Autonomous Vehicle Robotic Wheel Drive

The purpose of this project is to create a compact high torque/rpm robotic wheel drive for Bastian Solution's autonomous warehouse robot.

=Problem Definition=

Background
Bastian Solutions has a current WDS (wheel drive system) that uses too much horizontal space. The axial length of their current design is approximately 8 inches, with the motor contributing to most of the assembly's axial length. For their current motor to meet the required torque/rpm and power conditions, it uses a 4:1 planetary gearbox. A planetary gearbox was chosen because it has advantages over other gearboxes that work well in this application.

Pertinent planetary gearbox advantages:

 Naturally compact, but can supply high reduction ratios (reduces WDS axial length, most motors cannot supply the required torque without modification)   Inline or coaxial design allows input shaft to be inline with gearbox (saves radial space)   Generally have low backlash (vehicle must move with precision)   The nature of their design distributes internal gearbox loads more equally (WDS undergoes high loadings)   Long life and efficiency (gearboxes are expensive to replace and they add to mechanical energy loss)</li> </ul>

For them to add more features and to further optimize the vehicle design, they must decrease the volume of the WDS. They require the WDS to be capable of meeting two primary torque/rpm conditions as well as moving a payload up to 100 lbs.

Specifications
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Volumetric Constraints

The full WDS assembly must not exceed 4.4 inches in axial length nor 8 inches diametric length.

=Iterative Design=

Sub-Systems
Our first step was to break the WDS into 3 sub-systems:  Motor</li> </ul>  Gearbox</li> </ul>  Output Shaft</li> </ul>

Design Procedure
For this project there are quite a few known variables, but also quite a few unknown. The biggest constraint is the volumetric constraint of about 8 inches in diameter by 4.4 inches axially. The strategy is to find the thinnest motor who's diameter is at least half an inch smaller than the 8 inch diametric constraint. The motor has to be thin enough to leave room for the gearbox and bracket, while also being capable of providing a reasonable torque/rpm.To figure out what would be a "reasonable" torque/rpm from the motor, one of our members performed an analysis. The analysis showed the different torque/rpms the motor would need to provide using different gear reduction ratios to meet the torque/rpm requirements. This approach has allowed us to narrow our search and filter through good and bad motors.

Dimensional Analysis
Our torque and spatial requirements for our motor led us to investigate pancake-servo motors. Three brushless series pancake-servo motors from three different companies stood out: Printed Motor Work's motors, Koll-Morgen’s motors, and the Thin Gap's motors. Pancake-servo motors have a very high torque constant and a relatively low motor constant which allow for high torque at low voltages. This combination was essential for our needs. We found that a space constraint of less than 7.2” in diameter (referenced to our axle) would be critical to meet our space constraints, as well as an axial length of less than 3.4 inches for the motor and gearbox combination.

Power Analysis
To meet the power constraints, the team created a table in Excel that approximates electrical power required for a given torque, speed, and gearbox ratio. The inputs of the table were the motor constants, mechanical and electrical restrictions of the motor, and the power restrictions from the client. The table used the below governing equations:

$$I=\frac{(T+B\omega)}{k_t}$$,  $$V=\frac{(T+B\omega)R}{k_t}+k_e\omega$$

{Insert picture of graph from Excel}

The selection process involved recording a motor’s specifications into the table and then iteratively changing the gearbox ratio until the motor met every requirement. If the motor didn’t meet all of the requirements within a gearbox ratio range of 2:1-6:1, then a new motor was selected and the process repeated. Eventually, after many iterations, two motors became prime candidates; the TG5152 and the TG5153.

{Insert picture of graph from Excel}



Thermal Analysis
The next step was to perform a parametric study on the heat dissipation (W) of the motor in relationship to power supply and various torque and speed combinations. An electric motor will dissipate heat due to resistance in the coils and damping from lubrication. Considering the gearbox and bearings, the damping coefficient, B, was estimated to be 0.001- 0.003 (Nm∙s)/rad. A Matlab script (below) was written to produce the heat dissipation due to power-loss. {Insert Picture of Matlab Code}

The input values for the script were k_t, k_e, R, and B. To calculate the power-loss, the equations for Voltage and Current were used above along with the below electrical and mechanical power equations.

$$P_E=VI$$

$$P_m=\tau\omega$$

Gear Design and Selection
 Load Analysis</li> </ul>

According to the American Gear Manufacturer’s Association (AGMA), there are two primary formulas in practice when modelling gear life. The first is the fundamental formula for bending stress,

$$\sigma=\frac{W_tK_a}{K_v}\frac{P_d}{F}\frac{K_sK_m}{J}$$

where σ denotes the bending stress acting through the base line of a single gear tooth in mesh. The other variables constitute gear geometry, transmitted load, and a variety of other application factors, which can be referenced in Shigley and Mischke’s “Mechanical Engineering Design.”

The σ associated with the bending stress formula can be substituted for using the maximum allowable bending stress formula, given material and operating condition. This formula is as follows,

$$\sigma=\frac{S_tK_L}{K_TK_R}$$

where σ_all solves for the allowed bending stress, S_t represents the AGMA bending strength of a given material, and where the K factors adjust for life, temperature, and reliability accordingly. Once an allowable bending stress has been solved for, it is substituted into the bending stress equation to back-solve for W_t, which is the maximum allowable tangential load that the subjected gear can handle. As long as this value translates into a smaller torque load than what is required by the robotic wheel, the gear satisfies the prototype requirement.

The second primary formula accounts for pitting resistance, and is written as,

$$\sigma_c=C_p\sqrt{\frac{W_tC_a}{C_v}\frac{C_s}{Fd}\frac{C_sC_f}{I}}$$

where σ_c denotes the absolute value of contact stress allowable in preventing surface failure due to repetitive mesh. Like with the bending stress equation, σ_c is replaced with the allowable contact stress for a given material and operating condition. Allowable contact stress is written as,

$$\sigma_{c,all}=\frac{S_cC_LC_H}{C_TC_R}$$

where σ_(c,all) is the allowable contact stress between meshing teeth. Once again, W_t can be back solved to ensure torque requirements will not overload a chosen gear. It is assumed that multiple contact points on the gears within the planetary gearbox should be ignored, and only one contact point should be evaluated. Although, it is likely that the tangential load can be modeled as dispersed among points of contact.

 Material Analysis</li> </ul>

Because the allowable bending and contact stresses for a given gear geometry are dependent on material properties, an iterative approach is used to find an adequate material. S_t and S_c in the previous equations denote AGMA’s bending strength and fatigue strength respectively. It is important to note that these values are only used when analyzing gears, and are represented as the mean of a possible range of values. The design range for steel’s bending strength lies somewhere between the following two boundaries:

$$S_t=6235+174H_B-0.126H_B^2$$

$$S_t=-274+167H_B-0.152H_B^2$$

And a materials surface fatigue strength between:

$$S_t=27,000+364H_B$$

$$S_t=26,000+327H_B$$

where H_B represents the chosen material’s Brinell Hardness. Because each strength variable can lie within the AGMA ranges listed above, an average of the governing equations is used in this analysis. To satisfy the tangential loads solved for in the bending and pitting stress equations, a material with sufficient Brinell Hardness must be chosen. For the current prototype, Brinell Hardness of at least 250 is required, but a greater hardness is expected for final production attempts.

Casing design


 Spatial Analysis</li> </ul>

For our gearbox design to meet our spatial requirements we need to design our casing to be as slim as possible. We are hoping to incorporate our mounting bracket into our gearbox design so that we can eliminate the need for more bearings and space usage in our final assembly.

 Thermal Analysis</li> </ul>

The high torque and RPM operating range of our gearbox means that we will need to consider the thermal characteristics of our gear operation. Most planetary gear trains produced today are over 95% efficient, however that does mean that a certain portion of mechanical energy is translated into heat caused by frictional losses. Due to the prototype nature of our design we will simply estimate our inefficiency in order to determine how much energy must be dissipated away from our gearbox. If not managed, the heat generated by the gearbox can lead to damage to our gear teeth and ball bearings, severely reducing the life span of our gearbox. We are looking into using aluminum for the gearbox casing as this gives us much higher thermal conductivity and helps cool our gears and bearings.

<ul> <li>Manufacturability Analysis</li> </ul>

In order to deliver a cost-effective design we must simplify our part geometry and create a design that can be produced quickly and efficiently. To accomplish this we are focusing on a design that minimizes CNC tool changes and allows for use of larger cutting tools to reduce time spent fabricating. Using aluminum will also speed production as it is a soft metal and will process quickly on a mill for production.

Carrier Plate Design


<ul> <li>Fatigue Analysis</li> </ul>

Our design must be durable and not allow the carrier plate to crack after years of abuse. To accomplish this we will design for infinite lifespan with a loading safety factor. The primary concerns in designing for this is how we mitigate stress concentrations at the planet gear pins and our output shaft. We also want to minimize weight and rotational inertia so we can maximize our gearbox efficiency.

<ul> <li>Material Analysis</li> </ul>

Due to the reversed nature of our loading we need a fairly ductile material that won’t crack due to repeated loadings in multiple direction.

<ul> <li>Manufacturability Analysis</li> </ul>

Low cost and speed of production are critical for this piece, ideally we can create this from a flat cut profile to reduce fabrication costs.

Shaft Design Analysis
Stress and Fatigue Analysis

The durability of the input shaft (from motor to gearbox) and the output shaft (from gearbox to wheel) can both be analyzed using Stress – Life criteria. For the ease of presentation, some information is omitted from the following explanations.

To begin analyzing the shafts, the geometries and acting loads are evaluated. For example, bending stress (σ_x), torsion, and shear as a mode of torsion (τ_xy) are present in the output shaft.

The endurance limit of a mechanical element represents the stress a part can endure before succumbing to fatigue failure from cyclic loading. The unmodified version of this parameter is given as:

$$S_e^'=0.504S_{ut}$$

where S_ut is the ultimate tensile strength of a given material. Once the unmodified endurance limit is found, geometric characteristics of the shaft reduce that limit until the shaft’s unique endurance limit is found. A simplified version (no stress concentrations) of the endurance limit is calculated from,

$$S_e=S_e^'k_ak_bk_c$$

where k variables represent surface finish, part size, and load factors respectively. After utilizing the Shigley and Mischke text to solve for each variable, the unique endurance limit for the shaft is calculated and can be compared to the total stress within the component. Still ignoring all stress concentrations, the total stress (σ') is found by utilizing a simplified version of the governing von Mises equation:

$$\sigma^'=\sqrt{\sigma_x^2+3\tau_{xy}^2}$$

Comparing σ' to the calculated endurance limit and given yield strength (S_y) of the material, safety factors guarding against fatigue and yield failures are found as,

$$\eta_{fatigue}=\frac{S_e}{\sigma^'}$$

and,

$$\eta_{yield}=\frac{S_y}{\sigma^'}$$

Because the shaft should seldom or never be replaced during the life of the wheel assembly, “infinite” life is desired. Infinite life in this case is recognized as somewhere between $$10^6$$ and $$10^8$$ cycles, and is satisfied if $$\eta_{fatigue}$$ > 1. If the condition is not satisfied, a harder steel must be selected, or shaft geometry must be altered. Currently, many heat treated steels of at least 5/8 inch diameter are satisfying infinite life cycles for the prototype output shaft. The input shaft has yet to be analyzed.

Design Concept
We decided to go with a pancake gearbox and motor combo. We chose this concept because of its simplicity. However, the most difficult part is the gearbox design because stock pancake planetary gearboxes don't exist, so we have to design it ourselves.

=Team Members=

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