Round-a-BotHolonomic Inverted Pendulum Robot

Jeff FletcherSung-Tu Ho

Jason KloessRanjit Raj

Steve Vozar

ME 552 Design Review 3

24 November 2010

web: roundabot.wordpress.comemail: round-a-bot@umich.edu

Note: This document is best viewed as a dynamic HTML or PDF document, as it contains internal links as a part of the document structure. This document can be viewed as a webpage at: http://tinyurl.com/roundabot-DR3

IntroductionTechnical Specifications

Physical Design (3D CAD)Assembly Views

Motor shaft stackIdler wheel stackLower half assembly (base plate)Top electronics plateFull robot stack

Exploded ViewsOmni-wheel connectionsMotor Shaft StackIdler Wheel Stack AssemblyLower Half Assembly (Base Plate)Frame Mounts: Bar Stool Rings to Plates

Module CloseupsMathematical Modeling

Planar Mathematical ModelSimulink ModelLinearizationParasitic Effects

ActuatorsSensors

Controller DesignController Simulations

Rider Input ResponseOffset Response:Floor SlopeBump Disturbance while DrivingShove Response

Major ModulesChassisDrive-train

Motors/GearboxMotor Shaft and KeywayBearingsWheelsBall

Power SupplyMotor BatteryBattery State of Charge Sensor

Sensors

Inertial Measurement Unit (IMU)Encoder

Miscellaneous ElectronicsEmergency Stop

Voltage RegulatorPrinted Circuit Board (PCB)Wiring PracticesCircuit DiagramPCB Circuits

MicroprocessorComponent Verification

IMUAccelerometer testGyroscope test

MicroprocessorIssues to be Addressed

Motor ControllerTop Plate Rearrangement

ConclusionsBill of MaterialsEngineering DrawingsAppendix A - Simulation Parameter File

Introduction The aim of this project is to design a robot system that balances on a single spherical ball and can translate in two directions (forward/backward and left/right). The platform should be able to support the weight of a person (<200 lbs) at waist height (2.5-3ft). The system is a multi-directional inverted pendulum setup, as opposed to a Segway which is a single-directional inverted pendulum. This adds a second complexity to the modeling and controlling scheme.

Technical Specifications● Approximate ball size: 8.5 - 10in diameter (soccer - basket ball sized)● Approximate height of robot (atop ball): 2.5 - 3 ft (table-top height)● Cost: $1500● Accommodate a human rider up to 200lbs● Operational Environment: Indoors

○ Temperature: 15 - 25 ºC○ Range of surfaces for traction: Tile, Carpet, Concrete, Brick, etc.○ Able to traverse slopes up to 10º○ Can reject disturbances caused by small (~1cm) bumps○ Operational Range: ~1 hr continuous battery life with ability to easily recharge

(AC outlet)● Operatonal Speed: 2-3 mph translational● Safety Features:

○ Passive catch/kickstand system to avoid over tilt (training wheels)○ Low battery warning○ Emergency Stop

Physical Design (3D CAD)The 3D physical model, constructed in Solidworks, is detailed below with 3D assembly views, exploded views, and closeups/section views. The dimensioned drawings for all the fabricated parts are linked at the end of this document.

Assembly ViewsThe total robot stack was broken into four main assemblies in order to highlight the specific components and their layout: (i) motor shaft stack, (ii) idler wheel stack, (iii) lower half assembly: base plate, and (iv) top electronics plate. Each assembly is shown below with a description highlighting major areas and any changes from the previous design review presentation (DR2).

Motor shaft stackIn order to transfer torque from the motor to the omni-wheels and ultimately to the ball, a keyed shaft was connected to an adapter and then to the keyed omni-wheels (Rotacasters supplied by Kornylak). The following assembly was created including the following components:

● FIRST CIM motor● Bane Bots 16:1 planetary gear box● M4 x 80mm motor mounting bolt x4● Gearbox standoff x2● Bearing support (gearbox side and encoder side)● ½” ID bearing x2● Omni-wheel adapter slug (gearbox side and encoder side)● ½” ID shaft collar● 3/16” alignment dowel pins● US digital encoder● Encoder standoffs x2● #4-40 encoder mounting screws x2

The specific assembly method, alignment, and fastening of all the components is outlined in the exploded views section below. For the total robot lower half assembly, two of these motor shaft stacks are fixed to the base plate.

3D View of Motor Shaft Stack Assembly

Idler wheel stackSince only two motors are used the lower half assembly (positioned 90° apart), a third set of omni-wheels was positioned (135° from each motor shaft stack) in order to adequately balance the lower half assembly. This third set of omni-wheels is identical to the motor shaft assembly

which spacing and components with the exception of the motor, gearbox, and encoder parts. The same space, alignment, and fastening are used (further explained in the exploded view section below.

3D View of Idler Stack Assembly

Lower half assembly (base plate)Both motor shaft stacks and idler assemblies are attached to the base plate and placed on top of the ball. This base plate is the main component that supports the weight of the rest of the robot (bar stool frame and all upper components) as well as the rider. Due to this weight requirement, the plate is made of ¼” 6061 aluminum which has a high strength to weight ratio. It is fastened to the lower ring of the Chicago bar stool frame via four ½”-13 x 2” bolts. The batteries are also mounted to base plate and positioned with ⅜” sq PVC spacers. The top piece that goes over the batteries is slotted so that it fits length wise and is mounted to the ¼”-20 x 6” threaded rods.

3D View of Lower Half (Base Plate) Assembly

Top electronics plateThe second mounting plate holds all the electronic components to the upper ring of the frame via four ¼”-20 x 2” bolts. In order to electrically insulate the electronics from the frame and rider, the top electronics plate is made of ¼” high insulation PVC. The components mounted to this plate include:

● H-bridge motor servos x4● IMU sensor assembly: chip and PVC enclosure with metal Faraday wrapping (on

bottom)● PCB electronics chip● cRIO controller● E-stop assembly: button and bracket (mounted to top ring, not pictured)

3D View of Lower Half (Base Plate) Assembly

Full robot stackCombining all these assemblies the full robot is shown below, with and without a simulated human rider (for mass and moments of inertia properties). Some additional items are the thin plastic protective wrapping around the frame protecting the top electronics plate and the four casters attached to the bottom of the frame legs. The total weight of the robot turned out slighlty higher than originally expected: ~65 lb final vs. ~50 lb original. The total height is on par with the technical specifications though, 31” from ground to top of seat.

3D View of Full Robot Assembly with and without Rider

Exploded ViewsEach of the complex assemblies were further broken down and separated in order to visualize how each component is connected, aligned, and fastened to create the stack. The five major assemblies (i) omni-wheel connections, (ii) motor shaft stack, (iii) idler wheel stack, (iv) lower half assembly (base plate), and (v) bar stool frame are presented below.

Omni-wheel connectionsSince the omni-wheels have a significantly large center bore hole (22.25mm with 6.6mm keyway), we must use an adapter slug in order to accommodate the standard ½” shaft with ⅛” keyway supplied with the motor. This slug made from 1215 machinable steel and will be turned on the lathe to the correct OD, milled with an end mill to form the outer keyways and broached to from the inner keyway. In the assembly two slightly different slugs are made (keyway positions) in order allow for the 22.25° offset between the omni-wheel rollers. Two omni-wheels with this offset were selected in order to meet the 200lb loading specification (each omni-wheel rated to 100lb) and ensure at any position of the motor shaft, at least one roller surface is contacting the balancing ball. This helps decrease the dampening losses and maximize ‘no-slip’ torque transfer. The omni-wheels and slugs are co-centric aligned and placed between the bearings in the next motor shaft explosion.

Exploded View of the Omni-wheel Adapter Slug Components

Motor Shaft StackThe motor shaft stack is the drive unit of the lower assembly. All axial components (bearings, omni-wheels, encoder, etc.) are aligned via the motor shaft and keyway. This ensures co-centric alignment between these components and smooth torque transmission. The gearbox side bearing support is fastened to the motor via the four M4 x 80mm cap screws and counter-sunk to a flush surface.The whole motor shaft stack assembly is aligned (for slight flexure, support rotation about shaft, and planar offsets) via the two 3/16” x 1” steel dowel pins that protrude through the base plate.The assembly is then fastened with four 5/16” x 1” socket cap screws to the base plate each torqued to the bolts specifications. The gearbox standoffs and supporting ¼”-20 bolts will provide axial leveling since these bolts will help support the weight of the motor and gear box. Leveling this will reduce the deviation in axial straightness of the shaft and ensure proper torque transmission.

Exploded View of the Motor Shaft Stack Assembly

Idler Wheel Stack AssemblyAs mentioned before, the idler wheel will serve as a balancing support for the robot. The same aligning and fastening considerations mentioned for the motor stack will be used here since the assembly is identical less the motor and encoder. The shaft used is purchased separately from McMaster (as opposed to BaneBots) but is identical in size and material.

Exploded View of the Idler Wheel Stack Assembly

Lower Half Assembly (Base Plate)Two motor shaft stacks and one idler shaft stack will be mounted to the bottom of the base plate while one 12V drive battery and two 12V axillary batteries (24V parallel) will be mounted to the top surface. The bar stool frame will also be mounted to the top surface of the base plate. The heavy drive battery is centered on the base plate while with the axillary batteries are equidistant and parallel to the drive battery. This ensures an axial center of gravity (measured currently at 0.02” off center in x- and y-planar directions).

Exploded View of the Lower Base Plate Assembly

Frame Mounts: Bar Stool Rings to PlatesLastly, the lower half assembly and top electronics plate are mounted to the frame via two sets of four socket cap screws. Since only the nominal sizing of the stool is known (provided in the frame discussion in the major modules section below), we had to estimate various dimensions in the frame assembly including the tube diameter and each ring diameters. For this lack of information, the mounting places were left as slots to allow for error in our estimations. One additional item to the frame is a plastic protective wrapping that protects the top electronics plate from the user. This is attached with velcro to the frame legs to allow removal and quick placement.

Exploded View of the Frame and Support Plates

Module CloseupsIn order to better visualizes some of these components, some close-ups and a section are presented. Most of these assemblies have been discussed previously with the exception of the IMU mount. The IMU is mounted on the bottom surface of the top electronics plate so it is out of the way of the other electronics. It is sensitive to bumps and absolute positioned for calibration so it is mounted in a well protected space. The enclosure and IMU chip are centered on the top electronics plate. The enclosure is mounted via two #4-40 screws and the IMU is mounted with four #6-32 screws and two hex stand-offs. This will allow for leveling and calibration of the IMU.

Motor Shaft Vertical Section View

Close-up of Battery Fastening

Close-up of Motor/Gearbox Fastening

Exploded View of IMU Enclosure Mounting

Mathematical Modeling

Planar Mathematical ModelThis model is similar to the model presented by Lauwers et al. but uses a more convenient set of generalized coordinates. We are assuming the there is no slip between the ball and the wheels, and that all components are rigid bodies. We proceed to determine the equations of motion using the Lagrangian approach. The generalized coordinates are given by , defined as the angle of the cylinder from the vertical inertial axis, and , the angle of the ball from the initial alignment of the cylinder. See the figure below for an illustration. One can track the position of the robot from it’s original starting point by multiplying + by r, but as is generally small, and can grow quite large, when the robot is rolling, we can estimate the distance traveled by .

The positions of the center of mass of the ball and the center of of the mass of the cylinder with respect to a point a in an inertial reference frame A are (respectively):

And the angular velocities of the ball and cylinder with respect to Frame A are:

Thus potential energies of the ball and cylinder are given by

And the kinetic energies of the ball and cylinder are given by:

Taking the Lagrangian yields a set of two coupled differential equations, which can be written in the following form:

Where the Mass matrix is given by:

The Coriolis matrix is given by:

The Gravitational matrix is:

And the Damping matrix is:

Simulink Model

The closed-loop Simulink model is shown below:

Overall closed-loop Simulink Model

The nonlinear system model is shown below, and is based on the system of equations developed in the previous section. Details for the controller gain, sensor dynamics, and actuator dynamics are discussed in later.

Simulink block diagram of nonlinear planar model

LinearizationIn order to create a controller for the system based on linear control theory, the original

nonlinear model was linearized about the operating point of ( , ) = (0,0), and the following state-space representation was obtained:

where M1-4 are the entries of the Mass matrix from left to right, and top to bottom (English reading order). This linearization was verified for a step input on ball torque, the nonlinear and linear systems are shown below, showing that the linear and nonlinear systems are very close up to about 0.5 radians of cylinder offset, which is fine for our purposes:

The linearized system is a good approximation for the nonlinear system model, even up to cylinder angles of 1.5 radians.

Parasitic EffectsParasitic effects, such as motor backlash and saturation, sensor noise, quantization, and discretization are significant factors in what will become the real-world model of our system. The controller may command motor torques that are not achievable for our setup, and incorrect sensor readings can detrimentally affect the performance of our closed-loop system. This section describes how the parasitic effects were incorporated into the model.

Actuators

The model for the actuators, shown below, is used to implement the parasitic effects on the motors. In this case, the motor is subject to quantization and discretization effects from the D/A converter coming out of the microcontroller. These are modeled as a Quantizer and a Zero-Order hold in the model. The values for these parameters are taken from the resolution of the D/A converter (12 bit), and the output sample rate of the microcontroller (500Hz). Additionally, the motor driver has a limited current it can supply to the motor, which is modeled as a saturation block, and is taken from the driver specifications (70 Amps). Backlash and friction are modeled together, and lacking the proper specifications for friction of the motor from the vendor, are estimated. An estimate for the noise introduced into the command current is also added after the D/A converter from the microcontroller. A simple first-order filter is used before the quantizer to model the effects of filtering out the noise from the controller.

Simulink block diagram of actuator dynamics with included parasitic effects.

SensorsThe sensors in the system are also subject to a number of parasitic effects, particularly noise, quantization, and discretization issues.

The accelerometer (top row of the figure below) measures the cylinder angle, and in our system is subject to quantization and discretization effects due to the A/D converter. Its noise is determined from the accelerometer data sheet. The rate gyro measures (3rd row) the angular velocity of the cylinder directly, and as an analog sensor, the quantization and discretization effects come at the D/A converter interface as well. The noise for this sensor is taken from the gyro specs. The encoder (2nd row) for this system measures the wheel angle, and is a digital encoder, which is not subject to noise. However, it’s quantization is based on the resolution of the encoder, and the discretization effects are still controlled by the sample rate of the microcontroller. The angular velocity of the wheels is estimated by taking the derivative of the encoder data within the microcontroller. This is shown in the bottom row of the block diagram.

Simulink block diagram of sensor dynamics with included parasitic effects.

Controller DesignAn LQR controller for state feedback was designed using the linearized state-space model discussed earlier. The optimal gain matrix k was determined using the following inputs to the cost function:

Which places a high emphasis on maintaining a stable cylinder position, and a very low emphasis on maintaining a constant ball position (as the ball is allowed to roll where ever it likes). It should be noted that this controller was developed assuming an input of motor current to the system. However, motor controllers with this function are prohibitively expensive. In practice, we will use an H-bridge motor controller, which closely approximates a current-mode controller. However, in practice, system identification will have to be performed on the system when it is assembled, and a new controller will be created. Regardless, this controller is useful for showing that a controller can be build for the system given all the parasitic effects and actuator restrictions, even though a different controller will need to be used in practice.

Controller SimulationsThe following are a series simulations of real-world situations that are relevant to the operation of the Round-a-Bot robot, using the parameters of the model determined by the selected components to see if the robot can perform to specifications. The MATLAB file defining these parameters is shown in Appendix A. These situations often represent worst-case scenarios in which the robot is at the limit of its abilities, as it is operating at relatively high offset angles, and in the planar case, so only one actuator can work to balance the robot.

Rider Input ResponseThe following plot shows the results of a simulation of a rider leaning forward on the cylinder, causing the cylinder to pitch forward. This action is simulated by applying a torque to the cylinder equal to that created by the rider at the specified lean angle, which is . The plot shows simulation results for a 10° lean, which is approximately the upper limit for lean angle. As the rider is unlikely to jerk forward, the action is modeled as a ramp starting at t=2s, taking one second to reach full lean.

Simulation of system response to 10° rider lean. We can see that the controller does saturate current at 70A, but the robot is able to maintain balance. The steady-state ball speed of around 9 rad/s corresponds to about 2.6 mph, which is within our specified target speed range.

Offset Response:The following plot shows the system response to a 10° initial offset of the robot chassis from the vertical. The robot is able to correct itself after a brief surge in current, and settle to a steady ball position and zero-cylinder offset.

Simulation of system response to 10° initial offeset in cylinder angle.

Floor SlopeThe effect of a sloped floor can be determined by applying a torque to the ball equal to that applied to the ball when on a sloped surface, namely where ψ is the slope of the floor. In this case, we investigate the effect of a sudden disturbance step in slope to an angle of 7.5° at time t= 2s, followed by a reverse step in slope at t=5s on a robot at rest.

Simulation of system response to 7.5° step in floor slope, followed by a reverse step of the same magnitude.

The robot can remain upright, even when jostled by this slope disturbance. This is about the highest slope the robot can handle, so we fall a bit short of the 10° specification. However, for indoor use, this should suffice for most situations.

Bump Disturbance while DrivingThe effect of running over a bump while driving the robot was investigated by applying two large, equal but opposite slope disturbances over a small time span. In this case, a slope angle of 45° was applied for a duration of .05 seconds before being removed while the robot was being “driven” by the user at a lean angle of 5°.

Simulation of system response to hitting a bump while driving the robot at 1.1mph. We can see that there is a spike in current and torque at the time of the bump while the robot tries to regain its balance, and that ultimately it is successful in remaining upright.

Shove ResponseThe response of the robot from rest to a 200N push applied to the center of mass of the robot over 0.05s is shown below.

Simulation of system response to being pushed, from rest, at the center of mass for a duration of 0.05s with a force of 200N.

The robot has no problem remaining upright despite the shove. The motor current peaks just after the push and there is a small overall movement of the ball relative to the floor, which is what we would intuitively expect in this situation.

Major ModulesThe major modules of the system are defined as the chassis, drivetrain, power supply, sensors, and microprocessor. They are discussed in further detail in the following sections.

Chassis For the round-a-bot, we considered a stool to be the appropriate chassis. A chair might seem to be a more logical choice, but in this case, the person riding on it may have the tendency to lean on the backrest, thus decelerating the robot unintentionally. However in a stool, a person will usually sit straight up and therefore in a better position to control the motion of the round-a-bot. We wanted the stool to be similar to the ones in the lab. A simple design such as the stools in lab would ensure more space within the stool so that all the required components can be placed appropriately rather than placing them in a cramped space. This is really important to our project since we would like the batteries and the control circuit to be as far apart as possible. The stool must be of sufficient height, so that a person can conveniently sit on it. A stool of around 30 inches height, we felt would be ideal. The stool must be sufficiently strong as well as light. We decided that the stools in the lab are

of sufficient strength, and therefore, looked for stools similar to the ones in lab. In general, we looked for the lightest stool that would be capable of carrying a 200 pound person. Based on the above criterion, the ‘Chicago Bar stool,’ supplied by BudgetBarStools.com, was selected.

The Chicago-style bar stool provides the frame for our robot chassis.

● The dimensions of the stool :

○ Height: 30”○ Base area: 16x16”

● Weight of the stool: 16 pounds The ‘Chicago bar stool’ seems to have all what we wanted in a single package. Firstly, the stool was to the height we wanted, exactly 30 inches. The base area (16x16”) seemed appropriate. The design was simple, which would give us a lot of space to place our components. The two rings would conveniently act as good supports for the base plate and the electronic plate. Also the rings are only bolted to the frame and therefore can be moved around and positioned where we want it to be. The stool weighs only 16 lbs and is capable of carrying a 200 pound person. Apart from these primary advantages, another advantages is that the stool has a swivel seat. Therefore, the stool is capable of rotating 360 degrees. This will simplify our design, because for a normal stool, we would have to buy a swivel and then install it as well. Also, the stool comes with a cushion which will provide some riding comfort to the person who travels on the Round-a-Bot. The design of the mounting plates for the electronics and mechanical elements of the robot is discussed in the CAD section. Note that the exact dimensions of the stool are not given by the supplier, so some minor changes to the mounting plate design may occur when we are able to

explicitly measure the frame.

Drive-train

Motors/GearboxIn order to power the wheels and brushed DC motor and planetary gearbox will be used. A suitable motor was very difficult to come across; we needed to meet the following technical specifications to ensure proper operation while a rider is seated:

● High torque transmission >30 Nm● Low rotational speed: 200-350 RPM (2-3 MPH)● Low weight <10 lb each

For an ungeared motor, these torque and speed requirements are generally not practical to obtain. The lowest gear ratio for the motor was chosen in order to minimize the frictional losses and rotational inertia which highly affect the motor dead band (which was noted during simulation to significantly increase the current required recover from a large tilt). Our initial choice was a DeWalt Powerdrive Kit that uses an old 18.8V drill motor but had too high of gear ratio (~45:1). The DeWalt motor also claimed to have a 22,000 un-geared, no load speed which is very high for a brushed DC motor increasing the doubts we had that this motor could perform as well as it claimed. After further searching, more reasonable motor was chose: FIRST CIM motor:

FIRST CIM Motor from Bane Bots Motor SpecificationsPERFORMANCE

Model M4-R0062-12

Operating Voltage 6V - 12V

Nominal Voltage 12V

No Load RPM 5310

No Load A 2.7A

Stall Torque 343.27 oz-in 2424 mN-m

Stall Current 133A

Kt 2.58 oz-in/A 18.2 mN-m/A

Kv 443 rpm/V

Efficiency 65%

RPM - Peak Eff 4614

Torque - Peak Eff 45 oz-in 317.8 mN-m

Current - Peak Eff 19.8A

PHYSICAL

Weight : 46 oz (1304g)

Length - for motor : 4.32 in (109.6mm)

Diameter : 0 in (0mm)

Diameter : 2.6 in (66mm)

ShaftDiameter : 0.31 in (8mm)

Shaft Length : 1.4 in (35.6mm)

The gearbox chosen is structured to adapt this specific motor and has a more reasonable gear ration (16:1) but is still able to provide a stall torque of 38Nm.

P80 Planetary Gear Box 16:1

PHYSICAL PARAMETERS

Type Planetary

Reduction 16:1

Stages 2 - 4:1, 4:1

Gear Material Hardened Steel

Weight 41oz

Length 3.2 in (81.3mm)

Width (Square) 2.5 in (63.5mm)

Shaft Diameter 0.50 in (12.7mm)

Shaft Length 3.25 in (82.6mm)

Shaft Key 0.125 in (3.2mm)

Shaft End Tap #10-32

Base Plate Mounting Holes (12) 1/4-20

Shaft Stack Mounting Holes (4) M4 x 80mm

During assembly, the standard motor shaft will be replaced with the long shaft by removing the four M4 x 80mm screws and the shaft snap ring. The motor will be attache to the gearbox per the instructions provided from Bane Bots.

Motor Shaft and KeywayTo ensure proper torque transmission between the motor shaft and omni-wheels, keyway stress analysis was performed on the motor shaft keyway and omni-wheel keyways. In summary, the ⅛” square motor shaft keyway should be made of 18-8 steel with clearances noted in the tables below. The 6.6mm x 5.5 mm rectangular omni-wheel shaft keyway should made of 18-8 steel as well. The stress are much lower in the omni-wheel key so only the sizing and stress calculations are provided. An online resource [http:\\www.tribology-abc.com] was used to determine the stress values giving a certain size and maximum torque input. Tolerances were taked from the Machinery Handbook, found online using UM’s Miryln system.

Motor Shaft Stress CalculationsKeyway: Shaft size 5/16” thru 9/16” => Sq. = ⅛” , rect. = 3/32” Seat Depth: Shaft size 5/16” thru 9/16” => Sq. = 1/16” , rect. = 3/64” Key Stock Sizing

Tolerances:

Keyway Fit Tolerances

J = S - (M+D) + C Dimensions used for CalculatorFor diameter S = ½”: E = ⅛”, M = 0.079” = ½*(S - sqrt(S^2 - E^2)) Material Properties:Shaft: Chromium steel ETD150 => S_yield = 150 kPSI = 1034.214 MPaKey:

Material S_tensile (kPSI) min key size (in)

416 SS --- 3/16” undersized: -0.002”

316 SS 75 3/16” undersized: -0.002”

18-8 SS 75 1/8” (over or under): +0.002”, -0.002”

brass (360) 50 1/8” (standard): +/- 0.005”

Aluminum (6061) 38 1/8” (standard): +/- 0.005”

Alloy 8630 steel 90 1/8” (oversize): + 0.001”

High C: C1090-1095 90 1/16-1/8” (standard): +/- 0.0005

Plain Steel 55 1/16-1/8” (oversized): + 0.002”

Zinc Plated Steel 55 1/16-1/8” (oversized): +0.003”, -0.002”

Stress CalculationsDimensions: d - 0.5” = 12.7mm, T = 400 in-lb = 45.1939 Nm, L = 1” = 25.4mm**key width b,h = ⅛” = 3.125 , key seat = t1 = 1/16” = 1.5875 T_shear,key = F*S/(L*b) = 176.576 MPa = 25.61018 kPSIP_bearing = 179.903 MPa = 26.02006 kPSIS_torsional = 167.731 MPa = 24.32732 kPSI

**For FOS > 2, Alloy 8630 and High C keys are best choices, 18-8 is cheaper Adapter Slug Stress CalculationsModified from standard in Machinery Handbook to fit omni-wheel

KeywayShaft size 9/16” thru 7/8” => Sq = 3/16” , rect = 1/8” Seat Depth Shaft size 9/16” thru 7/8” => Sq = 3/32” , rect = 1/16” Key Stock SizingThe sizing is determined in metric since the wheel key seat giving in metric (will be milled down from 5/16” sq. key stock) Tolerances:

Dimensions used for Calculator:

J = S - (M+D) + C

For diameter S = 22.225m: E = 6.6mm, M = 0.501mm = ½*(S - sqrt(S^2 - E^2)) Material PropertiesShaft: Chromium steel ETD150 => S_yield = 150 kPSI = 1034.214 MPaKey: (found to be same materials as before) Stress CalculationsModified Dimensions: d - 7/8” = 22.225, T = 400 in-lb = 45.1939 Nm, L = 1” = 25.4mm**key width b,h = 6.6mm , keyseat = t1 = 2.75 T_shear,key = F*S/(L*b) = 24.26 MPa = 3.518 kPSIP_bearing = 116.448 MPa = 16.889 kPSIS_torsional = 46.186 MPa = 6.698kPSI **Stress are much less than smaller diameter shaft calculations above. For consistency, the same key stock material was used but it was also noted that the glass filled nylon would not fail from these calculated stresses as well.

BearingsThe bearings are used to provide proper rotational motion between the motor shaft and the wheel. Each shaft - wheel pair requires two bearings. Thus in total, six bearings are required. Since the primary type of load is radial and the maximum load acting on the bearing is not very high (about 50lbs), we selected a ball bearing (McMaster 6383K241).

● Flanged Open Ball Bearing specifications:○ Bore diameter = 0.5“ ○ Outer diameter = 1.375“○ Width = 0.5“○ Flange outer diameter = 1.5”○ Flange thickness = 1/16”○ Dynamic radial load = 450 lbs○ maximum rpm = 1000○ Bearing material = Steel

Bearing Calculations:It is important to determine the bearing life before we start using the bearings. It is also necessary to determine the fits and tolerances. The calculations are performed based on data and tables from the Machinery’s Handbook. Determining the C-Rating:This is the first step in the calculation since most of the other subsequent calculations are based on this value.

where D = ball diameter = 0.211 inchesZ = number of balls per row = 12i = number of rows of the balls= 1= angle of contact between the ball and the bearing surface = 0 degrees

Determination of fc:

= mean diameter = (0.5+1.375)/2 = 0.9375”D*cos( )/dm = 0.225”From the table, fc = 4517.5 (further calculation to determine fc was performed as the value of

D*cos( )/dm lies in between the two given values )Substituting the values,C = 4517.5*1*[12^(2/3)]*(0.211)^(1.8)C = 1437.25 lbf

Bearing life:This quantity indicates the number of cycles a bearing will last before failure. By definition it indicates 90% probability that a bearing will sustain till the calculated bearing life. Bearing life L_10 = (C/P)^3P = total load on the bearing = radial load + thrust loadAssuming a total load (Round-a-bot + person) = 300 lbsLoad on a single bearing = 300/6 = 50 lbsAssuming maximum tilt from vertical = 10 degrees

Load on a single bearing = 300*(sin 10)/ 6 = 8.682 lbs=> total load = 58.682 lbs = 1888.03 lbf => Bearing Life L_10 = (1437.25/1888.03)^3 = 441134 cycles.

● Bearing fits and tolerances:

(applied radial load)/(max radial load capacity) = 50/450 = 0.111From the above table, we see that the bearing can be considered as normally loaded.

from table 13, considering normal loading, for bore diameters less than 0.71” tolerance on the inner diameter = j5 = +0.0002” -0.0001” Tolerance on the housing:

from Table 14 of the Machinery’s Handbook section on bearing tolerances, the tolerance on the outer diameter = J6 = -0.0002” +0.0004” (values of j5 and J6 obtained from table 15)

Wheels The drive wheels are responsible for supporting the weight of the robot and rider, and for translating the motor rotation to the ball. In order to support the holonomic performance of the robot, they must also be able to roll in the axis normal to the wheel’s plane. The wheels were selected based on size, strength, torque transmission, and availability. Based on these criteria, Rotacaster wheels RW610 supplied by Kornylak have been selected. The wheels will be paired, as each wheel has an axial rating of 100 lbs, so we get a factor of 2.4 safety factor assuming a total load of 250 lbs. Additionally, the wheels have a radius about half that of the proposed ball, so they effectively act as a 2:1 gear with the ball. The width of the wheels is thin enough such that a pair can fit between the bearing mounts on the motor shaft. The shape of the rollers is such that no corners of the wheel can interfere with the ball surface, which would be a concern for rollers that were enclosed within the wheel casing, such as the Kornylak-supplied Transwheel series. The rollers are made of injection-molded polymers, which should provide decent friction with the ball, if paired with a good ball surface.

The specifications for the wheels are as follows:● Max Load: 100 lbs ● OD: 4.92"● Bore: 0.875" with 5/16” keyway● Width: 1.05"● Hub Material: Glass-Filled Nylon● Roller Material: Injection Molded Polymers● Price: $15

Double-wheels are available together, but they lack keyways for torque transmission. For the single wheel, the keyways are offset such that the rollers are staggered to create a smooth circular shape for the wheel profile, and preventing the rollers from losing contact with the ball surface. The 3D modeling section previously shows how the drive train, including the wheels, shaft, and motors are assembled.

BallThe ball was selected based on the following parameters:

● Diameter: 8 in● Weight: 10 lbs● Gripping surface to limit motor slip● Rigidity to support 200 lbs● Cost <$50

The selection of the ball is critical as it will determine the effectiveness of the torque transfer from the omni-wheels to the ground. In order to optimize this transfer, we must limit losses due to sliding and deformation of the ball. We determined that a medicine ball best meets all of these parameters. They come in various diameters and weights, are relatively inexpensive (around $20 depending on the model), have a rubber surface for acceptable grip, and are much more rigid than basketballs. Medicine balls can be found at all sporting goods stores, and we have chosen a locally available TKO 8 lb rubberized medicine ball.

Power SupplyThe selection of the batteries was mainly based on power requirements, weight and size concerns, and the charging and discharging cycles. Power requirements are the most important parameter based on which we selected our battery. We wanted a battery which would be capable of providing the maximum 133 A current, for the motors, at 12V. A separate set of batteries that can provide 24V to the cRIO since the current requirements are not so high for these set of batteries. Next, it was important to consider the weight of the battery as they are significantly heavier than the other components. It is also required that the batteries should not be physically very large, so that they do not occupy lot of the area on the base plate. Finally we have to look into the charging and discharging cycles of the batteries. To determine when exactly a battery can get completely discharged is a bit complicated. Also for a robot with

a person riding on it, we do not want to rely on this value. Therefore, it is necessary to plan beforehand, to consider the discharging cycle of the battery. The discharge depends on a few of different parameters; capacity, amount of current drawn, and temperature. Thus it is always better to have a factor of safety in this aspect and go for a battery which provides more than what we want. Charging time of a battery is also an important consideration. It should not be the case that a battery operates for one hour, and takes a day to recharge. Finally we have to consider the cost of the battery. While selecting components , there always exists some kind of trade-off. Cost plays an important role in deciding which choice to go with. We had the choice for going with Li-ion, Ni-MH or a sealed lead acid battery(SLA).The Li-ion and the Ni-MH batteries provide very good performance and are also very light. But because of our limited budget, we did not choose one of these batteries as this would really tighten the budget for the other components. The only choice left is a SLA battery. Among the SLA batteries, we looked for batteries that can provide us the required current at a specific voltage and have an adequate capacity. The weight and the physical size of the batteries were also considered.

Motor BatteryThe battery capacity calculations came from NexEnergy’s free online tool [http://www.nexergy.com/capacity-calculator.htm]. The ‘FIRST CIM Motor’ :

● Requires 12 V operating voltage.● 133 A peak stall current.

The capacity of the battery was calculated based on the following values: Required run time : 1 hrStandby current : 2.7APeak current duration: 30 secNumber of peaks per hour: 6The required capacity was calculated to be 10.3 Amp-hrs. Battery Selected: Powerstar Sealed Lead Acid (SLA) Battery

Battery Specifications:

● voltage:12 V● capacity: 15 Amp-Hrs● peak current: 210 Amps● size: 5.95x3.86x3.98”● weight: 9.26 lbs

Discharging Characteristics:The discharging characteristics of the battery was determined from the battery discharge vs time curve assuming a nominal operation of the battery at a specified C rating. The question was, how long would the battery last if the motor requires 4 A continuous current at 12 V

=> 4*2 amp(2 motors) = 8/15 = 0.533*CFrom the battery discharging cycle curve, at this rating, the battery would last for 1.5 hrs, with a 1V decrease in operating voltage.Charging Characteristics:The charging characteristics depends on the ‘state of charge’ before the battery charging cycle starts. Assuming that the battery is discharged to 50% of its charge before recharging, a 12V battery would require 4 hours to charge. Auxiliary BatteriesThese batteries power:

● NI cRIO● Inertial measurement unit (IMU)● Encoders● Other electronics circuitry.

The cRIO requires 24V. But does not require a very high current for operation. Thus two 12V batteries with relatively low capacity are required in order to power the cRIO and other control circuitry. Battery Selected: PowerStar Sealed Lead Acid Battery(SLA)

● Battery Specifications:

○ voltage: 12V○ Capacity: 3.5 Amp-Hrs○ Peak Current: 52.5 A○ Size: 5.28x2.64x2.62”○ Weight: 2.98 lbs

The relatively small size and weight of these two types of batteries when compared to other SLA batteries which provides us a lot of free space on the plate and does not influence the overall weight of the robot that much.

Battery State of Charge Sensor A simple LED Battery life indicator will be included for each battery. Each indicator had a green, yellow, and red indicator representing good, ok, and low voltage respectively. This will give the user a visual warning when battery life is low. Since low battery voltage could lead to improper performance, it is imperative that the user heed the warning of the LED.

Sensors

Inertial Measurement Unit (IMU)

The IMU is used to measure the orientation of the robot and feedback to the cRIO. Between the analogy output and digital output IMU, there are some factors used to compare. First, the price of the analogy one is half the price of digital one. Furthermore, the gyroscope and accelerometer are on the same board so they come aligned on the same coordinates. The Freedom IDG500/ADXL335 supplied by SparkFun most closely matches our requirements. In addition, the IMU will be contained in a Faraday cage style enclosure to reduce noise.

Specifications: ● Type: Five degrees of freedom IMU combo board ● Sensitivity: 300 mV/g for accelerometer and 2 mV/g/degree/sec for gyroscope● Output type: Analog ● Dimension: 20*23(mm)● Input voltage and current: 3.3V and 10.5mA

IMU FilterIn order to properly utilize the outputs of the combo board we have selected, we must design a filter to combine the preferable characteristics of the accelerometer and gyroscope. A low pass filter is added to the accelerometer to limit the sensor’s output beyond its bandwidth. A high pass filter modifies the gyro output to provide compensation for the drift associated with the gyro. The complimentary filter is implemented by designating the filtered gyro signal as the base signal and adding the filtered accelerometer signal to compensate for the gyro bias. The filter we have designed is based upon previous M-Way teams experiences with the same combo board. When we receive the combo board and are able to confirm it’s performance, we will modify the filter as needed. The filters are denoted as Ga for the accelerometer and Gg for the gyroscope;

EncoderBased on simulations, the a 512 PPR quadrature encoder is sufficient, and based on size and cost, the E7P provided by US Digital is suitable for our requirements. The encoder is mounted to a short turned down section on the end of the shaft to the specified 0.394” (10mm) disk hole. The encoder kit comes with a centering and mounting tool which will be used to attach the encoder once the rest of the motor shaft stack is complete.

Specifications:

● Type: US digital E7P● CPR:180 to 720● Number of channels:2 ● The input voltage and current:5V and 30mA

Miscellaneous Electronics

Emergency Stop

The emergency stop will be connected between servo and power supply and mounted to the frame. The emergency stop will interrupt the power supply to the H-bridge servos rather than the the current sent from the servo to the motors. Specifications:

● IDEC-AVW411-R● Max current: 40amp● Continuous current:10amp

Voltage Regulator

Since the supplied voltage is 5V for the encoder and 3.3V for IMU so we need to transform 24V battery voltage to 5V and 3.3V. The max input current of sensors is around 70 mA so a high power DC-DC converters are not required . Therefore, two voltage regulators will be used to drive the sensors. The regulators have 5V and 3.3V output voltage and 100 mA output current which is good enough for the sensors. Each is mounted to the printed circuit board discussed in the next section. Specifications:

● Type: 78L05 (5V) and LP2950(3.3V)● Output current:100mA● Input voltage range: 0 to 30V

Printed Circuit Board (PCB)In order to power the small sensors, two shaft encoders, and IMU chip, the 24V voltage supply for the cRIO must be reduced using a voltage regulator circuit presented below in the circuit diagrams. The PCB board will be used to connect the voltage regulators to the 24V supply.

● Type: Matching Printed Circuit Board with 550 Connect Points

Wiring PracticesSince the current flowing through parts of the robot is relatively high, proper wiring and connections must be ensured. This will also improve the performances of the main auxiliary 24V circuit by limiting EM noise and sudden drops in voltages. This voltage fluctuation concern was solved by separating the drive circuit and the main power circuit. Proper sizing of wiring and connections to battery terminals, motor controller terminals, and motor wires are provided in the

wire choice section below. High importance wiring practices:

● All sensors and controller cabling will be twisted pair, shielded wires. This will limit noise. ● The motor cables are separated from the wiring of cRIO, motor controller and sensors. ● Single ground point to eliminate ground loops.

Wire choice

AWG Gauge Max amps for chassis wiring Max amps for power transmission

7 89 30

8 73 24 The above table comes from Handbook of Electronic Tables and Formulas for American Wire Gauge. Since the limit of current in this table is very conservative and the power transmission of the robot is not very big, the gauge 8 wire is chose to connect with motors, servo and battery according to the max chassis wiring current limit and the 70A peak current of motors. For the low current circuit, the smaller 16 and 24 Ga wire will be suitable and is readily available in the lab.

Circuit Diagram

Since the voltage source of sensor should be kept constant to reduce the noise, separate batteries will be used; one used to drive the servo and motors, another two used to power the cRIO and sensors.

PCB CircuitsA PCB will be used to as stated before the mount the 24V to 3.5 and 5V voltage regulators. These will power both the motor shaft encoders and IMU chip. It is attached to the top electronics plate using four ⅛” bolts on the corners of the board. It should also be shielded using a Faraday cage style enclosure, discussed in the issues section below.

MicroprocessorThe microprocessor will be responsible for the following tasks:

● Control Algorithm● Sensor Feedback● PWM generator● Accelerometer and gyro filter

We have chosen to use the NI cRIO 9004 and module as our microprocessor. This device uses Labview as the front end GUI and flashes the developed software onto the cRIO FPGA. It requires a 24 V external power supply, draws 48 A of power, and includes 512 MB flash memory and 64 MB DRAM on-board. One advantageous feature of the cRIO is the modularity of the input and output cards. We can choose to place 8 channel analog or digital input or output modules into four slots on the cRIO chassis. For our application we will require one digital I/O module (NI 9401) and one analog input module (NI 9201). The digital I/O is a high sped, TTL logic module with a 10 MHz clock. The analog input module can receive up to a ± 10 V and has a sampling frequency of 500 kHz.The I/Os are arranged as follows;

Component Verification

IMUTo get an idea of the performance for the IMU, we performed some preliminary tests on an IMU combo board with 3 degrees of freedom:

Accelerometer test The IMU was perpendicular to the surface of earth and rotated the IMU by protractor to test the effect of gravity

As can been seen from the figure, when the rotational angle increase from 0 to 90, the gravity on the negative x direction increase and decrease on the minus y direction so the output voltage of accelerometer decrease in x direction and increase in the y direction.

Gyroscope testThe IMU was attached on the arm of motor and the arm was rotated to compare the variance of

angular velocity coming from encoder and output voltage of gyroscope.

As can been seen from the figure, the variance of gyroscope’s output voltage is the same as the variance of motor’s angular velocity.

MicroprocessorWe will use the inverted pendulum system from lab 3 to verify microprocessor performance and our ability to use the cRIO. The Labview VI has been created and we will begin testing the inverted pendulum system over the Thanksgiving break. We have confirmed that we can perform the getting started tutorials with the cRIO, which entails some simple I/O programming and Labview processing.

Issues to be Addressed

Motor ControllerThe motor controller has become a large stumbling block in this project. The Round-A-Bot falls into a void in terms of motor controllers; it is a small robot in terms of weight and size considerations, but requires a large current output for emergency maneuvers. As a result, we have had trouble finding a controller which can provide the needed current at a reasonable price and size. Our current solution is to use the Simple H motor controller; a simple controller capable of providing 20 A continuous, 48 A peak as a full H bridge controller. The controller has the option to function as a ganged half bridge, which can then provide 70 A peak in a single direction. While this meets our current requirements, it fails to meet our need for bi-directional rotation. The following diagram shows how a single Simple H bridge can be used as a full bridge or ganged half bridge;

We have three possible choices of action with the Simple H;

1. Once we receive the motor and build the robot, we may find that the full H bridge configuration provides enough power to drive the robot. In this case, we will use the controllers in their basic function.

2. If we must use the ganged half bridge, it may be possible to use two half bridges to manually create a full bridge. This option implies purchasing two motor controller, and borrowing two motor controllers for the duration of the project. Since we currently have access to the motor controllers which could be borrowed, we can test this method early on to determine if it is in fact an option.

3. The final option is to change the project goals. Instead of carrying a 200 lb person, we can limit the riding capacity to a smaller weight. This would reduce the torque requirement substantially and allow us to use the Simple H in full bridge configuration.

This Thanksgiving break, we will test option #2 with the borrowed motor controllers. If this options is possible, we will go ahead with the purchase of the Simple H motor controllers. If it is not, we will consider revising our requirements to match the Simple H full bridge capabilities.

Top Plate RearrangementThe top plate arrangement has become a topic of great debate. The issue is that noisy components (the motor controllers) are situated near the sensitive sensor circuits. We have three suggestions / ideas which we are considering. None of the ideas require additional on-line

ordering and a decision will be made over Thanksgiving break. Our possible solutions are;1. Use high quality wiring practices to limit the noise within the cables. This solution does

not address the issue of the noise source, but will attempt to limit the noise as much as possible within our current design. All sensitives circuitry will be separated from the motor controllers and use twisted pair, shielded cable in order to limit noise in the wiring. We could include ferrite beads at the motor controller to help suppress some of the noise in the motor signals also.

2. Install a Faraday cage around the motor controllers to limit noise radiation. We could install a wire cage around the motor controllers, which would limit the radiated noise. A cage is preferred over a box as it will still provide airflow over the motors and prevent overheating.

3. Rearrange the top and base plates. We could move the motor controllers to the base plate and the cRIO batteries to the top plate, essentially isolating the high powered and noisy motor circuit from the lower power and more sensitive controller circuit. This solution would have two benefits:

a. The noisy components would be placed on the base plate, essentially isolating the sensitive equipment from the larger noise sources.

b. The high power motor circuit would be entirely located on the base plate, limiting the length of high powered wires.

We will consider all three solutions over the Thanksgiving break and proceed with the chosen solution the following Monday.

ConclusionsThis report demonstrates that all major and minor component and material selections have been made. Based off of performance simulations using planar mathematical models, and a controller implemented in Simulink (including real-world parasitic effects), the robot meets the desired technical specifications. The fabricated and modified parts for this robot have been designed, and the engineering drawings have been completed. The Bill of Materials is completed and a link can be found at the end of this document. Preliminary component verification and testing is underway and will continue through the next week. Some minor changes to the design of the robot will be implemented following the design review meeting. The major issues are discussed in the previous section. However, none of them have a substantial impact on the majority of the Bill of Materials. and thus we are ready to begin placing orders for almost all components and raw materials.

Bill of MaterialsThe complete Bill of Materials for the Round-a-Bot can be found here: http://tinyurl.com/roundabot-BoM

Engineering DrawingsThe complete set of Engineering Drawings for fabricated parts and modified purchased components can be found here: http://www.umich.edu/~svozar/roundabot_drawings

Appendix A - Simulation Parameter File% Parameter file for planar model simulation

%Dynamic Model

g = 9.81; %[m/s^2] Gravitational acceleration

r = 0.254; %[m] Ball radius (10")

Mb = 3.629; % [kg] Ball mass (8lb)

Ib = 2/5*Mb*r^2; % [kg-m^2] Ball Moment of Inertia

mu_theta = 0.1; %Viscous Damping ball-ground

mu_phi = 0.1; %0.1; %Viscous Damping ball-body

Mc = 267.2/2.2; % [kg] Cylinder Mass

l = 32.41*.0254; % [m] Distance from CoM of ball to CoM of Cylinder

Ic = 78038/2.2*(.0254)^2 + Mc*l^2; % [kg-m^2]Cylinder Inertia

Gamma1 = Ib + Ic + Mb*r^2+Mc*r^2+Mc*l^2;

Gamma2 = Ib + Mb*r^2 + Mc*r^2;

%Actuator

r_wheel = 0.124968; %[m] wheel radius

GR = r/r_wheel; %wheel-to-ball gear ratio

Kt = 0.02104342; %[N-m/A]motor torque constant

GBR = 16; % gearbox ratio

curr_sat = 70; %[A] motor current saturation

deadband_width = 1; % [Nm] motor deadband width;

deadband_initial = 0; % [Nm] motor deadband initial output;

servo_amp_noise = 1e-6;

actuator_quant = 70/2^12; %Amps

%Sensors

sample_rate = 500; %[Hz]

encoder_PPR = 512*4;

gyro_quant = 2000/2^12; % mV

accel_quant = 1800/2^12; % mV

gyro_noise = 0.8; % mV (rms)

accel_noise = 45e-6/1^2; %V (rms) at 1Hz

S = 300; %[mV/g], Accelerometer Sensitivity

Vb = 1500; %[mV], Accelerometer Bias Voltage

Sg = 2.0 * 180/pi; % [mV/(rad/s)], Gyro Sensitivity