As noted in the preceding subsection, B´ezier curves provide smoothness throughout a curve between two points but precise blending of multiple curve segments and, hence, the determi- nation of a trajectory that is feasible for a vehicle between the endpoints of a complex curve may require further inter-segment constraints to be useful for path planning. Alternatively, Basis splines, denoted as B-splines in the literature, were developed by Schoenberg [58] and de Boor [59] and many others to provide a more open framework that could address these
types of constraints. The general equation for B-splines is: r(τ) = M X m=0 Nm,d(τ)pm, whereτ ∈[0, τmax)
pm≡ set ofM + 1 geometrically configured control points inRn Nm,d(τ)≡ set ofM + 1 basis functions of degreed
(3.3)
Note, unlike B´ezier curves, the degree of the B-spline basis functions and the number of control points is somewhat decoupled6. B-splines are a general framework for parametric splines. In fact, B´ezier curves are a special case of B-spline. See the text by Lyche and Mørken [60] for a comprehensive description. Like other parametrically based methods, a set of basis functions is used in conjunction with a set of geometrically configured control points. The linear combination of control points and basis functions constitutes the curve over a specified range of an independent parameter. Planar curves can be defined in inertial space as: r(τ) = [x(τ), y(τ)], where the parameter τ varies over the domain of the curve: 0≤τ ≤τmax. Unlike other basis function sets, B-spline curves are not necessarily defined
over the entire domain of the parameter. They use a finite sequence of points, within the parameter space, designated as knots in order to derive and anchor the basis functions. The basis functions, Nm,d(τ) are actually a set of piecewise of polynomials that are continuous
across each knot.
B-splines can be configured to replicate B´ezier curves. B-splines embed curve segment continuity within their algebraic structure in a way that is easily managed for trajectory specification. B-splines are formulated so that when basis functions have degree d, a curve can feature up to Cn−1 continuity between segments. The way the user defines the para- metric spacing of basis functions controls the degree of continuity. So, rather than use high order polynomials to generate highly complex curves, it becomes advantageous to make composites of piecewise continuous segments. Shape control of a curve can be controlled in a variety of manners through control point configuration or basis function distribution
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with respect to the independent parameter. In subsequent subsections, it will be shown how to constrain parametric curves of a specified order to fit path planning applications using B-splines. Additional control points7may be used in the B-spline formulation to more tightly control the curve shape in comparison to the B´ezier curve, which, as noted, requires an increase in polynomial order to include additional control points. Construction is still based on a linear combination of selected basis functions over an interval, typically defined as: [0,1) [59].
The B-spline basis has the following features that make it particularly suited to our application. The basis is structured to provide what is denoted as compact support so that any individual basis function only spans d+ 1 knots, Ni,d(τ) > 0 in the open interval of
knots, (τi, τi+d+1) and zero everywhere else. Consequently, moving control points outside of
this range will have no effect on a curve. The basis functions constitute a partition of unity whereby the sum of all functions for any parameter value in the full basis span of the knot vector = 1. The main advantage this approach provides is that the curve will reside within the convex hull of the control points (since each basis function is ≥ 0). The polynomial degree of these basis functions is not driven by the number of control points, except for a minimum associated with degree, so a large amount of control points does not imply high order polynomials. Continuity across curve segments is built into the basis functions and the degree of continuity across each knot span can be selectively controlled. Lastly, a large body of B-spline definition algorithms are available as this approach has been embraced by the computer graphics and animation community. B-splines permit a curve to be tailored by the manipulation of the positions of the control points in inertial space, the values of the knot sequence which anchor the basis functions within the parameter space, and the degree of the polynomials used to construct the basis functions. The only requirements for a viable knot vector are:
• For M + 1 control points and basis functions of degree d, there must be M+d+ 2 knots, since each basis function spans d+ 1 knots.
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• The first knot, τ0, must satisfy τ0 ≤ τmin, the minimum parameter over which the
curve is defined.
• The knots form an increasing sequence: τi ≤τi+1.
• The last knot, τM+d+2, must satisfyτmax≤τM+d+2.
When the knot sequence is strictly increasing, the functions do not span the entire space, so the curve is taken as the portion over which there is a basis for polynomials of degreed, which in our case is four. In our application, it is important to interpolate initial and final points. To accomplish this, the initial and final knots are repeatedd+ 1 times to constitute what is denoted as a clamped spline. We also uniformly space the internal knots to provide approximate uniform support in the determination of the splines and their relation to the control points. Note, B-spline basis functions are implicitly geometrically configured since they are related to the parameter range of the curve. By uniformly spacing the knots, the basis functions tend to be relatively equally spaced with respect to the curve parameter. While non-uniform knot spacing is feasible, without a priori information for the shape of a curve, it is not practical to develop spline curves in this manner because it yields a set of non-uniform basis functions that can result in curve distortions. Non-uniform knot spacing is best applied to curve refinement once a desired shape has been established. Consequently, our knot sequence starts at 0 and ends at M + 1−d. Between the starting knot (0) and maximum (M+ 1−d), knots are either repeated or increase with a unity interval. For ex- ample, a uniform knot sequence that supports 12 control points, uses quartic basis functions and interpolates segment endpoints is: [0,0,0,0,0,1,2,3,4,5,6,7,8,8,8,8,8]. The range of the parameter over which the B-splines form a basis in this case is [0,8), as illustrated in Figure 3.5 (for which there ared+ 1 non-zero functions across each knot span).
As mentioned earlier, an important property of B-splines is that they provide compact support, which means each basis function within a curve has a non-zero domain that is≤ the domain of the entire curve. This can be observed in the shape of the basis function illustrated on Figure 3.5. This permits local curve reshaping, while a change in most other bases (Bernstein, etc.) globally affects a curve. Compact support poses algorithmic
Figure 3.5: Quartic B-splines defined for clamped uniform knots over the range 0 to 8 (12 control points)
constraints with respect to curve definition but is generally a useful feature, especially if online re-planning is a consideration.