Velociraptor Final Design Report - zacharyjblum.com · Our final velociraptor was 19 cm tall and powered by two AA batteries that provided our motor with 2.627 Volts and allowed 0.54

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Velociraptor Final Design Report

Compiled by Zach Blum, Akash Modi, Ryan O’Rourke, and Marisa Tashima

ME 112: Mechanical Systems Design, Department of Mechanical Engineering

22 March 2017

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Team MARZ Dino VAIL 11:30

Zach Blum, Akash Modi, Ryan O’Rourke, Marisa Tashima

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Table of contents

1. Executive Summary…………………………………………………………………………….4

2. Background

2.1 Overview………………...………………………………………………………......6

2.2 Basic description of logistics……………………………………..…………………7

3. Basic Description of Design

3.1 Linkage, Leg, and Feet Design……………………………………………...………8

3.2 Kinematic Motion Design Goals…………………………………………...……...12

3.3 Motor Gearbox Goals……………………………………………………………. .13

3.4 Drivetrain Design………………………………………..………...........................14

3.5 Material and Adhesive Choice………………………………………………….…15

3.6 Assembly Design…………………………………………………………....….....17

3.7 Material and Adhesive Choice………………………………………………..…. 18

3.8 Final Adjustments…………………………………………………………….…. 18

4. Analysis of Performance

4.1 Motor and Transmission Performance……………………………..……………..19

4.2 Gear Analysis…………...………………………………………………………...21

4.3 Analysis of Power at Foot ………………………………………………………..21

4.4 Free Body Diagram……………………………………………………………….26

5. Gameday Performance…………………………………………………………………...….27

6. Redesign…………………………………………………………………………………...….28

7. Conclusion…………………………………………………………………………………....30

8. Bibliography………………………………………………………………………………….31

9. Appendix…………………………………………………………………………………….. 32

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1. Executive Summary

Figure 1 Here is the MARZ Dino in its finished form. It can transverse a variety of terrain with reliability and ease

For our ME 112: Mechanical Systems Design final project, our team worked to create a

lightweight, battery operated, bipedal robot. In order to meet the challenge requirements, it had

to stand less than 20 cm tall, walk at a speed of at least 3cm/ sec along a 1 meter section of

Meyer-Green, and resemble a dinosaur in both its gait and shape. Our robot was built to walk

consistently and repeatedly under the power of 2 - 3 AA batteries, a lightweight DC motor, and

an internal gear train. Our design utilized a large, spread-out footprint, a low center of gravity,

and a parallelogram linkage with two powered axles in order to walk well on various types of flat

terrain with ease.

Our final velociraptor was 19 cm tall and powered by two AA batteries that provided our

motor with 2.627 Volts and allowed 0.54 Amps of current to flow through. Our motor was

connected to a Tamiya 6-Speed motor box with a 196.7:1 reduction ratio and ran at an efficiency

of 60.95%. In addition to this gearbox, we implemented a custom built 3-geared drive train to

power our top and bottom axles. This propelled our 475 gram robot forward at a 7.7 cm / s

walking pace. The elongated feet with wide “U” shaped attachments provide a stable platform

for our dinosaur to walk on, and allowed it to stand on one leg without falling from side to side

or leaning dramatically one way or another. We prioritized crawler stability and found that these

“U” shaped feet were essential to the consistency and reliability we strived to create throughout

the design process.

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The placement of all of the heavy portions of our robot gave the walking robot a uniquely

low center of gravity that improved the stability of the dinosaur on different types of terrain. Our

team coupled the two heaviest parts of our dinosaur, the motor and the battery pack, together in

the center of our dinosaur as close to the ground as possible. This system allowed us to keep the

center of gravity of the 19 cm tall dinosaur just around 5 cm off the ground. To further decrease

our center of mass, we added a heavy AA battery (24 grams) and a slender, but dense screw (17

grams) to each foot of our dinosaur. In our testing, this added weight proved useful as it helped

our dinosaur easily balance and navigate its way across a path of rough cobble stones.

The parallelogram linkage system we employed was a simple four bar linkage system

made up of a 2 cm crankshaft directly connect to our motor. This crank was connected to a 5 cm

coupler (the leg of the dinosaur) and a 2 cm rocker that was connected directly to a second

powered axle 5 cm above the axle the motor crankshaft; the body of the dinosaur acted as a 5 cm

ground linkage. Our team chose this system because we believed it would give our robot

maximum stability and reliability when walking over different terrain. This linkage allowed our

feet to stay parallel to the ground throughout the entire walking cycle- an aspect that we hoped

would help our dinosaur stay balanced.

The final version of our dinosaur robot operated consistently over both smooth and rocky terrain.

Our design choices of a large spread out footprint, low center of gravity, and parallelogram

linkage system with two powered axles allowed us to complete the challenge on testing day. The

following sections of this paper go more in depth on each of the design details behind the

construction of our robot, as well as additional areas for improvement, and a full analysis of our

motor and transmission system.

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2. Background 2.1 Overview

Figure 2.1a Anatomical view of a velociraptor Figure 2.1b Design inspiration for a four bar parallelogram

[source: extinctanimals.org] linkage system [source: i.ytimg.com]

Our team understood the design challenge to be simply “build a bipedal robot that

resembled a bird or dinosaur.” For our robot we decided to create a bipedal velociraptor. More

than any other design requirement we wanted our dinosaur to walk consistently without fault

over any surface we would be required to test on. Our final design achieved this goal as our

dinosaur was able to walk at over two and a half times the required speed (~7.7 cm / s) over both

flat and cobblestoned surfaces.

Our robot resembles the anatomical shape of a velociraptor (figure 2.1a). However, we designed

the gait to be a bit different than that of the real creature in order to minimize power

consumption, weight, and other design ramifications. Our team felt that given our time

constraints, a complex linkage arrangement would result in a bipedal system that might not

operate well on adverse terrain- our design priority was to maximize our robot’s stability and

walking ability. With this in mind we chose to optimize a four bar parallelogram linkage system

that uses an equal length rocker and crank and an equal length ground and coupler. We were

inspired by the robot shown in figure 2.1b. We wanted to exercise a form follows function

approach that would result in a robot that had superior walking and stability capabilities.

With these design goals in mind and an emphasis on functionality, we made the design

decision to gear our powered bottom axle up to our top axle so that our dinosaur would run two

axles in perfect synchronization. This increased the reliability of our design by decreasing the

likelihood that one of the dinosaur’s two legs or one of its four links would be disrupted and

moved out of line by an unpredictable change internally or externally. In the same spirit we

decided to link our bottom linkages directly to the motor so that any tolerance issues in the

pinning system used to connect our linkage to the motor’s crankshaft would be minimized.

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Additionally, instead of creating feet that resembled the unique foot of a velociraptor we created

unique “U” shaped feet that allowed the dinosaur to easily stand on one leg during any part of its

foot’s cycle.

These design choices allowed us to create a dinosaur robot that was able to complete our

challenge requirements described in section 2.2 with ease.

2.2 Basic Description of Logistics

On presentation day, our dinosaur had to walk across two different surfaces of Meyer Green. In

order to successfully complete the challenge, the following requirements had to be met:

• be battery-powered;

• have a velocity of at least 3cm /sec;

• autonomously maintain a stable, straight trajectory over both smooth and cobblestone

pavement for at least 1 meter;

• exhibit physiologically accurate bird/ dinosaur bipedal motion; and

• measure less than 20 cm in height

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Three birds/ dino crawlers were tested at a time on either side of Meyer- Green. Once each

creature was released, it had to walk autonomously across the smooth surface for a meter and

then transition to the cobblestone pavement. Taking these design considerations into mind, we

worked to create a short, compact four bar robot. The next section details each of these design

choices more completely and illustrates the full build process our group went through to create

our robot.

3. Design Description

This section outlines the various design features of our velociraptor robot along with the rationale

for each decision. The pictures below illustrate the different features of our robot that are described in the

following pages. (figure 3.0a and 3.0b)

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Figure 3.0a A side view that features the 3 gear drive train, stabilizing weights, and parallelogram crank

mechanism.

Figure 3.0b a front view of our robot. You can see the added stabilizing weights, the battery back mounted on top of

the motor as well as the four cranks.

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3.1 Linkage, Leg, and Feet Design Our dinosaur relied on a four-bar parallelogram mechanism for movement. We chose this

linkage design because keeping the feet parallel to the ground during each moment of the

walking cycle was an important aspect of stability that we wished to achieve. Additionally, we

prioritized creating a dinosaur that would be able to reliably transverse all types of ground

surfaces and believed that this model would help us reach that goal. To begin the design of our

robot, our team’s first goal was to create a sturdy linkage that fit the output shaft of the Tamiya 6

Speed High Efficiency Gearbox on one end and insert into the legs of our dinosaur on the other.

We 3D printed this crank linkage because we wanted a material that was both easily

customizable and rigid. A CAD model of our crank is shown below in figure 3.1a.

Figure 3.1a A CAD model of our 3D printed crankshaft. One end fits over the motor output shaft while the other

connects directly to the leg of the dinosaur with shoulder screws. A hole was drilled in the long hub so that a pin

could be inserted between the crank and motor shaft.

A thick piece of wire fit through a small hole we had drilled in the crank and output shaft

and acted as a locking pin between the two. For the top axle, a similar pin mechanism was used

to secure the crank to the metal shaft. We added long hubs on either side of the crank to stabilize

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the entire connection and to limit side-to-side motion during walking. In the early design stage,

the length of the hubs and crankshaft were somewhat arbitrary. However, once we began to

optimize our robot to be as low-to-the ground and consolidated as possible, we created our

cranks to fit the narrowest body size possible with the limiting factor being our gearbox.

A pin mechanism secured our cranks to the rotating shafts while the other side connected

to the legs of the dinosaur with screw posts and duron bearings. Our robot’s legs were simple- a

single piece of laser cut duron with two holes 5 cm apart. Two cranks fit into each leg and

allowed it to rotate during the walking cycle. Although simple, these legs were designed to be

anatomically similar to that of a velociraptor (anitomicalanimals.org). Figure 3.1b shows the

outline of our leg while figure 3.1c shows the cut out duron equivalent.

Figure 3.1b illustration of our leg Figure 3.1c leg in fully assembled dinosaur

One of the most important features of our robot are its “U” shaped feet. These long,

skinny feet gave our dinosaur increased stability and helped center its balance during walking.

This design allowed the center of mass of the crawler to be directly above the center of the

polygon of support of each foot. One of the most difficult challenges in creating a bipedal robot

is designing feet that allow for balance on one foot. Our design was successful in allowing our

dinosaur to balance on one foot, contributing to its success on testing day. Figure 3.1d shows

how our robot looks balancing on one foot.

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Figure 3.1d dinosaur successfully balancing on one foot.

Our feet were cut out of duron, although this time, multiple layers were glued together to

increase surface area between the legs and feet. After multiple rounds of testing, our team

noticed that the connection between the feet and legs of the dinosaur began to break down due to

fatigue. In order to alleviate this problem, we fixed a square piece of duron at the intersection of

the foot and leg to prevent it from bending. (figure 3.1d)

figure 3.1e top down view of “U” shaped feet

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Figure 3.1f supporting duron strip was attached to the feet/ leg intersection to increase stability.

3.2 Kinematic Motion Design Goals One of the goals of this project was to create a dinosaur that walked in a similar fashion

as the living specimen. In order to learn more about the gait of a dinosaur, our group studied

several velociraptor walking simulations [Raptor] [T-Rex]. However, because our group chose to

prioritize stability with a four-bar parallelogram mechanism, our feet were limited to travel in a

semicircular trajectory. The gait of our dinosaur is illustrated in the 5 panels below.

Figure 3.2a dinosaur is about to begin walking figure 3.2b right foot is lifted, dinosaur is balanced on its left leg. Figure 3.2c

right foot is placed down, a step forward has been made.

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Figure 3.2d now the left leg is lifted, the dinosaur balances on its right leg figure 3.2e the left leg is placed down, another

step has been made.

Although our achieved gait was not perfectly biomimetic, our dinosaur was able to

successfully progress forward with a commanding stomp across both flat and cobbled ground

surfaces- fulfilling our teams’ design goals. Below shows the working model diagram of our

mechanism and the path of the coupler point (our feet). Further analysis of our gait are explained

later in this paper

Figure: 3.2f

Working Model representation of our parallelogram 4-bar linkage

3.3 Grashof Criteria and Gruebler’s Formula

Our four bar linkage with a parallelogram design meets the extreme case of Grashof Criteria. It is

a change-point mechanism linkage. This means that our linkage can sustain continuous motion.

Our system only has a single degree of freedom.

a is the longest bar = 4 cm

b is shortest bar = 2 cm

c and d are the intermediate‐length bars = 2 cm and 4 cm

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(2 cm + 4 cm) ? (2 cm + 4 cm)

(a+b) < (c+d) Grashof Mechanism

(a + b) > (c + d) Non‐Grashof Mechanism

(a + b) = (c + d) Extreme case: Change-Point Mechanism

In order to confirm that our linkage has only one degree of freedom, one can use the following

Gruebler’s Formula: DOF = 3(N-1)-2f1-2f2. In our 4-bar, there are 4 linkages (N=4). there are 4

type-1 joints (f1=4), and there are zero Type 2 joints. This expectedly outputs 1 degree of

freedom for our linkage system.

3.4 Drivetrain Design

In order to create our dinosaur’s rotating gait, we offset our crank pairs by 180 degrees

and chose to drive both the top and bottom axles. By driving the axles, we made sure that the

entire leg was moving at the same rate and that our parallelogram shape was maintained. We

used three ⅛” laser-cut acrylic gears to drive the top axles from the powered crank. We decided

to custom make these gears so that they would fit the profile of our design and mesh smoothly

when powered by the motor. The teeth on these gears were 40, 26 and 40 respectively. Proper

alignment of these gears was critical to the success of our dinosaur as any misalignment caused

the top axle to stop moving and led to jams in our parallelogram mechanism. We aided proper

alignment of these gears by creating a sturdy body supported by wooden dowels and tight

connections between our cranks, axles, and legs. A side view of our dinosaur is shown below

along with the three gears (figure 3.4a).

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Figure 3.4a Close up of our 3 gear drivetrain. The bottom 40 tooth gear transmitted power from the motor

shaft to the 26 tooth gear and finally to the top 40 tooth gear and axle.

3.5 Motor Gearbox Goals We used a Tamiya 6 Speed High Efficiency Gearbox equipped with a RE-260 motor to

power our dinosaur. This gearbox allowed us to pick and choose from a wide variety of gear

ratios: 11.6:1, 29.8:1, 76.5:1, 196.7:1, 505.9:1, and 1300.9:1. We initially chose to use the 76.5:1

ratio because we envisioned our dinosaur to be pretty compact and light. However, after talking

with the teaching team, we were advised to use a slightly higher ratio to give our dinosaur

enough torque to walk. We settled on the reducing ratio of 196.7:1. A schematic of our chosen

gear ratio is shown below.

Figure 3.5a

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We were slightly limited in the placement of our gearbox since our legs and attached cranks were

connected directly to the motor. Working with these constraints, we were able to place our

gearbox in the lower belly supported by our inner support dowels.

Figure 3.5b shows the motor placement in the underbelly of the dinosaur

One large problem that our team ran into during testing was failures in the gearbox. We

sheared the teeth off of the fifth gear (a blue 36/ 14 tooth gear) many times because when our

drive train mechanism failed (either the 3 laser cut gears got out of line, or our cranks shifted out

of phase, our legs would jam and stall the motor until it broke). This was an incredibly

frustrating problem for our group as each failure meant taking apart the entire robot, replacing

the broken gear and then reassembling the entire machine. We were finally able to overcome this

challenge by making a small change to our power supply. We went from powering our dinosaur

with 3 AA batteries to using only 2 and shorting the last connection. Our battery pack was placed

right above the gearbox to help keep our weight centered. A more detailed description of our

power and motor analysis are given later in this paper.

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Figure 3.5c shows the placement of the battery pack right on top of the gearbox

Figure 3.5d our shorted battery pack

3.6 Assembly Design In the early prototyping phase, the hardest part about creating and testing our dinosaur

was the constant assembly and disassembly processes. We relied on hot melt glue to hold our

parts together, but quickly found that it was messy and we usually ended up damaging our duron

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pieces. Once we had an idea of our design and the inner mechanics figured out, we recut and

printed entirely new parts (body, legs, feet, and cranks). To secure these pieces into place, we

sparingly used strong wood and super glues. Our final design was relatively easy to disassemble

as the dinosaur structure stayed rigid and secure while we could remove the legs and gearbox as

necessary.

3.7 Material and Adhesive Choice In terms of materials and adhesives, we decided to create the body, legs and feet of our

dinosaur out of Duron. The body did not require much structural support and we found that

Duron was easily cut-able and relatively light. While Duron was a good choice for our dinosaur

body, we could have chosen to use a stronger material for our feet and legs. As mentioned

earlier, after multiple rounds of testing, our wood glue- Duron connection between the feet and

legs began to fatigue and come apart. For a future build, we would consider 3D printing the legs

and feet of our dinosaur as one part to increase the structural rigidity between the two segments.

We chose to use 1/8” acrylic to create the gears in our 3 gear drive train because we were

looking for strong teeth and clean cuts. Our team had designed a previous project that used laser

cut gears with success so we opted to implement them again.

A steel rod was used as a top driven axle. Its rigidity helped our robot maintain steadiness

and consistency while walking. Wooden dowels were interspersed along the left and right sides

of the dinosaur’s body to add strength to the entire frame. They were cut to ensure a compact and

narrow final design.

3.8 Final Adjustments Near the end of our testing, our dinosaur functioned well on its back (no load) as well as

on smooth ground. However, once placed on concrete or carpet, it would attempt to walk and

end up falling backwards. In order to fix this problem of balance, we decided to add weights (2

batteries, 24 grams each and 2 bolts, 17 grams each for a total of 82 additional grams) to the tops

and insides of the legs. These added weights helped lower the center of gravity and we found that

these adjustments were the final touches we needed to finish our walking robot.

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Figure 3.8a shows the placement of our added weights. Each battery was 24g while each bolt weighed 17

grams

4. Analysis of Performance

4.1 Motor and Transmission Performance

After running our dinosaur at a nominal 4.5V (3 AA batteries), we realized the operating voltage

was too high for reasons discussed in section 3.5. We rolled back to a nominal 3V (2 AA

batteries) and found success. Batteries are rated at a certain voltage, but there is a noticeable

resistance in each battery that causes the actual voltage to be lower than the nominal voltage of

the battery. To find the actual operating voltage and current of the motor, we connected a

multimeter to our circuit in parallel for the voltage and in series for the current while our

dinosaur was walking. Thus, we found that our motor runs at a voltage of 2.627V and a mostly

constant current of 0.54A. Knowing the actual motor voltage, we can calculate the resistance of

both batteries. Using the equation:

Resistance of both batteries = (Nominal Voltage - Actual Voltage) / Operating Current

We can see that the resistance of the two batteries together is 0.6907 Ohm.

To begin our analysis of the locomotion of the dinosaur, we looked at how much power the

motor was giving to the legs. To characterize the motor, we found the three motor constants (R,

km, Tfriction) using the following equations:

V - iR - ⍵km = 0

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Tload = Tmotor - Tfriction

To find R, we look at the motor at stall. ⍵ = 0 at stall, so Equation 1 becomes R = istall/V. km

and Tfriction can be found by testing the motor under the no load condition. We can use Equation 1

to get km = (V-inlR)/⍵nl. At no load, Tload = 0, so Equation 2 becomes Tfriction = Tmotor = kminl.

Using the stall and torque values given for our motor at 3V tabulated by Pololu [Motor], we

determined the following motor constants:

R = 1.111 Ohm

km = 0.0029 Nm/A

Tfriction = 4.3175x10-4 Nm

Due to our parallelogram 4 bar linkage system, explained in detail in the section 3.2, the motor

operated at a near constant 0.54 A. We created plots (found in Appendix 4.1) to track the

operation of the motor under the characteristics of an input voltage of 2.627 V.

Figure 4.1a: Motor Efficiency vs. Current at 2.627 V

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Through this analysis, we found our motor to be operating at 60.95% efficiency as seen in Figure

4.1a, just below the maximum efficiency of 61.43%.

Given that a Duracell Coppertop battery’s capacity is 0.83 AH at a 1 A discharge rate,

[Discharge] we estimate our motor operating at 0.54 A to last for approximately 39 minutes. This

is significantly more time than the 19.5 seconds it would take our dinosaur to travel the whole

1.5m, allowing us many attempts before the batteries would fully discharge.

4.2 Gear Analysis

Our original design utilized 3 AA batteries to get more power out to the legs, but we ran into a

major problem with this choice. After assembling our dinosaur, we would test it to see if it

walked. When it walked properly, it moved about normally. As soon as something went wrong,

though, the dinosaur would fall, immediately stall, and we would hear the sharp snap of a gear

breaking. After reassembling and attempting several times, each with the same outcome of the

9th gear in the transmission shearing, we realized we needed to change our approach. After some

consultation between ourselves and instructors, we decided upon dropping down to 2 AA

batteries to slow things down and decrease stress on the gears. Gear Lewis stress is found using:

σLewis = (FtP/bJ)KmKoKv

Km, Ko, and Kv are factors that account for gear mounting quality, shocks, and gear speed,

making σLewis quite accurate. Our problem with the three battery setup was at stall. We found that

the maximum Lewis stress was on gear 9 at stall, confirming why it was that specific gear that

broke every time. With 3 batteries at stall, the Lewis stress on gear 9 was a massive 110 ksi, far

exceeding the 7 ksi tensile yield strength of ABS plastic [Thermoplastics] that we assumed to be

similar to the plastic used for the Tamiya gears. It was no wonder that the gear would break

instantly upon stalling.

Throttling down to 2 batteries solved two problems: it lessened the stress on the gears, but it also

caused the dinosaur legs to move slower, thus not causing them to jam up and stall the motor

when the dinosaur fell. With 2 batteries at our operating point, the maximum Lewis stress (once

again on gear 9) dropped down to 2.6 ksi, significantly below the 7 ksi limit.

4.3 Analysis of Power at Foot We were interested in finding the power output at the foot not only to see the average output

power over one step, but also to see how the power output varies in time over the step. The full

MATLAB code for this analysis is shown in Appendix 4.3.

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First, the angular velocity of the crank (ω2) is the angular velocity of the motor divided by the

gear ratio. Next, we analyzed the walking motion of the dinosaur on a force plate. By matching

the force plate data with slow-motion video, we found the initial crank angle with respect to

“ground” (the dinosaur body), which allowed us to coincide the force data in time with crank

motion in time. The original force plate data is shown below in Figure 4.3a, with the blue plot

representing the normal force (N) and the orange plot showing the tangential force (N) as a

function of time (ms).

Figure 4.3a: Normal and

tangential force on

dinosaur foot as a function

of time (ms)

As can be seen above, the walking motion consists of multiple trapezoid-shaped variations in

normal force. In order to analyze one step of the dinosaur, we zoomed in on one of the

trapezoids to generate Figure 4.3b below. The leftmost data represent the foot initially making

contact with the force plate, the relatively flat section represents the time at which the foot is in

contact with the ground, and the end represents when the foot elevates off of the ground.

Figure 4.3b:

Normal force

and

tangential

force as a

function of

time (ms) for

one step.

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After scaling this data so that the average force for a step is the weight of the dinosaur, we knew

the instantaneous force on the foot at each point in time. In order to calculate power at the foot,

we then needed the instantaneous velocity component. As mentioned earlier, in order to match

the velocity and force as a function of time, we needed to relate the crank angle as a function of

time. We used the following equation: 𝜃2 = ∫ 𝜔2𝑑𝑡′𝑡

0 and since the angular velocity of the crank

is constant over time, 𝜃2 = 𝜔2𝑡 + 𝜃2,𝑠𝑡𝑎𝑟𝑡.

We then needed to find the instantaneous velocity of the coupler point. The following method

was used instead of the graphical method examining the angle of the foot with time, since our

parallelogram 4-bar causes the feet to always be parallel to the floor. In order to find the

instantaneous coupler velocity, we first needed to introduce a fictitious linkage 7 with length r7 to

simplify calculations. This is shown below in Figure 4.3c.

Figure 4.3c. Diagram

showing naming

convention of the

linkages and angles.

Equations for

solving linkage

lengths and angles

[ME112 “Linkages

Introduction

Handout”]

From equation 4, we can isolate r7 to get the following equation:

𝑟7 = √𝑟12 + 𝑟2

2 − 2𝑟1 ∗ 𝑟2𝑐𝑜𝑠(𝜃2) . Similarly, 𝜓 = acos(𝑟7

2+𝑟42−𝑟3

2

2𝑟1𝑟2) and 𝛼 = acos(

𝑟72+𝑟1

2−𝑟22

2𝑟1𝑟7).

These equations allow us to calculate 𝜃3 and 𝜃4 using the following: 𝜃4 = 𝜋 − 𝛼 − 𝜓 and 𝜃3 =

atan2(r4 sin(𝜃4) − 𝑟2 sin(𝜃2) , r4 cos(𝜃4) − r2 cos(𝜃2) + r1). After this, we were able to

explicitly calculate 𝜔3 and 𝜔4. From ME112 “Mechanism Forces Power Transformations”

Handout, we have the following equations:

Isolating for 𝜔3 and 𝜔4, we get the following:

𝜔4 =𝑟2 cos(𝜃2)𝜔2+cot(𝜃3)∗(−𝑟2 sin(𝜃2)∗𝜔2)

− cot(𝜃3)∗𝑟4∗sin(𝜃4)−𝑟4cos(𝜃4) and 𝜔3 =

−𝑟2 sin(𝜃2)∗𝜔2+𝑟4 sin(𝜃4)∗𝜔4

𝑟3sin(𝜃3). Then, the

instantaneous velocity of the coupler in the frame of reference of the moving dinosaur is 𝑣𝑝2 =

𝜔3𝑟3. We were then able to plot the foot velocity as a function of crank angle, shown below in

Figure 4.3d.

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Figure 4.3d: Plot of speed of dinosaur foot as a function of crank angle over a portion of a step

Points 1 and 2 highlight the absolute extremes of this part of the step process. Point 1

corresponds to when the foot is at its maximum height and the coupler velocity (v=omega*r) is

in the same direction as the velocity of the dinosaur, causing a momentary spike in velocity

relative to the world. Importantly, this corresponds to a crank angle of about 270 degrees. From

slow-motion video footage of the dinosaur walking, this also corresponds to the moment in time

when the second foot begins to touch the ground. This causes a quick transition to state 2 in

which this second foot is moving at maximum speed in the direction opposing motion (bottom of

the coupler curve) which leads to a momentary drop in foot speed.

We proceeded to use trigonometry in order to decompose this velocity into its x and y

components, and this is shown below in Figure 4.3e for an arbitrary orientation of the linkages.

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Figure 4.3e: Trigonometry of instantaneous velocity of coupler (V2) in relation to linkages.

Using this angle φ, we can calculate the x and y components of velocity using the following

figure, Figure 4.3f.

Figure 4.3f. X and Y components of coupler velocity

We then used the following equations to calculate the components: cos(𝜑) =𝑉2𝑥

𝑉2 and thus 𝑉2𝑥 =

𝑉2cos(270° − 𝜃2). Similarly, 𝑉2𝑦= 𝑉2sin(270° − 𝜃2). Importantly, we added the walking

velocity of the dinosaur (0.077m/s) to get the final V2x in the world’s reference frame. It is useful

to note that while we could take this information and find a plot of x vs. theta2 and y vs. theta2,

this would simply tell us that each leg moves in a semicircular pattern every span of pi, and that

the combination of the two legs that are 180 degrees out of phase, leads to a circular pattern (as

we knew from our parallelogram four-bar linkage).

Finally, in order to find the output power for each data point of time and force, we took the inner

product [Fxi, Fyi]'*[Vxi, Vyi], where i represents each data point, vx and vy are called v2x and v2y

above respectively, and Fxi is the tangential force and Fyi is the normal force (both from the

force plate data). This gave the following plot, shown below in Figure 4.3g.

26

Figure 4.3g:

Power out of feet as

a function of 𝜃2

over a portion of

one step.

In this plot, we can again analyze what occurs at points 1 and 2 (same points as in the velocity

plot above). At state 1, as the velocity spikes and the second foot begins to make contact with

the ground, the power requirements drastically increase. Similarly, from state 1 to state 2, the

second foot achieves maximum negative velocity relative to the world while also incurring a

large force (since it is at the point in the cycle in which the foot is completely on the ground).

Then, the foot lifts off of the ground, momentarily providing negative power out which could be

recaptured if our foot had included a spring system.

While the power out does fluctuate a lot over the cycle, we can take the average power out to

find how much power the foot dissipates over one step. This Pout was calculated to be 0.4258.

First of all, this satisfies the condition that it is less than the power out of the motor, which was

0.8647. Second, this calculated output power at the feet should be less than an approximation of

power as 𝜇 ∗ 𝑤𝑒𝑖𝑔ℎ𝑡 ∗ 𝑣𝑥, where the weight is m*g (the mass of the dinosaur is 0.475kg) and 𝜇

is the coefficient of friction between duron and concrete (estimated as 0.62, the coefficient of

wood on concrete from http://www.engineeringtoolbox.com/friction-coefficients-d_778.html).

This approximation should give the maximum average power out over one step, and in fact, gave

Pout = 0.4803 which is expectedly higher than our calculated power. Lastly, one final

approximation can be given based on the torque out of the crank. This power approximation is

the following: Pout = 𝜔2 ∗ 𝑤𝑒𝑖𝑔ℎ𝑡 ∗ 𝑟2=0.5770 W, which is close to our calculated output

power.

4.4 Free Body Diagram In order to analyze the forces on the dinosaur, we began with a free body diagram of the full

dinosaur, shown below in Figure 4.4a.

27

Figure 4.4a:

Free body diagram of dinosaur

while on one leg.

Our design is relatively simple to analyze because the center of mass is over the one foot, which

is the only foot contacting the ground. Further, while the normal force and tangential force act

over the entire bottom surface of the foot, this diagram approximates the forces acting at a single

point in the middle of the foot. Assuming that the dinosaur achieves constant velocity, the

following equations will pertain: ∑𝐹𝑥 = 𝐹𝑝 − 𝐹𝑇 = 0 and thus 𝐹𝑝 = 𝐹𝑇 where Fp is the

propulsive force of the dinosaur from the motor and FT is the tangential force at the ground

(friction), which was determined as a function of crank angle in the above section. Similarly,

force balance in the y direction gives the following: ∑𝐹𝑦 = 𝐹𝑁 − 𝐹𝐺 = 0 and thus 𝐹𝑁 = 𝐹𝐺,

where FG is the weight of the dinosaur and FN is the normal force from the ground, found using

the force plate.

5. Game Day Performance On game day, our dinosaur successfully completed its walking challenges. It was able to

transverse over 1m of flat surface on Meyer-Green as well as amble across a cobblestoned path.

Our drive train system did not get stuck and our cranks reliably turned in their parallel paths. Our

team was extremely excited to see our robot stomp around the ground and we were pleased with

it reliable and stable performance. Since our dinosaur had a slight right-curving gait, it took several tries to reach the one

meter mark. However, we proceeded to tighten the screw posts on the right leg and it then

walked straight and experienced no problems on both the smooth and cobblestone surfaces.

28

6. Redesign For a production-ready prototype we would make several changes to our dinosaur.

Fundamentally, we would want a more permanent design. Due to significant amounts of trial and

error testing we were forced to place our motor and battery pack in easy to extract places and did

not secure them in such a way that would make them ready for a toy manufacturer to test. We

would start with finding a secure location in the body of the robot that would house our driving

motor and the two batteries we would want to use to power the device. Secondly, we would

move the gears that drive our top axle to the inside of our robot. When designing our dinosaur

we made the design choice to place our axle driving gears outside the body of our dinosaur. We

felt this added an element of uniqueness to our dinosaur. Unfortunately, this meant that when

altering any of our linkages the gears would be disrupted and had to be carefully maintained.

Figure 6.0a Figure 6.0b

The most significant changes we would have made would be in our linkage to motor

connection. We drilled holes through our 3D printed crankshaft and rocker. In the bottom axle

we put a pin directly through that hole and through the pre-drilled hole in the motor crankshaft.

We did a similar thing for our top axle, but had to self-drill a hole through the 6 mm mild steel

rod we chose to use as an axle. We used a very small steel post that we cut from the end of a

bungee cord. We wrapped a small layer of tape around the steel post so that we could squeeze it

through the holes in the link and the axle and then allow the tape to expand inside the holes

resulting in a tighter fit. However, even with this system we saw about 5 degrees of play in each

link. This was a tight enough fit to allow for a stable and consistent walk, but resulted in an

unusual gait at times and certainly contributed to power losses. By incorporating holes for

pinning our linkages to our axles that were predesigned and pre-made through 3D printing we

could have created a much tighter linkage fit that did not allow any play between linkage set up.

This would create a much more consistent gait for our dinosaur and make it much more effective

in its walking ability.

29

It was completely necessary to create a pin system to join the crankshaft our motor to the

crank linkage we 3D printed, but for the top axle it would have been possible to 3D print the

entire axle and dual linkage system so that it was one solid piece. This would have been very

difficult for our purposes because we were continually taking apart and rebuilding our dinosaur,

but in a situation where we were trying to manufacture a toy, a solid linkage system would be

excellent. It would completely destroy any inconsistencies in orientation of our linkages and

largely eradicate power losses caused by our top axle.

Also, we could have slightly improved the weight balance and the way we accomplished

weight balance in our dinosaur. In order to give our dinosaur appropriate balance and keep it

from falling backwards we were forced to add weight to the feet of our dinosaur. This ended up

working well, but ultimately looked thrown together and was not the best way to balance our

dinosaur. Given more time we could have redesigned the legs to hold the batteries from our

battery pack. This would have allowed us to free room in the body of our dinosaur that was taken

up by our battery pack and also distribute weight lower in the body of the robot.

Figure 6.0c

Finally, we could have improved the center of mass of our dinosaur. We achieved a

relatively low center of mass by adding weight to the feet of our dinosaur and by placing our

motor and battery back as low in the body as possible. We could have cut out a few centimeters

of height in the legs and designed a system to hold our battery pack and motor slightly lower.

However, I feel there are only small gains to be made in this situation as we strived at almost

every opportunity we were given to lower the center of mass of our dinosaur.

Our dinosaur would have been improved greatly by finalizing the design of our robot and

creating more permanent structures that we would not have to be constantly adapting, improving

our linkage and pinning system, redistributing weight and lowering the center of gravity. Given

more time and the ability to test and iterate through designs until we had a solid and permanent

design, we would have made all of these modifications.

30

7. Conclusion

The final iteration of our walking velociraptor walked consistently on concrete and over

the cobblestone at the testing location with somewhat realistic bipedal motion. It moved at about

7.7 cm/s, well above the required 3.0 cm/s threshold. Our team is pleased with the performance

of our final velociraptor design.

We think with future iterations we could have created a smoother and more biologically

accurate walking pattern for our velociraptor. We would improve our pinning system to develop

a smoother gait and change our simple, but reliable four bar linkage system to a more complex

linkage mechanism that would more accurately replicate the gait of a velociraptor.

We also would have created more permanent battery pack and engine mounts. We also

would have worked to conceal the stability weights we added to the feet. These helped lower the

center of gravity of our dinosaur and were a necessary addition. However, the added weights

were not part of our design aesthetic and that could have been improved.

The combination of a more permanent and more rigid body with a more complex and

robust linkage system would results in a longer lasting, more consistent and more biologically

realistic design. Yet, we feel that our velociraptor performed well in most aspects of performance

criteria and we were pleased to complete two of our three performance tests with perfect scores.

31

8. Bibliography

Discharge Tests "Discharge Tests of AA Batteries, Alkaline and NiMH." Discharge Tests of

Alkaline AA Batteries. N.p., n.d. Web. 18 Mar. 2017.

"Friction and Friction Coefficients." The Engineering ToolBox. N.p., n.d. Web. 22 Mar. 2017.

<http://www.engineeringtoolbox.com/friction-coefficients-d_778.html>.

ME112 “Linkages Introduction Handout”

ME112 “Mechanism Forces Power Transformations” Handout

Motor Specs "Pololu - Tamiya 72005 6-Speed Gearbox Kit." Pololu Robotics & Electronics.

N.p., n.d. Web.

Picture of Robot. Digital image. I.ytimg.com. N.p., n.d. Web. 18 Mar. 2017.

<https://i.ytimg.com/vi/kMnu_C7wMsg/maxresdefault.jpg>. 18 Mar. 2017.

Picture of Velociraptor. Digital image. Extinctanimals.org. N.p., n.d. Web. 18 Mar. 2017.

<http://www.extinctanimals.org/wp-content/uploads/2015/08/Pictures-of-Velociraptors.jp

g>.

Robot Walking. N.p., n.d. Web. 21 Mar. 2017.

<https://www.youtube.com/watch?v=4D96edLti7U>.

Thermoplastics Properties "Thermoplastics Physical Properties." Engineering Toolbox. N.p.,

n.d. Web. 18 Mar. 2017.

T-Rex Walking. N.p., n.d. Web. 21 Mar. 2017.

<https://www.youtube.com/watch?v=FR4AQetma2o>.

https://i.ytimg.com/vi/kMnu_C7wMsg/maxresdefault.jpg

http://www.extinctanimals.org/wp-content/uploads/2015/08/Pictures-of-Velociraptors.jp

https://www.youtube.com/watch?v=FR4AQetma2o

32

9. Appendices

Appendix 4.1

Motor Analysis Matlab %constants

RPM_to_rad_s = 2.*pi./60;

%Numbers for motor constants

V=3;

i_nl = 0.15

nl_rpm = 9400

i_s = 2.7

%Numbers for operating point

voltage = 2.627; %V

noLoadCurrent = 0.13; %A

noLoadRPM = 8250; %rpm

stallCurrent = 2.63; %A

%Motor Resistance

resistance = V./i_s

%Motor Constant

noLoadSpeed = noLoadRPM.*RPM_to_rad_s

wnl = nl_rpm.*RPM_to_rad_s;

k = (V - i_nl.*resistance)./wnl

frictionTorque = k.*i_nl

stallTorque = k.*i_s - frictionTorque

current = linspace(stallCurrent,noLoadCurrent);

speed = linspace(0,noLoadSpeed);

torque = linspace(stallTorque,0);

power = torque.*speed;

idealPower = voltage.*current;

efficiency = power./idealPower;

%torque vs speed plot

figure

plot(speed,torque)

title('Torque vs. Speed')

xlabel('Speed (rad/s)')

ylabel('Torque (Nm)')

hold on;

plot(722.3,.001197,'r*')

text(722.3,.001197,' (793.5,.001197)')

33

%torque vs current plot

figure

plot(current,torque)

title('Torque vs. Current')

xlabel('Current (A)')

ylabel('Torque (Nm)')

hold on;

plot(0.54,.001197,'r*')

text(0.54,.001197,' (0.54,0.001197)')

%power vs current plot

figure

plot(current,power)

title('Power vs. Current')


ylabel('Power (W)')

hold on;

plot(0.54,.8647,'r*')

text(0.54,.8647,' (0.54,.8647)')

%efficiency vs current plot

figure

plot(current,efficiency)

title('Efficiency vs. Current')


ylabel('Efficiency')

hold on;

plot(0.54,.6095,'r*')

text(0.54,.6095,' (0.54,.6095)')

Motor Analysis Plots

34

Appendix 4.2

Gear Transmission Matlab clear all; close all; clc;

%Handy constants, conversion factors

m_to_in = 39.37; %Meters to inches

N_to_lbf = 0.2248; %Newtons to pounds force

Nm_to_inlbf = N_to_lbf*m_to_in; %Nm to inchpounds torque

inprsec_ftprmin = 60.0/12; %inches/second to feet/minute

rpm_to_radps = 2*pi/60; %rotations/minute to radians/second

deg_to_rad = pi/180; %degrees to radians

watt_to_hp = 745.7; %Watts to horsepower

%Input to the transmission at the highest current (.54 A)

Tinput = .001197.*Nm_to_inlbf; %inch lbf

Rpm_in = 820; %rpm of input

%for all gears:

Pressure_angle = 20; %degrees pressure angle of gears

phi = deg_to_rad*Pressure_angle;

Pitch = 52.06;

FaceWidth = .09409;

%Gear teeth

N1 = 8;

N2 = 36;

N3 = 14;

N4 = 36;

N5 = 14;

N6 = 36;

N7 = 14;

35

N8 = 36;

N9 = 14;

N10 = 36;

%% FIRST Stage

Jlewis1_2 = 0.23; % Look up from chart, changes with N1, N2


%% SECOND Stage



%% THIRD Stage



%% Fourth Stage



%% Fifth Stage



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Calculated Values Start Here

%1. total speed ratio and torque, output speed, etc.

Omega_in = Rpm_in*rpm_to_radps; %input omega in radians/second

Watts_in = (Tinput/Nm_to_inlbf)*Omega_in; %input power in watts

ratio = (N2/N1)*(N4/N3)*(N6/N5)*(N8/N7)*(N10/N9)

Torque_out = Tinput*ratio

Omega_out = Omega_in/ratio;

cycles_per_second = (Rpm_in/ratio)/60

%2. Contact Ratios (CR) on each stage

% Want > 1.5 if possible.

% Stage 1 contact ratio

rp1 = 0.5*N1/Pitch;

rp2 = 0.5*N2/Pitch;

C = rp1+rp2;

Pbase = pi * cos(phi)/Pitch;

ra1 = 0.5*(N1+2)/Pitch;

ra2 = 0.5*(N2+2)/Pitch;

rb1 = rp1*cos(phi);

rb2 = rp2*cos(phi);

numerator = sqrt(ra1^2-rb1^2) + sqrt(ra2^2-rb2^2) - C*sin(phi);

ContactRatio1 = numerator/Pbase


36

rp3 = 0.5*N3/Pitch;

rp4 = 0.5*N4/Pitch;

C = rp3+rp4;


ra3 = 0.5*(N3+2)/Pitch;

ra4 = 0.5*(N4+2)/Pitch;

rb3 = rp3*cos(phi);

rb4 = rp4*cos(phi);




rp5 = 0.5*N5/Pitch;

rp6 = 0.5*N6/Pitch;

C = rp5+rp6;


ra5 = 0.5*(N5+2)/Pitch;

ra6 = 0.5*(N6+2)/Pitch;

rb5 = rp5*cos(phi);

rb6 = rp6*cos(phi);




rp7 = 0.5*N7/Pitch;

rp8 = 0.5*N8/Pitch;

C = rp7+rp8;


ra7 = 0.5*(N7+2)/Pitch;

ra8 = 0.5*(N8+2)/Pitch;

rb7 = rp7*cos(phi);

rb8 = rp8*cos(phi);


rp9 = 0.5*N9/Pitch;

rp10 = 0.5*N10/Pitch;

C = rp9+rp10;


ra9 = 0.5*(N9+2)/Pitch;

ra10 = 0.5*(N10+2)/Pitch;

rb9 = rp9*cos(phi);

rb10 = rp10*cos(phi);



%3. Lewis bending stresses

37

% First get tangential forces on each gear from torque

% Stage1

Ftan1 = Tinput/rp1; % Tangential tooth force in lbf

Km = 1.3; %from gear spreadsheet

Ko = 1.5; %from gear spreadsheet

%calculate Kv

v1 = rp1*(1/m_to_in)*Rpm_in*2*pi/60; %m/s

Qv = 9; %from gear spreadsheet

B = 0.25*(12-Qv)^(2/3);

A = 50+56*(1-B);

Kv1 = ((A+sqrt(200*v1))/A)^B;


Kv2 = ((A+sqrt(200*v2))/A)^B;

v3 = rp3*(1/m_to_in)*Omega_out; %m/s

Kv3 = ((A+sqrt(200*v3))/A)^B;


Kv4 = ((A+sqrt(200*v4))/A)^B;


Kv5 = ((A+sqrt(200*v5))/A)^B;


Kv6 = ((A+sqrt(200*v6))/A)^B;


Kv7 = ((A+sqrt(200*v7))/A)^B;


Kv8 = ((A+sqrt(200*v8))/A)^B;


Kv9 = ((A+sqrt(200*v9))/A)^B;


Kv10 = ((A+sqrt(200*v10))/A)^B;

SigmaLewis1 = (Ftan1*Pitch)*Km*Ko*Kv1/(FaceWidth*Jlewis1_2)


% Stage2

Ftan2 = Ftan1*(rp2/rp3); % Tangential force, stage 2



% Stage3

Ftan3 = Ftan2*(rp4/rp5);



% Stage4

Ftan4= Ftan3*(rp6/rp7); % Tangential force, stage 4

38



% Stage5

Ftan5= Ftan4*(rp8/rp9); % Tangential force, stage 5



Sigma_allow = 7000 % [Thermoplastics]

%For each gear compare with corresponding SigmaLewis stresses above

Appendix 4.3

clear all; close all; clc;

RPM_to_rad_s = 2.*pi./60;

w_motor = 11641; %rpm

motor_speed = w_motor*RPM_to_rad_s;

GR = 196.7;

w_crank = motor_speed/GR;

w2 = w_crank;

m_dino =0.475; %kg

g = 9.8; %m/s^2

weight = m_dino*g;

%read in data from force plate

%from matching up slow motion video to force plate data, theta 2 is

%~210degrees when the LED turns on for the third time, so start with this

%data point. Take data during one stride

theta2_start = 210*(pi/180); %radians

file = 'force plate data 3 columns better.xlsx';

time = xlsread(file,'A:A'); %upload from force plate data

time = 0.001*time; %s

time = time-time(1); %so time starts at 0s

time_max = time(length(time));

Fn = xlsread(file,'B:B'); %normal force [N]

Fn_step = xlsread(file,'B12:B80'); %entirety of foot on ground

Ft = xlsread(file,'C:C'); %shear/tangential force [N]

Ft_step = xlsread(file,'C12:C80'); %entirety of foot on ground

time_step = xlsread(file,'A12:A80'); %time over step

time_step = 0.001*time_step; %s

time_step = time_step-time(1);

Fn_step_avg = mean(Fn_step);

scale_factor = weight/Fn_step_avg;

Fn_step_scaled = Fn_step*scale_factor;

Ft_step_scaled = Ft_step*scale_factor;

39

r1 = 0.06; %ground [m]

r2 = 0.02; %crank

r3 = 0.05; %coupler

r4 = 0.02; %rocker

theta2 = w2*time +theta2_start; %rad

theta2_step = theta2(12:80);

r7 = sqrt((r1^2)+(r2^2)-2*r1*r2*cos(theta2_step));

input = ((r7.*r7)+(r4^2)-(r3^2))*(2*r1*r2)^-1;

psi = real(acos(input));

alpha = acos(((r7.^2)+(r1^2)-(r2^2))./(2*r1*r7));

theta4 = pi-alpha-psi;

theta3 = atan2(r4*sin(theta4)-r2*sin(theta2_step),r4*cos(theta4)-

r2*cos(theta2_step)+r1);

v_dino = 0.077; %measured velocity of crawler [m/s]

left = (1./(-cot(theta3).*r4.*sin(theta4)-r4.*cos(theta4)));

right1 = r2.*cos(theta2_step).*w2;

right2 = -cot(theta3).*r2.*sin(theta2_step).*w2;

right = (right1)+(right2);

w4 = left.*right;

w3 = (1./(r3.*sin(theta3))).*((-

r2.*sin(theta2_step).*w2)+(r4.*sin(theta4).*w4));

vp2 = w3*r3; %instantaneous coupler velocity in frame of dino

%plot speed of dino as a function of crank angle

v_world = v_dino+vp2;

figure

plot(theta2_step,v_world);

title('Speed of Dinosaur Foot as a Function of Crank Angle');

xlabel('\Theta_2 [rad]');

ylabel('Speed of Dinosaur Foot(in world reference frame) [m/s] ');

%from trigonometry

vp2x = (vp2.*cos((3*pi/2)-theta2_step));

vp2x = v_dino+vp2x; %subtract dino velocity to x component but not y for

world reference frame

vp2y = (vp2.*sin((3*pi/2)-theta2_step));

%calculate power

Pwr_out = zeros(69, 1);

for i = 1:1:69

Pwr_out(i) =[Fn_step_scaled(i),Ft_step_scaled(i)]*[vp2x(i);vp2y(i)];

%inner product

end

%compare to approximation

vx = mean(vp2x);

40

mu = 0.62; %duron (wood) on ground (concrete) from engineering toolbox

website

Pwr_approx = mu*weight*vx

feet_factor = 0.5; %from videos, other foot on ground most of time first foot

is

avg_pwr_out = mean(Pwr_out)*feet_factor

pwr_approx_2 = weight*w2*r2

figure

plot(theta2_step, feet_factor*Pwr_out);

hold on;

title('Power out of Feet over part of Step as a Function of Crank Angle');

xlabel('\Theta_2 [rad]');

ylabel('Power Out [W]');

set(gcf,'color','w');

Documents

Velociraptor Final Design Report - zacharyjblum.com · Our final velociraptor was 19 cm tall and powered by two AA batteries that provided our motor with 2.627 Volts and allowed 0.54