Introduction to Control:
How Its Done In Robotics
R. Lindeke, Ph. D.
ME 4135
Remembering the Motion Models:

As we found using the L-E approach, the
Required Joint Torque is:
 i  Di (q) qi  Ci (q, qi )  h(q)  b(qi )
Dynamical
Manipulator
Inertial Tensor
– a function of
position and
acceleration
Coupled joint
effects
(centrifugal and
coriolis) issues
due to multiple
moving joints
Gravitational
Effects
Frictional Effect
due to Joint/Link
movement
Lets simplify the model (a Bit)

This torque model is a 2nd order one (in position) lets look at it
as a velocity model rather than positional one then it becomes a
system of highly coupled 1st order differential equations

We will then isolate Acceleration terms (acceleration is the 1st
derivative of velocity)


v  q  D (q )   i  Ci (q, qi )  h(q )  b(qi ) 


1
i
Considering Control:

Each Link’s torque is influenced by each other links
motion



We say that the links are highly coupled
Solution then suggests that control should come
from a simultaneous solution of these torques
We will model the solution as a “State Space”
design and try to balance the torque-in with
positional control-out – the most common way it is
done!

But we could also use ‘force control’ to solve the control
problem!
The State-Space Control Model:
Inertial Coupling
Torque
+
+
+
D-1(q)
accel
Friction
Coriolis
Centrifugal
Effects
Gravitation
Effects
1/s
b
C
h
Vel
1/s
pos
Output
Positions
Kinematics
State Space Approach



The State Space General Model feeds out positional
kinematics based on the “torque (power) demand”
input
Notice: 1/s is the Laplace transform of a unit step
(torque) impulse
As you remember, Laplace transforms convert linear
differential equation sets into algebraic equation sets

once solved we need to do inverse Laplace transforms to
return to torque/position space
 In LaPlace models, S is a complex variable!
Ultimately how do we know How much Torque
to specify – and if we are ‘In Control’




In robots with moving joints, we set
targets for motion
We sense motion at the joint level
(using kinesethic sensors)
We study differences between where
we are and where we’re going as a
“Feedback” (ie. servo) error
Control means that we try to minimize
error (make error go to zero) when we
move toward a new location
Setting up a Real Control

We will (start) by using positional error to drive our torque
devices
+
+
Qd
Error
Ke
State Space Model,
Generalized
Joint Drive
Torque Needed
-
Feedback, Qa

This simple model is called a PE (proportional error)
controller
Q
PE Controller:

To a 1st approximation,  = Km*I


Torque is proportional to motor current
And the Torque required is a function of ‘Inertial’
(Acceleration) and ‘Friction’ (velocity) effects as
suggested by our L-E models
m
J eq q  Feq q
to the 1st approx:
K m I m  J eq q  Feq q
Setting up a “Control Law”


We will use the positional error (as drawn in the state model)
to develop our torque control
We say then for PE control:
  k pe ( d   a )

Here, kpe is a “gain” term that guarantees sufficient current
will be generated to develop appropriate torque based on
observed positional error
Using this Control Type:


It is a representation of the physical system of a
mass on a spring!
We say afer setting our target as a ‘zero goal’
that:
k pe * a  J   F 
a is a function
of the servo
feedback as a
function of time!
who's solution is:

a  e
 F
2J
t C e 12 t  C e  12 t 

1
2

Examining this ‘solution’:

The 1st term is a damping term for the motion
e

Ft

2J
The systems ‘natural frequency’ is given by:
 F
2
4 ke 


2 
J
J 
Studying the solution:

If (F2/4kpe)> J we are ‘over damped’ and the
system will never quite reach its target
considering “reasonable time”

If (F2/4kpe)= J we are ‘critically damped’ giving
the system ideal behavior

If (F2/4kpe)< J we are ‘under damped’ and the
system will over-shoot and oscillate about our
target at the system’s natural frequency – a
dangerous situation in robotics!
These behaviors make sense
(physically!):

Under High Friction Conditions (over
damped):


A system is hard to start but easy to stop
With High Moment of Inertia Conditions
(under damped):

A system is hard to start and it is hard to stop leading
to overshoot and possibly one that oscillates about the
target ‘forever’
Problems with PE Control:

First the so-called “Steady-State”
error – the torque goes to zero
when the target is hit!

Secondly, we may be out of balance
– the GAIN is not meeting our
Inertial vs. Friction balance leading
to overshoot or undershoot

Typically we will add a term to our
model to react to increasing speed so
we minimize overshoot
Dealing With Steady State Zero
Error


This is a gravitational issue: we
must add an ‘L’ or gravitational
term back to our Torque control
model
M2
Gravitational input is positionally
controlled:

L = -g (M1r1 + M2R)* Cos()
For a R Manipulator
with a payload on M2
M1
Solving the Overshoot
Problem:

Lets expand our control law:



We should include a term that reacts to the
velocity of the link –
But velocity is the derivative of the position
We will call this a proportional – derivative
controller (PD Control)
 d
  k pe  d   a   kd 
 dt

 L

State Space Model of PD:
+
+
Qd
Error
State Space Model,
Generalized
Ke
Kd
dQ/dt
Feedback, Qa
Joint Drive
Torque Needed
Q
Solving the PD Model:

Remembering:
  J eq q  Feq q


k




k


L

J


F



pe
d
a
d
eq
eq


k pe     J eq    Feq  kd   L
Leads to a Solution of the
form:
 e
 Feq  kd


t

2
J
eq 

.5 t
C1e
 C2 e
where:
 F  k 
eq
d
 
2

J eq

2
  4k 
   pe 
  J eq 

.5 t

Effect of Derivative Term:




It is observed to be a form of
“Active Friction”
It tends to slow down the link as it
moves faster when high errors
(being far from goal) are observed
Thus it can be thought of as a
brake on the motion
It is a component that anticipates
changes and provides very fast
response to these changes
Taking Care of Trouble:

We add an integrator to the model




To damp out oscillations from over shoot
To control effects due to environmental
perturbations
To damp out Wild data gyrations – typically due to
encoder errors
This leads to a model of control:

 (t )  k pe ( d   a  t )  ki  ( d   a  t )dt  kd 
t
t0
PID State Space Model:
ki dt
+
+
Qd
Error
State Space Model,
Generalized
Ke
Kd
dQ/dt
Feedback, Qa
Joint Drive
Torque Needed
Q
Developing Optimal Control



PID is most often found in the systems Arm
Joints
The components of Torque are functions of
POSE meaning the Jeq and Feq as well as L
factors change as one observes the robot
moving throughout the work envelope
We achieve control with kpe, kd and ki

if they are fixed values, we can expect critically
damped control at only a single (or very few)
position(s)
What is done:


We operate off critical PID on arm joints and PD
w/ gravitational compensation for remote and
wrist joints
Most controls use a form of adaptive control


Tables of gains applicable over certain geometries with
automatic changes as the manipulator moves about
the work envelope
We swap in and out the gain values such that we
minimize energy consumed by the drive:
min  u 2 dt
given by :

u  k pe   kd 
Another Idea:

Develop a Performance Index (PI) that
judges controller stability

This PI is an external measurement scheme
that using logic and comparisons between
desired and actual performance then adjusts
the model
State Model of Adjustable
Controller
Kinematic/
Kinetic Models
Physical
Parameters
Qd
+
+
Error
Controller w/ Adj.
Parameters
Control Input
-
Robot Sys.
Transfer
Functions
Drive Position/Torque
Calc. Drive
Modifications
Decision
Logic
Desired Drive
Performance
Index
Measure
Feedback, Qa
Actual Drive using
Separate Feedback
Sensors
Actual
Pos
Thus Ends our Introductory
Studies of “Robotics & Controls”

This is a rich and deep field of applied
Mechanical and Industrial Engineering

While we have deeply explored some topics,
others have only been scanned

I wish you well as you move forward in your
lifelong exploration of “AUTOMATION” and
its myriad of supporting technologies

I sincerely hope that you have all learned
something of this fascinating field and that
these lessons will prove to be valuable in
your careers!