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Remembering the Motion Models: As we found using the L-E approach, the
Required Joint Torque is:
( ) ( , ) ( ) ( )i i i i i iD q q C q q h q b q
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)
1( ) ( , ) ( ) ( )i i i i iv q D q C q q h q b q
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:
D-1(q) 1/s 1/sOutput
PositionsKinematics
b
C
h
+
+
+
Torque accel Vel pos
Friction
CoriolisCentrifugal
Effects
GravitationEffects
Inertial Coupling
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
This simple model is called a PE (proportional error) controller
+ KeError
State Space Model,Generalized Torque Needed
Feedback, Qa
Qd
+
-
QJoint Drive
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
to the 1st approx:
m eq eq
m m eq eq
J q F q
K I J q F 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:
Here, kpe is a “gain” term that guarantees sufficient current will be generated to develop appropriate torque based on observed positional error
( )pe d ak
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:
1 12 2 2
1 2
*
who's solution is:
pe a
F t t tJa
k J F
e C e C e
a is a function of the servo
feedback as a function of time!
Examining this ‘solution’:
The 1st term is a damping term for the motion
The systems ‘natural frequency’ is given by:
2
Ft
Je
2
24 ekFJJ
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
Gravitational input is positionally controlled: L = -g (M1r1 + M2R)* Cos()
For a R Manipulator with a payload on M2
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)
pe d a d
dk k L
dt
State Space Model of PD:
+ KeError
State Space Model,Generalized Torque Needed
Feedback, Qa
Qd
+
-
QJoint Drive
Kd
dQ/dt
Leads to a Solution of the form:
2 .5 .51 2
2
2
4
where:
eq d
eq
F ktJ t t
eq d pe
eq eq
e C e C e
F k k
J J
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:
0
( ) ( ) ( )t
pe d a i d a dtt k t k t dt k
PID State Space Model:
+ KeError
State Space Model,Generalized Torque Needed
Feedback, Qa
Qd
+
-
Q
Kd
dQ/dt
ki dt
Joint Drive
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:
2
given by
min
:
pe d
u dt
u k k
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
+ Controller w/ Adj.Parameters
Error Control Input
Feedback, Qa
Qd+
-
Drive Position/TorqueActualPos
PerformanceIndex
Measure
Robot Sys.Transfer
Functions
Desired Drive
Calc. Drive
Actual Drive usingSeparate Feedback
Sensors
DecisionLogic
Modifications
Kinematic/Kinetic Models
PhysicalParameters
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!