# TA-RRT*: Adaptive Sampling-Based Path Planning Using Terrain Analysis
## 5 Discussion of Real-World Scenarios
### Consideration of Kinematic Model  
To deploy the TA-RRT* algorithm on an actual mobility platform, it is essential to consider the kinematic model of the platform during operation. This discussion will focus on the two-axle model, which applies to most mobility platforms, including vehicles such as cars or bicycles, where the front axle is responsible for steering and the rear axle provides propulsion. Specifically, the bicycle model will be utilized as the basis for this analysis.  
For the real-world implementation of TA-RRT* in the two-axle model, it is necessary to account for the physical constraints of the mobility platform, such as its minimum turning radius, steering angle limits, and velocity constraints. In particular, during the trajectory planning phase of TA-RRT*, waypoints are derived using methods such as Pure Pursuit [41] or the Stanley algorithm [42]. The Stanley algorithm is a path tracking method originally developed for autonomous vehicles. It employs a nonlinear feedback control law to minimize the cross-track error and heading deviation, ensuring a smooth trajectory. To ensure these waypoints are feasible for the mobility platform, the physical constraints of the kinematic model--including the minimum turning radius, steering angle limits, and velocity constraints--must be considered. This involves evaluating the suitability of each derived waypoint and, if necessary, generating modified waypoints that comply with these physical constraints.  
\[\dot{x}=v\cos\theta,\ \ \dot{y}=v\sin\theta,\ \ \dot{\theta}=\frac{v}{L}\tan\phi,\]  
where \(\dot{x}\) and \(\dot{y}\) are the Cartesian coordinates of the vehicle's rear axle center, \(v\) denotes the vehicle's linear velocity, \(\theta\) represents the vehicle's orientation angle relative to the global coordinate frame, \(L\) is the distance between the front and rear axles (wheelbase) and \(\Phi\) is the steering angle of the front wheel.  
In particular, the following physical constraints are taken into account: the minimum turning radius, \(R_{\text{min}}\); the steering angle limit, \(|\Psi|\leq\Phi_{\text{max}}\); and the velocity constraint, \(v_{\text{min}}\leq v\leq v_{\text{max}}\). Here, \(R_{\text{min}}\) represents the minimum radius within which the mobility platform can execute a turn, given the steering angle limitations. This is defined by the following equation:  
\[R_{min}=\frac{L}{\tan\Phi_{max}}. \tag{4}\]  
Furthermore, the steering angle limit is defined as \(|\Phi|\leq\Phi_{\text{max}}\), where \(\Phi\) represents the steering angle of the front axle, constrained by the maximum value \(\Phi_{\text{max}}\). Additionally, the velocity is bounded by the acceleration and deceleration capabilities, expressed as \(v_{\min}\leq v\leq v_{\max}\). By taking these parameters into account, the feasibility of a given waypoint is evaluated, and a new waypoint closest to the original one is determined within the constraints of these limitations for operational use. In cases where a four-axle model is considered, the Double Ackermann Model may be applied.