Dynamic Steering

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

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Description

The Dynamic Steering block implements dynamic steering to calculate the wheel angles for Ackerman, rack-and-pinion, and parallel steering mechanisms. The block uses the steering wheel input torque, right wheel torque, and left wheel torque to calculate the wheel angles. The block uses the vehicle coordinate system.

If you select Power assist, 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.

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.

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 Location parameter to specify front or rear steering.

Setting	Implementation
Front	Front steering
Rear	Rear steering

Dynamics

To calculate the steering dynamics, the Dynamic Steering block models the steering wheel, shaft, steering mechanism, hysteresis, and, optionally, power assist.

Calculation	Equations
Steering column and steering shaft dynamics	$$J_1{\ddot{\theta }}_1={\tau }_{in}-b_2{\dot{\theta }}_1-{\tau }_{hys}$$ $$J_2{\ddot{\theta }}_2={\tau }_{eq}-b_3{\dot{\theta }}_2+{\tau }_{hys}-{\tau }_{fric}$$
Hysteresis spring damper	$$\begin{array}{l}\delta ={\theta }_1-{\theta }_2\\ \Delta \delta ={\delta }_{current}-{\delta }_{previous}\\ \\ {\tau }_{hys}=\left(b_1\dot{\delta }-k_1\delta \right)\left(1+\exp \left(-\frac{\left|\Delta \delta \right|}{\beta }\right)\right)\\ \\ \beta =\{\begin{array}{c}{\beta }_u\text{ when }\delta >0\\ {\beta }_l\text{ when }\delta \le 0\end{array}\end{array}$$
Optional power assist	$$\begin{array}{l}{\tau }_{ast}=f_{trq}(v,{\tau }_{in})\\ \\ J_1{\ddot{\theta }}_1={\tau }_{in}+{\tau }_{ast}-b_2{\dot{\theta }}_1-{\tau }_{hys}\\ J_2{\ddot{\theta }}_2={\tau }_{eq}+{\tau }_{ast}-b_3{\dot{\theta }}_2+{\tau }_{hys}-{\tau }_{fric}\end{array}$$

The illustration and equations use these variables.

J1	Steering wheel inertia
J2	Steering mechanism inertia
$${\theta }_1,{\dot{\theta }}_1,{\ddot{\theta }}_1$$	Steering wheel angle, angular velocity, and angular acceleration, respectively
$${\theta }_2,{\dot{\theta }}_2,{\ddot{\theta }}_2$$	Shaft angle, angular velocity, and angular acceleration, respectively
b1, k1	Hysteresis spring and viscous damping coefficients, respectively
b2	Steering wheel viscous damping coefficient
b3	Steering mechanism damping coefficient
τhys	Hysteresis spring damping torque
τfric	Steering mechanism friction torque
τeq	Wheel equivalent torque
τast	Torque assist
βu , βl	Upper and lower hysteresis modifiers, respectively
v	Vehicle speed
ƒtrq	Torque assist lookup table

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.

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

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.

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

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

Ports

Input

TrqIn — Torque

scalar

Torque, τ_in, in N·m.

TrqLft — Left wheel torque

scalar

Left wheel torque, τ_L, in N·m.

TrqRght — Right wheel torque

scalar

Right wheel torque, τ_R, in N·m.

VehSpd — Vehicle speed

scalar

Vehicle speed, v, in m/s.

Dependencies

To create VehSpd, select Power assist.

Output

Info — Bus signal

bus

Bus signal contains these block calculations.

Signal	Description	Unit
StrgWhlAng	Steering wheel angle	rad
StrgWhlSpd	Steering wheel angular velocity	rad/s
ShftAng	Shaft angle	rad
ShftSpd	Shaft angular velocity	rad/s
AngLft	Left wheel angle	rad
SpdLft	Left wheel angular velocity	rad/s
AngRght	Right wheel angle	rad
SpdRght	Right wheel angular velocity	rad/s
TrqAst	Torque assist	N·m
PwrAst	Power assist	W
PwrLoss	Power loss	W
InstStrgRatio	Instantaneous steering ratio	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

Rack and pinion (default) | Ackerman | 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.

Dependencies

This table summarizes the Type and Parametrized by parameter dependencies.

Type	Parameterized By	Creates 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 Pinion radius, PnnRadius
	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 Pinion radius, PnnRadiusTbl
Parallel	Constant	Steering range, StrgRng Steering ratio, StrgRatio
	Lookup table	Steering range, StrgRng Steering angle breakpoints, StrgAngBpts Steering ratio table, StrgRatioTbl

Parametrized by — Select parameterization

Lookup table (default) | Constant

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.

Dependencies

This table summarizes the Type and Parametrized by parameter dependencies.

Type	Parameterized By	Creates 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 Pinion radius, PnnRadius
	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 Pinion radius, PnnRadiusTbl
Parallel	Constant	Steering range, StrgRng Steering ratio, StrgRatio
	Lookup table	Steering range, StrgRng Steering angle breakpoints, StrgAngBpts Steering ratio table, StrgRatioTbl

Power assist — Specify power assist

on (default) | off

If you select Power assist, you can specify a torque assist lookup table, ƒ_trq, that is a function of the vehicle speed, v, and steering wheel input torque, τ_in.

$${\tau }_{ast}=f_{trq}(v,{\tau }_{in})$$

The block uses the steering wheel input torque and torque assist to calculate the steering dynamics.

Dependencies

Selecting Power assist creates the VehSpd input port and these parameters.

Power Assist	Parameters
on	Steering wheel torque breakpoints, TrqBpts Vehicle speed breakpoints, VehSpdBpts Assisting torque table, TrqTbl Assisting torque limit, TrqLmt Assisting power limit, PwrLmt Assisting torque efficiency, Eta Cutoff frequency, omega_c

Location — Select location

Front (default) | Rear

Use the Location parameter to specify front or rear steering.

Setting	Implementation
Front	Front steering
Rear	Rear steering

General

Track width, TrckWdth — Width

1 | 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.

Steering range, StrgRng — Range

1.25*pi (default) | scalar

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

Steering ratio, StrgRatio — Ratio

13.5 (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 front axis and rack, D — Distance

0.2 (default) | scalar

Distance between front 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.

Dynamics

Steering wheel inertia, J1 — Inertia

0.1 (default) | scalar

Steering wheel inertia, J₁, in kg*m^2.

Steering mechanism inertia, J2 — Inertia

0.01 (default) | scalar

Steering mechanism inertia, J₂, in kg*m^2.

Upper hysteresis modifier, beta_u — Upper hysteresis modifier

0.1 (default) | scalar

Upper hysteresis modifier, β_u, dimensionless.

Lower hysteresis modifier, beta_l — Lower hysteresis modifier

0.1 (default) | scalar

Lower hysteresis modifier, β_l, dimensionless.

Hysteresis viscous damping, b1 — Damping

0.001 (default) | scalar

Hysteresis damping, b₁, in N·m·s/rad.

Hysteresis stiffness, k1 — Stiffness

30 (default) | scalar

Hysteresis stiffness, k₁,in N·m/rad.

Steering wheel damping, b2 — Damping

1 (default) | scalar

Steering wheel damping, b₂, in N·m·s/rad.

Steering mechanism damping, b3 — Damping

0.001 (default) | scalar

Steering mechanism damping, b₃, in N·m·s/rad.

Initial steering angle, theta_o — Angle

0 (default) | scalar

Initial steering angle, θ₀, in rad.

Initial steering angular velocity, omega_o — Angular velocity

0 (default) | scalar

Initial steering angular velocity, ω_o, in rad/s.

Friction torque, FricTrq — Torque

0 (default) | scalar

Friction torque, τ_fric, in N·m.

Power Assist

Steering wheel torque breakpoints, TrqBpts — Breakpoints

[-100 0 100] (default) | 1-by-M vector

Steering wheel torque breakpoints, in N·m.

Dependencies

Selecting Power assist creates the VehSpd input port and these parameters.

Power Assist	Parameters
on	Steering wheel torque breakpoints, TrqBpts Vehicle speed breakpoints, VehSpdBpts Assisting torque table, TrqTbl Assisting torque limit, TrqLmt Assisting power limit, PwrLmt Assisting torque efficiency, Eta Cutoff frequency, omega_c

Vehicle speed breakpoints, VehSpdBpts — Breakpoints

[0 20] (default) | 1-by-N vector

Vehicle speed breakpoints, in m/s.

Dependencies

Selecting Power assist creates the VehSpd input port and these parameters.

Power Assist	Parameters
on	Steering wheel torque breakpoints, TrqBpts Vehicle speed breakpoints, VehSpdBpts Assisting torque table, TrqTbl Assisting torque limit, TrqLmt Assisting power limit, PwrLmt Assisting torque efficiency, Eta Cutoff frequency, omega_c

Assisting torque table, TrqTbl — 2D torque table

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

Assisting torque table, ƒ_trq, in N·m.

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 uses the steering wheel input torque and torque assist to calculate the steering dynamics.

Dependencies

Selecting Power assist creates the VehSpd input port and these parameters.

Power Assist	Parameters
on	Steering wheel torque breakpoints, TrqBpts Vehicle speed breakpoints, VehSpdBpts Assisting torque table, TrqTbl Assisting torque limit, TrqLmt Assisting power limit, PwrLmt Assisting torque efficiency, Eta Cutoff frequency, omega_c

Assisting torque limit, TrqLmt — Torque limit

100 (default) | scalar

Assisting torque limit, in N·m.

Dependencies

Selecting Power assist creates the VehSpd input port and these parameters.

Power Assist	Parameters
on	Steering wheel torque breakpoints, TrqBpts Vehicle speed breakpoints, VehSpdBpts Assisting torque table, TrqTbl Assisting torque limit, TrqLmt Assisting power limit, PwrLmt Assisting torque efficiency, Eta Cutoff frequency, omega_c

Assisting power limit, PwrLmt — Power limit

1000 (default) | scalar

Assisting power limit, in N·m/s.

Dependencies

Selecting Power assist creates the VehSpd input port and these parameters.

Power Assist	Parameters
on	Steering wheel torque breakpoints, TrqBpts Vehicle speed breakpoints, VehSpdBpts Assisting torque table, TrqTbl Assisting torque limit, TrqLmt Assisting power limit, PwrLmt Assisting torque efficiency, Eta Cutoff frequency, omega_c

Assisting torque efficiency, Eta — Efficiency

1 (default) | scalar

Assisting torque efficiency, dimensionless.

Dependencies

Selecting Power assist creates the VehSpd input port and these parameters.

Power Assist	Parameters
on	Steering wheel torque breakpoints, TrqBpts Vehicle speed breakpoints, VehSpdBpts Assisting torque table, TrqTbl Assisting torque limit, TrqLmt Assisting power limit, PwrLmt Assisting torque efficiency, Eta Cutoff frequency, omega_c

Cutoff frequency, omega_c — Cutoff frequency

200 (default) | scalar

Cutoff frequency, in rad/s.

Dependencies

Selecting Power assist creates the VehSpd input port and these parameters.

Power Assist	Parameters
on	Steering wheel torque breakpoints, TrqBpts Vehicle speed breakpoints, VehSpdBpts Assisting torque table, TrqTbl Assisting torque limit, TrqLmt Assisting power limit, PwrLmt Assisting torque efficiency, Eta Cutoff frequency, omega_c

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
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Version History

Introduced in R2018a