Steering System

Steering system for Ackermann and rack-and-pinion steering mechanisms

Since R2022b

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Libraries:
Vehicle Dynamics Blockset / Steering

Description

The Steering System block implements dynamic steering to calculate the wheel steer angles for rack-and-pinion mechanisms with friction^[4], compliance, and Ackermann steering features. The block uses the steering wheel input angle or torque input, vehicle speed, caster angle, and right and left wheel feedbacks to calculate the wheel steer 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 percentage Ackermann values to calculate the Ackermann steering effects, or a constant Ackermann percentage, where 100 percent means perfect Ackermann steering.

If you select the Power assist, Ackerman steering, or Kingpin moment^[5] parameters in the Input signals section, you can specify additional inputs for the external power assist torques, percent Ackermann values, or kingpin moments.

Use the Steered axle parameter to specify whether the front or rear axle is steered.

Setting	Implementation
Front	Front axle steering
Rear	Rear axle steering

Steering

Rack-and-Pinion

For rack-and-pinion steering, pinion rotation causes linear motion of the rack, which steers the wheels through the tie rods and steering arms.

To calculate the steered wheel 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	Pinion angle (steering shaft angle into pinion)
δL	Left wheel steer angle
δR	Right wheel steer angle
TW	Track width
r	Pinion radius
ΔP	Linear change in rack position from "straight ahead" position
D	Longitudinal distance between rack and steered axle
lrack	Rack length (distance between inner tie-rod ends)
larm	Steering arm length
lrod	Tie rod length

Ackermann Steering

For 100% (ideal) Ackermann steering, all wheels follow circular arcs with the same center point.

To calculate the steered wheel angles, the Steering System block uses these equations:

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

This table defines variables used in the equations:

δin	Pinion angle (steering shaft angle into pinion)
δL	Left wheel steer angle
δR	Right wheel steer angle
δAck	Ackermann steer angle
TW	Track width
WB	Wheel base
γ	Steering ratio: Ratio of pinion angle to Ackermann angle

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 Script

Ports

Input

VehSpd — Vehicle speed

scalar

Vehicle speed, v, in m/s, specified as a scalar. This is the magnitude of the vehicle CG longitudinal velocity vector.

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 is the angle for the left wheel and the second is the angle for the right wheel.

Dependencies

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

WhlAngFdk — Wheel steer angle feedback

1-by-2 vector

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

Dependencies

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

AngIn — Steering wheel angle input

scalar

Steering wheel angle input from driver, in radians, specified as a scalar.

Dependencies

To enable this port, select Steering inputs > Angle.

TrqIn — Steering torque input

scalar

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

Dependencies

To enable this port, select Steering inputs > Torque.

PwrAstTrq — Power assist torque

scalar

The torque value of power assist on the steering shaft, in N*m, specified as a scalar. The value is supplied externally into this port.

Dependencies

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

PctAck — Percent Ackermann value

scalar

The Ackermann value in percent, specified as a scalar. The value is supplied externally into this port.

Dependencies

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

TireFdk — Tire forces and moments feedback

1-by-12 vector

Tire forces and moments feedback from both right and left tires, specified as a 1-by-12 vector that contains the following values, in the order specified in this table:

Description	Unit
x-directional Force, Left	N
x-directional Force, Right	N
y-directional Force, Left	N
y-directional Force, Right	N
z-directional Force, Left	N
z-directional Force, Right	N
x-directional Moment, Left	N*m
x-directional Moment, Right	N*m
y-directional Moment, Left	N*m
y-directional Moment, Right	N*m
z-directional Moment, Left	N*m
z-directional Moment, Right	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 these signals:

Signal	Description	Unit
StrWhlAngFdk	Steering wheel angle	rad
AstTrq	Torque applied by power assist	N·m
AstPwr	Power applied by power assist	W
LftTieRodForce	Axial force in left tie rod	N
RghtTieRodForce	Axial force in right tie rod	N
LftKpM	Left kingpin moment	N·m
RghtKpM	Right kingpin moment	N·m
LftWhlAng	Left wheel steer angle	rad
RghtWhlAng	Right wheel steer angle	rad
LftWhlSpd	Left wheel steer angle velocity	rad/s
RghtWhlSpd	Right wheel steer angle velocity	rad/s
TrqIn	Torque applied by driver on steering wheel	N·m

AngLft — Left wheel steer angle

scalar

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

AngRght — Right wheel steer angle

scalar

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

Parameters

Block Options

Type — Steering type

Rack and pinion (default)

Set the steering type for the steering system.

Intermediate shaft type — Intermediate shaft type

Single Cardan joint (default) | Double Cardan joints

Select whether to model the intermediate shaft type using single or double Cardan joints.

Power assist — Option to model power assist

on (default) | off

Select this parameter to model power assist within the steering system.

Dependencies

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

Ackerman steering — Option to use Ackermann steering

on (default) | off

Select this parameter to set the Ackermann steering percentage within the Steering System block.

Dependencies

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

Input Signals

Power assist — External power assist torque

off (default) | on

Select this parameter to enable the PwrAstTrq port.

Ackerman steering — External Ackermann percent

off (default) | on

Select this parameter to enable the PctAck port.

Kingpin moment — Left and right kingpin moments

off (default) | on

Select this parameter to enable the LftKpM and RghtKpM ports.

Steered axle — Location of steering system

Front (default) | Rear

Select either the front or rear axle as the location of the Steering System.

Steering inputs — Steering wheel angle or torque

Angle (default) | Torque

Select either the AngIn or TrqIn port as the driver input.

General

Track width, TrckWdth — Track width

1 (default) | scalar

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

Steering angle input range, StrRng — Steering wheel range

1.25*pi (default) | scalar

Steering wheel angle from straight-ahead to either left or right lock, in rad, specified as a scalar. This parameter causes both steered wheels to remain within their designed steering range.

Steering angle input deadband width, DbWdth — Steering angle input deadband width

0 (default) | scalar

Steering wheel deadband angle, in radians, from left-engagement to right-engagement.

Steering wheel inertia, StrWhlInert — Steering wheel inertia

0.1 (default) | scalar

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

Steering shaft inertia, StrColInert — Inertia of steering shaft

0.01 (default) | scalar

Steering shaft moment of 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

0.2094 (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 tire radius, in m, specified as a scalar.

Overall steering ratio, OvrlStrRatio — Overall steering ratio

17.42 (default) | scalar

Overall steering ratio, specified as a scalar.

Steering angle breakpoints, StrgAngBpts — Steering wheel angle breakpoints

[-6.2832 -5.0265 -3.7699 -2.5133 -1.2566 0 1.2566 2.5133 3.7699 5.0265 6.2832] (default) | 1-by-11 vector

Steering wheel angle breakpoints, in rad, specified as a 1-by-11 vector. These breakpoints are used to parameterize either Ackermann value or rack gain.

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

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 parameterized by — Rack gain parameterization

Constant (default) | Lookup table

Whether to parameterize 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 parameterized 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 parameterized 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 length

0.5 (default) | scalar

Rack length (distance between inner tie rod ends), 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 axle and rack

0.2 (default) | scalar

Longitudinal distance between the steered axle and rack centerline, D, in m, specified as a scalar.

Efficiency of gears, Epsilon — Efficiency of gears

0.9 (default) | scalar

Efficiency of the rack and pinion mechanism, 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 the 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 joint, Alpha_c — Spatial angle for lower Cardan joint

2.7051 (default) | scalar

Spatial angle for the lower Cardan joint, 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

Rotation phase angle between the two joints, 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 the Power assist Block Option.

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 the Power assist Block Option.

Assisting torque table, TrqTbl — Torque assist table

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

Torque assist 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 applies the steering wheel input torque and torque assist to the pinion.

Dependencies

To enable this parameter, select the Power assist Block Option.

Assisting torque limit, TrqLmt — Torque assist limit

100 (default) | scalar

Torque assist limit, in N·m, specified as a scalar.

Dependencies

To enable this parameter, select the Power assist Block Option.

Assisting power limit, PwrLmt — Assist power limit

1000 (default) | scalar

Assist power limit, in watts, specified as a scalar.

Dependencies

To enable this parameter, select the Power assist Block Option.

Assisting torque efficiency, Eta — Torque assist efficiency

1 (default) | scalar

Torque assist efficiency, specified as a scalar.

Dependencies

To enable this parameter, select the Power assist Block Option.

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 the Power assist Block Option.

Ackerman Steering

Percent Ackerman parameterized by — Ackermann parameterization

Constant (default) | Lookup table

Whether to parameterize the Ackermann values as a constant value or by a lookup table.

Dependencies

To enable this parameter, select the Ackerman steering Block Option.

Percent Ackerman, PctAck — Percent Ackermann

100 (default) | scalar

Percent Ackermann, specified as a scalar.

Dependencies

To enable this parameter, select the Ackerman steering Block Option and set Percent Ackerman parameterized by to Constant.

Percent Ackerman table, PctAckTbl — Percent Ackermann table

ones(1,11)*100 (default) | 1-by-11 vector

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

Dependencies

To enable this parameter, select the Ackerman steering Block Option and set Percent Ackerman parameterized 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 N*m/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 — Torsional stiffness of steering shaft

30 (default) | scalar

Torsional stiffness of steering shaft, in N*m/rad, specified as a scalar.

Torsion bar damping coefficient, TorDamp — Torsional viscous damping in steering shaft

1 (default) | scalar

Torsional viscous damping in steering shaft, in N*m*s/rad, specified as a scalar.

Steering Torque Controller

Integral anti-windup gain, A [] — Integral anti-windup gain

0 (default) | scalar

Integral anti-windup gain, specified as a scalar.

Dependencies

To enable this parameter, select Steering inputs > Angle.

Derivative gain, D [] — Derivative gain

0.1 (default) | scalar

Derivative gain, specified as a scalar.

Dependencies

To enable this parameter, select Steering inputs > Angle.

Integral gain, I [] — Integral gain

5 (default) | scalar

Integral gain, specified as a scalar.

Dependencies

To enable this parameter, select Steering inputs > Angle.

Steering torque output saturation lower limit, OutSatLowLmt [Nm] — Output saturation lower limit

-500 (default) | scalar

Steering torque output saturation lower limit, specified as a scalar, in Nm.

Dependencies

To enable this parameter, select Steering inputs > Angle.

Steering torque output saturation upper limit, OutSatUpLmt [Nm] — Steering torque output saturation upper limit

500 (default) | scalar

Steering torque output saturation upper limit, specified as a scalar, in Nm.

Dependencies

To enable this parameter, select Steering inputs > Angle.

Proportional gain, P [] — Proportional gain

150 (default) | scalar

Proportional gain, specified as a scalar.

Dependencies

To enable this parameter, select Steering inputs > Angle.

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.

[4] Pfeffer, P. E., M. Harrer, and D. N. Johnston. “Interaction of Vehicle and Steering System Regarding On-Centre Handling.” Vehicle System Dynamics 46, no. 5 (May 2008): 413–28. https://doi.org/10.1080/00423110701416519.

[5] Reimpell, Jörnsen, Helmut Stoll, and Jürgen W. Betzler. The Automotive Chassis: Engineering Principles Chassis and Vehicle Overall, Wheel Suspensions and Types of Drive, Axle Kinematics and Elastokinematics, Steering, Springing, Tyres, Construction and Calculations Advice. 2nd ed. Oxford: Butterworth Heinemann, 2001.

Extended Capabilities

C/C++ Code Generation
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Version History

Introduced in R2022b

R2024a: New steering torque controller parameters

Starting in R2024a, the Steering System block includes these parameters that you can specify to define a steering torque controller:

• Integral antiwindup gain

• Derivative gain

• Integral gain

• Output saturation lower limit

• Output saturation upper limit

• Proportional gain