Effective Parametrization of Low Order Bézier Motion Primitives for ContinuousCurvature PathPlanning Applications
Abstract
:1. Introduction
Objectives Furthermore, Contributions
 Based on the findings of our literature review this is one of the few if not the first approach that apply third order Bézier curve motion primitives in path planning and takes into account the given requirements for initial and final position, orientation and curvature. The combined path using the proposed primitives has minimal required turns or oscillations between the initial and final configurations, since only a third order curve is used. Due to the small number of required parameters, the method is also suitable for use in path planning optimisers, since it reduces computational effort and improves convergence.
 We provide a new parametrization that allows an intuitive geometric interpretation of the curve and a simple algorithm to calculate its parameters. Parametrization is based on the initially available information, i.e., the position, orientation, and curvature requirements in the endpoints. The problem is solved by solving a system of two quadratic polynomial equations without the need for optimisation. It is therefore computationally very efficient.
 Practical directions for the construction of primitives and the related analysis of their performance are provided. The applicability of the proposed primitives is illustrated on typical pathplanning scenarios.
2. Bézier Curves as Motion Primitives
2.1. Bézier Curves
2.2. The Curvature of Bézier Curves
2.3. Parametrization of Motion Primitives
 The position of both endpoints, i.e., ${P}_{0}$ and ${P}_{3}$;
 The orientation in both endpoints, i.e., angles ${\phi}_{1}$ and ${\phi}_{3}$; and
 The curvature in both endpoints, i.e., ${\kappa}_{0}$ and ${\kappa}_{3}$.
2.4. Reconstruction of the Motion Primitive from the Boundary Conditions
 Utype and Rtype functions are increasing while Dtype and Ltype are decreasing;
 Only Utype functions are convex while Dtype, Rtype, and Ltype are concave.
 An increasing and a decreasing function can have at most one intersection;
 A decreasing convex function and a decreasing concave function can have at most two intersections (the same is true if both functions are increasing);
 Two decreasing concave functions (or any other of the three combinations decreasingconvex, increasingconcave, increasingconvex) can have more than two intersections.
 One solution. This is the most frequent case that gives a unique Bézier curve.
 Two solutions. This happen rarely because the boundary conditions given above are quite restricting. An example is given in Figure 2 (the intersection of a green and a blue line).
 Three solutions. This can only happen if one function is ${D}_{+}$ and the other is ${L}_{+}$ and the parameters of both parabolas are extremely restricted. Two intersections lie very close to both axes. The solutions with very low values of ${d}_{1}$ and/or ${d}_{3}$ often result in a highcurvature path near the endpoints and should be avoided.
 No solutions. This happens when the parabolas do not have intersections in quadrant 1. Examples in Figure 2: two green lines, a black line, and a red line. We have to also mention here the cases where the parabolas do not even enter quadrant 1—the vertex is negative and the leading coefficient ${l}_{1}$ (or ${l}_{3}$) is negative.
2.5. Basic Motion Primitives
2.5.1. CShaped Primitive
2.5.2. SShaped Primitive
 ${U}_{}$ parabola. The intersection exists if the point of entry of the ${U}_{}$ parabola into the first quadrant lies below the vertex of the ${L}_{+}$ parabola. If this is not the case, the intersection appears by increasing the curvature of the ${U}_{}$ parabola. An alternative solution is to change the sign of the ${L}_{+}$ parabola which turns the primitive to the loopshaped or Vshaped curve (shown in the bottom part of Figure 3).
 ${U}_{+}$ parabola. The intersection exists if the vertex of the ${U}_{+}$ parabola is lower than the point where parabola ${L}_{+}$ crosses from quadrant 1 to quadrant 2. It there is no intersection, one can be obtained by changing the curvature of the ${U}_{+}$ parabola. This then changes the primitive into Cshaped one.
2.5.3. LoopShaped Primitive and VShaped Primitive
3. Construction of the Path from Motion Primitives
 Waypoints or intermediate points denoted by the sequence ${W}_{0},{W}_{1},\dots {W}_{N}$ (${W}_{0}$ and ${W}_{N}$ are the initial and the final position, respectively);
 The orientation in the waypoints denoted by the sequence ${\theta}_{0},{\theta}_{1},\dots {\theta}_{N}$ (${\theta}_{0}$ and ${\theta}_{N}$ are the initial and the final orientation, respectively);
 The curvature in the intermediate points denoted by the sequence ${K}_{0},{K}_{1},\dots {K}_{N}$.
Algorithm 1: The algorithm for calculation of motion primitives from boundary conditions. 
In general, the solution of the system of equations in Equation (8)—or the intersection of parabolas in (9)—is obtained by following these steps:

3.1. The Algorithm for Proposing Suitable Orientations in Waypoints
3.2. The Algorithm for Proposing Suitable Curvatures in Waypoints
 ${d}_{3}$ follows directly from (12).
4. Examples and Comparisons
4.1. Example 1: A Path in a Given Corridor
4.2. Example 2: Environment with Random Obstacles—Comparison with an Existing Method
5. Discussion
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
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Algorithm  Length  Average ${\mathit{\kappa}}^{2}$  Max $\mathit{\kappa}$  Min $\mathit{\kappa}$ 

funnel alg [40]  16.1395  0.2415  0.8669  $1.0000$ 
Section 3.1 and Section 3.2  16.6174  0.6825  5.3447  $1.2397$ 
optimization  16.3910  0.2451  0.8394  $0.8666$ 
“filtering” of [40]  15.9238  0.2576  0.8493  $1.0266$ 
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Blažič, S.; Klančar, G. Effective Parametrization of Low Order Bézier Motion Primitives for ContinuousCurvature PathPlanning Applications. Electronics 2022, 11, 1709. https://doi.org/10.3390/electronics11111709
Blažič S, Klančar G. Effective Parametrization of Low Order Bézier Motion Primitives for ContinuousCurvature PathPlanning Applications. Electronics. 2022; 11(11):1709. https://doi.org/10.3390/electronics11111709
Chicago/Turabian StyleBlažič, Sašo, and Gregor Klančar. 2022. "Effective Parametrization of Low Order Bézier Motion Primitives for ContinuousCurvature PathPlanning Applications" Electronics 11, no. 11: 1709. https://doi.org/10.3390/electronics11111709