Abstract

This article describes the design and the development of a novel six-legged robotic walking machine named SphereWalker. The six legs are arranged into pairs, and each pair of legs is supported and actuated by a single spherical four-bar mechanism. Two of the four-bar mechanisms are operated in a synchronous fashion, while the middle one is operated at 180 deg out of phase with respect to the other two. A physical prototype has been built, a digital twin has been generated, an actuation and control system has been designed, and the technology has been patented.

Introduction

Motivation.

The inspiration for the creation of the SphereWalker hexapod walking machine came from the the first author’s family pet. In 1999, the first author acquired Geochelone sulcata or African spurred tortoise named Chomper as a family pet (see Fig. 1). The complex motion of the front legs of the tortoise and their ability to support the relatively large mass of the tortoise inspired the first author in 2006 to pursue the design of kinematic closed chains that could replicate the motion of the front foot. Initially, the use of open kinematic chains to generate the desired motion was investigated; however, it was soon found that four or more degrees-of-freedom were required for each foot. Simpler, lower degree-of-freedom solutions were sought; hence, the investigation shifted focus to kinematic closed chains. In Fall 2013, efforts began to identify kinematic closed-chain architectures that would effectively and efficiently replicate the front leg motion of Geochelone sulcata. The SphereWalker design presented here, based on a one degree-of-freedom spherical four-bar mechanism, is one of the more promising architectures that the authors have designed and investigated. Another promising design by Du and Larochelle, reported in Ref. [1], employs a two degrees-of-freedom S-C-U dual planar four-bar linkage. One linkage is used to actuate each foot of the hexapod. The result is a walking machine with 12 actuated degrees-of-freedom. In contrast, here, we present SphereWalker that has three actuated degrees-of-freedom.

Fig. 1
Fig. 1
Close modal

Human beings have long been curious about the behavior of the world’s wonderful creatures and have tried to understand and imitate them. The earliest walking machines were mechanical toys. Their legs were driven by cranks or cams from a source of rotary power, usually clockwork, and executed a fixed cycle [2]. One of the first documented walking mechanisms appeared in about 1870 and was based on a four-bar mechanism invented by the Russian mathematician P. L. Chebyshev as an attempt to imitate natural walking (Artobolevsky, 1964) [3]. In 1893, the first patents for legged systems were registered with the US Patent Office [3].

Walking animals, insects, and mechanical devices may be classified based on the number of legs that they have. There are bipeds, e.g., humans or birds; quadrupeds, e.g., mammals and reptiles; hexapods, e.g., insects; and octopods, e.g., spiders [3]. A hexapod robot is a mechanical device that walks on six legs. One example is RHex, a biologically inspired hexapod with compliant legs [4,5]. SphereWalker is another example of a biologically inspired hexapod device.

Project Overview and Goals.

The goal of the SphereWalker project is to design, simulate, manufacture, and test a hexapod walking machine whose leg pairs are actuated by spherical four-bar mechanisms that replicate the motion of the front leg of Geochelone sulcata. The idea was to design a single one degree-of-freedom crank-rocker spherical four-bar mechanism whose coupler link can be extended such that each end of the link supports one of the hexapod’s feet. Then, three identical mechanisms would be used in the device, each driving a pair of legs of the hexapod. Traditionally each leg of a hexapod is driven by at least one actuator [4,610]. By using a single one degree-of-freedom mechanism to drive a pair of legs, only three actuators are needed to drive all six legs, thereby reducing the cost, weight, and control complexity. Our expectation is that the resulting hexapod proves to be effective when navigating rough terrain, both indoors and out, and in applications where energy efficiency is paramount.

Related Works and Paper Outline.

Related hexapod walking machine works include the University of California Irvine Spider designed by Soh and McCarthy [10]. In Refs. [4,11], a biologically inspired hexapod with compliant legs named RHex is presented, and in Ref. [5], an open-loop controller is presented that enables RHex to climb stairs. Wait and Goldfarb [12] present a biologically inspired method for the control of the location of a robot hexapod. They build upon the WalkNet control structure to yield stable gaits. In Ref. [13], an adaptation strategy for adjusting, in real-time, the stride in a running hexapod’s gait is presented. In Ref. [14], the robustness of a neural network-based locomotion controller for a hexapod is studied. Finally, the computer-aided design tools that were created by the Robotics and Spatial Systems Lab at the Florida Institute of Technology and later by the Robotics and Computational Kinematics Innovation Laboratory at the South Dakota School of Mines & Technology and used to design SphereWalker are reported in Refs. [1517].

A survey on hexapod walking robots and gait planning is presented in Ref. [18]. Tedeschi and Carbone [19] present the design of a novel hexapod with wheels employed as feet [19] and a study of motion and path planning for hexapods in Ref. [20]. The design and analysis of a hexapod based on the classical walking mechanism of Theo Jansen can be found in Ref. [21]. Skaburskyte et al. present a study of gait stability for hexapods [22]. The kinematics, dynamics, and power consumption of a hexapod with a tripod gait is studied in Ref. [23]. Sahin and Stevensen study the dynamics and control of a hexapod with six legs that are actuated by dedicated planar mechanisms [24]. Obstacle avoidance and motion planning for the Octopus-III hexapod is studied in Ref. [25]. Additional works related to the design, actuation, gait, and control of hexapods include Refs. [26,68,27,28,9,2934].

The following sections provide an overview of the SphereWalker mechanism, the process of manufacturing and assembling the components of the SphereWalker, creation of a digital twin, simulation results of the digital twin, and acknowledgments.

Mechanism Overview

The SphereWalker, see Figs. 2 and 3, is composed of three spherical four-bar linkages each connected to aluminum base plates, which are in turn connected by two revolute joints. Each linkage is made of identical components although the central four-bar is rotated 180 deg about the vertical, and the mechanism is assembled in the other circuit. The SphereWalker invention was patented in the United States in 2015 [35]. Additional information, including photos, simulation videos, and video recordings of the physical prototype walking, may be found here [36].

Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal

The legs and feet that propel the SphereWalker are integral extensions of the coupler link; see Fig. 4 in which the input link is shown in green, the fixed link in yellow, the driven link in red, and the coupler link in blue. The two distal ends of the coupler link constitute SphereWalker’s legs, and its feet are the extreme ends of the link. Having each spherical four-bar mechanism operate a pair of legs and feet, as opposed to a single leg and foot, allows for the SphereWalker to always have three points of contact with the ground at any given time while only requiring three mechanisms instead of six, thereby reducing complexity and mass. Computer-aided design and analysis software for spherical four-bar mechanisms ([1517]) were utilized to design the SphereWalker mechanism such that the foot’s coupler curve closely resembled the motion of the foot of the walking Geochelone sulcata. Moreover, the two fixed axes of the spherical four-bar mechanism were oriented to replicate the fixed pivots in the anatomical shoulder complex of the Geochelone sulcata, one fixed axis being horizontal and the other being near vertical. Qualitative observational studies of the walking motion of Geochelone sulcata wee conducted by the investigators. Multidimensional sketches and feature notations of the foot’s motion were recorded. Dimensional synthesis for rigid-body motion generation was performed using sfb designer and sphinxcam-proe software tools. Design iterations were investigated until a mechanism satisfying the constraints and that produced the desired foot motion had been synthesized. The resulting mechanism is shown in Fig. 4. The mechanism’s link lengths and their ordering from the center of the sphere outward are presented in Table 1. Note that the link lengths listed are the spherical link lengths, that is, to say the angles between the revolute joint axes of each link. In addition, the coupler link, as shown in Fig. 4, has been extended into a $180deg$ arc with straight extensions to generate two feet. Nevertheless, the coupler link length remains $96.1deg$.

Fig. 4
Fig. 4
Close modal
Table 1

LinkLength ($deg$)Order
Fixedγ = 97.11
Inputα = 17.22
Outputβ = 22.82
Couplerη = 96.13
LinkLength ($deg$)Order
Fixedγ = 97.11
Inputα = 17.22
Outputβ = 22.82
Couplerη = 96.13

We utilized the spherical mechanism classification procedure in Ref. [37] to analyze the SphereWalker mechanism. That procedure requires determining four T parameters from the link lengths: T1 = γα + ηβ, T2 = γαη + β, T3 = η + βγα, and T4 = 2πηβγα. The four T parameters were found to be: T1 = 2.674, T2 = 0.115, T3 = 0.080, and T4 = 2.213. It was shown in Ref. [37] that a spherical four-bar mechanisms is Grashof if T1T2T3T4 > 0; therefore, the SphereWalker mechanism is Grashof. Furthermore, using the four T parameters and Table 2 of Ref. [37], we find that the SphereWalker mechanism is a crank-rocker linkage. The SphereWalker four-bar mechanism is also shown in Fig. 5 with the same color coding of for the links that was used in Fig. 4. In Fig. 5, a coordinate frame has been attached to the coupler link, denoted by the red and green axes, such that its origin coincides with the foot’s contact point with the ground. The coupler curves of the ground contact point are shown in yellow for both circuits, or assemblies, of the mechanism. In Fig. 4, the coupler curves of the three ground contact points nearest to the viewer are shown in blue.

Fig. 5
Fig. 5
Close modal

The central four-bar mechanism warrants further discussion. Because the two feet of each coupler link are on the great circle associated with the coupler link, the central four-bar was able to rotate to 180 $deg$ about the vertical with respect to the front and rear mechanisms. Moreover, the mechanism was assembled in the other circuit. This was done so that the SphereWalker would have three feet in contact with the ground at all times with the feet making similar motions, see Fig. 5. This was not a design constraint that was imposed on the synthesis process. It was a fortuitous occurrence that the synthesized mechanism had the desired foot motion on the coupler link’s great circle that enabled this simple and elegant solution to yield a hexapod driven by 3 one degree-of-freedom four-bar mechanisms.

Qualitative observational studies were performed on the physical prototype to verify that the design objectives and synthesis constraints were satisfied. Video recordings of the physical prototype walking, as well video simulations of the digital twin walking, may be found in Ref. [36]. It was concluded that the SphereWalker hexapod successfully satisfied the design objectives. The following sections provide an overview of the SphereWalker mechanism, the process of manufacturing and assembling the components of the SphereWalker, creation of a digital twin, actuation and testing of the assembled components, and simulation and control system design.

Manufacturing and Assembly

The main components of the three SphereWalker mechanisms, coupler, fixed link, driven link, and driving link, were manufactured using a computer numerically controlled (CNC) machine tool mill. mastercam software was used to plan the tool paths. A specific layout was drawn using mastercam to fit all of the components for all three mechanisms onto a single piece of 1 (in.) thick aluminum plate, see Fig. 6. After finalizing the layout, we selected the path in which the components were to be milled along with an appropriately sized flat end mill. We used a $3/8$ in. diameter cutting tool with an interlink offset space of $1/8$ in., which resulted in a $1/2$ in. spacing between links. The path was chosen to start with the smaller components at the interior of the layout and then to work its way outward to the larger components.

Fig. 6
Fig. 6
Close modal

The next manufacturing step was to drill holes for the bearings and gauge pins. Gauge pins were used to serve as the axles for the revolute joints. The bearing holes are located on the fixed, driven, and driving links. Two different programs were written in mastercam to manufacture the bearing holes, one for the fixed link and the other for the driving and driven links. The through holes were made two thousandths of an inch smaller in diameter than the bearings, so that the bearings could be press fitted into the links. Next, programs were written to manufacture the gauge pin holes in the coupler, fixed, and driving links. These holes were also made two thousandths smaller in diameter to accommodate press fitting of the gauge pins. This was the most challenging step of the manufacturing process because the gauge pin holes on the coupler and fixed link are in locations that are difficult to access.

Base plates were manufactured for each of the three segments of SphereWalker, and an assembled SphereWalker mechanism was mounted to each. The three base plates were fastened together using clevis joints with machine screws and nuts. The clevis joints were designed so that the connection could be made rigid by tightening the nut, or the connection could be made into a freely rotating revolute joint about the vertical by loosening the nut and inserting a nylon washer and bushing, see Fig. 7. Spherical feet were added to the distal ends of the legs to facilitate the generation of point contact with the ground. The assembled SphereWalker prototype is shown in Fig. 8, and video recordings of the physical prototype walking may be found in Ref. [36].

Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal

Digital Twin

A digital twin of SphereWalker was created using ptc creo and its embedded mechanism module [38]. The SphereWalker digital twin was placed on a fixed ground plane using a point contact model for the interaction between the feet and the ground, and the coefficient of friction between the feet and ground was set at 1. To facilitate the study of fixed base plate connections, revolute jointed base plate connections, and u-jointed base plate connections, the two revolute joints indicated in Fig. 7 were replaced by universal joints as shown in Fig. 9.

Fig. 9
Fig. 9
Close modal

The digital twin simulation utilized the mechanism module and the mechanism analysis tools within ptc creo. Three servo motors were added to the model, and six three-dimensional contacts with maximum friction were defined between each foot and the ground plane. A fixed coordinate frame attached to the ground was defined, and the motion of the SphereWalker was recorded with respect to this frame. Figure 10 shows the positive x direction of the fixed frame as being forward with respect to SphereWalker. The frame’s y-axis is positive upward, and the z-axis completes the right-handed frame and is oriented positive to the left with respect to SphereWalker.

Fig. 10
Fig. 10
Close modal

Digital Twin Simulation

To track the position of the digital twin in simulation, a coordinate frame was attached to the central base plate and oriented parallel to the fixed frame [38]. Simulations were performed to determine how the phase angle offset of the central walking mechanism effects the motion of SphereWalker. The hypothesis was that by operating the two outer mechanisms in phase with each other that SphereWalker could be made to walk straight, turn to the left, or turn to the right by varying the phase angle offset of the central mechanism with respect to the two outer mechanisms.

In the digital twin simulations, the angle of the three input links were set to a default position of 0 deg. To measure the effects of different phase angle offsets of the central mechanism, simulations were done at several starting input link angles: 0, 15, 30, 45, 60, 75, 90, 105, 120, 150, 210, 240, 270, 300, and 330 deg. Moreover, the u joints were locked, so that the two base plates were rigidly connected. Also, to ascertain if the initial standing configuration of SphereWalker impacted the hexapod’s motion, studies were done for the two different possible standing configurations, or initial conditions, of the legs. These two standing configurations are as follows: 1R2L = one foot on the right side and two feet one the left side contacting the ground as shown in Fig. 11 and 2R1L = two feet on the right side and one foot on the left side in contacting with the ground, see Fig. 12.

Fig. 11
Fig. 11
Close modal
Fig. 12
Fig. 12
Close modal

Simulation and analysis results for the SphereWalker walking on the ground with different initial angle offsets with respect to $0deg$ of the central motor shaft and different starting feet positions, 1R2L and 2R1L, were conducted. Measurements are made along the x, y, and z axes of a fixed reference frame attached on the ground.

Central Phase Offset: 0 deg.

The simulation results for the 0 deg phase angle offset of the central SphereWalker mechanism show that the SphereWalker walks forward in the X direction with a slight right turn in the Z direction with respect to the fixed frame. The SphereWalker turns more when started in the 2R1L starting configuration (see Fig. 13) than in the 1R2L configuration (see Fig. 14) as the Z positions of SphereWalker at t = 15 s are 55.94 in. and 53.92 in., respectively.

Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

Central Phase Offset: 120 deg.

The simulation results for the 120 deg phase angle offset of the central SphereWalker mechanism show that the SphereWalker walks forward in the X direction with a left turn in the Z direction with respect to the fixed frame. The SphereWalker turns less when started in the 2R1L starting configuration (see Fig. 15)than in the 1R2L configuration (see Fig. 16) as the Z positions of SphereWalker at t = 15 s are 16.48 in. and 24.64 in., respectively.

Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal

Central Phase Offset: 330 deg.

The simulation results for the 330 deg phase angle offset of the central SphereWalker mechanism show that the SphereWalker walks forward in the X direction with a slight left turn in the Z direction with respect to the fixed frame. The SphereWalker turns more when started in the 2R1L starting configuration (see Fig. 17) than in the 1R2L configuration (see Fig. 18) as the Z positions of SphereWalker at t = 15 s are 6.84 in. and 4.16 in., respectively.

Fig. 17
Fig. 17
Close modal
Fig. 18
Fig. 18
Close modal

Simulation Results Summary.

After analyzing the simulation results for 15 different phase angle offsets of the central mechanism performed for both the 1R2L and 2R1L starting configurations, it can be concluded that the phase angle offset of the middle mechanism has an influence on the walking direction. Note that a sampling of the simulation results are included here. Detailed results for all 30 of the simulations performed may be found in Ref. [38]. Referring to Tables 2 and 3, the differential in Z is calculated from the value of Z after 15 s of walking minus the value of Z at 0 s, which is 49.23 in. The SphereWalker turns right if the differential is positive, and it turns left if the differential is negative. The SphereWalker is generally able to turn right with the phase angle offset less than 105 deg and turn left with the phase angle offset equal to or greater than 120 deg, except for the 60 deg and 105 deg offsets in which the SphereWalker turns left in the 1R2L configuration and turns right in the 2R1L configuration. This can be seen by referring to Tables 2 and 3. Moreover, it is clear from the holistic analysis of the simulations that the walking heading direction of the walking SphereWalker hexapod may be controlled by varying the phase angle offset of the central mechanism from the outer two mechanisms.

Table 2

Simulation results: 1R2L

Offset angleDifferential in Z
(deg)(in.)Turn direction
04.69Right
152.40Right
308.84Right
453.68Right
60−1.48Left
752.61Right
904.33Right
105−3.77Left
120−24.64Left
150−30.09Left
210−9.03Left
240−19.98Left
270−22.15Left
300−14.20Left
330−4.16Left
Offset angleDifferential in Z
(deg)(in.)Turn direction
04.69Right
152.40Right
308.84Right
453.68Right
60−1.48Left
752.61Right
904.33Right
105−3.77Left
120−24.64Left
150−30.09Left
210−9.03Left
240−19.98Left
270−22.15Left
300−14.20Left
330−4.16Left
Table 3

Simulation results: 2R1L

Offset angleDifferential in Z
(deg)(in.)Turn direction
06.71Right
159.10Right
305.27Right
458.89Right
6011.07Right
7510.59Right
9012.14Right
1055.01Right
120−16.48Left
150−29.31Left
210−10.21Left
240−18.81Left
270−20.51Left
300−16.67Left
330−6.84Left
Offset angleDifferential in Z
(deg)(in.)Turn direction
06.71Right
159.10Right
305.27Right
458.89Right
6011.07Right
7510.59Right
9012.14Right
1055.01Right
120−16.48Left
150−29.31Left
210−10.21Left
240−18.81Left
270−20.51Left
300−16.67Left
330−6.84Left

Conclusions

This article described the design and the development of a novel six-legged robotic walking machine named SphereWalker. The SphereWalker has six legs that are arranged into pairs with each pair being supported and actuated by a single spherical four-bar mechanism. The design, manufacture, prototyping, actuation, and control of the SphereWalker was detailed. A prototype has been built that includes an on-board autonomous closed-loop feedback control system employing servo actuators. A digital twin was constructed, and dynamic walking simulations were performed. It was found that the heading direction of the walking hexapod may be controlled by varying the phase angle offset of the central driving mechanism with respect to the two outer driving mechanisms. Ongoing and planned future works are focused on studying a variety of gaits and the designs of feet to optimize the dynamic performance of SphereWalker for a variety of tasks and terrains.

Acknowledgment

We sincerely wish to acknowledge the efforts of several students that have contributed to the design and development of SphereWalker. Oliver Zimmerman, Jacob Sleight, Garrett Lee, Christina Lucas, Jennifor Mori, and Cassandra Scully all made significant contributions to the SphereWalker Project with respect to conceptual design, mechanism synthesis, CAD and CAM modeling, and physical prototyping. Xiaoyang Mao created the digital twin and designed the control system. This material is based upon work supported by the National Science Foundation under Grant No. 0422705. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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