dual`

Normal force on the right and left rear wheels

`$FzR=\left[\begin{array}{cc}F{z}_{rl}& F{z}_{rr}\end{array}\right]$`

Array – `[1x2]`

## Parameters

expand all

Options

Use the Vehicle track parameter to specify the number of wheels.

Vehicle Track SettingImplementation

`Single (bicycle)`

• Forces act along the center line of the axles.

`Dual`

Forces act at the axle hard-point locations.

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

Select to create input port `WhlAngF`.

Select to create input port `WhlAngM`.

Select to create input port `WhlAngR`.

Select to create input port `WindXYZ`.

Select to create input port `Mu`.

#### Dependencies

To enable this parameter, set Axle forces to `External longitudinal forces` or `External forces`.

Select to create input port `FExt`.

Select to create input port `MExt`.

Select to create input port `Fh`.

Specify to create input port `Mh`.

Specify to create input port `X_o`.

Specify to create input port `psi_o`.

Specify to create input port `xdot_o`.

#### Dependencies

To enable this parameter, set Axle forces to `External longitudinal forces` or `External forces`.

Specify to create input port `r_o`.

Specify to create input port `Y_o`.

Specify to create input port `AirTemp`.

Specify to create input port `ydot_o`.

Longitudinal

Number of wheels on the front axle, NF, dimensionless.

Number of wheels on the middle axle, NM, dimensionless.

Number of wheels on the rear axle, NR, dimensionless.

Vehicle mass, m, in kg.

Distance from vehicle CM to front axle, a, in m.

Distance from vehicle CM to middle axle, b, in m.

Distance from vehicle CM to rear axle, c, in m.

Vertical distance from vehicle CM to axle plane, h, in m.

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.

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.

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

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

#### Dependencies

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

• ```External longitudinal forces```

• `External forces`

Lateral

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 the geometric centerline to the center of mass, d, in m, along the vehicle-fixed y-axis. 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.

Front, middle, and rear track widths, wf, wm, and, wr, respectively, in m. Dimensions are `1-by-3`.

#### Dependencies

To enable this parameter, set Vehicle track to `Dual`.

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

#### 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.

Middle axle tire corner stiffness, Cym, in N/rad.

#### 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.

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

#### 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.

Front tire relaxation length, σf, in m.

#### Dependencies

To enable this parameter:

1. Set Vehicle track to one of these options:

• `Single 2-axle`

• `Dual 2-axle`

• `Single 3-axle`

• `Dual 3-axle`

2. Set Axle forces to one of these options:

• `External longitudinal velocity`

• `External longitudinal forces`

3. Do either of these:

• Select Mapped corner stiffness.

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

Middle tire relaxation length, σm, in m.

#### Dependencies

To enable this parameter:

1. Set Vehicle track to one of these options:

• `Single 2-axle`

• `Dual 2-axle`

• `Single 3-axle`

• `Dual 3-axle`

2. Set Axle forces to one of these options:

• `External longitudinal velocity`

• `External longitudinal forces`

3. 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 Vehicle track to one of these options:

• `Single 2-axle`

• `Dual 2-axle`

• `Single 3-axle`

• `Dual 3-axle`

2. Set Axle forces to one of these options:

• `External longitudinal velocity`

• `External longitudinal forces`

3. 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 tire 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.

Middle axle slip angle breakpoints, αmbrk, 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.

Middle axle tire corner data, Cymdata, 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 tire 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.

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

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

Yaw

Yaw polar inertia, in kg*m^2.

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, 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

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

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

• `External longitudinal velocity`

• `External longitudinal forces`

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

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

Vehicle chassis offset from the axle plane along the vehicle-fixed z-axis, in m. When you use the 3D visualization engine, consider using the offset to locate the chassis independently 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 this interval, consider clearing 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.

## Version History

Introduced in R2020a