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Page 344 Sellers et al. Intell Robot 2022;2(4):333­54 I http://dx.doi.org/10.20517/ir.2022.21

Figure 7. Illustration of the B-spline function. (A) The improved segmented B-spline curve. (B) The same path smoothed by the fundamen-
tal B-spline and the improved B-spline function.

{
1, ≤ ≤ +1
, ( ) = ; ∈ [0, 1] (13)
0, ℎ
Geometric continuity is the metric used to evaluate smoothing methods, which is defined by the tangent
2
unit and curvature vector at the intersection of two continuous segments [38] [Figure 7A]. To achieve con-
2
tinuity, the control points of B-spline curve to the path point, , is defined as

1 = − (1 + ) 2 −1
2 = − 2 −1
(14)
3 =
4 = + 2
5 = + (1 + ) 2 −1
where is smoothing length ratio = 1 / 2, and −1 defines the unit vector of −1 . represents the unit
vector of the line +1. The combined sum of 1 and 2 is the smoothed length. The half of the corner angle
is denoted as: Γ = /2. Using a knot vector of [0, 0, 0, 0, 0.5, 1, 1, 1, 1], the smoothing error distance and
maximum curvature within the smooth path can be expressed as:

2 sin Γ
= (15)
2
= 4 sin Γ (16)
2
3 2 cos Γ
Usingthepreviousequationsthesmoothingerrordistance canbedefinedbytheexistingmaximumcurvature
given by the robot:

2
= 2 tan Γ (17)
3
The improved B-spline model has specific advantages over the basic B-spline model, one of which is its ability
to smooth many different trajectories with various angles, as seen in Figure 7B. The curve produced by the im-
proved B-spline model is significantly closer to the original path than the original model. When considering
the constraints of the robot, the improved B-spline mode performs better in various degrees of angles. The
overall advantages of the improved B-spline model are as follows:

(1) The path generated is tangent and curvature continuity, so that the robot can have a smooth steering com-
mand, which can correct any discontinuity of normal acceleration and establish a safer path for the robot to
follow.