13.1.2.4 A car pulling trailers

An interesting extension of the simple car can be made by attaching one or more trailers. You may have seen a train of luggage carts on the tarmac at airports. There are many subtle issues for modeling the constraints for these models. The form of equations is very sensitive to the precise point at which the trailer is attached and also on the choice of body frames. One possibility for expressing the kinematics is to use the expressions in Section 3.3; however, these may lead to complications when analyzing the constraints. It is somewhat of an art to find a simple expression of the constraints. The model given here is adapted from [727].^13.4

Figure 13.6: The parameters for a car pulling trailers.

Consider a simple car that pulls $k$ trailers as shown in Figure 13.6. Each trailer is attached to the center of the rear axle of the body in front of it. The important new parameter is the hitch length $d_i$ which is the distance from the center of the rear axle of trailer $i$ to the point at which the trailer is hitched to the next body. Using concepts from Section 3.3.1, the car itself contributes ${\mathbb{R}}^2 \times {\mathbb{S}}^1$ to ${\cal C}$, and each trailer contributes an ${\mathbb{S}}^1$ component to ${\cal C}$. The dimension of ${\cal C}$ is therefore $k + 3$. Let $\theta _i$ denote the orientation of the $i$th trailer, expressed with respect to the world frame.

The configuration transition equation is

$$\begin{split}{\dot x}&= s \cos \theta_0 {\dot y}&= s \sin ... ... \theta_j) \right) \sin (\theta_{k-1} - \theta_k) . \end{split}$$ (13.19)

An interesting variation of this model is to allow the trailer wheels to be steered. For a single trailer, this leads to a model that resembles a firetruck [163].