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.

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

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.

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.

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.

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.

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).

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.

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