Efﬁcient Curvature-optimized G

-continuous Path Generation with

Guaranteed Error Bound for 5-axis Machining

Evgenia Selinger

and Lars Linsen

1,2

Jacobs University, Bremen, Germany

Westf

alische Wilhelms-Universit

at M

unster, M

unster, Germany

Keywords:

NURBS Curves, 5-axis Machining, G

-continuity.

Abstract:

Automated machining with 5-axis robots require the generation of tool paths in form of positions of the tool

tip and orientations of the tool at each position. Such a tool path can be described in form of two curves, one

for the positional information and one for the orientational information, where the orientation is given by the

vector that points from a point on the orientation curve to the respective point on the position curve. As the

robots need to slow down for sharp turns, i.e., high curvatures in the tool path lead to slow processing, our goal

is to generate tool paths with minimized curvatures and a guaranteed error bound. Starting from an initial tool

path, which is given in form of polygonal representations of the position and orientation curves, we generate

optimized versions of the curves in form of B-spline curves that lie within some error bounds of the input path.

Our approach ﬁrst computes an optimized version of the position curve within a tolerance band of the input

curve. Based on this ﬁrst step, the orientation curve needs to be updated to again ﬁt the position curve. Then,

the orientation curve is optimized using a similar approach as for the position curve, but the error bounds are

given in form of tolerance frustums that deﬁne the tolerance in lead and tilt. For an efﬁcient optimization

procedure, our approach analyzes the input path and splits it into small (partially overlapping) groups before

optimizing the position curve. The groups are categorized according to their geometric complexity and handled

accordingly using two different optimization procedures. The simpler, but faster algorithm uses a local spline

approximation, while the slower, but better algorithm uses a local sleeve approach. These algorithms are

adapted to both the position and orientation curve optimization. Subsequently, the groups are combined into a

5-axis tool path in form of two G

-continuous B-spline curves over the same knot vector.

1 INTRODUCTION

The sculptured surface machining technology has be-

come a widely used technology in manufacturing en-

gineering, e.g., for shaping metal and other solid ma-

terials. A necessary integral part is the generation

of a milling path. The milling machine cuts along a

given path to obtain the required shape of the work-

piece. For 5-axis milling, the cutter rotates around the

spindle axis, the spindle can move in all three spa-

tial directions, and the workpiece is ﬁxed to a table,

which can also move in two angular directions to al-

low the milling tool to have access to the workpiece

from different orientations. The ﬂexibility of 5-axis

machining allow for manufacturing of more compli-

cated workpieces and for higher quality. However, it

requires the control of the ﬁve axes. The ﬁrst three

axes represent the position of the milling tool in form

of xyz-coordinates, the other two are necessary to rep-

resent the orientation of the table to which the work-

piece is attached, i.e., the angle between the milling

tool and the table (see Fig. 1a).

The tool path for 5-axis machining is given in

form of a position curve of the tool tip and a respec-

tive orientation of the tool for each point on the posi-

tion curve. We assume that the orientation is given in

form of an orientation curve, where the implied orien-

tation is given by the vector that points from a point

on the orientation curve to the respective point on the

position curve. As sudden changes in the tool path

requires the robot to slow down to perform the re-

spective movements, it is desirable to have tool paths

with low curvature on both position and orientation

curve. Our goal is to minimize the overall manu-

facturing time, i.e., to generate a curvature-optimized

tool path for 5-axis machining. Our algorithms take

a given tool path and performs an optimization of po-

sition and orientation within given error bounds. As

Selinger, E. and Linsen, L.

Efﬁcient Curvature-optimized G

-continuous Path Generation with Guaranteed Error Bound for 5-axis Machining.

DOI: 10.5220/0006537400590070

In Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 1: GRAPP, pages

59-70

ISBN: 978-989-758-287-5

Figure 1: (a) 5-axis machining. (b) Orientational informa-

tion of the tool in form of lead and tilt.

this optimization is embedded into the manufacturing

workﬂow, it needs to be efﬁcient, as well.

The input to our algorithm is a sequence of pairs

(p,q) of points. The points p are the positions of

the milling tool on the workpiece, the points q are

the orientation points of the milling tool at that time.

When connecting the sequences of points p and q, re-

spectively, they describe the position and orientation

curves in polygonal representation. The orientation

vector q − p given as the difference of the position

and the orientation point can be directed at various an-

gles to the workpiece depending on the desired man-

ufacturing process. The angle is deﬁned by values of

lead and tilt. Lead is the angle between the normal

vector N of the workpiece’s surface and the orienta-

tion vector q − p of the milling tool in the direction

of movement F. Tilt is the angle between the nor-

mal vector N and the orientation vector q − p of the

milling tool perpendicular to the direction of move-

ment F (see Fig. 1b).

Our optimization procedure starts with the opti-

mization of the position curve. The curvature of the

curve shall be minimized within a given tolerance

band. This part of our approach is equivalent to the

optimization for 3-axis machining (Selinger and Lin-

sen, 2011). Then, we need to synchronize the repre-

sentation of the orientation curve with the new (op-

timized) representation of the position curve. Subse-

quently, we perform the optimization of the orienta-

tion curve. To get a desired result we use the restric-

tion that corresponding position and orientation con-

trol points have to form an orientation vector in the

same direction and length as neighboring initial vec-

tors with some small given allowed deviation of lead

and tilt (see Fig. 2). For the synchronization of the

orientation curve with the optimized position curve,

we need to build new orientation vectors starting at

the control points of the optimized position curve with

the direction depending on the direction of the given

neighboring orientation vectors with weights propor-

tional to the distance to them. Based on the given tol-

erances of lead and tilt, we built frustums around gen-

erated orientation points and restrict them lie within

these frustums.

As the optimization of the tool path is embedded

in the manufacturing process, it shall be efﬁcient. We

Figure 2: Black vectors are newly generated orientation

vectors starting from the optimized position control points.

Frustums (green) are built with respect to given tolerances

for lead and tilt (blue). Red curve is the resulting B-spline

orientation curve with the control polygon shown as dotted

dark-red lines.

propose a method that is based on local optimizations,

which are faster to compute than global computations.

Therefore, we ﬁrst split the toolpath into groups, han-

dle groups individually, and combine the results. The

handling of the individual groups is based on how

complicated their geometry is. Simple groups can be

handled with a simple spline approximation with suf-

ﬁcient quality. Complicated groups are handled using

a more complex sleeve algorithm. Section 3 describes

the overall processing pipeline, which outlines the re-

mainder of the paper.

The resulting position curve is a B-spline curve

C(u) :=

∑

j=0

(u), a ≤ u ≤ b (1)

where the interval [a,b] can be any, P

denotes the j-th

position control points, and N

denotes the j-th nor-

malized B-spline function of degree d. The B-spline

basis functions are deﬁned over a nonperiodic knot

vector

U = (a,··· , a

| {z }

d+1

,··· , u

m−d−1

,b,··· , b

| {z }

d+1

), (2)

where m = n + d + 1.

The resulting orientation curve shall also be rep-

resented as a B-Spline curve

B(u) :=

∑

j=0

(u), a ≤ u ≤ b

where Q

denotes the j-th orientation control point.

The B-spline basis functions N

and knot vector U

are identical to the position basis functions and knot

vector.

The B-spline curve constructions assure that the

curves are G

-continuous, which is a desired require-

ment for smooth tool paths for machining.

GRAPP 2018 - International Conference on Computer Graphics Theory and Applications

2 RELATED WORK

A lot of work has been done in 5-axis milling and one

of the main topics in this context is to ﬁnd a proper

position and orientation of the milling tool (Jun et al.,

2003). To make the best use of 5-axis machines they

have to solve complicated interference problems and

to determine the optimal tool orientations for com-

plex data machining. Gouging and tool collision (see

Fig. 3) are the main problems in 5-axis machining of

sculptured surfaces. Gouging denotes the removal of

the excess material in the area of the cutter contact

point due to the mismatch in curvatures between the

tool tip and part of the milling path. Collision is the

interference of the cylindrical part of the tool or tool

holder and the workpiece part. The goal is to calculate

an optimal slope of the milling tool based on a given

workpiece trajectory. We, on the other hand, assume

that a tool path is already given. Our goal is to opti-

mize the tool path respect to curvature within an error

bound of the given tool path. Some of existing papers

(Kim and Sarma, 2002; Lavernhe et al., 2007) are us-

ing kinematic constraints to calculate tool path such

as maximum feedrate and acceleration, and also con-

sider the shape of the tool tip, which are not available

for us on this step.

Figure 3: Gouging (left) and collision (right).

In some papers (Fleisig and Spence, 2001;

Langeron et al., 2004) it was suggested to ﬁnd posi-

tion P(u) and orientation Q(u) B-spline curves by in-

terpolating of position points P

and orientation points

with the restriction that the two B-spline curves sat-

isfy the criterion

∃H ∈ R so that ∀u,kQ(u) − P(u)k = H,

where H is the distance between position and orien-

tation B-spline curves. In our case, we want to get

B-spline curves which are generated using approxi-

mation technique and we are allowed to have a small

given tolerance for H.

The term of double NURBS curve was intro-

duced to deﬁne the desired result in industry. It in-

dicates that the output should be presented as two

NURBS curves with common knot vector (see (Li et

al., 2008; Yongzhang et al., 2007)). When such a dou-

ble NURBS is generated it is decomposed into simple

parts understandable by a CNC machine.

Selinger and Linsen (Selinger and Linsen, 2011) de-

scribe a method to obtain curvature-optimized B-

spline curves for 3-axis machining (see Section 5 for

details). For the 5-axes problem we introduce an ap-

proach that is based on the same ideas of local sleeve

approach and local G

-continuous spline approxima-

tion, but extends them to handle 5-axis tool paths.

3 OVERVIEW

As a part of our approach, we build upon two local ap-

proximation algorithms which we use depending on

the arrangement of the initial points. The polygonal

input curves are split into small groups of point se-

quences, to which the local algorithms are applied.

If one of the angles that are formed by three con-

secutive points of a group of points are smaller than

some threshold α

sharp

, we consider the group as being

complicated and apply the idea of threading splines

through 3D channels (see Section 5.1). This algo-

rithm provides us with a local solution with nearly

optimal curvature values. However, the optimiza-

tion involves linear programming methods, which are

computationally intense. For the non-complicated (or

simple) groups we use the idea of local non ratio-

nal cubic approximation (see Section 5.2). This algo-

rithm provides acceptable approximation results only

for such simple groups, but it is substantially faster

than the ﬁrst algorithm. Afterwards, we combine the

locally optimized groups to one global curve. Since

an average workpiece consists mostly of simple parts,

the computation times of our combined algorithm are

signiﬁcantly lower.

Based on these two local optimization algorithms,

we handle the 5-axis tool path optimization by ﬁrst

processing the position curve, then adjusting the ori-

entation curve representation, and ﬁnally processing

the orientation curve. The overall work ﬂow of our

approach is depicted in Fig. 4. The individual steps

are described as:

1. Split the sequence of initial position points into

small position groups, estimate the groups’ prop-

erties, and decide which algorithm to apply (see

Section 5).

2. For each position group, generate the local so-

lution using the appropriate algorithm, i.e., local

sleeve approach for complicated groups or spline

approximation for simple groups (see Section 5).

3. For each corresponding orientation group, gener-

ate new orientation vectors based on the obtained

position control points and given initial orienta-

Efﬁcient Curvature-optimized G

-continuous Path Generation with Guaranteed Error Bound for 5-axis Machining

Figure 4: Outline of the process.

tion vectors (see Section 4.2). Based on the deci-

sion made for the position curve, use a modiﬁed

version of the respective algorithm that was ap-

plied to the position group to get the solution for

the orientation group (see Section 6).

4. If we obtain desired results we continue with the

next position group.

5. In case a solution could not be found, we need to

detect where the algorithm failed in the orienta-

tion group and decide where we need to insert ad-

ditional control points to form a constellation that

can be solved. Insert respective additional knots

in the knot vector and recalculate position control

points according to new knots (see Section 6).

6. For orientation group we regenerate orientation

vectors according to changed position control

points and resolve.

7. If all groups have been processed, connect all

groups to double NURBS curve (see Section 7).

4 ALLOCATING GROUPS

4.1 Splitting into Position Groups

As a ﬁrst step, we need to partition the given se-

quence of position points into small curve segments.

The curve segments are represented as groups (or se-

quences) of consecutive input points. We want to

distinguish between simple and complicated groups.

As handling complicated groups will be done in a

more time-consuming processing step, it is desirable

to keep the number of complicated groups as small

as possible and to keep the complicated groups them-

selves as small as possible. A complex group is a

group that contains at least one sharp corner, i.e., an

angle smaller than a certain threshold α

sharp

. Conse-

quently, simple groups are those with no sharp cor-

ners. For complicated regions of the path we apply

the idea of threading splines through 3D channels (see

Section 5.1), for simple ones we use the idea of local

nonrational cubic approximation (see Section 5.2).

As processing time for groups increases superlin-

early with increasing group size, groups shall not ex-

ceed a certain upper limit of points n

max

. Also, very

long distances between consecutive points may make

it difﬁcult to optimize for curvature. Hence, if two

consecutive input points exhibit a distance larger than

a certain threshold d

max

, the two points shall belong to

two different groups. If two successive groups have

not been separated by the maximum-distance crite-

rion, the two groups are close together and, in general,

have not been split in an area of low curvature. In this

case, it is likely that the optimization of the ﬁrst group

generates a solution with the endpoint on the border

of the tolerance channel. This fact will make it hard

to produce a G

-continuous transition with low cur-

vature terms between the two groups. Thus, we want

the two groups to overlap, i.e., we want the groups to

share a small area. Consequently, we call this con-

nection an overlapping connection (see Fig. 5). De-

pending on which criterion caused the formation of a

group, we have different methods of how a group is to

be connected with the subsequent group when gener-

ating the overall curve. We distinguish between two

types of connections, namely a line connection and an

overlapping connection.

The algorithm of splitting points into groups was

described in (Selinger and Linsen, 2011).

4.2 Adjusting Orientation Groups

To solve the orientation part we ﬁrst need to create

new initial orientation points based on the already op-

timized of position curve and given initial data. The

GRAPP 2018 - International Conference on Computer Graphics Theory and Applications

Figure 5: Necessity for overlapping connection at critical

area.

location of the new orientation points have to be cal-

culated in such a way that these points and the corre-

sponding position control points form new orientation

vectors. Direction of the vectors is considered as the

combination of the directions of the two closest ini-

tial vectors proportional to the distance between the

optimized position control point and the initial posi-

tion points which form these closest vectors. Given

a position control point, we detect to which interval

of initial position points the position control point be-

longs. We distinguish three different cases, which are

depicted in Fig. 6. For each control point P

we com-

pute lines that go through the control point P

and are

perpendicular to the lines of the initial position curve

,...,B

. We compute those perpendicular lines for

the closest two initial position curve segments B

i−1

and B

i+1

. We determine whether the intersection of

the perpendicular lines with the initial position curve

segments belong to the initial position curve. The fol-

lowing cases occur (cf. Fig. 6):

• Point P

: one of the perpendiculars at point P

in-

tersects with line B

inside interval (solid black

line) and another one intersects with outside part

of line B

(dashed black line). Hence, points P

lies in the ﬁrst interval B

and the closest initial

orientation vector are V

• Point P

: two perpendiculars at point P

intersect

with the lines B

and B

not inside the inter-

vals. Then, we compute the distances of the inter-

section point to the control point P

(blue lines)

and P

relates to that segment with shorter dis-

tance.

• Point P

: two perpendiculars at point P

inter-

sect with lines B

and B

inside the intervals.

Then, we again make the decision based on the

distance as in the preceding case.

Figure 6: Control points determination to a proper interval.

Arrows V

,...,V

indicate initial orientation vectors, where

the starting points of these vectors are the initial position

points B

,..., B

. P

,..., P

are optimized control points of

the local position B-spline curve (red curve), blue lines are

perpendiculars to lines formed by B

,..., B

After we have found the relating initial position

curve segment we built a new orientation vector from

the position control points with the direction linearly

interpolated between the closest initial vectors in pro-

portion to the distance from them to the current posi-

tion control point.

5 POSITION CURVE

OPTIMIZATION

In this section we present the algorithm for optimizing

the position curve. This is equivalent to the optimiza-

tion for 3-axis machining. Consequently, this step

is equivalent to the approach described previously

(Selinger and Linsen, 2011). For a comprehensive

description of the entire approach, we brieﬂy describe

the main aspects of the algorithm. The main reason of

splitting data into groups is the superlinear time com-

plexity of the algorithm for complicated groups. We

have two types of connections between groups: lines

and overlapping. Separation of groups by line is ap-

plicable when the groups end at neighboring points

that are located in a distance more than some thresh-

old d

max

. Between these groups we can construct

a straight-line solution without spending any calcu-

lation time. Overlapping connections are necessary

when we have a large number of points close to each

other and collecting these points in one group might

increase running time of the algorithm signiﬁcantly,

so we have to separate them. On the other hand, sep-

arating these points in different groups might occur in

areas with high curvature and it is not possible to have

feasible local solutions. In this case, we have overlap-

ping connections by including the same points at the

end of the ﬁrst group and at the beginning of the sec-

ond group. Then, we can ﬁnd solutions locally and

Efﬁcient Curvature-optimized G

-continuous Path Generation with Guaranteed Error Bound for 5-axis Machining

the overlap assures a proper transition. For details we

refer to the literature (Selinger and Linsen, 2011).

5.1 Local Sleeve Approach for

Complicated Group

The handling of complicated groups is done by ex-

ecuting a local method based on the slefe approach

by Lutterkort and Peters (Myles and Peters, 2005;

Lutterkort and Peters, 1999a; Lutterkort and Peters,

1999b). Slefe is short for “subdividable linear efﬁ-

cient function enclosure”. The idea is to generate two

polygons that represent lower and upper boundaries

of a given B-spline curve. The construction can be

generalized to curves in space by applying the con-

struction in both coordinate-axis directions of the un-

derlying 2D domain, see Fig. 7. It is, then, referred

to as the sleeve approach. After construction of the

piecewise linear sleeve, it is sufﬁcient to constrain

this sleeve rather than the original nonlinear curve

to stay within the channel. Carefully formulated, this

approximation results in a linear feasibility problem

that is solvable using linear programming.

The construction of a sleeve around the spline

curve is equivalent to enclosing the spline curve with

linear pieces. However, in the channel problem, the

coefﬁcients of the spline curve are unknown and are

sought as the solution of the feasibility problem.

Figure 7: Sleeve ¯e, e, and channel.

Given a B-spline curve

C(u) =

∑

j=0

(u),

with control points P

. Let the B-spline basis func-

tions N

of degree d be deﬁned over a knot vector

). The control polygon l(u) of the spline is the

piecewise linear interpolant to the control points P

the Greville abscissa

∗

j+d

∑

i= j+1

/d,

i.e. l(u

) = P

The weighted second differences ∆

of the con-

trol points are deﬁned as

∆

= P

i+1

− P

, P

− P

i−1

∗

− u

∗

i−1

In addition, we deﬁne ∆

−

= min{0,∆

} and ∆

max{0,∆

Over the interval [u

∗

k+1

] the contribution of the

i-th B-spline to the distance between a spline and its

control polygon is captured by the non-negative and

convex functions

∗

) :=

(

∑

j=i

∗

− u

∗

, i > k,

∑

j=k

∗

− u

∗

, i ≤ k,

(3)

(4)

where

k and k are the indices of the ﬁrst and last ( at

most) d + 1 B-spline basis functions N

whose sup-

port spans u

∗

Then, we can formulate the restrictions for our

curve by

e(u) ≤ C(u) ≤ ¯e(u)

where

¯e(u) = l(u) + L (

∑

∆

∗

∑

∆

k+1,i

∗

k+1

)),

e(u) = l(u) + L (

∑

∆

−

∗

∑

∆

−

k+1,i

∗

k+1

)),

with u ∈ [u

∗

k+1

] and

L (a

) = a

∗

k+1

− u

∗

k+1

− u

∗

+ a

u − u

∗

k+1

− u

∗

The constraints force the sleeve, and thus the

spline, to stay inside the channel. We solve the

linear programming problem using a simplex ap-

proach. The target function we intend to minimize

∑

m−1

i=1

∑

j∈{x,y,z}

(∆

−

i, j

+ ∆

i, j

). By minimizing the ab-

solute second differences we are minimizing the cur-

vature of the spline. Sometimes the constraints cannot

be met. In this case, we need to insert further points

as described in Section 4.2.

5.2 Local G

- continuous Spline

Approximation for Simple Groups

Following the algorithm described in the approach

presented by Piegl (Piegl and Tiller, 1997) and its

GRAPP 2018 - International Conference on Computer Graphics Theory and Applications

modiﬁcations presented in previous work (Selinger

and Linsen, 2011) we generate a local approxima-

tion scheme for simple groups using two cubic B

ezier

curves to assure G

-continuous connections. Given a

sequence of input points q

,..., q

k+n

, we set the ﬁrst

control point of the ﬁrst curve and the last control

point of the second curve to q

and q

k+n

, respectively.

Using the notations of Fig. 8, we set

= q

, R

= q

k+n

The endpoint interpolation property assures C

continuity. To assure G

-continuity, we set

= P

+ αT

and R

= R

+ βT

where T

and T

are the start and end unit tangents.

The unit tangents T

and T

as well as the values for

α and β are deﬁned by the continuity constraints with

the preceding and succeeding group.

For G

-continuity at the group’s beginning we set

− 2P

+ P

= A,

where A is determined by the preceding group (or

A=0 for the ﬁrst group, i.e., if k=0). The respective

continuity requirements where the two B

ezier curve

stitch together are given by

= R

− P

= R

− R

− 2P

+ P

= R

− 2R

+ R

Figure 8: Local nonrational cubic curve approximation.

To check for validity, we build the tolerance chan-

nel and the sleeve as described in Section 5.1. If the

sleeve lies inside the tolerance channel, the approxi-

mation is sufﬁciently precise. If not, we do the steps

described in Section 6 and then try to solve the prob-

lem for the modiﬁed sequence.

6 ORIENTATION CURVE

OPTIMIZATION

We want to generate a smooth B-spline curve for the

orientation of the milling tool based on the obtained

position curve and the newly generated orientation

vectors. We can see in Fig. 9 that the orientation

curve repeats the sharp angles of the position curve,

but we want to solve the orientation problem indepen-

dently from position curve. Therefore, we transform

the problem of ﬁnding the orientation control points

in space to the problem on the unit sphere. To re-

formulate the problem we start with orientation vec-

tors which are given as the difference between posi-

tion control points and respective interpolated orien-

tation points. We can imagine that the orientation op-

timization problem is deﬁned on a sphere by moving

all position points to the sphere’s center (see Fig. 10).

After that we solve the problem with the same ap-

proach that was used for the position curve with the

modiﬁcation that the tolerance channel is replaced by

a tolerance frustum, see Fig. 11. The frustum repre-

sents the tolerances for lead and tilt in the orientation

vector. They can be speciﬁed independently. In the

end we ﬁnd the solution to the orientation curve by

adding the computed orientation vectors to the corre-

sponding position control points, i.e., by transforming

back from the sphere to the spatial representation. We

can do this operation without loss of continuity be-

cause by summing up corresponding control points of

the different B-spline curves we get a new B-spline

curve (see Fig. 12).

Figure 9: Green line connects initial position points, green

dashed vectors are initial orientation vectors, red line is a

B-spline position curve, blue line is control polygon of B-

spline position curve with control points P

,..., P

, blue vec-

tors V

,...,V

are generated new orientation vectors starting

from position control points.

To apply the local sleeve approach for compli-

cated groups and the spline approximation for simple

groups for solving the orientation curve problem we

have to modify a few parts. The main distinction from

the position curve optimization algorithm is using

the frustums to keep the control points inside them

(see Fig. 11). So, we need to add these constraints

additionally to the other restriction we have.

In some cases we might have an unfeasible orien-

tation group problem. To solve this problem we need

to achieve greater ﬂexibility of the desired B-spline

curve by adding an additional control points. We ﬁnd

a place in the orientation curve where the insertion of

Efﬁcient Curvature-optimized G

-continuous Path Generation with Guaranteed Error Bound for 5-axis Machining

Figure 10: Described algorithms use the sphere representa-

tion to solve the orientation part of the problem. Generated

orientation vectors V

,...,V

from the Fig. 9 are placed at

the origin of a sphere such that, control points P

,..., P

co-

incide. We can see that V

and V

are the same for this

example, because its have the same directions in Fig. 9.

Figure 11: Frustums are built around every vector to re-

strict control points Q

to lie inside these frustums. Con-

trol Points Q

lie inside matching frustums formed

around collinear vectors V

but they do not need to be

placed at the same position. Brown dashed line is a control

polygon Q

,..., Q

of green B-spline curve.

Figure 12: Obtained solution on the sphere is transformed

back to the initial problem using formula Q

= P

+ Q

Brown vectors are ﬁnal orientation of the tool, red line is a

position B-spline curve, green line is a orientation B-spline

curve, dashed brown line is a control polygon of orientation

curve. The two B-spline curves have the same knot vector.

a new additional control point is required. Usually

the solution fails in places where we have a longer

distance between some control points than in the rest

of the group. Thus, we want to insert new control

points in the middle of this distance. Then, we in-

sert a respective knot in the existing knot vector and

recalculate dependent control points for the position

sequence of the control points, build new orientation

vectors for the changed points, and resolve the orien-

tation group. Thus, even if we have to insert addi-

tional control points in the orientation curve, we keep

proper orientation vectors.

The procedure of the point insertion for position

B-spline curve C(u) as in Equation (1) with a knot

vector U as in Equation (2) is the following (Piegl

and Tiller, 1997):

Given a knot ¯u ∈ [u

k+1

) that we want to insert into

U to form the new knot vector U

= (u

= u

,...,u

k+1

= ¯u,u

k+2

= u

k+1

,...,u

m+1

= u

), we recal-

culate the control points using Equation (5) by

k−d

= P

k−d

= αP

+ (1 − α)P

i−1

, k − d + 1 ≤ i ≤ k,

k+1

= P

For i = k − d + 1,...,k we have

α =

¯u − u

i+d

− u

(5)

We need to generate a new control point in the middle

of the interval P

i−1

then build a new orientation vec-

tor from this point and frustum around it to get an ori-

entation control point at the desired place. To achieve

this, we set α =

. Hence, from Equation (5) we ob-

tain u =

i+d

. Afterwards, we recalculate points

i−d+1

,...,P

and try to solve the orientation problem

again.

7 COMBINING GROUPS

To complete the process we need to put all groups to-

gether in two B-spline curves with common knot vec-

tor. We have to assure G

-continuity at the connec-

tions. To fulﬁll G

-continuity requirements in case

of a line connection, we insert three additional points

lying on the line a

k+1

, where neighboring initial

points a

and a

k+1

belong to different groups. We ad-

just two groups by adding the three additional points

to the end of the ﬁrst group (i.e., after a

) and to the

beginning of the second group (i.e., before a

k+1

For an overlapping connection of two groups, we

need to distinguish whether the ﬁrst group is a com-

plicated or a simple one. If the ﬁrst group is a com-

plicated group, we calculate a new control point ly-

ing on the curve. We obtain this by double knot in-

sertion (see (Selinger and Linsen, 2011)), cutting off

the control points after this point, and adjusting the

knot sequence by removing knots after the inserted

GRAPP 2018 - International Conference on Computer Graphics Theory and Applications

ones. The second group will then start with the last

control point of the ﬁrst group. We did not mod-

ify the shape of the ﬁrst group’s curve. If the ﬁrst

group is a simple group, we delete the last few control

points until we ﬁnd a control point that was an input

point. Since B

ezier curves are endpoint interpolating,

we can start the second group with that control point.

When solving for the second group, we have to pick

the ﬁrst three control points of that group such that

-continuity is achieved. G

-continuity requires that

the last control point of the ﬁrst group has to be equal

to the ﬁrst control point of the second group, for G

continuity we need to have the proportional tangent

vector of the adjacent parts, and the G

-continuity

condition requires equality of the second derivatives

of the adjacent parts. Using notations of Fig. 13, we

obtain:

= R

k−1

− P

||P

k−1

− R

||R

k−2

− 2P

k−1

+ P

= R

− 2R

+ R

Figure 13: Connecting groups with G

-continuity.

The procedure of combining groups is described

in (Selinger and Linsen, 2011) in details for position

curves in 3-axis matching and it is the same for orien-

tation curve.

8 RESULTS AND DISCUSSION

In Fig. 14 we show a zoomed-in version of a real 5-

axis tool path that was optimized with our approach.

We can observe the correctness of the solution. Gen-

erated orientation vectors have similar directions as

the initial ones and control points of the orientation

curve stay inside the frustums.

Fig. 15 depicts the individual steps of the op-

timization of the 5-axis tool path for a 90

◦

turn.

Using initial data of orientation curve and control

points of the position curve, we build frustums around

the newly generated orientation vectors from position

control points and get a solution with orientation con-

trol points inside the corresponding frustum.

Next, we want to investigate the computation time

of our approach. For our experiments, we used a

workpiece which mostly consists of almost straight

Figure 14: Optimized 5-axis tool path.

Figure 15: Outline of the process of generating a 5-axis tool

path optimization for a 90

◦

turn. (a) Initial position and ori-

entation data. (b) Constructed frustums. (c) Generated ori-

entation vectors and curve. (d) Resulting B-spline curves.

parts, rounded parts, and turns as shown in Fig. 16

with 209 input pairs of position and orientation points.

Computation time for the 5-axis orientation algorithm

is 0.735 seconds with the tolerance of the channel be-

ing 0.0008 mm and lead and tilt tolerance being 0.1

◦

The respective 3-axis optimization when only consid-

ering the position curve takes 0.38 seconds with the

same tolerance of the channel. It takes less than dou-

ble time to compute double amount of initial points

because for almost straight areas of orientation parts

on the sphere we have a very simple solution which

we can get right away or we need to insert much less

(or even none) additional points when compared to

the position part.

Efﬁcient Curvature-optimized G

-continuous Path Generation with Guaranteed Error Bound for 5-axis Machining

Table 1: Maximum and average curvature of parts of the workpiece parts separated in groups by type of initial data: line,

simple parts with almost straight lines, rounded parts, and turns with small angles (see Fig. 16). In the ﬁrst test (upper row)

we use tolerances of lead and tilt equal to 0.5

◦

, in the second test (lower row) the tolerances are 1.2

◦

We calculate the average and maximum curvature (Selinger and Linsen, 2011) of the orientation B-spline curve separately for

different parts of the workpiece. Table 1 lists the computed curvature values. The upper row was calculated using tolerances

of lead and tilt equal to 0.5

◦

, the lower row using tolerances equal to 1.2

◦

. We can observe that for line and simple parts the

tolerances of lead and tilt do not inﬂuence the curvature because the solution is simple and control points are lying inside a

smaller frustum anyway. However, for rounded parts and turns we have high values of curvature and the higher values are

obtained for smaller angle tolerances, because the curve is forced to make sharper turns. These tests were performed on the

workpiece shown in Fig. 17.

line parts simple parts rounded parts parts with turn

av max av max av max av max

1 1.8e-10 9.4e-10 0.0018 0.0377 1.111 2.2009 18.615 329.435

2 1.8e-10 9.4e-10 0.0018 0.0377 1.105 2.0455 18.3175 311.257

Figure 16: Parts of the workpiece. (a) Rounded parts (b)

Turns and almost straight parts.

Figure 17: Parts of the workpiece. High curvature (higher

than one) of the orientation curve are highlighted by green

points, red line is a position B-spline curve, orange lines are

orientation vectors and orientation polygon. (a) Rounded

parts. (b) Turns.

We also calculate the deviation of the resulting

orientation vectors from the interpolated orientation

vectors in terms of lead and tilt. We denote by vec-

tor D the forward direction of the lead, where D is in

the forward direction of the curve, −D is the back-

ward direction of the lead. The vector R denotes the

right direction of the tilt, where R is computed by the

cross-product D × O, where O is the orientation vec-

tor of the curve. Left direction of the tilt is the di-

rection −R. For the lead, we compute the ratio of

the angle between the resulting orientation vector and

the plane that is formed by the left and right points

of the frustum, and the given angle of lead tolerance.

This ratio is 0 if the interpolated orientation vector

and the resulting orientation vector are equal, and 1 if

the maximum tolerance is used. The sign of the ratio

indicates whether the orientation has been changed in

forward or backward direction. Respectively for tilt,

we compute the ratio of the angle between the result-

ing orientation vector and the plane that is formed by

forward and backward points of the frustum, and the

given angle of tilt tolerance

We can observe that for part of the workpiece as

in Fig. 18 with allowed lead and tilt tolerance of 1

◦

the values of deviation are not very high (see Fig. 19).

However, if we decrease the allowed tolerance to 0.1

◦

we can see that deviation for some parts of the work-

piece can reach to 100% in one direction and after

that very quickly to the other direction (see Fig. 21).

This effect can be explained by the example shown

in Fig. 20. In Fig. 20, we show the orientation vectors

from Fig. 18 on the sphere. The ﬁrst few cyan orienta-

tion vectors and its frustums are almost identical. The

subsequent vectors of different red colors are start-

ing to slightly change the direction inside the same

frustums. The reason is to keep the orientation curve

straight. The resulting orientation vectors are located

in one plane, but the positions are slightly changing in

the direction of the subsequent frustum.

On the workpiece with rounded parts and turns the

deviation of the resulting vectors are shown in Fig. 22.

For the rounded part in Fig. 22a, we can see zoomed-

in picture in Fig. 23a, and for Fig. 22b the view from

the top is in Fig. 23b. In Fig. 23a, we can see that the

orientation vectors are frequently changing its posi-

tions with respect to the generated vectors due to the

minimization of the sum of the second differences. In

Fig. 23b, one can observe that the resulting deviation

vectors before the turn changed the side with respect

to the polygon formed by the centers of the frustums

in order to to decrease the curvature.

GRAPP 2018 - International Conference on Computer Graphics Theory and Applications

Figure 18: Part of the workpiece. Yellow points are control points of the position curve, blue are orientation control points.

Pink points are connection points between groups. Green lines are orientation vectors.

Figure 19: Diagram of lead (upper) and tilt (lower) devia-

tion of the orientation vectors for a given tolerance of 1

◦

for

both lead and tilt and the example in Fig. 18.

Figure 20: Resulting vectors and its frustums for problem

transformed into the sphere problem. All orientation vec-

tors (lines of different color) start from one position, which

is the center of the sphere.

Figure 21: Diagram of lead deviation of the resulting vec-

tors on the sphere for the example in Fig. 18. Color of the

points is the same as the color of the corresponding vector

in Fig. 20. Given tolerance is 0.1

◦

for lead and tilt.

Figure 22: Parts of the workpiece. Points are control points,

red curve is position B-spline curve, green curve is orienta-

tion B-spline curve. (a) Straight and rounded parts. Cyan

and orange vectors are unit resulting orientation vectors,

which indicate delay or overdrive of the resulting orienta-

tion vector in comparison to the to initial one. (b) Turns.

Dark purple and gray vectors are unit resulting orientation

vectors. Different colors indicate different sides of the poly-

gon formed by centers of the frustums.

Figure 23: Parts of the workpiece. Purple lines are inter-

polated orientation vectors. (a) Zoomed-in rounded part,

see Fig. 22a. Orange lines are resulting vectors with delay,

light blue are resulting vectors with overdrive. (b) View of

the turn from the top, see Fig. 22b. Gray and dark purple

lines are resulting vectors on opposite sides of the polygon.

As parameters for our experiments, we deﬁne dis-

tance measures as multiples of the tolerance of the

channel ε, which is half of the width of the chan-

nel. A good tradeoff in terms of computation time for

solving individual groups and for splitting and joining

overhead was found by setting the maximum number

max

of points per group to n

max

= 50. A good value

for the distance threshold d

max

for splitting into sep-

arate groups was found to be d

max

= 300ε. An angle

is reported as a sharp corner, iff its cosine is smaller

Efﬁcient Curvature-optimized G

-continuous Path Generation with Guaranteed Error Bound for 5-axis Machining

than cos(α

sharp

) = −0.85, because for corners that are

sharper than this value the oscillation increases signif-

icantly when we apply the local spline approximation

method.

9 CONCLUSIONS

We developed an algorithm for generating curvature-

optimized tool paths for 5-axis milling. The gener-

ated tool paths consist of a position B-spline curve

and an orientation B-spline curve. The position curve

can be handles like a 3-axis machining tool path. For

the orientation curve, the control points are restricted

to stay within a frustum formed by the lead and tilt

tolerances. To obtain an acceptable quality-speed ra-

tio, we combine two algorithms: a local version of

the sleeve-algorithm and a local spline approximation

with G

-continuity connections. Corresponding po-

sition and orientation point groups are solved by the

same algorithm. We solve the orientation problem on

the sphere to avoid any complications caused by loca-

tion of position points. The splitting of the original

orientation curve into groups is performed with re-

spect to the corresponding position point groups and

complexity of the groups. The local spline approxi-

mation algorithm is very efﬁcient and produces good

results for simple groups, while the local sleeve algo-

rithm is less efﬁcient but produces high-quality results

even for complicated groups.

The created algorithm can be used as a part of a

micro milling process. This approach provides new

possibilities for calculating velocity, acceleration, and

jerk proﬁles. The beneﬁts are that proﬁles are easier

to compute and that the milling can generally be per-

formed with a higher velocity. The latter is essential

for practical purposes, since the higher the velocity of

the tool tip is the faster is the manufacturing process.

ACKNOWLEDGMENTS

This work was supported by BMWi under grant num-

ber 5166/50258.

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