Cycle Dynamics

So let'start with a little sketch of our bicycle model and a list of ass The first and most obvious ass The second ass Correspondingly, we'll use the following ve M is the ve Here Z refers to the vehicle coordinate system, but anyways we ass So the motion of the ve An important ass In other words, the lateral motion and the yaw motion are determined by forces and in t Let's call the point of attack of the front wheel tire force the point And that of the rear wheel tire force the point B and let the distance between And C be the length LF and the distance between C and B the length LR. So that clearly LF plus LR is equal to the wheelbase L of the ve Moreover, we ass Here the parameter C alpha F represents the cornering stiffness of the combined front tires and C alpha R is the cornering stiffness of the combined rear tires. In our derivations, we'll also make the ass The tangent of delta is also approximately equal to delta and the cosine of delta is approximately equal to one. Finally, for the longitudinal motion of the vehicle, we just ass So we'll only apply the laws of motion for the lateral and yaw motion while for the longitudinal motion we'll just make the simple ass Now with all of these ass For the laws of motion, we'll make use of the coordinates of the inertial frame and those of the ve Let's begin with the lateral dynamics. By the center of gravity principle, we have that the mass of the vehicle times its acceleration in the y direction in the inertial system is equal to the since the only two forces acting on the ve So with the small angle approximation, this is approximately equal to the since the ve Substituting t Next, let's get to the yaw dynamics of the ve By the angular moment T And the front tire force has the lever arm LF, and only the component LF cosine delta turns in t And the other component of the front tire force along the x axis has no lever arm, so it drops out of the equation. Again, with the small angle approximation of cosine delta equal to one, we get t In order to complete our model, we need to derive expressions for the tire forces in equations one and two. And to t So in t At this point, we always have to pay close attention to the sign which we are using, and this sign depends on the direction we have ass And note that in t In order to determine the slip angles at the front and the rear tire, we need to look at the velocity geometry at the points And B. Here, the rear tire is actually the easier case because the tangent of the slip angle alpha r is just given as the ratio between the velocity component of point B in the y direction and the velocity component of the point B in the x direction. So we have the tangent of alpha r is equal to VBY divided by VBX. And again, we do a small angle approximation that the tangent alpha r is approximately equal to alpha r. For the case of the front tire, we actually need to look at the velocity components in the directions of the wheel coordinates, w And then the tangent of the slip angle alpha f is given as the ratio of the velocity component of the point A in the eta direction and the velocity component in the xi direction. So we get t From the laws of kinematics for the rigid body of the ve In terms of the longitudinal velocity of the ve Now let's continue t In order to get from the velocity components of the point A in the x direction and in the y direction to the xi and eta direction, we have to perform a coordinate transformation. In fact, the velocity components in the xi-eater directions are rotated by the steering angle delta compared to the velocity components in the x-y direction. So what we get are these expressions for vx i and va eta in terms of the steering angle delta and the velocity components va x and va y. And we again apply our small steering angle approximations where the cosine of delta is equal to 1 and the sine of delta is equal to delta. By substituting the velocities va x and va y into va xi and va eta, and then in turn substituting these velocity components into the front tire slip angle, and then the front tire slip angle into the front tire force, we get t With the ass Doing the analogous t So let's put it all together to obtain the state space representation of our linear bicycle model. What we have to do for t So in one, t Analogously substituting the expression for the rear tire force and the front tire force into the angular moment These two equations can now be written more compactly as a state space representation. To the second state, x2, is equal to psi, x3 is equal to omega, w And finally, the control input, u, is equal to the steering angle delta that appears here and here. Hence what we get is the state space representation in the yellow box, w Observe that t As a final remark, with these two differential equations, the model can be augmented by the global position to a nonlinear state space model and then analogous to the kinematic bicycle model, it also contains the global position of the ve