Lecture 12: Multivariable Control of Robotic
Manipulators
Part II
Reading: SHV Ch.8
Mechanical Engineering
Hanz Richter, PhD
Robust vs. Adaptive Control
A robust control system uses a fixed controller capable of “performing
well” (guaranteed stability+performance) despite uncertainties in plant
model parameters.
An adaptive system, in contrast, attempts to obtain estimates of
uncertain/ unknown plant parameters and contains self-adjusting control
parameters, which are calculated on the basis of the current estimates
of the plant parameters. That is, the structure of the controller is fixed,
but the control parameters (gains) are self-adjusted.
Adaptive systems were used in the 1950’s before they were fully understood.
Stability proofs and systematic procedures became available as
late as the 1980’s. Kumpati Narendra and Karl Åström are among the
most recognized researchers in this field (see textbooks by each).
Robot Adaptive Inverse Dynamics
Consider the manipulator dynamics again:
We know that an inner-loop feedback linearizing input of the form
results in a double integrator plant 
which
is easily stabilized by
decoupled PD loops.
Now suppose M(q), 
and g(q) are not known
exactly. We only
have current estimates 
and
which will be adjusted in real time by
the adaptive system.
Take the control law (drop the arguments q and q˙ for notational
simplicity):
Adaptive Inverse Dynamics...
Take the virtual control aq to be
where K0 and K1 are positive-definite diagonal matrices and
is the
tracking error. In class, we carry the full details of the derivation of the
adaptation law (how to update 
.
First, we recall the linear parameterization property :
This al lows us to isolate all parameters in vector . Defining the parameter
estimation error as 
, we show that the
dynamics of the tracking error
are given by
where A is made Hurwitz by choosing K0 and K1 to be diagonal with positive
non zero entries and 
.
Adaptive Inverse Dynamics...
Since A is Hurwitz, we can find a symmetric and positive-definite matrix
as the solution to the Lyapunov equation
ATP + PA = −Q
where Q is an arbitrary symmetric, positive-definite matrix.
Then we form a Lyapunov function for the adaptive system as
We compute its derivative and force it to be negative -semidefinite by
choosing the parameter adaptation law as follows
where 
is a symmetric positive-definite matrix and .
Adaptive Inverse Dynamics...
In summary, the control system is formed by a control law and a parameter
adaptation law:
Notes:
Since 
Qe (independent of
), it follows that it is zero at non-zero values
of 
. This implies
is really negative-semidefinite. Therefore, it
can
only be said that e is and are bounded. We can actually prove e
-> 0 by taking
extra steps (details developed in class). The parameter estimation error can
only be guaranteed to be bounded (sometimes good enough). A more serious
problem is the need to measure
in realtime and that
must be invertible at
all times.
Passivity-Based Robot Control
The inverse dynamics approaches seen before are characterized by
their attempt to cancel plant nonlinearities and transform the control
problem into a linear one.
An alternative is to apply direct nonlinear control techniques to
guarantee that the closed-loop system will be stable, without attempting
to linearize it.
Take the control input to be
where
with K and being diagonal matrices with positive nonzero entries.
Passivity-Based Control...
Substitution results in the closed-loop dynamics
The closed-loop system is nonlinear and coupled. Consider the Lyapunov
function
As we show in class, the derivative computes to
where Q is symmetric and positive-definite, making
negative-definite. The
error dynamics are therefore globally asymptotically stable.
Note that we still require exact knowledge of the inertia matrix, the C matrix
and the gravity term. The real advantage of the passivity method comes when
considering the robust and adaptive cases.
Robust Passivity-Based Control
As suming only estimates of the robot matrices are available, we take
Using the linear parameterization we can write the control as
As we show in class, we take 
, where
is our best
estimate of the parameters (nominal values) and
is a new control.
The uncertainity in must be bounded by a known constant
:
Then we use the variable-structure control or discontinuous control of
Eq.(8.98) to force the previous Lyapunov function to be negative-definite.
Robust Passivity-Based Control...
Note that this powerful control strategy requires only position and
velocity feedback (easily obtained from encoders) and an estimate of
parameter uncertainty ( bound).
In the final project, you will be implementing this controller assuming
im perfect
knowledge of the manipulator parameters.
Adaptive Passivity-Based Control
In this case, our estimates of M, C and g are not frozen, but will be
continuously adjusted by the system. The form of the control law is the same
as in the robust passivity-based approach:
We again obtain
The derivation of an adaptation law is done on the basis of the Lyapunov
function
As shown in class, the parameter adaptation law
results in a negative-semidefinite derivative .
Adaptive Passivity-Based Control...
Since 
, we can only conclude non-asymptotic stability at
this point. However, e is a square-integrable function, since the integral
of a quadratic function of e is a finite number , computable in closed form
as:
This implies that 
itself and
are themselves square-integrable (components
of e). Also, 
and the fact that V contains only non-negative
terms must mean that r, and
remain bounded. Therefore, since
, it must also be bounded, which concludes boundedness
of .
Adaptive Passivity-Based Control...
These two facts ( square integrable and
bounded can be used in
Barbalat’s Lemma (see p.311 in SHV) to conclude
-> 0 as 
.
We can go further and use a similar argument to conclude that
-> 0.
To use Barbalat’s Lemma, we need to establish that
is bounded, which
follows from the closed-loop equation
if we assume that 
is bounded.
Example: Two-Link Planar Robot
We de sign each of the above controllers to the 2-link planar manipulator.
The regressor Y (
) and the parameter vector
are listed in SHV,
p.271. We assume that the true value of each one of the parameters
is unity. For robust schemes, we take a random parameter deviation of
10% in each parameter. For adaptive schemes, we take a deviation of
90% as initial guess for the adaptation laws.
We first design an inverse dynamics controller to track a circular
trajectory using joint-space feedback, then using task-space feedback.
We examine the performance of the controller under off-nominal
conditions.
We the tune the controllers to track the same circular trajectory using
adaptive inverse dynamics, robust passivity-based control and adaptive
passivity-based control.
