[PPT] - ROBOTICS 01PEEQW Basilio Bona DAUIN Politecnico di Torino PowerPoint Presentation

SLIDE 1

ROBOTICS 01PEEQW

Basilio Bona DAUIN – Politecnico di Torino

SLIDE 2

Control – Part 1

SLIDE 3

Tasks

Two types of tasks can be defined:

1. Tasks that do not require an interaction with the environment

(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 task requirements

Sometimes it is sufficient to move the joints from a specified value to another specified value without following a specific geometric path

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 of task

Basilio Bona - DAUIN - PoliTo 3 ROBOTICS 01PEEQW - 2014/2015

( )

i

q t ( )

i f

q t

SLIDE 4

Motion control

In particular, the motion control problem consists in generating the time functions of the actuating torques/forces, such that the TCP motion follows a specified task in the cartesian space, fulfilling the specifications on transient and steady-state response requirements Control schemes can be developed for:

Joint space control Task space control

considering that the task description is usually specified in the task space, while control actions are defined in the joint space

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SLIDE 5

Joint space control

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The Inverse Kinematics block transforms the desired task space positions and velocities into desired joint space reference values. The Transducer measures 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 moves the robot joints

SLIDE 6

Task space control

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In this case, the Transducer must measure the task space quantities in order to compare them with desired ones. Usually this is not an easy task, since it requires sensors that are environment- aware; the most used ones are digital camera sensors (vision-based control) or

ther types of exteroceptive sensors (infra-red, ultra-sonic, ...).

Otherwise, one must rely on the direct kinematics to estimate the task space pose Torques are always applied to the joints, so Inverse Kinematics is hidden inside the Controller block

Controller Actuator Gearbox Robot Transducer

d

p p

SLIDE 7

Joint space control architectures

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Two main joint space control architectures are possible Decentralized control or independent joint control: each i-th joint motor has a local 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 The 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 robustness The 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

( )

i

q t ( )

i

q t ɺ

SLIDE 8

Decentralized control

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Joint space …

controller 2 controller 6 joint 1 reference joint 2 reference joint 6 reference

Decentralized Joint Control

1( )

q t

Task space

controller 1

2( )

q t

6( )

q t

SLIDE 9

Decentralized control

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DLV-11J EPROM RAM CPU

Disk Teach pendant Other Terminal Interface

COMPUTER ROBOT CONTROL

µG

D/A D/A Amplifier Amplifier Encoder Encoder Motor 6 Motor 1 Reference angles T=28 ms T=0.875 ms T=0.875 ms

PUMA Control

µG

SLIDE 10

Decentralized control

The control is based on the classical model of the electrical motor, with the addition of a gearbox Inertia of the mechanical part and linear friction effects are taken into account, but the elasticity of the various parts is neglected The motor model is approximated in the usual way, i.e., neglecting the electrical dynamics (electrical time constant much larger than the mechanical time constant) The overall control is the “sum” of the individual controllers, one for every joint

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SLIDE 11

Motor and gearbox model (rigid body assumptions)

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Robot Gearbox Friction Inertia

,

m r

N; ω τ

Motor Inertia Friction

, ′ ′ ′

m r

N ; ω τ

′

m

τ

m

τ

′ N r = N

Gearbox = Geartrain

r m

τ ω ′ ′

r m

τ ω

Output power Input power

SLIDE 12

Motor and gearbox model

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τ′ η

gearbox efficiency

d d

b m b m

t Γ ω β ω +

p

τ

m r

ω τ

m m

ω τ

Motor side Joint side

SLIDE 14

Gearbox model

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Motor side Joint side

θ′ θ ρ′ ρ

ρ θ ρθ ρ ω ρω ′ ′ = ′ ′ = N r N ρ ρ = = ′ ′

m r

ω τ ′ ′

m r

ω τ

Ideal gearbox:

1 η =

GEARBOX Power in Power out

m r

ω τ ′ ′

m r m r

ω τ ηω τ ′ ′ =

r r m m

r r τ τ ω ω ′ = ′ =

SLIDE 15

CC motor equations – 1

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d d

a a a a a e m m m a a m a p m m m m m m m m r m p

L i v R i E K i E k K k i K i i K t

τ φ ω τ

ω ω τ τ τ β ω β φ φ φ τ Γ θ ω τ τ θ Γ = − − = ′ ′ = = ′ ′ = = ′ = ′ ′ ′ = + = + ′ ′ ′ = − ɺɺ ɺ ɺ k k K K

ω τ

′ ≈ ⇒ ≈

MOTOR SIDE

SLIDE 16

CC motor equations – 2

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JOINT SIDE

2

( ) ( ) ( ) ( )

r r m p m m m m m m m m m m m m m m m

r r r r r r r r τ τ τ τ τ ω β ω τ ω β ω Γ Γ τ β ω Γ ω ′ = ′ ′ −   ′ ′ ′ = − −       ′ = − −     ′ = − + = ɺ ɺ ɺ

2

1 1 ( ) 1 1 1 1 1 1 ( )

r r m p m b m b m m b m b m m b m b m

r r r r r r r r τ τ τ τ τ ω β ω τ ω β ω Γ Γ Γ τ ω β ω ′ =   =             ′ ′ = +   = + + +            +      ′ ′ =   +   + ɺ ɺ ɺ

MOTOR SIDE

SLIDE 17

Control equations

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( ) ( , ) ( ) ( )

T e

+ + + + = H q q C q q q B q q g q J F τ ɺɺ ɺ ɺ ɺ

1 1 1

( ) ) ) ( (

n n n ij j ijk j k bi i i fi ri j j k

H q q h g q q β τ τ

= = =

+ + + + =

∑ ∑∑

q q q ɺ ɺɺ ɺ

component-wise joint torques

1 1

( ) ( ) ( ) ( )

r n n n i ij j ijk j k bi i i fi j i i j i k i

q H q H q q h g q β τ

≠ = =

+ = + + + +

∑ ∑∑

q q q q τ ɺɺ ɺɺ ɺ ɺ ɺ

Inertial torques Coriolis & centripetal torques Friction, gravity & external torques

SLIDE 18

Control equations

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1 1

( ) ( ) ( ) ( )

n n n ij j ijk j k ii i bi i fi j i j k ri i

H q h q q g H q q τ τ β

≠ = =

+ = + + + +

∑ ∑∑

q q q q ɺɺ ɺ ɺ ɺ ɺɺ ( )

M i ci gi f r ii i bi i i i

H q q τ τ τ τ β τ + + + + + = q ɺɺ ɺ

Modelled torques “Disturbance” torques

SLIDE 19

From single motor model to robot control equation

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mi mi mi i i i i i i

q q r r q r θ ω ω = = = ′ ′ ′ ɺ ɺ ɺ ɺ

Gearbox transformation Equation seen at the joint side Structured disturbance

n ri ii i bi i ij j Mi ci gi fi j i mi mi bi bi Mi ci gi fi i i mi mi bi bi di i i

H q H q r r r r q τ β τ τ τ τ ω β τ τ τ τ ω β Γ ω Γ τ ω

≠

= + + + + + ′ + + + + ′ = ′ + + = ′ + +

∑

ɺɺ ɺɺ ɺ ɺ ɺ

SLIDE 20

From single motor model to robot control equation

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Equation seen at the motor side since

( )

2

1 1 1

ri ri bi mi bi mi di i i i

r r r Γ β ω τ τ ω τ ′ ′ + + ′ = = ɺ

( )

ri mi pi mi mi mi mi mi

τ τ τ τ Γ β ω ω ′ ′ ′ ′ = = + ′ ′ − − ɺ

and

2 2

1 1 1

mi ri pi bi mi mi bi mi mi di i i i

r r r τ τ τ β β τ Γ ω Γ ω           ′ ′ ′ ′ = + =  + + + +               ′ ɺ

we obtain Total inertia Total friction

SLIDE 21

From single motor model to robot control equation

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mi pi di ti mi ti mi di

τ τ τ ω β ω τ Γ ′ ′ ′ ′ ′ ′ ′ = + = + + ′ ɺ

p d t m t m m d

′ ′ ′ ′ ′ ′ ′ ′ = + = + + B τ τ τ Γ ω ω τ ɺ

= + +

t′

Γ

t′

B

2

1

bi mi i

r Γ Γ       +       

2

1

bi mi i

r β β       +       

Motor side

SLIDE 22

From single motor model to robot control equation

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Joint side

mi pi di ti mi ti mi di

τ τ τ ω β ω Γ τ = + = + + ɺ

p d t m t m d m =

+ = + + B τ τ τ Γ ω ω τ ɺ

= + +

t

Γ

t

B

( )

2 bi i mi

r Γ Γ +

( )

2 bi i mi

r β β +

SLIDE 23

Block diagram of open-loop CC motor (motor side)

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For 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 have

( ) ( ) ( ) ( )

t t m m d m a

s s s s K i

τ

β ω τ τ τ Γ + = − =

+ – + 1

a a

R sL + 1

t t

s β Γ + Kω Kτ 1 s

a

v

a

i

m

τ

d

τ

m

ω

m

θ – E

r

τ

1 ( ) ( )

m m

s s s θ ω =

SLIDE 24

Block diagram of open-loop CC motor (motor side)

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The armature circuit inductance is small and usually can be neglected

a a a a m

v R L i Kωω − = ≈ ⇒

m a m a a m m d t m t m a a a a d t t m

i K v R K K R R R v s K K K K

τ ω τ ω τ τ τ

τ τ ω τ τ ω β ω τ β ω Γ Γ = − = = + +          − = + +              ɺ

SLIDE 25

Block diagram of open-loop CC motor (motor side)

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friction torque

armature losses armature e.m.f. torque m a t a t m m t m a a m a

R K K K K R K K R i K K i

ω ω ω τ ω τ ω τ τ

β β ω ω β ω ω ′ = + ≈ ≪ ≪

since

SLIDE 26

Block diagram of open-loop CC motor (motor side)

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– +

( )

G s

ω

1 s

a

v

d

τ

m

ω

m

θ

d

K

1 1

a t a m a d

R R s v K K K K K

τ ω ω τ ω

Γ ω τ      + = −    ′ ′ ′    

a t a d

R T K K R K K

τ ω τ

Γ = ′ =

where

( ) ( )

1 1 G s K sT

ω ω

= ′ +

SLIDE 27

Matrix formulation (joint side)

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( ) ( , ) ( ) ( ) ( )

T e r m m p

+ + + + = ≡ = − H q q C q q q Bq g q J q F R τ τ τ τ τ ɺɺ ɺ ɺ ɺ

where

0 ; 0 ;

i m i a ai a i i i

R R K r K Kτ

τ ω ω

                      = = =                             R K K ⋱ ⋱ ⋱ ⋱ ⋱ ⋱

2 m m m a a m ω

= + R R K v R K q τ ɺ

2

( ) ( )

m p m m m m m m m m

= + = + R R q B q R q B q τ Γ Γ ɺɺ ɺ ɺɺ ɺ

Motor side Joint side Motor side Joint side and Lagrange Equation

SLIDE 28

Matrix formulation (joint side)

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Then, we have

( )

2 2

( ) ( , ) ( ) ( )

m m m m e m a a ω

+ + + + + + + + = H q R q C q q B R K B q g q J q F R K v Γ ɺɺ ɺ ɺ

T

( ) M q F

c

u

Mass matrix Friction matrix Command input

( )

( , ) ( , ) = + h q q C q q F q ɺ ɺ ɺ

Often we use this symbol to indicate the velocity dependent terms Gravity Interaction

SLIDE 29

Matrix formulation (joint side)

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( ) ( , ) ( ) ( )

T e c

+ + + = M q q h q q g q J q F u ɺɺ ɺ ( ) ( , ) ( )

c

+ + = M q q h q q g q u ɺɺ ɺ ( ) ( , )

c

+ = M q q h q q u ɺɺ ɺ

Gravity and interaction No interaction No gravity, no interaction Control Design Problem

...?

c =

u

SLIDE 30

t c

s + = F u Γ ω

disturbance The model is diagonal, i.e., naturally decoupled Each joint can be controlled by local controllers

SLIDE 31

Decentralized joint control

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– +

( )

G s

ω

1 s

a

v

d

τ

m

ω

m

θ

d

K

D

K

( )

t

K s

– +

r

v e

This is the proportional velocity controller

r velocity compensator

m d d d t

s T G s K G s s sT

ω

ω τ Γ = = − = − +

SLIDE 33

The closed-loop system

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– +

( )

G s

ω

′ 1 s

r

v

d

τ

m

ω

m

θ

d D

K K

Closed loop – +

( )

G s

ω

1 s

a

v

d

τ

m

ω

m

θ

d

K

Open loop

SLIDE 34

The closed-loop system

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Time constant is reduced

T T α <

Disturbance gain is reduced

d d D

K K K →

Design parameter when

1

t

K =

– +

( )

G s

ω

1 s

a

v

d

τ

m

ω

m

θ

d

K

D

K

– +

r

v e

SLIDE 35

Position compensator

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– +

( )

G s

ω

1 s

a

v

m

ω

m

θ

D

K

– +

e

P

K

– +

r

θ

d

τ

d

K

1 K θ ≈ K θ

t

K 1

t

K ≈

Controller 1

SLIDE 36

Position compensator

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1 2

( ) ( ) ( )

m r

s K G s s s s T K θ θ α = = + +

2 2

( ) 1 ( ) ( ) ( )

m d t

s G s s s s T K τ Γ θ α = = − + +

D P D P a t

K K K K K K TK R

τ ω

Γ = = ′

where Second-order TF with

1 1 · 2 2 1 ·

a D P t D P n a t

K K T K R K K K K K K K R

τ ω τ τ

α α ζ Γ ω Γ ′ = = = =

Configuration dependent

SLIDE 37

Design parameters

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( )

2

1 ( )

t b m b ii

r H t Γ Γ Γ Γ     = +        = q The damping coefficient and the natural frequency are inversely proportional to the square root of the inertia moment, that may vary in time when the angles vary Since 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 in such a way that the damping ratio is satisfactory, e.g., no overshoot appears in the step response

,max t

Γ ζ ,

P D

K K

SLIDE 38

– +

( )

G s

ω

1 s

a

v

m

ω

m

θ

P D

K sK ′ ′ +

– +

r

θ e

d

τ

d

K

An alternative

( ) ( ) ( )

a P D

v t K e t K e t ′ ′ = + ɺ

3 2

( ) / ( ) ( )

m D P D r a t a t P

s K K s K K G s s R s s T K K R

τ τ

Γ α θ Γ θ ′ ′ ′ + = = ′ + +

4 2

( ) 1 1 ( ) ( )

m d t P a t

s G s s s s T K K R

τ

τ Γ α θ Γ       = = −     ′  + +    

A zero appears Controller 2

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SLIDE 39

– +

( )

G s

ω

1 s

a

v

m

ω

m

θ

P D

K sK ′ ′ +

– +

r

θ e

d

τ

d

K

Another alternative

– +

D

K

Controller 3

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SLIDE 40

Comparison

Controller 2 Controller 3 Controller 1

( ) ( ) ( )

a D P D m

v t K K e t K t ω = − ( ) ( ) ( ) ( )

a P D r D m

v t K e t K t K t ω ω ′ = + −

( )

( ) ( ) ( ) 1 1 ( )

a D P D D r D D D m

v t K K e t K K t K K K t ω ω ′ ′ ′ ′ = + − +

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SLIDE 41

Practical Issues

Practical issues are considered that impose some constraints on the control algorithms:

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1. Saturating actuators
2. Elasticity of the structure and of the gearboxes
3. Nonlinear friction effects at joints and gearboxes
4. Sensors or amplifiers with finite band

SLIDE 42

Saturating Actuators

Linearity saturation saturation

( ) u t ( ) y t

( ) y t ( ) u t

It is a nonlinear effect, difficult to be considered a-priori

max max min max min min

( ) ( ) ( ) ( ) ( ) y u t u y t ku t u u t u y u t u           > = ≤ ≤  < IF IF IF

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SLIDE 43

Elasticity of the structure

Although we have considered rigid bodies, the elasticity is a phenomenon that limits the closed loop band We cannot design controllers that are “too fast” without taking explicitly into consideration some sort of elastic model. Recall that when we use a simplified model

( ) ( )

t m t m

t k t Γ θ θ + = ɺɺ

the proper structural resonance (or natural) frequency is

t r t

k ω Γ =

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SLIDE 44

Elasticity of the structure

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SLIDE 45

Nonlinear friction

total

( ) v t ( ) f t

( ) f t ( ) v t

A nonlinear effect between velocity and friction force

viscous coulomb stiction

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SLIDE 46

Nonlinear friction

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Nonlinear static friction models include:

Coulomb + viscous Static model that includes stiction and “Stribeck effect”: friction decreases with increasing velocity for v < vs (Stribeck velocity)

sign

c

F(v) F (v) v β = +

( )

sign

s s

v v c s c

F(v) F F F e (v) v

δ

β

−

      = + − +      

SLIDE 47

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 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.

47 ROBOTICS 01PEEQW - 2014/2015 Basilio Bona - DAUIN - PoliTo