JPH0658603B2 - Trajectory interpolation method for automatic machines - Google Patents

Description

The present invention relates to a method for interpolating a discrete point sequence input as a passing point of a tool with a smooth curved trajectory in an automatic machine such as an NC device.

[Conventional technology]

As a method of giving trajectory information of automatic position control devices such as NC machine tools and industrial robots, a method of giving a discrete point sequence on the trajectory (PTP trajectory information) and a method of giving the trajectory as continuous information (CP Orbit information).

Comparing the two, the PTP trajectory information has a much smaller amount of information, the trajectory can be easily generated, the trajectory can be partially corrected or added, and the speed can be easily set. There are advantageous attributes.

However, in order to actually generate a trajectory by the automatic position control device, it is finally necessary to give a continuous position command to the servo system.

Therefore, in order to generate the trajectory of the automatic position control device based on the PTP trajectory information, it is necessary to convert the PTP trajectory information into the CP trajectory information. This conversion is called a trajectory interpolation method.

Conventionally, various methods have been proposed as a trajectory interpolation method that connects a plurality of point sequences with a smooth curve such that the first derivative and the second derivative are continuous.

As a well-known conventional orbital interpolation method,
There are polygonal line interpolation method, circular arc interpolation method, and spline function interpolation method.

[Problems to be solved by the invention]

Among the trajectory interpolation methods listed above, the polygonal line interpolation method interpolates between two points with a straight line, and is the simplest trajectory interpolation method. In the trajectory obtained by this interpolation method, the position is continuous, but the first derivative, that is, the velocity is generally discontinuous.

The circular arc interpolation method interpolates three points with the same circular arc, and the position, velocity, and acceleration are continuous on the same circular arc.
However, generally, the position is continuous at the contact points of the two arcs, but the first derivative or velocity is discontinuous.

The spline function interpolation interpolates between discrete point sequences by an nth power function, and at this time, continuity up to the (n-1) th derivative is guaranteed. However, spline function calculation requires a large amount of calculation time when the number of points is large, because simultaneous linear equations of an n × n matrix must be solved when a sequence of n points is given for the entire trajectory. And
Therefore, even the NC control device cannot process in real time. Therefore, off-line operation is performed by an external operation device, the result is decomposed into a conventional NC language program, and N
The C controller was moving.

The present invention has been made in view of such conventional problems, and an object thereof is to perform trajectory interpolation while calculating a spline function in real time.

[Means for solving problems]

In order to achieve this object, the interpolation method of the present invention uses the first quadratic curve and the (m + 1) th point passing through the m-th point and points before and after it in the point sequence given as the passing point of the tool. A second quadratic curve passing through the first and second points, and the m-th point and the m + 1-th point as the starting point and the ending point, respectively, and the first-order and second-order differential coefficients at the starting point are the first quadratic curves. Of the second quadratic curve whose linear and quadratic derivatives at the end point are equal to those of the second quadratic curve, and based on the spline function, the m-th point and m + 1 are derived.
It is characterized in that the trajectory is interpolated to and from the second point.

[Action]

In the present invention, as a passage point of the tool, a sequence of n points P ₀ (X ₀ , Y ₀ , Z ₀ ), P ₁ (X ₁ , Y ₁ ,
Z ₁ ), ... P _n (X _n , Y _n , Z _n ) is given, and a quadratic curve passing through P _m-1 , P _m , and P _{m + 1} is represented by a parameter t.

x _{m (t)} ⁼ at _{a x t 2 + b x t} + c x y m (t) = a x t 2 + b y t + c y z m (t) = a z t 2 + b z t + c z Here, the parameter t, t = -1 passes through the point P _m-1 , t = 0 passes through P _m , and t = 1 passes through the point P _{m + 1} . Note that only the x-axis will be considered to simplify the description. Therefore, the following relationship holds for the quadratic curve x _m .

_{x m (-1) = a x} -b x + C x = X m-1 x m (0) = c x = X m x m (1) = a x + b x + c x = X m + 1 From this _a x, b _x, when obtaining the _{c x,} the _{a x = (X m-1} + X m + 1) / 2-X m b x = (X m + 1 -X m-1) / 2 c x = X m. Therefore, in −1 ≦ t ≦ 1, x _m (t) = {(X _m−1 + X _{m + 1} ) / 2−X _m } t ² + (X _{m + 1} −
X _m−1 ) / 2 · t + X _m . Similarly, when a quadratic curve passing through the points P _m , P _{m + 1} , and P _{m + 2} is obtained, in −1 ≦ u ≦ 1, x _{m + 1} (u) = {(X _m + X _{m + 2} ) / 2−X _{m + 1} } u _{^{2 + (X m + 2 -X}} m
/ 2 · u + X _{m + 1} . Here, the interpolation curves of the sections P _m and P _{m + 1} are defined as follows.

First, sections P _m and P _{m + 1} (0 ≦ t ≦ 1, −1 ≦ u ≦ 0)
Then, x _m (t) = {(X _m−1 + X _{m + 1} ) / 2−X _m } t ² + (X _{m + 1} −
X _m−1 ) / 2 · t + X _m x _{m + 1} (u) = {X _m + X _{m + 2} ) / 2−X _{m + 1} } u ² + (X _{m + 1} −X _m ).
/ 2 · u + X _{m + 1 When the} above equation is replaced with u = t−1 and rewritten, x _{m + 1} (t) = {(X _m + X _{m + 2} ) / 2−X _{m + 1} } (t−1) ² + (X
_{m + 2 -X m) / 2} · (t-1) + to become _{X m + 1.} Here, the interpolation curve S _m (t) of the sections P _m and P _{m + 1}
Where S _m (t) = x _m (t) (1/2 + 1/2 cosπt) + x
_{m + 1} (t) (1 / 2-1 / 2 cosπt) (see FIG. 1).

Similarly, the interpolation curve S _m-1 (t) of the sections P _{m + 1} and P _m is S _m-1 (t) = x _m-1 (t) (1/2 + 1/2 cosπt) + x _m (t) (1 / 2 + 1/2 cos
πt) x _m-1 (t) = {(X _m-2 + X _m ) / 2-X _m-1 } t ² + (X _m −
_{X m-2) / 2 ·} t + X m-1 x m (t) = {X m-1 + X m + 1) / 2-X m} (t-1) 2 + (X m + 1 -X
_m−1 ) / 2 · (t−1) X _m . Here, the primary differential coefficient and the secondary differential coefficient at the point P _m are obtained.

Since S _m-1 (t) passes through P _m when t = 1, Next, when S _m (t) is similarly obtained, S _m (t) is t =
Since it passes through the point P _m at 0, From the above, it can be seen that S _m-1 (t) and S _m (t) are connected continuously at the point P _m with both the first derivative and the second derivative.

Although the above description is for the x-axis only, the same holds true for the y-axis and the z-axis.

If this is rearranged again, the points P ₀ , (X ₀ , Y ₀ ,
Z ₀ ), P ₁ (X ₁ , Y ₁ , Z ₁ ), ... P
_{When n} (X _n , Y _n , Z _n ) is given, the interval P _m ,
The interpolation curve S _m of P _{m + 1} can be obtained as follows.

However, 0 ≦ t ≦ 1, 1 ≦ m ≦ n−2.

_{x m (t) = {(} X m-1 + X m + 1) / 2-X m} t 2 + (X m + 1 -
_{X m-1) / 2 ·} t + X m y m (t) = {(Y m-1 + Y m + 1) / 2-Y m} t 2 + (Y m + 1 -
_{Y m-1) / 2 ·} t + Y m z m (t) = {(Z m-1 + Z m + 1) / 2-Z m} t 2 + (Z m + 1 -
Z _m-1 ) / 2 · t + Z _m x _{m + 1} (t) = {(X _m + X _{m + 2} ) / 2-X _{m + 1} } (t-1) ² + (X
_{m + 2 -X m) / 2} · (t-1) + X m + 1 y m + 1 (t) = {(Y m + Y m + 2) / 2-Y m + 1} (t-1) 2 + (Y
_{m + 2 -Y m) / 2} · (t-1) + Y m + 1 z m + 1 (t) = {(Z m + Z m + 2) / 2-Z m + 1} (t-1) 2 + (Z
_{m + 2 -Z m) / 2} · (t-1) + Z m + 1 S xm (t) = x m (t) (1/2 + 1 / 2cosπt) + x m + 1 (t) (1 / 2-1 / 2cosπ
_{t) S ym (t) =} y m (t) (1/2 + 1 / 2cosπt) + y m + 1 (t) (1 / 2-1 / 2cosπ
_{t) S zm (t) =} z m (t) (1/2 + 1 / 2cosπt) + z m + 1 (t) (1 / 2-1 / 2cosπ
t) Therefore, in the sections P _m and P _{m + 1} , the point P
_m-1 (X _m-1 , Y _m-1 , Z _m-1 ), point P _m (X
_m , Y _m , Z _m ), point P _{m + 1} (X _{m + 1} , Y _{m + 1} ,
Z _{m + 1} ), point P _{m + 2} (X _{m + 2} , Y _{m + 2} , Z
_The interpolation curve can be obtained from the data of only 4 points of ( _{m + 2} ). However, for the sections P ₀ and P ₁ , X ₀ (t),
A quadratic curve determined by Y ₀ (t) and Z ₀ (t) is divided into sections P _n−1 ,
Regarding P _n , X _n-1 (t), Y _n-1 (t), Z _n-1
Each quadratic curve determined by (t) is used as an interpolation curve for that section.

〔Example〕

Next, an example in which the present invention is applied to an NC device will be described.

The points P ₀ (X ₀ , Y ₀ , Z ₀ ), P ₁ (X
₁ , Y ₁ , Z ₁ ), ... Position information of P _n (X _n , Y _n , Z _n ) and the division number k of how many straight line segments are divided in the sections P _m and P _{m + 1.} input.

t = 0, 1 / k, 2 / k, ... n / k, ... k / k
Can be sequentially substituted into the equation (1) to obtain S _x , S _y , and S _z .

By performing the linear interpolation connecting these points with the NC control device, it is possible to perform a machining operation such as cutting with a smooth interpolation curve passing through the points P ₀ , P _1, ... P _n .

〔The invention's effect〕

As described above, in the present invention, a smooth spline curve is obtained and interpolation is performed by sequentially performing operations only on four points in the vicinity of the current position among a plurality of point sequences. Therefore, the amount of calculation becomes constant regardless of the number of input points, and the sequential processing becomes possible. Therefore, it becomes possible to perform processing while calculating online, and it is possible to save the time and cost for creating a program by an off-line CAD or tape creator as in the past.

[Brief description of drawings]

FIG. 1 is a diagram for explaining the interpolation of a sequence of points according to the method of the present invention.

Claims (1)

[Claims] 1. A sequence of n + 1 points P ₀ (X ₀ , Y ₀ , Z ₀ ), P ₁ (X ₁ , Y ₁ ,
Z ₁ ), ... P _n (X _n , Y _n , Z _n ) of the m-th (1 ≦ m ≦ n−2) point P _m (X _m , Y _m ,
Z _m ) and the points P _m-1 (X _m-1 , Y _m-1 ,
Z _m−1 ), P _{m + 1} (X _{m + 1} , Y _{m + 1} ,
The first quadratic curve passing through Z _{m + 1} ) and the m + 1-th point P _{m + 1} (X _{m + 1} , Y _{m + 1} , Z _{m + 1} ) and the points P _m (X _m , Y _m , Z _m ), P _{m + 2} (before and after). X _{m + 2} ,
Y _{m + 2} , Z _{m + 2} ) and the second quadratic curve is derived, and the m-th point and the m + 1-th point are set as the start point and the end point, respectively, and the first and second derivatives at the start point are the first quadratic coefficient. The following equation S _xm which is equal to that of the curve and whose primary and quadratic derivatives at the end point are equal to those of the second quadratic curve
(t), S _ym (t), S _zm (t) Based on the spline function represented, in the automatic machine characterized by performing the interpolation of the trajectory between the m-th point and the m + 1-th point Trajectory interpolation method. _{S xm (t) = x m} (t) (1/2 + 1 / 2cosπt) + x m + 1 (t) (1 / 2-1 / 2cosπ
_{t) S ym (t) =} y m (t) (1/2 + 1 / 2cosπt) + y m + 1 (t) (1 / 2-1 / 2cosπ
_{t) S zm (t) =} z m (t) (1/2 + 1 / 2cosπt) + z m + 1 (t) (1 / 2-1 / 2cosπ
t) However,