GRRC Technical Report 2010-01
Improving the Electromechanical
Performance of Robot Arms on
Unmanned Ground Vehicles
Author: William Brown
Advisor: Prof. A. Galip Ulsoy
12/20/2009
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Abstract
The purpose of this report is to examine potential methods of improving the performance of
robot arms and unmanned ground vehicles (UGVs) with respect to reliability, utility, and efficiency. The
report begins with a brief overview of what a UGV is, how UGVs are used, how often they fail, and why
they fail. This is followed by a literature review. The report examines a handful of simple proposed
solutions to problems with current UGVs and focuses specifically on a procedure that alters the
mechanical design of a robot arm to incorporate springs optimized to reduce the maximum motor
torque required to perform a specified maneuver. By reducing the maximum motor torque required,
the reliability and utility of the motors, and by extension – the entire UGV, are increased. Initial results
indicate that this procedure can reduce maximum required torque for a basic maneuver by 47%. The
report concludes with a section proposing possible future objectives.
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I.
Introduction and Background
Unmanned ground vehicles (UGVs) were once seen only in a hobbyist’s garage. Today they are
being used more and more for a variety of commercial and military purposes. They are frequently used
in the military, or by emergency responders, to go places that are inconvenient, too dangerous or
impossible for people to reach directly. Their uses and designs vary greatly. Vehicles have been
designed to go places where humans cannot, such as hundreds of meters down oil drilling pipes to
inspect wells. Unmanned ground vehicles were used extensively to help rescue workers search the
rubble of the World Trade Center [1]. The famous Spirit and Opportunity rovers are exceptional
examples of unmanned ground vehicles that remotely wandered the Martian surface collecting large
amounts of data that will be examined for years to come. At the presidential inauguration in February,
UGVs were driven underneath every bus to check for bombs. UGVs have also played a larger and more
important role in the armed forces in recent years; with over 5,000 currently deployed by the US Army
[26]. In Iraq and Afghanistan, UGVs of a variety of designs are used to inspect potential improvised
explosive devices (IEDs), caves and buildings [2]. Hundreds of soldiers are alive today because a UGV
found or detonated an IED instead of a soldier [26]. In every one of these examples, UGVs were used to
accomplish something that would have been impossible or extremely dangerous for a human to do.
What exactly is an unmanned ground vehicle? A report by Nguyen-Huu & Titus provides an
excellent overview of UGVs and how they fail [3]. A UGV can broadly be described as mechanized
equipment that moves on the ground that does not carry a human being [4]. If the UGV contains
sensing equipment and is capable of interacting with its surroundings it could be considered a groundbased mobile robot [5]. UGVs that we will be considering in the remainder of this paper fall into this
category. UGV designs can vary greatly depending on the intended function of the machine. However,
in general a UGV is comprised of the following subsystems [6]:
Sensors: Sensors allow the robot to perceive its environment.
surroundings is essential for reliable, effective and safe operation.
Proper perception of the UGVs
Platform: This is essentially the electromechanical part of the UGVs and is analogous to the skeleton and
muscles on an animal. The platform provides the power, locomotion, structure, and physical utility for
the UGV.
Control: How the robot perceives its surroundings and follows human commands is a function of
controller design. Designs vary widely – they can be as basic as a design that lets the human controller
interpret all sensor data and then follow the human instructions exactly. On the other extreme, a
controller could be built so that the UGV operates autonomously for extended periods – interpreting its
surroundings and making decisions on its own.
Interface: How the robot interacts with human commands. UGVs often have many degrees of freedom
and depending on the control system, every one of them may need to be controlled by the human.
Communication: How the robot sends and receives information is very important. If the robot is
operating within range of a base, a fiber optic cable could be used. However, cables can snag on things
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and limit the range of the robot. In the case of military robots, wireless communication is necessary.
Furthermore, the data must be both accurate and privileged.
System Integration: The configuration of the overall system. Increasingly well designed robotic systems
will become more reliable and therefore able to operate more autonomously.
Reliability Issues with UGVs and Common Failure Modes
Because UGVs are a very young industry and generally operate in uncertain environments they
suffer from a number of reliability issues. This is very much in contrast to aircraft and automobiles –
both decades old industries that not only put a premium on vehicle reliability, but also have optimized
their vehicles for very specific operating conditions. In many ways UGVs are in a place now that
computers were twenty years ago. Like computers twenty years ago, the early history of UGVs occurred
mostly by hobbyists. If their computer/UGV broke, a hobbyist would open it up and use whatever was
convenient as a method to fix the problem. Furthermore, they could perform these repairs whenever
they wanted. This is not considered acceptable in any industry. People would never buy a car if it
broke down every other week of routine driving, and people would certainly never board an airplane if it
broke down after a few flights. UGVs have not yet developed this level of reliability, in fact, they break
down frequently.
Technical papers authored by Carlson and co-workers are some of the most extensive works on
the reliability of UGVs to date. Carlson’s thesis [5] summarizes and includes the data collected and
published in three previous papers [7][8][9]. These papers describe both how often UGVs fail and in
what manner. One particularly useful piece of information to gauge reliability is the mean time between
failures (MTBF). Table 1 shows reliability statistics for three robots in both an initial study (top) and
follow up study (bottom.)
Note: % of Usage is the percentage of total time that the robot was used in fields.
Table 1. Summary of results from CRASAR Reliability [9]
A major conclusion of the Carlson study is that the MTBF is usually between 6 and 20 hours.
This was also corroborated by anecdotal comments during our visit to the US Army Joint Robotics Repair
and Fielding (JRRF) facility in Michigan. This would not be a large problem if UGVs were only owned by
hobbyists, who could fix them whenever they wanted or had the time. This is no longer the case and it
becomes less so every year. Every year the army deploys more and more UGVs (Fig. 1). These vehicles
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are placed in admittedly difficult environments, but rarely last more than twenty hours of active duty
before requiring some form of maintenance or repair. When these vehicles are being repaired they are
not in the field of duty where they are needed to assist soldiers. Furthermore, as more UGVs are
deployed, each UGV technician will have more work to do, as there will be more vehicles needing
repairs at any given time.
Figure 1: Improvements in military UGV reconnaissance, and the dramatic increase of UGVs deployed
in recent years [10].
Qualitative Comparison Between Army UGVs and Mars Rovers
Interesting examples of UGVs that have shown to be reliable well beyond anyone’s expectations
are the mars rovers, Spirit and Opportunity (Fig. 2). It was hoped that the rovers would remain
functional for a three month mission. Amazingly, both rovers lasted for multiple years! The obvious
question then is: why have these UGVs worked for years, while the average one can only last for a dozen
hours without breaking. There are many answers to this question. First of all, the mars rovers were
designed from the ground up for the very specific mission of exploring the Martian surface [11]. This is a
major contrast to the Talons and Packbots being used by the US Army, which are off the shelf vehicles,
with a few customizable features. Second, because there is no way to repair the rovers on mars, the
remote handlers are extremely careful with them [12]. They know exactly what the robots are capable
of doing and are very careful to make sure that the robot never does anything it was not built to do.
Army UGVs do not experience this form of care. Operators will routinely operate the robots in a way
that pushes the robot beyond what it was designed to do. This can happen intentionally (trying to lift
something that is too heavy) or from difficulty using the UGV interfaces. NASA mission controllers have
hours to make decisions and plan the rovers’ next moves. Army UGV operators have to make those
same decisions in real time. The Mars rovers also have also been able to take advantage of redundant
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systems. Despite losing control of a wheel, NASA engineers found that by running the rover in reverse it
could still be maneuvered. If an analogous situation were to occur in a Talon or Packbot, the operator
would not have time to come up with a workaround solution and would instead take the robot to a
maintenance facility.
Figure 2. Artist’s depiction of a Mars Exploration Rover [13]
No doubt, there are lessons to be learned about reliability from the mars rovers. Interfaces
could be improved a lot, possibly by using haptics, which would allow operators to control their vehicle
more easily and have a better sense of the limitations of the UGV. Finally, the UGVs could be built with
either redundant systems, or some sort of fall back system, so that when a part of the UGV failed, the
vehicle would still be able to limp back. The UGVs could be designed more specifically for certain
environments instead of the one size-fits-all approach that is currently used. The difficulty is in being
able to implement these improvements without each robot costing as much as a mars rover.
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II.
Review of Literature on Reliability Relating to UGVs
One of the most daunting problems of improving the reliability of UGVs is the complexity of the
system. Each subsystem is complicated and can suffer from a lack of robustness in its design. If any of
the numerous subsystems fail, the UGV may be crippled and in need of repair. Because the general
reliability problem is so complicated, it is essential to break the problem down into more manageable
pieces. In this section I will only look at methods used to improve the reliability of the electromechanical design. To do this I will examine two general areas of the literature. First I will examine
what research has already been done on the design and reliability of ground vehicles and robotic
manipulators. The second area will examine some of the techniques used by other industries to ensure
reliability in their products. There is an extensive amount of literature in this area, especially from the
aircraft literature where reliability is of utmost importance.
Reliability and Design Techniques in the field of Robotics and UGVs
“Reliability-Based Design Optimization of Robotic System Dynamic Performance”
One of the major causes of failure in robotics is that the robot is not being used in a manner
consistent with its design. One way this happens is simply due to uncertainty. Instead of starting with
an exact configuration (certainty), reliability based design optimization (RBDO) represents that
configuration by a probabilistic distribution. RBDO then looks at the solution over the probabilistic
configuration. This information is often more useful than the solution to an exact configuration because
in the real world things are never known exactly. It is important that the robot work over the range of
uncertainty and not just one certain configuration.
Designers can predict the dynamic performance of a robot using the dynamic capability
equations (DCE). These equations can be used to predict how the end effector of the robot can
accelerate or apply forces and torques given a single configuration. Either of these descriptions (as well
as several others [14][15][16][17]) could be used to describe the dynamic capability of the robot arm.
The key limitation of this approach is that the DCE is only found for a single configuration at a time. By
systematically performing DCE calculations for a large number of configurations that fill the feasible
space, an overall measure of dynamic performance can be determined.
Alternatively, an overall measure of dynamic performance can be calculated by combining RBDO
methods to the DCEs and using a probabilistic density function to characterize the feasible space [18].
This approach yields a number of advantages over testing many points within the feasible space. One
advantage is that the number of feasible operational points will rise exponentially as the number of
degrees of freedom increases. Evaluating enough points to adequately fill in this space would require
immense computational time compared to a probabilistic distribution. Secondly, the solutions from a
discretized approach can still miss the optimal values found by the RBDO approach. This is illustrated
clearly in Figure 3.
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Figure 3. Contours of Optimal Torque Values [18]. Figure 3a was generated by taking samples within
the feasible space and solving DCEs, yet misses the true optimum value found in Figure 3b utilizing the
probabilistic appoach.
RBDO in this setting can also be used for the selection of actuator sizes. While a certain sized
actuator would be required to achieve a certain dynamic performance at any point, it is generally not
practical to have more than one motor per degree of freedom. The RBDO appoach offers an effective,
and conservative, way to view the entire operational space of the robot at once - providing an effective
way to select actuators optimized for reliability. Results from Bowling’s paper confirm the expected
result that larger motors would be required in a system optimized for reliability [18].
“Concept of Intelligent Mechanical Design for Autonomous Mobile Robots”
The reliability of robots can also be improved by utilizing the “concept of intelligent design.” [19]
Nassiraei notes that all aspects of a robot system are becoming more complicated and that future
robots are going to have to be much more complicated than the robots of today in order to achieve
their objectives. These complexities may lead to an increase in weight and power consumption, but will
most certainly lead to a loss of reliability and an increase in cost. To mitigate these effects, the electromechanical system or platform must be kept as simple as possible. This approach, however, does not
take into account the relationship between mechanical design and controllability. Ultimately, there
needs to be a balance where the design has enough complexity to achieve a desired performance, yet
remain simple enough to keep cost low and reliability high.
Nassiraei proposes the use of Intelligent Mechanical Design (IMD), where IMD is a design that is
“self-controllable, reliable, feasible, compatible … and solves the functionality-usability tradeoff in an
optimal way.” [19] This can be formulated as an optimization problem. Self-controllability, reliability,
feasibility, and compatibility essentially define a set of constraints on the design space. The curve of
optimal solutions to the functionality-usability tradeoff is the same as looking at the Pareto set.
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Figure 4. Approach to Mechanical Design and related factors [19]
Nassiraei continues by tabulating a list of mechanical design principles (MDP). Although these
are mostly common sense, some are worth mentioning specifically and have been paraphrased here:
1) To design mechanical parts for a UGV, one has to know the environmental niche, the desired
behaviors of the robot, and the design of robot mechanics (Fig. 4).
2) The mechanics (and sensors and actuators) must be complicated enough to perform the desired
tasks, but not any more complicated than that.
3) Take advantage of the robots environmental niche.
These techniques are all good ideas to follow and should be utilized during the design of UGVs. The
coupling between mechanical design and controller design, and the role of controllability in design
are also discussed in the work of Peters [32].
Reliability Techniques from other Industries
The robotics industry is very young and as such has not yet generated a large volume of
literature on reliability based design as applied to robotic systems or UGVs. This stands in stark contrast
to the automotive, aircraft, or manufacturing industries. Therefore, this section of the paper will
examine some relevant literature related to reliable design found in other industries and discuss how
the techniques discussed could be applied to robotics.
Consistent utilization of reliable design techniques have made airplanes one of the safest ways
to travel. Mavris describes how a robust design simulation can be used on an aircraft design to optimize
the variables prior to the construction of a prototype[20]. In the introduction to his paper, Mavris has a
figure that captures the purpose of using a robust design (Fig. 5).
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Figure 5. Improvements to the variation of the Lift to Drag ratio by using a robust design (Design 2)
compared to a standard design (Design1.) [20]
In design optimization an optimal value will be found for a function at an optimal point in the
design space. However, if the design is not performing exactly at that optimal point there is no way to
know how well the design is actually behaving. Furthermore, it is unlikely that the design will be
operating at the precise optimal point due to potential variations arising from manufacturing, the
environment, the operator or other sources. This is demonstrated in Figure 5 by design 1. Design 1 is
excellent at the exact middle of the operational range but its performance degrades considerably
everywhere else. A robust design is one that might not be excellent at any point, but is acceptable at all
points. Design 2 in Figure 5 shows this type of behavior. Although, it is not quite as good as design 1 in
the central region of operation, it is much better than design 1 in the fringes of the operational range.
By examining the distributions on the y-axis of the figure we see that design 1 shows a large degree of
variation, while design 2 is almost constant over the operational range. Clearly, design 2 is a much more
robust design.
The main content of Mavris’s paper describes how an aircraft can be optimized by subjecting it
to various realistic duty cycles. It is an optimization problem where the amount of fuel consumed needs
to be minimized over a large range of possible flight paths, while at the same time satisfying all of the
safety requirements and other design constraints [20]. Applying this technique to UGVs should be
relatively straightforward. The biggest difference would likely arise from the fact that a UGV (unlike an
airplane) would experience more sudden and intense interactions with the environment. This is simply
the nature of ground vehicles. It should be possible to take a UGV model and simulate it on a virtual
bump course in the same manner as a military vehicle.
Another technique that could be useful for improving the reliability of UGVs would be
incorporating derivative free methods into RBDO. Derivative free methods outperform optimization
methods that utilize derivative information when the function being optimized has lots of local minima.
A derivative method relies on going “downhill” until the optimal solution is reached. However, it can get
stuck in a local minimum. If the problem has lots of local minima, it would be very difficult for such a
gradient approach to show global convergence. A derivative free approach will not get stuck in a local
minimum. Rather it uses an algorithm to test different points, selecting new points that are more
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optimal than the previous points. A disadvantage of derivative free methods is that they cannot know if
they are at an actual minimum and thus will terminate at a different location every time. Youn
describes the benefits of using the Eigenvector Dimension Reduction as an algorithm to perform the
derivative free optimization - specifically how it can be used to assess quality and reliability
simultaneously without requiring more computational power [21]. These techniques could be useful in
optimizing UGVs due to the highly nonlinear nature of non-holonomic ground vehicles and robotic
manipulator arms. It is also possible (and commonly done in optimization problems) to perform a
derivative free approach first and then use the results to perform a gradient based optimization. This
way a true minimum can be determined that is very likely to also be the global minimum.
Compared to standard design optimization, RBDO can be extremely computationally expensive
due to evaluating distributions instead of points. The situation is exacerbated as the system being
optimized grows in complexity. UGVs are certainly very complex systems. Thus, it would be greatly
beneficial if there was a method of performing RBDO in a way that was less computationally expensive.
Du takes a multidisciplinary design optimization framework and utilizes both system and subsystem
uncertainty analysis to estimate the mean and variance of system performance [22]. As a UGV is a
prime example of a multidisciplinary system it is a perfect candidate for this type of approach.
These are just a few examples from the literature whose methods could be applied to improving
the electromechanical design of UGVs. Some other promising resources include a paper by Sanchez
[23], and two authoritative texts by Phadke [24], and Park [25].
Example Problems and Anecdotal Solutions
Based on interactions with members of the armed forces stationed at the JRRF, and personal
interactions with two commonly used UGV platforms, I can say with certainty that there is a lot that can
be done to improve the reliability of UGVs. According to the repair technicians at the JRRF, the most
common causes of failure were electrical. In one UGV, this was especially true because of the
abundance of loose electrical connections and the rats nest of wires. This problem is actually twofold:
the wiring is subject to failure because of the large number of wire connections, and it is very difficult to
repair the robots quickly because of how many wires there are. One solution to this problem would be
to replace some of the wires with easily replaceable circuit boards. We also learned that the robots are
routinely used beyond their design specifications and that some of this can be attributed to an
inadequate human-UGV interface. Although operating a UGV became easier with more training, it was
much harder than operating a car and arguably more difficult than operating a helicopter. One major
difficulty was that the UGVs had more degrees of freedom than it was possible to control at any given
time. The Talon has fewer degrees of freedom than the PackBot, which arguably made it easier to use.
Another problem is that the operator had to control each joint independently as opposed to
commanding the end effector to move and letting the UGV calculate the intermediate joint angles.
Apparently, Army operators prefer controlling each joint as opposed to just controlling the end effector.
I believe that these difficulties could be reduced by doing any combination of the following:
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1) Implement a control system that allows for the simultaneous control of most (if not all) degrees
of freedom.
2) Develop a control logic that is intuitive to the point that controlling the end effector is
preferable to controlling each joint independently.
3) Create a haptic interface that allows the operator to feel when the arm is obstructed or at the
limits of its motion.
4) Develop a control structure that allows for more autonomous operation.
Implementing these improvements should make it easier to operate a UGV which would in turn
make it less likely that the UGV suffered because of inadequacies in the user interface.
The chassis of one UGV contains much of the sensitive electronics as well as the drive motors for
each tread. Because water and sand can wreak havoc on electronics, the chassis is built to be
waterproof, allowing the UGV to drive through puddles without short circuiting. However, this has the
unfortunate side effect of not allowing adequate ventilation. The temperatures generated by the drive
motors, electronics, and desert sun can make the UGV overheat, leaving it inoperable until it cools
down. Although many solutions exist, two are relatively simple and cheap. The chassis has no internal
temperature sensor. Adding one would give the operator a warning when the UGV was getting too hot.
This would give the operator a chance to run the UGV less vigorously – or if that is not possible – have it
move to a safe area where it can cool down. The other option would be the installation of heat sinks on
the outside of the chassis. These could keep the electronics cooler without sacrificing the waterproof
aspects of the case.
Another problem with this chassis design is that a dent in the case caused by the UGV running
over a rock can short circuit the electronics. In a redesign, this problem could be taken care of using the
type of RBDO discussed in the previous section. Optimization of the distance between the bottom of
the chassis and the circuit board would take into account the dents likely to occur from operating on
rocky terrain. In the meantime, a much cheaper workaround would be to add a thin layer of insulating
material between the chassis and the electronics. This way, when a dent in the chassis impinges on the
electronics no short circuiting occurs. These are just a few of the ways that the reliability of UGVs could
be improved with relatively simple solutions.
Another idea promoted in a paper by Qi et al. is the concept of using springs to reduce the
forces exerted by joints. [27] The object of his paper is to describe a method where a human operator
can train for work in zero or low gravity environments by wearing a “suit” outfitted with springs that can
partially or completely cancel out the forces created by gravity. Taking inspiration from this idea we will
consider adding torsional springs as a mechanical design parameter in parallel to the joints on a robot
arm. By reducing the gravitational load the performance of a robot arm can be improved by reducing
the maximum required motor torques. We will explore this concept in detail in the remainder of the
report.
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III.
Model Development, Optimization and Results
It is clear that there is a great deal of work that can be done to improve UGVs and that there are
many different ways to implement these improvements. Our specific focus is to improve the reliability,
utility and efficiency of the electromechanical design of a robot arm attached to a UGV. Of these three
tasks the one that we have worked on the most so far is reliability. DC motors will wear out relatively
quickly if operated near or at their stall torque. Therefore, if a relatively simple redesign of the
mechanical system can result in an arm where the motors need not use so much torque, then the UGV
will have been made more reliable. Alternatively, if a smaller torque is needed to lift a given load, then
the total load that can be lifted increases, thus the utility of the UGV will have been increased. In the
subsequent analysis we will minimize the maximum required motor torque over the course of a
prescribed maneuver.
For the initial stages of analysis it is prudent to start with a very simple model. This will allow for
the general concepts to be developed and demonstrated without requiring the complexity and time
commitment of a more complete model. Thus, we will begin our model development with the simplest
robot arm that will capture the necessary behavior. The arm we developed is a single link, single joint
arm. The joint end is attached to a stationary horizontal base while the other end has an idealized end
effector that is capable of instantaneously attaching itself to a load. The joint in our model includes a DC
motor, represented as an idealized torque source, and a worm drive transmission. It is necessary to
include the worm drive in our model because worm drive components are very common in robotic arms
due to their large speed reduction ratio in a compact space, and their ability to hold a joint in place in
the presence of external loads (e.g., gravity) with no motor input. A device that shows this second
characteristic is considered non-backdrivable. The fact that the transmission can support a load torque
without any torque from the motor shows that the transmisive behavior of the worm drive is highly
nonlinear. A paper by Dohring et al. derives the equations of motion for a backdrivable transmission
beginning with a wedge model for the worm gear interface [28]. Although, this is not directly applicable
to our objective, it provides an excellent introduction into worm gear dynamics. A more comprehensive
paper by Yeh and Wu uses Dohring’s paper as a starting point and derives the equations of motion for a
non-backdrivable worm drive transmission [29]. We draw heavily from this paper to derive the
equations that describe our model.
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Figure 6. A sketch of our simple robot arm. τw and τg are torques acting on each side of the worm
drive. Ci is the torque transmission coefficient and it depends on the geometry of the worm as well as
the operating condition defined in [29]. Angles θg and φ are defined with respect to the horizontal.
For consistency and clarification all terms that contain the subscript g correspond to the
load/gear side of the joint (e.g., Jg is the total inertia acting on the load side) and all terms with the
subscript w correspond to the motor/worm side of the gear (e.g., τw is the total torque acting on the
worm). The following equations take advantage of two critical assumptions: that the worm drive has no
backlash and that
𝐶1 >
−𝐽 𝑤 𝑖𝑤𝑔
𝐽𝑔
(1)
Where iwg is the constant speed ratio of the worm drive.
𝜃𝑤 = 𝑖𝑤𝑔 𝜃𝑔
(2)
C1 and C2 are constants that depend on the angles and friction coefficients inherent to the worm drive
design. For more information on them, refer to the paper by Yeh and Wu [29].
The first assumption is rarely true in real systems, but even in systems where backlash is present, the
backlash is a minor transient effect and for most large, slow, and non-precise motions its influence is
negligible. The second assumption is valid provided that the inertia of the load is much larger than the
inertia of the motor, which is reasonable for many systems.
In order to ultimately reduce the maximum torque experienced by a motor, we propose the
addition of a linear torsional spring attached to the arm joint in parallel with the worm gear. The spring
has two design variables: a spring stiffness, k, and a neutral angle, φ, where the spring exerts no torque.
15
The addition of this component adds a torque to the load side of the worm drive. We can define the
combined load torque as:
𝜏𝑔 = 𝑀𝑔 𝑔 cos 𝜃𝑔 + 𝑘(𝜑 − 𝜃𝑔 )
(3)
Where Mg is the weighted average of the mass of the arm and the load, while g is the acceleration due
to gravity.
Our analysis depends on knowledge of the trajectory of the arm. Since there are an infinite
number of possible trajectories for even a simple, single link model, we had to prescribe one that
seemed both simple and practical. We propose a maneuver where the arm begins at rest in a vertical,
“stowed” position. It then lowers to a horizontal position where it stops and attaches a load to the end
effector. Finally, the arm lifts the load back to the original, “stowed” vertical position. This is illustrated
in Figure 7.
 g [rad/s/s]
g [rad/s]
g [rad]
Angle
2
1
0
0
1
2
3
4
5
Time, t [s]
Angular Velocity
6
7
8
0
1
2
3
6
7
8
0
1
2
6
7
8
1
0
-1
4
5
Time, t [s]
Angular Acceleration
5
0
-5
3
4
Time, t [s]
5
Figure 7. The prescribed maneuver - the arm starts in a raised position, lowers to the
horizontal, attaches a load at time t=2s, and raises the load up to the starting vertical position.
Because we have defined the trajectory, we have knowledge about the position, angle and
acceleration for all times of interest.
Because we have prescribed a trajectory we know position, velocity, and acceleration for all times. This
allows for the motor torque to be calculated at any time by using simple algebraic equations rather than
16
the differential equations that would be required if we were to solve for the trajectory, given the motor
torques.
0
𝑖𝑓
𝜃𝑔 𝐴𝑁𝐷 𝐹𝑓 < 𝜇𝐹𝑛
𝜏𝑤 ≤
𝐽𝑤 𝑖𝑤𝑔 + 𝐶1 𝐽𝑔 𝜃𝑔 − 𝐶1 𝜏𝑔
𝑖𝑓
𝜏𝑤 >
𝜏𝑤 =
𝜏𝑤 >
𝐽𝑤 𝑖𝑤𝑔 + 𝐶2 𝐽𝑔 𝜃𝑔 − 𝐶2 𝜏𝑔
𝑖𝑓
𝜏𝑤 ≤
𝐽 𝑤 𝑖𝑤𝑔
𝐽𝑔
𝐽 𝑤 𝑖𝑤𝑔
𝐽𝑔
𝐽 𝑤 𝑖𝑤𝑔
𝐽𝑔
𝐽 𝑤 𝑖𝑤𝑔
𝐽𝑔
𝜏𝑔 𝐴𝑁𝐷 𝜃𝑔 > 0
𝑂𝑅
𝜏𝑔 𝐴𝑁𝐷 𝜃𝑔 < 0
(4)
𝜏𝑔 𝐴𝑁𝐷 𝜃𝑔 > 0
𝑂𝑅
𝜏𝑔 𝐴𝑁𝐷 𝜃𝑔 < 0
Ff and Fn are the friction and normal forces on the worm-gear interface respectively. μ is the coefficient
of static friction internal to the worm drive. The first equation is the case where the arm is stuck – it is
not moving, nor is it about to. The second equation corresponds to load driven case. This can loosely be
thought of as a situation where the motor is removing energy from the system. The third equation
corresponds to a motor driven case. This can loosely be thought of as a situation where the motor is
adding energy to the system. The second and third equations can also hold in situations where the arm
+
is currently not moving but is accelerating such that in the immediate future 𝜃𝑔 ≠ 0. In this situation,
+
𝜃𝑔 can be substituted for 𝜃𝑔 in the above equations to determine τw. The torque required of the motor
can then be found from
𝜏𝑚 = 𝜏𝑤 − 𝑏𝜃𝑔 𝑖𝑤𝑔
(5)
Where b is the coefficient of viscous damping acting on the rotating motor shaft.
Recall that our objective is to minimize the maximum motor torque required to perform a prescribed
maneuver. From the trajectory and equations of motion described above we can find a value for the
motor torque for every moment in time that depends on our design variables k an φ. We then consider
the maximum value of this function with respect to time to be our objective function
𝜏𝑚 _𝑚𝑎𝑥 𝑘, 𝜑 = 𝑚𝑎𝑥 𝜏𝑚 𝑡, 𝑘, 𝜑
(6)
This eliminates the variable t, and we are left only with the maximum motor torque as a function of our
design variables. Finally, we utilize Matlab’s optimization function fmincon with its default parameters
to find the minimum maximum motor torque and the corresponding optimal design values, k* and φ*.
Our results are promising but not surprising (Fig. 8). Using our prescribed procedure and
specified system values we were able to reduce the maximum motor torque required to perform the
maneuver from 11.8N-m to 6.3N-m – a 47% reduction in effort. A quick static analysis reveals that if the
motor loaded a spring to assist it with lifting a load, a maximum 50% reduction in effort could be
17
achieved. It would not be possible to reduce the effort more than 50% in the static case because it
would take more than 50% of the original effort simply to load the spring. We believe that our results
are reasonable because the limiting maximum static reduction and our dynamic results are close.
Motor Torque
12
10
8
6
m [N-m]
4
2
0
-2
-4
Optimal Spring Values - Max Torque = 6.3 N-m
No Spring - Max Torque = 11.8 N-m
-6
-8
0
1
2
3
4
Time, t [s]
5
6
7
8
Figure 8. Optimization of our spring has reduced the maximum required torque of the motor from
11.8N-m to 6.3N-m, a 47% improvement. By reducing the maximum torque on the motor we have
effectively increased the motors utility and reliability and by extension the utility and reliability of the
entire UGV.
In order to check our procedure we attempt to solve what could be considered the inverse
problem. Whereas, in our optimization procedure we prescribed a trajectory and used that information
to determine the motor torque values for all time, here we use values for motor torque as a function of
time as well as the optimal design values k* and φ* in order to solve for a trajectory. This problem is a
nonlinear differential equation and we can use numerical integration software such as Matlab’s ode45
to solve it. If the solution to the differential equation is the same as the trajectory that we originally
prescribed then we know that it is much less likely that we have made a mistake in our calculations.
Due to the severe nonlinear behavior and the large magnification through the transmission, the solution
is very sensitive to integration errors and can quickly become unstable. With sufficient tweaking of
integration tolerances and parameters it is possible to make the solution converge to the original
trajectory, thus our procedure is checked. It is important to note that this simply shows that our
equations are consistent and not ill formed, it does not verify the accuracy of our model. To do that we
would need to perform some experimental testing.
18
IV.
Opportunities for Future Work
There is great potential to expand on this project. So far we have demonstrated that by adding
springs to the joints of the robot arm we can reduce the required maximum motor torque significantly
and thus make the robot arms more useful and more reliable. Our current setup is limited by the
specific optimization objective and by the simplicity of the model. Addressing these issues should
provide a significant contribution to the field of mobile robotics.
Thus far, minimizing the maximum motor torque has been our only objective. There are a
number of other objectives that would also be worth considering either alone or as a combined
objective. Minimizing the maximum required power would be an example of another useful objective
function. Recall that we had three objectives: improve the reliability, utility and efficiency. Thus far, we
have not addressed the issue of efficiency. DC motors have characteristic efficiency curves and operate
most efficiently at certain torques [30]. Thus, one future objective would be to append these equations
onto out current model and optimize the spring design parameters such that the least amount of energy
was used to perform a given maneuver – maximizing efficiency. In addition to looking at each one of
these optimization problems separately, a combined problem may prove to be the most practical
approach and merits consideration.
The current optimization procedure (as well as those listed above) is dependent on a prescribed
trajectory. An alternative approach would be to optimize the arm to maximize its “utility space”. A
code could be developed to calculate the force that can be delivered to the end effector at different
operating conditions. It could then be determined how many points in the operational space could
supply a sufficient force to the end effector. This subset of the operational space would be considered
the utility space. The utility space could then be maximized be changing the spring design parameters
[31].
Our model is highly limited in its simplicity. Although it is sufficient to demonstrate certain
optimization techniques a single link robot arm is not very useful in real life. Real robots almost always
have two or more links. Our model would benefit greatly by being expanded such that it could model
specific robot arms, like those on the Talon and Packbot. Ideally, our model could handle an arbitrary nlink robot arm and take a DH table as an input.
The fact that our current procedure depends on a specific maneuver is problematic because it
only allows for the optimization of that specific maneuver. Although this may be convenient for an
assembly robot, the value of an arm on a UGV is that it can perform many different maneuvers.
Therefore, our results would be of much more use if the prescribed trajectory were analogous to an
automotive “drive cycle” rather than a specific maneuver. Getting a quality drive cycle is always a
challenge and would require substantial input from people who routinely use these vehicles.
Another limitation of the current procedure is the fact that there is no way to account for
variations in the model or the maneuver. Although a drive cycle will help greatly with variation to the
19
trajectory, variations in actual systems need to be taken into account in the model and procedure.
Doing so will effectively incorporate reliability based design optimization techniques.
In addition to the directions for expansion mentioned above, there are several other long term
project objectives. Any one of these would increase the complexity of the problem drastically. So far
we have only considered the electromechanical plant of the robot arm. Ultimately, it could be desirable
if we performed an analysis and optimization of the combined design optimization of the plant and
controller [32]. We have only considered the spring parameters as design variables. In the future we
may want to optimize more of the parameters describing the system such as the lengths of the links or
the angles of the worm drives. One of the main reasons for mechanical UGV failure is overheating; a
phenomenon that we have not considered directly thus far. Including the heat output of the motors as
well as other thermal effects could further increase the utility of our procedure.
20
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