Vehicle Body 6DOF Three Axles

Three-axle vehicle tractor body with translational and rotational motion

Since R2020b

Libraries:
Vehicle Dynamics Blockset / Vehicle Body

Description

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

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

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

Parameter	Input Port	Description
Front hitch forces	`FhF`	Hitch force applied to the body at the front hitch location, FhF_x, FhF_y, and FhF_z, in the vehicle-fixed frame
Front hitch moments	`MhF`	Hitch moment at the front hitch location, MhF_x, MhF_y, and MhF_z, about the vehicle-fixed frame
Rear hitch forces	`FhR`	Hitch force applied to the body at the rear hitch location, FhR_x, FhR_y, and FhR_z, in the vehicle-fixed frame
Rear hitch moments	`MhR`	Hitch moment at the rear hitch location, MhR_x, MhR_y, and MhR_z, about the vehicle-fixed frame

Inertial Loads

To analyze the vehicle dynamics under different loading conditions, use the Inertial Loads parameters. You can specify these loads:

Tractor front
Cab overhead
Tractor frame left and frame right
Cab left and cab right
Tractor rear

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

The illustrations provide the load locations and vehicle parameter dimensions. The table provides the corresponding location parameter sign settings.

Top down and side views of vehicle showing load locations

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 axle to load, along the vehicle-fixed x-axis
Vehicle centerline to load, along the vehicle-fixed y-axis
Front axle to load, along the vehicle-fixed z-axis

Load	Parameter	Example Location
Tractor front	Distance vector from front axle, z1R	`z1R(1,1)<0` — Forward of the front axle `z1R(1,2)>0` — Right of the vehicle centerline `z1R(1,3)>0` — Above the front axle suspension hardpoint
Cab overhead	Distance vector from front axle, z2R	`z2R(1,1)>0` — Rear of the front axle `z2R(1,2)<0` — Left of the vehicle centerline `z2R(1,3)>0` — Above the front axle suspension hardpoint
Tractor frame left	Distance vector from front axle, z3R	`z3R(1,1)>0` — Rear of the front axle `z3R(1,2)<0` — Left of the vehicle centerline `z3R(1,3)>0` — Above the front axle suspension hardpoint
Tractor frame right	Distance vector from front axle, z4R	`z4R(1,1)>0` — Rear of the front axle `z4R(1,2)>0` — Right of the vehicle centerline `z4R(1,3)>0` — Above the front axle suspension hardpoint
Cab left	Distance vector from front axle, z5R	`z5R(1,1)>0` — Rear of the front axle `z5R(1,2)<0` — Left of the vehicle centerline `z5R(1,3)>0` — Above the front axle suspension hardpoint
Cab right	Distance vector from front axle, z6R	`z6R(1,1)>0` — Rear of the front axle `z6R(1,2)>0` — Right of the vehicle centerline `z6R(1,3)>0` — Above the front axle suspension hardpoint
Tractor rear	Distance vector from front axle, z7R	`z7R(1,1)>0` — Rear of the front axle `z7R(1,2)>0` — Right of the vehicle centerline `z7R(1,3)>0` — Above the front axle suspension hardpoint

Equations of Motion

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

The block considers the rotation of a body-fixed coordinate frame about a flat earth-fixed inertial reference frame. The origin of the body-fixed coordinate frame is the vehicle center of gravity of the body.

The block uses this equation to calculate the translational motion of the body-fixed coordinate frame, where the applied forces [F_x F_y F_z]^T are in the body-fixed frame, and the mass of the body, m, is assumed to be constant.

$\begin{array}{l} {\bar{F}}_{b} = [\begin{matrix} F_{x} \\ F_{y} \\ F_{z} \end{matrix}] = m ({\dot{\bar{V}}}_{b} + \bar{ω} \times {\bar{V}}_{b}) \\ {\bar{M}}_{b} = [\begin{matrix} L \\ M \\ N \end{matrix}] = I \dot{\bar{ω}} + \bar{ω} \times (I \bar{ω}) \\ I = [\begin{matrix} I_{x x} & - I_{x y} & - I_{x z} \\ - I_{y x} & I_{y y} & - I_{y z} \\ - I_{z x} & - I_{z y} & I_{z z} \end{matrix}] \end{array}$

To determine the relationship between the body-fixed angular velocity vector, [p q r]^T, and the rate of change of the Euler angles, $[\begin{matrix} \dot{ϕ} & \dot{θ} & \dot{ψ} \end{matrix}]^{T}$ , the block resolves the Euler rates into the body-fixed frame.

$[\begin{matrix} p \\ q \\ r \end{matrix}] = [\begin{matrix} \dot{ϕ} \\ 0 \\ 0 \end{matrix}] + [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos ϕ & \sin ϕ \\ 0 & - \sin ϕ & \cos ϕ \end{matrix}] [\begin{matrix} 0 \\ \dot{θ} \\ 0 \end{matrix}] + [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos ϕ & \sin ϕ \\ 0 & - \sin ϕ & \cos ϕ \end{matrix}] [\begin{matrix} \cos θ & 0 & - \sin θ \\ 0 & 1 & 0 \\ \sin θ & 0 & \cos θ \end{matrix}] [\begin{matrix} 0 \\ 0 \\ \dot{ψ} \end{matrix}] \equiv J^{- 1} [\begin{matrix} \dot{ϕ} \\ \dot{θ} \\ \dot{ψ} \end{matrix}]$

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

$[\begin{matrix} \dot{ϕ} \\ \dot{θ} \\ \dot{ψ} \end{matrix}] = J [\begin{matrix} p \\ q \\ r \end{matrix}] = [\begin{matrix} 1 & (\sin ϕ \tan θ) & (\cos ϕ \tan θ) \\ 0 & \cos ϕ & - \sin ϕ \\ 0 & \frac{\sin ϕ}{\cos θ} & \frac{\cos ϕ}{\cos θ} \end{matrix}] [\begin{matrix} p \\ q \\ r \end{matrix}]$

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

$\begin{array}{l} {\bar{F}}_{b} = [\begin{matrix} F_{x} \\ F_{y} \\ F_{z} \end{matrix}] = [\begin{matrix} F_{d}_{_{x}} \\ F_{d}_{_{y}} \\ F_{d}_{_{z}} \end{matrix}] + [\begin{matrix} F_{g}_{_{x}} \\ F_{g}_{_{y}} \\ F_{g}_{_{z}} \end{matrix}] + [\begin{matrix} F_{e x t}_{_{x}} \\ F_{e x t}_{_{y}} \\ F_{e x t}_{_{z}} \end{matrix}] + [\begin{matrix} F_{F L}_{_{x}} \\ F_{F L}_{_{y}} \\ F_{F L}_{_{z}} \end{matrix}] + [\begin{matrix} F_{F R}_{_{x}} \\ F_{F R}_{_{y}} \\ F_{F R}_{_{z}} \end{matrix}] + [\begin{matrix} F_{M L}_{_{x}} \\ F_{M L}_{_{y}} \\ F_{M L}_{_{z}} \end{matrix}] + [\begin{matrix} F_{M R}_{_{x}} \\ F_{M R}_{_{y}} \\ F_{M R}_{_{z}} \end{matrix}] + [\begin{matrix} F_{R L}_{_{x}} \\ F_{R L}_{_{y}} \\ F_{R L}_{_{z}} \end{matrix}] + [\begin{matrix} F_{R R}_{_{x}} \\ F_{R R}_{_{y}} \\ F_{R R}_{_{z}} \end{matrix}] \\ {\bar{M}}_{b} = [\begin{matrix} M_{x} \\ M_{y} \\ M_{z} \end{matrix}] = [\begin{matrix} M_{d}_{_{x}} \\ M_{d}_{_{y}} \\ M_{d}_{_{z}} \end{matrix}] + [\begin{matrix} M_{e x t}_{_{x}} \\ M_{e x t}_{_{y}} \\ M_{e x t}_{_{z}} \end{matrix}] + [\begin{matrix} M_{F L}_{_{x}} \\ M_{F L}_{_{y}} \\ M_{F L}_{_{z}} \end{matrix}] + [\begin{matrix} M_{F R}_{_{x}} \\ M_{F R}_{_{y}} \\ M_{F R}_{_{z}} \end{matrix}] + [\begin{matrix} M_{M L}_{_{x}} \\ M_{M L}_{_{y}} \\ M_{M L}_{_{z}} \end{matrix}] + [\begin{matrix} M_{M R}_{_{x}} \\ M_{M R}_{_{y}} \\ M_{M R}_{_{z}} \end{matrix}] + [\begin{matrix} M_{R L}_{_{x}} \\ M_{R L}_{_{y}} \\ M_{R L}_{_{z}} \end{matrix}] + [\begin{matrix} M_{R R}_{_{x}} \\ M_{R R}_{_{y}} \\ M_{R R}_{_{z}} \end{matrix}] + {\bar{M}}_{F} \end{array}$

Calculation	Implementation
Load masses and inertias	The block uses the parallel axis theorem to resolve the individual load masses and inertias with the vehicle mass and inertia. $J_{i j} = I_{i j} + m ({\| R \|}^{2} δ_{i j} - R_{i} R_{j})$
Gravitational forces, F_g	The block uses the direction cosine matrix (DCM) to transform the gravitational vector in the inertial-fixed frame to the body-fixed 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. $\begin{array}{l} \bar{w} = \sqrt{{(\dot{x} - w_{x})}^{2} + {(\dot{y} - w_{y})}^{2} + {(w_{z})}^{2}} \\ F_{d x} = - \frac{1}{2 T R} C_{d} A_{f} P_{a b s} (^{\bar{w}) 2} \\ F_{d y} = - \frac{1}{2 T R} C_{s} A_{f} P_{a b s} (^{\bar{w}) 2} \\ F_{d z} = - \frac{1}{2 T R} C_{l} A_{f} P_{a b s} (^{\bar{w}) 2} \end{array}$ Using the relative airspeed, the block determines the drag moments. $\begin{array}{l} M_{d r} = - \frac{1}{2 T R} C_{r m} A_{f} P_{a b s} (^{\bar{w}) 2} (a + c) \\ M_{d p} = - \frac{1}{2 T R} C_{p m} A_{f} P_{a b s} (^{\bar{w}) 2} (a + c) \\ M_{d y} = - \frac{1}{2 T R} C_{y m} A_{f} P_{a b s} (^{\bar{w}) 2} (a + c) \end{array}$
External forces, F_in, and moments, M_in	The external forces and moments are input via ports FExt and MExt.
Suspension forces and moments	The block assumes that the suspension forces and moments act on these hardpoint locations: F_FL, M_FL — Front left F_FR, M_FR — Front right F_ML, M_ML — Middle left F_MR, M_MR — Middle right F_RL, M_RL — Rear left F_RR, M_RR — Rear right

The equations use these variables.

$x, \dot{x}, \ddot{x}$	Vehicle CM displacement, velocity, and acceleration along the vehicle-fixed x-axis
$y, \dot{y}, \ddot{y}$	Vehicle CM displacement, velocity, and acceleration along the vehicle-fixed y-axis
$z, \dot{z}, \ddot{z}$	Vehicle CM displacement, velocity, and acceleration along the vehicle-fixed z-axis
φ	Rotation of the vehicle-fixed frame about the earth-fixed X-axis (roll)
θ	Rotation of the vehicle-fixed frame about the earth-fixed Y-axis (pitch)
ψ	Rotation of the vehicle-fixed frame about the earth-fixed Z-axis (yaw)
F_FLx, F_FLy, F_FLz	Suspension forces applied to the front left hardpoint along the vehicle-fixed x-, y-, and z-axes
F_FRx, F_FRy, F_FRz	Suspension forces applied to the front right hardpoint along the vehicle-fixed x-, y-, and z-axes
F_MLx, F_MLy, F_MLz	Suspension forces applied to the middle left hardpoint along the vehicle-fixed x-, y-, and z-axes
F_MRx, F_MRy, F_MRz	Suspension forces applied to the middle right hardpoint along the vehicle-fixed x-, y-, and z-axes
F_RLx, F_RLy, F_RLz	Suspension forces applied to the rear left hardpoint along the vehicle-fixed x-, y-, and z-axes
F_RRx, F_RRy, F_RRz	Suspension forces applied to the rear right hardpoint along the vehicle-fixed x-, y-, and z-axes
M_FLx, M_FLy, M_FLz	Suspension moment applied to the front left hardpoint about the vehicle-fixed x-, y-, and z-axes
M_FRx, M_FRy, M_FRz	Suspension moment applied to the front right hardpoint about the vehicle-fixed x-, y-, and z-axes
M_MLx, M_MLy, M_MLz	Suspension moment applied to the middle left hardpoint about the vehicle-fixed x-, y-, and z-axes
M_MRx, M_MRy, M_MRz	Suspension moment applied to the middle right hardpoint about the vehicle-fixed x-, y-, and z-axes
M_RLx, M_RLy, M_RLz	Suspension moment applied to the rear left hardpoint about the vehicle-fixed x-, y-, and z-axes
M_RRx, M_RRy, M_RRz	Suspension moment applied to the rear right hardpoint about the vehicle-fixed x-, y-, and z-axes
F_extx, F_exty, F_extz	External forces applied to the vehicle CM along the vehicle-fixed x-, y-, and z-axes
F_dx, F_dy, F_dz	Drag forces applied to the vehicle CM along the vehicle-fixed x-, y-, and z-axes
M_extx, M_exty, M_extz	External moment about the vehicle CM about the vehicle-fixed x-, y-, and z-axes
M_dx, M_dy, M_dz	Drag moment about the vehicle CM about the vehicle-fixed x-, y-, and z-axes
I	Vehicle body moments of inertia
a, b, c	Distance of the front, middle, and rear axles, respectively, from the normal projection point of the vehicle CM onto the common axle plane
h	Height of the vehicle CM above the axle plane
d	Lateral distance from the geometric centerline to the center of mass along the vehicle-fixed y-axis
hh_f, hh_r	Height of the front and rear hitches, respectively, above the axle plane along the vehicle-fixed z-axis
dh_f, dh_r	Longitudinal distance of the front and rear hitches, respectively, from the normal projection point of the vehicle CM onto the common axle plane
hl_f, hl_r	Lateral distance from center of mass to the front and rear hitches, respectively, along the vehicle-fixed y-axis
w_F, w_M, w_R	Front, middle, and rear track widths, respectively
C_d	Air drag coefficient acting along the vehicle-fixed x-axis
C_s	Air drag coefficient acting along the vehicle-fixed y-axis
C_l	Air drag coefficient acting along the vehicle-fixed z-axis
C_rm	Air drag roll moment acting about the vehicle-fixed x-axis
C_pm	Air drag pitch moment acting about the vehicle-fixed y-axis
C_ym	Air drag yaw moment acting about the vehicle-fixed z-axis
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 vehicle-fixed x-, y-, and z-axes
W_x, W_y, W_z	Wind speed along inertial X-, Y-, and Z-axes

Examples

Three-Axle Tractor Towing a Three-Axle Trailer

Simulates three-axle tractor towing a three-axle trailer for commercial trucking applications. Implements hitch subsystem, sinusoidal wave steering or braking test, and axle torque applied to tractor rear wheels.

Open Model

Ports

Input

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FSusp — Suspension forces on vehicle
`3-by-6` array

Suspension longitudinal, lateral, and vertical suspension forces applied to the vehicle at the hardpoint location, in N, specified as a 3-by-6 array.

$F S u s p = [\begin{matrix} F_{F L x} & F_{F R x} & F_{M L x} & F_{M R x} & F_{R L x} & F_{R R x} \\ F_{F L y} & F_{F R y} & F_{M L y} & F_{M R y} & F_{R L y} & F_{R R y} \\ F_{F L z} & F_{F R z} & F_{M L z} & F_{M R z} & F_{R L z} & F_{R R z} \end{matrix}]$

Array Element	Axle	Track	Force Axis
`FSusp(1,1)`	Front	Left	Vehicle-fixed x-axis (longitudinal)
`FSusp(1,2)`	Front	Right
`FSusp(1,3)`	Middle	Left
`FSusp(1,4)`	Middle	Right
`FSusp(1,5)`	Rear	Left
`FSusp(1,6)`	Rear	Right
`FSusp(2,1)`	Front	Left	Vehicle-fixed y-axis (lateral)
`FSusp(2,2)`	Front	Right
`FSusp(2,3)`	Middle	Left
`FSusp(2,4)`	Middle	Right
`FSusp(2,5)`	Rear	Left
`FSusp(2,6)`	Rear	Right
`FSusp(3,1)`	Front	Left	Vehicle-fixed z-axis (vertical)
`FSusp(3,2)`	Front	Right
`FSusp(3,3)`	Middle	Left
`FSusp(3,4)`	Middle	Right
`FSusp(3,5)`	Rear	Left
`FSusp(3,6)`	Rear	Right

MSusp — Suspension moment on vehicle
`3-by-6` array

Suspension longitudinal, lateral, and vertical suspension moments applied about the vehicle at the hardpoint location, in N, specified as a 3-by-6 array.

$M S u s p = [\begin{matrix} M_{F L x} & M_{F R x} & M_{M L x} & M_{M R x} & M_{R L x} & M_{R R x} \\ M_{F L y} & M_{F R z} & M_{M L y} & M_{M R y} & M_{R L y} & M_{R R y} \\ M_{F L z} & M_{F R z} & M_{M L z} & M_{M R z} & M_{R L z} & M_{R R z} \end{matrix}]$

Array Element	Axle	Track	Moment Axis
`MSusp(1,1)`	Front	Left	Vehicle-fixed x-axis (longitudinal)
`MSusp(1,2)`	Front	Right
`MSusp(1,3)`	Middle	Left
`MSusp(1,4)`	Middle	Right
`MSusp(1,5)`	Rear	Left
`MSusp(1,6)`	Rear	Right
`MSusp(2,1)`	Front	Left	Vehicle-fixed y-axis (lateral)
`MSusp(2,2)`	Front	Right
`MSusp(2,3)`	Middle	Left
`MSusp(2,4)`	Middle	Right
`MSusp(2,5)`	Rear	Left
`MSusp(2,6)`	Rear	Right
`MSusp(3,1)`	Front	Left	Vehicle-fixed z-axis (vertical)
`MSusp(3,2)`	Front	Right
`MSusp(3,3)`	Middle	Left
`MSusp(3,4)`	Middle	Right
`MSusp(3,5)`	Rear	Left
`MSusp(3,6)`	Rear	Right

FExt — External forces acting on vehicle
`vector`

External forces on the vehicle, in N, specified as a 1-by-3 or 3-by-1 vector.

$FExt = F_{e x t} = [\begin{matrix} F_{e x t_{x}} & F_{e x t_{y}} & F_{e x t_{z}} \end{matrix}] o r [\begin{matrix} F_{e x t_{x}} \\ F_{e x t_{y}} \\ F_{e x t_{z}} \end{matrix}]$

Array Element	Force Axis
`FExt(1,1)`	Vehicle-fixed x-axis (longitudinal)
`FExt(1,2)` or `FExt(2,1)`	Vehicle-fixed y-axis (lateral)
`FExt(1,3)` or `FExt(3,1)`	Vehicle-fixed z-axis (vertical)

MExt — External moments acting on vehicle
`vector`

External moments acting on the vehicle, in N·m, specified as a 1-by-3 or 3-by-1 vector.

$MExt = M_{e x t} = [\begin{matrix} M_{e x t_{x}} & M_{e x t_{y}} & M_{e x t_{z}} \end{matrix}] o r [\begin{matrix} M_{e x t_{x}} \\ M_{e x t_{y}} \\ M_{e x t_{z}} \end{matrix}]$

Array Element	Force Axis
`MExt(1,1)`	Vehicle-fixed x-axis (longitudinal)
`MExt(1,2)` or `MExt(2,1)`	Vehicle-fixed y-axis (lateral)
`MExt(1,3)` or `MExt(3,1)`	Vehicle-fixed z-axis (vertical)

Fh — Hitch force on the body
`array`

Hitch force applied to the body at the hitch location, Fh_x, Fh_y, Fh_z, 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.

Mh — Hitch moment about body
`array`

Hitch moment at the hitch location, Mh_x, Mh_y, Mh_z, 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.

WindXYZ — Wind speed
`array`

Wind speed, W_x, W_y, W_z along inertial X-, Y-, and Z-axes, in m/s, specified as a 1-by-3 or 3-by-1 array.

AirTemp — Ambient air temperature
`scalar`

Ambient air temperature, T_air, in K, specified as a scalar.

Dependencies

To enable this port, under Environment, select Air temperature.

Output

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Info — Vehicle body information
bus

Vehicle body information, returned as a bug signal containing the following values.

Signal						Description	Value	Units
`InertFrm`	`Cg`	`Disp`	`X`			Vehicle CM displacement along the earth-fixed X-axis	Computed	m
			`Y`			Vehicle CM displacement along the earth-fixed Y-axis	Computed	m
			`Z`			Vehicle CM displacement along the earth-fixed Z-axis	Computed	m
		`Vel`	`Xdot`			Vehicle CM velocity along the earth-fixed X-axis	Computed	m/s
			`Ydot`			Vehicle CM velocity along the earth-fixed Y-axis	Computed	m/s
			`Zdot`			Vehicle CM velocity along the earth-fixed Z-axis	Computed	m/s
		`Ang`	`phi`			Rotation of the vehicle-fixed frame about the earth-fixed X-axis (roll)	Computed	rad
			`theta`			Rotation of the vehicle-fixed frame about the earth-fixed Y-axis (pitch)	Computed	rad
			`psi`			Rotation of the vehicle-fixed frame about the earth-fixed Z-axis (yaw)	Computed	rad
	`FrntAxl`	`Lft`	`Disp`	`X`		Front left axle displacement along the earth-fixed X-axis	Computed	m
				`Y`		Front left axle displacement along the earth-fixed Y-axis	Computed	m
				`Z`		Front left axle displacement along the earth-fixed Z-axis	Computed	m
			`Vel`	`Xdot`		Front left axle velocity along the earth-fixed X-axis	Computed	m/s
				`Ydot`		Front left axle velocity along the earth-fixed Y-axis	Computed	m/s
				`Zdot`		Front left axle velocity along the earth-fixed Z-axis	Computed	m/s
		`Rght`	`Disp`	`X`		Front right axle displacement along the earth-fixed X-axis	Computed	m
				`Y`		Front right axle displacement along the earth-fixed Y-axis	Computed	m
				`Z`		Front right axle displacement along the earth-fixed Z-axis	Computed	m
			`Vel`	`Xdot`		Front right axle velocity along the earth-fixed X-axis	Computed	m/s
				`Ydot`		Front right axle velocity along the earth-fixed Y-axis	Computed	m/s
				`Zdot`		Front right axle velocity along the earth-fixed Z-axis	Computed	m/s
	`MidlAxl`	`Lft`	`Disp`	`X`		Middle left axle displacement along the earth-fixed X-axis	Computed	m
				`Y`		Middle left axle displacement along the earth-fixed Y-axis	Computed	m
				`Z`		Middle left axle displacement along the earth-fixed Z-axis	Computed	m
			`Vel`	`Xdot`		Middle left axle velocity along the earth-fixed X-axis	Computed	m/s
				`Ydot`		Middle left axle velocity along the earth-fixed Y-axis	Computed	m/s
				`Zdot`		Middle left axle velocity along the earth-fixed Z-axis	Computed	m/s
		`Rght`	`Disp`	`X`		Middle right axle displacement along the earth-fixed X-axis	Computed	m
				`Y`		Middle right axle displacement along the earth-fixed Y-axis	Computed	m
				`Z`		Middle right axle displacement along the earth-fixed Z-axis	Computed	m
			`Vel`	`Xdot`		Middle right axle velocity along the earth-fixed X-axis	Computed	m/s
				`Ydot`		Middle right axle velocity along the earth-fixed Y-axis	Computed	m/s
				`Zdot`		Middle right axle velocity along the earth-fixed Z-axis	Computed	m/s
	`RearAxl`	`Lft`	`Disp`	`X`		Rear left axle displacement along the earth-fixed X-axis	Computed	m
				`Y`		Rear left axle displacement along the earth-fixed Y-axis	Computed	m
				`Z`		Rear left axle displacement along the earth-fixed Z-axis	Computed	m
			`Vel`	`Xdot`		Rear left axle velocity along the earth-fixed X-axis	Computed	m/s
				`Ydot`		Rear left axle velocity along the earth-fixed Y-axis	Computed	m/s
				`Zdot`		Rear left axle velocity along the earth-fixed Z-axis	Computed	m/s
		`Rght`	`Disp`	`X`		Rear right axle displacement along the earth-fixed X-axis	Computed	m
				`Y`		Rear right axle displacement along the earth-fixed Y-axis	Computed	m
				`Z`		Rear right axle displacement along the earth-fixed Z-axis	Computed	m
			`Vel`	`Xdot`		Rear right axle velocity along the earth-fixed X-axis	Computed	m/s
				`Ydot`		Rear right axle velocity along the earth-fixed Y-axis	Computed	m/s
				`Zdot`		Rear right axle velocity along the earth-fixed Z-axis	Computed	m/s
	`Hitch`	`Disp`	`X`			Hitch offset from the axle plane along the earth-fixed X-axis	Computed	m
			`Y`			Hitch offset from the axle plane along the earth-fixed Y-axis	Computed	m
			`Z`			Hitch offset from the axle plane along the earth-fixed Z-axis	Computed	m
		`Vel`	`Xdot`			Hitch velocity along the earth-fixed X-axis	Computed	m/s
			`Ydot`			Hitch velocity along the earth-fixed Y-axis	Computed	m/s
			`Zdot`			Hitch velocity along the earth-fixed Z-axis	Computed	m/s
	`Geom`	`Disp`	`X`			Vehicle chassis offset from the 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 the axle plane along the earth-fixed Z-axis	Computed	m
		`Vel`	`Xdot`			Vehicle chassis offset velocity along the earth-fixed X-axis	Computed	m/s
			`Ydot`			Vehicle chassis offset velocity along the earth-fixed Y-axis	Computed	m/s
			`Zdot`			Vehicle chassis offset velocity along the earth-fixed Z-axis	Computed	m/s
`BdyFrm`	`Cg`	`Vel`	`xdot`			Vehicle CM velocity along the vehicle-fixed x-axis	Computed	m/s
			`ydot`			Vehicle CM velocity along the vehicle-fixed y-axis	Computed	m/s
			`zdot`			Vehicle CM velocity along the vehicle-fixed z-axis	Computed	m/s
		`AngVel`	`p`			Vehicle angular velocity about the vehicle-fixed x-axis (roll rate)	Computed	rad/s
			`q`			Vehicle angular velocity about the vehicle-fixed y-axis (pitch rate)	Computed	rad/s
			`r`			Vehicle angular velocity about the vehicle-fixed z-axis (yaw rate)	Computed	rad/s
		`Acc`	`ax`			Vehicle CM acceleration along the vehicle-fixed x-axis	Computed	gn
			`ay`			Vehicle CM acceleration along the vehicle-fixed y-axis	Computed	gn
			`az`			Vehicle CM acceleration along the vehicle-fixed z-axis	Computed	gn
			`xddot`			Vehicle CM acceleration along the vehicle-fixed x-axis	Computed	m/s^2
			`yddot`			Vehicle CM acceleration along the vehicle-fixed y-axis	Computed	m/s^2
			`zddot`			Vehicle CM acceleration along the vehicle-fixed z-axis	Computed	m/s^2
		`DCM`	Direction cosine matrix				Computed	rad
	`Forces`	`Body`	`Fx`			Net force on the vehicle CM along the vehicle-fixed x-axis	Computed	N
			`Fy`			Net force on the vehicle CM along the vehicle-fixed y-axis	Computed	N
			`Fz`			Net force on the vehicle CM along the vehicle-fixed z-axis	Computed	N
		`Ext`	`Fx`			External force on the vehicle CM along the vehicle-fixed x-axis	Input	N
			`Fy`			External force on the vehicle CM along the vehicle-fixed x-axis	Input	N
			`Fz`			External force on the vehicle CM along the vehicle-fixed x-axis	Input	N
		`FrntAxl`	`Lft`	`Fx`		Front left axle velocity along the earth-fixed Y-axis	Computed	N
				`Fy`		Lateral force on the left side of the front axle left along the vehicle-fixed y-axis	Computed	N
				`Fz`		Normal force on the left side of the front axle along the vehicle-fixed z-axis	Computed	N
			`Rght`	`Fx`		Longitudinal force on the right side of the front axle along the vehicle-fixed x-axis	Computed	N
				`Fy`		Lateral force on the right side of the front axle left along the vehicle-fixed y-axis	Computed	N
				`Fz`		Normal force on the right side of the front axle along the vehicle-fixed z-axis	Computed	N
		`MidlAxl`	`Lft`	`Fx`		Longitudinal force on the left side of the middle axle along the vehicle-fixed x-axis	Computed	N
				`Fy`		Longitudinal force on the left side of the middle axle along the vehicle-fixed x-axis	Computed	N
				`Fz`		Normal force on the left side of the middle axle along the vehicle-fixed z-axis	Computed	N
			`Rght`	`Fx`		Longitudinal force on the right side of the middle axle along the vehicle-fixed x-axis	Computed	N
				`Fy`		Lateral force on the right side of the middle axle left along the vehicle-fixed y-axis	Computed	N
				`Fz`		Normal force on the right side of the middle axle along the vehicle-fixed z-axis	Computed	N
		`RearAxl`	`Lft`	`Fx`		Longitudinal force on the left side of the rear axle along the vehicle-fixed x-axis	Computed	N
				`Fy`		Lateral force on the left side of the rear axle left along the vehicle-fixed y-axis	Computed	N
				`Fz`		Normal force on the left side of the rear axle along the vehicle-fixed z-axis	Computed	N
			`Rght`	`Fx`		Longitudinal force on the right side of the rear axle along the vehicle-fixed x-axis	Computed	N
				`Fy`		Lateral force on the right side of the rear axle left along the vehicle-fixed y-axis	Computed	N
				`Fz`		Normal force on the right side of the rear axle along the vehicle-fixed z-axis	Computed	N
		`Hitch`	`Fx`			Hitch force applied to the body at the hitch location along the vehicle-fixed x-axis	Computed	N
			`Fy`			Hitch force applied to the body at the hitch location along the vehicle-fixed y-axis	Computed	N
			`Fz`			Hitch force applied to the body at the hitch location along the vehicle-fixed z-axis	Computed	N
		`Tires`	`FrntTires`	`Lft`	`Fx`	Front left tire force along the vehicle-fixed x-axis	Computed	N
					`Fy`	Front left tire force along the vehicle-fixed y-axis	Computed	N
					`Fz`	Front left tire force along the vehicle-fixed z-axis	Computed	N
				`Rght`	`Fx`	Front right tire force along the vehicle-fixed x-axis	Computed	N
					`Fy`	Front right tire force along the vehicle-fixed y-axis	Computed	N
					`Fz`	Front right tire force along the vehicle-fixed z-axis	Computed	N
			`MidlTires`	`Lft`	`Fx`	Middle left tire force along the vehicle-fixed x-axis	Computed	N
					`Fy`	Middle left tire force along the vehicle-fixed y-axis	Computed	N
					`Fz`	Middle left tire force along the vehicle-fixed z-axis	Computed	N
				`Rght`	`Fx`	Middle right tire force along the vehicle-fixed x-axis	Computed	N
					`Fy`	Middle right tire force along the vehicle-fixed y-axis	Computed	N
					`Fz`	Middle right tire force along the vehicle-fixed z-axis	Computed	N
			`RearTires`	`Lft`	`Fx`	Rear left tire force along the vehicle-fixed x-axis	Computed	N
					`Fy`	Rear left tire force along the vehicle-fixed y-axis	Computed	N
					`Fz`	Rear left tire force along the vehicle-fixed z-axis	Computed	N
				`Rght`	`Fx`	Rear right tire force along the vehicle-fixed x-axis	Computed	N
					`Fy`	Rear right tire force along the vehicle-fixed y-axis	Computed	N
					`Fz`	Rear right tire force along the vehicle-fixed z-axis	Computed	N
		`Drag`	`Fx`			Drag force on the vehicle CM along the vehicle-fixed x-axis	Computed	N
			`Fy`			Drag force on the vehicle CM along the vehicle-fixed y-axis	Computed	N
			`Fz`			Drag force on the vehicle CM along the vehicle-fixed z-axis	Computed	N
		`Grvty`	`Fx`			Gravity force on the vehicle CM along the vehicle-fixed x-axis	Computed	N
			`Fy`			Gravity force on the vehicle CM along the vehicle-fixed y-axis	Computed	N
			`Fz`			Gravity force on the vehicle CM along the vehicle-fixed z-axis	Computed	N
`Moments`	`Body`	`Mx`			Body moment on the vehicle CM about the vehicle-fixed x-axis	Computed	N·m
		`My`			Body moment on the vehicle CM about the vehicle-fixed y-axis	Computed	N·m
		`Mz`			Body moment on the vehicle CM about the vehicle-fixed z-axis	Computed	N·m
	`Drag`	`Mx`			Drag moment on the vehicle CM about the vehicle-fixed x-axis	Computed	N·m
		`My`			Drag moment on the vehicle CM about the vehicle-fixed y-axis	Computed	N·m
		`Mz`			Drag moment on the vehicle CM about the vehicle-fixed z-axis	Computed	N·m
	`Ext`	`Mx`			External moment on the vehicle CG about the vehicle-fixed x-axis	Computed	N·m
		`My`			External moment on the vehicle CG about the vehicle-fixed y-axis	Computed	N·m
		`Mz`			External moment on the vehicle CG about the vehicle-fixed z-axis	Computed	N·m
	`Hitch`	`Mx`			Hitch moment at the hitch location about vehicle-fixed x-axis	Computed	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	Computed	N·m
`FrntAxl`	`Lft`	`Disp`	`x`		Front left axle displacement along the vehicle-fixed x-axis	Computed	m
			`y`		Front left axle displacement along the vehicle-fixed y-axis	Computed	m
			`z`		Front left axle displacement along the vehicle-fixed z-axis	Computed	m
		`Vel`	`xdot`		Front left axle velocity along the vehicle-fixed x-axis	Computed	m/s
			`ydot`		Front left axle velocity along the vehicle-fixed y-axis	Computed	m/s
			`zdot`		Front left axle velocity along the vehicle-fixed z-axis	Computed	m/s
	`Rght`	`Disp`	`x`		Front right axle displacement along the vehicle-fixed x-axis	Computed	m
			`y`		Front right axle displacement along the vehicle-fixed y-axis	Computed	m
			`z`		Front right axle displacement along the vehicle-fixed z-axis	Computed	m
		`Vel`	`xdot`		Front right axle velocity along the vehicle-fixed x-axis	Computed	m/s
			`ydot`		Front right axle velocity along the vehicle-fixed y-axis	Computed	m/s
			`zdot`		Front right axle velocity along the vehicle-fixed z-axis	Computed	m/s
`MidlAxl`	`Lft`	`Disp`	`x`		Middle left axle displacement along the vehicle-fixed x-axis	Computed	m
			`y`		Middle left axle displacement along the vehicle-fixed y-axis	Computed	m
			`z`		Middle left axle displacement along the vehicle-fixed z-axis	Computed	m
		`Vel`	`xdot`		Middle left axle velocity along the vehicle-fixed x-axis	Computed	m/s
			`ydot`		Middle left axle velocity along the vehicle-fixed y-axis	Computed	m/s
			`zdot`		Middle left axle velocity along the vehicle-fixed z-axis	Computed	m/s
	`Rght`	`Disp`	`x`		Middle right axle displacement along the vehicle-fixed x-axis	Computed	m
			`y`		Middle right axle displacement along the vehicle-fixed y-axis	Computed	m
			`z`		Middle right axle displacement along the vehicle-fixed z-axis	Computed	m
		`Vel`	`xdot`		Middle right axle velocity along the vehicle-fixed x-axis	Computed	m/s
			`ydot`		Middle right axle velocity along the vehicle-fixed y-axis	Computed	m/s
			`zdot`		Middle right axle velocity along the vehicle-fixed z-axis	Computed	m/s
`RearAxl`	`Lft`	`Disp`	`x`		Rear left axle displacement along the vehicle-fixed x-axis	Computed	m
			`y`		Rear left axle displacement along the vehicle-fixed y-axis	Computed	m
			`z`		Rear left axle displacement along the vehicle-fixed z-axis	Computed	m
		`Vel`	`xdot`		Rear left axle velocity along the vehicle-fixed x-axis	Computed	m/s
			`ydot`		Rear left axle velocity along the vehicle-fixed y-axis	Computed	m/s
			`zdot`		Rear left axle velocity along the vehicle-fixed z-axis	Computed	m/s
	`Rght`	`Disp`	`x`		Rear right axle displacement along the vehicle-fixed x-axis	Computed	m
			`y`		Rear right axle displacement along the vehicle-fixed y-axis	Computed	m
			`z`		Rear right axle displacement along the vehicle-fixed z-axis	Computed	m
		`Vel`	`xdot`		Rear right axle velocity along the vehicle-fixed x-axis	Computed	m/s
			`ydot`		Rear right axle velocity along the vehicle-fixed y-axis	Computed	m/s
			`zdot`		Rear right axle velocity along the vehicle-fixed z-axis	Computed	m/s
`Hitch`	`Disp`		`x`		Hitch offset from the 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 the axle plane along the vehicle-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	Computed	m/s
`Pwr`	`PwrExt`				Applied external power	Computed	W
`Pwr`	`Drag`				Power loss due to drag	Computed	W
`Geom`	`Disp`		`x`		Vehicle chassis offset from the 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 the axle plane along the vehicle-fixed z-axis	Input	m
	`Vel`		`xdot`		Vehicle chassis offset velocity along the vehicle-fixed x-axis	Computed	m/s
			`ydot`		Vehicle chassis offset velocity along the vehicle-fixed y-axis	Computed	m/s
			`zdot`		Vehicle chassis offset velocity along the vehicle-fixed z-axis	Computed	m/s
	`Ang`		`Beta`		Body slip angle, β $β = \frac{V_{y}}{V_{x}}$	Computed	rad

Vb — Vehicle velocity along vehicle-fixed frame
`vector`

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

pqr — Vehicle angular velocity about vehicle-fixed frame
`vector`

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

DCM — Direction cosine matrix
`array`

Direction cosine matrix, in rad, returned as an array.

Euler — Euler angles
`array`

Euler angles, φ, θ, and ψ, respectively, in rad, returned as an array.

Xe — Vehicle position in inertial reference frame
`vector`

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

Ve — Vehicle velocity in inertial reference frame
`vector`

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

Parameters

expand all

Block Options

Input Signals

Hitch forces — Create hitch force input port
`off` (default) | `on`

Select to create an input port, Fh, for the hitch forces.

Hitch moments — Create hitch moment input port
`off` (default) | `on`

Select to create an input port, Mh, for the hitch moments.

Chassis

Vehicle mass, m — Mass
`2000` (default) | `scalar`

Vehicle mass, m, in kg.

Longitudinal distance from center of mass to front axle, a — Distance from center of mass to front axle
`1.4` (default) | `scalar`

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

Top down and side views of vehicle showing load locations

Longitudinal distance from center of mass to middle axle, b — Distance from center of mass to middle axle
`1.6` (default) | `scalar`

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

Top down and side views of vehicle showing load locations

Longitudinal distance from center of mass to rear axle, c — Distance from center of mass to rear axle
`1.8` (default) | `scalar`

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

Top down and side views of vehicle showing load locations

Lateral distance from geometric centerline to center of mass, d — Distance from geometric centerline to center of mass
`0` (default) | `scalar`

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

Vertical distance from center of mass to axle plane, h — Distance
`.35` (default) | `scalar`

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

Top down and side views of vehicle showing load locations

Longitudinal distance from center of mass to hitch, dh — Longitudinal distance from CM to hitch
`1` (default) | `scalar`

Longitudinal distance from the CM to the hitch, dh, in m.

Top down and side views of vehicle showing load locations

Dependencies

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

Longitudinal distance from center of mass to hitch, hl — Lateral distance from CM to hitch
`0` (default) | `scalar`

Lateral distance from the CM to the hitch, hl, in m.

Top down and side views of vehicle showing load locations

Dependencies

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

Vertical distance from hitch to axle plane, hh — Distance from hitch to axle plane
`0.1` (default) | `scalar`

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

Top down and side views of vehicle showing load locations

Dependencies

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

Initial position in the inertial frame [Xeo,Yeo,Zeo], Xe_o — Initial position
`[0,0,0]` (default) | `vector`

Initial position of the vehicle in the inertial frame, Xe_o, in m.

Initial velocity in body axes [xdot_o,ydot_o,zdot_o], xbdot_o — Initial velocity
`[0,0,0]` (default) | `vector`

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

Initial Euler orientation [roll, pitch, yaw], eul_o — Initial Euler rotation
`[0,0,0]` (default) | `vector`

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

Initial body rotation rates [p,q,r], p_o — Initial rotation rate
`[0,0,0]` (default) | `vector`

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

Chassis inertia tensor, Iveh — Inertia
`[430 0 0; 0 1900 0; 0 0 2100]` (default) | `array`

Vehicle inertia tensor, I_veh, in kg*m^2. Dimensions are 3-by-3.

Track widths [front,rear], w — Widths
`[1.9,1.9,1.9]` (default) | `vector`

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

Top down view of track widths and CM offset from centerline.

Inertial Loads

Tractor Front

Mass, z1m — Tractor front mass
`0` (default) | `scalar`

Mass, z1m, in kg.

Distance vector from front axle, z1R — Tractor front distance from front axle
`[-.25,.125,.15]` (default) | `vector`

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

Array Element	Description
`z1R(1,1)`	Front suspension hardpoint to load, along the vehicle-fixed x-axis
`z1R(1,2)`	Vehicle centerline to load, along the vehicle-fixed y-axis
`z1R(1,3)`	Front suspension hardpoint to load, along the vehicle-fixed z-axis

For example, this table summarizes the parameter settings that specify the load location.

Example Location	Sign
Forward of the front axle Right of the vehicle centerline Above the front axle suspension hardpoint	`z1R(1,1)` < 0 `z1R(1,2)` > 0 `z1R(1,3)` > 0

Example Location

Sign

Forward of the front axle
Right of the vehicle centerline
Above the front axle suspension hardpoint

z1R(1,1) < 0
z1R(1,2) > 0
z1R(1,3) > 0

Inertia tensor, z1I — Tractor front inertia
`[1.4,-.2,.1;-.2,1.4,.1;.1,.1,2.25].*0` (default) | `array`

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

$z 1 I = [\begin{matrix} I_{x x} & I_{x y} & I_{x z} \\ I_{y x} & I_{y y} & I_{y z} \\ I_{z x} & I_{z y} & I_{z z} \end{matrix}]$

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

x-axis along the vehicle-fixed x-axis
y-axis along the vehicle-fixed y-axis
z-axis along the vehicle-fixed z-axis

Cab Overhead

Mass, z2m — Cab overhead mass
`0` (default) | `scalar`

Mass, z2m, in kg.

Distance vector from front axle, z2R — Cab overhead distance from front axle
`[1.4,0,.8]` (default) | `vector`

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

Array Element	Description
`z2R(1,1)`	Front suspension hardpoint to load, along the vehicle-fixed x-axis
`z2R(1,2)`	Vehicle centerline to load, along the vehicle-fixed y-axis
`z2R(1,3)`	Front suspension hardpoint to load, along the vehicle-fixed z-axis

For example, this table summarizes the parameter settings that specify the load location.

Example Location	Sign
Rear of the front axle Left of the vehicle centerline Above the front axle suspension hardpoint	`z2R(1,1)` > 0 `z2R(1,2)` < 0 `z2R(1,3)` > 0

Example Location

Sign

Rear of the front axle
Left of the vehicle centerline
Above the front axle suspension hardpoint

z2R(1,1) > 0
z2R(1,2) < 0
z2R(1,3) > 0

Inertia tensor, z2I — Cab overhead inertia
`[1.4,-.2,.1;-.2,1.4,.1;.1,.1,2.25].*0` (default) | `array`

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

$z 2 I = [\begin{matrix} I_{x x} & I_{x y} & I_{x z} \\ I_{y x} & I_{y y} & I_{y z} \\ I_{z x} & I_{z y} & I_{z z} \end{matrix}]$

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

x-axis along the vehicle-fixed x-axis
y-axis along the vehicle-fixed y-axis
z-axis along the vehicle-fixed z-axis

Tractor Frame Left

Mass, z3m — Tractor frame left mass
`0` (default) | `scalar`

Mass, z3m, in kg.

Distance vector from front axle, z3R — Tractor frame left distance from front axle
`[.75,-.5,.4]` (default) | `vector`

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

Array Element	Description
`z3R(1,1)`	Front suspension hardpoint to load, along the vehicle-fixed x-axis
`z3R(1,2)`	Vehicle centerline to load, along the vehicle-fixed y-axis
`z3R(1,3)`	Front suspension hardpoint to load, along the vehicle-fixed z-axis

For example, this table summarizes the parameter settings that specify the load location.

Example Location	Sign
Rear of the front axle Left of the vehicle centerline Above the front axle suspension hardpoint	`z3R(1,1)` > 0 `z3R(1,2)` < 0 `z3R(1,3)` > 0

Example Location

Sign

Rear of the front axle
Left of the vehicle centerline
Above the front axle suspension hardpoint

z3R(1,1) > 0
z3R(1,2) < 0
z3R(1,3) > 0

Inertia tensor, z3I — Tractor frame left inertia
`[5,-.1,-2;-2,9,.1;-.1,.1,6].*0` (default) | `array`

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

$z 3 I = [\begin{matrix} I_{x x} & I_{x y} & I_{x z} \\ I_{y x} & I_{y y} & I_{y z} \\ I_{z x} & I_{z y} & I_{z z} \end{matrix}]$

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

x-axis along the vehicle-fixed x-axis
y-axis along the vehicle-fixed y-axis
z-axis along the vehicle-fixed z-axis

Tractor Frame Right

Mass, z4m — Tractor frame right mass
`0` (default) | `scalar`

Mass, z4m, in kg.

Distance vector from front axle, z4R — Tractor frame right distance from front axle
`[.75,.5,.4]` (default) | `vector`

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

Array Element	Description
`z4R(1,1)`	Front suspension hardpoint to load, along the vehicle-fixed x-axis
`z4R(1,2)`	Vehicle centerline to load, along the vehicle-fixed y-axis
`z4R(1,3)`	Front suspension hardpoint to load, along the vehicle-fixed z-axis

For example, this table summarizes the parameter settings that specify the load location.

Example Location	Sign
Rear of the front axle Right of the vehicle centerline Above the front axle suspension hardpoint	`z4R(1,1)` > 0 `z4R(1,2)` > 0 `z4R(1,3)` > 0

Example Location

Sign

Rear of the front axle
Right of the vehicle centerline
Above the front axle suspension hardpoint

z4R(1,1) > 0
z4R(1,2) > 0
z4R(1,3) > 0

Inertia tensor, z4I — Tractor frame right inertia
`[5,-.1,-2;-2,9,.1;-.1,.1,6].*0` (default) | `array`

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

$z 4 I = [\begin{matrix} I_{x x} & I_{x y} & I_{x z} \\ I_{y x} & I_{y y} & I_{y z} \\ I_{z x} & I_{z y} & I_{z z} \end{matrix}]$

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

x-axis along the vehicle-fixed x-axis
y-axis along the vehicle-fixed y-axis
z-axis along the vehicle-fixed z-axis

Cab Left

Mass, z5m — Cab left mass
`0` (default) | `scalar`

Mass, z5m, in kg.

Distance vector from front axle, z5R — Cab left distance from front axle
`[1.25,-.5,.4]` (default) | `vector`

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

Array Element	Description
`z5R(1,1)`	Front suspension hardpoint to load, along the vehicle-fixed x-axis
`z5R(1,2)`	Vehicle centerline to load, along the vehicle-fixed y-axis
`z5R(1,3)`	Front suspension hardpoint to load, along the vehicle-fixed z-axis

For example, this table summarizes the parameter settings that specify the load location.

Example Location	Sign
Rear of the front axle Left of the vehicle centerline Above the front axle suspension hardpoint	`z5R(1,1)` > 0 `z5R(1,2)` < 0 `z5R(1,3)` > 0

Example Location

Sign

Rear of the front axle
Left of the vehicle centerline
Above the front axle suspension hardpoint

z5R(1,1) > 0
z5R(1,2) < 0
z5R(1,3) > 0

Inertia tensor, z5I — Cab left inertia
`[5,-.1,-2;-2,9,.1;-.1,.1,6].*0` (default) | `array`

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

$z 5 I = [\begin{matrix} I_{x x} & I_{x y} & I_{x z} \\ I_{y x} & I_{y y} & I_{y z} \\ I_{z x} & I_{z y} & I_{z z} \end{matrix}]$

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

x-axis along the vehicle-fixed x-axis
y-axis along the vehicle-fixed y-axis
z-axis along the vehicle-fixed z-axis

Cab Right

Mass, z6m — Cab right mass
`0` (default) | `scalar`

Mass, z6m, in kg.

Distance vector from front axle, z6R — Cab right distance from front axle
`[1.25,-.5,.4]` (default) | `vector`

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

Array Element	Description
`z6R(1,1)`	Front suspension hardpoint to load, along the vehicle-fixed x-axis
`z6R(1,2)`	Vehicle centerline to load, along the vehicle-fixed y-axis
`z6R(1,3)`	Front suspension hardpoint to load, along the vehicle-fixed z-axis

For example, this table summarizes the parameter settings that specify the load location.

Example Location	Sign
Rear of the front axle Right of the vehicle centerline Above the front axle suspension hardpoint	`z6R(1,1)` > 0 `z6R(1,2)` > 0 `z6R(1,3)` > 0

Example Location

Sign

Rear of the front axle
Right of the vehicle centerline
Above the front axle suspension hardpoint

z6R(1,1) > 0
z6R(1,2) > 0
z6R(1,3) > 0

Inertia tensor, z6I — Cab right inertia
`[5,-.1,-2;-2,9,.1;-.1,.1,6].*0` (default) | `array`

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

$z 6 I = [\begin{matrix} I_{x x} & I_{x y} & I_{x z} \\ I_{y x} & I_{y y} & I_{y z} \\ I_{z x} & I_{z y} & I_{z z} \end{matrix}]$

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

x-axis along the vehicle-fixed x-axis
y-axis along the vehicle-fixed y-axis
z-axis along the vehicle-fixed z-axis

Tractor Rear

Mass, z7m — Tractor rear mass
`0` (default) | `scalar`

Mass, z7m, in kg.

Distance vector from front axle, z7R — Tractor rear mass distance from front axle
`[2,0,.25]` (default) | `vector`

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

Array Element	Description
`z7R(1,1)`	Front suspension hardpoint to load, along the vehicle-fixed x-axis
`z7R(1,2)`	Vehicle centerline to load, along the vehicle-fixed y-axis
`z7R(1,3)`	Front suspension hardpoint to load, along the vehicle-fixed z-axis

For example, this table summarizes the parameter settings that specify the load location.

Example Location	Sign
Rear of the front axle Right of the vehicle centerline Above the front axle suspension hardpoint	`z7R(1,1)` > 0 `z7R(1,2)` > 0 `z7R(1,3)` > 0

Example Location

Sign

Rear of the front axle
Right of the vehicle centerline
Above the front axle suspension hardpoint

z7R(1,1) > 0
z7R(1,2) > 0
z7R(1,3) > 0

Inertia tensor, z7I — Tractor rear inertia
`[1.4,-.2,.1;-.2,1.4,.1;.1,.1,2.25].*0` (default) | `array`

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

$z 7 I = [\begin{matrix} I_{x x} & I_{x y} & I_{x z} \\ I_{y x} & I_{y y} & I_{y z} \\ I_{z x} & I_{z y} & I_{z z} \end{matrix}]$

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

x-axis along the vehicle-fixed x-axis
y-axis along the vehicle-fixed y-axis
z-axis along the vehicle-fixed z-axis

Aerodynamic

Longitudinal drag area, Af — Vehicle cross-sectional area
`2` (default) | `scalar`

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

Longitudinal drag coefficient, Cd — Air drag coefficient
`.3` (default) | `scalar`

Air drag coefficient, C_d, dimensionless.

Longitudinal lift coefficient, Cl — Air lift coefficient
`.1` (default) | `scalar`

Air lift coefficient, C_l, dimensionless.

Longitudinal drag pitch moment, Cpm — Pitch drag
`.1` (default) | `scalar`

Longitudinal drag pitch moment coefficient, C_pm, dimensionless.

Relative wind angle vector, beta_w — Wind angle
`[0:0.001:0.01]` (default) | `vector`

Relative wind angle vector, β_w, in rad.

Side force coefficient vector, Cs — Side force drag
`[0:0.01:0.1]` (default) | `vector`

Side force coefficient vector coefficient, C_s, dimensionless.

Yaw moment coefficient vector, Cym — Yaw moment drag
`[0:0.001:0.01]` (default) | `vector`

Yaw moment coefficient vector coefficient, C_ym, dimensionless.

Environment

Absolute air pressure, Pabs — Pressure
`101325` (default) | `scalar`

Environmental air absolute pressure, P_abs, in Pa.

Air temperature, Tair — Ambient air temperature
`273` (default) | `scalar`

Ambient air temperature, T_air, in K.

Dependencies

To enable this parameter, clear Air temperature.

Gravitational acceleration, g — Gravity
`9.81` (default) | `scalar`

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

Simulation

Longitudinal velocity tolerance, xdot_tol — Tolerance
`.1` (default) | `scalar`

Longitudinal velocity tolerance, xdot_tol, in m/s.

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

Geometric longitudinal offset from axle plane, longOff — Longitudinal offset
`0` (default) | `scalar`

Vehicle chassis offset from the axle plane along the body-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.

Geometric lateral offset from center plane, latOff — Lateral offset
`0` (default) | `scalar`

Vehicle chassis offset from center plane along the body-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.

Geometric vertical offset from axle plane, vertOff — Vertical offset
`0` (default) | `scalar`

Vehicle chassis offset from the axle plane along the body-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 Euler angles, wrapAng — Selection
`on` (default) | `off`

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 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.

Extended Capabilities

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Version History

Introduced in R2020b

Vehicle Body 6DOF Three Axles

Description

Inertial Loads

Equations of Motion

Examples

Three-Axle Tractor Towing a Three-Axle Trailer

Ports

Input

FSusp — Suspension forces on vehicle 3-by-6 array

MSusp — Suspension moment on vehicle 3-by-6 array

FExt — External forces acting on vehicle vector

MExt — External moments acting on vehicle vector

Fh — Hitch force on the body array

Dependencies

Mh — Hitch moment about body array

Dependencies

WindXYZ — Wind speed array

AirTemp — Ambient air temperature scalar

Dependencies

Output

Info — Vehicle body information bus

Vb — Vehicle velocity along vehicle-fixed frame vector

pqr — Vehicle angular velocity about vehicle-fixed frame vector

DCM — Direction cosine matrix array

Euler — Euler angles array

Xe — Vehicle position in inertial reference frame vector

Ve — Vehicle velocity in inertial reference frame vector

Parameters

Block Options

Hitch forces — Create hitch force input port off (default) | on

Hitch moments — Create hitch moment input port off (default) | on

Chassis

Vehicle mass, m — Mass 2000 (default) | scalar

Longitudinal distance from center of mass to front axle, a — Distance from center of mass to front axle 1.4 (default) | scalar

Longitudinal distance from center of mass to middle axle, b — Distance from center of mass to middle axle 1.6 (default) | scalar

Longitudinal distance from center of mass to rear axle, c — Distance from center of mass to rear axle 1.8 (default) | scalar

Lateral distance from geometric centerline to center of mass, d — Distance from geometric centerline to center of mass 0 (default) | scalar

Vertical distance from center of mass to axle plane, h — Distance .35 (default) | scalar

Longitudinal distance from center of mass to hitch, dh — Longitudinal distance from CM to hitch 1 (default) | scalar

Dependencies

Longitudinal distance from center of mass to hitch, hl — Lateral distance from CM to hitch 0 (default) | scalar

Dependencies

Vertical distance from hitch to axle plane, hh — Distance from hitch to axle plane 0.1 (default) | scalar

Dependencies

Initial position in the inertial frame [Xeo,Yeo,Zeo], Xe_o — Initial position [0,0,0] (default) | vector

Initial velocity in body axes [xdot_o,ydot_o,zdot_o], xbdot_o — Initial velocity [0,0,0] (default) | vector

Initial Euler orientation [roll, pitch, yaw], eul_o — Initial Euler rotation [0,0,0] (default) | vector

Initial body rotation rates [p,q,r], p_o — Initial rotation rate [0,0,0] (default) | vector

Chassis inertia tensor, Iveh — Inertia [430 0 0; 0 1900 0; 0 0 2100] (default) | array

Track widths [front,rear], w — Widths [1.9,1.9,1.9] (default) | vector

Inertial Loads

Mass, z1m — Tractor front mass 0 (default) | scalar

Distance vector from front axle, z1R — Tractor front distance from front axle [-.25,.125,.15] (default) | vector

Inertia tensor, z1I — Tractor front inertia [1.4,-.2,.1;-.2,1.4,.1;.1,.1,2.25].*0 (default) | array

Mass, z2m — Cab overhead mass 0 (default) | scalar

Distance vector from front axle, z2R — Cab overhead distance from front axle [1.4,0,.8] (default) | vector

Inertia tensor, z2I — Cab overhead inertia [1.4,-.2,.1;-.2,1.4,.1;.1,.1,2.25].*0 (default) | array

Mass, z3m — Tractor frame left mass 0 (default) | scalar

Distance vector from front axle, z3R — Tractor frame left distance from front axle [.75,-.5,.4] (default) | vector

Inertia tensor, z3I — Tractor frame left inertia [5,-.1,-2;-2,9,.1;-.1,.1,6].*0 (default) | array

Mass, z4m — Tractor frame right mass 0 (default) | scalar

Distance vector from front axle, z4R — Tractor frame right distance from front axle [.75,.5,.4] (default) | vector

Inertia tensor, z4I — Tractor frame right inertia [5,-.1,-2;-2,9,.1;-.1,.1,6].*0 (default) | array

Mass, z5m — Cab left mass 0 (default) | scalar

Distance vector from front axle, z5R — Cab left distance from front axle [1.25,-.5,.4] (default) | vector

Inertia tensor, z5I — Cab left inertia [5,-.1,-2;-2,9,.1;-.1,.1,6].*0 (default) | array

Mass, z6m — Cab right mass 0 (default) | scalar

Distance vector from front axle, z6R — Cab right distance from front axle [1.25,-.5,.4] (default) | vector

Inertia tensor, z6I — Cab right inertia [5,-.1,-2;-2,9,.1;-.1,.1,6].*0 (default) | array

Mass, z7m — Tractor rear mass 0 (default) | scalar

Distance vector from front axle, z7R — Tractor rear mass distance from front axle [2,0,.25] (default) | vector

Inertia tensor, z7I — Tractor rear inertia [1.4,-.2,.1;-.2,1.4,.1;.1,.1,2.25].*0 (default) | array

Aerodynamic

Longitudinal drag area, Af — Vehicle cross-sectional area 2 (default) | scalar

Longitudinal drag coefficient, Cd — Air drag coefficient .3 (default) | scalar

Longitudinal lift coefficient, Cl — Air lift coefficient .1 (default) | scalar

Longitudinal drag pitch moment, Cpm — Pitch drag .1 (default) | scalar

Relative wind angle vector, beta_w — Wind angle [0:0.001:0.01] (default) | vector

Side force coefficient vector, Cs — Side force drag [0:0.01:0.1] (default) | vector

Yaw moment coefficient vector, Cym — Yaw moment drag [0:0.001:0.01] (default) | vector

FSusp — Suspension forces on vehicle
`3-by-6` array

MSusp — Suspension moment on vehicle
`3-by-6` array

FExt — External forces acting on vehicle
`vector`

MExt — External moments acting on vehicle
`vector`

Fh — Hitch force on the body
`array`

Mh — Hitch moment about body
`array`

WindXYZ — Wind speed
`array`

AirTemp — Ambient air temperature
`scalar`

Info — Vehicle body information
bus

Vb — Vehicle velocity along vehicle-fixed frame
`vector`

pqr — Vehicle angular velocity about vehicle-fixed frame
`vector`

DCM — Direction cosine matrix
`array`

Euler — Euler angles
`array`

Xe — Vehicle position in inertial reference frame
`vector`

Ve — Vehicle velocity in inertial reference frame
`vector`

Hitch forces — Create hitch force input port
`off` (default) | `on`

Hitch moments — Create hitch moment input port
`off` (default) | `on`

Vehicle mass, m — Mass
`2000` (default) | `scalar`

Longitudinal distance from center of mass to front axle, a — Distance from center of mass to front axle
`1.4` (default) | `scalar`

Longitudinal distance from center of mass to middle axle, b — Distance from center of mass to middle axle
`1.6` (default) | `scalar`

Longitudinal distance from center of mass to rear axle, c — Distance from center of mass to rear axle
`1.8` (default) | `scalar`

Lateral distance from geometric centerline to center of mass, d — Distance from geometric centerline to center of mass
`0` (default) | `scalar`

Vertical distance from center of mass to axle plane, h — Distance
`.35` (default) | `scalar`

Longitudinal distance from center of mass to hitch, dh — Longitudinal distance from CM to hitch
`1` (default) | `scalar`

Longitudinal distance from center of mass to hitch, hl — Lateral distance from CM to hitch
`0` (default) | `scalar`

Vertical distance from hitch to axle plane, hh — Distance from hitch to axle plane
`0.1` (default) | `scalar`

Initial position in the inertial frame [Xeo,Yeo,Zeo], Xe_o — Initial position
`[0,0,0]` (default) | `vector`

Initial velocity in body axes [xdot_o,ydot_o,zdot_o], xbdot_o — Initial velocity
`[0,0,0]` (default) | `vector`

Initial Euler orientation [roll, pitch, yaw], eul_o — Initial Euler rotation
`[0,0,0]` (default) | `vector`

Initial body rotation rates [p,q,r], p_o — Initial rotation rate
`[0,0,0]` (default) | `vector`

Chassis inertia tensor, Iveh — Inertia
`[430 0 0; 0 1900 0; 0 0 2100]` (default) | `array`

Track widths [front,rear], w — Widths
`[1.9,1.9,1.9]` (default) | `vector`

Mass, z1m — Tractor front mass
`0` (default) | `scalar`

Distance vector from front axle, z1R — Tractor front distance from front axle
`[-.25,.125,.15]` (default) | `vector`

Inertia tensor, z1I — Tractor front inertia
`[1.4,-.2,.1;-.2,1.4,.1;.1,.1,2.25].*0` (default) | `array`

Mass, z2m — Cab overhead mass
`0` (default) | `scalar`

Distance vector from front axle, z2R — Cab overhead distance from front axle
`[1.4,0,.8]` (default) | `vector`

Inertia tensor, z2I — Cab overhead inertia
`[1.4,-.2,.1;-.2,1.4,.1;.1,.1,2.25].*0` (default) | `array`

Mass, z3m — Tractor frame left mass
`0` (default) | `scalar`

Distance vector from front axle, z3R — Tractor frame left distance from front axle
`[.75,-.5,.4]` (default) | `vector`

Inertia tensor, z3I — Tractor frame left inertia
`[5,-.1,-2;-2,9,.1;-.1,.1,6].*0` (default) | `array`

Mass, z4m — Tractor frame right mass
`0` (default) | `scalar`

Distance vector from front axle, z4R — Tractor frame right distance from front axle
`[.75,.5,.4]` (default) | `vector`

Inertia tensor, z4I — Tractor frame right inertia
`[5,-.1,-2;-2,9,.1;-.1,.1,6].*0` (default) | `array`

Mass, z5m — Cab left mass
`0` (default) | `scalar`

Distance vector from front axle, z5R — Cab left distance from front axle
`[1.25,-.5,.4]` (default) | `vector`

Inertia tensor, z5I — Cab left inertia
`[5,-.1,-2;-2,9,.1;-.1,.1,6].*0` (default) | `array`

Mass, z6m — Cab right mass
`0` (default) | `scalar`

Distance vector from front axle, z6R — Cab right distance from front axle
`[1.25,-.5,.4]` (default) | `vector`

Inertia tensor, z6I — Cab right inertia
`[5,-.1,-2;-2,9,.1;-.1,.1,6].*0` (default) | `array`

Mass, z7m — Tractor rear mass
`0` (default) | `scalar`

Distance vector from front axle, z7R — Tractor rear mass distance from front axle
`[2,0,.25]` (default) | `vector`

Inertia tensor, z7I — Tractor rear inertia
`[1.4,-.2,.1;-.2,1.4,.1;.1,.1,2.25].*0` (default) | `array`

Longitudinal drag area, Af — Vehicle cross-sectional area
`2` (default) | `scalar`

Longitudinal drag coefficient, Cd — Air drag coefficient
`.3` (default) | `scalar`

Longitudinal lift coefficient, Cl — Air lift coefficient
`.1` (default) | `scalar`

Longitudinal drag pitch moment, Cpm — Pitch drag
`.1` (default) | `scalar`

Relative wind angle vector, beta_w — Wind angle
`[0:0.001:0.01]` (default) | `vector`

Side force coefficient vector, Cs — Side force drag
`[0:0.01:0.1]` (default) | `vector`

Yaw moment coefficient vector, Cym — Yaw moment drag
`[0:0.001:0.01]` (default) | `vector`

Absolute air pressure, Pabs — Pressure
`101325` (default) | `scalar`

Air temperature, Tair — Ambient air temperature
`273` (default) | `scalar`

Gravitational acceleration, g — Gravity
`9.81` (default) | `scalar`

Longitudinal velocity tolerance, xdot_tol — Tolerance
`.1` (default) | `scalar`

Geometric longitudinal offset from axle plane, longOff — Longitudinal offset
`0` (default) | `scalar`

Geometric lateral offset from center plane, latOff — Lateral offset
`0` (default) | `scalar`

Geometric vertical offset from axle plane, vertOff — Vertical offset
`0` (default) | `scalar`

Wrap Euler angles, wrapAng — Selection
`on` (default) | `off`

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