Industrial Robots Industrial Robots Control Control Part 2 Control - - PowerPoint PPT Presentation

Industrial Robots Industrial Robots

Control Control – Part 2 Part 2 Control Control Part 2 Part 2

Introduction to centralized control

Independent joint decentralized control may prove inadequate when

the user requires high task velocities ⇒ structured disturbance torques (Coriolis, centrifugal) greatly influence the robot behavior motors are of direct drive type ⇒ since non gearboxes are present, their beneficial effect is absent and nonlinear effects and coupling effects become important

In these cases the disturbances torques may cause large errors

• n the reference trajectory tracking

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Introduction to centralized control

Since in these cases it is not possible to sufficiently reduce the effects due to the disturbance torques, it is more convenient to try to cancel such torques, adopting control algorithms that use nonlinear compensation terms We call this architecture “centralized” since the applied joint command torques are function also of the other joint positions and velocities. The approach is not “local” anymore, as happens in the decentralized architecture, where each joint controller uses

• nly the local joint information (position and velocity)

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Centralized control

In the centralized control architectures the robot is considered as a MIMO system, with n inputs (the joint torques) and n ( h j i i i ) h i i h h h

• utputs (the joint positions ) that interact with each other

according to the nonlinear dynamic equations of the robot model model The centralized control algorithms shall take into account this dynamic model, and usually they have a nonlinear form The main centralized control architecture is called inverse inverse dynamics dynamics, since the command torques are computed from the robot dynamic equation and from the knowledge of the joint variables (positions and velocities), i.e., as a solution of an inverse dynamic problem

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Inverse dynamics control

( ) ( , )

c

+ = M q q h q q u

• Consider again the following simplified dynamic model of the robot

( ) ( , )

c

q q q q

We can compute a control command as a function of the dynamic model, where the inertia and disturbance terms are approximate

• where the inertia and disturbance terms are approximate
• h

Reference acceleration + + +

q q

c

u

r

q

• M

ROBOT ROBOT –

q

c c

a

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( )

r c c

− + q u M a h

• Additional acceleration

command signal

Approximate linearization

Hence, developing the previous equation, we have:

• ( )

( , ) ( )

r c

+ = − + M q q h q q M h q a

• (

)

1 1

( ) ( ) ( , ) ( )

r c − −

− − − = q M q M M q h q q q a h

• (

)

( , ) ( , ) Δ − = h q q h h q q

• 1( )

( )

−

= + M q M I E q

( )

( , ) ( , ) q q q q

• 1

( ) ( )

−

= − E q M q M I

Here we try to invert the inertia matrix Here we try to invert the inertia matrix Here we try to send to zero the disturbance

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Here we try to send to zero the disturbance terms

Conclusions

( )

, ) , , (

r c c r

= + − q v a q q q q η

• If we can cancel this part,

the system becomes linear and decoupled, but unstable Structured disturbance

( )

1

, , , ( ) ( ) ( ) ( , )

r r c c

Δ

−

= − − q q q q a E q M q h q a q η

• Approximation

i i i d l Approximation in in inertia model Coriolis model

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In general

If we are able to compute the exact model of the nonlinear dynamical system, we can build the following control architecture, where ac is a suitable acceleration control signal, whose

definition will allow to obtain asymptotic stability and other performances

ROBOT

q

u

( ) M q

+

ac

ROBOT

q

• ( )

M q

+ +

( , ) h q q

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Exact linearization and decoupling

The control scheme above performs an exact system linearization The resulting system is decoupled: it consists in n double integrators; the i‐th component ac i of the new acceleration signal g p

c,i

g influences only the behavior of the i‐th joint component qc,i that is independent of the other joints motion q

ROBOT

• u

( ) M q

+ +

ac

∫ ∫

ac

q q

• ( , )

h q q

• q

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State variable representation

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Error variable representation

This term is equivalent to the injection into the system of a structured nonlinear disturbance that can make it unstable in spite of the control design

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Controller design

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Controller design

h

• Nonlinear Inner Loop

+

q

• q
• h

ROBOT ROBOT

+ + –

q q

c

u

r

q

• M

, M h

q

c

a

Linear Outer Loop

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Inner loop – outer loop (nonlinear linearizing control)

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Exact linearization

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Exact linearization

• ( , )

h q q

• ROBOT

ROBOT + + +

q

• q

c

u

r

q

• ,

M h ( ) M q

–

c

a

Inner Loop

1 s 1 s

1

q

1

q

• +

r

q

• 1

s 1 s

• –

c

a

n

q

n

q

• c

n n

Control design

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PD outer loop control design

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PD outer loop control design

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PD outer loop control design

Inner Loop

• Inner Loop

( , ) h q q

• ROBOT

+ + + –

q

• q

c

u

r

q

• ,

M h ( ) M q

c

a

+

q

• Outer Loop

D

−K

+ + + + –

q

r

q

P

−K

+ –

r

q

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PID outer loop control design

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Inner Loop

PID outer loop control design

h

• Inner Loop

+ + +

r

q

• q

q

c

u

M

ROBOT

–

c

a

a PID controller for the outer loop

Outer Loop

D

K −

+ + + –

K −

+

r

q

• q

P

K −

+ –

I

K s −

r

q

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PID outer loop control design

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Practical aspects of exact linearization

Exact linearization hypothesis implies the capacity to compute on‐ line the model matrices

Dynamic models must be perfectly known, without errors or approximations The model matrices must be computed online; at every sampling time (approx. 1 ms) the inverse dynamics equations must be solved Software and hardware architectures must be able to do

• solved. Software and hardware architectures must be able to do

so Unmodeled dynamics (e g elastic vibrations) are not taken into Unmodeled dynamics (e.g., elastic vibrations) are not taken into account

In practice it is impossible to satisfy all these assumptions at In practice it is impossible to satisfy all these assumptions at the same time

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Approximate linearization

In practice one may adopt a control scheme based on the imperfect compensation of the robot inverse dynamics There are different possibilities to do so: p

• 1. Only a part of the robot dynamics is computed and used in the

controller (usually the dominant one and/or that is best known), leaving to the outer loop the task of guaranteeing the overall stability and the reference trajectory tracking

• 2. A feedforward compensation of the robot dynamics; it uses the

reference values of the desired trajectory instead of the real measured joint variables and velocities to build the model measured joint variables and velocities to build the model matrices

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Approximate linearization

• 3. Robust control techniques; these are advanced techniques that

allow to overcome the effects of the approximations and other i d d b h l i i f h i errors introduced by the real‐time computation of the inverse dynamics 4 Adaptive control techniques; they provide an online estimation

• 4. Adaptive control techniques; they provide an online estimation
• f the true model parameters that are successively used in the

controller controller

Also for the outer loop controller it is possible to adopt more complex algorithms than the simple PD controller. complex algorithms than the simple PD controller.

A PID controller will be able to cancel the steady state effects of any additive constant disturbance that the inner loop cannot y p cancel

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Approximate linearization

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Approximate linearization

Under the assumption that only an approximate compensation is possible, the external control law becomes: ( ) ( , )

c c

= + u M q a h q q

• ;

M M h h

• Where

represents an estimate of the true matrices. These matrices could be both the best available approximation of the true matrices, or the result of an “a‐priori” decision that i lifi h d l simplifies the model

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Inner loop

Approximate linearization

h

• Inner loop

+ + +

q q

c

u

r

q

• M

ROBOT

–

c

a

We assume to use a PD controller for the outer loop

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Approximate linearization

There are some inverse dynamics architectures with approximate linearization that include approximate linearization, that include

• 1. Independent joint control
• 2. PD control with gravity compensation
• 3. Inverse dynamics feedforward control (also called computed

torque method)

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Independent joint control

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Inner loop

Independent joint control

Inner loop

• r

q

• q

q

ROBOT

• M

− +

c

a

Outer loop

c

+ + + –

D

K −

r

q

• q

+ –

P

K −

r

q

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PD control with gravity compensation

Exact knowledge of the gravity terms

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PD control with gravity compensation

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Inner loop

PD control with gravity compensation

Inner loop

( )

g q

τ

• +

–

q

• q

c

u

ROBOT

c

a

Outer loop + + + –

D

K −

r

q

• q

+ –

P

K −

r

q

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Computed torque method

The computed torque method represents an solution midway between the decentralized control and the inverse dynamics bet ee t e dece t a ed co t o a d t e e se dy a cs control It is mainly used when the reduced processing power cannot y p g p afford the complete inverse dynamics implementation When the real‐time constraints are strict, the feedforward term can be approximated

Considering only the diagonal terms of the inertia matrix and the it ti l ff t th t d i t h th l iti gravitational effects, that are dominant when the velocities are small Computing off‐line the feedforward torque terms and store them in Computing off line the feedforward torque terms and store them in a mass memory; this is possible when cyclic trajectories are performed again and again

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Inner loop

Computed torque method

( , )

r r

h q q

• Inner loop

( )

r

M q

+ + +

q q

c

u

r

q

• ROBOT

– Inner Loop

c

a

Outer loop + + + –

r

q

• q

D

K −

+ –

r

q

P

K −

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