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M. Karweit
ROBOTICS
Robotics is the technology of robots. What is a robot? From the Robot Institute of
America, "a robot is a reprogrammable, multifunctional manipulator designed to move material,
parts, tools, or specialized devices through variable programmed motions for the performance of a
variety of tasks." And, from those who must deal with the present capabilities of robots, " a robot
is a one-armed, blind idiot with limited memory and which cannot speak, see, or hear."
Whatever the definition, it doesn't take a lot of imagination to realize that the potential for
more sophisticated robots is enormous. In some cases, the potential is primarily economic, e.g.,
using robots on automotive assembly lines to reduce labor costs. In other cases, the potential
exists in performing tasks that are not humanly possible, e.g., carrying out activities in hostile
environments--the ocean floor, areas of high-level nuclear contamination. And then there’s the
potential for medical applications—the possibility surgery through three small holes rather than a
40cm long slit; or heart repair on a beating, functioning heart; or an operation performed by a
specialist surgeon 1000 km away. These are the challenges for robotics engineers.
"Reprogrammable, multifunctional" are the operational words. There are many automated
machines which are designed to carry out single tasks. But that's all that they can do. Candydispensing machines, toasters, CD-changers are examples. And these automated machines can be
very efficient because they can be optimally designed to perform their single task. The idea of a
robot is that it can be reconfigured to perform different tasks. As a result, they tend to be much
less efficient than automated machines.
Computers and robots are quite similar. They are both programmable and can be
configured to carry out a variety of tasks. The difference between the two, however, lies in what
kinds of tasks can be handled. Computers can carry out only mathematical operations. True, they
can be interfaced with other physical devices, but then they become controllers (perhaps for
robots). Robots, on the other hand, carry out tasks on tangible things: " materials, parts, tools,
specialized devices. . ." Dealing with tangible things as opposed to abstract things (such as
mathematical operations) makes the robot a far more complicated device.
Actually robots and computers have a common heritage--a programmable loom. The idea
came in 1801 from Joseph Jacquard, a stone mason from Lyon, France. The Jacquard loom was a
device whose weave pattern was determined by a sequence of perforated cards. A single
machine could produce a wide variety of textile weaves simply by punching holes in cards and
properly sequencing them. One card at a time, the loom would detect the hole configuration, and
appropriately raise or lower the warp threads to generate the textile pattern. So the idea of a
programmable machine is almost 200 years old. The concept of a robot, however, is much newer.
"Robot" was first used in a Czech play which opened in London in 1921. The word comes from
the Czech robota which means serf or one of subservient labor.
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II Robot characteristics
What types of robots are there? It’s useful to classify robots in terms of the type of task
that they’re designed to perform. Recall, that a robot is reprogrammable, so that, with some sort
of reconfiguring, it should be able to carry out a range of similar tasks. In fact it is useful to group
robots in terms of the tasks they can perform:
1) “pick and place”, e.g., moving a piece from a conveyor belt into a box.
2) continuous path control, e.g., welding a seam along the edges of adjacent pieces of steel
3) sensory—capable of 1) or 2), but with the added feature of being able to “feel” the piece
on the conveyor belt or sense the track of the desired weld path or feel the resistance of the
material.
Basically “pick and place” means carrying out an action at location A, moving to B, carrying out
another action at B. The route for getting from A to B is unimportant. This is the simplest type of
robot. All you have to do is figure out mechanisms to perform the actions at A and B, then pick
from a variety of mechanisms to get from A to B. Don’t misunderstand. These robots are not so
easy to design. The real question is just how flexible do you want the robot to be? Would you
want the same robot to be able to place eggs in crates from a conveyor belt and also move
automobile engines from a test stand to an assembly line? The more flexible, the less efficient.
But a different robot for each slightly different task is costly. Where’s the tradeoff? It’s an
engineering question.
“Continuous path control” can be much more complicated. These robots must exhibit
continuous, precise control of its actuator(s) as they move along their specified path from A to B.
Both the path and the speed along the path can be important. Robotic welders obviously must
follow the intersection of the pieces it is trying to join, but it must also travel at a constant rate to
produce a uniform weld joint. We will see later the problems that this requirement poses.
Tasks can be performed in one of two ways: “open loop”, i.e., no feedback, deterministic;
or “closed loop”--with feedback control. Robots can be designed for “Pick and place” and
“continuous control” tasks as open loop. An example of this might be picking up a bottle on a
conveyor belt. But if a bottle is too far away from where it is supposed to be, a robotic arm will
miss the bottle. To address that problem, we could design the robot to have some sort of “vision”
or mechanical sensor to detect where the bottle actually is. Then, we can use that position
information to define the action of the robotic arm. Such a system would be "closed loop",
because the motion of the arm is being determined by the bottle's position. The arm "senses"
where the bottle is.
Sensory control is yet another level of complication in robots. In some cases the control
can be simple, e.g., reflection-actuated direction control which allows a robot to follow a yellow
line on the floor. Here, for example, three light sensors might detect the reflectivity of the line on
the floor. If the middle sensor “sees” the line, the robot is on the right track. If the left sensor
detects the line, then the robot is too far to the right. So how do you use this information in
programming a robot to follow the line? One way would be to link the sensor output to a steering
device—turn left, go straight, turn right. But how much left or right? The robot could oscillate
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rather wildly along the line. What might you build in to a control system to prevent this from
happening? Sophistication in handling the sensor output? Maybe a different sensor system
altogether? Think about this. There’s always two issues: What is the sensory signal you will
use? What will you do with the signal?
[If you’re going to be a good scientist or engineer, always ask questions: How could such
a task be accomplished? Is there a better way of performing the task? Is there a different
technology that might be better suited for the task? Then, of course try to answer them. Just like
a pianist, the more you practice your art, the better you can become. Over time your questions
will become more insightful, your solutions more elegant. ]
As computers become more powerful, more robots will probably be using television
cameras as sensory devices. In these cases, the real sensing consists of image processing—the
extraction of information from images. Image processing can be used to determine, for example,
the orientation of a part on an conveyor belt, so that the grasping mechanism can be properly
rotated to pick the piece up. Or it can detect the beginning and end of a seam to be welded. But
all this must be done in real time. Edge detection, pattern recognition, and image enhancement
are techniques of image processing.
Not all robots are entirely preprogrammed to run autonomously. Some are designed to be
controlled by a person, e.g., a robotic surgical scalpel. Let’s look at some of the issues pertaining
to such a device. First, the system must provide a sense of “touch” to the operator. How hard is
the scalpel being pushed against the tissue? The classical example is whether a system could
allow an operator to “feel” when a scalpel passing through jello would detect the presence of a
grape. Jello and grapes have consistencies not too different from living tissues. Second, the
system might provide error correction. In long operations, a surgeon’s hand might begin to
tremble. If that surgeon were operating via a robotic scalpel, the system could cancel out those
tremors for far steadier procedures. Third, the system must provide instant feedback. A
surgeon/operator needs to know the scalpel’s position and resistance to motion as it is moved, not
later. Robotic surgery exists today when the surgeon/operator is in the same room as the patient.
But it does not exist today when the surgeon/operator is performing from some remote site.
Why? Because transmission delays interrupt that instantaneous feel of the scalpel. (Incidentally,
the study and development of machines with senses of touch and kinesthesia is called haptics.)
III.
Measures of performance
Robots move. Some are stationary and move only their appendages; others are mobile
and, in addition to moving their arms, move from place to place. How well they move is
extremely important, because how well they move determines what they're good for. We can
identify a number of characteristics that affect performance.
First, there is the working volume, the space within which the robot operates. Since a
robot is built with mechanical linkages, there is almost always a spatial limit to its tasks. Usually
the bigger the volume, the better, because a larger working volume would suggest that a wider
variety of tasks might be performed. But there is a tradeoff. Mobile robots obviously will have a
larger spatial range than stationary robots, but their control is much more difficult and expensive.
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Second, we might ask how quickly can the robot perform a task, i.e., what is the speed
and acceleration of its actuators. Can it move 20 objects per minute or 200 objects per minute?
Sometimes speed is not an issue. For example, in robotic welding, the rate at which a seam can
be welded is usually not limited by the speed of the robot, but rather the speed at which a good
joint can be made. But, for other things, like testing the pins on a 186-pin microprocessor chip,
speed is money.
Third, repeatability. Will the mechanism always return to the same point under the same
control conditions? This is not just an issue with robots. Almost anything mechanical has some
"slop" in its system. An experimenter always "dials in" a value with a knob on an instrument
from the same direction, say, on a high-precision potentiometer. This is because there is often a
little play between the indicator on the knob and its action. In robots, the problem is usually
worse, because there is typically a sequence of links, each having play. Further, loads may affect
repeatability. That is, a fixed control condition may produce different positions depending on the
load. This is a tough problem because it may also depend on an arm's orientation.
Then there is resolution. What is the smallest "step" the robot can take? Suppose we have
a single-arm robot whose arm tip position is set by a stepper motor which can be programmed in
one-degree increments. If the arm is 1 meter long, the resolution will be no better than 1.7 cm. Is
that good enough? It depends on the application. What if I told you I wanted to remove your
appendix with a robotic scalpel whose resolution was 1.7cm?
Finally, we can list accuracy. Accuracy is the difference between the actual position of
the robot (or arm) and the programmed position. A robot might be accurate, but not repeatable-i.e., over a series of attempts at reaching the desired position, the robot would, on average, cluster
around the target, but have substantial positional error on each attempt. Or a robot might be
inaccurate, but reapeatable--it would constantly miss the target, but its attempts would cluster
tightly together at some other location.
These five attributes of robotic behavior are issues for all robots. They are affected by
design, by materials, by loads, by construction quality. But these are the details of how robots are
put together. How can we specify the positions--from one link to the next? Here we must turn to
kinematics and dynamics.
III Kinematics and dynamics
Kinematics is the physics of motion. It's prescribes what mechanically goes on. But it is
not concerned with forces or momentum. That's the realm of dynamics. Dynamics is the more
difficult subject. We'll touch on dynamics a little later, but for now we'll focus on kinematics.
Robots have varying abilities to move. Some might only be able to extend or retract.
Others are able to move in any direction and simultaneous rotate its actuator arm. What we're
specifying here are degrees of freedom--the number of independent motions a robot can make.
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There are two sets of motions: translation and rotation. Translation is motion from point
A to point B. It can take place in one, two, or three dimensions. Rotation is motion at a fixed
point. Again, it can take place with respect to one, two, or three independent axes. Between the
two types of motion--translation and rotation--all motions can be accounted for. Full threedimensional motion requires 6 degrees of freedom--3 translation, 3 rotation. A robotic arm that
could only extend or contract and, say, twist its actuator on the axis of the arm would have only
two degrees of freedom--1 translation, 1 rotation. More degrees of freedom implies more
versatility and larger working volume. However more degrees of freedom, also implies higher
costs, less accuracy, lower repeatability. Consequently, robots are usually designed having the
fewest number of degrees of freedom to do the job.
Robots are usually constructed of rigid members, whose motions with respect to one
another are afforded through the joints that hold them together. It is the properties of these joints
that the give robots the mechanical maneuverability.
Basically, there are only two types of joints: prismatic and revolute. Prismatic joints
slide, e.g., a square cylinder in a square tube. Revolute joints or hinge joints rotate. So revolute
joints allow angles between members to change. OK, but what about ball-and-socket joints?
Well, a ball-and-socket joint can be thought of as three hinge joints. A ball-and-socket joint has
three degrees of freedom in rotation. So its action can be duplicated by the combined action of
three hinge joints. A cylindrical shaft in a cylindrical tube can rotate and extend. It's easy to see
that this action could be replicated with one prismatic joint and one hinge joint.
One can construct a variety of mechanical joints. We've mentioned a few--ball-andsocket, cylindrical shaft in cylindrical shaft. But the motions of all joints can always be reduced
to a combination of the two basic types: prismatic and revolute. The reason we want to reduce
their actions to combinations of the primitive types is so we can mathematically describe them
more easily. We'll see this a little later.
We've mentioned the actions of mechanical joints. That's motion within the robot. Let's
not forget the motion of the robot itself. Again, whether the robot turns on an axis, moves on
wheels, or swims through water, the actions can be reduced to the primitive forms.
Now that we've touched on the conceptual ideas of robots, we must turn to the messy
stuff--the mathematics that describes their actions. We won't delve too deeply into this topic
because the mathematics can get very complicated very quickly. But we can develop some
methods for prescribing the positions of the actuator with respect to the position of the base of the
robot. Here, the mathematics is not very difficult; but it can be cumbersome because there are
many different ways we might want to represent position.
What coordinate system(s) should we use. There are many to choose from. The most
familiar is the "cartesian" system (x,y,z) in which position is specified by the three distances x,y,z
from the origin. But we could also consider a cylindrical system (r,,z)--consisting of two
distances and one angle, or a spherical system (r,,) consisting of one distance and two angles.
Our decision will depend, in part, on the frame of reference in which the task or object can most
easily be represented, and, in part, on the frame of reference in which the mechanical actions of
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the robot are most conveniently represented. A robot consisting of a fixed base and an extensible
arm that can be rotated in a horizontal plane, would most naturally described by an (r,,z) system.
We say most natural because the choice of the coordinate system reflects the action of the robot:
one action is spatial, the other is angular.
Now suppose, at the end of this robotic arm, there is a tool which moves in and out, up
and down, and back and forth, i.e., three spatial actions. Here, most naturally we would use an
(x,y,z) system. So, we might want an (r,,z) for the arm, an (x,y,z) system for the tool tip, and
maybe another (x,y,z) system to describe the actuator position in "world" coordinates. So, several
different frames of reference might be useful. To specify absolute position of the tool tip, its
position with respect to the arm must be added to the position of the arm. How do we bolt all of
this together?
Here is a schematic of the situation. There are three coordinate systems: (x,y,z), (r',',z'),
and (x'',y'',z''). The robot is presented here as three heavy lines; the tool is represented as a ball.
The positions of the robot base, the robot arm, and the tool are given in italics.
z''
z'
z
y
(x'',y'',z'')
x'
' (r',',z')
'
x
(x,y,0)
r'
We want to prescribe the position of the tool in the world coordinate system (x,y,z). It's not so
simple. Basically we have to get from one coordinate system to the next. This is accomplished
through the operation of coordinate transformation. Basically there are two things you can do
with coordinate systems: you can translate them, and you can rotate them. If you look at our
robot example, you will see that the x'' system (the reference frame of the actuator) is a
translation of the x' system (the reference frame of the arm). And the x' system is a translation
and rotation of the x system (the world reference frame).
For a moment, let's simplify a bit. In two dimensions, if we want to relate a position x', y'
in a coordinate system x' to a system x , where the x' system is rotated with respect to the x
system through an angle , we can write down the equations as
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x = x' cos  - y' sin 
y = x' sin  + y' cos 
 cos   sin   x'   x 
     . (See the appendix on matrix algebra.)
In matrix form, this would be 
 sin  cos   y '   y 
The 2 x 2 matrix is called the rotation matrix. This set of equations can also be represented as R
x' = x. So, this equation describes a rotation transformation. What about a coordinate
translation? If k is the vector representing the translation of a coordinate system x, then x' = x k. If there's both rotation and translation, then we can combine the coordinate transformations
into a single equation as x = R x' - k. Of course, we can also extend these ideas to three
dimensions.
We've implemented rotation and translation above as two separate--and different-operations: a matrix multiplication and a vector addition(subtraction). It turns out we can
combine these two operations into a single homogeneous coordinate transformation. Consider
the following matrix equation.
 x   e11 e21 e31 k x   x' 
  
 
 y   e12 e22 e32 k y   y ' 
 x = H x' .
 z e
e23 e33 k z   z ' 
13
  
 
1  0
1
0
0
1
  
 
H is called the homogeneous transformation matrix and can be viewed as having four
k
 R
 . R is a 3 x 3 rotation matrix; k is a 3 x 1 translation vector. So, H can be
parts: H = 
0 0 0 1 
used for any coordinate transformation because it encompasses rotations, translations, and
combinations of the two.
D. Transformations
1) General coordinate transformation from x’ to x is x = Bx’ + p , where B is a rotation
matrix and p is a translation vector
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 B p
2) More conveniently, one can create an augmented matrix A  

000 1 
which allows the above equation to be expressed as x = A x’.
3) Coordinate transformations of multilink systems are represented as
x0 = A01 A12A23. . .A(n-1)(n)xn
E. Dynamics
1) Velocity, acceleration of end actuator
a) power transmission
b) actuator
i) solenoid –two positions , e.g., in, out
ii) motor+gears, belts, screws, levers—continuum of positions
iii) stepper motor—range of positions in discrete increments
Example of a 2D robotic link having three solenoids to determine geometry. All members
are linked by pin joints; members A,B,C have two states—in, out—controlled by in-line
solenoids. Note that the geometry of such a link can be represented in terms of three binary digits
corresponding to the states of A,B,C, e.g., 010 represents A,C in, B out. Links can be chained
together and controlled by sets of three bit codes.
A
B
C
A
B
C
A
B
C
A
A
B
B
A
C
C
A
A
B
B
B
C
C
C
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2) Problems
a) joint play, compounded through N joints
b) accelerating masses produce vibration, elastic deformations in links
c) torques, stresses transmitted depending on end actuator loads
IV Control and programming
1) Position of end actuator
a) multiple solutions
2) Trajectory of end actuator—how to get end actuator from point A to B
a) programming for coordinated motion of each link
b) problem—sometimes no closed-form solution
Let's explore what would seem to be a really simple robotics problem. Suppose we have a
two-segment robotic arm that is attached to a base and that can move in only two dimensions, x
and y. Each arm segment is the same length, L1 = L2 ; and the motion of each arm segment is
determined by a stepper motor that can position the arm at any prescribed angle. The problem is
to program the two stepper motors so that the robot arm tip, i.e., the end of L2 , follows a path at y
= const.
L2
y
2
L1
1
Two-segment robotic arm
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We'll assume that each stepper motor can be controlled as  (t )  0  t   t 2 where 0 is the
angular position at the beginning, and  and  are parameters to establish the angular velocity and
acceleration over time. And a programming segment will constitute time t moving from 0 to 1.
So, by setting values for 0, , and  for each motor, the time-behavior of the arm tip can be
generated one program segment at a time.
The geometry is easy: assuming that the base of the robotic arm is fixed at (0,0), the position of
the arm tip is
x = L1 (cos 1 + cos 2)
y = L1 (sin 1 + sin 2)
If we want the arm tip to move, we must rotate the stepper motors which prescribe the angles 1
and 2. As 1 and 2 change, so does the position of the arm tip. In fact, using the chain rule of
elementary calculus we can deduce the relationship between the rates of change of x and y and
d
the rates of change of 1 and 2. Using a dot above a variable to denote
, we calculate that
dt

x   L1 sin 1 1  sin  2 2



y  L1 cos1 1  cos 2 2  0 .
Setting the latter equation equal to zero gives us the condition that the y-position of the arm tip
does not change, i.e., y = constant. These equations give the simple result:
cos1
. This says that the rate of change of 2 is proportional to the rate of change of 1,
cos 2
but the rate of proportionality depends on the values of 1 and 2. In fact, as simple as this
robotics problem is, there is no closed-form solution. That is, we cannot produce an explicit
recipe which will ensure that, as 1 changes, 2 will change in such a way that the y-position of
the arm tip is constant.
2   1
Since there is no mathematical solution, we must try to find an engineering solution—an
approximation. And within this approximation we will have to accept some minor variations in y.
Further, we might not be able to find even an approximate solution that will work throughout the
entire range of motion of the arm. So, moving the arm tip through its maximum range of x might
have to be accomplished through a sequence of program steps that define different rates of
changing 1 and 2.
Recall, that each motor can be programmed to rotate as  (t )  0  t   t 2 ,. These
parameters are all we have to work with to make the arm move correctly. To get a visual feel for
how an arm tip moves as a function of the angular speeds of stepper motors, investigate the
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robotic arm simulation at http:\\www.jhu.edu\virtlab\robot.htm. Use the simulation to try out
the strategies outlined below.
Assuming that the arm tip is initially at the desired value of y, what strategies can one use
to move the tip along x, say in the interval from xbegin to xend, while keeping y = const.? There are
three straightforward ones.
The simplest approach is to calculate the rates of change of 1 and 2 for y = const. for the
initial values of 1 and 2 , i.e., determine values for a1 and a2 (here, the subscripts refer to the
 cos( 01 )
motors) such that 1 
and 1 = 2 = 0. This will ensure that the arm tip will least start
 2 cos( 0 2 )
along a trajectory of y = const. If , over time, the arm tip exceeds some allowable y, reprogram
the (t) equation for another step using the final values of 1 and 2 from the previous step as
initial values for the next step.
A few trials with the robotic arm simulator will demonstrate that this technique isn’t very
good. The principal problem is that 1(t) and 2(t) will produce y(t) = const. only at the beginning
of the interval over which the arm tip is to move. The farther the arm tip moves away from this
initial position, the farther the tip will deviate from its initial y-position. In other words, the
condition for y = const. is established for the beginning of the interval but is hoped to hold for the
entire interval.
dy
 0 , not just at the beginning of
dt
the interval, but rather as an average over the whole interval. In this case, the arm tip would
begin to move with y  const. But as the tip moved farther into the interval, things would get
better, not worse—as in the first case. To produce this case we need only to program the rates of
change of 1 and 2, i.e., 1 and 2, so that the arm tip at the end of the time period would be at
the desired end point.
A better solution might be to consider the condition for
That is, we don’t need to make an explicit calculation for
dy
 0 . If the arm tip begins
dt
dy
 0 on average must be met. Here, the
dt
hope again is that the arm tip will remain within some allowable error y. Again, try this
technique with the robotic arm simulation. You should find it to be a significant improvement
over the first method. But it still may not allow you to traverse the tip from one extreme value of
x to the other.
and ends at the same value of y, then the condition of
What we really need are forms for 1(t) and 2(t) that will take into account the changing
condition for y = const. as the arm tip moves. To do this, we must first get an idea of how that
condition changes with x. So, let’s set up a hypothetical problem and do some calculations.
Assume that each arm is length 100 and assume that y-position that we’re interested in traversing
is y = 85. Then an initial position for the arm could be 1= 2 = 25.10. This yields
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y=100*(sin(1) + sin(2)) = 85. To see the condition for y=const., let’s plot
as a function of 1, with
d 1
1 .
dt
M. Karweit
d 2
cos(1 ) d1

dt
cos( 2 ) dt
First, why do we choose this particular plot? There are four variables involved: 1 ,
d 1
d 2
, and
. But 2 can be determined through geometry by 1. And we really don’t care
dt
dt
d 1
d 2
about the specific values of
and
. We care only about the value of their ratio. So one
dt
dt
d 1
1 .
of these rates can be picked arbitrarily; for example
dt
2 ,
d 1
d 2
to
changes
dt
dt
approximately linearly with 1 , i.e., the graph is almost a straight line. This means that, if the
d 1
1 ), the second motor should uniformly
first stepper motor rotates at a constant rate (say,
dt
accelerate as 1 is swept out. That we can do with our simple programming capabilities.
Now this is an interesting graph. It says that the ratio of
d 1
d 2
to
at the beginning of a programmed
dt
dt
segment, and assign an acceleration term to 2 (t) so that the appropriate ratio exists at the end of
the traverse, the middle of the traverse will pretty well take care of itself.
So, if we start out with the appropriate ratio of
See if you can figure out how to do this with Robotic Arm simulator. Remember, pick
some angle through which stepper motor one should operate. This will determine 1, since a
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Johns Hopkins University
What is Engineering?
M. Karweit
program segment will operate from t =0 through t=1. For this 1, calculate an 2 , so that the
d 1
d 2
ratios of
to
are correct for this beginning point. At the end of the time segment (if 1 =
dt
dt
0), the value of 1 at t = 1 will be 0 + 1. At this new position, calculate what 2 should be.
Then calculate the coefficient 2 so that, by the time t =1, the correct value of 2 will be attained.
d 1
d 2
The coefficient 2 produces the linear change in the ratio of
to
that is suggested by the
dt
dt
plot. Understand that this recipe is still an approximation to the solution. The arm tip will not
traverse at y = const. But it will be close—maybe close enough for the particular task. So, this is
an engineering solution, not an analytic solution. And it demonstrates one aspect of the field of
robotics.
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