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Industrial RobotsIndustrial Robots
ControlControl –– Part 1Part 1Control Control Part 1Part 1
Introduction to robot control
The motion control problem motion control problem consists in the design of control algorithms for the robot actuators
In particular it consists in generating the time functions of the li d i h h h TCP igeneralized actuating torques, such that the TCP motion
follows a specified task specified task in the cartesian space, fulfilling the specifications on transient and steady state responsespecifications on transient and steady‐state response requirements
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Tasks
T f t kt k b d fi dTwo types of tasks tasks can be defined:1. Tasks that do not require an interaction with the environment
(f i ) h i l i TCP f ll i(free space motion); the manipulator moves its TCP following cartesian trajectories, with constraint on positions, velocities and accelerations due to the manipulator itself or the taskaccelerations due to the manipulator itself or the task requirements
Sometimes it is sufficient to move the joints from a specified value ( )q tSometimes it is sufficient to move the joints from a specified value to another specified value without following a specific geometric path
0( )iq t
( )i fq t
2. Tasks that require and interaction with the environment, i.e., where the TCP shall move in some cartesian subspace while subject to forces or torques from the environment
We will consider only the first type only the first type of task
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Motion control
In particular the motion control problem motion control problem consists in generating the time functions of the generalized actuating torques, such that the TCP motion follows a specified task in the cartesian space, fulfilling the specifications on transient and steady‐state
iresponse requirements
Control schemes can be developed for:Joint space controlpTask space control
considering that the task description is usually specified in theconsidering that the task description is usually specified in the task space, while control actions are defined in the joint space
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Joint space control
Th I Ki ti bl k t f th d i d t k itiThe Inverse Kinematics block transforms the desired task space positions and velocities into desired joint space reference values.
The Transducermeasures the value of the joint quantities (anglesThe Transducermeasures the value of the joint quantities (angles, displacements) and compares them with the desired ones, obtained, if necessary, from the desired cartesian quantities.
The Controller uses the error to generate a (low power) signal for the Actuator that transforms it in a (high power) torque (via the Gearbox) that
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moves the robot joints
Task space control
Controller Actuator Gearbox Robotdp p
Transducer
In this case, the Transducermust measure the task space quantities in order to
Transducer
In this case, the Transducermust measure the task space quantities in order to compare them with desired ones.
Usually this is not an easy task, since it requires environment‐aware sensors; y y , q ;the most used ones are digital camera sensors (vision‐based control) or other types of exteroceptiveexteroceptive sensors (infra‐red, ultra‐sonic, ...).
Otherwise one uses the direct kinematics to estimate the task space pose
Torques are always applied to the joints, so Inverse Kinematics is hidden inside
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the Controller block
Joint space control architectures
T i j i t t l hit tj i t t l hit t iblTwo main joint space control architectures joint space control architectures are possible
Decentralized control or independent joint control: each i‐th joint motor has a p j jlocal controller that takes into account only local variables, i.e., the joint position and velocity The control is of SISO type usually based on a P PD or PID architecture
( )iq t ( )
iq t
The control is of SISO type, usually based on a P, PD or PID architectureThe controller is designed considering only an approximated model of the i‐th joint. This scheme is very common in industrial robots, due to its simplicity, modularity and robustnessThe classical PUMA robot architecture is shown in the following slideThe classical PUMA robot architecture is shown in the following slide
Centralized control: there is only one MIMO controller that generates a command vector for each joint motor; it is based on the complete model of the manipulator and takes into account the entire vector of measured positions and velocities
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and velocities
Decentralized control
joint 1reference
Decentralized Joint Control 1
( )q tcontroller 1
joint 2 ( )q tJoint spacecontroller 2
joint 2referenceTask space 2
( )q t
…
controller 6
joint 6reference 6
( )q t
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Decentralized controlTeach
DiskTeach
pendant OtherTerminalPUMA Control
μG D/A Amplifier Motor 1
DLV-11J
EPROM
EncoderT=0.875 ms
EPROM
RAM
InterfaceReference
anglesT=28 ms
CPUD/A Amplifier Motor 6μG
EncoderT=0.875 ms
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COMPUTER ROBOT CONTROL
Motor and gearbox model (rigid body assumptions)
Gearbox Geartrain
GearboxFriction
′N
r =
Gearbox = Geartrain
′N
RobotInertia,m rN; ω τ
τFriction
m
Inertia ,′ ′ ′m rN ; ω τ
Motor′mτ
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r mτ ω′ ′r mτ ω Output powerInput power
Motor and gearbox model
bβ+
mω
i aLaR N bΓ τ
m p= +rτ τ τ
aiaa N b m
avei E mΓ ′N
′τ τ
mβ
mτ′mω
pτ
mβm p
′ ′ ′= −rτ τ τ
′
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p′τ
Losses in geared motor
Motorside
Jointside
Armature circuit
Motor inertia Gearbox Joint
inertiaaEi
m rω τ′ ′
a av i
m rω τ
m mω τ
ddda a a a
L i R it
+ ηgearboxefficiency
ddm m m mt
Γ ω β ω′ ′+voltage drop
efficiency
dΓ ω β ω+
pτ ′
db m b mtΓ ω β ω+
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pτ
Gearbox model
ρ θ ρθρ ω ρω
′ ′ =′ ′
Nr
Nρρ
= =′ ′ω τ
Joint side ρ
ρ ω ρω= Nρm rω τ
Ideal gearbox: 1η =
θGEARBOX
Power in Power out
M t id
θ′
′
GEARBOX
m rω τ′ ′
m r m rω τ ηω τ′ ′=
Motor side ρ′
m rω τ′ ′
rτ τ ′=r r
m
rτ τω
ω′
=
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m r
CC motor equations – 1
d
MOTOR SIDE
dda a a a a
L i v R i E
K it
φ
= − −
e
m m
K i
E k Kφ
ωω ωφ
φ=
′ ′= =k k K K′ ⇒m m
m a ak i K i
ω
τττ
φ′ ′= =′
k k K Kω τ′≈ ⇒ ≈
mai
Kτ
τ=
′ ′ ′p m m m m m m m m
r m p
τ β ω βτΓ θ ω
τ τθ Γ′ ′ ′= + = +
′ ′ ′= −
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r m p
CC motor equations – 2
GEARBOX SIDE
′1
MOTOR SIDE
( )r r
m p
r
r
τ ττ τ
′=′ ′−
⎡ ⎤=
1
1( )
r rrτ τ′ =
+
( ) ( )m m m m m
m m m m m
r
r r r
τ ω β ω
τ ω β ω
Γ
Γ
⎡ ⎤′ ′ ′= − −⎢ ⎥⎣ ⎦⎡ ⎤′= − −⎢ ⎥⎣ ⎦
( )
1m prτ τ
τ ω β ωΓ⎡ ⎤= ⎢ ⎥⎣ ⎦
= +
+ +2( )
m m m m m
m m m m mr r Γτ β ωω⎣ ⎦′= − +
1 1 1
m b m b m
m b m b m
rτ ω β ω
τ ω β ω
Γ
Γ
⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎟⎜⎢ ⎥′ ′= +⎟⎜
+ +
⎛⎟⎢ ⎥⎜ ⎟
⎞⎟⎜+ ⎟⎜ ⎟⎜ ⎟
2
1 1( )
m b m b m
m b m b m
r r rβ
Γτ ω β ω
⎟⎢ ⎥⎜ ⎟⎝ ⎠ ⎟⎣ ⎦
′ ′=
⎜⎝
+
⎟⎠
+2( )
m b m b mr rβ
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Control equations
( ) ( , ) ( ) ( ) Te
+ + + + =H q q C q q q B q q g q J F τ
component-wise joint torques
( ) )) ((n n n
ij j ijk j k bi i i fi iH q qh gq qβ τ τ+ + + + =∑ ∑∑q q q1 1 1
( ) )) ((ij j ijk j k bi i i fi ri
j j k
q q gq qβ= = =
+ + + +∑ ∑∑q q q
1 1
( ) ( ) ( )( )r
n n n
i ij j ijk j k bi i i fij i
ij
ik
iq H qH q qh gqβ τ
≠ = =
+ =+ + + +∑ ∑∑q q q q τ
I ti l t Coriolis & centripetal Friction, gravity
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Inertial torques Co o s & ce t petatorques
ct o , g a ty& external torques
Control equations
( ) ( ) ( )( )n n n
H q h q q gH q q ττβ+ =+ + + +∑ ∑∑q q qq1 1
( ) ( ) ( )( )ij j ijk j kii i bi i fi
j i j krii
H q h q q gH q q ττβ≠ = =
+ =+ + + +∑ ∑∑q q qq
( )Mi i i fii i bi ii i
H q q τ τ ττβ τ+ + ++ + =q( )Mi ci gi f rii i bi ii i
q q τ τ ττβ τ+ + ++ +q
Modelled torques “Disturbance” torques
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From single motor model to robot control equation
θ′ ′ ′mi mi mi
i i ii i i
q qr r
qr
θ ω ω= = =
′ ′ ′
Gearbox transformationGearbox transformation
n
H q H qqτ β τ τ τ τ= + + + + ++∑
StructuredStructured
ri ii i bi i ij j Mi ci gi fij i
mi mi
H q H qqτ β τ τ τ τ
ωβ τ τ τ τΓ
ω≠
= + + + + +
′+ + +
+
′= + +
∑
StructuredStructureddisturbancedisturbancebi bi Mi ci gi fi
i i
mi mi
r rβ τ τ τ τ
ωβ
Γ
ωΓ
+ + +=
′
+ +
′mi mi
bi bi dii ir r
βΓ τ= + +
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Equation seen at the joint sidejoint side
From single motor model to robot control equationEquation seen at the motor sidemotor sideEquation seen at the motor sidemotor side
since
( )1 1 1Γ β′ ′′( )2ri ri bi mi bi mi di
i iir rr
Γ βωτ τ ω τ′ ′+ +′= =
( )ri mi pi mi mi mi mi miτ τ τ τ Γ β ωω′ ′ ′ ′= = +′ ′− −
and
1 1 1β βΓ Γ
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟′ ′ ′ ′+ ⎜ + +⎜ + +′
we obtain
2 2mi ri pi bi mi mi bi mi mi diii irr r
τ τ τ β β τΓ ωΓ ω⎟ ⎟′ ′ ′ ′= + = ⎜ + +⎜ + +⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠′
Total inertia Total friction
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From single motor model to robot control equation
mi pi di ti mi ti mi diτ τ τ ω β ω τΓ′ ′ ′ ′ ′ ′ ′= + = + +′
′ ′ ′ ′ ′ ′ ′ ′= + = + +Bτ τ τ Γ ω ω τ Motor sideMotor sidep d t m t mm d
= + = + +Bτ τ τ Γ ω ω τ Motor sideMotor side
= + +t′Γ
t′B
⎛ ⎞ ⎛ ⎞2
1bi mi
ir
Γ Γ⎛ ⎞⎟⎜ ⎟⎜ + ⎟⎜ ⎟⎟⎜⎝ ⎠
2
1bi mi
ir
β β⎛ ⎞⎟⎜ ⎟⎜ + ⎟⎜ ⎟⎟⎜⎝ ⎠
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i⎝ ⎠ i⎝ ⎠
From single motor model to robot control equation
mi pi di ti mi ti mi diτ τ τ ω β ωΓ τ= + = + +
Joint sideJoint side= + = + +Bτ τ τ Γ ω ω τ Joint sideJoint sidep d t m t m dm
= + = + +Bτ τ τ Γ ω ω τ
= + +t
ΓtB
( )2bi i mirΓ Γ+ ( )2
bi i mirβ β+
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Block diagram of open‐loop CC motor (motor side)For simplicity we drop the prime ′ symbolFor simplicity we drop the prime ′ symbol
for the motor side quantities, and we consider the generic i-th motor
Taking the Laplace transform of the involved variables, we haveg p ,
( ) ( ) ( ) ( )t t m m d
s s s sβ ω τ τΓ + = −( ) ( ) ( ) ( )t t m m d
m aK iτ
βτ =
1( ) ( )s sθ ω
+dτ
( ) ( )m ms s
sθ ω=
+
–+
1
a aR sL+
1
t tsβ Γ+
Kτ
1s
av
ai m
τ mω
mθ–
rτ
Kω
E
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ω
Block diagram of open‐loop CC motor (motor side)
Th t i it i d t i ll d ll b l t d The armature circuit inductance is small and usually can be neglected
0 v RL i K ω=≈ ⇒0aa a a m
v RL i Kωω− =≈ ⇒
τm
ai
Kτ
τ=
ma a mv R K
K ωτ
τω− =
m d t m t m
a a aR R R
K
τ τ ω β ω
β
Γ
Γ
= + +⎡ ⎤⎛ ⎞⎟⎜⎢ ⎥a a a
a d t t mv s K
K K Kωτ τ τ
τ β ωΓ ⎟⎜⎢ ⎥⎟− = + +⎜ ⎟⎢ ⎥⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦
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Block diagram of open‐loop CC motor (motor side)
R βa tR
K K KKω ω ω
τ
β′ = + ≈
a tR
Kβ
since
friction torque
a tm m
KK ω
τ
ω ω
t ma a mR i K
K i ω
β ωω
armature losses armature e.m.f.torque m
aK iτ
τ
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Block diagram of open‐loop CC motor (motor side)
⎛ ⎞ 11a t a
m a d
R Rs v
K K K K Kτ ω ω τ ω
Γω τ
⎛ ⎞⎟⎜ ⎟+ = −⎜ ⎟⎜ ′ ′ ′⎟⎜⎝ ⎠
dτ
⎝ ⎠
a tR
TK K
Γ=
′dτ
Ka
d
K KR
KK
τ ω
=where
– θ
dK Kτ
+ ( )G sω
1s
av m
ωmθ
( ) ( )1
G s =
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( ) ( )1G s
K sTωω′ +
Matrix formulation (joint side)Lagrange
( ) ( , ) ( ) ( )
( )
Te
r m m p
+ + + + =≡ = −H q q C q q q Bq g q J q F
Rτ
τ τ τ τ
LagrangeEquation
( )r m m p
where
2= +R R K v R K qτm m m a a m ω+R R R qτ
2( ) ( )m p m m m m m m m m
= + = +R R q B q R q B qτ Γ Γp
Motor side Joint sideMotor side Joint side
0 0 0 00 0
KKK
⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥
and
0 0 ; 0 0 ; 0 0
0 00 0 0 0
im i a
ai a
i i
iR R
Kr
KKτ τ ωω
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦
R K K
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0 00 0 0 0⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
Matrix formulation (joint side)Then we haveThen, we have
Mass matrix Friction matrix
( ) ( )
( )M q F
( ) ( )( )2 2( ) ( , )
( ) ( )m m m mω+ + + + + +
+ + =
H q R q C q q B R K B q
g q J q F R K v
ΓT( ) ( )
e m a a+ +g q J q F R K v
uGravity
Interaction cu
Command input
Often we use this symbol to indicate
Interaction
( )( , ) ( , )= +h q q C q q F q
ythe velocity dependent terms
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( )( ) ( )
Matrix formulation (joint side)
( ) ( ) ( ) ( )T+ + +M h J F
Gravity and interaction
( ) ( , ) ( ) ( )Te c
+ + + =M q q h q q g q J q F u
No interaction
( ) ( , ) ( )c
+ + =M q q h q q g q u
No gravity, no interaction
( ) ( , )c
+ =M q q h q q u
Control Design ProblemControl Design Problem ...?c=u
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Decentralized joint control
A i f i li it
( ) ( , )c
+ =M q q h q q u
Assuming, for simplicity
( ) ( , )c
q q q q
If …
( ) diagonal 2 2( )m m m m m
⇒ + ≈R I H q R RΓ Γ
small ( , )C q q q
Th Then …
t d c+ + =q Fq uΓ τ 2
t m m=RΓ Γwith
disturbance
( )t cs + =F uΓ ω
The model is diagonal, i.e., naturally decoupled
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g , , y pEach joint can be controlled by local controllers
Decentralized joint controlThis is the proportional velocity controller
dτ
This is the proportional velocity controlleror velocity compensator
d
KdKreference voltage
–
+( )G sω
1s
av m
ωmθ
DK
–
+rv e
( )K ( )tK s
Transducer T.F. ( )K s K≈
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(tachimetric sensor)( )t t
K s K≈
Open loop vs Closed loop
( )( )
( ) (1 )m Ds K
G sv s K s Tω
ωα
α′= =
′ +
( ) 1( )
( ) (1 )ms
G sv s K sTω
ω= =
′ + ( ) (1 )rv s K s Tω α+( ) (1 )
av s K sTω +
( )( ) ( )
( ) (1 )m
d d
s TG s K G s
s sTω
ωτ Γ
= = − = −+
( )( ) ( )m d d
s K K TG G
ω α α′ ′
( ) (1 )d ts sTτ Γ +
( )( ) ( )
( ) (1 ) (1 )m d d
dd D t
G s G ss K s T K s Tω
ω α ατ Γ′ ′= = − = − = −
′ + +
1Kωα′
= <
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1D t
K K Kω
α = <′ +
The closed‐loop system
dτ
K
dτ –
( )G s1a
v mω
mθ
dK
dK
K
+( )G sω s
Open loop
DK
–
+( )G sω
′ 1s
rv
mω
mθ
s
Closed loop
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The closed‐loop system
Time constant is reduced T Tα <
KDisturbance gain is reduced
dd
D
KK
K→
Design parameterd
τ
when 1dK = dK
–
( )G sω
1s
av m
ωmθ
DK
+rv e
+( )
s–
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Position compensator
dτController 1
dK
–
( )G1a
vm
ωmθ
K+e
K+rθ
+( )G sω sD
K
–
PK
–
tK
1tK ≈
1K θ ≈
K θ
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θ
Position compensator
( )θ1 2
( )( )
( )m
r
s KG s
s s s T K
θθ α
= =+ +
2 2
( ) 1( )
( ) ( )ms
G ss s s T Kτ Γ
θ
α= = −
+ +( ) ( )d ts s s T Kτ Γ α+ +
D P D PK K K K K
K τ= =where
a t
KTK Rω Γ
= =′
Configuration dependent
Second-order TF with
1 1·
2 2
K K
T K R K K Kτ ω
α αζ
Γ
′= =
with 2 2
1
a D P t
D P
T K R K K K
K K KK
τ
τ
α α Γ
ω = =
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·n
a t
KR
ωΓ
= =
Design parameters
The damping coefficient and the natural frequency are inversely proportional to the square root of the inertia moment, that may vary p p q , y yin time when the angles vary
2
1t b mr
Γ Γ Γ⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜ ⎟⎝ ⎠
( )( )b iiH tΓ⎝ ⎠
= q
Since the damping coefficient and the natural frequency are often usedSince the damping coefficient and the natural frequency are often used as control specifications, we can design a controller computing the maximum inertia moment and adjusting the two gains int
Γ ,P DK Kmaximum inertia moment and adjusting the two gains in
such a way that the damping ratio is satisfactory, e.g., no overshoot appears in the step response
,maxt
ζ,P D
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An alternative
dτ
KController 2
+
dK
–
+( )G sω
1s
av
mω
mθ
P DK sK′ ′+
–
+rθ e
( ) ( ) ( )a P Dv t K e t K e t′ ′= +
3 2
( ) /( )
( )m D P Ds K K s K K
G sR
τ
Γθθ
′ ′ ′+= =
′
A zero appears
3 2( )
( )r a t a tPs R s s T K K Rτ
Γ α Γθ ′+ +
( ) 1 1( )m
sG
θ ⎛ ⎞⎟⎜ ⎟⎜ ⎟4 2( )
( )m
d t P a t
G ss s s T K K Rτ
τ Γ α Γ⎜= = − ⎟⎜ ⎟⎜ ′ ⎟+ + ⎟⎜⎝ ⎠
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Another alternative
dτ
KController 3
++
dK
–+
( )G sω
1s
av
mω
mθ
P DK sK′ ′+
–
+rθ e
–+D
K
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Comparison
Controller 1
( ) ( ) ( )t K K t K t( ) ( ) ( )a D P D mv t K K e t K tω= −
Controller 2
( ) ( ) ( ) ( )a P D r D mv t K e t K t K tω ω′= + −
Controller 3Controller 3
( )( ) ( ) ( ) 1 1 ( )a D P D D r D D D mv t K K e t K K t K K K tω ω′ ′ ′ ′= + − +
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Practical Issues
1.1. Saturating actuatorsSaturating actuators
2.2. Elasticity of the structureElasticity of the structure
33 Nonlinear friction at jointsNonlinear friction at joints3.3. Nonlinear friction at jointsNonlinear friction at joints
4.4. Sensors or amplifiers with finite band Sensors or amplifiers with finite band 4.4. Sensors or amplifiers with finite band Sensors or amplifiers with finite band
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Saturating ActuatorsIt is a nonlinear effect, difficult to be considered a-priori
( )y t
It is a nonlinear effect, difficult to be considered a priori
saturation
( )y t( )u t
( )u t
( )y t( )u t
Li it
saturation
Linearity
( )t⎧⎪ IF
max max
min max
( )
( ) ( ) ( )
y u t u
y t ku t u u t u
⎪⎪⎪⎪⎨⎪
>= ≤ ≤
IFIF
min min( )y u t u
⎪⎪⎪⎪⎩ <IFBasilio Bona 41ROBOTICA 03CFIOR
Elasticity of the structure
Although we have considered rigid bodies, the elasticity is a phenomenon that limits the closed loop bandphenomenon that limits the closed loop band
We cannot design controllers that are “too fast” without taking li itl i t id ti t f l ti d lexplicitly into consideration some sort of elastic model.
Recall that when we use a simplified model
( ) ( ) 0t m e mt k tθ θΓ + =
the proper structural resonance (or natural) frequency is
er
kω
Γ=
tΓ
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Elasticity of the structure
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Nonlinear friction
A li ff t b t l it d f i ti fA nonlinear effect between velocity and friction force
total( )f t
istiction
( )
( )f t( )v tviscous
coulomb
stiction
( )v t
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Nonlinear friction
Nonlinear static friction models include:Coulomb + viscous
signc
F(v) F (v) vβ= +
Static model that includes stiction and “Stribeck effect”: friction decreases with increasing velocity for v < vs (Stribeckvelocity)
( ) i
s
s
vvF( ) F F F ( )
δ
β−
⎡ ⎤⎢ ⎥⎢ ⎥+ +( ) signs
c s cF(v) F F F e (v) vβ⎢ ⎥= + − +
⎢ ⎥⎢ ⎥⎣ ⎦
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Finite pass‐band in sensors and amplifiers
Sensors and amplifiers are often modelled as simple gains, while in the real world they have a finite pass‐band, nonlinearities, saturations, etc.
These effects must be taken into account when the simulated and realThese effects must be taken into account when the simulated and real behaviours differ.
Fortunately, very often the band of sensors and amplifiers is much wider than the final closed loop band of the controlled system.
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