University of California at Berkeley

Mechatronics Design

ME 102B

SphOrb: Design Process forGyro Propulsion System

Group 10

Authors:Aaron WienkersBen HightowerSina BigdeliMauricio BaioccoShengji Yang

Supervisor:Dr. Liwei Lin

May 10, 2013

Abstract

A sphere is defined as “a round solid surface, with every point on its surface equidis-tant from its center.” In a robotics sense, a spherical structure can, without constraints,rotate in any direction and is stable in all positions. Due to the symmetrical and self-contained features of a sphere, this robot provides a safe, minimalist, and friendlylook with no sharp corners, making it ideal in family settings. As an entirely enclosedsphere protects electronics from the elements, its form is optimal for other applications,such as situations too dangerous or small for human beings to get involved with, suchas burning structures, clogged pipes, and gutters. As such, a maneuverable spherewith the capability to traverse remote terrain and provide the operator with informa-tion about the spheres surroundings would see great demand in a range of fields andapplications.

The SphOrb aims to fulfill this niche. It is a self-contained sphere that stores andtransfers momentum through the use of dual counter-rotating gyros. Since the mo-mentum is stored inside, the geometry and traction of the outside terrain has minimalimpact on the maneuverability of the sphere. In order to provide the optimal maneu-verability, high speed motors must be used to spin the gyros, while a high torque servowas used to apply the torque to the gyros to facilitate turning and motion. A complexcontrol scheme was implemented in the microcontroller to adjust for feedback and noisefrom the sensors, pendulum, and environment. The amalgamation of these tasks hasproduce a product and exceeds the needs of the customers.

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Contents

1 Introduction 4

2 Specifications 42.1 Functional Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Engineering Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Quality Function Deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Concept Generation 53.1 Concept 1: Internal Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Concept 2: Gyro Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Concept 3: Pendulum Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Concept Selection 8

5 Parameter Analysis 95.1 Mechanical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.1.1 Sphere Size and Obstacle Clearance . . . . . . . . . . . . . . . . . . . 95.1.2 Gyro Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.1.3 Gyro Housing Clearance . . . . . . . . . . . . . . . . . . . . . . . . . . 115.1.4 Axle Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.1.5 Servo Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.1.6 Motor Mount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.1.7 Gyro Shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.2 Electrical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2.1 Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2.2 Battery Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Final Design 176.1 Mechanical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.2 Software Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.2.1 Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2.2 Program Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 20

7 Plan of Action 227.1 RASIC Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2 Gantt Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 Problem Analysis 23

9 Test Results 25

10 Future Improvements 2710.1 Bearing Support for Gyros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.2 Mechanical Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.3 Servo Housing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.4 Control Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11 Conclusion 29

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List of Figures

1 The Quality Function Deployment (QFD) Chart for SphOrb . . . . . . . . . 62 Sketches of Concept 1 Brainstorming. . . . . . . . . . . . . . . . . . . . . . . 73 Sketches of Concept 2 Brainstorming. . . . . . . . . . . . . . . . . . . . . . . 74 Sketches of Concept 3 Brainstorming. . . . . . . . . . . . . . . . . . . . . . . 85 Pughs Matrix for Gyro Propulsion. . . . . . . . . . . . . . . . . . . . . . . . . 86 Plot of Necessary Velocity to Clear an Obstacle . . . . . . . . . . . . . . . . . 107 Moment of Inertia for Gyro . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Analytical Simulations to determine gyro yielding and deflection conditions. 129 Shaft Deflection Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1310 The complete Circuit Diagram of the Electrical System. . . . . . . . . . . . . 1611 Gyro CAD Drawing and Force Diagram . . . . . . . . . . . . . . . . . . . . . 1712 Detailed gyroscope assembly view. . . . . . . . . . . . . . . . . . . . . . . . . 1813 Tabulated Parts List and Costs . . . . . . . . . . . . . . . . . . . . . . . . . . 1914 RASIC Chart for the SphOrb Team Members . . . . . . . . . . . . . . . . . . 2415 Gantt Chart progress up to April 27th, 2013. . . . . . . . . . . . . . . . . . . 2516 The Free, Undamped Response compared to the Forced Feedback . . . . . . . 2617 SphOrb demonstration on May 3rd, 2013. . . . . . . . . . . . . . . . . . . . . 2718 The Final Mechanical Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2819 The Future SphOrb Final Design with intended Shell. . . . . . . . . . . . . . 29

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1 Introduction

The SphOrb takes full advantage of the spheres geometry. The SphOrb is a self-containedsphere that is controlled via remote controller or under autonomous control, with the op-tional addition of various sensory attachments and augmentations inside the clear housingsuch as a camera or global positioning unit. Two high powered gyros are utilized to movethe ball. A torque is applied to the gyros using a servo, resulting in gyroscopic precession,thus changing the spheres center of gravity. By manipulating the placement of the centerof gravity of the sphere with respect to the contact surface, movement and control can beachieved. Under remote control, the sphere is controlled in real time, able to move in anydirection specified by the controller; under autonomous control the sphere will attain feed-back from an infrared sensor to maneuver within its environment. This prototype should beable to perform under typical household conditions, whereas future models could be robustall-terrain robots with the capability of reconnaissance and geolocation mapping.

2 Specifications

The SphOrb is 13” in outer diameter, with approximately 12.5” available on the insidediameter for housing the gyros and motors. The mechanism is an axle going from pole topole through the sphere, and the contact surface is the equator of the sphere. Detailedspecifications are evidenced below:

2.1 Functional Specifications

• Mechanism must be a sphere that drives itself with most directions and various speeds

• Mechanism must be driven by the precession from the gyros spinning within the shell

• Mechanism must be controlled wirelessly and with no externally wired connection

• Mechanism must be powered by a rechargeable battery

• Mechanism must be able to drive on regular incline and climb a single step

• Mechanism must be able to get out from narrow or slightly inhibited situations

• Mechanism must allow the user to see certain processes in action

• Mechanism must be remotely controlled through a simple to use RC interface

• Mechanism must be easily portable

• Mechanism must be safe for operation around humans

2.2 Engineering Specifications

• Mechanism must be able to drive at 3-6 ft/s

• Mechanism must be able to operate for 20 minutes continuously

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• Mechanism must be able to climb up a single 3 inch high step

• Mechanism will use RC as command input

• Mechanism must fit in a space 13x13x13 inches

• Materials for the mechanism will include polycarbonate, aluminum, and machine steel

2.3 Quality Function Deployment

An assessment of customer needs and engineering specifications was accomplished throughthe use of a Qualify Function Deployment chart. The main goals are maneuverability, dura-bility, movement on an incline, inexpensive, movement over different terrains, and speed.These customer needs are translated into engineering specifications through high poweredmotors for the gyros, a tough outer shell, a fracture resistant outer shell, using fewer customparts, using high powered servos to turn, and having frictionless bearings. After weighingthe different goals accordingly, the most important engineering specification found is theneed for high powered motors for the gyros, as that would greatly affect maneuverabilityand ability to maneuver over different terrain.

There are not many competitors for SphOrb; the closest commercially available match isa product called “Sphero.” Sphero is also a remote controlled sphere. However, that iswhere the similarities between SphOrb and Sphero end. Sphero is smaller, at 3 inches indiameter. As such, its mobility is very limited compared to SphOrb. Similarly, Sphero doesnot have the option for expadability like SphOrb does. While Sphero is intended as a toyfor children, SphOrb can be utilized in the field for reconaissance and scouting missions.Finally, the mode of propulsion for SphOrb allow for much greater accelerations comparedto Sphero. The results of the QFD analysis are summarized in the figure below.

3 Concept Generation

Through the course of coming up with the SphOrb idea, many other concepts were createdand discarded. Each concept brought something different to the table, be it through thephysics involved or the difficulty in fabricating the prototype. The three major conceptsdescribe different ways to automate the movement of a spherical vehicle either through theuse of an inner wheeled vehicle, an axle and pendulum, or the use of dual gyros. In the end,the gyros were chosen due to the novelty and ingenuity of the design in comparison to theothers, which have been done numerous times previously.

The design team conceptualized designs through brainstorming. The following idea gener-ators were utilized during the brainstorming process:

i Combine: The product combines movement speed with precision control.

ii Reverse or Rearrange: The augmentations on the axle can be reversible so forward isbackwards and vise versa.

iii Adapt: An enclosed sphere makes the movement less dependent upon outside obstaclesbecause the momentum is stored inside the sphere.

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Figure 1: The Quality Function Deployment (QFD) Chart for SphOrb

iv Substitute: Wheels are substituted for the sphere so it is omnidirectional and moremaneuverable.

v Eliminate or Minimize: A universal augmentation port is not necessary and makes theproduct bulkier.

vi Modify or Magnify: The sphere should be able to go over a variety of terrain, so thediameter of the sphere should be large compared to intended obstacles.

These ideas were used to generate the following three possible concepts:

3.1 Concept 1: Internal Vehicle

This concept uses wheels in the base of the sphere to make the robot move. Counter-rotationof the wheels will allow the sphere to turn, while rotation in the same direction will allowthe sphere to move forward or backwards. The apparatus will be forced into the spherethrough the use of a spring, which will double as a mount for possible extensions.

Augmentations will be added through the use of magnets. A bearing with a magnet willbe on the top inside and top outside of the sphere, yielding a mount for augmentations likecameras. Other auxiliary sensors and tools can be put on as well in the future.

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Figure 2: Sketches of Concept 1 Brainstorming.

3.2 Concept 2: Gyro Propulsion

This concept uses the idea of gyroscopic precession in order to shift the center of mass ofthe sphere to move it. When a torque is applied to the gyros, the resultant torque that thesystem feels will be orthogonal to the original applied torque. A pair of servos will applythe torque to the gyros. The gyros will spin in opposite directions, and turning will beaccomplished through applying torque synchronously, while equal and opposite use of theservos will yield forward and backwards movement, and therefore accelerations.

Figure 3: Sketches of Concept 2 Brainstorming.

Sensors and augmentations can be added to either side of the axle, as that is the only partof the spherical shell that does not move when the sphere moves.

3.3 Concept 3: Pendulum Drive

This concept uses a pendulum to shift the center of mass of the sphere to get it to turn.Shifting the pendulum to the right would cause the sphere to turn to the right, and vice

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Figure 4: Sketches of Concept 3 Brainstorming.

versa. The main axle of the sphere would be powered by dual in line electric motors in orderto have the sphere move forwards and backwards.

4 Concept Selection

Multiple different ideas were generated as to how this design could accomplish all of theplanned features. Pughs method was utilized to whittle down these ideas, specifically theangular momentum storage system.

Figure 5: Pughs Matrix for Gyro Propulsion.

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As evidenced by Table 5, the best design for the internal energy storage is the dual gyrostoring and transferring angular momentum. The criteria used are as follows:

(a) Manufacturability: The part must be easy to manufacture. It should be DFMA.

(b) Size Constraint: There are certain size considerations with this aspect of the design.The momentum storage system should be thin such that it does not take up much spaceinside the sphere.

(c) Stability: The momentum storage system is supplying the momentum necessary to movethe sphere, so it should be stable. Additionally, oscillations inherent to pendulums willbe damped by a feedback control.

(d) Aesthetic: The entire apparatus should be pleasing to the eye.

(e) Cost: The momentum storage system should be affordable.

(f) Ease of Use: The momentum storage system should not hinder the movement of thesphere, and it should be easy to control.

5 Parameter Analysis

5.1 Mechanical Design

5.1.1 Sphere Size and Obstacle Clearance

The overall sphere size was limited by the availability of pre-made hollow and translucentspheres. The largest one found that could be manipulated easily was a hamster ball witha 13 inch outer diameter. However, even at this size, the internal mechanism is unable tofit, so a custom acrylic sphere should be obtained for future models. This sphere with a13 inch outer diameter is used in the following calculations as a lower bound estimate ofthe obstacle clearance, as future spheres would yield better performance. This large radiusallows for excellent obstacle clearance at a range of velocities. For a sphere of radius r,mass m, radius of gyration k, initial velocity v1, final velocity v2, and obstacle height h, theequations of motion reduce to the following:

Before impact,

H1 =mk2v1

r+mv1(r − h)

After impact,

H2 =mk2v2

r+mv2r

but for an instantaneous impact, ∆H = 0

v2 = v1

(1− rh

k2 + r2

)Conservation of Energy dictates that ∆KE +∆PE = 0 so that,

−1

2m(k2 + r2)

v22r2

+mgh = 0

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which then becomes,

v1 =r

k2 + r2 − rh

√2gh(k2 + r2)

0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

14Velocity Necessary for Obstacle Clearance

Vel

ocity

(fe

et/s

ec)

Obstacle Height (inches)

Figure 6: Plot of Necessary Velocity to Clear an Obstacle

As evidenced on the graph, in order to clear a 3 inch tall obstacle, a velocity of 4.3 feet/secondis required, which is well within the engineering specifications required. Additionally, at themaximum projected speed, the sphere is able to clear an obstacle of height 4.5 inches. Thisequation is less valid for obstacle heights that approach the radius of the sphere because thegeometry of the impact of the sphere and obstacle would not allow for the movement of thecenter of gravity of the sphere. Thus, the maximum obstacle height clearance of 4.5 inchesis well within the envelope of attainability.

5.1.2 Gyro Shape

All gyroscopes are cylindrical in shape due to the rotational symmetry about the verticalaxis. However, a hollow tube was found to be the ideal shape for the gyroscope due tothe increased moment of inertia. The following equations show the rationale for using ahollow tube, where I is the moment of inertia of the cylinder, Ω is the angular velocity, mis the mass, r1 is the outer radius, and r2 is the inner radius (applicable only to the hollowtube):

KE =1

2IΩ2

For a solid shaft design,

I =1

2mr21

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For the hollow tube gyro design,

I =1

2m(r21 + r22)

Thus, for any given fixed outer radius, the hollow tube would allow a reduction in mass whilestill retaining a high moment of inertia, yielding a high kinetic energy. A light gyroscopeis prudent because the servos must be powerful enough to apply the torque to the spinninggyroscope. Additionally, as total ball mass increases, linear acceleration decreases given aconstant servo torque input. A range of inner radiuses is used to generate the followingfigure and determine the optimal inner radius of the gyroscope.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

1

2

3

4

5

6

7

8Moment of Inertia as Function of Wall Thickness

Inner radius (inches)

Mom

ent o

f Ine

rtia

(lb

in2 )

Figure 7: Plot of the moment of inertia of the gyroscope as a function of the inner radius.The outer radius is a constant 1.8 inches.

From the plot and design considerations, the optimal inner radius was determined to be 1.5inches. This retains over half of the maximum moment of inertia possible while reducingthe weight of the gyroscope dramatically, thus maximizing the potential acceleration forconstant servo torques.

5.1.3 Gyro Housing Clearance

There must exist some clearance between the gyroscope and the external housing in orderto ensure that the parts do not grind and destroy each other. However, this clearancewill be diminished due to the rotation of the gyroscope: the centrifugal force acting onthe gyroscope will cause the edges to bow outwards, possibly hitting the inner edge of theexternal housing and causing damage. Thus simulation was performed in SolidWorks tofind the maximum deflection of the gyroscope under a variety of target angular velocities,Ω. For safety, a Von Mises stress test was also performed to find the stress at which the

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gyroscope will yield. The simulation is shown in the following figure. The centripetal forcewas calculated according to

Fc =mv2

r= mΩ2r

(a) Deflection of the gyroscope. Note that the scale isexaggerated so the maximum deflection is 4.1·10−5mm.

(b) Von-Mises on Gyro

Figure 8: Analytical Simulations to determine gyro yielding and deflection conditions.

As discerned from the simulation (Figures 8(a) and 8(b)), the allowable clearance, with afactor of safety built in, for the gyroscope and the external housing at the design rpm is15000 rpm. For safety considerations in testing, a tiered trial system will be used, testingthe SphOrb at each increment of 500 rpm.

5.1.4 Axle Design

The axle of the sphere will be supporting all of the load of the gyroscopes, the servos, andassorted attachments. As such, it is a critical point of failure that must be analyzed. Themain shaft will rotate according to the motion of the ball, which is approximately 3-6 ft/s.Due to this small velocity, the dynamic solicitation can be neglected, so a static calculationis valid.

Admissible axial tension with a safety coefficient of 4 is given by σadm = σe/4 In whichσe is the endurance limit of the shaft. For the 12L14 steel, σe = 270 MPa. Therefore,σadm = 67.5 MPa= 67.5 N/mm2

Considering that the 13 inch main shaft with hold a central load of about 8 kg, the handlingtension will be

σ∗ =√3τ2

Andτ = Mf/Wf

where, for a circular shaft, Wf = πd3/32 For a safe project design, σ∗ < σadm Thus, thediameter must satisfy at least

d = 2.17 3

√Mf

σadm

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A maximum moment of 6478.5 N-mm gives the final relation,

d = 2.173

√6478.5

67.5= 10 mm

Figure 9: Deflection simulation of the main shaft under load. Note that the scale is exag-gerated so the maximum deflection is 3.4 · 10−1mm.

Therefore, a shaft of 0.5 inch (12.7 mm) in diameter will be more than enough. A simulationdone in SolidWorks is shown in Figure 9 with the maximum deflection.

5.1.5 Servo Torque

Since the gyros will be spinning at high angular velocities, they will be quite rigid in space, soa high torque servo must be used to act against the gyros and consequently tilt the hangingplatform. Observing that the servos effect the angular acceleration of the sphere directlythrough similarity of the mechanisms (acting on the gyros and gravity-originating torquesfrom the pendulum-mass system), the problem can be simplified with only assumptions ofefficiency. Modeling the apparatus as a single pendulum with the mass being equal to themass of the entire vehicle (assuming the plastic sphere is of negligible mass), the equationsof motion of the system become

4 · τservo = mgrcm sin θ +marcm cos θ

a =grcm sin θ

rballWhere a is the acceleration of the central axle of the ball. The torque is divided by 4 becausethere are two servos being used, and each of the servos are geared to increase the torque bya factor of two. Solving for Torque in terms of acceleration,

τservo =ma

4

(rball +

√g2r2cm − a2r2ball

g

)

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Aiming for a linear acceleration of a ≃ 3 ft/s2, and with a pendulum mass of m = 5 lb andrball = 6.5 in, rcm = 5 in, the necessary pendulum tilt angle is θ = 7. This corresponds toτservo = 209 oz · in. Two 240 oz · in servos will be sufficient to drive the pendulum, resultingin a linear acceleration of a = 3.61 ft/s2 assuming the control movement gyros are spinningfast enough to be modeled as perfectly rigid relative to each other so that the applied torqueis entirely transferred via the gyros and pendulum to a linear acceleration.

5.1.6 Motor Mount

The outrunner motor drives the hollow cylindrical gyros. Since the gyros will be rotatingat high angular velocities and the whole apparatus will be tilted by the servos, the motormust be firmly attached to the bottom housing. From basic gear principles, the torque andangular velocity are related thusly:

Wm = Tmωm = Tgωg

where m is for the motor and g is for the gear. Thus the gyro should spin at about 5000rpm to produce enough precession torque to make the sphere move.

τpx = 2τp cos θtilt

For higher rpms, it will require more torque to tilt the gyro, so the gear attached to the con-nector must be bigger. Considering that the angular velocity of the gyros is 1 rad/s:

ωmdm = ωgdg → ωm = ωg31.90

15.80∼= 2ωg = 2

1

s

According to specifications, each motor will require 3.5V to operate, resulting in an energyoutput:

Tg =Wm

ωg= 690J

Dividing this value into the four screws, symmetrically distributed from the centroid withradius ri = 0.25in, it is possible to obtain:

Fi =Mri∑

r2i=

Tg

4r= 27.2kN

The pressure on each screw of diameter d = 0.125in is

Pi =Fi

Ai= 3.4MPa < σMaxsteel

Therefore, the chosen screws can stand the applied load under all expected conditions.

5.1.7 Gyro Shaft

The gyro shaft connects the gear to the gyro housing itself. Since the servos torque thecontrol movement gyros to tilt the undercarriage and move the vehicle, this is a critical

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component of the design. Repeated use may fatigue the shaft, so Goodmans fatigue criteriaare utilized to find a factor of safety η for the current design:

n−1 =σa

Se+

σm

Sut

n−1 =16

πd3

[√[4(KfMa)2 + 3(KfsTa)2]/Se +

√[4(KfMm)2 + 3(KfsTm)2]/Sut

]From Shigleys Tables A-15-8, Kts = 1.65.

Kfs = 1 + q(Kts − 1) = 1.423

and Kt = 0 with Kf = 1. Solving, the factor of safety on the steel shaft is ηfatigue = 1.3,which is low for final design specifications, but enough for a proof-of-concept design aspresented here.

5.2 Electrical Design

The Electrical Systems were simulated and the results are presented below. The wiring andcomponent diagram are displayed in Figure 10.

5.2.1 Motor

Due to the high torque and speed requirements and differential loadings felt by the motorsdue to the outrunner housing connections, adequate motors must be specified. Given thetarget angular velocity of 10000 RPM from previous analysis above, and the desired supplyvoltage of 7.4V, a motor must be found whose cruise torque exceeds that from frictionalbearing and vibrational losses, in order to be in equilibrium at 10000 RPM. Since thefrictional torque is given by

τr = Fcfd/2

for an applied load F . Using cf = 0.002 for the specific ball bearings, and an applied loadF = mgyrog, the frictional loss is τf = 0.0024 N-m at speed. Thus, the required motormust output at least 2.5 Watts. From the target spin speed, a 1350 kv (RPM/V) motorshould be sought. A 1450 kv brushless outrunner motor was chosen since peak efficiencyoccurs below the maximum (stall) speed. The cruise torque at 10000 RPM for this motor isτ = 0.329N-m determined from the given power output P = 345W running at 7.4V.

5.2.2 Battery Life

To meet the engineering specification of a 20 minute battery life, all power consumersmust be considered. To simplify analysis, the sensors and microcontroller will be assumednegligible. The sustaining current required for each gyro when spinning at 10000 RPMis

If = τfω/V

given τf in the previous analysis above. This component of current draw becomes If =0.33A. The no-load current at cruise RPM for the specific motor is given to be 4A, so

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Figure 10: The complete Circuit Diagram of the Electrical System.

that the cruise current required at 10000 RPM is 4.33A for each gyro. Each of the servosdraw 0.73A given P = τω = 5.13W. Therefore, the total current load is I = 10.12A. Ina dual-battery configuration, each battery must then have a capacity greater than 1.5 Ahof continuous draw. A 6Ah, 7.4V (Two LiPo Cell) battery was chosen. Each of the twobatteries will be placed in parallel to give a total of 12 Ah. Assuming no acceleration ordeceleration of the gyros, these batteries will run the electrical systems for 1 hour and 10minutes.

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6 Final Design

6.1 Mechanical Design

As can been seen in Figures 11(a) through 12, the steel control movement gyros are housedin the cylindrical housings on either side of the main axle shaft. The outrunner motors arepinned to the inside of the gyroscopes. The shafts of the motors are supported throughbearings at the top of the aluminum housing to transfer the load from the motor shaft tothe housing of the gyros. The aluminum housing is held in place via two 14 inch aluminumclamps and 1 inch aluminum angles. Steel shafts connect the control movement gyros tothe swing arms, with belt-driven gears connecting the swing arm to the servos to actuatevehicular movement. Offset servos on the bottom bed allow room for the smaller gear.The entire swing-arm assembly is lock-fitted together to create a rectangular bed. TheArduino is also placed on the top of the bed, while the electronic speed controllers and thebatteries are placed on the sides and bottom of the bed to maximize the moment arm of thependulum and thus minimize the required deflection angle required. Other sensors, such asthe infrared sensor and camera, are placed on the top of the swing assembly over the shaftconnection.

(a) Gyro assembly. (b) Applied Torque vs. Precessional Torque

Figure 11: Gyro CAD Drawing and Force Diagram

As the control movement gyros are spinning, the servos will apply a torque of 240 oz-in at 6V, which is doubled by the gearing ratio, to tilt the gyroscope assembly. Due to conservationof angular momentum, as the gyroscope assembly is tilted, it produces a counteractingtorque that is perpendicular to the applied torque, known as precession torque. Sincethe full assembly is only free to rotate about the center shaft, the assembly will thereforedisplace the center of gravity of the ball, initiating motion that is proportional to the tiltangle of the control movement gyroscopes, and directly proportional to the applied servotorque. Banking is accomplished by torquing the gyros in the same direction, such that thereaction force acts along the pendulum laterally. Turning in place can also be accomplishedby differentially speeding up and/or braking the gyroscopes to cause the entire vehicle to

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(a) Front view of SphOrb assembly. (b) Lateral view of SphOrb assembly.

Figure 12: Detailed gyroscope assembly view.

spin about the vertical axis. By conservation of angular momentum about the vertical axis,the resultant angular velocity of the sphere is proportional to the difference in δω of thetwo control movement gyros. Figure 11(b) illustrates the angular momentum transfer viadifferential precessional torques.

The final CADed internal assembly design is displayed in Figure 18.

All of the requisite parts have been procured and prototyping has begun. The parts are sum-marized in Table 13. For a complete listing of each part and drawing, see Appendix I.

6.2 Software Design

Arduino’s C++ variant was used to program the microcontroller, and data collection anddisplay was accomplished using Java. Various auxiliary filtering and feedback subfunctionswere made to ensure software efficiency.

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Figure 13: Tabulated Parts List and Costs

6.2.1 Control Scheme

The control scheme of SphOrb is semi-autonomous but will also include code for entirelyautonomous movement. From a high level, SphOrb will be controlled by a handheld remotecontrolled joystick and point-of-view camera to an onboard Arduino UNO R3. Control inputfrom the RC transmitter as a PWM signal is filtered and then used in a PID feedback loopas the reference signals, to mitigate the oscillatory behavior typical to a pendulum whensubject to a sudden and constant torque. Finally, this signal is differenced to combine thetwo signals into two coordinated gyro tilt signals (Coordinated gyro tilting corresponds tobanking, compared to equal and opposite gyro tilting which accelerates SphOrb). This canbe written as

ϑf − ϑb = α

ϑf + ϑb

2= β

where ϑ is the gyro actuation angle and α and β are the reference acceleration and bankinput angles. This implies that

ϑf,b =[±α

2+ β + 135

]· 180/270

where the translation and scaling factors limit the range to 0 ≤ ϑ ≤ 180.

Input for the PID controller comes from a motion processing unit with three degrees ofaccelerometers and gyroscopes. Implementing a Karman algorithm, the (α, β, φ) inclinationof the hanging pendulum is determined, thus providing the values to be compared to thereceiver with the PID feedback. The PID coefficients determined by using Ziegler-Nicholstuning, are listed in Table 6.2.1.

Gyro Speed κP κI κD

Low 1.25 0.3 0.25High 1.15 0.3 0.25

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Limits for the proportional feedback coefficient were determined by simulation. The gov-erning control model for this system as determined from Newtonian mechanics is

θ = −g

rθ +

τamr2

where τa is the torque applied by the servo, r is the length to the undercarriage center ofgravity, and m is the mass of the gyro. For a proportional controller, the input τa can bemodeled as τa = Kpθ. Thus, the form of the controller becomes

θ + θ

(g

r− Kp

mr2

)= 0

For a second order system, the stability requirements are such that the coefficients in frontof each term are all the same sign. Therefore(

g

r− Kp

mr2

)> 0

so that Kp < gmr A suitable value of 1.15 was chosen for Kp.

To complete the semi-autonomous control system, an interrupt command is defined contin-gent on the distance given by an inboard infrared sensor. If impact with a wall would beimminent (defined given a maximum constant deceleration), a control override takes overonly forward acceleration control, and slowly backs up until it is free from the danger ofcollision.

An extension from interrupting manual control is the entirely autonomous patrolling. At apreset distance from the wall, a 90 degree turn will be performed. Looped until regainingmanual control again, SphOrb would patrol the perimeter of a building.

6.2.2 Program Implementation

The software with PID control feedback and RC interface has been written and rigorouslytested in computer trials interfacing with the electrical hardware. The pseudo-code is pre-sented below:

Define Variables (at Neutral Angle or 0 Degrees)

Setup

Semi-Manual loop

//Gyro Speed

if Channel (Sw. A) is On

Ramp up two gyro motors to Speed indicated by Channel (Left)

else

Ramp down two gyro motors

end

//Gyro Servo Control

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Read RX Channel for Forward/Backwards==alpha // Using a 50 microsecond quick PWM

Read RX Channel for Left/Right Banking==beta // sampling code.

Map each of the channels to (-2*MaxDeg,+2*MaxDeg)

Apply Mean/Median hybrid 2nd order filter to alpha and beta

// Uses six historic PWM values. Averages each pair to get three values

// and then take the Median of those values to soften out RX noise.

//Sensors

Read Gyroscope Sensor Data (Collects Linear acceleration and Angular Velocity)

Apply 2+ Hz Lowpass 1st order Bessel filter to steeply attenuate

vibration-induced noise from the two spinning control movement gyros

Coalesce Acceleration and Velocity to determine inclination angle and

apply a Karman filter to reduce sensor precession.

// Additional software was uploaded onto the Digital Motion Processor (DMP)

// located on the Gyro/Accel MPU so that Arduino processing time could minimized.

Read IR Sensor Data

if signal < 3 feet // PID Control to include conditional for speed requirement

Override Manual Control

Gyro Servos to Neutral (Ramp down quickly)

while signal < 5 feet

Reverse Gyro Servos slowly

end

Initiate Turn to avoid Obstacle and proceed

end

//PID Control

Implement PID feedback control

Reference Signals: alpha, beta

Feedback Input: Pitch, Roll

Output: alpha_n, beta_n

// This acts to dampen control input and stabilize pendulum oscillation

Define tilt of gyros:

th_front = alpha/2 + beta

th_back = -alpha/2 + beta

Check and limit range on th_front and th_back

Autonomous loop

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Read IR Sensor Data

Read Gyro Sensor Data (bank angle)

PID Control on Gyro Sensor Data to mitigate pendulum oscillations

Proceed Straight ahead slowly

if signal < 1 foot

Bank left

Reverse (Mirror) Gyro Servo Positions

Bank right

Return Gyro Servo Positions to Normal

Delay to complete 90 degree turn

Write Bank angle to 0

end

//Write Servo Inputs to hold desired torque on the gyros

Write th_b and th_f to Control Servos

Write Gyro Speed PWM signal to both gyro motors (through ESC)

//Bluetooth Data Retrieval

Send Gyroscope, IR, and feedback data to PC via bluetooth

// PC collects data, using ’Processing’ to display.

7 Plan of Action

Different group members were in charge of different components of the design and prototyp-ing. Tasks were split up under two domains: Mechanical engineering and Electrical/Softwareengineering. Sina, Ben, and Mauricio were in charge of the mechanical design, while Aaronwas in charge of the electrical engineering and software components. Mechanical engineer-ing tasks include CADing the design and simulating it on the computer, machining theparts, and making calculations to ensure the mechanical components meet the specifica-tions required by the project. The overall design and specifications were completed byMarch 2, 2013, and the parts were machined and finished by April 23, 2013. The electri-cal/programming tasks included wiring the motors and microcontroller, programming themicrocontroller and Digital Motion Processor (DMP), specifying the motors and ESCs, andattaching and integrating the sensors as well as regulating power to each of them. Theelectrical components were specified by March 2, 2013, and the wiring and low level pro-gramming was finished by April 23, 2013.

The project was completed within the allotted three months, to be completed by the UCBerkeley Mechanical Engineering Design Exposition on May 3, 2013. The maximum budget

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for the research and development of the prototype was set at $1200. Once a prototypewas built, further modifications and refinements were made to the manufacturing processto reduce costs for the consumer. The target selling price is $300, with material coststaking up $150. This was not achieved in the prototype since parts needed to be replaceddue to design oversights. As the design and manufacturing processes are refined, this targetprice can be readily reached.

Further details on the proposed direction and work delegation can be evidenced in theRASIC Chart and the Gantt Charts.

7.1 RASIC Chart

A responsibility assignment matrix was performed to describe the roles of the team membersand the tasks that need to completed for the project. According to the RASIC model,R corresponds to the person who is responsible for achieving the task. The accountableperson is the team member who must approve the work done by the person designated asresponsible for completing the task. The accountable person is designated as A. Supportersare designated as S, and help those responsible to complete the task. I refers to those whoare kept up-to-date on the progress of the designated task. C corresponds to the person whocan be consulted on certain subjects, and aid in the completion of the task. The RASICchart for the SphOrb project is listed below in Figure 14.

7.2 Gantt Chart

The Gantt Chart illustrates the intended project schedule for the SphOrb. Key engineeringtasks are designated on the vertical axis, and the important work dates for the SpringSemester are listed at the top. The ordering of the tasks shows chronological dependency,as split up between the Mechanical Group and the Electrical Group. The Gantt Chart hasbeen completed up to March 11, 2013, as can be seen in Figure 15.

8 Problem Analysis

In order to properly execute this project, much research and development has been accom-plished before the prototype was built. As such, all of the various component specs weredetermined. This includes the motors, the servos, the gears, the gyros, and the axle sinceeach is interdependent on each other due to the size constraint of the ball. These com-ponents all depended upon the size and weight of the gyro, which in turn depended uponthe size and weight of the overall sphere. The size of the sphere is fixed as the 13” outerdiameter plastic hollow sphere that is readily available. Thus, the overall weight of SphOrbhas been estimated so that the specifications of the other components can be set. To ensurethat the gyros will be able to move the sphere, research was done as to how gyroscopicprecession works, and how that precessional torque would affect the center of gravity ofthe entire sphere, which consequently accelerates the device. Research was also done todetermine how an input to an electric motor would output the desired RPM and torqueto drive the gyros as well as how the rotation speed would depend upon the load that the

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Figure 14: RASIC Chart for the SphOrb Team Members

motor undergoes in order to spec the components. The full list of references can be foundin the Bibliography.

Now that the project has been properly specified and CADed, work on the prototype mustbegin. The choice of materials for the undercarriage and other axial supports must be de-cided upon. Furthermore, proper machining process must also be chosen. This informationwill be gathered through online research into the materials and through talking with thestaff in the Berkeley Machine Shop and utilizing their expertise.

The most problematic aspect of this project is also the most novel: the gyroscopic propulsionsystem. Changing the center of mass of the sphere using gyroscopic precession works intheory, but very few applications have actually been built previously. Numerous problemsmust be overcome for the project to be successful, including proper specs of the motors,gyros, and servos. As such, much novel research, development, and calculations must bedone before the prototypes can be fabricated. Some of these potential problems can beanalyzed through the use of the simulation toolkits available in SolidWorks; the design canbe simulated under typical operating conditions to gauge the feasibility before work on the

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Figure 15: Gantt Chart progress up to April 27th, 2013.

actual prototype can start. If the gyroscopic motion is too ambitious for undergraduatestudents to pursue, the sphere can be driven by a rotating axle and a mass on a servo canbe rotated to change the center of mass of the sphere. This would preserve the intent of theuse of the gyros, but remove an additional layer of complexity.

This project could also fail if we get too ambitious with the auxiliary tools and features.Thus, the extraneous features could be toned down, like the additional augmentations thatcould be placed in the sphere. The base model of a remote controlled spheroid robot iscomplex, especially with the kinematics and stability required for movement of the robot.The base model to be demoed at the open house will be a remote controlled sphere thatattendees could move around.

9 Test Results

Preliminary tests were successful in displaying the proof of concept of utilizing control move-ment gyros for vehicular movement. The tests were highly indicative of how a spherical robotwould move under applied control configurations, albeit sans an actual spherical enclosure.It was impossible to find a factory-made hollow sphere that is large enough under student

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0 2 4 6 8 10 12 14−40

−20

0

20

40Free Response

Pitc

h (D

egre

es)

0 2 4 6 8 10 12 14−40

−20

0

20

PID Feedback Response

time

Pitc

h (D

egre

es)

Figure 16: The Free, Undamped Response compared to the Forced Feedback

budgets; one could be custom made and purchased for $500, but this was outside the budgetof this project. Perhaps industry sponsors could alleviate the financial burden.

In typical test setups, the main axle was held up on both sides by a person. With thegyros off, lurching the undercarriage forwards and backwards showcased how the servoswould move the gyros in response to their position. However, since the gyros were off, thiscaused minimal damping and thus it took a long time for the undercarriage to return to itsoriginal position. When the gyros were turned on, any disturbance felt by the undercarriagewas quickly damped out within 2 to 3 oscillations, displaying the effective control code.Position data from the accelerometer was transmitted by Bluetooth to the computer, wherethe oscillatory forwards and backwards motion can be readily viewed. This PID Feedbackresponse can be seen in Figure 16 where the initial step input occurred at t = 1 second.

Furthermore, when a differential torque was applied to the moving gyros, the undercarriagewould move forward, which would move the center of mass of the apparatus. Thus, hadthe axle been connected to a sphere, the vehicle would move forwards. This shifting of thecenter of mass can be felt holistically by the volunteer. Torquing the gyros in sync wouldbank the undercarriage to the side, which can again be felt by the volunteer holding theaxle. In a spherical enclosure, this would cause the vehicle to turn in place, thus allowingfor increased mobility.

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Figure 17: SphOrb demonstration on May 3rd, 2013.

10 Future Improvements

The concept of a spherical robot that uses the unique advantages of control movement gyroswas presented here. Linear forward and backward movement was displayed through theoppositional torquing of the gyros, while banking, and subsequently turning, was displayedthrough synchronized torquing of the gyros. A stable PID controller was developed toallow the undercarriage to return to its steady state position after approximately threeoscillations.

However, as with any novel design, important improvements can still be made for futureversions. Weaknesses present in the mechanical design affected subsequent areas of theelectronic design, and are summarized below.

10.1 Bearing Support for Gyros

Each gyro and housing weighs approximately 2 lbs, and is supported by a 5/16in steel rodthat is force-fit into a bearing in the undercarriage arm and is attached to a gear, as seen inFigure 11(a). Bearings of this size are built for neither strength nor high speed applications.The large mass of the gyros coupled with the weaknesses of the bearings caused the bearingsto slip out of the undercarriage. Problems associated with this phenomenon include the gearsmoving too close to each other and grinding, the L-clamps on the gyro moving too close tothe undercarriage arm, and the gyros sagging such that they are no longer parallel to theground.

There are two proposed solutions that will be included in future design revisions. First, alarger gyro shaft will be used. This will allow for less sag due to the comparatively immense

27

weight of the gyros. Second, this will also necessitate the use of a larger bearing that iscapable of handling increased loads without premature failure.

Figure 18: The Final Mechanical Design.

10.2 Mechanical Vibrations

The set of gyros must be operated at high angular velocities to maximize the precessiontorque generated. However, the gyros are not perfectly symmetric, so when they rotatethey generate unwanted vibrations. As expected, at higher angular velocity, the amplitudeand frequency of the vibrations are increased. This causes problems with the mechani-cal components which tend to loosen after repeated vibrations. Unfortunately, vibrationalanalysis was one aspect that was sorely overlooked during the design process since there arescarce undergraduate courses that expand upon the subject matter.

This problem was somewhat remedied by the introduction of rubber washers to dampen thevibration at critical junctions. For example, a rubber washer was used on the connectionbetween the gyro housing and the gyro shaft, thus isolating the source of vibration from themechanism. Further improvements would include remaking the gyros with smaller tolerancesto reduce the asymmetric vibrations and the use of spring dampers to further isolate thevibrational sources from the rest of the mechanism.

10.3 Servo Housing

The servos are only held in place through friction; they are wedged in between two platesof aluminum which are tightened by 3 screws. This allows ease of access to swap outdifferent servos and tighten the belt that connects the servo gear to torque the gyro shaft.

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Nonetheless, under a heavy applied load, such as when the gyros are rotating at a highrpm, or after repeated use, the servo system will fail in one of two modes: the servos willeither slip out of the clamp housing, or the belt will slip by one notch and thus changethe equilibrium position of the gyro, yielding subsequent errors when trying to control thevehicle.

In order to correct for this problem, rubber washers were placed in between the aluminumplates to increase the friction between the servo and the housing to hold it in place better.Subsequent designs will make use of the four mounting holes on the servo so that they canbe attached without slippage.

10.4 Control Code

Due to time constraints, the control code received less attention that it should have beenallotted. Thus, the PID controller was tuned holistically using trial-and-error to determinegains that would allow for a stable control system. The values chosen for KP, KI, andKD were sufficient at specific rpm settings of the gyros. However, these are static gains,and thus will not be appropriate for a range of power settings of the gyro. In the future,dynamic gains can be utilized, and the system can be modeled and optimized for criticaldampening.

Figure 19: The Future SphOrb Final Design with intended Shell.

11 Conclusion

Spherical robots display a range of advantages through their unique maneuverability andgeometry. The spherical form factor possesses no sharp edges, allowing the robot to move in

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any direction while maintaining optimal stability. The entire mechanism is self-contained,making it ideal for work in dangerous areas that would compromise other, more delicaterobots. This paper proposes a novel method to store and utilize angular momentum, yieldingsignificant advantages over other methods. Compared to traditional methods, the high rpmgyro can output unparalleled energy in short periods of time via stored angular momentumwhen acted upon by an internal torque.

Spherical robots are not well understood, and remain a continued area of research. Inorder for spherical robots to perform in the field, their mobility needs to be improved andundesirable dynamics need to be mitigated. Manipulating angular momentum internallyhas been shown to significantly enhance the mobility of spherical robots, but adds layers ofcomplexity to the challenging system.

Nonetheless, practical manifestation of these advantages in a prototype would revolutionizemany areas of robotics. Future research effort will be directed towards the mechanicaldesign, addressing the difficult control issues present in this prototype. Such a unit wouldsee demand in a wide variety of markets, as its functions would easily be optimized toapplications in home, educational institutions, commercial, and government use.

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References

[1] Gregory C Schroll. Dynamic Model of a Spherical Robot from First Principles. PhDthesis, Colorado State University, 2010.

[2] Gregory C Schroll. Design of a Spherical Vehicle with Flywheel Momentum Storage.2008.

[3] Andrew Rodgers. Controllable Gyroscopic Propulsion Apparatus.

[4] Eric Laithwaite. Gyro Propulsion System.

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