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Lesson 5: Lateral Dynamics of Bicycle Model

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Introduction to Self-Driving Cars

Universidade de Toronto

4.8 (258 classificações) | 12K alunos inscritos

Curso 1 de 4 no Carros autoguiáveis Programa de cursos integrados

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Welcome to Introduction to Self-Driving Cars, the first course in University of Toronto’s Self-Driving Cars Specialization. This course will introduce you to the terminology, design considerations and safety assessment of self-driving cars. By the end of this course, you will be able to: - Understand commonly used hardware used for self-driving cars - Identify the main components of the self-driving software stack - Program vehicle modelling and control - Analyze the safety frameworks and current industry practices for vehicle development For the final project in this course, you will develop control code to navigate a self-driving car around a racetrack in the CARLA simulation environment. You will construct longitudinal and lateral dynamic models for a vehicle and create controllers that regulate speed and path tracking performance using Python. You’ll test the limits of your control design and learn the challenges inherent in driving at the limit of vehicle performance. This is an advanced course, intended for learners with a background in mechanical engineering, computer and electrical engineering, or robotics. To succeed in this course, you should have programming experience in Python 3.0, familiarity with Linear Algebra (matrices, vectors, matrix multiplication, rank, Eigenvalues and vectors and inverses), Statistics (Gaussian probability distributions), Calculus and Physics (forces, moments, inertia, Newton's Laws). You will also need certain hardware and software specifications in order to effectively run the CARLA simulator: Windows 7 64-bit (or later) or Ubuntu 16.04 (or later), Quad-core Intel or AMD processor (2.5 GHz or faster), NVIDIA GeForce 470 GTX or AMD Radeon 6870 HD series card or higher, 8 GB RAM, and OpenGL 3 or greater (for Linux computers).

Avaliações

4.8 (258 classificações)

5 stars
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4 stars
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DC

Feb 25, 2019

Great course, very good overview of the field, excellent/challenging/rewarding projects. Highly recommended!

MA

Feb 25, 2019

Excellent education and supporting material! It is fully worthy of doing this course.

Na lição

Module 4: Vehicle Dynamic Modeling

The first task for automating an driverless vehicle is to define a model for how the vehicle moves given steering, throttle and brake commands. This module progresses through a sequence of increasing fidelity physics-based models that are used to design vehicle controllers and motion planners that adhere to the limits of vehicle capabilities.

Lesson 1: Kinematic Modeling in 2D11:24

Lesson 2: The Kinematic Bicycle Model8:05

Lesson 3: Dynamic Modeling in 2D10:33

Lesson 4: Longitudinal Vehicle Modeling11:14

Lesson 5: Lateral Dynamics of Bicycle Model7:49

Lesson 6: Vehicle Actuation9:51

Lesson 7: Tire Slip and Modeling10:52

Ministrado por

Steven Waslander
Associate Professor
Jonathan Kelly
Assistant Professor

In the previous video,

we covered longitudinal dynamics for the vehicle.

In this lesson, we will cover the dynamic modeling of

a four-wheel car based on the bicycle modeling approach.

By the end of this video,

you will be able to;

build a dynamic model of a car using the kinematic bicycle model as a starting point,

and represent it in a standard state-space form suitable for lateral control design.

Let's get started. We would like to extend our kinematic bicycle model

to a dynamic model by relaxing the no slip condition and force for the kinematic model.

Recall that in the full dynamic bicycle model,

we maintain two components of the motion: the first

in the longitudinal direction in the direction of the heading,

and the second in the lateral direction perpendicular to the heading.

Specifically, for the lateral vehicle model,

we are interested in modeling the rotation rate of

the vehicle based on the moments that affect the vehicle while moving.

To start modeling the lateral dynamics of the bicycle model,

the following assumptions will be made: first,

the forward longitudinal velocity is assumed constant.

This is done to decouple our lateral and longitudinal dynamic models,

which simplifies our modeling task greatly,

but does lead to modeling inaccuracies when accelerating or decelerating out of curves.

Second, as with the kinematic bicycle model,

the left and right wheels for

both front and rear axles are lumped into a single wheel each.

So, this assumption converts the four wheels to two wheels bicycle model.

Finally, other nonlinear effects such as suspension movement,

road inclination, and aerodynamic forces are assumed to be negligible.

In practice, these effects can have a significant impact on the tire forces that occur.

So, this is again,

a limiting assumption in some cases,

but it's sufficient for our purposes.

We will use the vehicle center of gravity as the reference point for

the dynamic model as it simplifies the application of Newton's second law.

We can define the total acceleration in the inertial frame as a_y,

and this includes the lateral acceleration in the body frame y double dot,

and the centripetal acceleration from rotation of the vehicle,

omega squared R. These expressions can be

rewritten in terms of the slip angle rate of change, beta dot,

and the heading rate of change,

psi dot using the definition of the slip angle and the fact that V equals omega R,

and omega is equal to psi dot respectively.

The model formulation for lateral dynamics can now be formed with the only two

forces affecting the dynamics being the lateral forces on the front and rear tires.

The vehicle longitudinal velocity is defined by V as before and

the mass is m. For the angular accelerations, psi double dot,

the moments produced by the tire forces act in

opposite directions and combine with the inertial term,

I_z times psi double dot to form this second-order equation.

The parameters, L_f and L_r define the distances to each tire from the cg.

One of the most important components in vehicle dynamics modeling or the tires.

Tire forces are generally hard to predict exactly,

and tire models tend to be nonlinear and empirically identified.

We'll explore some of

the more common tire models in detail in the final video in this module.

Fortunately however, for normal driving conditions,

a simple linear approximation can be used to model tire force generation.

This approximation is really only valid for small slip angles,

and the tire forces are modeled as varying linearly with slip angles specifically.

As we use the linear tire model in our control design,

we must make sure to not exceed

this small angle assumption by sticking to non-aggressive driving maneuvers.

In order to use these linear tire models,

we need to define the front and rear side slip angles,

alpha f and alpha r. They are defined

in exactly the same way as the vehicle slip angle beta,

but are defined relative to the direction of

the wheel and the vehicle velocity at the wheel center.

The cornering stiffness of a tire is defined by

its ability to resist deformation while the vehicle corners.

This diagram shows the relationship between

the lateral tire force and the tire slip angle.

The slope of the line at zero is known as the cornering stiffness coefficients, C_y.

As can be seen in the plot for small slip angles,

typically less than eight degrees,

the relation is almost a linear relation.

We can introduce C_f and C_r as linearized

cornering stiffness of the front and rear wheels for the bicycle model.

Therefore, the relation between the lateral forces acting on

the front and rear wheels will be the product of the cornering stiffness,

C_y, and the slip angle,

alpha for each tire.

We can rewrite the tire slip angles in terms of the vehicle slip angle beta,

the steering angle delta,

the longitudinal velocity V,

and the yaw rate.

The previously derived dynamic equations for the lateral bicycle model can

now be modified by substituting in the tire force slip equations.

This leads to the following pair of

coupled ordinary differential equations that

govern the dynamics of the vehicle in the lateral direction.

Note that the left-hand side variables represent the rate of change of

the vehicle slip angle and the rate of change of the yaw rate.

Conveniently and because of

the approximations and assumptions made throughout this derivation,

the resulting dynamic lateral vehicle model is linear.

We can define a state vector,

X_lat that includes the lateral position;

y, the slip angle; beta,

the yaw angle; psi,

and the yaw rates; psi dot.

The system can now be expressed in standard state-space form as follows;

X dot lat equals A_lat times,

X_lat, plus B_lat times delta.

The system dynamics matrices are A_lat and B_lat

which are time-invariant if the forward speed V is kept constant.

The main input to the system is the driver steering angle command, delta.

This state space representation is very useful when we are designing

different control strategies such as PID or MPC for lateral control.

The linearity of this model also makes it

suitable for state estimation with Kalman filters,

which you'll learn more about in course two.

Let's summarize this video.

We first learned how to extend

our kinematic bicycle model to become a dynamic bicycle model,

and then we derived a lateral dynamics of

the bicycle model and expressed it in state space form.

In the next video,

we'll explore and build models for the vehicle actuation system including a throttle,

break, and steering. See you there.

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