Robotics 2017 09 Control 2 - Home@ · PDF fileInverse dynamics feedforward control (also...

ROBOTICS

01PEEQW

Basilio Bona

DAUIN – Politecnico di Torino

Control – Part 2

Introduction to centralized control

� Independent joint decentralized control may prove

inadequate when

� the user requires high task velocities, since structured

disturbance torques (Coriolis, centrifugal) greatly influence the robot behavior

� motors are of direct drive type, since no gearboxes are

present, cancelling their beneficial effect. Nonlinearities and

coupling effects become important

� In these cases the disturbance torques may cause large

errors on the reference trajectory tracking that the

independent control architecture is unable to reduce

Basilio Bona - DAUIN - PoliTo 3ROBOTICS 01PEEQW - 2016/2017

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 of the all other joint positions

and velocities.

The approach is not “local” anymore, as happens in the

decentralized architecture, where each joint controller uses

only the local joint information (position and velocity)


� In the centralized control architectures the robot is considered

as a MIMO system, with n inputs (the joint torques) and n

outputs (the joint positions) that interact with each other

according to the nonlinear dynamic equations of the robot

model

� The centralized control algorithms shall take into account this

dynamic model, and usually they present a nonlinear structure

� The main centralized control architecture is called inverse

dynamics, since the command torques are computed from the

robot model dynamic equation and from the knowledge of the

joint variables (positions and velocities), i.e., as a solution of a

model-based inverse dynamic problem

Centralized control


Inverse dynamics control


( ) ( , )c

+ =M q q h q q uɺɺ ɺ

Consider again the following simplified dynamic model of the robot

+

+

+

–

qɺ

qcu

ca

rqɺɺ

hɵ

�M ROBOT

We can compute a control command as a function of the dynamic model,

where one tries to cancel the inertia and other terms

�( )r cc− +qu M a hɺ≐ ɺ

ɵ

Reference acceleration

Additional acceleration

command signal

�( )r cc− +qu M a hɺ≐ ɺ

ɵ

Approximate linearization


Hence, developing the previous equation, we have:

�( ) ( , ) ( )r c

+ = − +M q q h q q M hq aɵ

ɺɺ ɺɺɺ

� ( )1 1( ) ( ) ( , )( )r c

− −− −−=q M q M M q h q qq a hɵ

ɺɺ ɺɺɺ

( )( , ) ( , )∆− =h q q h h q qɵ

ɺ ɺ�

�

1

1

( ) ( )

( ) ( )

−

−

= +

= −

M q M I E q

E q M q M I

Here we try to invert the inertia matrix

Here we try to send to zero the disturbance

terms

Conclusions


( ),) , ,(rc cr

= +−q q q aq a qɺɺɺɺ ɺ ɺɺη

( ) 1, , , ( ) ( )( ) ( , )rr c c

∆−= −−qq q q a E q M q h qa qη ɺɺɺ ɺɺ ɺ

Structured disturbance

Approximation

in inertia modelApproximation in

Coriolis model

Conclusions


( ),) , ,(rc cr

= +−q q q aq a qɺɺɺɺ ɺ ɺɺη

If we can cancel the structured

disturbance, the system becomes

linear and decoupled, but unstable

( )r c r c c

= − − = ⇒ =⇒q q a q q a e aɺɺ ɺɺ ɺɺ ɺɺ ɺɺ

The chains of double integrators make the system unstable

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

( , )h q qɺ

q

qɺ

u( )M q +

+

ac

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 influences only the behavior of the i-th joint component

qc,i that is independent of the other joints motion


ROBOT

( , )h q qɺ

q

qɺ

u( )M q +

+

ac

∫ ∫ac

qqɺ

Exact linearization

+

+

+

–

qɺ

qcu

ca

rqɺɺ

Inner Loop

( , )h q qɺ

,M h

ROBOT

( )M q

1

s

1

s

1

s

1

s

⋯

1q1

qɺ+

–

ca

rqɺɺ

nq

nqɺ

⋯

Control design


State variable representation


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

Controller design


Controller design


+

+

+

–

qɺ

qcu

ca

rqɺɺ

hɵ

�M ,M h

ROBOT

Linear Outer Loop

Nonlinear Inner Loop

Inner loop – outer loop (nonlinear linearizing control)


Exact linearization


Exact linearization

+

+

+

–

qɺ

qcu

ca

rqɺɺ

Inner Loop

( , )h q qɺ

,M h

ROBOT

( )M q

1

s

1

s

1

s

1

s

⋯

1q1

qɺ+

–

ca

rqɺɺ

nq

nqɺ

⋯

Control design


PD outer loop control design




D−K

P−K

+

+

+

–

+

–

rq

rqɺ

Outer Loop


+

+

+

–

qɺ

qcu

ca

rqɺɺ

Inner Loop ( , )h q qɺ

,M h

ROBOT

( )M q


PID outer loop control design


+

+

+

–

DK−

PK−

+

+

+

–

+

–I

K

s

−

+rqɺ

rq

rqɺɺ

h

qɺ

qcu

ca

M

a PID controller for the outer loop

Inner Loop

Outer Loop


ROBOT




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 so

� Unmodeled dynamics (e.g., elastic vibrations) are not taken

into account

In practice it is impossible to satisfy all these assumptions at

the same time



� 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:

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

matrices



3. Robust control techniques; these are advanced techniques

that allow to overcome the effects of the approximations and

other errors introduced by the real-time computation of the

inverse dynamics

4. Adaptive control techniques; they provide an online

estimation of the true model parameters that are successively

used in the controller

� Also for the outer loop controller it is possible to adopt more

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 cancel





� Under the assumption that only an approximate compensation

is possible, the external control law becomes:


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

simplifies the model

�( ) ( , )c c= +u M q a h q q

ɵɺ

� ;M M h hɵ

≃ ≃

Where

hɵ

+

+

+

–

qɺ

qcu

ca

rqɺɺ

�M ROBOT

Inner loop

We assume to use a PD controller

for the outer loop




� There are some inverse dynamics architectures with

approximate linearization, that include

� PD control with gravity compensation

� Independent joint control

� Inverse dynamics feedforward control (also called computed

torque method)


PD control with gravity compensation


Exact knowledge of

the gravity terms

Outer loop

Inner loop

+

+

+

+

–

+

–

–

qɺ

qcu

ca

DK−

PK−

rqɺ

rq



ROBOT

e

eɺ

( )gqτ



(A)









Independent joint control




+

+

+

–

+

–

qɺ

q

DK−

PK−

rqɺ

rq



ROBOT

Outer loop

Inner loop

e

eɺ

c c= −u a

Computed Torque Method

� The computed torque method represents an solution midway

between the decentralized control and the inverse dynamics

control

� It is mainly used when the reduced processing power cannot

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

gravitational effects, that are dominant when the velocities are

small

� Computing off-line the feedforward torque terms and storing

them in a mass memory or look-up table; this is possible when

cyclic trajectories are performed again and again






+

+

+

–

+

+

+–

+–

qɺ

q

rqɺ

rq

DK−

PK−

cu

ca

rqɺɺ



ROBOT

Outer loop

e

eɺ

Feedback linearization – the general framework








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