6DOF rigid vehicle body to calculate translational and rotational motion
Vehicle Dynamics Blockset / Vehicle Body
The Vehicle Body 6DOF block implements a six degreesoffreedom (DOF) rigid twoaxle 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.
To analyze the vehicle dynamics under different loading conditions, use the Inertial Loads parameters. Specifically, you can specify these loads:
Front powertrain
Front and rear row passengers
Overhead cargo
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 vehiclefixed xaxis
Vehicle centerline to load, along the vehiclefixed yaxis
Front suspension hardpoint to load, along the vehiclefixed zaxis
Load  Parameter  Example Location 

Front  Distance vector from front axle, z1R 

Overhead  Distance vector from front axle, z2R 

Row 1, left side  Distance vector from front axle, z3R 

Row 1, right side  Distance vector from front axle, z4R 

Row 2, left side  Distance vector from front axle, z5R 

Row 2, right side  Distance vector from front axle, z6R 

Rear  Distance vector from front axle, z7R 

To determine the vehicle motion, the block implements calculations for the rigid body vehicle dynamics, wind drag, inertial loads, and coordinate transformations. The bodyfixed and the vehiclefixed are the same coordinate systems.
The Vehicle Body 6DOF block considers the rotation of a bodyfixed coordinate frame about a flat earthfixed inertial reference frame. The origin of the bodyfixed coordinate frame is the vehicle center of gravity of the body.
The block uses this equation to calculate the translational motion of the bodyfixed coordinate frame, where the applied forces [F_{x} F_{y} F_{z}]^{T} are in the bodyfixed frame, and the mass of the body, m, is assumed constant.
$$\begin{array}{l}{\overline{F}}_{b}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m\left({\dot{\overline{V}}}_{b}+\overline{\omega}\times {\overline{V}}_{b}\right)\\ \\ {\overline{M}}_{b}=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=I\dot{\overline{\omega}}+\overline{\omega}\times (I\overline{\omega})\\ \\ I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]\end{array}$$
To determine the relationship between the bodyfixed angular velocity vector, [p q r]^{T}, and the rate of change of the Euler angles, $$[\begin{array}{ccc}\dot{\varphi}\text{\hspace{0.05em}}\text{\hspace{0.17em}}& \dot{\theta}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}& \dot{\psi}\end{array}{]}^{T}$$, the block resolves the Euler rates into the bodyfixed frame.
$$\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{c}\dot{\varphi}\\ 0\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& \mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right]\left[\begin{array}{c}0\\ \dot{\theta}\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& \mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right]\left[\begin{array}{ccc}\mathrm{cos}\theta & 0& \mathrm{sin}\theta \\ 0& 1& 0\\ \mathrm{sin}\theta & 0& \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{c}0\\ 0\\ \dot{\psi}\end{array}\right]\equiv {J}^{1}\left[\begin{array}{c}\dot{\varphi}\\ \dot{\theta}\\ \dot{\psi}\end{array}\right]$$
Inverting J gives the required relationship to determine the Euler rate vector.
$$\left[\begin{array}{c}\dot{\varphi}\\ \dot{\theta}\\ \dot{\psi}\end{array}\right]=J\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\text{\hspace{0.17em}}=\left[\begin{array}{ccc}1& (\mathrm{sin}\varphi \mathrm{tan}\theta )& (\mathrm{cos}\varphi \mathrm{tan}\theta )\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& \frac{\mathrm{sin}\varphi}{\mathrm{cos}\theta}& \frac{\mathrm{cos}\varphi}{\mathrm{cos}\theta}\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]$$
The applied forces and moments are the sum of the drag, gravitational, external, and suspension forces.
$$\begin{array}{l}{\overline{F}}_{b}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=\left[\begin{array}{c}{F}_{d}{}_{{}_{x}}\\ {F}_{d}{}_{{}_{y}}\\ {F}_{d}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{g}{}_{{}_{x}}\\ {F}_{g}{}_{{}_{y}}\\ {F}_{g}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{ext}{}_{{}_{x}}\\ {F}_{ext}{}_{{}_{y}}\\ {F}_{ext}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{FL}{}_{{}_{x}}\\ {F}_{FL}{}_{{}_{y}}\\ {F}_{FL}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{FR}{}_{{}_{x}}\\ {F}_{FR}{}_{{}_{y}}\\ {F}_{FR}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{RL}{}_{{}_{x}}\\ {F}_{RL}{}_{{}_{y}}\\ {F}_{RL}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{RR}{}_{{}_{x}}\\ {F}_{RR}{}_{{}_{y}}\\ {F}_{RR}{}_{{}_{z}}\end{array}\right]\\ \\ {\overline{M}}_{b}=\left[\begin{array}{c}{M}_{x}\\ {M}_{y}\\ {M}_{z}\end{array}\right]=\left[\begin{array}{c}{M}_{d}{}_{{}_{x}}\\ {M}_{d}{}_{{}_{y}}\\ {M}_{d}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{ext}{}_{{}_{x}}\\ {M}_{ext}{}_{{}_{y}}\\ {M}_{ext}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{FL}{}_{{}_{x}}\\ {M}_{FL}{}_{{}_{y}}\\ {M}_{FL}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{FR}{}_{{}_{x}}\\ {M}_{FR}{}_{{}_{y}}\\ {M}_{FR}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{RL}{}_{{}_{x}}\\ {M}_{RL}{}_{{}_{y}}\\ {M}_{RL}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{RR}{}_{{}_{x}}\\ {M}_{RR}{}_{{}_{y}}\\ {M}_{RR}{}_{{}_{z}}\end{array}\right]+{\overline{M}}_{F}\end{array}$$
Calculation  Implementation 

Load masses and inertias  Block uses parallel axis theorem to resolve the individual load masses and inertias with the vehicle mass and inertia. $${J}_{ij}={I}_{ij}+m({\leftR\right}^{2}{\delta}_{ij}{R}_{i}{R}_{j})$$ 
Gravitational forces, F_{g}  Block uses direction cosine matrix (DCM) to transform the gravitational vector in the inertialfixed frame to the bodyfixed frame. 
Drag forces, F_{d}, and moments, M_{d}  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. $$\begin{array}{l}\overline{w}=\sqrt{{({\dot{x}}_{b}{w}_{x})}^{2}+{({\dot{x}}_{y}{w}_{x})}^{2}+{({w}_{z})}^{2}}\\ \\ {F}_{dx}=\frac{1}{2TR}{C}_{d}{A}_{f}{P}_{abs}{(}^{\overline{w}}\\ {F}_{dy}=\frac{1}{2TR}{C}_{s}{A}_{f}{P}_{abs}{(}^{\overline{w}}\\ {F}_{dz}=\frac{1}{2TR}{C}_{l}{A}_{f}{P}_{abs}{(}^{\overline{w}}\end{array}$$ Using the relative airspeed, the block determines the drag moments. $$\begin{array}{l}{M}_{dr}=\frac{1}{2TR}{C}_{rm}{A}_{f}{P}_{abs}{(}^{\overline{w}}(a+b)\\ {M}_{dp}=\frac{1}{2TR}{C}_{pm}{A}_{f}{P}_{abs}{(}^{\overline{w}}(a+b)\\ {M}_{dy}=\frac{1}{2TR}{C}_{ym}{A}_{f}{P}_{abs}{(}^{\overline{w}}(a+b)\end{array}$$ 
External forces, F_{in}, and moments, M_{in}  External forces and moments are input via ports

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

The equations use these variables.
$$x,\dot{x},\ddot{x}$$  Vehicle CM displacement, velocity, and acceleration along the vehiclefixed xaxis 
$$y,\dot{y},\ddot{y}$$  Vehicle CM displacement, velocity, and acceleration along the vehiclefixed yaxis 
$$z,\dot{z},\ddot{z}$$  Vehicle CM displacement, velocity, and acceleration along the vehiclefixed zaxis 
φ  Rotation of the vehiclefixed frame about the earthfixed Xaxis (roll) 
θ  Rotation of the vehiclefixed frame about the earthfixed Yaxis (pitch) 
ψ  Rotation of the vehiclefixed frame about the earthfixed Zaxis (yaw) 
F_{FLx}, F_{FLy}, F_{FLz}  Suspension forces applied to front left hardpoint along the vehiclefixed x, y, and zaxes 
F_{FRx}, F_{FRy}, F_{FRz}  Suspension forces applied to front right hardpoint along the vehiclefixed x, y, and zaxes 
F_{RLx}, F_{RLy}, F_{RLz}  Suspension forces applied to rear left hardpoint along the vehiclefixed x, y, and zaxes 
F_{RRx}, F_{RRy}, F_{RRz}  Suspension forces applied to rear right hardpoint along the vehiclefixed x, y, and zaxes 
M_{Fx}, F_{Fy}, F_{Fz}  Suspension moments applied to vehicle CM about the vehiclefixed x, y, and zaxes 
F_{extx}, F_{exty}, F_{extz}  External forces applied to vehicle CM along the vehiclefixed x, y, and zaxes 
F_{dx}, F_{dy}, F_{dz}  Drag forces applied to vehicle CM along the vehiclefixed x, y, and zaxes 
M_{extx}, M_{exty}, M_{extz}  External moment about vehicle CM about the vehiclefixed x, y, and zaxes 
M_{dx}, M_{dy}, M_{dz}  Drag moment about vehicle CM about the vehiclefixed x, y, and zaxes 
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 
w_{F}, w_{R}  Front and rear track widths 
γ  Road grade angle 
C_{d}  Air drag coefficient acting along vehiclefixed xaxis 
C_{s}  Air drag coefficient acting along vehiclefixed yaxis 
C_{l}  Air drag coefficient acting along vehiclefixed zaxis 
C_{rm}  Air drag roll moment acting about vehiclefixed xaxis 
C_{pm}  Air drag pitch moment acting about the vehiclefixed yaxis 
C_{ym}  Air drag yaw moment acting about vehiclefixed zaxis 
A_{f}  Frontal area 
R  Atmospheric specific gas constant 
T  Environmental air temperature 
P_{abs}  Environmental absolute pressure 
w_{x}, w_{y}, w_{z}  Wind speed along the vehiclefixed x, y, and zaxes 
W_{x}, W_{y}, W_{z}  Wind speed along inertial X, Y, and Zaxes 
[1] Gillespie, Thomas. Fundamentals of Vehicle Dynamics. Warrendale, PA: Society of Automotive Engineers (SAE), 1992.