CNC technology marks the core of modern manufacturing, and CNC interpolation module is one of the most important numerical control technology modules. Avery important feature of the CNC is to implement the feed rate that consists in producing the set points based on a NC program. In the high speed machining, the feed rate is restricted by the velocity, acceleration, and jerk. And the NURBS curve is a free curve, due to the many advantages of NURBS curves, it can be well applied to the CNC feed rate interpolation. The algorithm can get more smooth feed rate curves, which makes better use of kinematical characteristics of the machine. Finally, according to each machine axis capability, one can use the feed rate control method which is verified by simulation analysis and processing to test this method. The results show that the algorithm can effectively control the speed, acceleration and jerk.

1. International Journal of Research in Engineering and Science (IJRES)
ISSN (Online): 2320-9364, ISSN (Print): 2320-9356
www.ijres.org Volume 3 Issue 8 ǁ August. 2015 ǁ PP.25-32
www.ijres.org 25 | Page
Aplanning algorithm offive-axis feedrate interpolation based
on drive and jerk constraints
Li Dong, Zhang Liqiang*, Chen Yang
(College of Mechanical Engineering, Shanghai University of Engineering Science, No.333 Longteng
Road, Songjiang District, Shanghai 201620, China)
Abstract: CNC technology marks the core of modern manufacturing, and CNC interpolation module is
one of the most important numerical control technology modules. Avery important feature of the CNC is
to implement the feed rate that consists in producing the set points based on a NC program. In the high
speed machining, the feed rate is restricted by the velocity, acceleration, and jerk. And the NURBS curve
is a free curve, due to the many advantages of NURBS curves, it can be well applied to the CNC feed rate
interpolation. The algorithm can get more smooth feed rate curves, which makes better use of
kinematical characteristics of the machine. Finally, according to each machine axis capability, one can
use the feed rate control method which is verified by simulation analysis and processing to test this
method. The results show that the algorithm can effectively control the speed, acceleration and jerk.
Keywords: CNC, interpolation, feed rate, jerk
I. Introduction
In modern CNC systems, demand for machining new curves/surfaces designed by CAD systems
keeps increasing. In order to machine a curve in conventional CNC systems, CAD/CAM systems usually
segment a curve into a huge number of small linear/circular blocks and send them to CNC systems. It has
been reported that the accuracy and efficiency of machining can be improved, using a ‘parametric
interpolator’[1, 2]
.In the servo drive to simultaneously achieve higher levels of processing efficiency and
accuracy, it is desirable to make best use of the dynamic capabilities of the machine tool, itsactuators, and
servo drives. Actuator limitations necessarily constrain the smooth velocities, accelerations and jerks of
five-axis, numerically controlled (NC) machine tool.
Several researchers have proposed parameter interpolators for the Bézier/B-spline and implicit
curves. These interpolators mainly rely on parameter approximation methods by Taylor’s expansions for
only the desired feed rate. The parametric interpolator generates a curved cutter path directly without
segmentation contour processing, and that may cause feed rate discontinuities among blocks.
In order to resolve these problems, Altintas [3]
introduced a new quintic spline interpolation
technique for more improvement of relationship between the parameter to the actual arclength, and
Farouki[4]
introduced interpolators for new PH curves. The closed-form reductions of the arc-leng thin
tegral for such curves make it possible to eliminate feed rate fluctuations caused by parameterization
errors and obtain smooth feed rate profiles.
Feed rate planning is a complex problem, which can be solved by using different methods and taking
into account different constraints. In robotics, Bob row et al.[5]
and Shin and McKay[6]
proposed at wo
pass it erative algorithm based on the constraints inter section principle. The idea is ostart from the
beginning of the tool path and to go as fast as possible even if some constraints are not respected. Then in

2. Aplanning algorithm of five-axis feed rate interpolation based on drive and jerk constraints
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the reverse pass, the procedure is repeated with the additional constraints which the feed rate is lower
than the forward pass. Finally, acorrective algorithm is applied to connect both passes with velocity and
acceleration constraints. The two pass algorithms werere used and improved. Renton and
Elbestawi[7]
researched on the velocity and acceleration limits determination. Timar et al[8]
used
polynomial parametric curves on which they could obtain a closed form solution for the feed rate
planning problem with axis acceleration constraints. Dong and Stori[9]
tried to prove the optimality of the
two pass algorithm. All the previously cited article stook velocity and acceleration constraints of the
drives. But it is well known that jerk is an important parameter which should be considered as well. And
in many high speed machine operations jerk is the parameter which constraints the feed rate variations.
The effect of jerk limitation on the mechanical structure was researched in detail by Barre et al[10]
. It
is clear that jerk has to be limited to reduce the frequency content of the trajectory and to avoid exciting
the natural modes of the structure. However, several articles are dealing with the tangential jerk only
(third derivative with respect to the time of the tool/ work piece movement). This can be interesting for
man ipulators but to avoid vibrations of a machine tool structure, each axis jerk(third derivative with
respect to the time of the axis movement) limit has to be considered too. Liu et al.[11]
modified the feed
rate profile to take into account the jerk and the natural frequencies of the machine tool. This method
should be applied carefully to control the contour error generated.
But the main problem is that the constraint on the predefined profile limits only the tangential
derivatives. In practice each axis has its own limitations; furthermore with linear and rotary drives it is
impossible to make the link between tangential jerk and axis jerk due to the non-linear kinematical
transformation.
Jerk limits to determine a specific minimum time track issues into the feeding speed profile greatly
increases the complexity. The size of the problem becomes larger, increasing the jerk limitation and
optimization space is no longer one-dimensional, and is suitable for line search or scanning techniques.
Although previous significant work has been done or feed minimum time trajectory generation capacity
optimization with an ax to consider rate, relatively little progress has been made in optimizing the
development of technology to efficiently obtain the best solution to restrict the feed rate question, jerk
limitation.
While velocity and machine capabilities of the axes’ drives are necessary constraints in any such
attempt to exploit the dynamic capabilities of a machine to the fullest, it is also compatible to impose
constraints on the jerk experienced by the system. Without jerk constraints, the acceleration profile that
results from the optimization would be discontinuities. And the discontinuities correspond to step
changes in the force result limit of the drive, rising to large contouring errors, exciting vibrations in the
transmission and bearing elements of the drives, producing noise during operation and, in general,
accelerating wear in the system.
The goal of this paper is to present a feed rate planning solution for five-axis structures with jerk and
drive constraints on each axis. A decoupled algorithm is used to detach the geometrical problem to the
temporal interpolation. In addition, the mathematical formula used in here allows treatment of rotation in
the same way. First the geometrical work is performed. Then the kinematical transformation is used to
obtain the joint movements. After that, the real-time interpolation is carried out with the use of a
constraint intersection principle and a formulation which will be detailed. Finally the result is the
presented axis set points inspecting velocity, acceleration and jerk constraints of each drive.

3. Aplanning algorithm of five-axis feed rate interpolation based on drive and jerk constraints
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II. Feed rate planning algorithm
2.1 Driving characteristics of machine
Using the formula for the definition of federate along the trajectory curve (Equation 1), it is
necessary to express the geometryr multiplied by a function of the motion r. Then, the motion is
decoupled from the geometry. And this is available for linear and rotary axes. Furthermore, the
acceleration A and jerk J of the drives are calculated identically by Equations 2 and 3.
( )
( ( )) ( )
dr u du
V u t r u
dt dt
   (1)
( ( ))
( ( ))
dV u t
A u t
dt
 (2)
2
2
( ( ))
( ( ))
d V u t
J u t
dt
 (3)
If the curve parameters u is strictly regulated by the arc length parameterization, then the velocity,
acceleration and jerk can be simply obtained in Equations 4, 5 and 6.
( )
( ) ( )
j
j jdr u du
V u r u
dt dt
   (4)
2
2
2
( )
( ) ( )( ) ( )
j
j j jdV u du d u
A u r u r u
dt dt dt
    (5)
2 3
3
2 3
( )
( ) ( )( ) 3 ( )( )( ) ( )
j
j j j jda u du d u du d u
J u r u r u r u
dt dt dt dt dt
      (6)
( )r u , ( )r u , ( )r u are the geometrical derivatives with respect to the displacement u along the tool
path. They are known as soon as the geometrical treatment of the tool path is realized. The drives of the
machine of the velocity, acceleration and jerk of each individual are limited. For a 5_axis machine tool,
the velocity constraints for each discretized point j along the tool path are presented in the equation 7.
￣
max,
max,
max,
max,
max,
x
y
z
a
c
v
v
v
v
v
 
 
 
 
 
 
  
≤
j
s
j
s
j
s
j
s
j
s
X
Y
Z
A
C
 
 
 
 
 
 
 
  
≤
max,
max,
max,
max,
max,
x
y
z
a
c
v
v
v
v
v
 
 
 
 
 
 
  
(7)
Because it is usually used in the machine characteristics, all the constraints are set to be symmetrical.
Then the following set of inequalities is calculated respectively velocity, acceleration and jerk limits.
Thenotation | | is on behalf of the absolute value of each scalarterm.
max,( )j
jV u V
,
max,( )j
jA u A
,
max,( )j
jJ u J
(8)
The purpose of this algorithm is to be calculating the next reachable point with fixed ΔT known all
the previous points. The constraints proposed in [1] are done by intersecting. Each constraint can be

4. Aplanning algorithm of five-axis feed rate interpolation based on drive and jerk constraints
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reduced to a polynomial inequality when use the discretization. Solving the inequality, a fixed interval
over each constraint is verified. The intersection of all these intervals gives time interval solutions [smin,
smax] to all constraints which are respected in this step. This interval can be zero then the feed rate is too
high at the entrance of a sharp curvature area. The minimum and maximum allowable solutions smin and
smax are preserved.
2.2 Detailed algorithm
For a discretized algorithm, two discretizations are conceivable. A geometrical discretization in Δu
or in Δs and a discretization in time Δt. The problem of the discretization in Δs or Δu is sending the set
points to the controller with a fixed frequency eventually. So at really low feed rate you need a small Δ
sincrements to be closed to the desired command frequency therefore at high feed rate many useless
points will be computed. Fore more, with a fixed Δs the evaluation of the acceleration and jerk will be
really fluctuant. That is why afixed Δt has been chosen for the algorithm that presented here. The velocity,
acceleration and jerk are obtained as follow:
1
1
j j
j
u u
u
t






,
1
1
j j
j
u u
u
t





 

,
1
1
j j
j
u u
u
t





 

(9)
In order to show clearly how the algorithm works, it has been used in a really simple example. The
tool path is a straight line which aims to achieve the programmed feedrate from the rest. The feed rate
will be limited only by the programmed feed rate and the jerkof the axis which are respectively 5 m/min
and 5 m/s3
.Tofind the switching point where smin has to be chosen instead of smax a formulation is used.
Using the discretization, every constraint can be calculated in a polynomial inequations (10)–(12). The
functions qA0-2, qJ0-3 can be calculated in equations in equations (4)-(6) and (9). Those equations count on
the positions uj, uj-1, uj-2calculated in the previous iterations.
11
max, max,,
j jj
i iu j
u u
V r V
t


  

 (10)
2
max, max,1 2 1 1 0i ij A j A AA u q u q q A       (11)
3 2
max, max,1 3 1 2 1 1 0i ij J j J j J JJ u q u q u q q J          (12)
There is no need to try to prove the optimality of the solution, because it depends on the time step Δt
and selects a calculation scheme of the discrete derivatives. The designed algorithm is not optimal but it
is powerful. It gives a solution, which is very close to mathematical the best solution. A better solution
could be obtained by choosing points in the middle of the [smin,smax] interval, but the computation load
will be complex.
III. Simulation and experimental results
The experiment is carried out on a Self-developed five-axis CNC machining system which is
presented in Figure 1. The machine is controlled by a series of parameters which allows the measurement
of the position and velocity of each axis during the movement. The cycle time of the position control loop
is 6 ms. The kinematical characteristics of the machine are given in Table 1.

5. Aplanning algorithm of five-axis feed rate interpolation based on drive and jerk constraints
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Fig. 1Double blade five-axis machining experiment
X Y Z A C
Vmax
(m/min)
30 30 30 15 20
Amax
(m/s2
)
2.5 3 2.1 0.83 0.83
Jmax
(m/s3
)
5 5 50 5 100
Table1: Machine tool drive limits
The tool path is shown in Figure 2. The NURBS format is very suitable for that purpose and CAM
software offer the possibility to generate NURBS programs. This format facilitates the fee drate
interpolation process as there is no need to modify the geometry. And that is why many research papers
start from that format. Here the only thing to do is to find the relation between the path parameter r and
the arc length u of the curve. The velocity, acceleration and jerk of each axis are computed based on the
measurement of the position set points.
Fig. 2NUBRS tool path
The velocity, acceleration and jerk of each axis are computed based on the measurement of the
position set points. It allows to get rid of the noise generated by the mechanical transmission which can

6. Aplanning algorithm of five-axis feed rate interpolation based on drive and jerk constraints
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be seen on the linear encoders. To make a fair comparison, the feed rate planning is realized with Δt = 6
ms and the derivatives are computed in the same manner using the derivative scheme of the Equation 6.
Fig. 3Without drive and jerk constraints
Fig. 4 Underjerk and drive constrains
The algorithm is designed to be used off-line but the computation time is satisfactory considering
that the algorithm is developed in Matlab environment. The results of the algorithm are presented in
Figure 3 and 4. The comparison with the measurement is made on CNC machine. We can see the feed
rate is mainly limited with the programmed constraint of X-, Y-, Z-, A-, C-axes. In the picture, one can
see the vibration of the velocity, acceleration and jerk without constraints are higher than those with
constraints. One can see that after the discretized algorithm, the curve of velocity, acceleration and jerk
relatively become smoother. It is important that the curvature of a larger area of the track curves of tool

7. Aplanning algorithm of five-axis feed rate interpolation based on drive and jerk constraints
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feed rate is reduced accordingly, so that it can effectively avoid the constraints on quality of machine
parts and damage to structure of machine. From the two pictures, one can see no matter what are under
the limit, which illustrates the effectiveness of the algorithm in control of velocity, acceleration and jerk.
IV. Conclusion
High speed machining involving the high velocity and acceleration, and both do harm to the
machine and the surface quality of the work piece. In order to solve this problem, it is necessary to
control each axis kinematics parameters (speed, acceleration and especially acceleration), which
associated with the tool and the work piece movement. The control of these parameters always takes
much in the productive cost, and that does not take the function of the characteristic machine. This paper
proposes a unified and efficient solution to minimize the processing time with the machine kinematics
performance better use.
For any given five-axis machine structure, this paper derived the relationship between velocity,
acceleration and jerk of NURBS curve in five-axis machine generated for each axis. And according to a
given machine axes maximum feed rate and maximum acceleration constraints, it can get a feasible
extent of maximum feed rate and acceleration. On the basis above, we can obtain the maximum extent of
feed rate and acceleration under the combined effects with various constraints.
Our method is based on a decoupled approach which separates the problem of geometrical treatment
of the program med tool path and of feed rate in terpolation. In the first stage, the local rounding of the
geometry is achieved according to well-known strategies. The novelty in solving the global problem lies
in the treatment performed for the feed rate in terpolation considering the previously defined geometry.
Several examplesin3to5-axisdemonstratethatthealgorithm is efficient and that the jerk of each axisis
respected. The results are compared with the measure ements made on the five-axis milling machine.
Finally, the example demonstrates that the proposed method could be widely used.
V. Acknowledgement
The research is sponsored by the Innovation Program of Shanghai University of Engineering
Science for Graduate Students (No.15KY0107).
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