Path Planning Placing Bezier Curves on Corners (BC)

Another path planning method (BC) adds quadratic Bezier curves on the corner area around W_j, j 2 {2, . . ., N 1}. The quadratic Bezier curves are denoted as

jQðtÞ ¼P2

k¼0B²_kðtÞ^jQ⁰_k intersects the j-th bisector. The first and the last control point,^jQ⁰₀ and^jQ⁰₂are constrained to lie withinGj1andGj, respectively. Within each segment Gi, another Bezier curve is used to connect the end points of ^jQ withC²continuity and/orW₁with slope ofc0and/orW_Nwithcf. Hence,^jQis the curve segments of even number index:

2ðj1ÞPðtÞ ¼^jQðtÞ; j2 f2;. . .;N1g

The degree ofⁱPis determined by the minimum number of control points to satisfy C²continuity constraint, independently:

Lettcdenote the Bezier curve parameter corresponding to the crossing point of

jQ(t) on the bisector, such that

jQðtcÞ ¼Wjþdjb^j: (12)

Letyjdenote the angle of the tangent vector at the crossing point fromX-axis:

jQ_ðtcÞ ¼ ðj^jQ_ðtcÞjcosyj; j^jQ_ðtcÞjsinyjÞ: (13) The notations are illustrated in Fig.4. Due to the constraint of^jQ⁰₀and^jQ⁰₂within Gj1andGj, the feasible scope ofyjis limited to the same direction asW_jþ1is with respect to^bj. In other words, ifW_jþ1is to the right of^bj, thenyjmust point to the right of^bj, and vice versa.

Given^jQ⁰₀,^jQ⁰₂,dj, andyj, the other control point^jQ⁰₁ is computed such that the crossing point is located atWjþdj^bj and the angle of the tangent vector at the crossing point isyj. Since each control point is two-dimensional, the degrees of freedom of^jQ(t) are six. Sincedjandyjare scaler, representing^jQ(t) in terms of

jQ⁰₀,^jQ⁰₂,dj, andyjdoes not affect the degrees of freedom. However, it comes up with an advantage for corridor constraint. If we compute^jQ⁰₁ such as above, the points computed by applying the de Casteljau algorithm such that two subdivided curves are separated by thej-th bisector are represented as^jQ⁰₀and^jQ⁰₂as described in the following. The two curves are constructed by f^jQ⁰₀; ^jQ¹₀;^jQ²₀g and f^jQ²₀;^jQ¹₁;^jQ⁰₂g. We can test if the convex hull off^jQ⁰₀;^jQ¹₀;^jQ²₀g lies withinGj 1and if that off^jQ²₀; ^jQ¹₁;^jQ⁰₂g lies withinGjin (26), instead of testing that of f^jQ⁰₀;^jQ⁰₁;^jQ⁰₂g. (Note that^jQ⁰₁is not constrained to lie within corridor as shown in Fig.4.) So, the convex hull property is tested for tighter conditions against the corridor constraint without increasing the degrees of freedom.

In order to compute^jQ⁰₁, the world coordinate frameTis transformed and rotated into the local frame^jTwhere the origin is at the crossing point,^jQ(tc) andXaxis is codirectional with the tangent vector of the curve at the crossing point,^jQ_ðtcÞ.

Y

3 Piecewise Bezier Curves Path Planning with Continuous Curvature Constraint 39

Let us consider the subdivision ratio, t^∗ 2 (0, 1) such that the location of ^jQ²₀ computed by applying the de Casteljau algorithm with it is the crossing point. In other words,t^∗has^jQ²₀be at the origin with respect to^jTframe. Fig.5illustrates the control points of^jQ(t) with respect to^jTframe. Note that^jQ²₀is at the origin by the definition of^jTandt^∗.^jQ¹₀ and^jQ¹₁ are on theXaxis by the definition of^jT and Remark1. Let the coordinates of the control points be denoted as Q⁰_i ¼ ðxi;y_iÞ, i2{0, 1, 2}, where all coordinates are with respect to^jT.

Lemma 2.Given djand yj, for ^jQ(t) to intersect j-th bisector with the crossing point determined by the djand (12), and the tangent vector at the point determined by theyjand (13), it is necessary that y0y20.

Proof. Let (x(t), y(t)) denote the coordinate of ^jQ(t) with respect to ^jT. By the definition of^jT and Remark 1,Q(t) passes through the origin with tangent slope of zero with respect to^jT. That is,x(tc)¼0,y(tc)¼0 andyðt_ cÞ ¼0. Suppose that y0¼y(0)<0. Sincey(t) is a quadratic polynomial,yðtÞ_ >0 and€yðtÞ<0 fort2[0,tc).

Subsequently,yðtÞ_ <0 andyðtÞ€ <0 fort2(tc, 1]. Thus,y2¼y(1)<0 andy0y2>0.

Similarly, ify0>0 theny1>0. Ify0¼0 thenyðtÞ ¼_ 0 fort2[0, 1] andy2¼0.

Thus,y0y2¼0. □

We are to calculate^jQ⁰₁ depending on whethery0y2is nonzero. For simplicity, superscript j is dropped from now on. Without loss of generality, suppose that y0<0 andy2<0.Q²₀is represented as

Q²₀¼ ð1tÞQ¹₀þtQ¹₁

by applying (1). So the coordinates ofQ¹₀ andQ¹₁can be represented as

Q¹₀ ¼ ðat;0Þ; Q¹₁¼ ðað1tÞ;0Þ; a>0 (14) wherea>0 is some constant. Applying (1) withi¼0 andj¼1 and arranging the result with respect toQ⁰₁by using (14) gives

Q⁰₁¼

Similarly, applying (1) withi¼1 andj¼1 yields

whereaandt^∗are obtained by equations (15) and (16):

t¼ 1 determined, hence,Q(t) is infeasible. That is, (17) ends up with Lemma2.

Ify0¼y2¼0 then all control points of^jQare onXaxis (see proof of Lemma2).

In the geometric relation of control points and the points computed by applying the de Casteljau algorithm as shown in Fig.6, we obtain

x0¼ ðaþbÞt x2¼ ðaþgÞð1tÞ

a¼bð1tÞ þgt

(18)

where a > 0, b > 0, g > 0 are some constants. Using (18), Q⁰₁¼ ðx1;0Þ is represented in terms of arbitraryt2(0, 1):

x1¼ 1

The constraints imposed on the planned path are formulated as follows:

l The beginning and end position oflareW₁andW_N.

3 Piecewise Bezier Curves Path Planning with Continuous Curvature Constraint 41

l i1PandⁱP,8i2{2,. . ., 2N3} areC²continuous at the junctions.

i1Pn_i1¼ⁱP0 (22a)

ni1ðⁱ¹Pn_i1ⁱ¹Pn_i11Þ ¼niðⁱP1ⁱP0Þ (22b) ni1ðni11Þðⁱ¹Pn_i12ⁱ¹Pn_i11þⁱ¹Pn_i12Þ

¼niðni1ÞðⁱP₂2ⁱP₁þⁱP₀Þ (22c)

l The crossing points are bounded within the corridor.

jdjj< 1

2 minðwj1;wjÞ; 8j2 f2;. . .;N1g (23)

l yjhas the same direction asW_jþ1is with respect to^bj. modðffðWjþ1W_jÞ ff^b_j;2pÞ>p

)modðyj ff^bj;2pÞ>p; (24a)

modðffðWjþ1WjÞ ff^bj;2pÞ<p

)modðyj ff^bj;2pÞ<p (24b)

l jQ⁰₀and^jQ⁰₂with respect to^jTsatisfies Lemma2.

y0y20 (25)

wherey0andy2are with respect to^jT.

l jQ⁰₀and^jQ¹₀lie withinGj1.^jQ⁰₂and^jQ¹₁lie withinGj.

jQ⁰₀ 2Gj1;^jQ¹₀2Gj1;^jQ⁰₂2Gj;^jQ¹₁2Gj (26)

l {ⁱP₁,. . .,ⁱP_ni1} always lie within the area ofGi.

iP12Gi; . . .;ⁱPni1 2Gi; i2 f1;3;. . .;2N3g (27) The free variables are, for j 2 {2, . . .,N 1}, Q ¼{^jQ0, ^jQ2}, d ¼{dj}, u¼{yj}, andL¼{l0,lf}. The degrees of freedom is 6N10. The variables are computed by minimizing the constrained optimization problem:

min

Q;d;y;L J¼X^N1

i¼1

Ji (28)

42 J.‐W. Choi et al.

subject to (23), (24,b), (25), (26), and (27), whereJiis the cost function ofⁱP(t), defined in (11). Notice that the convex hull property is tested for^jQ¹₀and^jQ¹₁of the divided curves instead of^jQ⁰₁in (26). Thus, it comes up with tighter conditions for curves against the corridor constraint.