**Current and Previous
Research Areas**

**(See Publications By
Category)**

**Iterative Learning and Repetitive
Control**

Work in these fields started with
visiting scholar Sun Jian-Guo in 1983, and also resulting in a first
paper by Richard Middleton, doctoral student, and Prof. Graham
Goodwin at the University of Newcastle in Australia, submitted in
1984, the year many people consider as the start of the field (there
were precursors). This year saw independent publications of similar
ideas coming from four continents, including the first two
publications by Arimoto. Work accelerated so that there are now
approximately 180 publications in this area by the research group.

*Iterative Learning Control (ILC):
*Feedback control systems do not do what you ask them to do.
Classical control theory treats steady state frequency response of
linear systems, and a command of one frequency produces an output of
the same frequency but with the wrong amplitude and the wrong phase.
In addition there is error during the transient phase and there is
error from disturbance effects that often are the same every time the
same command is given. Iterative learning control aims to eliminate
all such errors in following a specific command repeatedly, by
adjusting the command given to the feedback control system based on
the error observed in the previous run. Initial motivation of the
field came from robots doing repetitive tracking operations in
manufacturing. Experiments done by the research group on a robot at
NASA Langley Research Center decreased the tracking error of the
robot following a high speed command, by a factor of 1000 in
approximately 12 cycles for learning.

*Repetitive Control (RC):* RC has
two main focuses, one is to converge to zero tracking error in a
feedback control system when given a periodic command, and the second
is to converge to zero error for a constant command in a feedback
control system that is subject to a periodic disturbance of known
period. RC is analogous to ILC except that it looks back to the error
observed in the last period of the command and/or disturbance to make
adjustments to the current command. Applications include computer
disk drives, eliminating 60Hz related ripple in rectified DC voltage
in physics particle accelerators, non-circular machining, vibration
isolation of fine pointing equipment on spacecraft containing
reaction wheels or control moment gyros, eliminating velocity
variations from imperfect gearing in copy machine belt drives,
eliminating periodic sensor error in belt steering in copy machines,
etc.

**Kinematics, Intelligent Mechanisms,
Morphing Mechanisms**

Ferdinand Freudenstein, the so-called
Father of Modern Kinematics, and my neighbor in the ME department at
Columbia, and I co-advised doctoral student Meng Sang Chew in a
thesis applying optimal control theory to the design of cam follower
systems. This work received a best paper award from the Mechanisms
Committee of ASME. A series of publications have been produced since
that time.

*
Mechanisms and Repetitive Control:*
Important contributions often occur at the interface between fields.
Many mechanisms perform periodic motions, and very often they are
actuated using a feedback control system whose objective is to
maintain constant input rotation rate. A series of publications study
the use of RC

- To maintain the constant input velocity in spite of the variation in load or inertia during a rotation.
- To allow one to fix the effects of
inaccuracy in the manufacture of the mechanism, such as a cam –
*Intelligent Mechanisms* - To change the mechanism
characteristics to modify and improve performance, again as in a cam
–
*Morphing Mechanisms*

**System Identification, Damage
Detection in Structures**

Research in this area started around
1987 working with Dr. Jer-Nan Juang at NASA Langley together with my
Columbia doctoral students, and continuing with Prof. Minh Phan (then
at NASA, later at Princeton, and now at Dartmouth) and Dr. Lucus
Horta (NASA) after they finished their doctorates. Structural
dynamics identification research also includes collaboration with
Prof. Raimondo Betti (Columbia). Many of the results of the early
part of this research can be seen in the textbook, *Applied System
Identification *by Jer-Nan Juang, Prentice Hall, 1994.

*System Identification:* Before
one can design a control system one needs a model of the plant to be
controlled. If the plant hardware is available one can make
input-output experiments to collect data that allows you to create a
mathematical model. There are many subtleties in this process and as
a result it is still somewhat of an art. With Dr. Jer-Nan Juang,
Prof. Minh Q. Phan and Dr. Lucas Horta, we developed the OKID (and
SOCIT) algorithm, Observer Kalman Filter Identification algorithm. It
is distributed by NASA and it was ordered by most of the major
aerospace companies in the US. The approach develops the steady state
Kalman filter from input-output data first, and from this result it
backs out a state space model of the system. Of course this is
backward from what one normally expects to do, to find the system
equations, then make some assumptions on the noise levels. Here one
gets the Kalman gain directly from the data without making
assumptions. For systems that have the properties assumed by the
Kalman filter, this order of identification has the advantage of
whitening the residuals. An additional optimization step (in
Optimized Identification) is also treated in order to minimize output
prediction error. This is particularly useful in eliminating biases.

*Structural Dynamic Models and Damage
Detection in Structures:* A second objective in the research is to
develop structural dynamics models from identified state space
models, i.e. obtaining mass, damping, and stiffness matrices.
Identifying structures such as suspension bridges from earthquake
response data, or identifying structural dynamics of a large
spacecraft in orbit, can mean finding these matrices since they have
extra physical meaning. In addition they can be used to identify
damage and damage location in structures. The field of damage
detection in this sense is still in its infancy.

**Robot Time Optimal Control, Time
Optimal Path Planning**

Research on time optimal control of
robots started in 1984 with Prof. Hans Georg Bock. Prof. Bock, now
heading IWR (Interdisciplinary Center for Scientific Computing) at
the University of Heidelberg is the creator of numerical optimization
algorithms that made the solution of robot optimal control problems
possible with ease in 1984. An early focus for the research was a
specific robot on a press chain in the Mercedes Benz assembly line.
It was the slowest to accomplish its task, and therefore speeding it
up could allow the whole press chain to be operated at a higher rate
with increased productivity. During this phase, the Kuka robotics
company of Augsburg, Germany was involved.

First the properties of time optimal paths of robots was studied for ideal situations of polar, elbow, and SCARA robots. The properties are very interesting physically, showing how Coriolis and Centrifugal effects are used to have the motor of one link help that of another. The same phenomena appear in other optimization objectives, e.g. energy optimal control. A series of publications examine each step needed to go from an idealized situation to the practical situation with realistic motor constraints, etc.

**Robotics in Space**

With ex-doctoral student, Dr. Robert
Lindberg (then of NRL, later VP or Orbital Sciences Corp., and now
President of American Institute of Aerospace), research on robotics
in space started very early, with the first publication in 1985. Two
of the publications appear as the first two papers in the first book
reprinting basic papers on the field produced by the Robotics
institute at Carnegie Mellon (Xu and Kanade, *Space Robotics:
Dynamics and Control*, Kluwer 1993).

*Forward and Inverse Kinetics: *The
main issues are the fact that the mass on the end of a space
manipulator arm on a spacecraft, e.g. the shuttle, can be a
significant percentage of the mass of the shuttle, and the shuttle is
not fixed to an point in space. So moving the load on the tip end of
the robot, moves and rotates the shuttle at the base of the robot.
Nevertheless, one can have the load move to a new location and have
the shuttle un-rotated when the load gets there. The forward and
inverse kinematics problem for ground based robots, become forward
and inverse kinetics (or dynamics) problems in space.

*Optimal Path Planning for Robots
Mounted on Satellites:* Additional research treated optimal path
planning problems for robots in space with then doctoral student,
Volker Schulz, and Prof. Bock at the University of Heidelberg. The
optimization can be based on minimum disturbance to the zero gravity
environment of the shuttle.

**Walking, Hopping, Somersaulting
Robots**

Co-author and ex-doctoral student
(shared with Bock at Heidelberg) Dr. Katja Mombaur initiated a series
of research publications to search for the existence of open loop
stable walking of robots, and to develop software that could adjust
robot design parameters and find associated periodic gaits to produce
such walking. She used point feet to prevent the shape of the foot
having any stabilizing influence. Open loop stable walking means the
following. Compute a periodic gait for a walking robot. This
corresponds to periodic torque histories at the joints of the robot.
If these periodic torques are applied to the links, it is open loop
stable if the resulting motion has the property that one can disturb
the state away from the periodic gait and it will recover naturally –
without any feedback from any sensor, without any vision or knowledge
of what is happening. She was able to demonstrate that robots can
have open loop stable walking motions, and also extended it to open
loop stable hopping, open loop stable somersaults, etc.

**Satellite Attitude Dynamics**

I was a doctoral student of Prof.
Robert E. Roberson who did the first study of satellite attitude
stabilization and control for The RAND Corporation in about 1950,
seven years before the first satellites went into orbit. He was the
chief person behind gravity gradient stabilization which acts like
the moon to keep one side of the satellite facing the earth all of
the time, because of the difference in the gravitational force from
the near to the far side of a satellite. Spin stabilization is
another way to passively make a satellite handle the small torque
disturbances in orbit, and alternatively one can spin a wheel inside
the spacecraft. Gravity gradient stabilization of gyrostat satellite
combines the advantages of both methods. Publications in this area
complete the picture of the equilibria and stability of gyrostat
satellites under gravity torques.

**Concepts of Degree of
Controllability and Degree of Observability, Criteria for Sensor and
Actuator Placement**

Controllability and observability, like
stability, are binary concepts. Either a system is controllable or it
is not. Motivated by the need to have appropriate criteria to decide
where to place sensors and actuators in large flexible spacecraft, a
series of publications develop definitions of the Degree of
Controllability (DOC), and the Degree of Observability (DOO).

**Energy Optimal Control of Subways**

With support from the New York City
Transit Authority, we developed methods to calculate the way in which
a motorman should accelerate and decelerate a subway train so that it
gets to the next station at the desired schedule time and does so
using minimum electrical energy. This presented an interesting and
novel optimal control problem that was challenging numerically.
Methods were extended to do create feedback optimal control of the
nonlinear system, that could adjust for such things as a desired
change in the transit time in order to get back on schedule. The
approach to subway operations was tested on unused track on the
Culver Line, then it was tested in revenue service on the Flushing
Line which is electrically isolated. In some places there are speed
restrictions which prevent use of the results. In those sections of
the line where they could apply the approach they saved 18%
electrical energy, and averaged over the whole line they saved 11%,
and it was implemented. This corresponds to a savings of $34 million
per year in current (2009) dollars.

**Feedback Control of Plasma Physics
Tokomaks**

Tokomaks are gradually reaching the
engineering design stage, and people are starting to consider the use
of feedback control and Kalman filtering.