Lecture Notes 5.A1

Fundamental Principles in Motion Control

Naomi Ehrich Leonard

Department of Mechanical and Aerospace Engineering

Princeton University, Princeton NJ 08544

1 Motion from Shape Change

A single fundamental principle can be used to describe the way a variety of both living and

engineered systems move. In each case, the system performs a desired maneuver by repeating

a cyclic change in the shape of its \body". A classic example is a falling cat that wiggles its

body so that it lands feet �rst on the ground. The cat falls under the force of gravity, but

it is the wiggling of the body that produces the cat's reorientation as needed. These body

wiggles are cyclic changes in the cat's shape. Consider that at the instance of the beginning

of the fall, the cat is upside down and its spine is bent forward. The cat bends �rst to one

side, then backward, then to the other side and �nally forward again (see Figure 1). At the

end of this cycle, the cat's shape is as it was initially but its orientation has changed; it has

turned over so that it is now falling feet �rst (Kane and Scher 1969)!

Many other animals repeat cyclic shape changes to get where they are going. For example,

a snake wiggles its body in a repetitive manner so that after every cycle it looks the same

in shape but it has made some progress in moving forward. Bacteria swimming in water are

faced with a challenge that is analogous to our trying to swim around in molasses (because of

their small size, bacteria see the water as very viscous!). They can't simply paddle and then

glide. Essentially, as soon as they stop moving their \paddle", they stop moving altogether.

Just like a snake they must slither along, changing their body shape repetitively (e.g., using

body hairs called cilia, as in Figure 2, or turning a corkscrew-like tail) so that they can keep

going. After every cycle of shape change, their shape will look as it did originally, but their

position and orientation will be di�erent.

In order to produce motion, the animal must change its shape in a non-trivial way. If the

animal changes its body into a particular shape and then changes back to its original shape

by going through the identical contortion in reverse, then we say the animal has made a

reciprocal shape change (Purcell 1977). Reciprocal shape changes generally do not produce

a net motion in the animal. This is because the animal is essentially retracing its steps and

any motion that was e�ected in the �rst half of the shape change is precisely undone in the

second half. It is instead necessary for the animal upon returning back to its original shape

to choose a means to get there that is not precisely the reverse of what has already been

done. In this way, although there may be some backtracking in the direction of travel, the

1 c by Naomi Ehrich Leonard, 2000

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Figure 1: Falling cat. (Ref. 1)

net motion of the animal will not be zero.

Consider an astronaut oating around in zero gravity who would like to turn around

but has nothing to grab on to. To turn, the astronaut must change his body shape in a

non-reciprocal way (analogous to the falling cat). Suppose the astronaut decides to use only

his arms to turn himself around. Figures 3 and 4 below each show an overhead view of

the astronaut performing a sequence of arm motions. In each shape change sequence, the

initial and �nal shapes of the astronaut are the same. However, the sequence in Figure 3 is

a reciprocal shape change and leads to no net turn of the astronaut, whereas the sequence

in Figure 4 is a non-reciprocal shape change producing a net rotation of the astronaut.

A useful way to graphically represent a reciprocal and a non-reciprocal shape change

is to make a plot of one shape variable versus another shape variable. In the case of the

astronaut, one shape variable is the angle �1 between the left arm and the centerline of the

body and another shape variable is the angle �2 between the right arm and the centerline

of the body (see Figure 5). Figure 6 shows a plot of �1 versus �2 that corresponds to the

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Figure 2: Protozoan with cilia. (Ref. 2)

A B C D E

Figure 3: Overhead view of astronaut performing reciprocal shape change.

reciprocal shape change of Figure 3. Figure 7 shows a plot of �1 versus �2 that corresponds

to the non-reciprocal shape change of Figure 4. Note that the area inside the plot in Figure 6

is zero, whereas the area inside the plot of Figure 7 is nonzero. A nonzero area inside this

\shape change" plot is the characteristic feature of a non-reciprocal shape change and is

necessary for a net motion.

2 Control

Control is the ability to a�ect the behavior of a dynamical system. You see control every-

where, not only in biological systems but also in man-made systems of all kinds. Animals

and humans exert control over their bodies to move around, to speak, to take some kind

of action, e.g., to pick up and eat food. Humans can also control man-made systems. For

example, a person drives a car, and that person can control the speed of the car by applying

appropriate pressure on the accelerator. We call this manual speed control. An alternative is

to build in to man-made systems automatic control. For example, cruise control on the car

provides automatic speed control; the cruise controller automatically operates the accelerator

A B C D E

Figure 4: Overhead view of astronaut performing non-reciprocal shape change.

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�2�1

Figure 5: Identi�cation of shape variables on astronaut.

�2

180

180

90

90

A,E B,D

C

0

0

(degrees)

(deg

rees

)

�1

Figure 6: Shape change plot corresponding to astronaut's reciprocal shape change.

to maintain a prescribed car speed. Similarly, the water level in your toilet bowl, the heating

of your room or house, street and traÆc lights, CD players, elevators, power plants, airplanes,

spacecraft (and much, much more!) are all controlled to some degree automatically.

In order to provide automatic control to any system, it is clearly necessary to determine

rules or instructions for how the controller will direct the system to behave. For example, how

much and how fast should one program the cruise controller to push down on the accelerator

when attempting to a get a car up to the prescribed speed? How can this be done so that

in every instance, the car speed will not overshoot the prescribed value (potentially leading

to a speeding ticket for the driver)? How can this be done so that the ride is comfortable

for the driver and passengers (e.g., jerky motions are avoided)?

Designers of controllers address these questions using a range of mathematical tools.

One of the prime means of studying a dynamical system, such as a car, is to create a

mathematical model of the system (a set of equations) using principles from physics. With

the mathematical model, the designer can then study how the system will behave under

various operating conditions (bumpy road, rain, snow, etc.) with various controller actions

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�2

180

180

90

90

A,E B

C

0

0

(degrees)

(deg

rees

)

�1

D

Figure 7: Shape change plot corresponding to astronaut's non-reciprocal shape change.

(fast or slow, heavy or light pressure on the accelerator). Ideally, one would like to be able

to use the model to back out an appropriate set of instructions for the controller that will

precisely achieve a prescribed system behavior (e.g., an appropriate controller action on the

accelerator for a prescribed speed-up maneuver). We typically call the action of the controller

the \input" to the system and the response or behavior of the system the \output". Figure 8

shows a schematic of a system with controller.

DesiredSpeed

(Command)

AccelerationAction

(Input)

MeasuredSpeed

(Output)

CruiseControl

Car

(System)(Controller)

Figure 8: Schematic of system with open-loop controller.

In Figure 8, we neglected to include the fact that the controller may have some informa-

tion as to how the system is behaving at any given moment. For example, on a car, there is

a sensor to measure the actual speed of the car. This measurement is, therefore, available to

be integrated into the algorithm that the cruise controller uses to �gure out how to operate

the accelerator. For example, the cruise controller might be programmed to compare (e.g.,

subtract) the measured speed from the desired speed at any given moment in time and use

the resulting speed error to come up with a plan as to how much to push down (if the error

is positive) or let up (if the error is negative) on the accelerator at that moment. We call this

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use of measurement data \feedback control" because we are \feeding back" the measurement

to the controller. This is illustrated in the modi�ed schematic of a system with feedback

control shown in Figure 9.

DesiredSpeed

(Command)

AccelerationAction

(Input)

MeasuredSpeed

(Output)

CruiseControl

(Controller)

Car(System)

Figure 9: Schematic of system with closed-loop controller.

With a perfect model of the dynamical system, prediction of the behavior of the system

would be exact. In this case, feedback control would be excessive (e.g., measurement of the

car's speed would be redundant since the speed would already be known by prediction),

and the simpler \open-loop control" of Figure 8 would suÆce. However, perfect models

are extremely rare, and feedback is useful particularly to guard against uncertainty and

disturbances (wind gusts, bumps in the road, etc.). Often, a controller will combine open-loop

control and feedback control. In this context, open-loop control provides a kind of planning,

i.e., what one would do were there perfect knowledge of the system and its environment,

and feedback control provides correction for errors in assumptions about the system and its

environment.

3 Control of Motion in Biology and Robotics

It is natural and rewarding to propose a uni�ed framework for studying control of motion

(locomotion) in biology and in robotic systems. In biology, control is carried out often

without a lot of conscious instruction. We do not think hard about what it is we must do to

walk across the room - we simply walk. However, electrochemical signals are being sent from

our brain to our muscles to make us walk. These signals provide the control instructions for

e�ective walking. In fact, it has been hypothesized that these instructions actually combine

open-loop and feedback control. That is, we walk as a result of a combination of motion

planning (open-loop control), such as planning to walk in a straight line towards the door,

and reaction to sensed (measured) changes in the environment (feedback control). The

appearance of a pebble on the oor or a chair in the way that we didn't originally see will

ultimately makes us change our path to the door.

A deeper appreciation of natural control instructions has many practical applications.

For instance, it can help us to determine how to provide arti�cial means of motion control

when there has been an accident to the natural controller (e.g., prosthesis). Understanding

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how biological systems move themselves about also helps us to predict more general patterns

in their behavior. For example, once we know what bacteria do to swim, we can calculate

how much energy it takes them to move a given distance. This can be used to determine

feeding strategies; for example, if the available food provides only half the calories that it

will take for the bacteria to swim over to fetch it, then the bacteria will not take the trouble

to do so. Understanding patterns in bacteria behavior may help in better understanding and

coping with disease.

In engineered systems, unlike biological systems, the control instructions must be con-

sciously prescribed (most particularly in the case of automatic control), i.e., they must be

programmed into the system. In an e�ort to make this prescription, much can be learned

from existing controllers in biological systems that already e�ectively perform a wide variety

of complex motions. A uni�ed framework of locomotion in biology and robotics allows us,

for example, to consider synthesizing snake-like or bacteria-like or falling-cat-like motion!

Thus, we can think up and design controllers for robots to move like snakes through holes

and around corners to defuse bombs or to take pictures of someone who has been trapped

where a human cannot see. We can build submarines that perform bacteria-like strokes to

maneuver through ancient underwater archeological wrecks. We can enable a spacecraft to

reorient itself to point a telescope much as the falling cat ips itself over.

The essential physical requirement for these robotic systems is that they have the ability

to change their shape. With this, we can prescribe a controller to change their shape in a

way that is motivated by the shape changes observed and studied in the biological systems

and formalized in the uni�ed framework.

For example, consider the spacecraft reorientation problem. Imagine for a moment that

the spacecraft is ellipsoidal with length greater than depth in turn greater than height.

Suppose that the spacecraft is equipped with three internal wheels as shown in Figure 10,

such that each wheel is allowed to spin about a di�erent direction as shown. We assume

that there is a motor attached to each wheel and we can program an on-board computer to

drive (control) each of the motors to any speed we choose.

2�

�3

1�

Figure 10: Spacecraft with internal wheels.

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From physics (conservation of angular momentum), we know that if we spin the �rst

wheel at a constant speed in the counterclockwise direction then the spacecraft will respond

by spinning at a constant speed in the clockwise direction around that same axis (i.e., the

spacecraft's longest axis). Similarly, we can control the speed of the spacecraft spin in the

other two directions by directing the other two wheels to spin at prescribed speeds. We call

spin about the spacecraft's longest axis \roll", spin about its intermediate axis \pitch" and

spin about its shortest axis \yaw".

Now our job is to e�ect a desired spacecraft reorientation maneuver using changes in

the angles of the wheels. The wheel angles correspond to the spacecraft's shape and so we

consider producing net reorientation maneuvers with non-reciprocal shape changes (wheel

rotations). Suppose for the moment that we are to only use the �rst and second wheels

(perhaps, the third wheel is broken). Then, we have two shape variables, the rotation angle

of the �rst wheel �1 and the rotation angle of the second wheel �2. A non-reciprocal shape

change (analogous to the astronaut shape change of Figure 4) would suggest to (1) rotate the

�rst wheel in the clockwise direction by � degrees, then (2) rotate the second wheel in the

clockwise direction by � degrees, then (3) rotate the �rst wheel in the counterclockwise di-

rection by � degrees and �nally (4) rotate the second wheel in the counterclockwise direction

by � degrees. The corresponding shape-change plot is qualitatively the same as Figure 7,

and the resulting maneuver would be for the spacecraft to turn about its third axis. If a

reciprocal shape change were performed analogous to Figure 3, i.e., the �rst wheel is rotated

in the clockwise direction, the second wheel is rotated in the clockwise direction, the second

wheel is rotated in the counterclockwise direction and �nally the �rst wheel is rotated in the

counterclockwise direction, then there would be no net spacecraft reorientation maneuver.

A starting point for investigating and synthesizing motions of general interest is to

develop a framework for describing motion from shape change as described above. The

framework involves geometry and a kind of algebra called matrix algebra.

References

1. Kane, T.R. and M.P. Scher [1969], A Dynamical Explanation of the Falling Cat Phe-

nomenon, Int. J. Solids Structures, 5, 663{670.

2. Purcell, E.M. [1977], Life at Low Reynolds Number, American Journal of Physics,

45:1, 3{11.

3. Satir, P. [1974], How Cilia Move, Scienti�c American, 231, 45{52.

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