Duty Factor Vs Stability Margin Graph

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OPTIMAL GAIT ANALYSIS OF QUADRUPED ROBOT By

Mohd Syahmi Bin Jamludin

Dissertation submitted in partial fulfillment of The requirements for the

Bachelor of Engineering (Hons) (Mechanical Engineering)

DECEMBER 2010

Universiti Teknologi PETRONAS Bandar Seri Iskandar

31750 Tronoh Perak Darul Ridzuan

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CERTIFICATION OF APPROVAL

Optimal Gait Analysis of Quadruped Robot by

Mohd Syahmi Bin Jamludin

A project dissertation submitted to the Mechanical Engineering Programme

Universiti Teknologi PETRONAS in partial fulfillment of the requirement for the

Bachelor of Engineering (Hons) (Mechanical Engineering)

Approved:

……….

(Rosmawati Binti Mat Zain) Project’s Supervisor.

UNIVERSITI TEKNOLOGI PETRONAS TRONOH, PERAK

December 2010

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CERTIFICATION OF ORIGINALITY

This is to certify that I am responsible for the work submitted in this project, that the original work is my own except as specified in the references and acknowledgements, and that the original work contained herein have not been undertaken or done by unspecified sources or persons.

………..

(MOHD SYAHMI BIN JAMLUDIN)

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ABSTRACT

Quadruped robot, types of robot which use four legs instead of wheel or track as its locomotion system need to have an optimal gait so that it can be program using the gait as the reference. There are a wide variety of quadruped gait source from the movement of an actual four legged animals called natural gait and also an artificial gait, which are gaits that are not used by the animals but possible as the optimal gait. To be a statically stable gait, the gaits have to contains several properties such as moving in a very low speed manner and able to keep at least 3 feet on the ground for the whole gait cycle.

Only six gaits satisfied the above limitation. These gaits called the creeping gait. Out of six only one identified as the optimum gait that can keep the projection of its center of gravity inside the boundary of polygon of stability while two of the gait can be use as an alternative gait as it can be a statically stable gait given the very high duty factor or discontinuous condition.

By using the property such as dimension of a quadruped robot, a simulator has been made to simulate the gaits behavior. The position of the center of gravity is observed to determine the condition of the gaits with respect to static stability. The results shown that gaits 1423 is statically stable while gait 1243 is statically stable in discontinuous mode.

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ACKNOWLEDGEMENT

On top of everything, I would like to express my utmost appreciation to the Most Almighty God as for His Blessing and Assistance that I had successfully went through the ups and downs with a strong heart in completion of this project. Alhamdulillah, all praises to Him that I have been able to stay on the planned course and complete my Final Year Project.

I would like to express my most gratitude to my supervisor of this Final Year Project, Ms. Rosmawati Binti Mat Zain for being together with me through the difficulties and challenges faced to achieve the objective of this project. Her assistance, expert guidance, advice and suggestions are very essential for completing this project.

I would also like to thank my beloved family for the spiritual support. Your assistance had really driven me forward in completion of this project. Last but not least, I would like to convey my gratitude to my colleagues, friends and to everyone who has contributed directly or indirectly in completing this project.

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APPENDIX 4: Example for Sequential PTP Control . . . . 36 APPENDIX 5: Duty Factor versus Stability Margin Graph . . . 38 APPENDIX 6: Polygon of Stability for Quadruped Creeping Gait 1423 . 39 APPENDIX 7: Polygon of Stability for Quadruped Creeping Gait 1243 . 40 APPENDIX 8: Polygon of Stability for Quadruped Creeping Gait 1342 . 41 APPENDIX 9: Polygon of Stability for Quadruped Creeping Gait 1324 . 42 APPENDIX 10: Polygon of Stability for Quadruped Creeping Gait 1234 . 43 APPENDIX 11: Polygon of Stability for Quadruped Creeping Gait 1432 . 44 APPENDIX 12: Gait Simulation (Gait 1423) . . . 45 APPENDIX 13: Gait Simulation (Gait 1243) . . . 62 APPENDIX 14: Gait Simulation (Gait 1243 discontinuous gait) . . 45

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LIST OF FIGURES

Figure 2.1.1: Example the Different Between Symmetrical

And Asymmetrical Gaits . . . 7

Figure 2.2: Quadruped Robot Prototype . . . 11

Figure 3.2: Support Polygon, Statically Stable and Unstable Cases . . 16

Figure 3.3: Formula for Stability Margin . . . 17

Figure 3.4: Numbering Of the Feet . . . 18

Figure 3.5a: Gait1423 . . . 19

Figure 3.5b: Gait1243 . . . 19

Figure 3.5c: Gait1342 . . . 19

Figure 3.5d: Gait1324 . . . 20

Figure 3.5e: Gait1234 . . . 20

Figure 3.5f: Gait1432 . . . 20

Figure 4a: Front View of the Robot . . . 22

Figure 4b: Torque and Angle . . . 23

Figure 4c: Constraint . . . 24

Figure 4d: Workspace . . . 25

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CHAPTER 1 INTRODUCTION

1.1 Project Background

Gait is a pattern of movement of the animal or human when there are moving from one place to another [1]. When researching about the pattern of movement or the gaits of certain animal, what we concern is the movement of the animals’ legs, hands or even bowel movement according to how the animal move or which part of the body that the animal use to move. The pattern is then recorded in a certain medium as example a wave graph to represent the gait of the animal [2].

Gait analysis on the other hand is the complete series of study about the animal movement using a certain method [3]. Location of center of gravity (CoG) against polygon of stability will be used to a list of different gaits to identify the optimal gaits in term of static stability so that the gait can be use to program a quadruped robot.

While the total number of theoretically possible quadruped gaits is quite large, only six gaits have the property that they can be executed while keeping at least three feet on the ground at all times. These gaits called creeping gaits seem to be well suited for low speed locomotion since they permit a quadruped to remain statically stable during most of the gait cycles [4].

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It is important for a robot that mimics the movement of an animal such as a quadruped robot to have an optimal gait for it to move effectively or in the other word, a movement that is guaranteed in stability & flexible to the terrain that it will be making. In order to achieve this, the result of this study will be represented in a graph form so it will be informative and easy to understand so that it can be use to asses, plan and also can be use as a reference to the quadruped robot

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1.2 Problem Statement

Robot that use wheel as its movement mechanism has less satisfactory maneuverability and mobility on rugged terrain. When running on an uneven terrain, wheeled robot will produce a vibration that affecting the robot’s lifespan and stability [5]. To overcome the weakness of using wheeled robot, a legged robot is introduced and this project is focused on gait cycles that are suitable for quadruped robot (4-Legged Robot)

A walking robot, in this scope is a quadruped robot need to have efficient gait as its model for it to mimic the animal movement considering the stability of the robot and many other aspect

Legged robots are expected as the attractive tool to transport in various environment such as rough terrain, nuclear reactors, aerospace etc [6]. The stability of their motion is one of most important problem and, especially, the gait change should be well considered not to lose the stability [6].

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1.3 Objective

1. To synthesis types of creeping gaits (biomechanics movement) of quadruped to perform gait analysis.

2. To perform gait analysis and identify the optimal walking gait which is the statically stable gait where the center of gravity lies within the boundary polygon at all time.

1.4 Scope of Study

As been suggested in the objective, the scope of study is limited to the gaits analysis of the quadruped or the four legged gaits only. The reason why is the research is limited to only the gaits of the quadruped is because in this research various quadruped creeping gaits will be analyze with aspect of their static stability for it to become an optimal gait for a quadruped robot so other type of gaits beside quadruped gaits will not be discussed

The study is about the static stability of the gaits. The stability of the gaits will be analyzed and being compared to each other. Other properties such as energy consumption and energy optimization of the gaits might also be discussed but not as the main argument.

The static stability will be represent by the projection of center of gravity inside the boundary polygon in two dimensional drawing.

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CHAPTER 2 LITERATURE REVIEW

In order to fully understand the project, a study to determine the best method to perform the gait analysis has been made to make sure the result of the research will be more accurate and reliable to be use as the reference to the quadruped robot. Also a method to make a comparison between those gait to identify the optimal gait and the method to make sure the result of the study is valid and easy to be understood.

2.1 Gait

The history has made a long story back in the time where animal being observed by naked eye without any tool or equipment as one method of the gaits analysis. With today’s technology, it is done smoothly using the aids of the computer and other tool to help the process of observing the animal gaits more accurate [2].

Fundamental to the locomotion of animals is that they move by lifting their legs and placing them at new positions [7]. While walking, the legs should be coordinated with respect to stability, propulsion and energy efficiency [7]. The coordinated manner of lifting and placing the legs is called a gait [7].

Most animals use a variety of gaits, selecting gait based on speed, terrain, the need to maneuver, and energetic efficiency. Different animal species may use different gaits due to differences in anatomy that prevent use of certain gaits, or simply due to evolved innate preferences as a result of habitat differences

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2.1.1 Classification of gaits

Theoretically the total number of possible gait event sequences for a quadruped is 5040, but only a very small portion of them are suitable as gaits and used by animals [4]. A gait is characterized by the sequence in which the legs are lifted and placed [7]. The lifting or placing of a leg is called an event of the gait, and the sequence in which the legs are lifted and placed is called a gait event sequence [7].

There are a variety of the gaits types where it can be differentiates by the species of the animals and also the types of the animal movement. Generally gaits are classed as symmetrical and asymmetrical based on the limb movement [8] but the main criteria that classified the gait usually are the number of legs the animal use to move [8]. In this study, only the gaits of a quadruped or four legged animal will be studied to determine the optimal gaits for the quadruped robot.

In a symmetrical gait, a pair of left and right leg move alternately where in asymmetrical gait, the two limbs move together [9]. An asymmetrical gait sometimes termed as a leaping gaits due to the presence of a suspended phase [9]. Figure 2.1.1 on the next page shows the different between symmetrical gait and asymmetrical gait.

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Figure 2.1.1: Example the different between symmetrical and asymmetrical gaits From: Hildebrand, M. (1989). Vertebrate locomotion an introduction how does an animal's body move itself along

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2.1.2 The Creeping Gaits

Based on research by McGhee & Frank (1968) [4], while the total number of theoretically possible quadruped gaits is quite large, only six gaits have the property that they can be executed while keeping at least three feet on the ground at all times [4].

These gaits are call the creeping gaits, and it seem to be well suited for low speed locomotion since they permit the quadruped to remain statically stable during most of the locomotion cycles [4].

The walking patterns of creeping gaits can be classified into two types – continuous and discontinuous [3]. In the continuous gaits, the body translation and the swing of each leg are occurred simultaneously. The discontinuous gaits are similar to the continuous crawl gait but it is different in the body motion. In the discontinuous gait all four feet are on the ground for the body translation and the body is stopped for the leg swing motion, hence it increases the stability [3].

2.1.3 Gaits and stability

The meaning of stability in this research, when the robot is walking with constant velocity, stability means that the robot is not toppled or has minimum vibration. For example, if the robot is going to be used as a vision recording device on unknown place, of course the viewer does not accept wobbled and fluctuated view.

While walking or running, a quadruped robot has to remain balanced in order to avoid unwanted body motion or falling [7]. Gaits are classified, depending on the strategy used in order to maintain balance, into statically and dynamically stable gaits [7]. The strategy chosen is related to speed, as slower walking gaits, i.e. creeping gaits, are generally statically stable whereas faster gaits are dynamically stable [7]. The main difference is that dynamically stable gaits remain balanced by moving whereas statically

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stable gaits remain balanced by relying on the support area formed by the legs in ground contact [4] [7]. An analogy is found with the difference of riding a tricycle and a bicycle [7]. When riding a tricycle it can be driven arbitrarily slowly without falling while a bicycle can be hard to balance at low speeds whereas it is easier at higher speeds [7].

A common assumption in statically stable walking is to neglect the effect of inertial forces acting on the robot [7]. This is motivated by the relatively slow speed of statically stable gaits, in which case, gravitational forces are more dominating than motion dependent forces [7]. Static stability requires a walking system to have a minimum of three feet on the ground with the center of gravity located the convex polygon formed by the legs at all time [4] i.e. the area formed by the feet in ground contact, otherwise there would be an uncompensated moment acting around an edge of the support area that could cause the robot to tip over [7]. In this case of quadruped, only creeping gaits can satisfy the above limitation. However, if the vertical projection of the center of gravity is sufficiently close to an edge of the support area, a small momentum of the robot, an external force or uncertainties in the exact position of the center of gravity, may be sufficient to tip the robot over.

A minor failure would be when the loss of stability results in a disruption of the gait in order to regain balance, for instance, by changing the timing or sequence in which the feet are lifted or set down, or by shifting the body in an unplanned manner [7]. A more severe failure is when the robot tips over on its side, risking damage to itself and its surroundings [7]. In order to reduce the risk of the robot losing stability while walking, a proper selection of gait can be used in the gait planning, in order to avoid the robot from become instable.

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For an ideal legged locomotion robot, the minimum number of leg required to archive a gait with a strictly positive static stability margin at all times is equal to three. If the time required to transfer a leg contact point to a new position is greater than zero, then the number is equal to four [4]. This is because the minimum number of legs required to the robot to form a support polygon is three. If the robot are about to lift its leg to move it to a new location, at the time the robot lift it leg, the number of leg that are still intact with the ground is only two, thus not sufficient to form the support polygon. Thus, the minimum four legs are needed. The four legs robot can lift one of its legs to move and have another three legs to form a support polygon and this condition can only be achieved if the robot lift only one leg at one time and land that leg before starting to lift another leg.

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2.2 Quadruped Legged Robot

Robots are finding applications in areas, such as cleaning, running errands, and assisting handicapped or elderly [7]. Mobile robots are also being developed to be used in areas that are inaccessible and/or dangerous for humans, for instance in demining, maintenance in hazardous environments, military, and exploration of volcanoes and space [7]. Quadruped robot is inspired by the 4-legged animal such as mammals, reptiles and some type of insect that are adapted into 4-legged robot. Having a quadruped robot has an advantage because the robot can use many type of gaits on different terrain [6].

The dimensions and properties of the quadruped robot prototype that mention is based on the research that has been made by Mr. Yee Yuan Bin (2009), UTP, Design Of Control System For Quadruped Robot (4-Legged) [10]. Please refer to FIGURE 2.2 below. It weighs about 1.93 kg where each leg weighs about 0.38 kg. The length and width of the robot’s body are 0.14 m with 0.1 m height.

Figure 2.2: Quadruped Robot Prototype

Quadruped robot prototype designed by Mr. Yee Yuan Bin (2009), UTP, Design of Control System for Quadruped Robot (4-Legged) [10]. All the dimension, calculation and analysis is done base on the property of this robot.

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CHAPTER 3 METHODOLOGY 3.1 Project Planning

The block diagram on the next page shows how the project is being conduct. The first step of this project is to do a literature review to review the critical points of current knowledge including substantive findings as well as theoretical and methodological contributions to a particular topic

The second step is to gather dimension and other properties from the quadruped robot.

The objective of this step is to know the characteristic of the quadruped robot. The limitation that the robot has and also how the robot moves including the number of degree of freedom and maximum and minimum reach of the robot.

The third step is to sketch the polygon of stability of for the six creeping gaits. The sketches then being compared by observing the position of center of gravity in each sixteen steps for all six creeping gait. Gaits that have the center of gravity inside the boundary polygon at all time the simulated in the next step of this project which is by using Microsoft Excel spreadsheet.

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3.1.1 Flowchart of Project Planning FYP

Figure 3.1.1 Flowchart of Project Planning FYP START

Literature Review

Gather Dimension & Other Properties from the Quadruped Robot

Sketch polygon of stability for the six creeping gaits and choose gait that have

the cog inside the polygon of stability

Build a simulator for the chosen gait by using Microsoft Excel spreadsheet

END

Draw a conclusion from the simulated gait by comparing the position of the center of

gravity for the gaits in the simulator

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3.1.2 Gantt Chart

Final Year Project I Gantt Chart

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Final Year Project II Gantt Chart

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3.2 Statically Stable Gait

For a statically stable gait, the vertical projection of the center of gravity (CoG) onto a horizontal plane, is kept within the support area at all times In the absence of any inertial or external forces and if the ground is sufficiently rigid, the robot can remain stable as long as the CoG is within the support area [4]. A necessary condition for static stability is that the robot has at least three legs on the ground at all times. This is necessary in order to form an area of support that can contain the projection of CoG within its borders [7]. In Figure 3.2 below an example is given for a four-legged robot. In the left part of the figure, three legs provide support and the projection of center of gravity is located inside the support area such that the robot is statically stable. The foot placements in the right part project the center of gravity outside the support area, which leads to instability due to a tipping moment caused by gravity.

Figure 3.2: Support Polygon, statically stable and unstable cases.

Statically Stable Statically Unstable

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3.3 Static Stability Margin

The magnitude of the static stability margin at time t for the event is equal to the shortest distance from the vertical projection of the center of gravity to any point on the boundary of the support polygon [4]. If the event is statically stable, the case where the projection of the center of gravity lies inside the support polygon boundary, then the value for static stability margin is a positive value. On the other hand, if the event is statically unstable, the case where the projection of the center of gravity lies outside the boundary of the support polygon, the value for static stability margin will become a negative value.

To calculate the stability margin need to a variable β, Duty Factor which express the fraction of time that one leg is in contact with the ground during one event of the gait [7]. Higher value of β indicates that high time given for one leg in contact with the ground or low velocity of movement. Figure 3.3 below show the formula for stability margin for creeping gait as discussed by R. B. McGhee & A. A. Frank, 1968, University of Southern California, Optimum Quadruped Creeping Gaits [4].

Gait Stability Margin

1423 Β -

1342 β -

1243 β -

1432 Β-1

1324 Β-1

1234 -1

Figure 3.3: Formula for stability Margin

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3.4 Numbering the Quadruped Gait

The polygon and the feet of the quadruped are projected as if the robot is observed in top view. The manners on how the leg of the robot is numbered are shown in Figure 3.4 below:

Figure 3.4: Numbering of the feet

Number 1 represent the left front leg, number 2 represent the right front leg, number 3 represent the left hind leg and number 4 represent the right hind leg. Gaits are usually name by the sequence of which leg is lift during the gait cycle. Creeping gaits may be either singular or non singular. For the non singular gaits, the leg is lifted one by one following the gait sequence. In singular gaits, there are legs that are lifted simultaneously. The analysis will concentrate on non singular gait as in the case of static stability need minimum three leg touching the ground, thus only one leg can be lifted at one time.

DIRECTION OF TRAVEL

2 – Right Front

4 – Right Hind 3 – Left Hind

1 – Left Front

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3.5 Type of Creeping Gaits

The six quadruped creeping gait used in this paper are 1423, 1243, 1342, 1324, 1234 &

1432 where the numbers means the sequence of leg that are lifted while the quadruped move. For example creeping gait 1423 means if the gait is started by lifting leg number 1 (left front leg), it is then followed by lifting leg number 4 (right hind leg), then followed by leg number 2 (right front leg) and then followed by leg number 3 (left hind leg) before continue to leg 1 as the cycle repeated. The graphical presentation of these gaits can be seen below.

1423 (Crawl gait)

Cycle 25% 50% 75% 100%

Left Front

Right Hind

Right Front

Left hind

1243

Cycle 25% 50% 75% 100%

Left Front

Right Hind

Right Front

Left hind

1342

Cycle 25% 50% 75% 100%

Left Front

Right Hind

Right Front

Left hind

Figure 3.5a: Gait 1423

Figure 3.5b: Gait 1243

Figure 3.5c: Gait 1342

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1324

Cycle 25% 50% 75% 100%

Left Front

Right Hind

Right Front

Left hind

1234

Cycle 25% 50% 75% 100%

Left Front

Right Hind

Right Front

Left hind

1432

Cycle 25% 50% 75% 100%

Left Front

Right Hind

Right Front

Left hind

Figure 3.5d: Gait 1324

Figure 3.5e: Gait 1234

Figure 3.5f: Gait 1423

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CHAPTER 4

RESULTS AND DISCUSSION

A few of important variables of the robot are calculated so that an analysis of the robot can be done. All the calculation, dimension and specification of the quadruped robot are based on the work done by Mr. Yee Yuan Bin (2009), UTP, Design of Control System for Quadruped Robot (4-Legged) [10]. Please refer to Appendix 1.

Below are the parameters that been used:

Material: Aluminium

Weight of the robot = weight of the legs + weight of the body + weight of motors + other equipment (wire, Battery etc)

= 1.930 kg

Weight in Newton = (1.930 kg) (9.81)

=18.9333 N Torque

There are 2 different servo motor used for different joint name Joint A and Joint B Torque Joint A :BizChip SV003 = 0.13kg m

=0.13 x 9.81

=1.2753 N m Torque Join B : Hitec HS 322HD = 0.037kg m

=0.037 x 9.81

=0.3630 N m

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Dimension

Referring to Figure 4a below,

Link A: 100mm length; 56mm width; 38mm thick Link B: 150mm length; 30mm width; 7mm thick

Figure 4a: Front view of the robot

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Vertical Force

Referring to Figure 4b below:

Figure 4b: Torque and Angle

Vertical Force for One leg =

=

=8.1975+0.8276

=9.025 N

Total vertical forces when 4 leg on the ground = 9.025 N x 4

=36.1 N Total vertical forces when 3 leg on the ground =9.025 N x 3

=27.075 N Joint A

Joint B

Link A

Link B

Weight

Torque A

Torque B

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Work Envelope

Work envelope of a robot arm, manipulator or leg is the boundary in space where the tool tip can reach. Constraint and limitation because of thickness of the material or the maximum extension of the robot the tool tip restricted only inside the boundary of the work envelope. Figure 4c below shows the constraint and the maximum extension of the tool tip, the robot leg cannot rotate further because it has reached its limit.

Figure 4c: Constraint CONSTRAINT

MAXIMUM EXTENSION

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Figure 4d below shows the work envelope of the tool tip on the X-Y plane. Please refer to Appendix 2 & 3 for work envelope of the tool tip on the X-Y plane from other view with dimension. Gait analysis need to be adjusted based on the work envelope boundary shown in the figures before programming the robot.

Figure 4d: Workspace

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4.1 Justification

The sketch and graph generated need to be justified so that the stable and unstable gait can be differentiated. The center of gravity is examined and any gaits that have CoG lays outside of the polygon boundary at any of its step/phase is considered unstable

4.1.1 Justification of Sketch of Polygon Boundary and Center Of Gravity

Referring to Appendix 6, Polygon of Stability Quadruped Creeping Gait 1423, all the Center of Gravity lays within the boundary lines of stability polygon indicating that the gait is statically stable and the gait is taken to be used for generating the graph of polygon boundary and Center of Gravity

Referring to Appendix 7, Polygon of Stability Quadruped Creeping Gait 1243, nearly all the Center of Gravity lays within the boundary lines of stability polygon indicating that the gait is statically stable but some of the CoG lays too close to the boundary for example for step 10 and 14 but the gait is still taken to be used for generating the graph of polygon boundary and Center of Gravity as none of the CoG lays outside the boundary. The gait however has a very high tendency to tip-over if the robot experiences even a small force in the direction of the CoG.

Referring to Appendix 8, Polygon of Stability Quadruped Creeping Gait 1342, nearly all the Center of Gravity lays within the boundary lines of stability polygon indicating that the gait is statically stable but some of the CoG lays too close to the boundary for example for step 6 and 10 but the gait is still taken to be used for generating the graph of polygon boundary and Center of Gravity as none of the CoG lays outside the boundary.

The gait however has a very high tendency to tip-over if the robot experiences even a small force in the direction of the CoG as the CoG lies so close to the boundary of the polygon meaning that the magnitude of stability margin of the gait is very low.

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Referring to Appendix 9, Polygon of Stability Quadruped Creeping Gait 1324, not all the Center of Gravity lays within the boundary lines of stability polygon indicating that the gait is statically unstable. The CoG in step 5, 6, 9 and 11 is outside the boundary line. Therefore the gait is not taken to be used for generating the graph of polygon boundary and Center of Gravity

Referring to Appendix 10, Polygon of Stability Quadruped Creeping Gait 1234, not all the Center of Gravity lays within the boundary lines of stability polygon indicating that the gait is statically unstable. The CoG in step 5, 6 and 11 is outside the boundary line.

Therefore the gait is not taken to be used for generating the graph of polygon boundary and Center of Gravity

Referring to Appendix 11, Polygon of Stability Quadruped Creeping Gait 1432, not all the Center of Gravity lays within the boundary lines of stability polygon indicating that the gait is statically unstable. The CoG in step 5, 6, and 13 is outside the boundary line.

Therefore the gait is not taken to be used for generating the graph of polygon boundary and Center of Gravity

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4.2 Gait Simulation (Gait 1423)

Referring to Appendix 12, 16 steps of the crawling gait is simulated where the time period for each step is where x is the step number while T is the period to complete the gait.

The gait shown in those figures is considered non-singular continuous gait as there are no lifting of two or more leg simultaneously and the body translation and the swing of each leg are occurred simultaneously.

As one of the creeping gait, the gait perform by the crawling gait of a quadruped will form a polygon which have at least three point touching the ground at the same time.

The projection of the CoG always lies inside the polygon and the quadruped can be assume stable at all time during the gait period if the CoG always lies inside the polygon boundary.

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4.3 Gait Simulation (Gait 1243)

Referring to Appendix 13, 16 steps of the 1243 gait is simulated where the time period for each step is where x is the step number while T is the period to complete the gait.

The gait shown in those figures is considered non-singular continuous gait as there are no lifting of two or more leg simultaneously and the body translation and the swing of each leg are occurred simultaneously.

The CoG lies inside the boundary on most of the phase but on step 9, 10, 14 and 15 the CoG is located slightly outside the boundary of the polygon making the event is not statically stable at that particular time. As observed on Figure 4.3on Appendix 5, graph Duty Factor versus Stability Margin, gait 1243 and 1342 only produce a positive value of stability margin on high duty factor or very low speed locomotion. This is the reason for the CoG on those steps lies outside the boundary of stability polygon and producing the negative value of stability margin.

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4.4 Gait Simulation (Gait 1243 discontinuous gait)

Referring to Appendix 13, 16 steps of the 1243 gait (discontinuous) is simulated where the time period for each step is where x is the step number while T is the period to complete the gait.

The gait shown in those figures is considered non-singular discontinuous gait as there are no lifting of two or more leg simultaneously and all four feet are on the ground for the body translation and the body is stopped for the leg swing motion, hence it increases the stability.

The CoG lies inside the boundary polygon on all 16 steps meaning that the magnitude of stability margin is positive in all steps. The gait can be considered as statically stable because of above reason.

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CHAPTER 5 CONCLUSION

For a quadruped to run smooth and steady, an optimal gait analysis is needed.

Techniques such as using polygon of stability to determining statically stable gait helps to identify the optimal gait for quadruped where the optimal gait should be statically stable to provide balanced to the walking quadruped.

Statically stable means the robot will not topple when walking in a very slow manner without any loads. To be declare as statically stable gait, the gait have to have a certain properties such as moving in a very slow velocity and also keeping at least three leg in tact with the ground for the whole cycle. There are six different gait combinations for very slow movement called the creeping gaits that has the properties to project a three points boundary forming the stability polygon.

Out of these six however, as can be observed by in Appendix 6 – 11, only three can be determine as statically stable, gait 1423, 1342 and 1243. Gait 1423 has the biggest value of stability margin as it manages to keep the CoG inside the polygon of stability on all phase hence the optimal gait. Gait 1342 and 1243 can be use as alternative gait as the gaits can be statically stable for very low speed locomotion where the value of duty factor is very high. But for the discontinuous 1243 gait, as being proven by the gait simulator can be statically stable at all time provided that the body move only when all four legs is touching the ground.

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REFERENCES

1. P. L. Buono & M. Golubitsky, 2001, Mathematical Biology

2. David P. Gibson, Neilll W Campbell & Barry T. Thomas. 2003, University Of Bristol, Quadruped Gait Analysis Using Sparse Motion Information

3. Roy B. Davis, Sylvia Ounpuu and Peter A. DeLuca., 2003, Analysis Of Gait 4. R. B. McGhee & A. A. Frank, 1968, University of Southern California, Optimum

Quadruped Creeping Gaits

5. Manish Saggar, Thomas D’Silva, Nate Kohl and Peter Stone. 2007, Autonomous Learning Of Stable Quadruped Locomotion

6. Seiji Masakado,Takayuki Ishii & Kazoo Ishii, 2005, Kyushu Institute Of Technology, A Gait-Transition Method For A Quadruped Walking Robot

7. Freyr Hardarson, 2002, Royal Institute of Technology, KTH, Stability Analysis And Synthesis Of Statically Balanced Walking For Quadruped Robots

8. Dragos & Huosheng Hu, 2005, University Of Essex, Evolving Locomotion Gaits For Quadruped Walking Robot

9. Hildebrand, M. 1989, Vertebrate locomotion an introduction how does an animal's body move itself along

10. Yee Yuan Bin, Jan 2009, Universiti Teknologi PETRONAS, Design Of Control System For Quadruped Robot

11. David P. Gibson, Neill W. Campbell and Barry T. Thomas. 2003, Quadruped Gait Analysis Using Sparse Motion Information

12. Michael Hardt and Oscar Von Stryk. 2002, Increasing Stability In Dynamic Gaits Using Numerical Optimization

13. Minkyu Won, Tae Hun Kang and Wan Kyun Chung. 2009, Gait Planning For Quadruped Robot Based On Dynamic Stability: Landing Accordance Ratio

14. Robotworx, 9 August 2010, <http://www.robots.com/faq.php?

question=work+envelope>

15. Industrial Electric, 9 September 2007, <http://www.industrial- electricity.com/9_Point-to-Point_Robot_Control.html>

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16. S R Deb. 2001, Robotics Technology and Flexible Automation

17. R. Tomoviv, 1961, A General Theoretical Model Of Creeping Displacement

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APPENDIX 1

Figure 2.2: Quadruped Robot

Quadruped robot designed by Mr. Yee Yuan Bin (2009), UTP, Design of Control System for Quadruped Robot (4-Legged) [10]. All the dimension, calculation and analysis is done base on the property of this robot.

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APPENDIX 2

Figure 4e: Work envelope

Figure 4e shows the work envelope for the tip of the robot’s leg. The tip of the leg can only reach the shaded areas which are from 60 mm from the body and the maximum of 310 mm from the body

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APPENDIX 3

Figure 4f: Work envelope from top view

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APPENDIX 4

Figure 4g: Example for Sequential PTP Control

A robot leg that have 3 servomotors at 3 different joint, joint 1, joint 2 and joint 3. By using Sequential PTP Control, servomotor at joint 1 runs first followed by the servomotor at joint 2 and then servomotor at joint 3 to complete the movement of the robot leg from point A to point B. The time to complete the movement is equal to the total time taken for each of the servomotor to complete their movement.

Joint 1

Joint 2

Joint 3

Point A

Point B

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Figure 4h: Example for Path Generated by Linear interpolation Versus Sequential PTP Control.

On the left is the path generated by Linear Interpolation and on the right is the path generated by sequential control. As can be seen from the figure the path generated from linear interpolation is smooth compared to the rough path generated by the sequential method

Joint 1

Joint 2

Joint 3

Point A

Point C

Joint 1

Joint 2

Joint 3

Point A

Point C

a) Linear Interpolation b) Sequential

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APPENDIX 5

Figure 4.3: Duty Factor versus Stability Margin graph

-1.40 -1.20 -1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40

0 0.2 0.4 0.6 0.8 1

Duty Factor Vs Stability Margin Graph

1432 1342 1243 1432 1324 1234

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APPENDIX 6

Polygon of Stability for Quadruped Creeping Gait 1423

STEP 0 STEP 1 STEP 2 STEP 3 STEP 4

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APPENDIX 7

Nearly all the Center of Gravity lays within the boundary lines of stability polygon indicating that the gait is statically stable but some of the CoG lays too close to the boundary for example for step 10 and 14 but the gait is still taken to be used for generating the graph of polygon boundary and Center of Gravity as none of the CoG lays outside the boundary. The gait however has a very high tendency to tip-over if the robot experiences even a small force in the direction of the CoG.

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APPENDIX 8

Nearly all the Center of Gravity lays within the boundary lines of stability polygon indicating that the gait is statically stable but some of the CoG lays too close to the boundary for example for step 6 and 10 but the gait is still taken to be used for generating the graph of polygon boundary and Center of Gravity as none of the CoG lays outside the boundary. The gait however has a very high tendency to tip-over if the robot experiences even a small force in the direction of the CoG as the CoG lies so close to the boundary of the polygon meaning that the magnitude of stability margin of the gait is very low.

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APPENDIX 9

Not all the Center of Gravity lays within the boundary lines of stability polygon indicating that the gait is statically unstable. The CoG in step 5, 6, 9 and 11 is outside the boundary line. Therefore the gait is not taken to be used for generating the graph of polygon boundary and Center of Gravity.

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APPENDIX 10

Not all the Center of Gravity lays within the boundary lines of stability polygon indicating that the gait is statically unstable. The CoG in step 5, 6 and 11 is outside the boundary line. Therefore the gait is not taken to be used for generating the graph of polygon boundary and Center of Gravity.

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APPENDIX 11

Not all the Center of Gravity lays within the boundary lines of stability polygon indicating that the gait is statically unstable. The CoG in step 5, 6, and 13 is outside the boundary line. Therefore the gait is not taken to be used for generating the graph of polygon boundary and Center of Gravity.

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APPENDIX 12 Gait Simulation (Gait 1423)

Gait Simulation at Step 0

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APPENDIX 13 Gait Simulation (Gait 1243)

Gait Simulation at step 0

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Gait Simulation at step 1 (Gait 1243)

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APPENDIX 14

Gait Simulation (Gait 1243 discontinuous gait) Gait Simulation at step 0

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Gait Simulation at step 1 (Gait 1243 discontinuous gait)

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Duty Factor Vs Stability Margin Graph

TABLE OF CONTENTS

Duty Factor Vs Stability Margin Graph