# 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, adjusted by percentage Ackerman. 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 ideal wheel angles, the block uses these equations.

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

After the block calculates the ideal wheel angles, it uses the Ackerman percentage to adjust the outside wheel angle.

`${\delta }_{o}={\delta }_{i}-{p}_{Ack}\left({\delta }_{i}-{\delta }_{Ack}\right)$`

The outside wheel angle depends on the turn direction.

• Right turn

• Outside angle, δo, is left wheel angle, δL

• Inside angle, δi, is right wheel angle, δR

• Left turn

• Outside angle, δo, is right wheel angle, δR

• Inside angle, δi, is left wheel angle, δL

The illustration and equations use these variables.

 δin Steering angle δL Left wheel angle δR Right wheel angle δo Outside wheel angle δi Inside wheel angle pAck Ackerman percentage 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

### Input

expand all

Use the parameter to specify a steering angle range. By default, the value is set to 1.25*pi, which limits the steering angle to a range of -1.25*pi to 1.25*pi.

Ackerman percentage, δin, in percent.

#### Dependencies

To create this input port:

• Set Type to `Ackerman`.

• On the Ackerman Steering pane, select Input percent Ackerman.

### 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.

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 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 wheel angle input range, in rad. The block limits the steering wheel input 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```.

Ackerman Steering

Select to create `PctAckIn` input port.

#### Dependencies

To enable this parameter, set Type to `Ackerman`.

Constant value of percent Ackerman, in percent.

#### Dependencies

To enable this parameter:

• Set Type to `Ackerman`

• Set Parametrized by to `Constant`

• Clear Input Percent Ackerman

Table of percent Ackerman values as a function of the steering angle, δin, in percent.

#### Dependencies

To enable this parameter:

• Set Type to `Ackerman`

• Set Parametrized by to `Constant`

• Clear Input Percent Ackerman

 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.

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

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