US 2019/0317468 A1 · AN EQUIVALENT-PLANE CROSS-COUPLING CONTROL METHOD

Abstract

The invented equivalent-plane cross-coupling control method belongs to high-precision and high-efficiency intelligent multi-axis CNC (Computer Numerical Control) machining filed, featured a three-axis cross-coupling controller based on the equivalent plane which can be used for improvement of the three-dimensional contour-following accuracy. This method first find the foot point from the actual motion position to the desired contour using a tangential back stepping based Newton method. Then, establish an equivalent plane which containing the spatial contouring-error vector by passing through the actual motion position and the tangential line at the foot point. After that, estimate the three-dimensional contouring error in a scalar form, thus controlling the signed error using a PID based two-axis cross-coupling controller. Finally, calculate the three-axis control signals according to the geometry of the equivalent plane, thus realizing the three-axis contouring-error control by using the well-studied two-axis contour controllers.

An equivalent-plane cross-coupling control method;

Its characteristic is that this method approximates the foot point from actual motion position to the desired contour using a tangential back stepping based Newton algorithm, and establishes an equivalent plane where the contouring-error scalar instead of vector can be obtained so that the PID controller based two-axis CCC can be utilized to constrain the spatial contouring error, thus improving the contour-following accuracy of three-axis CNC systems;
The detail procedures are as follows:
Step One:
Establishment of the equivalent plane
Denote the equation of the desired contour as C=C(u), where u is the curve parameter;
In addition, the desired and actual motion positions are denoted as R=[rx,ry,rz] and P=[px, py, pz], respectively;
The parameter corresponding to the desired position is denoted as ur;
Define a tangential error dt(u) on a point of the desired contour C(u) as the projection of the vector C(u)−P on the tangential direction at the position of C(u), and it is computed as:
d
t

(
u
)
=
(
C

(
u
)
-
P
)
·
C
′

(
u
)

C
′

(
u
)

(
1
)
where C′(u) is the first-order derivative of C(u) with respect to u, and ∥ ∥ means the Euclidean norm;
The contouring error is defined as the orthotropic distance from the actual motion position to the desired contour;
Therefore, the defined tangential error dt(u) must be zero when C(u) happens to be the foot point from P to the desired contour;
As a consequence, the foot-point parameter, denoted as uf can be obtained by solving the formula dt(u)=0;
To deal with this problem, the tangential back-stepping point parameter ub is first calculated by projecting the tangential error to the desired contour as:
u
b
=
u
r
-
(
C

(
u
r
)
-
P
p
)
·
C
′

(
u
r
)

C
′

(
u
r
)

2
(
2
)
Then, the parameter ub is taken as the initial value of the Newton method so as to find the solution uN of dt(u)=0 by:
u
N
=
u
b
-
(

C
′

(
u
b
)

2
·
(
C

(
u
b
)
-
P
)
·
C
′

(
u
b
)
)
(

C
′

(
u
b
)

4
+

C
′

(
u
b
)

2
·
(
C

(
u
b
)
-
P
)
·
C
″

(
u
b
)
-
(
(
C

(
u
b
)
-
P
)
·
C
′

(
u
b
)
)
·
(
C
′

(
u
b
)
·
C
″

(
u
b
)
)
)
(
3
)
At last, judge whether the convergence of the Newton method;
If |dt(uN)|<|dt(ub)|, indicating that it is convergent, take the solution as the foot-point parameter uf, i.e. let uf=uN;
Otherwise, applying the tangential back stepping method again at ub to obtain the foot-point parameter uf;
To summarize, the foot-point parameter ur is calculated by the following equation:
u
f
=
{
u
N
,

d
t

(
u
N
)

<

d
t

(
u
b
)

u
b
-
(
C

(
u
b
)
-
P
)
·
C
′

(
u
b
)

C
′

(
u
b
)

2
,

d
t

(
u
N
)

≥

d
t

(
u
b
)

(
4
)
Establish the equivalent plane by passing through the actual motion position P and the tangential line of the desired contour at the approximated foot point C(uf);
By doing this, the normal vector nE of the equivalent plane can be computed by:
n
E
=
(
C

(
u
f
)
-
P
)
×
C
′

(
u
f
)

C
′

(
u
f
)

(
5
)
Where x means the outer production;
The equivalent-plane horizontal axis, denoted as XE, is got as the intersection direction of the equivalent plane and the original XY plane;
Furthermore, the equivalent-plane vertical axis, denoted as YE is taken as a direction that perpendicular to XE and nE;
Their computation method is:
{
X
E
=
n
E
×
[
0
0
1
]
T

n
E
×
[
0
0
1
]
T

Y
E
=
X
E
×
n
E

X
E
×
n
E

(
6
)
Step Two:
Contouring-error calculation and cross-coupling control in the equivalent plane;
In the established equivalent plane, calculate the contouring error in a scalar form;
The XE-direction and YE-direction tracking errors from P to C(uf), denoted as ex,E and ey,E, respectively, are computed as:
{
e
x
,
E
=
(
C

(
u
f
)
-
P
)
·
X
E
e
y
,
E
=
(
C

(
u
f
)
-
P
)
·
Y
E
(
7
)
Thus, the estimated contouring error {circumflex over (ε)} is:
{circumflex over (ε)}=Cx,E·ex,E+Cy,E·ey,E (8)
Where Cx,E and Cy,E are XE-direction and YE-direction cross-coupling gains, respectively, and they are obtained by:
{
C
x
,
E
=
-
sin

(
θ
)
C
y
,
E
=
cos

(
θ
)
(
9
)
θ is the included angle of vectors C′(uf) and XE, and
θ
=
arctan


(
C
′

(
u
f
)
·
Y
E
C
′

(
u
f
)
·
X
E
)
;
Take the estimated signed contouring errors {circumflex over (ε)} as the control object, and the PID controller based two-axis CCC is utilized to control it;
The output control signal of the CCC at the time of t is thus obtained as:
U
c

(
t
)
=
k
p

ɛ
^
+
k
i

∫
0
t

ɛ
^

dt
+
k
d

d

ɛ
^
dt
(
10
)
where kp, kt, and kd are proportional, integral, and differential gains, respectively;
According to Uc(t), the XE-direction and YE-direction control signals, denoted by Δx,E and Δy,E, respectively, are computed as:
{
Δ
x
,
E
=
C
x
,
E
·
U
c
Δ
y
,
E
=
C
y
,
E
·
U
c
(
11
)
Step Three:
Calculation of the three-axis control signals in the real space;
According to the geometry relationship between XE/YE and the spatial X/Y/Z axes, calculate the coupling gains from the two axes of the equivalent plane to the three axes of the real three-dimensional space as:
{
k
x
,
x
=
X
E
·
[
1
0
0
]
T
k
x
,
y
=
X
E
·
[
0
1
0
]
T
k
y
,
x
=
Y
E
·
[
1
0
0
]
T
k
y
,
y
=
Y
E
·
[
0
1
0
]
T
k
y
,
z
=
Y
E
·
[
0
0
1
]
T
(
12
)
where kx,x, kx,y are gains from XE axis to X and Y axes, respectively, and ky,x, ky,y, and ky,z are gains from YE axis to X, Y, and Z axes, respectively;
After that, the X-axis, Y-axis, and Z-axis control signals, denoted by Δx, Δy, and Az, respectively, are calculated as:
{
Δ
x
=
k
x
,
x

Δ
x
,
E
+
k
y
,
x

Δ
y
,
E
Δ
y
=
k
x
,
y

Δ
x
,
E
+
k
y
,
y

Δ
y
,
E
Δ
z
=
k
y
,
z

Δ
y
,
E
(
13
)
By adding the obtained Δx, Δy, and Δz to the control signals of X-axis, Y-axis, and Z-axis position loops, the equivalent-plane cross-coupling control can hence be realized, which can reduce the three-axis spatial contouring error effectively.

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An equivalent-plane cross-coupling control method

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An equivalent-plane cross-coupling control method

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Abstract

Claims1