Bicycle Model

Implement a single track 3DOF rigid vehicle body to calculate longitudinal, lateral, and yaw motion

Description

The Bicycle Model block implements a rigid two-axle single track vehicle body model to calculate longitudinal, lateral, and yaw motion. The block accounts for body mass, aerodynamic drag, and weight distribution between the axles due to acceleration and steering. There are two types of Bicycle Model blocks.

Block	Implementation
Bicycle Model - Velocity Input	Block assumes that the external longitudinal velocity is quasi-steady state so the longitudinal acceleration is approximately zero. Since the motion is quasi-steady, the block calculates only lateral forces using the tire slip angles and linear cornering stiffness.
Bicycle Model - Force Input	Block uses the external longitudinal force to accelerate or brake the vehicle. Block calculates lateral forces using the tire slip angles and linear cornering stiffness.

To calculate the normal forces on the front and rear axles, the block uses rigid-body vehicle motion, suspension system forces, and wind and drag forces. The block resolves the force and moment components on the rigid vehicle body frame.

Ports

Input

WhlAngF — Wheel angle

scalar

Front wheel angle, in rad.

FxF — Force Input: Total longitudinal force on the front axle

scalar

Longitudinal force on the front axle, Fx_F, along vehicle-fixed x-axis, in N.

Bicycle Model - Force Input block input port.

FxR — Force Input: Total longitudinal force on the rear axle

scalar

Longitudinal force on the rear axle, Fx_R, along vehicle-fixed x-axis, in N.

Bicycle Model - Force Input block input port.

xdotin — Velocity Input: Longitudinal velocity

scalar

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

Bicycle Model - Velocity Input block input port.

Output

Info — Bus signal

bus

Bus signal containing these block values.

Signal						Description	Value	Units
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	rad
	FrntAxl	Disp	X			Front wheel displacement along the earth-fixed X-axis	Computed	m
			Y			Front wheel displacement along the earth-fixed Y-axis	Computed	m
			Z			Front wheel displacement along the earth-fixed Z-axis	0	m
		Vel	Xdot			Front wheel velocity along the earth-fixed X-axis	Computed	m/s
			Ydot			Front wheel velocity along the earth-fixed Y-axis	Computed	m/s
			Zdot			Front wheel velocity along the earth-fixed Z-axis	0	m/s
	RearAxl	Disp	X			Rear wheel displacement along the earth-fixed X-axis	Computed	m
			Y			Rear wheel displacement along the earth-fixed Y-axis	Computed	m
			Z			Rear wheel displacement along the earth-fixed Z-axis	0	m
		Vel	Xdot			Rear wheel velocity along the earth-fixed X-axis	Computed	m/s
			Ydot			Rear wheel velocity along the earth-fixed Y-axis	Computed	m/s
			Zdot			Rear wheel velocity along the earth-fixed Z-axis	0	m/s
	Hitch	Disp	X			Hitch offset from axle plane along the earth-fixed X-axis	Computed	m
			Y			Hitch offset from center plane along the earth-fixed Y-axis	Computed	m
			Z			Hitch offset from axle plane along the earth-fixed Z-axis	Computed	m
		Vel	Xdot			Hitch offset velocity from axle plane along the earth-fixed X-axis	Computed	m
			Ydot			Hitch offset velocity from center plane along the earth-fixed Y-axis	Computed	m
			Zdot			Hitch offset velocity from axle plane along the earth-fixed Z-axis	Computed	m
	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	rad
		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	rad/s
		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	rad/s
		DCM	Direction cosine matrix				Computed	rad
	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	Input	N
			Fy			Hitch force applied to body at the hitch location along the vehicle-fixed y-axis	Input	N
			Fz			Hitch force applied to body at the hitch location along the vehicle-fixed z-axis	Input	N
		FrntAxl	Fx			Longitudinal force on front wheel, along the vehicle-fixed x-axis	Computed	N
			Fy			Lateral force on front wheel along the vehicle-fixed y-axis	Computed	N
			Fz			Normal force on front wheel, along the vehicle-fixed z-axis	Computed	N
		RearAxl	Fx			Longitudinal force on rear wheel, along the vehicle-fixed x-axis	Computed	N
			Fy			Lateral force on rear wheel along the vehicle-fixed y-axis	Computed	N
			Fz			Normal force on rear wheel, along the vehicle-fixed z-axis	Computed	N
		Tires	FrntTire	Fx		Front tire force, along the vehicle-fixed x-axis	Computed	N
				Fy		Front tire force, along the vehicle-fixed y-axis	Computed	N
				Fz		Front tire force, along the vehicle-fixed z-axis	Computed	N
			RearTire	FxFx		Rear tire force, along the vehicle-fixed x-axis	Computed	N
				Fy		Rear tire force, along the vehicle-fixed y-axis	Computed	N
				Fz		Rear tire force, along the vehicle-fixed z-axis	Computed	N
		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	Disp	x			Front wheel displacement along the vehicle-fixed x-axis	Computed	m
			y			Front wheel displacement along the vehicle-fixed y-axis	Computed	m
			z			Front wheel displacement along the vehicle-fixed z-axis	Computed	m
		Vel	xdot			Front wheel velocity along the vehicle-fixed x-axis	Computed	m/s
			ydot			Front wheel velocity along the vehicle-fixed y-axis	Computed	m/s
			zdot			Front wheel velocity along the vehicle-fixed z-axis	0	m/s
		Steer	WhlAngFL			Front left wheel steering angle	Computed	rad
			WhlAngFR			Front right wheel steering angle	Computed	rad
RearAxl	Disp	x			Rear wheel displacement along the vehicle-fixed x-axis	Computed	m
		y			Rear wheel displacement along the vehicle-fixed y-axis	Computed	m
		z			Rear wheel displacement along the vehicle-fixed z-axis	Computed	m
	Vel	xdot			Rear wheel velocity along the vehicle-fixed x-axis	Computed	m/s
		ydot			Rear wheel velocity along the vehicle-fixed y-axis	Computed	m/s
		zdot			Rear wheel velocity along the vehicle-fixed z-axis	0	m/s
	Steer	WhlAngRL			Rear left wheel steering angle	Computed	rad
		WhlAngRR			Rear right wheel steering angle	Computed	rad
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	Computed	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	rad

Signal			Description		Value	Units
PwrInfo	PwrTrnsfrd	PwrFxExt	Externally applied longitudinal force power		Computed	W
		PwrFyExt	Externally applied lateral force power		Computed	W
		PwrMzExt	Externally applied roll moment power		Computed	W
		PwrFwFx	Longitudinal force applied at the front axle power		Computed	W
		PwrFwFy	Lateral force applied at the front axle power		Computed	W
		PwrFwRx	Longitudinal force applied at the rear axle power		Computed	W
		PwrFwRy	Lateral force applied at the rear 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

xdot — Vehicle body longitudinal velocity

scalar

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

ydot — Vehicle body lateral velocity

scalar

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

psi — Yaw

scalar

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

r — Yaw rate

scalar

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

Parameters

Longitudinal

Number of wheels on front axle, NF — Front wheel count

2 (default) | scalar

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

Number of wheels on rear axle, NR — Rear wheel count

2 (default) | scalar

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

Vehicle mass, m — Vehicle mass

2000 (default) | scalar

Vehicle mass, m, in kg.

Longitudinal distance from center of mass to front axle, a — Front axle distance

1.4 (default) | scalar

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

Longitudinal distance from center of mass to rear axle, b — Rear axle distance

1.6 (default) | scalar

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

Vertical distance from center of mass to axle plane, h — Height

0.35 (default) | scalar

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

Longitudinal distance from center of mass to hitch, dh — Distance from CM to hitch

1 (default) | scalar

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

Dependencies

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

Vertical distance from hitch to axle plane, hh — Distance from hitch to axle plane

0.2 (default) | scalar

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

Dependencies

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

Initial inertial frame longitudinal position, X_o — Position

0 (default) | scalar

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

Initial longitudinal velocity, xdot_o — Velocity

0 (default) | scalar

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

Dependencies

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

• External longitudinal forces

• External forces

Lateral

Front tire corner stiffness, Cy_f — Stiffness

12e3 (default) | scalar

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

Dependencies

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

1. Set Axle forces to one of these options:

   • External longitudinal velocity

   • External longitudinal forces

2. Clear Mapped corner stiffness.

Rear tire corner stiffness, Cy_r — Stiffness

11e3 (default) | scalar

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

Dependencies

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

1. Set Axle forces to one of these options:

   • External longitudinal velocity

   • External longitudinal forces

2. Clear Mapped corner stiffness.

Initial inertial frame lateral displacement, Y_o — Position

0 (default) | scalar

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

Initial lateral velocity, ydot_o — Velocity

0 (default) | scalar

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

Yaw

Yaw polar inertia, Izz — Inertia

4000 (default) | scalar

Yaw polar inertia, in kg*m^2.

Initial yaw angle, psi_o — Psi rotation

0 (default) | scalar

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

Initial yaw rate, r_o — Yaw rate

0 (default) | scalar

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

Aerodynamic

Longitudinal drag area, Af — Effective vehicle cross-sectional area

2 (default) | scalar

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

Longitudinal drag coefficient, Cd — Air drag coefficient

.3 (default) | scalar

Air drag coefficient, C_d. The value is dimensionless.

Longitudinal lift coefficient, Cl — Air lift coefficient

.1 (default) | scalar

Air lift coefficient, C_l. The value is dimensionless.

Longitudinal drag pitch moment, Cpm — Pitch drag

.1 (default) | scalar

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

Relative wind angle vector, beta_w — Wind angle

[0:0.01:0.3] (default) | vector

Relative wind angle vector, β_w, in rad.

Side force coefficient vector, Cs — Side force coefficient

[0:0.03:0.9] (default) | vector

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

Yaw moment coefficient vector, Cym — Yaw moment drag

[0:0.01:0.3] (default) | vector

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

Environment

Absolute air pressure, Pabs — Pressure

101325 (default) | scalar | scalar

Environmental absolute pressure, P_abs, in Pa.

Air temperature, Tair — Temperature

273 (default) | scalar

Environmental absolute temperature, T, 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.

Nominal friction scaling factor, mu — Friction scale factor

1 (default) | scalar

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

Dependencies

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

1. Set Axle forces to one of these options:

   • External longitudinal velocity

   • External longitudinal forces

2. Clear External Friction.

Simulation

Longitudinal velocity tolerance, xdot_tol — Tolerance

.01 (default) | scalar

Longitudinal velocity tolerance, in m/s.

Nominal normal force, Fznom — Normal force

5000 (default) | scalar

Nominal normal force, in N.

Dependencies

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

• External longitudinal velocity

• External longitudinal forces

Geometric longitudinal offset from axle plane, longOff — Longitudinal offset

0 (default) | scalar

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

Geometric lateral offset from center plane, latOff — Lateral offset

0 (default) | scalar

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

Geometric vertical offset from axle plane, vertOff — Vertical offset

0 (default) | scalar

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

Wrap Euler angles, wrapAng — Selection

off (default) | on

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

• Track the total vehicle yaw rotation.

• Avoid discontinuities in the vehicle state estimators.

References

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

Version History

Introduced in R2018a