Kinematic Steering

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

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Libraries:
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.

Setting	Block 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.

Setting	Block Implementation
Constant	Steering mechanism uses constant parameter data.
Lookup table	Steering mechanism implements tables for parameter data.

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 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.

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

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

$${\delta }_o={\delta }_i-p_{Ack}({\delta }_i-{\delta }_{Ack})$$

The outside wheel angle depends on the turn direction.

• Right turn

  • Outside angle, δ_o, is left wheel angle, δ_L

  • Inside angle, δ_i, is right wheel angle, δ_R

• Left turn

  • Outside angle, δ_o, is right wheel angle, δ_R

  • Inside angle, δ_i, is left wheel angle, δ_L

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.

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

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

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

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

AngIn — Steering angle

scalar

Steering angle, δ_in, in rad.

Use the Steering range, StrgRng 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.

PctAckIn — Ackerman percentage

scalar

Ackerman percentage, δ_in, in percent.

Dependencies

To create this input port:

• Set Type to Ackerman.

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

Output

Info — Bus signal

bus

Bus signal contains this block calculation.

Signal	Description	Variable	Unit
InstStrgRatio	Instantaneous steering ratio	γ	NA

AngLft — Left wheel angle

scalar

Left wheel angle, δ_L, in rad.

AngRght — Right wheel angle

scalar

Right wheel angle, δ_R, in rad.

Parameters

Type — Select steering type

Ackerman (default) | Rack and pinion | Parallel

To specify the steering type, use the Type parameter.

Setting	Block 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.

Parametrized by — Select parameterization

Constant (default) | Lookup table

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

Setting	Block Implementation
Constant	Steering mechanism uses constant parameter data.
Lookup table	Steering mechanism implements tables for parameter data.

Location — Select location

Front (default) | Rear

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

Normalization factor, NrmlFctr — Adjust the steering angle

scalar

Factor, Nrm_Fctr, 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.

To adjust the steering ratio or pinion radius, click Normalize.

Steering Type	Normalization
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	Block updates the Pinion radius, PnnRadius parameter to using the normalization factor, NrmFctr.

General

Track width, TrckWdth — Width

1 (default) | scalar

Track width, TW, in m.

Dependencies

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

Wheel base, WhlBase — Base

1.524 (default) | scalar

Wheel base, WB, in m.

Dependencies

To create this parameter, set Type to Ackerman.

Deadband, Db — Deadband

0 (default) | scalar

Deadband steering angle before pinion engages the gear, in rad.

Steering range, StrgRng — Steering wheel angle input range

1.25*pi (default) | scalar

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

Steering ratio, StrgRatio — Ratio

100 (default) | scalar

Steering ratio, γ, dimensionless.

Dependencies

To create this parameter:

• Set Type to Ackerman or Parallel.

• Set Parametrized by to Constant.

Steering angle breakpoints, StrgAngBpts — Breakpoints

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

Steering angle breakpoints, in rad.

Dependencies

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

Steering ratio table, StrgRatioTbl — Table

[13.5000 13.3750 13.2500 13.1250 13.0000 13.0000 13.0000 13.1250 13.2500 13.3750 13.5000] (default) | vector

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, StrgArmLngth — Length

0.1 (default) | scalar

Steering arm length, l_arm, in m.

Dependencies

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

Rack casing length, RckCsLngth — Length

0.5 (default) | scalar

Rack casing length, l_rack, in m.

Dependencies

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

Tie rod length, TieRodLngth — Length

0.248 (default) | scalar

Tie rod length, l_rod, in m.

Dependencies

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

Distance between axis and rack, D — Distance

0.2 (default) | scalar

Distance between axis and rack, D, in m.

Dependencies

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

Pinion radius, PnnRadius — Radius

0.0057 (default) | scalar

Pinion radius, r, in m.

Dependencies

To create this parameter:

• Set Type to Rack and pinion.

• Set Parametrized by to Constant.

Pinion radius table, PnnRadiusTbl — Table

[0.0055 0.0055 0.0056 0.0057 0.0057 0.0057 0.0058 0.0057 0.0056 0.0055 0.0055] (default) | vector

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

Input Percent Ackerman — Create PctAckIn input port

off (default) | on

Select to create PctAckIn input port.

Dependencies

To enable this parameter, set Type to Ackerman.

Percent Ackerman, PctAck — Percent Ackerman constant

100 (default) | scalar

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

Percent Ackerman table, PctAckTbl — Percent Ackerman table

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

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

Dependencies

To enable this parameter:

• Set Type to Ackerman

• Set Parametrized by to Constant

• Clear Input Percent Ackerman

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 R2018a