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# Steering System

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

Since R2022b

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

SettingImplementation

Front axle steering

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.

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

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:

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

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

## Ports

### Input

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

Wheel caster angle, τL, 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.

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.

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

#### Dependencies

To enable this port, select Steering inputs > Angle.

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

#### Dependencies

To enable this port, select Steering inputs > Torque.

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.

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.

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:

DescriptionUnit
x-directional Force, LeftN
x-directional Force, RightN
y-directional Force, LeftN
y-directional Force, RightN
z-directional Force, LeftN
z-directional Force, RightN
x-directional Moment, LeftN*m
x-directional Moment, RightN*m
y-directional Moment, LeftN*m
y-directional Moment, RightN*m
z-directional Moment, LeftN*m
z-directional Moment, RightN*m

#### Dependencies

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

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

#### Dependencies

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

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

#### Dependencies

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

### Output

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

SignalDescriptionUnit

`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

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

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

## Parameters

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Block Options

Set the steering type for the steering system.

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

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

#### Dependencies

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

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

Select this parameter to enable the PwrAstTrq port.

Select this parameter to enable the PctAck port.

Select this parameter to enable the LftKpM and RghtKpM ports.

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

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

General

Track width, TW, in m, specified as a 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 wheel deadband angle, in radians, from left-engagement to right-engagement.

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

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

Kingpin offset, in m, specified as a scalar.

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

Hub lead, in m, specified as a scalar.

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

Overall steering ratio, specified as a scalar.

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

Caster angle, in rad, specified as a scalar.

#### Dependencies

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

Rack and Pinion

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

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

#### Dependencies

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

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, in m, specified as a scalar.

Rack length (distance between inner tie rod ends), in m, specified as a scalar.

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

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

Efficiency of the rack and pinion mechanism, ɛ, specified as a scalar.

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

Single Cardan Joint

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, 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, 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, in rad, specified as a scalar.

#### Dependencies

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

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, in N·m, specified as a 1-by-M vector.

#### Dependencies

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

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

#### Dependencies

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

Torque assist table, ƒtrq, 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, τin:

${\tau }_{ast}={f}_{trq}\left(v,{\tau }_{in}\right)$.

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.

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

#### Dependencies

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

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

#### Dependencies

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

Torque assist efficiency, specified as a scalar.

#### Dependencies

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

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

#### Dependencies

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

Ackerman Steering

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

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

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

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

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

Maxwell linear damping coefficient, specified as a scalar.

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

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

Steering Torque Controller

Integral anti-windup gain, specified as a scalar.

#### Dependencies

To enable this parameter, select Steering inputs > Angle.

Derivative gain, specified as a scalar.

#### Dependencies

To enable this parameter, select Steering inputs > Angle.

Integral gain, specified as a scalar.

#### Dependencies

To enable this parameter, select Steering inputs > Angle.

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, specified as a scalar, in Nm.

#### Dependencies

To enable this parameter, select Steering inputs > Angle.

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.

## Version History

Introduced in R2022b

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