Data processing: generic control systems or specific application – Specific application – apparatus or process – Robot control
Reexamination Certificate
1998-08-24
2001-07-24
Grant, William (Department: 2786)
Data processing: generic control systems or specific application
Specific application, apparatus or process
Robot control
C700S250000, C700S063000, C318S568120
Reexamination Certificate
active
06266578
ABSTRACT:
BACKGROUND
The present invention relates to motion control, and more particularly to an improved microcontroller based control system for apparatus such as a robotics mechanism.
In the past, non-linear control problems required complicated intermediate steps involving linearization of differential equations. The linearization resulted in controls that were not exact, so that the required trajectory was not precisely followed by the mechanical system. Also, the computation of control force using linearized equations required powerful microprocessors.
Controls for mechanical systems (such as robotic systems, structural systems, etc.), have been developed using linear feedback, non-linear feedback, or non-linear cancellation. However, none of these systems provide a closed form, real-time control force signal for any prescribed trajectory. The required feedback and the computation of linearized equations make the control force signal computationally expensive and imprecise. It is believed that the only prior art disclosing a closed form, real-time solution to the control problem is the doctoral thesis of Hee-Chang Eun, entitled
On the Dynamics and Control of Constrained Mechanical and Structural Systems.
However this work was limited to trajectories that are not dependent on measurement parameters, and had a restrictive limitation on the weighting of control forces. Also, Hee-Chang's work requires {1,2,3,4} inversion of matrices, which is computationally restrictive.
SUMMARY
The present invention meets the needs by a control system wherein closed form control forces are determined in real time and applied to satisfy specific trajectory requirements of a physical plant which can be a nonlinear mechanical system. Generally, the present invention provides a system and method for determining exact control forces to be applied to any nonlinear mechanical system so that certain physical points in the system follow user defined trajectory requirements. As used herein, trajectory requirements are mathematical relationships between measurement quantities. The trajectory requirements can have functional dependence on a measurement vector such as position in space, velocity of the measurement vector and time; and the user also has the ability to apply weighting factors to the control forces. Further, fast computation of control forces that includes matrix manipulation is enabled using any {1,4} inverse of a computed matrix that is derived from dynamic attributes and measurements of the system. Moreover, the weighted control force resources are minimized at each instant of time, not just in an overall sense for the complete trajectory. Thus the invention provides a closed form, real-time solution to the problem of the control of non-linear mechanical and structural systems.
In one aspect of the invention, a control system is provided for operating a physical plant to follow predetermined desired trajectory requirements, wherein forces acting on the elements can include inertial forces, externally applied forces, and control forces, the plant having interrelated mechanical elements and a response characteristic described by M(x,t){umlaut over (x)}=F(x,{dot over (x)},t) in generalized coordinates x(t) wherein F is exclusive of control forces. The control system includes a control computer; means for storing in the computer values corresponding to M and F of the response characteristic; means for measuring a p-vector y(t) related to x(t); means for specifying the trajectory requirements as h
i
(y,{dot over (y)},t)=0, i=1,2, . . . , s and/or g
i
(y,t)=0, i=1,2, . . . , (m−s); means for transforming and storing in the computer the trajectory requirements as {dot over (h)}=−ƒ
(1)
(h,t) and/or {umlaut over (g)}=−ƒ
(2)
(g,{dot over (g)},t), wherein a fixed point h=0=g={dot over (g)}is asymptotically stable within a domain of attraction that includes deviations from the trajectory requirements; means for determining control forces as
F
c
=
G
*
(
[
D
(
h
)
-
f
(
1
)
(
h
,
t
)
D
~
(
g
)
-
f
(
2
)
(
g
,
g
.
,
t
)
]
-
[
∂
h
∂
y
.
∂
g
∂
y
]
[
0
D
~
(
ϕ
)
]
-
(
[
∂
h
∂
y
.
∂
g
∂
y
]
[
C
∂
ϕ
∂
x
]
)
M
-
1
F
)
,
G* being any {1,4} inverse of BM
−1
; and means for driving the plant by the control forces, thereby to generate the desired trajectory.
The response characteristic of the plant can be nonlinear. The control system can further include means for storing a positive definite matrix N for weighting of control forces; and wherein the means for determining control forces is further responsive to the matrix N, whereby
F
c
=
N
-
1
/
2
G
*
(
[
D
(
h
)
-
f
(
1
)
(
h
,
t
)
D
~
(
g
)
-
f
(
2
)
(
g
,
g
.
,
t
)
]
-
[
∂
h
∂
y
.
∂
g
∂
y
]
[
0
D
~
(
ϕ
)
]
-
(
[
∂
h
∂
y
.
∂
g
∂
y
]
[
C
∂
ϕ
∂
x
]
)
M
-
1
F
)
.
G* being any {1,4} inverse of B(N
½
M)
−1
.
The trajectory transformation equations {dot over (h)}=−ƒ
(1)
(h,t) and {umlaut over (g)}=−ƒ
(2)
(g,{dot over (g)},t) can be globally asymptotically stable at the point h=0=g={dot over (g)}. The p-vector can be linearly related to the response vector x(t). Alternatively, the means for specifying the p-vector further includes means for specifying a k-subvector being linearly related to the system response vector x(t); means for specifying a (p-k)-subvector having components being nonlinear functions of elements of the vector x(t) and possibly the time t; and means for determining physical values of the elements of the vector x(t) that are in the (p-k)-subvector.
In another aspect of the invention, a method for controlling the plant to follow predetermined desired trajectory requirements includes the steps of:
(a) providing a control computer,
(b) determining and storing in the computer a response characteristic of the plant as a matrix of force elements being related to a matrix of mass elements and acceleration in a vector of generalized coordinates x(t);
(c) specifying a measurement p-vector y(t) related to x(t);
(d) specifying the trajectory requirements as h
i
(y,{dot over (y)},t)=0, i=1,2, . . . , s and/or g
i
(y,t)=0, i=1,2, . . . , (m−s);
(e) transforming and storing in the computer the trajectory requirements as {dot over (h)}=−ƒ
(1)
(h,t) and/or {umlaut over (g)}=−ƒ
(2)
(g,{dot over (g)},t), wherein a fixed point h=0=g={dot over (g)} is asymptotically stable within a domain of attraction that includes deviations from the trajectory requirements;
(f) determining and storing in the computer measurements of the p-vector;
(g) determining control forces as
F
c
=
G
*
(
[
D
(
h
)
-
f
(
1
)
(
h
,
t
)
D
~
(
g
)
-
f
(
2
)
(
g
,
g
.
,
t
)
]
-
[
∂
h
∂
y
.
∂
g
∂
y
]
[
0
D
~
(
ϕ
)
]
-
(
[
∂
h
∂
y
.
∂
g
∂
y
]
[
C
∂
ϕ
∂
x
]
)
M
-
1
F
)
,
G* being any {1,4} inverse of BM
−1
;
(h) applying the control forces; and
(i) repeating steps (f-h) for generating a control trajectory of the plant corresponding to the trajectory requirements.
The method can include the further step of specifying a positive definite matrix N for weighting of control forces, and wherein the step of determining control forces uses
F
c
=
N
-
1
/
2
G
*
(
[
D
(
h
)
-
f
(
1
)
(
h
,
t
)
D
~
(
g
)
-
f
(
2
)
(
g
,
g
.
,
t
)
]
-
[
∂
h
∂
y
.
∂
g
∂
y
]
[
0
D
~
(
ϕ
)
]
-
(
[
∂
h
∂
y
.
∂
g
∂
y
]
[
C
∂
&varp
Grant William
Rao Sheela
Sheldon & Mak
LandOfFree
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