Two to Four: Bicycle model for Car
Lateral dynamics
For the control of lateral motion of the vehicle, it is critical to model the lateral dynamics. Lateral motion control is used in various ADAS features like lane centering, lane keeping etc. The lateral dynamics models are also imperative in modeling vehicle behaviours during the lateral manoeuvres and can be used to study and design the system and components.
In order to calculate accelerations and velocities in different directions of interest, it is necessary to define the axis systems. Generally axis system can be Earth fixed or Vehicle fixed.
Earth Axis System
Earth fixed axis system has X,Y (horizontal) and Z (vertical) as principal directions in global coordinates. The following figure shows a vehicle location and other properties like heading angle in the earth or global coordinates.
Vehicle Coordinate System
This is fixed to the vehicle. X is longitudinal direction, Y is lateral, Z is vertical. Origin is at CG of the vehicle. The rotational motion along vehicle X is called roll.
Bicycle Model
A lot of handling vehicle dynamics models are available of various complexities and accuracy. One of the simplest and most commonly used model is the bicycle model. The term bicycle is because both the front wheels are taken as single entity and also both the rear wheels making it a two wheel
model.
The term “bicycle model” is however a misnomer. It is not used to model bicycle handling dynamics. In fact, it would be a really bad model for bicycle dynamics as it doesn’t account for roll.
Kinematic bicycle model
(x, y) is the vehicle’s center of mass. ψ is the current heading of the car (heading angle) and v is the speed of the vehicle. lf, lr is the distance from the center of the mass to the front and rear axle respectively. β is the angle of v with respect to the car axis(sideslip angle of the vehicle as we are taking it at CG). The front wheel steering angle is δf and the car acceleration is a. For simplicity, we assume it is a front-wheel-drive car and we will write δf as δ for now.
The equations now are:
x_dot = v * cos(𝛽 +ψ )
y_dot = v* sin(𝛽 + ψ)
ψ_dot = v*sin(𝛽) /lr
v_dot = a
L = lf+lr
tan(𝛽) = lr * tan(𝛿)/ L
Once the equations are presented the time rate of change equations can be presented in the form of four state variables [x,y,ψ,v] as:
x(t+1)=x(t)+x_dot*dt
y(t+1)=y(t)+y_dot*dt
ψ(t+1)=ψ(t)+ψ_dot*dt
v(t+1)=v(t)+v_dot∗dt
Here the input variables are acceleration a (which can be negative for deceleration) and the steering angle δ
Kinematic model works is a very simple model and can work well in some controls problem. It is computationally inexpensive and also easy to parameterise- so portable
Dynamic bicycle model
The next model in the fidelity chain is a dynamic bicycle model. Now instead of body side slip angles, we would assume tire slip angles (angle from tire speed to tire orientation). The analysis also would be in vehicle coordinates rather than global. Also the longitudinal velocity of the vehicle is u and lateral velocity is v. The yaw rate (rate of change of angular velocity around Z axis) is r and δf is the front steering angle.
Before we can proceed to derive the dynamic equations,
Understeer, Oversteer
The simple geometry would yield the front and rear slip angles as:
where αf and αr are the front and rear sideslip angles. Please note that for above we have assume the vehicle to be front steered only and the assumption of small angle holds such that tan(angle) = angle
The above equations can be re written as
This equation implies that the steering angle required to make the turn consists of two parts the static part L/R is known as ackermann angle and the dynamic part which is the difference between front and rear tire slip angles.If the front tire slip angle is larger than the rear slip angle, this condition is termed understeer. This implies that the steering angle must be larger than the Ackermann angle to maintain a constant radius turn at nonzero speed. If the rear slip angle is greater than the front slip angle, the front steering angle is less than the Ackermann angle; a condition termed oversteer. Finally if the front and rear slip angles are equal, the steering angle is equal to the Ackermann angle and the condition is termed neutral steer. The understeer and oversteer of the vehicle are some the most important characteristics that define the static stability of the vehicle.
Cornering Stiffness
The side forces or cornering forces acting at the vehicle tires, Fyf and Fyr, are known to be related to the tire slip angles.
Although the relationship between tire slip angle and lateral force is non linear as shown in the figure,it can be assumed to be linear for small angles.
Where Cα is called Cornering Stiffness. The cornering stiffness depends on many tire geometric and material properties, and for a given tire it depends on the vertical load, Fz, and the inflation pressure. Also for the bicycle model, the cornering stiffness is for the entire axle and hence twice the single tire value.
Also, for the analysis presented here to be valid, the following assumptions
must hold:
- the radius of the turn, R, must be large compared to the vehicle wheelbase, L= a+b, and the vehicle track, t;
- the left and right steer angles of the front wheels must be approximately the same
- the slip angles of the front wheels are equal, as are the slip angles of the rear wheels
- Small sideslip angles so that tan(angle)= angle
Now that we have defined the terms and assumptions, we can derive the dynamic bicycle model equations. From the Newton’s law, the forces acting on the vehicle in Y direction can be summarised as:
Where
Fyf , Fyr : lateral forces on front and rear wheel
m : mass of the vehicle
u0 : velocity in x direction
β : sideslip angle at CG
v : lateral velocity
r: angular velocity about z direction (yaw rate)
Also taking the moment equation around CG for roll
where
a, b : distance from front and rear wheel to the center of gravity of the vehicle respectively
Iz : moment of inertia about z axis
Assuming linear tires, the forces can be defined as
where
Cαf , Cαr : cornering stiffness for front and rear wheels respectively
Combining all the equations into a state space form
Now we define two additional variables y, the lateral displacement and ψ , the yaw angle relative to the car
And the matrices would take the form
So the four state variables in this analysis are [y,v,ψ,r]. This dynamic model can be used to calculate the future state of the vehicle for the given inputs. This is for the straight path. If the road is curved, the yaw angle of the road can be used to convert the road coordinates to vehicle coordinates using the rotation matrix.
Even though it is much more complicated than the kinematic one, the dynamic model is also a simplified model as we made a lot of assumptions like linear tires, no roll, small angles etc. However it is still a powerful tool for processes like the lateral controller design for Lane Keeping which does not require extensive vehicle modeling. It can also provide a directional system level response to the parameters available. For component design and system level vehicle design, a full scale multi-body dynamics model is required which is usually done in softwares like MSC Adams etc.
References
“Kinematic and Dynamic Vehicle Models for Autonomous Driving Control Design by Jason Kong” , Mark Pfeiffer , Georg Schildbach, Francesco Borrelli
“Vehicle Dynamics Course Notes “ — University of Michigan
Written while listening to When Chai met Toast