3.3.2. Terminal Posture Constraints

The vehicle positions in the corridor entrance and exit are expected to be on the corridor centerline, while the vehicle yaw angles are expected to be parallel to the corridor heading orientations, which makes it quite difficult to find an exact solution. Thus, we constrain the vehicle's final state to be near the expected posture. By taking the discrete waypoints as a reference, the terminal constraint can be expressed as:

$$\mathfrak{x}(1) = \mathfrak{x}\_{\mathfrak{o}}(1), \mathfrak{x}(N) = \mathfrak{x}\_{\mathfrak{o}}(N) \pm \Delta \mathfrak{x},\tag{11}$$

$$y(1) = y\_o(1), \\ y(N) = y\_o(N) \pm \Delta y,\tag{12}$$

$$
\varphi(1) = \theta\_{\text{enter}} \,\varphi(N) = \theta\_{\text{exit}} \pm \Delta\theta\_{\prime} \tag{13}
$$

where *θ*enter and *θ*exit are the orientations of the corridor entrance and exit. The sizes of Δ*x*, Δ*y*, Δ*θ* are expected to be appropriate to guarantee that the vehicle is within the corridor.

#### 3.3.3. Vehicle Kinematics Constraints

Vehicle operation follows the kinematics relationship. Hence, the vehicle kinematics constraints are considered here. By taking the discrete waypoints as a reference, the kinematics constraints in the space domain can be approximately expressed as follows:

$$\mathbf{x}(k+1) = \mathbf{x}(k) + \Delta \mathbf{s}(k) \cos(\boldsymbol{\varrho}(k)),\tag{14}$$

$$y(k+1) = y(k) + \Delta s(k)\sin(\varphi(k)),\tag{15}$$

$$
\varphi(k+1) = \varphi(k) + \frac{\Delta s(k)\tan(\varphi(k))}{L}.\tag{16}
$$

#### 3.3.4. Vehicle Speed Constraints

We consider the constraint of vehicle speed here. By taking the discrete waypoints as a reference, the vehicle speed constraint is as follows:

$$
v\_{\min} \le v(k) \le v\_{\max}.\tag{17}$$

#### 3.3.5. Actuator Range Constraints

There are both longitudinal and lateral actuator constraints in a vehicle's operations. We ignore the limitation of longitudinal actuators here for the fact that a vehicle's normal operation in a narrow corridor would never reach its acceleration/deceleration limits. Though the longitudinal actuator limitation is ignored here, we would restrict the longitudinal motion parameters for comfort. This would be discussed in a later subsection. For the later actuator, its range could not be ignored since an adequate steering angle is necessary in a narrow turning area. By taking the discrete waypoints as a reference, the constraint of steering is as follows:

$$-\mathfrak{a}\_{\text{max}} \le \mathfrak{a}(k) \le \mathfrak{a}\_{\text{max}}.\tag{18}$$

#### 3.3.6. Tire Side-Force Constraints

A big tire slip angle caused by a large side force would lead to vehicle model invalidation. To avoid this situation, the side force related to the vehicle speed and path curvature is constrained by the formula *<sup>v</sup>*<sup>2</sup> *<sup>R</sup>* ≤ *μg*, where *R* is the curvature radius, *μ* is the side-force coefficient, and *g* is the gravity coefficient. Since the vehicle follows the Ackerman steering principle, *R* is dependent on the formula *R* = *<sup>L</sup>* tan(*α*). By taking the discrete waypoints as a reference, the final expression of the side-force avoidance constraint is as follows:

$$\frac{v(k)^2 \tan(\alpha(k))}{L} \le \mu \text{g.}\tag{19}$$
