## 3 Obstacle Avoidance Model by Harmonic Potential Field with Curvature Constraint
### Harmonic Potential Function and Streamlines
Then, the normalized velocity and the corresponding acceleration of each point on the streamline in the uniform flow are given by (5) and (6). \[u_{N}=\frac{u}{U},\ v_{N}=\frac{v}{U}, \tag{5}\]  
\[a_{x}=\frac{\delta u_{N}}{\delta x}u_{N}+\frac{\delta u_{N}}{\delta y}v_{N}, \quad a_{y}=\frac{\delta v_{N}}{\delta x}u_{N}+\frac{\delta v_{N}}{\delta y}v_ {N} \tag{6}\]  
Furthermore, curvature and deviation of curvature can be derived by velocity and acceleration as  
\[ \begin{split}&\kappa=\frac{u_{N}a_{y}-v_{N}a_{x}}{\left(u_{N} ^{2}+v_{N}\right)^{3/2}}\\ &=-2 Ay\sqrt{A^{2}+2 AU(-x^{2}+y^{2})+U^{2}\left(x^{2}+y^{2}\right) ^{2}}\\ &\times\frac{\left[A^{2}+2 AU\left(x^{2}+y^{2}\right)+U^{2}\left( -3x^{4}-2x^{2}y^{2}+y^{4}\right)\right]}{\left(x^{2}+y^{2}\right)\left[A^{2} +2 AU(-x^{2}+y^{2})+U^{2}\left(x^{2}+y^{2}\right)^{2}\right]^{2}}\\ \end{split}  \tag{7}\]\[\dot{\kappa}=\frac{\delta\kappa}{\delta x}\frac{dx}{dt}+\frac{\delta \kappa}{\delta y}\frac{dy}{dt}=\frac{\delta\kappa}{\delta x}u_{N}+\frac{\delta \kappa}{\delta y}v_{N}\] \[=\frac{24 AUxy\left[ \begin{array}{c}A^{4}\left(x^{2}-y^{2}\right) ^{4}-U^{4}\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right)\\ -2A^{3}U\left(x^{2}+y^{2}\right)^{2}+2 AU^{3}\left(x^{2}+y^{2}\right)^{4}\\ \left(x^{2}+y^{2}\right)^{2}\left[A^{2}+2 AU(-x^{2}+y^{2})+U^{2}\left(x^{2}+y \right)^{2}\right]^{3}\end{array} \right]}}{\left(x^{2}+y^{2}\right)^{2}\left[A ^{2}+2 AU(-x^{2}+y^{2})+U^{2}\left(x^{2}+y^{2}\right)^{2}\right]^{3}}\]  
From the above equations, given a start position and a circle with a known radius at the origin, a streamline (integral curve) can be derived by numerical integration of the velocity vector field that specifies the tangent of the path or the robot heading at each point. In addition, the higher order path characteristics such as curvature are easily computable as well, making the path to be followed more predictable. That is, the velocity vector field of streamlines serves as a vector field for the guidance of the robot's motion everywhere in the obstacle-free region. Therefore, in path planning applications, streamlines provide a pool of systematic paths that have an explicit or analytic vector field for the tangent to the path as the path specification, and their higher order path properties such as curvature could be easily computed as well. However, there are several drawbacks for robots to purely follow streamline paths. First of all, we assume that the curvature of unicycle kinematics (1) is constrained by its upper bound. The curvature of streamline it follows has a larger curvature as the distance to the obstacle gets closer. For example, Figure 5 depicts that streamline starting from (\(-5\), \(0.2\)) exhibits the maximum curvature. Second, the paths with a smaller curvature are longer and keep an unnecessary distance with the obstacle. Moreover, for paths with an initial position further than the radius of the obstacle in the direction of the \(x\)-axis, it is not necessary to follow a streamline path because a robot can pass the obstacle straightly, such as the paths with a start position further than the radius of obstacle with the \(x\)-axis in Figure 5. Therefore, we provide an improved streamline-based approach in the following section.