Aplanning algorithm offive-axis feedrate interpolation based on drive and jerk constraints

7/17/2019 Aplanning algorithm offive-axis feedrate interpolation based on drive and jerk constraints

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International Journal of Research in Engineering and Science (IJRES)

ISSN (Online): 2320-9364, ISSN (Print): 2320-9356

www.ijres.org Vol ume 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 highspeed 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 foronly 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



Aplanning algorithm of five-axis feed rate interpolation based on drive and jerk constraints


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 withrespect 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 thetransmission 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.





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 duV 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 udt dt

(4)

22

2

( )( ) ( )( ) ( )

j j j jdV u du d u

A u r u r udt dt dt

(5)

2 3

32 3

( )( ) ( )( ) 3 ( )( )( ) ( ) j

j j j jda u du d u du d u J u r u r u r udt 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

j A u A,

max,( ) j j J 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





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 uu

t

,

1

1

j j

j

u uu

t

,

1

1

j j

j

u uu

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 u j, u j-1, u j-2calculated in the previous iterations.

11max, max,,

j j ji iu j

u uV r V

t

(10)

2max, max,1 2 1 1 0i i j A j A A A u q u q q A (11)

3 2max, max,1 3 1 2 1 1 0i i j J j J j J J J 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 itis 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 [s min,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.





Fig. 1Double blade five-axis machining experiment

X Y Z A C

Vmax

(m/min)30 30 30 15 20

Amax (m/s

2)

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





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





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 agiven 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).

References

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[2] Shpitalni M, Koren Y, Lo C-C. Real time curve interpolators. Comput-Aid Des. 1994. 26(11):832 – 8.

[3] Erkorkmaz K, Altintas Y. High speed CNC system design. Part I. Jerklimited trajectory generation and quintic splineinterpolation. Int JMach Tools Manufact. 2001. 41:1323 – 45.

[4] Farouki RT, Shah S. Real-time CNC-interpolators for Pythagorean – Hodograph curves. Comput-Aid Geom Des. 1996.

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[9] S.D.Timar,R.T.Farouki,T.S. Smith, C.L. Boyadjieff. Algorithms for time optimal control of CNC machines along curved

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Aplanning algorithm offive-axis feedrate interpolation based on drive and jerk constraints

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