Placing a robot on track greatly expands its reach envelope Trapezoidal and S-Curve Motion Profiles

1. TECHNOLOGY
WHITEPAPER
March 2014 CYCLETIMES FORTRACKMOTION ROBOTIC CELLS
4881 RUNWAY BOULEVARD
ANN ARBOR, MICHIGAN
48108 USA
PH: 734.214.0000
EMAIL: info@us.gudel.com
WEB: www.gudel.com
GUDEL, INC. ©2014

2. GÜDEL, INC. TECHNOLOGY WHITEPAPER
GUDEL, INC. ©2014PAGE 01
CYCLETIMES FORTRACKMOTION™
ROBOTIC CELLS
Placing a robot on a track greatly expands its reach
envelope, potentially allowing it to tend many more
machines or to transfer materials across a much
greater distance. For example, instead of three ro-
bots each tending a machine at separate locations,
a single robot on a track could potentially tend all
three, saving money and increasing efficiency. Or, a
single material handling robot on a rail could replace
multiple robots with intermediate handoff locations.
MINIMIZING CYCLETIME:
GENERAL PRINCIPLES
The overall cycle time of a cell is defined as the time
it takes to produce a single unit. There are many
factors that affect the cycle time of a robotic cell.
1) Cell Layout. The layout of the workcell will
have a major effect on the overall cycle time. The
placement of the robot and other equipment, such
as machine tools, conveyors, and pallet racks, should
be done with the sequence of operations in mind.
For a work cell utilizing aTrackmotion™ LinearAxis,
first try to minimize the distance that the robot must
move along the track between one step and the next.
For a work cell with multiple operations, place each
operation as close to the previous operation
as possible.
2) Robot Motions. Reducing or removing robot mo-
tions that are unnecessary or excessive is the next
step. Examine intermediate points in the robot path
to determine if movements and rotations can be re-
duced or eliminated.
Some robot controllers allow for easy optimization
of robot motions. For example, changing the path ac-
curacy settings to coarse instead of fine for motions
which do not require precision allows for increased
speed.
Where allowable, increase the acceleration/deceler-
ation and velocity settings to their maximum levels,
except where this might be detrimental to the payload
(such as for fragile, long, or large parts).This includes
both robot movements and the 7th axis movements.
(Ensure that 7th axis acceleration and velocity settings
are within design parameters, as will be described be-
low.)
Finally, attempt to perform more than one robot op-
eration in parallel instead of in sequence. For example,
opening an end effector during an approach motion
instead of waiting until the motion has stopped.

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TRACKMOTION CYCLETIME: MOVING
FROM POINT ATO POINT B
For a robot on a track, minimizing the distance be-
tween operations is the first step to reduce overall
cycle time. Once this has been done, there are a
number of factors which will determine how quickly
the robot is able to move through its sequence of
operations.
The carriage motion profile1
describes the speed
and acceleration of the robot carriage over time.
To move from point A to point B, the carriage ac-
celerates to a certain velocity, maintains that travel
velocity for some amount of time (greater than or
equal to zero), and then decelerates to a stop. On
the speed-time graph this looks like a trapezoid, so
this is known as a“trapezoidal” motion profile. If the
travel distance is sufficiently long, greater than the
“critical distance,” the carriage will reach its maxi-
mum velocity (or rated speed). This maximum ve-
locity is a function of several factors, including the
rated speed of the servo motor, the rated speed of
the gearbox, and the pinion size. For longer tracks,
increasing the maximum velocity of the 7th axis can
significantly decrease the cycle time of your cell.
Determining Maximum Velocity. For a Trackmo-
tion™ linear axis driven by a Güdel rack & pinion
system, the maximum velocity is determined simply
by the effective pitch radius of the pinion gear and
the maximum angular velocity at which it is able to
turn.
1 See Appendix A
Vrack = wpinion ∙rpinion
The pinion pitch radii are 25.47mm for the Güdel
TMF-1 and TMF-2, 33.96mm for the TMF-3, and
42.44mm for the TMF-4.
Gearbox rating & ratio: Each gearbox will have a max-
imum allowable input speed. For the Güdel AE060
gearbox (used for TMF-1 and TMF-2) this is 6000
RPM (628.3 rad/s). For the AE090 (TMF-3) and
AE120 (TMF-4) gearboxes it is 4500 RPM (471.2
rad/s). Also, each gearbox is available in ratios from
2:1 to 24:1. Dividing the maximum input speed by
the gear ratio i will provide the maximum output
speed at which the pinion gear will turn.
woutput =
winput
i
Therefore, in order to increase the maximum ve-
locity of a given system, choose a smaller gearbox
ratio. (However, this will have an inverse effect on
maximum acceleration, as described below.)
Based on the information above, the maximum
possible velocity (using a gearbox ratio 2:1 and
the maximum gearbox input RPM) is 8.0 m/s for the
TMF-1, TMF-2, and TMF-3, and 10.0 m/s for TMF-
4. However, it is not advisable that the gearbox be
operated at maximum input speed. Also, finding a
servo motor which can provide the required torque
at such high speeds may be problematic, if not im-
possible.

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GUDEL, INC. ©2014PAGE 03
Therefore it is safe to assume that the actual max-
imum velocity will be no more than about 60-70%
of those theoretical values. Finally, keep in mind that
choosing a small gearbox ratio (such as 2:1) will have
a negative effect on the maximum possible accelera-
tion, as described below.
Determining Maximum Acceleration. Deter-
mining the maximum possible acceleration of the
carriage requires quite a bit more information. Ac-
cording to Newton’s famous equation, acceleration
equals force divided by mass. The forces acting upon
the carriage include both the force between the pin-
ion and the rack as well as various frictional forces:
the rolling resistance of the rollers, friction from the
guideway scrapers and lubrication system, etc. The
mass of the system includes the carriage itself, the
robot on the carriage, as well as everything else that
is connected to it (cables, connectors, etc.) Also,
because the carriage involves rotational elements
(rollers, motor shaft, couplings, gearbox elements,
pinion),there is also rotational momentum and rota-
tional friction which must be factored into the equa-
tions of acceleration.
For the sake of simplicity, let us focus on the three
main variables affecting maximum carriage accelera-
tion: robot mass, servo motor torque, and gearbox
ratio.
Robot Mass: For most Trackmotion™ systems, the
most significant contribution to mass will be the ro-
bot. Therefore, choosing the smallest robot which
can perform the required tasks is essential for max-
imizing the acceleration of the system. This also ap-
plies to any robot riser or payload attached to the
carriage. The lighter the overall system, the more
quickly it can accelerate with a given applied force.
Servo Motor Torque: There are a wide range of servo
motors which are compatible with Güdel gearboxes.
Each motor will have its own speed-torque charac-
teristics, usually illustrated by a speed/torque graph.
For electric motors, maximum torque will decrease
as the speed of the output shaft increases.
Servo motors typically have a “peak” or intermittent
torque capacity, as well as a rated torque capacity,
for continuous operation, for a given speed.The mo-
tor can operate for an indefinite period within the
“continuous” zone without overheating. Therefore
the motor’s RMS speed and torque must lie within
the continuous region.
To increase the acceleration of the carriage sys-
tem, choose a servo motor with a high intermittent
torque rating. (Keep in mind that the maximum ser-
vo motor torque must not exceed the maximum
torque rating of the gearbox coupling, or slipping
will occur.)

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GUDEL, INC. ©2014 PAGE 04
Gearbox Ratio: Because the gearbox reduces the
rotational velocity going from the motor to the pin-
ion, it has the effect of increasing the output torque
by a corresponding amount.
toutput = tinput ∙ i
Therefore, choosing a higher gearbox ratio allows
for a higher output torque for a given motor speed.
However, as we saw above, increasing the gearbox
ratio has the effect of reducing the maximum veloc-
ity! There is a trade-off between maximum velocity
and maximum acceleration for a given servo motor.
Balancing Speed and Acceleration. What is the
ideal tradeoff between speed and acceleration for
a particular application? As mentioned above, at
travel distances greater than the critical distance,
the robot carriage will peak at its maximum veloc-
ity Vmax . The critical distance is a function of the
maximum velocity and acceleration of the system:
dcr = V
2
max
a
Short DistanceTravel: For travel distances less than
this critical distance, where the robot carriage nev-
er achieves its maximum velocity before decelerat-
ing, the travel time is simply a function of distance
and the rate of acceleration:
t = 2
√
For short distances (below the critical distance), in-
creasing the maximum velocity has no effect on the
travel time! Therefore, when travel distances are
short enough such that the robot does not reach
its maximum speed, the strategy to reduce travel
time is to increase acceleration by choosing a servo
motor with higher torque, and/or by choosing a
higher gearbox ratio. (Keep in mind that maximum
acceleration can also be improved by lightening the
payload on the carriage, so look for ways to reduce
mass wherever possible).
Long Distance Travel: For distances above the critical
distance, the travel time is a function of distance,
maximum velocity and acceleration.
t = d + Vmax
Vmax a
For travel distances above the critical distance, the
first component of this equation will be the larger
factor in the travel time. Therefore, once the criti-
cal distance has been exceeded, the first strategy to
reduce cycle time is to increase maximum velocity
by choosing a servo motor with a greater maximum
rated speed, and/or by choosing a lower gearbox ra-
tio. Once an attempt has been made to increase the
maximum velocity, further cycle time improvements
may be achieved by increasing acceleration (as de-
scribed above).
WORKFLOW SCHEDULING
For robotic cells in which there is more than one
operation being performed on each part, there are
several decisions which can be made affecting cy-
cle time. First, there is a decision as to whether
a forward or a reverse cycle will be implemented.
Secondly, implementing a dual gripper instead of a
single gripper can greatly reduce cycle time. Last-
ly, the number of parallel machines simultaneously
performing each operation must be determined for
optimum efficiency.
Forward or Reverse Cycle. In a “forward cycle”,
the cycle begins with all machines empty.The robot
picks up a part from the input buffer,carries it to the
first machine, waits for the first machine to finish,
then carries the part to the next machine (and so
on until all processes are completed and the part is
dropped off at the output buffer).
d
a

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In a“reverse cycle”,the cycle begins with all machines
except the first one full (the first machine must be
empty in order to receive an incoming part). The
robot picks up a raw part from the input buffer and
carries it to the first machine. Then the robot trav-
els to the last machine, picks up a finished part, and
transfers it to the output buffer. Then the robot goes
to the second-to-the-last machine,picks up a finished
part,and transfers it to the last machine. This process
continues until the robot returns to the beginning.
The reverse cycle allows for more machines to be
occupied by parts simultaneously, which increases
efficiency. However, it increases the overall move
distance required by the robot per part.
So, for single-gripper robots, where the processing
time for each machine is always less than the time
required for the robot to travel from that machine
to the next, the forward cycle is preferable. In all
other situations,the reverse cycle is more efficient.2
Dual-Gripper EOAT. Implementing a dual-gripper
EOAT can greatly reduce the overall cycle time for
most systems in which machine processing time is
greater than the robot travel time. A dual-gripper
EOAT allows for all machines to be occupied at the
beginning of each cycle, and it reduces the overall
travel distance per part. The cycle begins with all
machines occupied and both grippers empty. The
robot picks up a part from the input device with the
first gripper and then travels to the first machine.
2
Dawande, M., Geismar, H. N., Sethi, S. P., & Sriskandarajah, C. S. (2005).
Sequencing and Scheduling in Robotic Cells: Recent Developments. Journal of
Scheduling, 8, 387-426.
The EOAT rotates and picks up the completed part
at machine #1 with the empty gripper, the EOAT
rotates and drops off the new part from the first
gripper. Then the robot travels to machine #2, the
EOAT rotates and picks up the finished part with
the empty gripper, the EOAT rotates and then
drops off the part from the first machine. This se-
quence continues until the robot drops off a fin-
ished part to the output device. It then returns
to the beginning and starts the sequence over.
Parallel Machines. In situations where one oper-
ation has a longer duration than another, the over-
all efficiency of the cell can be increased by running
more than one of the slower operation in parallel,
but having multiple machines. For example, if oper-
ation #1 takes 10 seconds per part, but operation
#2 takes 30 seconds per part, there will always be
wasted time as the robot waits for operation #2 to
complete, limiting the overall capacity of the system.
However, this wasted time can be reduced by adding
additional machines performing the slower operation.
CONCLUSION
There are a number of factors affecting the over-
all cycle time of a robotic cell, from the layout to
the selection of components. Making the work cell
as compact as possible, so that robot and track
movements are minimized, is the first step. Choos-
ing the lightest robot available which can perform
the necessary tasks will allow for more rapid car-
riage acceleration.A Güdel application engineer can
assist in the selection of a servo motor and gear-
box ratio combination which will provide the cor-
rect tradeoff of velocity and acceleration. Finally,
where parts are being moved through multiple op-
erations, a dual-gripper EOAT should be considered.

7. GÜDEL, INC. TECHNOLOGY WHITEPAPER
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APPENDIX A:TRAPEZOIDAL AND
S-CURVE MOTION PROFILES
Trapezoidal Motion Profile. The trapezoidal mo-
tion profile is based on the assumption of constant
acceleration and deceleration.
Looking at a graph of both velocity and acceleration
versus time,you can see that there are three distinct
travel phases: a first phase of constant acceleration,
a second phase of constant velocity (zero acceler-
ation), and a third phase of constant deceleration.
Looking at the acceleration graph, you can see that
the transition between these phases is instanta-
neous—the change in acceleration occurs instantly
between each phase. The “jerk” is the rate change
of acceleration. For trapezoidal motion, the jerk is
infinite.
There are drawbacks to designing a system with ex-
tremely high jerk, namely the introduction of vibra-
tions or oscillations. For a given load, the higher the
jerk (or change in acceleration), the more power-
ful the vibrations will be, and the larger the number
of vibrational modes that will be excited. This may
result in increased settling time or reduced accura-
cy if the vibration frequency matches resonances in
the mechanical system. Vibrations will also lead to
greater stresses on the system and reduced life due
to material fatigue. Therefore another motion pro-
file, known as the S-curve Profile, is often preferred.
S-Curve Profile. The S-curve consists of 7 distinct
phases of motion (as opposed the 3 phases of the
trapezoidal profile). Phase 1 starts the load from rest
at a linearly increasing acceleration until it reaches
the maximum acceleration. In phase 2, the profile
accelerates at this maximum acceleration rate until
it must start decreasing as it approaches maximum
velocity. In phase 3 the acceleration linearly decreas-
es until it reaches zero. In phase 4 the velocity is
constant, at which point the profile decelerates in a
matter symmetric to phases 1-3.
Since trapezoidal profiles spend all their time at full
acceleration or full deceleration, they are faster than
the S-curve profile. But if the “all on / all off” ap-
proach causes an increase in settling time, the ad-
vantage is lost. Many times, even a small amount of
smoothing between full acceleration and zero accel-
eration can substantially reduce induced vibration.
The specific choice of the form of the S-curve will
depend on the mechanical nature of the system and
the desired performance specifications.
Credit for the written content of this document is Don
Bromley, Applications Engineer, Güdel, Inc.

8. 4881 RUNWAY BOULEVARD
ANN ARBOR, MICHIGAN
48108 USA
PH: 734.214.0000
EMAIL: info@us.gudel.com
WEB: www.gudel.com
Copyright ©2014. All rights reserved. No portion of this whitepaper can be reproduced
without approval from the management at Güdel Inc.