Steering System

Steering system for Ackerman, rack-and-pinion, and parallel steering mechanisms

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• Vehicle Dynamics Blockset / Steering

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Description

The Steering System block implements dynamic steering to calculate the wheel angles for rack-and-pinion mechanisms with friction, compliance, and Ackerman steering features. The block uses the steering wheel input angle or torque, vehicle speed, caster angle, and right and left wheel feedbacks to calculate the wheel angles. The block uses the vehicle coordinate system.

If you select the Power assist parameter, you can specify a torque assist lookup table that is a function of the vehicle speed and steering wheel input torque. The block uses the steering wheel input torque and torque assist to calculate the steering dynamics. If you select the Ackerman steering parameter, you can specify a lookup table of percent Ackerman values to calculate the Ackerman steering effects. Otherwise, the block calculates the wheel angles based on a perfect 100 percent Ackerman.

If you select the Power assist, Ackerman steering, or Kingpin moment parameters in the Input signals section, you can specify additional inputs for the external power assist torques, percent Ackerman values, or kingpin moments.

Use the Location parameter to specify front or rear steering.

Setting	Implementation
Front	Front steering
Rear	Rear steering

Steering

Rack-and-Pinion

For ideal rack-and-pinion steering, the gears convert the steering rotation into linear motion.

To calculate the steering angles, the block uses these equations.

$$\begin{array}{l}{l}_1=\frac{TW-l_{rack}}{2}-\Delta P\\ \\ l_2{}^2=l_1{}^2+D^2\\ \\ \Delta P=r{\delta }_{in}\\ \\ \beta =\frac{\pi }{2}-{\tan }^{-1}\left[\frac{D}{l_1}\right]-{\cos }^{-1}\left[\frac{l_{arm}{}^2+l_2{}^2-l_{rod}{}^2}{2l_{arm}{l}_2}\right]\end{array}$$

The illustration and equations use these variables.

δin	Steering wheel angle
δL	Left wheel angle
δR	Right wheel angle
TW	Track width
r	Pinion radius
ΔP	Linear change in rack position
D	Distance between front axis and rack
lrack	Rack casing length
larm	Steering arm length
lrod	Tie rod length

Ackerman

For ideal Ackerman steering, the wheel angles have a common turning circle.

To calculate the steering angles, the block uses these equations.

$$\begin{array}{l}\cot ({\delta }_L)-\cot ({\delta }_R)=\frac{TW}{WB}\\ \\ {\delta }_{Ack}=\frac{{\delta }_{in}}{\gamma }\\ \\ {\delta }_L={\tan }^{-1}\left(\frac{WB\tan ({\delta }_{Ack})}{WB+0.5TW\tan ({\delta }_{Ack})}\right)\\ {\delta }_R={\tan }^{-1}\left(\frac{WB\tan ({\delta }_{Ack})}{WB-0.5TW\tan ({\delta }_{Ack})}\right)\end{array}$$

The illustration and equations use these variables.

δin	Steering angle
δL	Left wheel angle
δR	Right wheel angle
δvir	Virtual wheel angle
TW	Track width
WB	Wheel base
γ	Steering ratio

Ports

Input

VehSpd — Vehicle speed

scalar

Vehicle speed, v, in m/s, specified as a scalar.

CstrAng — Wheel caster angle

1-by-2 vector

Wheel caster angle, τ_L, in radians, specified as a 1-by-2 vector. The first element represents the angle of the left wheel and the second element represents the angle of the right wheel.

Dependencies

To enable this port, clear Input signals > Kingpin moment.

WhelAngFdk — Wheel angle feedback

1-by-2 vector

Wheel angle feedback, in radians, specified as a 1-by-2 vector. The first element represents the angle feedback for the left wheel and the second element represents the angle feedback for the right wheel.

Dependencies

To enable this port, clear Input signals > Kingpin moment.

StrWhlAng — Steering angle input

scalar

Steering angle input, in radians, specified as a scalar.

Dependencies

To enable this port, select Steer inputs > Angle.

StrWhlTrq — Steering torque input

scalar

Steering torque input, in N*m, specified as a scalar.

Dependencies

To enable this port, select Steer inputs > Torque.

PwrAstTrq — External power assistant torque

scalar

External power assistant torque, in N*m, specified as a scalar.

Dependencies

To enable this port, select Input signals > Power assist.

PctAck — External percent Ackerman value

scalar

External percent Ackerman value, in N*m, specified as a scalar.

Dependencies

To enable this port, select Input signals > Ackerman steering.

TireFdk — Tire forces and moments feedback

1-by-6 vector

Tire forces and moments feedback, specified as a 1-by-6 vector that contains the following values, in order:

Description	Unit
x-directional Force	N
y-directional Force	N
z-directional Force	N
x-directional Moment	N*m
y-directional Moment	N*m
z-directional Moment	N*m

Dependencies

To enable this port, clear Input signals > Kingpin moment.

LftKpM — Left kingpin moment

scalar

Left kingpin moment, in N*m, specified as a scalar.

Dependencies

To enable this port, select Input signals > Kingpin moment.

RghtKpM — Right kingpin moment

scalar

Right kingpin moment, in N*m, specified as a scalar.

Dependencies

To enable this port, select Input signals > Kingpin moment.

Output

Info — Vehicle dynamics information

bus

Vehicle dynamics information, returned as a bus signal that contains:

Signal	Description	Unit
StrgWhlAng	Steering wheel angle	rad
StrgWhlSpd	Steering wheel angular velocity	rad/s
ShftAng	Shaft angle	rad
ShftSpd	Shaft angular velocity	rad/s
AngLft	Left wheel angle	rad
SpdLft	Left wheel angular velocity	rad/s
AngRght	Right wheel angle	rad
SpdRght	Right wheel angular velocity	rad/s
TrqAst	Torque assist	N·m
PwrAst	Power assist	W
PwrLoss	Power loss	W
InstStrgRatio	Instantaneous steering ratio	NA

LftWhlAng — Left wheel angle

scalar

Left wheel angle, δ_L, in radians, returned as a scalar.

RghtWhlAng — Right wheel angle

scalar

Right wheel angle, δ_R, in radians, returned as a scalar.

Parameters

Block Options

Type — Steering type

Rack and pinion (default)

Steering type for the steering system.

Intermediate shaft type — Intermediate shaft type

Single Cardan joint (default) | Double Cardan joints

Whether to model the intermediate shaft type using single or double cardan joints.

Power assist — Whether to model power assist

on (default) | off

Select to model power assist in the steering system.

Dependencies

To enable this parameter, in the Input signals section, clear Power assist.

Ackerman steering — Whether to use Ackerman steering

on (default) | off

Select to use Ackerman steering in the steering system.

Dependencies

To enable this parameter, in the Input signals section, clear Ackerman steering.

Input Signals

Power assist — Specify external power assistance torque

off (default) | on

Select this parameter to enable the PwrAstTrq port.

Ackerman steering — Specify external percent Ackerman value

off (default) | on

Select this parameter to enable the PctAct port.

Kingpin moment — Specify right and left wheel angles

off (default) | on

Select this parameter to enable the LftKpM and RghtKpM ports.

Location — Location of steering system

Front (default) | Rear

Select the front or rear axle as the location of the steering system.

Steer inputs — Wheel angle or torque

Angle (default) | Torque

Specify wheel angle or wheel torque steering input.

General

Track width, TrckWdth — Track width

1 (default) | scalar

Track width, TW, in m, specified as a scalar.

Steering range, StrgRng — Steering range

1.25*pi (default) | scalar

Steering range, in rad, specified as a scalar. The block limits the wheel angles to remain within the steering range.

Steering wheel inertia, StrWhlInert — Steering wheel inertia

0.1 (default) | scalar

Steering wheel inertia, in kg*m², specified as a scalar.

Steering column inertia, StrColInert — Steering column inertia

0.01 (default) | scalar

Steering column inertia, in kg*m², specified as a scalar.

Kingpin offset, KngpnOfst — Kingpin offset

0.075 (default) | scalar

Kingpin offset, in m, specified as a scalar.

Kingpin inclination angle, Lambda — Kingpin inclination angle

0.20943951023932 (default) | scalar

Kingpin inclination angle, in rad, specified as a scalar.

Hub lead, HbLead — Hub lead

4.00E-05 (default) | scalar

Hub lead, in m, specified as a scalar.

Static loaded radius, StcLdRadius — Static loaded radius

0.4 (default) | scalar

Static loaded radius, in m, specified as a scalar.

Overall steer ratio, OvrlStrRatio — Overall steer ratio

17.42 (default) | scalar

Overall steer ratio, specified as a scalar.

Steering angle breakpoints, StrgAngBpts — Steering angle breakpoints

[-6.2832 -5.0265 -3.7699 -2.5133 -1.2566 0 1.2566 2.5133 3.7699 5.0265 6.2832] (default)

Steering angle breakpoints, in rad, specified as a 1-by-11 vector.

Dependencies

To enable this parameter, set one of these parameters to Lookup table:

• Rack and pinion > Rack gain parameterized by

• Ackerman steering > Percent Ackerman parameterized by

Caster angle, CstrAng — Caster angle

0 (default) | scalar

Caster angle, in rad, specified as a scalar.

Dependencies

To enable this parameter, select Input signals > Kingpin moment.

Rack and Pinion

Rack gain parametrized by — Rack gain parametrization

Constant (default) | Lookup table

Whether to parametrize the rack gain as a constant value or by using a lookup table.

Rack gain, RckGn — Rack gain

0.062 (default) | scalar

Rack gain, in m/rev, specified as a scalar.

Dependencies

To enable this parameter, set Rack gain parametrized by to Constant.

Rack gain table, RckGnTbl — Rack gain table

ones(1,11)*0.0057*2*pi (default) | 1-by-11 vector

Rack gain table, in m/rev, specified as a 1-by-11 vector.

Dependencies

To enable this parameter, set Rack gain parametrized by to Lookup table.

Steering arm length, StrgArmLngth — Steering arm length

0.1 (default) | scalar

Steering arm length, in m, specified as a scalar.

Rack casing length, RckCsLngth — Rack casing length

0.5 (default) | scalar

Rack casing length, in m, specified as a scalar.

Tie rod length, TieRodLngth — Tie rod length

0.248 (default) | scalar

Tie rod length, l_rod, in m, specified as a scalar.

Distance between axis and rack, Dst — Distance between front axis and rack

0.2 (default) | scalar

Distance between the front axis and rack, D, in m, specified as a scalar.

Efficiency of gears, Epsilon — Efficiency of gears

0.9 (default) | scalar

Efficiency of the gears, ɛ, specified as a scalar.

Pinion inertia, PnInert — Pinion inertia

0.1 (default) | scalar

Pinion inertia, in kg*m², specified as a scalar.

Single Cardan Joint

Spatial angle for the single Cardan joint, SptlAng — Spatial angle for single cardan joint

2.6178 (default) | scalar

Spatial angle for the single cardan joint, in rad, specified as a scalar.

Dependencies

To enable this parameter, set Intermediate shaft type to Single Cardan joint.

Double Cardan Joints

Spatial angle for the upper Cardan joint, Alpha_b — Spatial angle for upper cardan joint

2.6178 (default) | scalar

Spatial angle for the upper cardan joint, in rad, specified as a scalar.

Dependencies

To enable this parameter, set Intermediate shaft type to Double Cardan joints.

Spatial angle for the lower Cardan joints, Alpha_c — Spatial angle for lower cardan joints

2.7051 (default) | scalar

Spatial angle for the lower cardan joints, in rad, specified as a scalar.

Dependencies

To enable this parameter, set Intermediate shaft type to Double Cardan joints.

Edge view angle between the planes of the two joints, Sigma_bc — Edge view angle between planes of the two joints

0.2618 (default) | scalar

Edge view angle between the planes of the two joints, in rad, specified as a scalar.

Dependencies

To enable this parameter, set Intermediate shaft type to Double Cardan joints.

Phase angle, Gamma — Phase angle

0.2618 (default) | scalar

Phase angle, in rad, specified as a scalar.

Dependencies

To enable this parameter, set Intermediate shaft type to Double Cardan joints.

Power Assist

Steering wheel torque breakpoints, TrqBpts — Steering wheel torque breakpoints

[-100 0 100] (default) | 1-by-M vector

Steering wheel torque breakpoints, in N·m, specified as a 1-by-M vector.

Dependencies

To enable this parameter, select Power assist.

Vehicle speed breakpoints, VehSpdBpts — Vehicle speed breakpoints

[0 20] (default) | 1-by-N vector

Vehicle speed breakpoints, in m/s, specified as a 1-by-N vector.

Dependencies

To enable this parameter, select Power assist.

Assisting torque table, TrqTbl — Assisting torque table

[0 -100;0 0;0 100] (default) | M-by-N matrix

Assisting torque table, ƒ_trq, in N·m, specified as an M-by-N matrix.

The torque assist lookup table is a function of the vehicle speed, v, and steering wheel input torque, τ_in:

$${\tau }_{ast}=f_{trq}(v,{\tau }_{in})$$.

The block uses the steering wheel input torque and torque assist to calculate the steering dynamics.

Dependencies

To enable this parameter, select Power assist.

Assisting torque limit, TrqLmt — Assisting torque limit

100 (default) | scalar

Assisting torque limit, in N·m, specified as a scalar.

Dependencies

To enable this parameter, select Power assist.

Assisting power limit, PwrLmt — Assisting power limit

1000 (default) | scalar

Assisting power limit, in N·m/s, specified as a scalar.

Dependencies

To enable this parameter, select Power assist.

Assisting torque efficiency, Eta — Assisting torque efficiency

1 (default) | scalar

Assisting torque efficiency, specified as a scalar.

Dependencies

To enable this parameter, select Power assist.

Cutoff frequency, CutOmege [rad/s] — Cutoff frequency

200 (default) | scalar

Cutoff frequency, in rad/s, specified as a scalar.

Dependencies

To enable this parameter, select Power assist.

Ackerman Steering

Percent Ackerman parametrized by — Ackerman parametrization

Constant (default) | Lookup table

Whether to parametrize the Ackerman values as a constant value or by using a lookup table.

Dependencies

To enable this parameter, select Ackerman steering.

Percent Ackerman, PctAck — Percent Ackerman

100 (default) | scalar

Percent Ackerman, specified as a scalar.

Dependencies

To enable this parameter, select Ackerman steering and set Percent Ackerman parametrized by to Constant.

Percent Ackerman table, PctAckTbl — Percent Ackerman table

ones(1,11)*100 (default)

Percent Ackerman table, specified as a 1-by-11 vector.

Dependencies

To enable this parameter, select Ackerman steering and set Percent Ackerman parametrized by to Lookup table.

Friction and Compliance

Sealing stiffness, SlgStf — Sealing stiffness

1.5e4 (default) | scalar

Sealing stiffness, in N*m/rad, specified as a scalar.

Upper boundary friction, UpprFric — Upper boundary friction

1 (default) | scalar

Upper boundary friction, in N, specified as a scalar.

Pressure change due to friction boundary increase, PrsFric — Pressure change due to friction boundary increase

1e-5 (default) | scalar

Pressure change due to friction boundary increase, in N/bar, specified as a scalar.

Maxwell element stiffness, MaxStf — Maxwell element stiffness

10000 (default) | scalar

Maxwell element stiffness, in Nm/rad, specified as a scalar.

Maxwell element upper boundary friction, MaxUpprFric — Maxwell element upper boundary friction

0.2 (default) | scalar

Maxwell element upper boundary friction, in N, specified as a scalar.

Maxwell linear damping coefficient, MaxDamp — Maxwell linear damping coefficient

1 (default) | scalar

Maxwell linear damping coefficient, specified as a scalar.

Torsion bar stiffness coefficient, TorStf — Torsion bar stiffness coefficient

30 (default) | scalar

Torsion bar stiffness coefficient, in N*m/rad, specified as a scalar.

Torsion bar damping coefficient, TorDamp — Torsion bar damping coefficient

1 (default) | scalar

Torsion bar damping coefficient, in N*m*s/rad, specified as a scalar.

Model Examples

Double Lane Change Reference Application

Simulate a full vehicle dynamics model undergoing a double lane change maneuver standard ISO 3888-2. Use for vehicle dynamics ride and handling analysis and chassis controls development, including yaw stability and lateral acceleration limits.

Open Example

References

[1] Crolla, David, David Foster, et al. Encyclopedia of Automotive Engineering. Volume 4, Part 5 (Chassis Systems) and Part 6 (Electrical and Electronic Systems). Chichester, West Sussex, United Kingdom: John Wiley & Sons Ltd, 2015.

[2] Gillespie, Thomas. Fundamentals of Vehicle Dynamics. Warrendale, PA: Society of Automotive Engineers, 1992.

[3] Vehicle Dynamics Standards Committee. Vehicle Dynamics Terminology. SAE J670. Warrendale, PA: Society of Automotive Engineers, 2008.

Extended Capabilities

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

Version History

Introduced in R2022b