Design of

Omar Eduardo López Botello,Roble

Facultad de Ingeniería MecáNicolá

omar.lopezbt@uanl.edu.mx,

Abstract - Biomimicry studies the naturprocesses in order to learn from them and thThis work takes this concept tmechanical/structural design for a hexapod rpropose some suggestions for its constructbased on the anatomy and biomechanic charinsects, whose great motor stability is a veryin exploration robots. Once these features account, it was possible to design a robot easyto manufacture, further it was made a kinemmodel which was able to predict accuratelbehavior, achieving a highly precise contrcorrect selection of actuators and mconstruction. Finally, a functional prototype built demonstrating the viability of the prothis work.

Keywords: Biomimetics, Biomechanics, InsectsKinematics, Dynamics.

1. INTRODUCTION Nowadays, the engineering looks for

devices to solve the problems that the huday. A good example of this pursuit is therobots, which by their size and their moveable to reach places that humans catheirselves, or that can be even dangerous. Fgood starting point to obtain a robot decharacteristics is the biomimetics [1].

A well known characteristic of the stability provided by their bilateral symjointed legs. These qualities along with tallow them to be resistant to the difficultiesin their environment; these make them the pfor imitation in the robotics industry.

Therefore, through an extensive recharacteristics, causes and consequences ofof the insects and taking advantage of theswas possible to propose the movements robot as well as a mechanical structure for

f a Hexapod Robot based on Insects

Mónica Lizeth Balboa García, Patricia del Carmedo, Angel Rolando Rivas Velázquez

ánica y Eléctrica, Universidad Autónoma de Nueás de los Garza, Nuevo León, México. , monica.balboagr@uanl.edu.mx, pzambran@fi

arivas@fime.uanl.mx

re’s designs and hen emulate them. to propose a robot and also, to tion and control, racteristics of the

y desirable quality were taken into

y to assemble and matic and dynamic

ly the robot legs ol and evenly, a aterials for its of the robot was

oposed models by

s, Hexapod Robot,

r more accurate uman faces every e clearly need of ements might be annot reach by For this reason, a esign with these

insects is their mmetry and their their exoskeleton that are exposed perfect specimen

esearch on the f the movements se capabilities, it for the hexapod

r its construction,

having thereby a well founded basthe best characteristics of those orga

It was necessary to optimize thorder to reduce costs and to omanufacture and assemble. Further,final design, the mathematical mode

The mathematical models, basedynamics of the robot allow the pand position involved in the locallowing the accurate reproduction oof the insects, which leads to an eas

2. IDEAL MECHANICAL STRUCON AN ANALYSIS OF THE IN

2.1. Thorax A possible structure for a robot w

identified characteristics on the genthe insects. Their body consists otheir distinctive feature is a exoskeleton. These three fundamhead, the thorax and the abdomen “mention that for this study; only tbecause this particular segment is of the locomotion.

Figure 1: Regions of an i

men Zambrano

evo León, San

ime.uanl.mx,

sis to build a robot with anisms.

his first ideal proposal in obtain a robot easy to , based on this optimized els were made. d on the kinematics and prediction of the motion comotion of this robot, of the natural locomotion ier control of the device.

CTURE PROPOSED BASED NSECTS ANATOMY.

was deduced through the neral corporal structure of on a segmental plan and

hard and articulated mental segments are the “Fig. 1” [2]. It is worth to the thorax is considered the principal responsible

insect body.

2010 Electronics, Robotics and Automotive Mechanics Conference

978-0-7695-4204-1/10 $26.00 © 2010 IEEE

DOI 10.1109/CERMA.2010.47

347

The thorax is also divided in 3 segmentmeso- and meta-thorax and in most of thethese support a pair of legs. The transversally can be simplified as a structurin “Fig. 2” and laterally like the structure drBesides, it was concluded that for the pscheme, it was possible to work with subdivisions were similar since they preseonly on winged insects, which is not the csince only were taken into account the locomotion of walking and running insethis, the structure shown in “Fig. 4”demarcating the three characteristic segmenthorax in order to identify the carrier partlegs.

2.2. Basic structure of the appendices

Typically, each leg of the insects segments joint together through one or twfixed on a membrane. These segmentstrochanter, femur, tibia, tarsus, pretarsus “Fi

2.2.1 Coxa. The coxa is usually in truncated cone and it articulates proximallywall in three possible ways “Fig. 6”:

Figure 2: Structural representation of the thorax of transversally.

Figure 3: Structural representation of the thorax of

laterally.

Figure 4: Ideal proposed model for the thorax of th

Figure 5: Typical leg of an insec

ts known as pro-, e insects each of thorax, viewed

re like the drawn rawn in “Fig. 3”. purposes of this a thorax whose

ent modifications case of this study

capabilities and ects. Considering ” was designed nts of the insects’ t of each pair of

consists of six wo condyle joints s are the coxa, ig. 5” [3].

the form of a y with the thorax’

f an insect viewed

f an insect viewed

he hexapod robot.

ct.

1) A single articulation with t2) A second articulation with 3) Rigid pleural and sternal arIt was chosen the third articulat

robot since it is the most aptly to acof movements without increaconsidering that the first articulpromotes a move with greater freedof the robot coxa and the second tythe leg movement. Finally, it washown in the “Fig. 7”.

2.2.2 Trochanter and Femur. Tsegment articulated with the carticulation moving only in a vertstrongly fixed to the femur, whichany movement between them. Therthe trochanter and femur of the roboin the ideal mechanical design,complexity of the system. It is wofemur, in most adults, is the most rdue to the stresses generated at thait was proposed the structure shofemur (or trochanter and femur).

2.2.3 Tibia. Normally is a longmuch as the femur with which isarticulation allowing a vertical movinsects, the head of the tibia is cuback against the femur being paralthe degree of freedom “Fig. 5”. designed the structure presented in t

Figure 6: Types of thorax-coxa joint. The awith narrows.

Figure 7: Ideal proposed model for the

Figure 8: Ideal femur proposed mod

the pleuron. the trochantin.

rticulations. ion type for the hexapod ccomplish the coxa needs asing its complexity, ation type, although it

dom, it exceeds the needs ype restricts considerably as proposed a structure

The trochanter is a small coxa by a dicondylic tical plane and also it is h does not permit almost refore it was decided that ot were only one segment , avoiding unnecessary orth to mention that the robust segment of the leg at point. Considering this own in “Fig. 8” for the

g segment, sometimes as s joined by a dicondylic vement “Fig. 9”. In most

urved so that it can bend lel to it, thereby limiting

Based on this, it was the “Fig. 10”.

articulation points are marked .

coxa of the hexapod robot

del for a hexapod robot.

348

Figure 9: Trochanter-femur joint (without m

Figure 10: Ideal tibia proposed model for a he 2.2.4 Tarsus and Pretarsus. The tarsus itwo to five tarsites (or tarsomeres) which aof real segments because of the absence basal tarsomere articulates with the distal enonly one condyle [4]. The pretarsus membranous base which supports a medpillow and a pair of claws. There are also othe pretarsus that allow insects to hold to which they move, for example, through thesuction cups-like structures that allow it tosurfaces such as glass. For the structural dewas decided to take the tarsus and pretasegment, making the function of a foot stability to the overall structure of the leg “designed with a single condyle articulationecessarily controllable due to the greadamping that just the shape can providestructure, which also allows the robot adaptability to the terrain on which it would

2.2.5 Final configuration of the leg.designs presented above it was made an proposed structural design for the legs of arobot which can be visualized in “Fig. 1made an assembly of the proposed structurwhole hexapod robot (thorax and legs) show

Figure 11: Ideal tarsus and pretarsus proposed model

movement).

exapod robot.

s subdivided into are differentiated of muscles. The nd of the tibia by

consists of a dian lobe-shaped other structures in

the substrate on e development of o hold on smooth sign of the leg, it

arsus as a single providing some

“Fig. 11”. It was on, which is not at capability of e to the general to have certain

d walk

Based on the assembly of the

an ideal hexapod 12”. Also it was ral design for the wn in “Fig. 13”.

for a hexapod robot.

Figure 12: Leg assembly composed of the

Figure 13: Hexapod assembly formed by

3. IDEAL MOVEMENTSLOCOMOTION BASED CHARACTERISTICS OF T

3.1. Walking movements. On the description of the legs m

next terms [5]: • Protraction: Complete movemen

leg, relative to its articulation with• Promotion: Movement of the

protraction. • Retraction: Backwards movemen

articulation between the time thground and the time it is raised.

• Remotion: The corresponding mthe retraction.

• Adduction: The movement of the • Abduction: The movement of t

body. • Levation: The raising of the leg o

the protraction. • Depression: Lowering of the leg, • Extension: An increase in th

segments of the leg. • Flexion: A decrease in the angle b

the leg. 3.2. Mechanism of walking.

A leg may act as a strut “Fig. 14depending on its inclination angNevertheless, the leg is not just a simhas intrinsic muscles which also cbody by the flexion or extension extended anteriorly, flexion of the forwards “Fig. 16”, while, in a straightening the joints will push t16”.

coxa, femur, tibia and tarsus.

y the thorax and the six legs.

S FOR THE ROBOT ON THE LOCOMOTION

THE INSECTS.

movements are used the

nt forwards of the whole h the body.

coxa resulting in the

nt of the leg relative to its he foot is placed on the

movement of the coxa for

coxa towards the body. he coxa away from the

or a part of the leg, part of

or a part of the leg. he angle between two

between two segments of

4” or as a lever “Fig. 15” gle about to the body. mple, rigid strut or bar. It can apply forces on the

of the leg. If a leg is joints will pull the body leg directed backwards, the body forwards “Fig.

349

Figure 14: Leg acting as a strut.

Figure 15: Leg acting as a lever.

Figure 16: Extension and flexion of the coxa-trochantjoints.

When the insect (or the robot) starts to m

leg is completely protracted due to the promotion and the extension of all its segmAt this first stage, the leg acts as a strut rmovement. In order to accomplish the forwthe coxal remotion is started producing thwhich produce a lever effect and that conseinsect forwards adding the effect of the flerelative to the femur and the femur relative 17”. This movement continues until the legto the long axis of the robot and once that iposition, the leg start to apply a strut effectleg extension, tends to push the body forwar

Figure 17: Protraction and retraction of t

eral and femur-tibial

move, the forward maximal coxal

ments “Fig. 17”. retarding forward ward movement, he leg retraction

equently pulls the exion of the tibia to the coxa “Fig.

g is at right angle it has passed this t which, aided by rds.

the leg.

3.3 Pattern of leg movements of tEach step comprises a period of

is swung forwards with the foot off of retraction, as the leg moves back the foot on the ground. The pattevaries with the speed of movementhe insect may have most of its feetthe time and the legs are protracteR3 R2 R1 L3 L2 L1 etc. (where R left, and 1, 2 and 3 the fore, respectively) “Fig. 18” or more irreg

At higher stepping rates, three legs of one side and the middle lemore or less simultaneously, so thaon a triangle formed by the other protracted, the other is retracted, sosupported on three legs “Fig. 19” [by the fact that the body is slungcenter of gravity is low “Fig. 20”.

It is worth to mention that higincreases on the frequency with wand in the length of each stride [traduced on a hexapod robot thprogramming of the actuators of eac

Figure 18: Stepping p

Figure 19: Hexapod movem

Figure 20: Stabil

the insects. protraction, when the leg

f the ground, and a period relative to the body with

rn of the stepping often nt. At the lowest speeds, t on the ground nearly all d singly in the sequence and L indicate right and middle and hind legs

gularly. legs, the fore and hind

eg of the other are lifted at the insect is supported three legs. As one set is

o that the insect is always [3]. Stability is enhanced

g between the leg so the

gher speeds result from which the legs are moved

[6] [7], and this can be hrough an appropriated ch leg.

pattern.

ment pattern.

lity

350

4. OPTIMIZATION OF THE IDEAL MECHANICAL STRUCTURE FOR THE CONSTRUCTION OF A HEXAPOD ROBOT.

4.1. Design criteria. In order to simplify and reduce costs, the first proposal of

design was modified, and finally with the optimized structure it was made the mechanical (kinematic and dynamic) model easily having now a system much more simple. It should be noted that even with the optimizations made on the ideal model to obtain a more practical design, the essence of the insects’ movements and their anatomy are not lost. The principal criteria used to modify the structure are:

a) There must be a considerable saving of material, achieving a loose in robot’s weight and costs reduction.

b) The characteristics that make the robot be alike an insect must prevail.

c) Each of the joints must be adequate for the usage of an actuator, which in this case is going to be a servomotor.

d) In order to optimize costs and practicality in the kinematic and dynamic analysis and consequently, in the control, the degrees of freedom of each extremity must be limited to 3.

e) Each design piece must be easy to manufacture. 4.2. Adjustments made to the ideal design. 4.2.1 Thorax: Firstly, it was necessary to find a new thorax that would ease to manufacture, which lead to the conclusion that the thorax should be divided in two parts, leading in turn to a weight reduction. To preserve the thorax segmentation which allows the easy identification and placement of the appendages, the thorax was designed with six protuberances, each one of them able to form joints with the coxa. 4.2.2 Coxa: The movements that the coxa performs mainly are protraction and retraction, promoting the movements of extension and remotion of the appendices through the thorax-coxa joint. Likewise, the coxa articulates distally with the femur, promoting its movements of elevation and depression. Consequently, it is necessary to place two servomotors at this joint in order to achieve these movements, which lead to the complete modification of the structure but without losing its functionality. This design provides well support of the actuators, and also, with the material removal made it was possible to have the necessary space for the communication wires. All this description can be seen in “Fig. 21”, showing also the actuators already assemble.

Figure 21: Thorax-coxa assembly.

4.2.3 Femur: Because of the adjustments made to place a servomotor in the coxa-femur joint, now is only necessary to adjust the segment of the femur for its proper coupling with the axis of the actuator which will provide the movement. In order to have a femur easier to manufacture there were made the necessary optimizations on the ideal model of femur. Also, the new femur was design so that it is not carrying any actuator, but it proximal and distal ends are properly design to fit the actuators, this is because the femur is the central part of the limb, which is subjected to the greatest concentration of stresses during locomotion, that’s why this part must be the strongest. “Fig. 22” shows the coxa-femur joint which, as can be seen, allows a wide range of motion only limited by the actuator.

4.2.4 Tibia: As mentioned above, the femur has not support for the actuator that provides the movement to the femur-tibia joint, so the tibia was designed to be able to hold this actuator. For this design, it was necessary to make the tibia even more robust than the femur “Fig. 23”, contrary to that presented by the ideal model because for this model is the one that supports more stresses than any other segment, due to the modifications made in order to place the servomotors. Finally, due to considerations made for the optimization of the model taking into account the cost and the easy manufacturing according to current market conditions, it was decided to not take the femur-tarsus joint for the optimized model. In “Fig. 23”, it can be observed the final form and structure of the extremity of the hexapod robot.

Figure 22: Coxa-femur assembly.

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Figure 23: Assembly of the Extremi “Fig. 24” shows the complete final desig

robot, along with the corresponding servomotors already coupled. By means ofthe reduction of degrees of freedom, it reduce the complexity of the kinematianalysis of the legs, allowing a simpliequations and consequently, a simple progrthem.

5. RIGID BODY KINEMATICS.

In kinematics [8] there are two fundaminverse and direct kinematics [9] “Fig. 25”.

Figure 24: Hexapod assembly.

Figure 25: Comparison between inverse and dire

ity.

gn of the hexapod model of the

f this design and was possible to

ic and dynamic ification on the ramming through

mental problems:

ect kinematics.

5.1. Direct kinematics. Direct kinematics determines the

of the robot final effectors respesystem placed in the robot’s basejoints and geometric parameters methods to obtain the direct kinemathe methods of Denavit-Hartenberggeometrical method. 5.1.1 Denavit-Hartenberg algoHartenberg algorithm was implemlegs of the robot through the 16 algorithm in order to determine theIn “Table 1”, it is possible to noticethat conform the Denavit-Hartinformation was obtained throughorigins and distance, represented in 5.1.2 Geometrical method. This on the geometry of the mechanismis generally used in simple structurdevelopment becomes too extensivetwo dimension model, so that it wevaluate the three angles involved imethod. These angles were used to lengths, the end effector position, z). The angles θ2 and θ3 are defioperating the femur and the tibia,The angle θ1 is driven by the sethorax-coxa joint and it can be seesystem and the variables to finmathematical analysis of the prosimplest way to accomplish this trigonometric identities using now Thus, it was possible to obtain the f

Figure 26: Origins and distances of the syste

TABLE 1: Denavit-HartenJoint θ d

1 θ 1 0 2 90 L13 θ 2 0 4 θ 3 0

e position and orientation ect to a fixed reference e through values of the

[9]. There are several atics; this work develops g algorithm [10] and the

orithm. The Denavit-mented in each of the six

steps that compose this eir rigid body kinematics. e the values for each joint tenberg matrix, whose h values of the system’s

“Fig. 26”.

method is mainly based m, and its implementation res, because otherwise its e. First, it was obtained a

was easier to analyze and in the calculations of this calculate, along with the which is given by (x, y, ined by the servomotors , respectively “Fig. 27”.

ervomotor placed on the n in “Fig. 29”. Once the

nd are established, the oblem was started. The

analysis is the use of right triangles “Fig. 28”.

following equations (1):

em obtained by D-H algorithm.

nberg Matrix. a α 0 90

1 0 -90 L2 0 L3 0

352

sin cos 90 sin l sin 90 cos sin cos 90 cos

Figure 27: Movement limits.

Figure 28: Representation by right triangles.

Figure 29: Values to considerate for the inverse kinematic analysis.

5.2. Inverse kinematics.

With the inverse kinematics it is obtained the configuration that the robot must adopt to achieve a known end effector position and orientation to a certain spatial location, finding the coordinate values θ = (θ1, θ2, ... , θn) that the joints of the robot must hold [8]. The resolution of the kinematic problem by geometric methods is the appropriate procedure for the analysis of the legs because of the few degrees of freedom that the robot has. This is based on finding sufficient number of geometric relationships in which intervene the coordinates of the end effector, i.e. the

last point of the leg, joint coordinates and the physical dimensions of the elements.

On “Fig. 29” it is possible to see which are the links and joints to analyze. Once the values of the end effector position and the lengths of each of the links are known, by trigonometry and the law of cosines, the joints angular values required to reach a specific position were obtained, as described in (2).

θ

θ 180 arccos 2

θ 180 2180 arccos 2

5.3. Jacobian matrix.

Kinematic modeling of a robot looks for relations between joint variables, position (usually expressed as Cartesian coordinates), and the orientation of the end effector of the robot. This relationship does not take into account the forces and torques acting on the robot (actuators, load, friction, etc) and that can origin the movement of the robot. However, it must allow, in addition to the relationship between the joint and end coordinates the relationship between their respective derivatives. Thus, the robot control system should establish the speeds that should be set on each joint (through their respective actuators) to get the end effector to develop a specific time path, for example, a straight line at constant speed. For this and other purposes, it is useful to have the relationship between joint velocities and coordinates of the end effector position and orientation. The relationship between velocity vectors is obtained through the so-called Jacobian matrix [8]. The most direct method to obtain the relationship between joint velocities and the last point of the leg is the differentiation of the equations for the direct kinematic model, obtaining with this the matrices of the equations (3).

·

(1)

(2)

(3)

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6. ROBOT DYNAMICS.

The dynamic model of a robot is aimed at the understanding of the relationship between the movement of the robot and the forces involved in the same [11]. 6.1. Newton-Euler formulation.

Obtaining the dynamic model of a mechanism, and in particular of a robot, is mainly based on the approach of the balance of forces established by Newton’s second law or its equivalent for rotational movements, known as Euler’s law (4).

Proper implementation of these equations leads to a recursive formulation in which it can be obtained the position, as well as the velocity and the acceleration of the link i constrained to the base of the robot from the corresponding link i-1 and the relative motion of the joint i. Thus, starting from link 1, link n is reached. With these data we now can proceed to obtain the forces and moments acting on link I, constrained to the base of the robot from those corresponding to link i+1, running this way all the links from link n to link 1. To perform dynamic modeling using Newton-Euler formulation [12] a series of 10 steps where conducted, between those who were involved first, the D-H matrix obtained above and matrices of rotation, angular velocity and acceleration and the forces applied to the different segments of the extremities, thus obtaining the torque applied to the joint, as shown in the following equations (5).

7. CONCLUSIONS

Through this work it was possible to analyze the features that make the biomechanics and the structure of the insects being the optimal for its emulation in the robotics industry and thus, achieve a proposal of a mechanical structure similar to that of these animals in order to be able to make movements such as those made in the nature.

It is worth to mention that the complexity of the systems of nature is such that sometimes is difficult to fully emulate

them, because the materials and the manufacture processes to accomplish this objective could not be accessible. Even though, nowadays there are not the necessary tools to obtain a dispositive fully similar to the insects, it was possible to make a simplification of the designs that nature has made within the perfect evolution of its systems and obtain a very useful prototype of a hexapod robot based on the lessons of design that the nature offer by itself.

Mathematical (kinematic and dynamic) modeling allows an accurate reproduction of insects’ locomotion providing the robot, the necessary characteristics to achieve a proper way to imitate these creatures.

Also, the equations obtained from this model allow a better control of the robot, which was verified by constructing its working prototype.

Some of the next steps around this work are the addition of a fourth degree of freedom by leg, not necessarily controllable, but with a mechanical/structural design that allows a general damping of the structure and an even more accurate control for its proper adaptation to different terrain through the robot's dynamic equations, using for feedback some force sensors.

Finally, it is essential to point out the meaning of nature as a source of inspiration within the science because it is a huge example of efficient natural engineering design and of course, of continuous improvement, as it is the evolution.

8. REFERENCES

[1] Benyus, Janine M. “Biomimicry: Innovation Inspired by Nature”, 1997.

[2] The University of Nebraska Department of Entomology. "O. Orkin Insect zoo". http://insectzoo.msstate.edu/Students/basic.structure.html (Consulted in January 2009).

[3] Biewener, Andrew A. “Animal Locomotion. Oxford University Press”, 2003.

[4] P. J. Gullan,Peter Cranston. “The Insects: An Outline of Entomology”, 2005.

[5] Hughes, G.M. (1952). The coordination of insect movements. I. The walking movements of insects. Journal of Experimental Biology, 29, 167-84.

[6] Friedrich Pfeiffer,Teresa Zielińska. “Walking: biological and technological aspects”, 2004.

[7] Full, R.J. &Tu, M.S. (1990). Mechanics of six-legged runners. Journal of Experimental Biology. 148, 128-46.

[8] Crag, John J. “Introduction to Robotics Mechanics and Control”, 1989. Editorial: Addison-Wesley

[9] Barrientos, Antonio; Peñin, Luis Felipe; Balaguer, Carlos; Aracil, Rafael. “Fundamentos de Robótica” (in Spanish), 1997. Editorial: McGraw Hill.

[10] Hartenberg, R.S.; Denavit, J. “Kinematic synthesis of linkages”, 1964. Editorial: McGraw Hill.

[11] Michael Brady, John M. Hollerbach, Timothy L. Johnson, Tomás Lozano Pérez, Matthew T. Mason. “Robot Motion; Planning and Control” Editorial: MIT Press, 1982.

[12] Thomas R. Kurfess, “Robotics and Automation Handbook”, Editorial: CRC Press, 2005.

(4)

(5)

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