[MUSIC] In the last lesson, we developed the kinematic bicycle model to capture vehicle motion with steering rates and velocity inputs. In this lesson, we will move into the realm of dynamic modeling. To start taking into account the forces and moments acting on the vehicle. This type of model can lead to higher fidelity predictions. Than are possible with kinematic models. This higher fidelity, however, comes at the cost of higher computational complexity. So both kinematic and dynamic models have their uses in self driving development. Dynamic modeling is likely very familiar to many of you. So if you're comfortable making free body diagrams and applying Newton's equations, Don't hesitate to jump forward in the course. If you need a refresher, or haven't done dynamic modeling before, this introductory video is for you. [SOUND] So why do we need dynamic modeling? In vehicle modeling, when the vehicle is moving and turning at higher speeds, or when the road is slippery, the assumptions defined by the no slip condition may no longer hold. As the forces exerted on the vehicle cause the tires to slip over the pavement. Modeling the balance of forces during slip conditions can expand the set of driving conditions for which our model remains an accurate prediction of motion. Also, many sub systems in the car have similar conditions where a hard kinematic constraint is not correctly captured the evolution of that sub system, and dynamic models are needed there as well. An example would be the drive train for which the balance of torques is needed to capture the connection from throttle position to wheel torque through the engine and transmission systems. By explicitly modeling the balance of forces in moments acting on the vehicle, we can determine the accelerations the vehicle is experiencing and use them to model our vehicle motion. We now turn to the task of defining the dynamic modelling process for a rigid body. To build a typical dynamic model, we can follow the following steps. First we set up the coordinate frames to be used in the model. For example, the body frame and the inertial frame we discussed before. Next, we break down the dynamic system into lumped dynamic elements. These elements combine potentially distributed aspects of the system into discrete or lumped elements. In the case of a spring mass damper, the lumped elements would be the spring, the mass, and the damper. We also define a model for each lumped element. For example, the linear spring provides a force proportional to a displacement for a resting position. Next, we sketch the free body diagram for each rigid body in the list of elements. And properly name and model all the forces and moments acting on the body. Finally, by using Newton's second law, we write the mathematical equations that define our dynamic model, summing all forces along each axis for translational dynamics. In all moments about each rotational axis for rotational dynamics. The result is an ordinary differential equation describing the motion of our rigid body and this is our dynamic model. Let's follow our four step process for a one dimensional translational system, a rolling cart. This figure shows the cart of mass m whose position we'd like to track. We start by defining the coordinate frame for the position of the cart, denoted with x. Then we identify the rigid body, which is in this case, is the cart of mass m. Next, we draw a free body diagram and define all of the forces acting on the carts. In this case, there were three forces acting on the cart, f1 pulling the cart to the right, and f2 and f3 pulling the cart to the left. Finally, we apply Newton's second law in the x direction to form the dynamic model. The resulting dynamic equation describing the motion of the carts Is m times the acceleration in the x direction denoted by x double dot is equal to the summation of f1, f2, and f3. Note that f2 and f3 have negative signs due to the fact that they point in the negative x direction. The same process can be applied for a shock absorber in a car The dynamic model of a shock absorber helps the engineers to design and tune the vehicle suspension for better ride comfort and driveability. To model this component, we can go through the same four steps that we introduced in the previous slides. We again set up the coordinate system, which in this case is one coordinate y in the vertical direction. The identified of rigid bodies or lump dynamic elements. In the case of a real physical system, there is notion of lump element very important. The shock absorber relies on it's spring and hydraulics cylinder with flow restriction to absorb shocks. We model this system as three separate long elements A mass a spring and hydraulic cylinder or damper. We use a linear spring and damper model where the spring resist displacement in y and damper resist the y velocity. The third step is sketching the free body diagram. Normally we show the forces Or torques along with the velocities and directions in this diagram. Finally, we form the dynamic equation, which in this case is the famous second order spring mass damper equation. There is no variation in this process to handle rotational or torsional systems. Rotational systems are very common in the automobile with examples like the internal combustion engine chaps, gear boxes, [INAUDIBLE] converters and tires. In a rotational system, we sum the [INAUDIBLE] about each access of rotation and lump. Inertia is J, torsional springs and dampers in the same way as in the translational case. We refer to the rotational acceleration of a rigid body as alpha. Let's model the rotational dynamics of a wheel, using our four-step modeling process. First, we set up the coordinate system. One coordinate in the rotational direction is enough to represent the wheel's motion. We defined theta as the angular position of the wheel, with theta dot as its rotation rate and theta double dot, its angular acceleration. Next, we formed the lumped dynamic elements. The tire model has rotational inertia J due to its rotating mass. And torsional stiffness k and damping b, defined by the material properties of the tire in the wheel hub. We drive the wheel with the a drive torque from the vehicles drive shaft and resist this with a load torque coming from the tires interaction with the road's surface. Now we sketch the freebody diagram. The drive torque and low torque are defined in opposite directions as the load resists the drive torque's acceleration of the wheel. We can now form the dynamic model for the wheel system. The difference between the drive torque and the load torque define the net torque acting on the wheel. While the rotational spring and the damper from the lump element models resist the acceleration. The final dynamic model describes how the wheels position changes when subject to this net torque. Dynamic models of vehicles are useful for multiple applications. They can be used to improve state estimation methods when fusing sensor data to track motion. They can be used to aid in controller design to track a desired trajectory or path. And they can help self-driving engineers define the limits of vehicle performance to avoid from planning unsafe trajectories that a car cannot track. However, a full three-dimensional model of the vehicle, taking into account body roll and pitch. And arbitrarily inclined roads, and different forces and moments at each tire makes for a very complex model. Instead, it's possible to separate our model into two 2D models that split our vehicle control into a steering control and a throttle and brake control problem. Full 3D vehicle modeling is a fascinating field. And we've introduced some links in the supplementary materials if you'd like to learn more. For these reasons, we'll build a separate longitudinal and lateral dynamic model for our self-driving car. The longitudinal model considers a vehicle traveling on an inclined road. We restrict the vehicle motion to the XZ plane. There are several forces acting on the vehicle body and tires, including the Traction force, the Rolling resistance, Aerodynamic force, and Gradient resistance force due to gravity. In the next video we'll go through the Longitudinal Vehicle Dynamics model, in more detail. Similarly, the Lateral Vehicle Dynamics model, can be developed from motion in the xy plane. When looking at the vehicle from the top down or birds eye view. In this 2D model there is also several forces in the lateral direction acting on the vehicle such as the slip forces and centrifugal forces, this topic would be discuss in the future lesson as well in this video we cover the basic of 2D dynamic model and apply it to a server and wheel subsystem. These models are helpful in determining how those systems evolved over time. In the next lesson we will cover the vehicle longitudinal dynamics. And study the drive train components that provide tweak to the drive wheels. Will see there. [MUSIC] [SOUND]
