Analysis and Construction of a Bipedal Walking Robot

Sandeep Alankar Jerry Chen

sandeep.alankar4@gmail.com sponge8bob@gmail.com Theo Kadela Elizabeth Petrov Diana Voronin 180497@pcti.mobi elizabeth.petrov@gmail.com dianavoronin5k@gmail.com

Brian Lai* brianlai3@gmail.com

21 July 2017

New Jersey Governor’s School of Engineering and Technology 2017

*Corresponding Author

Abstract — The objective of this project was to geometrically model and create a miniature robot that can walk on two feet and to test the feasibility of using the 11 Holy Numbers in future bipedal robots. The purpose was to imitate human motion and to take advantage of the benefits, such as flexibility in terms of accessible terrain and the ability to add on arm-like limbs. The structure of the small-scale bipedal robot was based off Theo Jansen’s “11 Holy Numbers”, a set of ratios between the lengths of parts and connectors used in creating an artificial leg that could replicate the movement of a human leg [1]. However, Jansen’s “beasts” had many legs to work with, so the 11 Holy Numbers were modified to fit the design of a leg found in a bipedal toy created by Gakken

cgfcgccbgcdFigure 1: Bipedal toy robot

Otonano Kagaku Mook [2] and shown in Figure 1. The next step was to find a way to viably scale the model up into a larger bipedal robot. Ultimately, the results revealed that Jansen’s Strandbeest mechanics could be adapted to utilize the benefits of bipedalism.

I. Introduction

Bipedal robotics has been around since the early 1970s. A lot has changed the past 3 decades, but the overall goal has stayed the same: create a robot that acts like a human to help perform tasks only humans can do.

In the past few years, robots such as Honda’s Asimo and Boston Dynamic’s Atlas bipedal robots have made the news for their very advanced human-like movement, being able to jump and even give out refreshments. However, these robots cost millions of dollars to produce and would not be accessible to the average household.

Theo Jansen, a Dutch artist, looks at robotics from a different perspective. Instead of using expensive servos, motors, and custom-made parts, he uses PVC pipe, zipties, and water bottles. His robots have one axle of rotation, and the legs are able to mimic animal leg movement due to their unique geometry. The ratios of the parts in the leg geometry

1

are called the 11 Holy Numbers, which will be explored in depth throughout this paper.

To make bipedal robots more available in every household, they need to be cheap and efficient. Therefore, inspiration can be taken from Jansen’s cheap and clever design to create an affordable bipedal robot, something that will be further explored in this project.

II. Background

A. Mechanics of Bipedal Robotics

Engineers have attempted to design bipedal robots due to their advantages in mobility and wide range of uses. However, constructing bipedal robots is challenging because as a bipedal robot moves, the center of mass also moves in space. Therefore, in order for bipedal robots to stay balanced while achieving forward motion on only two feet, the two supports must be shifted back and forth so that the center of mass continues to fall over the surface area covered by the feet. As such, the following project explores the physics of balancing the robot’s center of gravity, the structural design of the legs and body, the most effective way to power the robot’s movement, and the size of the support pads located on the robot’s feet necessary to keep it from falling.

To keep the robot balanced, its center of mass must fall over the surface area covered by the feet at each point in its motion [3]. Balancing the robot can be facilitated by enlarging the surface area of the robot’s support pad, or foot. SolidWorks, a 3D design software, can simulate the assembled robot’s center of mass, allowing precise analysis of whether or not the model can stay upright when walking.

Existing sets of ratios which define the length of the leg members in Jansen’s Strandbeests, or the the set of ratios used in Gakken Otonano Kagaku Mook’s toy, are useful starting points when designing bipedal legs. However, modifications are necessary to optimize the robot’s movement. GeoGebra, a 2D analysis software, can simulate how certain changes affect the dynamics of the leg, and more specifically, the path the foot traces out while going through one cycle of motion. There are many criteria to consider, such as rotational direction, stride length, maximum vertical lift, and details about how the foot contacts the ground. Ultimately, the optimal structure results in the best combination of these traits.

B. 11 Holy Numbers, Center of Mass, ZMP

Bipedal robots are generally more applicable and feasible than traditional wheeled robots as they have more mobility and can interact freely with the

Figure 2: diagram of center of mass and Zero motion point when bipedal robot walks

physical world by traveling on various surfaces with greater agility. Two factors are particularly important to consider in constructing a bipedal robot that can achieve optimal balance, speed, and mobility: zero moment point (ZMP) and center of mass (COM). A robot’s ZMP is essential for balance because it is the point at which the foot needs to touch the ground to ensure zero net force and net torque upon impact [4]. COM is the mean position of mass at any given point in the robot’s motion and is also the point on which any uniform force act [5].

In the bipedal robot presented in this paper, analysis of the leg geometry is focused on COM, which is better suited for a statically stable gait. In contrast, ZMP is suited for a dynamic gait. The bipedal robot presented in this paper has a statically stable gait, or a manner of walking in which stability does not drastically change during motion, due to its slow speed and large feet. The bipedal robot’s COM is repositioned in space every time it lifts one of its feet. Therefore, it is essential that at each point in the robot's motion, the COM falls over the surface area covered by the feet [6]. Additionally, the shift in support of COM from one leg to the other must be timed correctly. In order to ensure the support of COM, the legs in the bipedal robot were modeled after those in Jansen’s Strandbeests, or multi-legged structures powered by wind. Jansen developed a set of ratios called the 11 Holy Numbers to dictate the length of the 11 members in each leg in his Strandbeests. Though they are named the “11 Holy Numbers,” there are actually 13 relevant numbers. The two “extra” numbers

2

deal with the central axis and offset axis that make up the crankshaft-not the actual leg. Since bipedal robots can not rely on the multi-legged nature of the Strandbeests to ensure balance, the 11 Holy Numbers ratio must be modified to produce the optimal footpath for a bipedal robot. Thus, an optimized version of the 11 Holy Numbers was applied to a scaled up original toy robot.

II. Procedure

A. Strandbeest Leg Path Mapping

When the leg members in the toy model were measured, the set of ratios was found to be generally similar to the ratio of the members in Jansen’s original 11 Holy Numbers. However, member M, or the axis of rotation, which is shown in Figure 3, had a much shorter relative length in the toy model. Therefore, in order to find the optimal leg geometry for a bipedal robot, the paths of the leg were traced as the axis of rotation changed while all other members remained Figure 3: Strandbeest leg with axis of rotation, M constant. 10 different lengths of member M were analyzed, with each length incremented by 0.1-units within the interval of 1-unit to 2-units. These lengths have arbitrary units since they are based on a set of ratios rather than finite measurements. Gradually incrementing member M resulted in 10 different paths to analyze. The red line in Figure 3 depicts the footpath traced using one length of member M. Analyzing the 10 paths traced revealed how the axis of rotation changes the leg path and what shape the leg path approaches as the axis of rotation increases in size. It was concluded

that a short axis of rotation results in an inefficient, flat walking cycle. However, a large axis of rotation resulted in an extreme maximum height, which would cause imbalance in the robot by displacing its center of mass. Therefore, it was decided that the ideal bipedal robot would need to cover a large horizontal distance and a short vertical distance. B. Definition of an Optimal Leg Path

It was determined that the optimal leg

geometry would produce a leg path that covers a long horizontal distance but a relatively short vertical distance. To create these criteria, the design motives of both the toy model designers and Jansen were explored. Jansen was able to design the feet in his Strandbeests with a longer axis of rotation, and thus a higher maximum foot height and step length. Strandbeests can walk by maximizing the step length. Additionally, the maximum foot height does not negatively affect the Strandbeests’ ability to balance because Strandbeests are multi-legged, meaning that at any given time, multiple feet are on the ground, offering greater stability than a bipedal design in which only one foot is on the ground at any given time. Finally, the high maximum foot height also makes it easier for the Strandbeests to navigate the sandy land of the beaches . Previous research has shown that a step-over gait, in which the swinging leg reaches a higher maximum height, is effective for bipedal robots that must navigate unpredictable terrain such as sand. A higher foot lift ensures the most stable vertical force upon impact with the ground because tangential force is eliminated and small obstacles are avoided. In contrast, the toy designers likely chose a smaller relative axis of rotation because unlike a traditional Strandbeest, the toy only has two legs to support itself on and is therefore more vulnerable to tipping over. Thus, at the cost of stride length and step height, it was designed to take safer and more conservative steps due to its increased danger of falling. C. Why Members A, L, and K Were Specifically Chosen for Testing

In order to determine the leg geometry that

would result in the ideal foot path, two animated models were created in GeoGebra. One model utilized the proportions of the members in the toy and the other utilized Jansen’s 11 Holy Numbers. To design each model in GeoGebra, line segments were

3

created and connected together to form the leg geometry. Then, the leg geometry was attached to an

Figure 4: Strandbeest leg showing axis of rotation animated angle that rotated from 0 degrees to 360 degrees to simulate the rotation of a hip. Finally, the footpath was traced as the leg moved. Each leg member length was controlled by a slider. Therefore, it was easy to change the lengths in the GeoGebra model to match the proportions of the measurements taken from the toy model.

Due to the complexity of the leg geometry, it was difficult to visualize the overall effect modifying each member had on the footpath. Therefore, tests were run on the GeoGebra model utilizing the 11 Holy Numbers. In these tests each member was changed incrementally while the rest of the members were held constant. This showed how specific changes to each part impacted the final result.

Figure 5: Resulting leg paths from changing axis of rotation

A problem was encountered that led to isolating members A, L, and K for analysis. The feet in the toy model moved in the favored direction of counterclockwise (forwards). However, when the measurements of the toy model were applied to the GeoGebra model, the foot moved clockwise (backwards). This confirmed that human error likely affected the measurements of the toy. Additionally, applying small changes at increments as small as

0.1-units to members A, L, and K, which were near the back of the leg and closer to the axis of rotation, caused dramatic changes in the footpath. Thus, to determine the optimal leg proportions, different combinations of lengths of only members A, L, and K were tested in the animated GeoGebra model utilizing the 11 Holy Numbers ratio. D. Finding the Optimal Leg Paths

The possible leg paths were analyzed when

changing the lengths of members A, L, and K because these members were found to most

dramatically affect the shape and direction of the leg

Figure 6: Optimal leg paths path. 27 Combinations of three sizes (small:

75%, medium: 100%, large: 125%) were tested for each member. Each size was a percentage of the original length of the member as derived from Jansen’s 11 Holy Numbers ratio [see Appendix A]. For each footpath of each combination, the maximum horizontal and vertical heights, whether the foot moved clockwise or counterclockwise, whether the foot moved faster on the top or bottom of its path, and the slope of the bottom of the foot path, were recorded. Each footpath was also visually mapped.

The combination that produced the second most optimal leg path (shown in orange in Figure 6) was the medium size for all three members. With this combination the maximum horizontal length was 8.7-units and the maximum vertical height was 2.1-units. The optimal leg path (shown in red in Figure 6) was constructed from a medium member A, large member L, and medium member K. With this combination the maximum horizontal length was 8.5-units and the maximum vertical height was 1.7-units.

Both of the most optimal leg paths displayed the ideal characteristics of the foot moving counterclockwise and faster on the top of its path. Additionally, both combinations traced a line with a near zero slope at the bottom of the footpath. However, the ratio of maximum horizontal length to maximum vertical height was higher for the path

4

shown in red. While the ratio of the orange path was 4.1, the ratio of the red path was 5. It was expected that the 11 Holy Numbers ratio would result in one of the most optimal leg paths. However, by testing various combinations of the three most variable members, a slightly more optimal leg path was found and used to build the bipedal robot presented in this paper.

The second step to determining the set of ratios which would lead to the optimal footpath was determining the optimal length of member M, the axis of rotation. As explained in Strandbeest Leg Mapping, located in Section 2, Procedure, it was observed that shortening the axis of rotation led to a more optimal footpath. Thus, 10 footpaths were traced in GeoGebra with the most optimal combination of members A, L, and K and increments of 0.1-units from 1-unit to 2-units for member M. After analyzing the 10 footpaths traced in GeoGebra when changing the length of the member M, the axis of rotation, it was determined that flatter paths such as the paths shown in pink in Figure 5 would be most optimal. Specifically, these paths were the result of reducing the axis of rotation to 1.28-units, 1.38-units, 1.48-units, and 1.58-units from the original 1.92-units length. All proportions determined to result in the most optimal path were then scaled up to align with the mm lengths of SolidWorks. The measurements were multiplied by 5.58mm, resulting in axes of rotation ranging from 7.14mm and 8.82mm

When the determined measurements were applied to the 3D model in SolidWorks and a motion study was done, the heel of the foot followed the paths predicted by GeoGebra in Figure 5. However, the angle at which the pad of the foot would be tilted up was not considered, and SolidWorks showed that it became steep to the point where the toe was almost pointing straight upwards. This posed a new balance problem, so M, the axis of rotation, was adjusted. The axis of rotation was changed because it was known that small changes in M were predictable and only significantly changed the maximum vertical height of the foot path, which was what was causing the foot to become steep. Therefore, four lengths of the axis of rotation were tested: 10.71mm, 8.04mm, 5.36mm, 2.68mm. In the SolidWorks motion study, the tilt of the foot at the highest position in its movement was observed. Since it was observed that an axis of rotation that was 2.68mm led to the minimum tilt, 2.68mm was used as the axis of rotation.

E. Foot Design

When making a bipedal robot, it is important to design the foot to support the leg. For example, the foot design needs to be flat, and have a large surface area so that the robot will not fall over. It the foot is small or rounded, the robot’s center of mass will not fall over the surface area supported by the foot and the robot will tip over. The foot pad design started with a simple 40mmx60mm rectangle, an estimate made from the dimensions of the Gakken robot’s foot.

How the foot pad was connected to the geometry of the leg was then examined. A modular design was chosen. The first iteration of the foot pad consisted of 3 columns of 4 U-shaped components that allow a 2mm rod to be snapped into place. On the bottom of the inside Figure 7: Foot pad and connecting rod

geometry of each leg, the triangular piece shown on the right in Figure 7 was adapted to also have a snap-in component. This modular design allowed for the foot pad to be connected in different locations in the leg geometry for testing purposes. The other side of the footpad was connected to the knee, creating a hinge (which is explained further in the next section). The piece was also lengthened to allow a greater tilt of the footpad, which is covered in the next section. Ultimately, it was decided that the same foot pad position displayed in the Gakken toy robot, with

Figure 8: Final foot design the leg geometry being connected to the inside back corner of the footpad, was the most optimal. Some of the unused U-shaped connectors were also cut from

5

the foot pad to reduce the weight and clutter. This final position and how the modular components fit together is shown in Figure 8. F. Creating a Natural Foot Tilt

In the process of the designing the foot, a challenge came up: if the foot is always flat and does not tilt side to side, the center of mass will not fall over the surface area covered by the footpad at every point in the leg’s motion. Therefore, the foot needs to tilt in order to shift to a position in which the center of mass is supported by the leg that is contacting the ground at that point in the motion [7]. When one foot is on the ground, the footpad needs to tilt up so that the center of mass falls over the normal of the grounded foot. When the foot is lifted up and suspended in the air, the footpad will naturally tilt down due to the circular rotation of the assembly. Ultimately, the robot should shuffle it's weight from one leg to another while moving forward. As explained in Leg Geometry Assembly, located in Section 3, Construction, each leg is composed of two geometrical sub-assemblies connected by threaded rods and nuts. One side of the foot is connected to the leg geometry subassembly. However, foot tilt is achieved by connecting the side of the foot not connected to the leg geometry to the knee. The knee is the intersection of connectors J and K as shown in Figure 10. As explained in Crankshaft Assembly, located in Section 3, Construction, the crankshaft is composed of two axes of rotation: the central axis and the offset axis. The leg geometry subassembly is attached to the central axis. However, since the knee is attached separately to the offset axis, it spins at a different angle than the leg itself.

Figure 9: Crankshaft

Successful foot tilt was confirmed by examining the full rotation of one leg in a Solidworks motion study. In this motion study, when the foot goes up and is suspended in air, the inner edge of the footpad tilts down. When the foot is on the ground,

the inner edge of the foot pad tilts up. Thus, the robot successfully shifts its weight onto the foot. G. Rotation of Legs Relative to Each other

After the first leg was created, another

flipped version of the leg was created in Solidworks. Then both sub-assemblies were connected together. One leg was positioned at the highest most location in its rotation cycle, and the other leg was positioned at the lowest most location in its rotation cycle. The central axis in the crankshaft was then locked so that when one leg was rotated, the other leg rotated as well. H. Incorporation of a Motor

Both Jansen’s Strandbeests and Gakken’s toy were wind-powered, which was deemed a limitation because wind is not a consistent source of strong power. Thus, the plan was to attach a battery-powered motor to the bipedal robot However, in the end, this was not possible because the right type of motor was not available. Looking at the toy robot, it could be seen that without the windmill in the center, there would be space to put a motor and battery. The center of a robot would have been a particularly convenient spot to put the motor because the weight it adds would not throw off the center of mass. A motor could have been attached to the frame of the robot in such a way that the worm gear on the motor spun a gear attached to the central axis, moving the legs of the robot. A worm gear would have been used because it is effectively a gear with only one tooth; every time the worm gear goes through a full rotation, the gear it is contacting only rotates by one tooth. This would have been necessary to slow down the rotation of the central axis, as the motor’s natural, zero load output is approximately 8,750 RPM. If the legs cycled at anywhere near this speed, the robot would not be able to walk properly. The coin cell batteries and switch were other parts of the circuit that would have had to been balanced, and it’s likely that they would have been placed on opposite feet to maintain balance. The wires would not have been difficult to deal with as there was enough open space in the robot to snake them through, and the motor would have been attached using superglue. The first motor used did not succeed though because it was not powerful enough to turn the central axis. The second motor was more powerful, but was too big to easily attach to the frame. It also spun too fast, which meant that a gear

6

would have to be used to reduce the rate of rotation. Due to time constraints, it was decided to manually operate the robot with a crankshaft.

III. Construction

A. Leg Geometry Assembly

Figure 10: Final leg geometry Each leg is composed of two leg geometries,

with one leg geometry shown in Figure 8. The parts were 3D printed with PLA and the holes were drilled out and hammered to the right size. To connect the two geometries that compose one leg, threaded rods and nuts were used as the support rods connecting the geometries. These threaded rods are 2mm in diameter and 25mm in length. The 3D printed parts were then positioned onto the rods and the nuts were used to hold the pieces in place. The foot pad was attached using the snap-in components and a rod. All the nuts were then fastened to the rods with super glue.

B. Crankshaft Assembly

The crankshaft, which is displayed in Figure 9, is composed of eight rotating ovals, through which run the central axis segments and offset axis segments. The eight rotating ovals are grouped into four rotating pairs. The two rotating ovals in each rotating pair are connected by one segment of the offset axis. The four rotating pairs are connected by central axis segments. As determined by GeoGebra analysis, the central axis and offset axis are the optimal distance of 2.68mm apart.

One important consideration for the bipedal robot design is that the two legs must move at different phases to mimic a walking motion. If the two legs moved in sync, this would result in a hopping motion. Since the rotating oval pairs are

offset, neither the rotating oval pairs nor the offset axis segments rotate in sync. Therefore, the legs also do not move in sync.

Since the offset segments are made of circular rods in circular holes, the leg members connected to the central axis are free to move. In contrast, the central axis, which is turned by a system of rotating gears, is a square rod locked into centered square holes, meaning that all crankshaft segments rotate in sync. Ultimately such a design means that each leg will move at a different time, resulting in a walking motion that still allows freedom for each member to move around the central axis. C. Frame

In the Solidworks model, six individual

triangle pieces were utilized and were connected with three rods to make up the frame. However, when the physical 3D printed parts were assembled, the constraints set in the SolidWorks model were found to interfere with the model and thus the 3D printed assembly had to be altered. When the construction of the 3D printed assembly was started by incorporating all six frame pieces, it was too wide. If the left sub-assemblies were to be attached to the frame in their respective places as they were located in Solidworks, the legs would be too far apart, altering the target center of mass path [8]. If the model traveled wider than the target path, it would result in an imbalanced frame and leg geometries. Then the model would most likely tip over after the first step. To combat imbalance, several ideas were formulated, but the idea of removing two of the frame pieces was more widely accepted for success. When two of the frame components were removed, the center of mass path was tightened horizontally and no excess frame extended outside the legs. Taking away the two frame components also moved the legs closer together, thus combatting the imbalance that tended to result from the alternating footpaths. An additional result was minimal displacement of the center of mass throughout the walking cycle.

7

Figure 11: Final robot frame

IV. Results/ Discussion

Analysis in GeoGebra and SolidWorks made clear the set of ratios which would lead to the optimal footpath. There are some criteria in the data that can be examined to immediately rule out many of the sets of ratios. For example, any set of lengths that causes the foot to travel clockwise would make the robot walk backwards, and any set that caused an extreme spike in the path of the foot would be physically impossible to balance. Other criteria, such as the steepness of the bottom of the path where the foot would be in contact with the ground, the relative speed of the foot at certain points along its path of motion, and the length of the stride compared to the maximum vertical height of the foot all affected the decision for which set of ratios was the most optimal. Paths that were flat at the bottom and had a near zero slope were preferred over those that were inclined because it would unbalance the robot if the feet pushed against the ground too much. Additionally, it was preferred that the foot move relatively faster while in the air rather than on the ground. Then, the center of mass would be shifted to either side for a shorter amount of time in each stride. For the remaining sets, ratio of the length of the stride to the maximum vertical height of the foot was calculated. A higher ratio was considered to be most optimal because a longer stride would result in a faster overall speed for the robot and a smaller vertical height would result in a smaller shift in the center of mass.

Unsurprisingly, Jansen’s original numbers had the second highest ratio of the remaining sets, 4.14. Thus, the analysis agrees that the 11 Holy Numbers lead to an optimal path. However, the set of ratios that kept A and K the same and increased the length of L by 25% had the highest ratio, 5.00. Thus,

it was decided that 4.87 units, 1.25 units, and 7.94 units were the best lengths for parts A, L, and K, respectively. These proportional lengths translated to 123.70mm, 31.75mm, and 201.68mm respectively. Finally, the most optimal axis of rotation, or member M, was reduced by 75% from the 11 Holy Numbers length. The final most optimal length for the axis of rotation was deemed to be 2.68mm. A. Finite Element Analysis

The bipedal walking robot presented in this paper presents a simple mechanical design which can be scaled up and applied to larger walking robots. Although stress on the parts of the leg due to external loads is not a concerning factor in the small scale bipedal robot presented in the paper, larger bipedal robots may prove useful for efficiently carrying heavy loads. Larger bipedal robots face new challenges since as volume increases at a higher relative rate than surface area, the pressure on supporting parts will rapidly increase, eventually leading to the robot no longer being able to support itself. This problem may be resolved by using stronger, denser materials, such as steel. However, using steel would severely increase the mass of the robot, causing it to require a significantly larger amount of power.

It is essential to use Solidworks Simulation and Finite Element Analysis (FEA) to analyze the effect of external loads on the mechanics of the bipedal walking robot design presented in this paper. FEA is a Solidworks feature that allows users to create a mesh that represents the geometry of the 3D model. Users can also impose various controls at vertices, edges, faces, components, and beams to specify certain features of the model such as fixtures, contact sets, and external loads. After running a simulation study, Solidworks then displays the stresses, strains, and displacement experienced by the model.

Several problems were encountered when attempting to perform FEA on the model. For example, the most logical place on the model in which to position an external load would be the flat platform resting on the frame of the robot. However, when a force of 200 N was exerted on the frame, FEA failed and caused Solidworks to crash. Efforts to resolve this issue included modifying the coarseness of the mesh, changing the number of contact points, changing where the force was located on the frame, changing the material, and changing the model, but all of these resulted in the software crashing. Upon

8

further investigation, incorrect placement of fixtures or contact sets were likely to blame. Placing a fixture on a model in SolidWorks Simulation allows a user to specify which points of the model should be fastened to a plane and therefore stay in place when a force is exerted. Selecting contact sets allows a user to specify where parts in an assembly contact and how these parts interact with each other. Due to the complexity of the model, there are many possible combinations of fixture and contact set locations and interactions. Thus, a greater understanding of how fixtures and contact sets should be placed is needed to create a successful simulation study.

The only study found to be successfully performed involved a force of 200 N evenly distributed over the leg of the model as shown by the brown arrows in Figure 12. Additionally, the leg was simulated to be made of steel, a material likely to be

Figure 12: Finite Element Analysis performed on individual bipedal robot leg

used if the model were to be scaled up. One fixture was placed at the tip of the bottom segment of the leg and one contact set was used. As shown, the frame suffers essentially no stress, but the foot and top triangle pieces feel around 1.088e108 newtons of stress where they are more vertically oriented. Connector C, which is attached to both of these pieces, surprisingly suffers very little stress. This indicates that it may be necessary to reinforce the foot and the part of the top triangle that often times lines up vertically with the back of the foot.

It was determined that the placement of the force on the frame of the leg was what caused the initial simulation study to fail. In many settings, robots such as these are used to transport heavy loads

that would otherwise be much more difficult or inefficient move. Heavy loads should then be placed on the frame of the robot to ensure stability. Placing the load on the leg for the purpose of this simulation was deemed a similar enough positioning to adequately determine the stresses that would generally be exerted on the leg due to a load. However, in the future in order to determine the cause of failure in the simulation study, each part of the model assembly would be isolated and FEA would be performed separately on each part. In this way the part causing the failure could be determined.

B. Center of Mass Analysis

In order to see if the bipedal robot

maintained its balance while walking, center of mass analysis was performed to see how it reacted to the walking cycle. If the robot’s center of mass was greatly displaced from its center, then the leg geometry would need editing to better deal with the robot’s weight and walking speed. Once center of mass analysis was performed, it showed that the center of mass was in the middle of the chassis and not directly above one foot as in the bipedal model. Placement of the center of mass over the middle of the chassis would cause the robot to lose balance while walking. As observed in the Gakken toy model, the feet of the toy were extremely close together to ensure that the center of mass was above one of the feet at each point in the motion. Therefore, the crankshaft of the bipedal robot presented in this paper was shortened, bringing the feet closer together to provide the robot with more support and balance. Once the necessary changes were made, center of mass analysis was done again and when one foot was airborne, the center of mass correctly moved above the foot that was grounded while the other was airborne, meaning that the robot would indeed stay balanced when walking.

C. Future Applications

The construction of a bipedal robot that operates based on Jansen’s Holy Numbers ratio creates new possibilities for engineers to build humanoid, bipedal robots that can operate with speed, mobility, and efficiency. Robots built with the Holy Numbers in mind simplify the assembly process and avoid complicated coding and programming of bipedal robot legs. Usually modern bipedal robots have low speed, require complex control, and are expensive to construct. Bipedal robots that contain

9

the 11 Holy Numbers ratio will be lightweight, mobile, and require no overly complicated programming. Legs based off of Jansen’s ratio are light and can move with agility with the use of a motor to propel the robot forward over rough terrains or elevated surfaces. In addition, robots built with the Holy Numbers ratio have a stable center of mass and maintain their balance while walking. Depending on the power of the motor and the material used for construction, bipedal robots in the future possess the ability to truly integrate into the modern environment. The bipedal robot presented in this paper also has the possibility to introduce mechanical engineers to the use of the 11 Holy Numbers in other constructions to provide efficiency and power.

Bipedal robots have many potential applications in people’s homes and in industry. By imitating human movement, robots possess the ability to do complicated human like tasks such as walk up stairs and over rough terrain.

V. Conclusions

The final design decisions for the bipedal

robot presented in this paper are as follows. It was determined that the best ratios for the leg are Theo Jansen’s 11 Holy Numbers, except with part L increased by 25% relative the to 11 Holy Numbers lengths, and the axis of rotation reduced by 75% relative to the 11 Holy Numbers axis of rotation. GeoGebra and SolidWorks demonstrated that the determined set of ratios would result in the best stride and foot path. Finite elemental analysis showed the the heel of the foot was under the most pressure, while the entirety of the frame was relatively unburdened. Moreover, center of mass analysis confirmed that the robot’s center of mass shifted correctly over each foot while the other was lifted. When the 3D printed parts arrived, more modifications had to be made. The feet were too far apart, meaning that the robot would have a hard time shifting its weight from one foot to the other, so two of the six frame components were removed in order to move the legs closer together. A motor could also have theoretically added. It would have been placed in the center of the frames to avoid throwing off the center of mass.

Figure 13: Final 3D-printed model

10

Appendix A:

11

VI. References

[1] JANSEN - KLANN LINKAGE COMPARISON. [Online]. Available: http://www.mechanicalspider.com/comparison .html [Accessed 5 Jul. 2017]. [2] Amazon.com. (2017). Theo Jansen method Biped Walking Robot Kit by Gakken Otonano Kagaku Mook: author: 9784056068511: Amazon.com: Books. [Online]. Available: https://www.amazon .com/Jansen-method-Walking-Gakken - Otonano/dp/4056068518/ref= sr_1_13?ie=UTF8&qid=1498702127&sr=8-13&keywords=strandbeest [Accessed 3 Jul. 2017]. [3]Khan Academy. (2017). Khan Academy. [Online]. Available: https://www. khanacademy.org/science/physics/linear -momentum /center-of-mass/a/what-is-center-of-mass [Accessed 7 Jul. 2017]. [4] Vukobratovic, M. and Borovac, B. (2017). Zero-Moment Point-Proper Interpretation. [Online]. Available: http:// ww.ausy.tu- -darmstadt.de/uploads/ Research/LocomotiveSeminar/ vukobratovic-Note 2000.pdf [Accessed 5 Jul. 2017]. [5] Hall, N. (2017). Center of Gravity. [Online]. Grc.nasa.gov. Available: https://www.grc.nasa.gov/www/k-12/airplane /cg.html [Accessed 6 Jul. 2017].[3] Mechanicalspider.com. (2017). [6] Sohn, K., Oh, P. and Jun, Y. (2017). Humanoid robots walking on grass, sands and rocks - IEEE Xplore Document. [Online]. Ieeexplore.ieee.org. Available: http://ieeexplore.ieee.org/stamp /stamp.jsp?arnumber=6556367 [Accessed 5 Jul. 2017]. [7] Erbatur, K. and Kurt, O. (2017). Natural ZMP Trajectories for Biped Robot Reference Generation - Semantic Scholar. [Online].

Semanticscholar.org. Available: https://www.semanticscholar.org/paper/Natural-ZMP-Trajectories- for-Biped-Robot-Reference-Erbatur-Kurt/8b2df2d6bfeaca6a7757f44e724eb56a7042c417 [Accessed 5 Jul. 2017]. [] Kurtus, R. (2017). Center of Gravity - Physics Lessons: School for Champions. [Online]. School-for-champions.com. Available: [2] Amazon.com. (2017). Theo Jansen method Biped Walking Robot Kit by Gakken Otonano Kagaku Mook: author: 9784056068511: Amazon.com: Books. [Online]. Available: https://www.amazon.com/Jansen-method-Walking-Gakken -Otonano/dp/4056068518/ref=sr_1_13?ie=UTF8&qid=1498702127&sr=8-13&keywords=strandbeest [Accessed 3 Jul. 2017]. [] Vukobratovic, M. and Borovac, B. (2017). Zero-Moment Point- Proper Interpretation. [Online]. Available: http:// www.ausy.tu-darmstadt.de/uploads/ Research/LocomotionSeminar/vukobratovic-Note2000.pdf [Accessed 5 Jul. 2017]. [] Elysium-labs.com. (2017). Zero Moment Point (ZMP) | Elysium-Labs. [Online]. Available: http://www.elysium-labs .com /robotics-corner/learn-robotics/biped-basics/zero-moment-point-zmp/ [Accessed 6 Jul. 20

http://www.school-for-champions.com/science/gravity_center .htm#.WV_l7igrIuV [Accessed 6 Jul. 2017]. [8] Elysium-labs.com. (2017). Zero Moment Point (ZMP) | Elysium-Labs. [Online]. Available: http://www.elysium-labs .com /robotics-corner/learn-Robotics/biped-basics/zero-moment-point-zmp/ [Accessed 6 July 2017] Otonano/dp/4056068518/ref=sr_1_13?ie=UTF8&qid=1498702127&sr=8-13&keywords=strandbeest [Accessed 3 Jul. 2017]. [] Vukobratovic, M. and Borovac, B. (2017). Zero-Moment Point- Proper Interpretation. [Online]. Available: http:// www.ausy.tu-darmstadt.de/uploads/ Research/LocomotionSeminar/vukobratovic-Note2000.pdf [Accessed 5 Jul. 2017]. [] Elysium-labs.com. (2017). Zero Moment Point (ZMP) | Elysium-Labs. [Online]. Available: http://www.elysium-labs .com /robotics-corner/learn-robotics/biped-basics/zero-moment-point-zmp/ [Accessed 6 Jul. 2017].

12