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).
Lesson 4: Longitudinal Vehicle Modeling
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Del curso dictado por University of Toronto
Introduction to Self-Driving Cars
255 calificaciones
De la lección
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.
Conoce a los instructores
Steven WaslanderAssociate Professor
Aerospace Studies
Jonathan KellyAssistant Professor
Aerospace Studies
In this lesson, we will go over the concept of the vehicle longitudinal dynamics,
and the power train component models needed to generate torques on the tires.
By the end of this video,
you'll be able to: define the balance of
dynamic forces acting on a vehicle in the longitudinal direction,
describe typical power train component models for an internal combustion engine vehicle,
and connect the typical power train component models to
form a full vehicle model for longitudinal motion.
These models will help us build
longitudinal controllers in the next module in this course.
So, let's get started.
The longitudinal vehicle dynamic model is simply
based on the dynamics of the vehicle that generate forward motion.
Here is a typical vehicle longitudinal motion on an inclined road.
The forces acting on the vehicle or the front and rear tire forces Fxf and Fxr.
The aerodynamic drag force F aero and the rolling resistance Rxf and Rxr.
There's also the force due to gravity mg sine alpha where alpha is the local road slope.
Based on Newton's second law,
the longitudinal tire forces of the front and rear tyres Fxf and Fxr,
which come from the vehicle power train must overcome
the resistance forces such as the aerodynamic force F aero,
the gravitational force mg sine alpha,
and the rolling resistance of the front and rear tires, Rxf and Rxr.
The imbalance between these forces defines the acceleration of
the vehicle in the longitudinal direction denoted by X double dot.
The longitudinal dynamics equations can further be simplified by grouping the front
and rear tire forces as Fx and the front and rear rolling resistance forces as Rx.
Further, we can assume moderate road inclinations,
which means that the small angle approximation can be applied.
So, the sine of alpha is approximately equal to alpha.
Our basic dynamic model for the longitudinal motion of a car is then as
follows: Mx double dot is equal to Fx minus F aero,
minus Rx, and minus mg alpha.
Here MX double dot is the inertial term that
defines the vehicles longitudinal acceleration.
Fx is the traction force generated by the power train,
and F aero, Rx,
and mg alpha make up the total resistance forces acting on the vehicle,
which we'll refer to as F load.
Note that we still need to develop models for each of the forces in this equation and
define how they connect to the throttle and
break inputs that our autonomous system will apply.
Through the rest of this video,
we will develop these models,
and we'll go into more depth in subsequent videos on specific aspects such as,
power train, break and tyre force modelling.
Let's now build some simple models for the resistance forces on the car.
A vehicles longitudinal motion is
resisted by aerodynamic drag rolling resistance and the force due to gravity.
We've already built a model for the effects of gravity,
so let's move on to aerodynamics.
The aerodynamic drag can typically be modeled as dependent on air density,
frontal area of the vehicle,
the vehicles coefficient of friction,
and the current speed of the vehicle.
Given a fixed vehicle shape and standard atmospheric pressure,
we can define a simple lumped coefficient of the aerodynamic drag C alpha,
and multiply it by the velocity squared to get the drag force.
For rolling resistance, we have a similar model which can depend on the normal force,
tire pressure and characteristics, and vehicle speed.
If we again assume
nominal operating conditions and drop the second-order terms for simplicity,
we can arrive at a linear rolling resistance model,
where Cr1 is our linear rolling resistance coefficient.
In both cases, these are
basic approximate models that are a good starting point for controller design.
In practice, the fidelity of the model used depends on
the accuracy required of the controller or the simulation environment.
Now that we have models for the resistance forces acting on the vehicle,
let's look more closely at the dry forces created by the vehicles power train.
The force generated to conquer the resistance forces comes from the power train,
and can be modeled as being generated by a series of components.
The power generated due to the combustion in gasoline, or diesel engines,
or electrochemical reactions in batteries for electric vehicles or hybrid vehicles,
flow through the drive line and ultimately provide a torque to the wheels.
The drive line refers to the sequence of components between the engine and the wheels,
and typically includes the torque converter or clutch,
the transmission or gearbox,
and finally the differential.
The breaks are also included in this figure,
and can provide a resistance torque on the wheel hub directly.
Because of the direct connection between wheel and engine when in gear,
it is possible to model the relationship between
the wheel speed and the engine speed as a kinematic constraint.
The wheel rotation speed omega w
varies according to the torque converters turbine speed omega t,
through several gear ratios including
the torque converter, transmission, and differential.
This combined gear ratio is denoted GR
and changes depending on the state of the power train components.
So, what gear the transmission is in, for example?
The engine speed is equal to the turbine speed and can be used interchangeably.
The vehicle forward velocity is also proportional to
the wheel angular speed times the tire effective radius,
which we will mostly model as a fixed tire radius.
It can however be modified for higher fidelity modeling based on
expected deformation of the tire due to forces and moments on the car.
Assuming the effective tire radius is known,
we can write that the longitudinal vehicle speed x dot is equal to
the tyre radius R effective times the wheel speed omega w. So,
if we can model the dynamics of the engine speed,
we can then relate it directly to the vehicle speed through these kinematic constraints.
This expression for velocity can be differentiated to give
longitudinal acceleration in terms of the engine rotational acceleration.
Now, let's go through the dynamic equations of each of
the power train components to build a dynamic model of the overall power train.
The wheel is the intersection between the torques coming from the power train side,
and the torques acting from the external resistance forces.
Therefore, we can present the wheel dynamics by
the following first-order differential equation: to calculate
the wheel angular speed omega w.
Here T wheel is the wheel torque generated by the power train.
We can solve for the wheel torque using this differential equation,
if the tyre force Fx is known,
which can be calculated from the vehicle longitudinal dynamics derived earlier.
The wheel torque T wheel is actually the combination of
the brake torque and the output torque of the transmission or gearbox,
as is visible on the left.
We'll look at break modelling in a later video,
and we'll focus on the power train for this video.
The torque applied to the transmission is referred to as the turbine torque T sub t,
and comes from the torque converter which couples the engine to the transmission.
Recall the turbine angular speed omega T of
the torque converter is related to the wheel angular speed by the gear ratio GR.
We can therefore define a similar ordinary differential equation for
the turbine angular speed as our transmission dynamic model.
Also, we can substitute in our expression for T wheel from the wheel dynamics model.
Next up is the torque converter.
In practice, that torque converter has
complex dynamics because it has multiple and pillars and
fluid going through the impeller which
allows for the coupling and decoupling of the engine.
When coupled, we can assume
the turbine angular speed is almost the same as the engine speed.
Therefore, we replace the turbine speed with the engine angular speed in
the transmission model to form a dynamic model that includes the torque converter.
Finally, we can define the full power train model by including the engine dynamics.
The engine inertia term is equal to the torque produced by the engine from
the combustion process minus the turbine torque from the torque converter,
which still includes a dependence on the tire force Fx.
If we return to the original longitudinal vehicle model and solve for the tire force,
we see a further dependence on the engine rotational acceleration,
which can be grouped with the terms from the engine,
transmission, and wheel inertias.
We now form the complete power train model and define
an effective power train inertia as the sum of all the individual component inertias,
which we'll call J sub
e. The power train model simplifies down to the following equation,
which balances the engine torque T engine with
the total load torque T load to drive
the engine acceleration and thereby the vehicle longitudinal acceleration.
This final equation achieves what we set out to do,
as it shows the dynamics of the whole power train system all the way
from the engine to the external resistance forces acting on the vehicle.
We do still need to relate the engine torque to the accelerator pedal position,
and the brake torque to the brake pedal position,
which we'll cover in the video on Actuator Modelling.
To summarize in this video,
we defined a compact models for the vehicle longitudinal dynamics and resistance forces.
We described typical power train components and derived dynamic models for them,
and we combine these subsystems enforce models into
a single unified longitudinal dynamic model
which is suitable for speed control development.
In the next video,
we will investigate the details of lateral dynamics in modelling
to aid in the development of steering controllers. See you then.
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