# Kinematic Steering

Kinematic steering for Ackerman, rack-and-pinion, and parallel steering mechanisms

• Library:
• Vehicle Dynamics Blockset / Steering

## Description

The Kinematic Steering block implements a steering model to determine the left and right wheel angles for Ackerman, rack-and-pinion, and parallel steering mechanisms. The block uses the vehicle coordinate system.

To specify the steering type, use the Type parameter.

SettingBlock Implementation

Ackerman

Ideal Ackerman steering. Wheel angles have a common turning circle center.

Rack and pinion

Ideal rack-and-pinion steering. Gears convert the steering rotation into linear motion.

Parallel

Parallel steering. Wheel angles are equal.

To specify the type of data for the steering mechanism, use the Parametrized by parameter.

SettingBlock Implementation

Constant

Steering mechanism uses constant parameter data.

Lookup table

Steering mechanism implements tables for parameter data.

Use the parameter to specify front or rear steering.

SettingImplementation

Front steering

Rear steering

### Steering Types

Ackerman

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

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

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

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

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.

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

The illustration and equations use these variables.

 δin 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

Parallel

For parallel steering, the wheel angles are equal.

To calculate the steering angles, the block uses this equation.

${\delta }_{R}={\delta }_{L}=\frac{{\delta }_{in}}{\gamma }$

The illustration and equations use these variables.

 δin Steering wheel angle δL Left wheel angle δR Right wheel angle γ Steering ratio

## Ports

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### Output

expand all

Bus signal contains this block calculation.

SignalDescriptionVariableUnit

InstStrgRatio

Instantaneous steering ratio

γ

NA

Left wheel angle, δL, in rad.

Right wheel angle, δR, in rad.

## Parameters

expand all

To specify the steering type, use the Type parameter.

SettingBlock Implementation

Ackerman

Ideal Ackerman steering. Wheel angles have a common turning circle center.

Rack and pinion

Ideal rack-and-pinion steering. Gears convert the steering rotation into linear motion.

Parallel

Parallel steering. Wheel angles are equal.

#### Dependencies

This table summarizes the Type and Parametrized by parameter dependencies.

TypeParameterized ByCreates Parameters

Ackerman

Constant

Track width, TrckWdth

Wheel base, WhlBase

Steering range, StrgRng

Steering ratio, StrgRatio

Lookup table

Track width, TrckWdth

Wheel base, WhlBase

Steering range, StrgRng

Steering angle breakpoints, StrgAngBpts

Steering ratio table, StrgRatioTbl

Rack and pinion

Constant

Track width, TrckWdth

Steering range, StrgRng

Steering arm length, StrgArmLngth

Rack casing length, RckCsLngth

Tie rod length, TieRodLngth

Distance between front axis and rack, D

Lookup table

Track width, TrckWdth

Steering range, StrgRng

Steering angle breakpoints, StrgAngBpts

Steering arm length, StrgArmLngth

Rack casing length, RckCsLngth

Tie rod length, TieRodLngth

Distance between front axis and rack, D

ParallelConstant

Steering range, StrgRng

Steering ratio, StrgRatio

Lookup table

Steering range, StrgRng

Steering angle breakpoints, StrgAngBpts

Steering ratio table, StrgRatioTbl

To specify the type of data for the steering mechanism, use the Parametrized by parameter.

SettingBlock Implementation

Constant

Steering mechanism uses constant parameter data.

Lookup table

Steering mechanism implements tables for parameter data.

#### Dependencies

This table summarizes the Type and Parametrized by parameter dependencies.

TypeParameterized ByCreates Parameters

Ackerman

Constant

Track width, TrckWdth

Wheel base, WhlBase

Steering range, StrgRng

Steering ratio, StrgRatio

Lookup table

Track width, TrckWdth

Wheel base, WhlBase

Steering range, StrgRng

Steering angle breakpoints, StrgAngBpts

Steering ratio table, StrgRatioTbl

Rack and pinion

Constant

Track width, TrckWdth

Steering range, StrgRng

Steering arm length, StrgArmLngth

Rack casing length, RckCsLngth

Tie rod length, TieRodLngth

Distance between front axis and rack, D

Lookup table

Track width, TrckWdth

Steering range, StrgRng

Steering angle breakpoints, StrgAngBpts

Steering arm length, StrgArmLngth

Rack casing length, RckCsLngth

Tie rod length, TieRodLngth

Distance between front axis and rack, D

ParallelConstant

Steering range, StrgRng

Steering ratio, StrgRatio

Lookup table

Steering range, StrgRng

Steering angle breakpoints, StrgAngBpts

Steering ratio table, StrgRatioTbl

Use the parameter to specify front or rear steering.

SettingImplementation

Front steering

Rear steering

Factor, NrmFctr, that the block uses to adjust the steering ratio, γ or pinion radius, r. The block can only normalize if you have Parametrized by set to Constant.

Steering TypeNormalization

Ackerman

Parallel

Block updates the Steering ratio, StrgRatio parameter to the normalized value, γnrm, specified by this equation.

${\gamma }_{nrm}=\frac{1}{Nr{m}_{Fctr}}$

Rack and pinion

General

Track width, TW, in m.

#### Dependencies

To create this parameter, set Type to Ackerman or Rack and pinion.

Wheel base, WB, in m.

#### Dependencies

To create this parameter, set Type to Ackerman.

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

Steering ratio, γ, dimensionless.

#### Dependencies

To create this parameter:

• Set Type to Ackerman or Parallel.

• Set Parametrized by to Constant.

#### Dependencies

To create this parameter, set Parametrized by to Lookup table.

Steering ratio table, γ, dimensionless.

#### Dependencies

To create this parameter:

• Set Type to Ackerman or Parallel.

• Set Parametrized by to Lookup table.

Rack-and-Pinion

Steering arm length, larm, in m.

#### Dependencies

To create this parameter, set Type to Rack and pinion.

Rack casing length, lrack, in m.

#### Dependencies

To create this parameter, set Type to Rack and pinion.

Tie rod length, lrod, in m.

#### Dependencies

To create this parameter, set Type to Rack and pinion.

Distance between front axis and rack, D, in m.

#### Dependencies

To create this parameter, set Type to Rack and pinion.

#### Dependencies

To create this parameter:

• Set Type to Rack and pinion.

• Set Parametrized by to Constant.

Pinion radius table, r, in m.

#### Dependencies

To create this parameter:

• Set Type to Rack and pinion.

• Set Parametrized by to Lookup table.

## 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.