SIMPACK User Meeting Eisenach, 2004
Modelling and Control of a Machine Tool Using Co-Simulation Alexandra Ratering Peter Eberhard
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
Outline introduction the Lambda Kinematic modelling • mechanical and mathematical model • equations of motion
flatness • introduction to the concept of flatness • flatness analysis of the machine tool
controller design • linear cascaded P-PI control • flatness based control
simulation • simulation environment • comparison of control concepts
conclusions Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
demand for highly productive machine tools
Introduction
(high velocities, large scale movements, …)
development of • new machine types (parallel and hybrid kinematics) • new technologies (linear drives, lightweight structures, …)
problem • assumption of independently controllable drives questionable • elastic properties become important • nonlinearities become important
solution hierarchical controllers combining position control with methods of active vibration damping Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
the Lambda Kinematic
Machine Tool
• processing of plate-like workpieces (esp. wood processing) • scissors-like kinematic for movement in xy-plane • movement along z-axis realized by a serial axle • high dynamics: support velocities up to 2 m/s, max. acceleration 9 m/s2 • implemented motion control system: two independent P-PI controllers for each drive built at the Institute of Machine Tools (IfW) at the University of Stuttgart
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
problems with • overall performance • vibrations during processing F 1st step: development of a better position controller
Animation of Motion
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
y
Mechanical Model A
TCP
C
multibody system model R P
• 4 rigid bodies P, H, R, A
β γA
γR
• 2 degrees of freedom
x H
• 1 kinematic loop: DAE system • minimal coordinates q m = [xP xH ]
• joint coordinates q t = [xP xH γ R γ A ]
procedure • cut kinematic loop at joint C • Newton-Euler equations for each branch of open tree structure • algebraic constraint as loop closing condition Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
Newton-Euler equations for the open tree structure
Equations of Motion
generalized gyroscopic generalized Boolean and centrifugal forces applied forces matrix mass matrix
motor forces
&&t + k (q t , q& t ) = Q a (q t , q& t , t ) + H ⋅ u + Q r M (q t ) ⋅ q
generalized constraint/ reaction forces
Qr = GT ⋅ λ implicit constraint equations
x p − xH + l R cos(γ R ) + l A cos(γ A ) g(q t ) = =0 y P − y H + l R sin (γ R ) − l A sin (γ A )
g& ≡
∂g ⋅ q& t = G (q t ) ⋅ q& t = 0 ∂q t
equations of motion in minimal coordinates explicit constraint equations qt = qt (q m ) can be calculated analytically
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
i = R, A γ i = γ i (q m ), ⇒ γ&i = γ&i (q m , q& m ), γ&& = γ&& (q , q& , q i i m m && m ), but resulting ODE-formulation is complicated and for flatnessbased controller not necessary!
Flatness
definition
x& = f (x, u ),
x(0 ) = x 0 ∈ ℜ n , u ∈ ℜ m
is (differentially) flat, if there exists
(
)
y = Φ x, u, u& ,K, u (α ) , dim(y ) = dim(u )
( ) u = Ψ (y , y& ,K, y β )
for which (locally) x = Ψ1 y , y& ,K, y ( β )
( +1)
2
y is called a flat output of the system
in other words: all states and inputs of a system can be calculated as an algebraic function of the flat output and a final number of its derivatives
open loop control
yd
inverse system
ud
machine tool
! y = yd
… sufficient for ideal system Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
input: u = [u1 u 2 ]
Flatness Analysis
states: x = [q m q& m ] = [xP x& P xH x& H ] flat output: y = q m = [xP xH ]
x& = f (x, u ) x = Ψ1 (y , y& ) ⇔ y = [xP xH ] u = Ψ 2 (y , y& , &y& )
γ R , γ A , γ&R , γ& A , γ&&R , γ&&A
… auxiliary variables
y1
d/dt
y&1
d/dt
y& 2
d/dt
system is flat, hence controllable and observable singularities: • 2 possible positions • not reachable due to construction
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
&y&1
x2 , x4 γ&R , γ& A
x1 , x3 γ R, γ A y2
d/dt
u1 , u2 γ&&R , γ&&A &y&2
Linear Controller • preliminary: implemented P-PI controller (for a single support, i = P, H) 1/s xid + -
Kv
+
-
Kp/Tn Kp
Fz + +
Kf
ui
support i
num. diff.
- equivalent to an incomplete PID controller - tuned experimentally - equal Kp,Kv and Tn values for both supports - system performance varies over the workspace
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
xi
Flat Controller Design with Feedback Linearization
• static state feedback
u = u(y , y& , v )
¾ linear system with new input v = &y& = q && m
• PD controller, PID controller
(
)
(
)
(
)
v = &y& d + K 1 ⋅ y& d − y& + K 2 ⋅ y d − y + K 3 ⋅ ∫ y d − y dt
∫
¾ linear, asymptotic stable error dynamics &e& + K 1 ⋅ e& + K 2 ⋅ e + K 3 ⋅ e dt = 0 ¾ tuning e.g. with pole placement … w
d
trajectory planing
yd
v controller
state feedback
u
machine tool
- homogenous system performance over the whole workspace - simple tuning of PID controller - robustness? Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
x
Flat Controller Design with Feedforward Linearization • feedforward linearization (Hagenmeyer, Delaleau 2003)
(
u = u y d , y& d , v
)
¾ exact linearization when being on the desired trajectory • PD controller, PID controller
(
)
(
)
(
)
v = &y& d + K 1 ⋅ y& d − y& + K 2 ⋅ y d − y + K 3 ⋅ ∫ y d − y dt ¾ stabilizing in the vicinity of the desired trajectory w
d
trajectory planing
yd controller
v
feedforward linearization
u
machine tool
x
z=x
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
- better robustness expected against measurement noise, parameter uncertainties …
Why Co-Simulation? Co-Simulation + implementation of control and other non-mechanical model parts ¾ well established control design and simulation tools ¾ easy to adapt/change ¾ offline trajectory generation ¾ real time code generation ¾…
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
simulation of complex MBS ¾ nonlinear dynamics ¾ kinematic loops ¾ elastic bodies ¾…
ability to efficiently test MBS with different control concepts etc. and vice versa
control • easy to change/adapt • real-time code (dSPACE …) desired trajectory (generation offline)
Simulation Environment
positions, velocities, …
SIMPACK
SIMULINK
motor forces, …
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
process forces
Open- vs. Closed-Loop Control
with 30% disturbance of the moment of inertia of the rod and the cantilever in the state feedback calculation
closed-loop
0.8
0.8
0.6
0.6
yTCP [m]
yTCP [m]
open-loop
0.4
0.2
0.2 2.6
0.4
2.8
3
xTCP [m]
3.2
3.4
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
2.6
2.8
3
xTCP [m]
3.2
: trajectory : simulation result
3.4
Linear vs. Nonlinear Controller
trajectory 0.8 -3
0.6
x 10
0.2 -2
4 2 0 -2 -4 0 -3 x 10 4 2 0 -2 -4 0
∆xP [mm]
0.4
-1
0
xTCP [m]
1
2
with 30% disturbance of the moment of inertia of the rod and the cantilever in the state feedback calculation
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
∆xH [mm]
yTCP [m]
position error of the supports
10
20
10
20
time [s]
: linear controller : flatness-based controller
Simulation with Process Forces
trajectory 0.8 0.6
∆xP [µm]
2
0.4 0.2 -2
-1
0
xTCP [m]
1
2
simplified model of the milling process
Fc = kc (v f )(sin (Ωzt ) + 1) in feed direction FcN = kcN (v f )(sin (Ωzt ) + 1) in z-direction
-6
0
2
x 10
5
10
15
20
25
5
10
15
20
25
-6
0
-2
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
x 10
-2
∆xH [µm]
yTCP [m]
position error of the supports
time [s]
with 30% disturbance of the moment of inertia of the rod and the cantilever in the state feedback calculation workspace
Feedforward and Feedback Linearization section of workspace
1
0.6
0.6
0.55 0.4
yTCP [m]
yTCP [m]
0.8
0.2 0
0.2
: trajectory
0.4
0.6
xTCP [m]
0.8
: feedback linearization : feedforward linearization
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
0.5
0.45
0.4
-10
-5
xTCP [m]
0
5 -3 x 10
with 30% disturbance of the moment of inertia of the rod and the cantilever in the state feedback calculation workspace
Feedforward and Feedback Linearization large initial error of the supports
1
0.05
∆xP [m]
0
-0.05
0.6
-0.1
0.4
0 0.2
0.05
0.1
0.05
0.1
0.4 0
0.2
: trajectory
0.4
0.6
xTCP [m]
0.8
: feedback linearization : feedforward linearization
∆xH [m]
yTCP [m]
0.8
0.2 0 0
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
time [s]
with 30% disturbance of the moment of inertia of the rod and the cantilever in the state feedback calculation workspace
Feedforward and Feedback Linearization position error of the supports
1
1
∆xP [µm]
0.6 0.4
-6
0
-1
0.2
1 0
0.2
: trajectory
0.4
0.6
xTCP [m]
0.8
: feedback linearization : feedforward linearization
∆xH [µm]
yTCP [m]
0.8
x 10
2
3
4
1
2
3
4
-6
0
-1
Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
x 10
1
time [s]
conclusions • • • • •
flatness-based controller feasible physically insightful simple controller design improves performance significantly co-simulation with SIMPACK and MATLAB/SIMULINK very useful
outlook • model enhancement (elastic MBS, …) • controller enhancement (active vibration damping, hierarchical control) • validation of model and simulation results at the real machine Institute B of Mechanics Prof. Dr.-Ing. Peter Eberhard University of Stuttgart
