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Acknowledgements
All gratitude is to Almighty Allah, who is the source of knowledge and wisdom, the most Gracious, the most Merciful, who blessed me with the courage and energy to complete this project successfully. This thesis is the end of my voyage in obtaining my bachelors degree in Computer and Information sciences. During this period there were some people whose words of encouragement and appreciation always helped me to get through. I would like to express deepest gratitude to my project supervisor Dr. Muhammad Arif for his continuous support and motivation. None of this would have been possible without his generous supervision, perceptiveness and patience throughout the duration of this project. I find myself very fortunate to work under his supervision and gain from his knowledge and acumen. My appreciation goes to the “Signal Processing” Panel Members who monitored my work and took effort in reading and providing me with valuable comments on various project documents. Here, I would also like to acknowledge my family and friends who have always been there to cheer and support me. I owe lots of gratitude to my late grandfather who shaped my vision and taught me the good things that really matter in life. The happy memory of my grandfather still provides a persistent inspiration for my journey in this life.
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Table of Contents
List of Figures ............................................................................................................... iv List of Tables ................................................................................................................ vi
iv
- Chapter 1 Introduction................................................................................................ Abstract ........................................................................................................................vii
- 1.1 Bipedal Walking Models
- 1.2 Human body dynamics
- 1.2.1 Human Body Structure
- 1.2.2 Bodies in Planar Motion
- 1.2.3 Impulse and Momentum
- 1.2.4 Energy Transfers
- 1.2.5 Three Dimensional Motion
- 1.3 Problem Statement
- 1.4 Methodology
- 1.5 Scope
- 1.6 Thesis Organization
- Chapter 2 Literature Survey.....................................................................................
- 2.1 History of Biped Robots
- 2.2 Steps involved in Walking Gait Generation
- Chapter 3 Two Dimensional Biped Walking Models
- 3.1 Two Link Straight-Legged Model
- 3.1.1 The Model
- 3.1.2 Equations of Motion
- 3.1.3 Heel Strike
- 3.1.4 Stride Function
- 3.1.5 Analysis of the Model
- 3.1.6 Results and Discussion
- 3.1.7 Summary
- 3.2 Five Link Kneed Model with Upper Body
- Chapter 4 Three Dimensional Biped Walking Models iii
- 4.1 The 3D Straight-Legged Walking Model
- 4.1.1 The Model
- 4.1.2 Model Analysis
- 4.1.3 Results and Discussions
- 4.1.4 Summary
- Chapter 5 Animations................................................................................................
- Chapter 6 Conclusion and Future Recommendations
- References
- Appendix A: Basic Dynamics
- Appendix B: Human Body Segments Properties
- Figure 1-1: Static walking [12] List of Figures
- Figure 1-2: Dynamic walking [12]
- Figure 1-3: The three primary planes of a standing person [9]......................................
- Figure 1-4: Frontal view of human skeleton [9]
- Figure 1-5: Dynamic human model and its degrees of freedom [16]
- Figure 1-6: General plane motion of a rigid object B parallel to the (e 1 ,e 2 ) plane [9].
- Figure 1-7: A rigid body in planar motion [9]
- Figure 1-8: A rigid body in planar motion [9]
- Figure 1-9: Planar rotation of reference frame B with respect to E [9]
- Figure 1-10: A rigid body undergoing three-dimensional motion [9]
- Figure 3-1: A typical passive walking step [4]
- Figure 3-2: Single step at a long-period gait cycle for =0.009 rad
- Figure 3-3: Single step at a short-period gait cycle for =0.009 rad
- Figure 3-4: Several steps at a long-period gait cycle for =0.009 rad........................
- Figure 3-5: Several steps at a short-period gait cycle for =0.009 rad
- Figure 3-6: Leg angles versus time for =0.0 rad.......................................................
- Figure 3-7: Leg angles versus time for =0.0020 rad.................................................
- Figure 3-8: Leg angles versus time for =0.0021 rad.................................................
- Figure 3-9: Leg angles versus time for =0.0142 rad.................................................
- Figure 3-10: Leg angles versus time for =0.0143 rad...............................................
- Figure 3-11: Leg angles versus time for =0.0151 rad...............................................
- Figure 3-12: Leg angles versus time for =0.0085 rad...............................................
- Figure 3-13: Leg angles versus time for =0.0105 rad...............................................
- Figure 3-14: Leg angles versus time for =0.0089 rad...............................................
- Figure 3-15: Leg angles versus time for =0.009 rad.................................................
- Figure 3-16: Leg angles versus time for =0.0140 rad...............................................
- Figure 3-17: Leg angles versus time for = 0.0020 rad with l 1 units
- Figure 3-18: Leg angles versus time for = 0.0020 rad with l 1.2units
- Figure 4-1: 3D Passive Dynamic Biped Model [4]
- Figure 4-2: Configuration angles versus time for =0.0 v
- Figure 4-3: Configuration angles versus time for =0.0
- Figure 4-4: Configuration angles versus time for =0.0031
- Figure 4-5: Configuration angles versus time for =0.0031
- Figure 4-6: Configuration angles versus time for =0.0032
- Figure 4-7: Configuration angles versus time for =0.0032
- Figure 4-8: Configuration angles versus time for =0.0100
- Figure 4-9: Configuration angles versus time for =0.0100
- Figure 4-10: Configuration angles versus time for =0.
- Figure 4-11: Configuration angles versus time for =0.105
- Figure A-1: A system of particles (B) moving with respect to E [9]
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Abstract
“The human body is a machine whose moments are directed by the soul.” Rene‟ Descartes wrote in the early seventeenth century [9]. The objective of this project is to effectively study the phenomenon of bipedal walking and help visualizing rather difficult mathematical biped walking models. In order to successfully build and study humanoid models, one needs to understand the human body structure and behavior. The attempt to simulate a human body leads to a better understanding of it. Human walking exploits natural dynamics of the body to produce an energy efficient and stable walk. Under this project, a few existing biped models are investigated. The models are analyzed through simulation and animation techniques. Since humans move in three dimensions, the obvious approach is to build a complete model having all degrees of freedom in three dimensions. The more the degrees of freedom the more likely is the gait human like. But increasing DOFs also makes model dynamics more complex and difficult to understand. According to the literature survey carried out under this project, no humanoid model to this date has been reported that can be declared as a complete human model involving all natural degrees of freedom in three dimensions. In order to get some insight in human walking, one can start from rather simpler models having fewer degrees of freedom. Bipedal model analysis is a good choice in order to gain some understanding of human walking. Therefore, we started our research with the simplest two link model in two dimensions. The model analysis is supported with simulation and animation. Then the same model is studied in three dimensions and supported with simulation and animation controls. Both these models are passive. Passive models have gravity as the only source of energy. No active control is used to generate and stabilize a walk. The last model investigated is a five link model in two dimensions. This model has active control. A PD controller is used to instigate and sustain a stable walk. The three models are acutely analyzed. In the end some important conclusions are drawn about the sensitiveness of gait stability to model parameters.
Chapter 1 Introduction
The contents in this chapter include an overview of biped walking models and human body dynamics. The problem statement for the project and the methodology adopted to attack the problem are also explained. Furthermore, a part of this chapter focuses on project scope. The chapter concludes with outlining the organization for the rest of the thesis.
1.1 Bipedal Walking Models
Human motion is a topic of great interest in many fields. In order to have some insight in human walking, biped models are the best choice to be studied because they posses human like gait. Biped models have two legs. Biped locomotion involves walking, running and standing on two legs 0. Different models have different degrees of freedom (DOFs) depending upon the model characteristics. The more the degrees of freedom the more likely is the gait human like. But increasing DOFs also makes model dynamics more complex and difficult to understand. There are basically two types of walking models. These are active models and passive models. Active models have no gravity vector as input to cause motion. Instead the effect of gravity is reproduced by a controller that helps to instigate and sustain a walk. Active models can be static or dynamic (explained shortly in detail). Static models have their centre of gravity inside the support region whereas dynamic models can have their centre of gravity outside for short intervals of time. In active models, because of the absence of a gravity source, we always need a counter balance mechanism to provide constant energy pumping and it is achieved through the implementation of a controller. Passive walking models have a gravity vector as input to perform necessary movement. Passive models are always dynamic. With static walking, slow walking speeds are achieved. With dynamic walking, faster speeds can be achieved. Static Walking Models Static walking models are statically stable i.e. the model will remain in a stable position even if the motion stops. For a static walking model to be stable, the projection of its centre of mass (or centre of gravity) on the ground must be contained within the foot support area (Figure 1-1). In single support phase, having only one leg
supporting foot will either get off the ground or get pressed against the ground, resulting in instability [12]. Passive Dynamic Walking Models There is another special kind of models known as passive dynamic walking models. The term passive means that the energy required to carry out a walking step comes from no external source other than gravity. Dynamic is used to classify the type of stability associated with bipedal walking. Quadruples, as compared to humans, are statically stable because the projection of the centre of mass is always inside the support region defined by the three feet that are on the ground at any time. But this is not the case with humans. During human walking, the ground projection of the center of mass is constantly making excursions outside the support area defined by the one foot on the ground. Therefore, in order to assure gait stability in humans, the body should be in constant motion and thus repeatedly interrupting the falling motion with a successive step. This is called dynamic stability. Dynamic gaits result in faster walking speeds. In literature, the passive dynamic walking is simply referred as passive walking. [13]
1.2 Human body dynamics
In this section, complete human body structure is explained. The coordinate system used for human motion representation, human skeleton, skeletal muscles and the degrees of freedom for human body are discussed. Dynamics is a branch of classical mechanics that deals with the effects of forces on the motion of objects. Some important dynamics principles that are used to analyze human motion are also explained in this section.
1.2.1 Human Body Structure
Humans and animals possess a unique physical structure that enables them to stand up against the pull of gravity. Human and animals utilize contact forces to create movement and motion. The biggest part of the human body is the trunk; comprising on the average 43% of total body weight. Head and neck account for 7% and upper limbs 13% of the human body by weight. The thighs, lower legs, and feet constitute the remaining 37% of the total body weight. There are 206 bones in human body. Bone is a facilitator of movement and protects the soft tissues of the body. The human skeleton would collapse under the action of gravity if it were not pulled on by skeletal
muscles. Approximately 700 muscles pull on various parts of the skeleton. About 40% of the body weight is composed of muscles. Skeletal muscles act on bones using them as levers to lift weights or produce motion. A lever is a rigid structure that rotates around a fixed point called the fulcrum. In the body each long bone is a lever and an associated joint is a fulcrum. The levers can alter the direction of an applied force, and the speed of movement produced by a force. In human body there are muscles that run across several joints and then there are some that act on a single joint. For example, the muscles controlling the motion of the toes are attached to the bone of the leg and not to the thigh bone because when the knee joint is flexed, if attached to the thigh bone, these muscles would be bound under the knee joint and would not be able to serve the toes. [9] Notation for Human Motion A Cartesian coordinate system originating at the centre of gravity of human body in the standing posture is used to describe the spatial position of various parts of the human body. There are three primary planes of a standing person whose directions are indicated by the directions of the coordinate axis (Figure 1-3). The plane made up of x 1 and x 3 axes passing through the hip bone, at a right angle to the long axis of the body, divides the body into superior and inferior sections is known as the transverse plane_._ The plane that passes through the x 1 and x 2 axes of the coordinate system and divides the body into anterior and posterior sections is known as the frontal plane. The plane made up of x 2 and x 3 axes which divides the body into left and right sections is known as the sagittal plane. There are four common angular movements of the various parts of the human body. These are flexion, extension, adduction, and abduction. Flexion and extension are movements that occur parallel to the sagittal plane and adduction and abduction occur parallel to the frontal plane. Flexion is rotational motion that brings that brings two bones closer to each other such as flexion of leg. Extension is the rotational motion opposite to flexion. If the movement of extension continues past the anatomical position, it is called hyperextension. Abduction and adduction are the movements of the limbs in the frontal plane. The movement away from the longitudinal axis of the body is known as abduction whereas the moving back towards the axis is adduction. For example, swinging the arm to the side is an example of abduction whereas bringing it back is adduction. There is another kind of movement in human body parts known as rotation. Rotation
The Human Skeleton
Figure 1-4: Frontal view of human skeleton [9]
The human skeleton is basically composed of 206 bones. The skeleton is divided into two parts: the axial and the appendicular parts. The axial skeleton shapes the longitudinal axis of the human body. It comprises of the skull bones, bones of the vertebral column, ribs and sternum. There are approximately 420 different skeletal muscles acting on acting on it. The weight of the head, trunk and the upper limbs is transmitted to the lower limbs at the hip joint by the axial skeleton. The appendicular skeleton consists of the bones of the upper and lower limbs and the supporting elements (girdles) that connect them to the trunk. There are 126 bones in the appendicular skeleton and approximately 300 muscles acting on the bones to cause movement or to sustain a certain pause. The long bone of thigh, the femur, is the longest and the heaviest bone in the body. The lower leg has two bones, tibia and fibula. At knee joint the end of femur articulates with tibia. The fibula is slender in comparison with the tibia and is excluded from the knee joint. The other end of fibula is connected to the ankle joint. The fibula does not transfer any weight to the ankle and the foot. There is a fibrous membrane present between the two bones of the lower leg to stabilize their position and provide additional surface area for muscle attachment. High impact activities drastically increase the loads carried by the bones of the lower leg. Usually the strong muscles and mobile joints act as shock absorbers, damping the intensity of the peek load transmitted to the bone. Human joints are basically divided into three categories based on the range of motion permitted at the joint. Synarthrosis is a kind of immovable joints. For example, joints found between the bones of the skull and between the teeth. The second group known as amphiathrosis consists of joints that allow for slight movements. For example distal articulation between tibia and fibula is an example of this kind of joint. The joints that allow considerable motion of articulating bones are called diarthrosis or freely moving joints. These joints are typically found at the end of the long bones, such as those of the leg and arm. Examples are hip, knee and ankle joints. The degrees of freedom provided by a joint vary from joint to joint. [9] The Skeletal Muscles Another important feature of human body is muscles. Muscles are composed of bundles of long and thin cells called muscle fibers. Any contraction of the muscle exerts a pull on the bone to which it is attached. Muscles have a stabilizing influence on articulations. Skeletal muscles must be stimulated by the central nervous system
displacements that specify completely the displaced or deformed position of the body or system. A particle that moves in three dimensional space has three translational displacement components as DOFs, while a rigid body would have at most six degrees of freedom including three rotations. In general, a rigid body in n-dimensions has n (n+1)/2 degrees of freedom (n translations + n (n-1)/2 rotations). In 1-, 2- and 3- dimensions, we have one, three, and six degrees of freedom. In three dimensions, the six DOFs of a rigid body are sometimes described using these nautical names: Moving up and down (heaving) Moving left and right (swaying) Moving forward and backward (surging) Tilting up and down (pitching) Turning left and right (yawing) Tilting side to side (rolling) 0 Applying rigid body analysis to human motion allows us to represent human body as a chain of interconnected links. A system with several bodies has a combined DOF that is the sum of the DOFs of the bodies. The DOFs for a 2D and 3D human model are tabulated in Table 1-1 and supported by Figure 1-5. The arrows in Figure 1- indicate the joint positions and the axes of rotational DOFs. Table 1-1: DOFs of a dynamic human model [16] Joint Rotational DOFs 3D Skeleton Model
Rotational DOFs 2D Skeleton Model Head 1 1 Neck^3 Shoulder 2 1 Elbow 2 1 Wrist 2 - Waist 3 1 Hip 3 1 Knee 1 1 Ankle 2 1
Figure 1-5: Dynamic human model and its degrees of freedom [16]
1.2.2 Bodies in Planar Motion
In reality, the mass of a body is distributed more or less uniformly throughout the body rather than being concentrated at few points. In the analysis of moment and motion, various body segments of human body are such as head, thighs and forearms can be reasonably assumed as rigid. A rigid body is a solid object such that the distance between any two of its material points remains constant during resting state or in motion. The significance of rigid body analysis to human dynamics is that the human body can be reasonably well represented by an interconnected chain of rigid links in the analysis of upper and lower limb movement. The motion of a body is called planar when all particles in the body move in parallel planes. [9] In planar motion parallel to ( e 1 , e 2 ) plane, the angular velocity ω of a rigid object B with respect to reference frame E is defined as the time rate of change of angle ( ) between a straight line in the rotating body B in the ( e 1 , e 2 ) plane, taken counterclockwise. = d /dt e 3 (1-1)
= d /dt = d 2 /dt 2 e 3 (^) = e 3 (1-5)
When angular acceleration is in the positive e 3 direction, then the rate of rotation increases in the counterclockwise direction. The moment of momentum of a rigid object is called angular momentum. For rigid objects that are undergoing planar motion in a plane of symmetry of the object, angular momentum with respect to the centre of mass is given as:
H c = Ic e 3 (1-6) where H c^ denotes the angular momentum with respect to the centre of mass I c^ is the mass moment of inertia of the object with respect to the centre of mass. It is a measure of resistance of the object to the changes in rate of rotation. If a point of the object, say point O, is fixed on earth and the object rotates around O, the angular momentum with respect to point O is given by the relation:
H o = Io e 3 (1-7)
The parameter I o, the mass moment of inertia with respect to point O that is fixed in E, is related to I c^ by the following equation:
I o = m r +^2 I c (1-8) where r is the distance between the centre of mass of the object and point O According to the law of conservation of angular momentum:
Io e 3 = M o (1-9)
Ic e 3 = M c (1-10)
The right hand side of these equations refers to the resultant external moment acting on the object with respect to the fixed point O and the centre of mass respectively. The principle of conservation of angular momentum relates the changes in the rate of rotation to the resulting moment acting on an object. [9] Applications to Human Body Dynamics The most fundamental step in the analysis of movement and motion is drawing a free- body diagram showing all external forces acting on an object. If the free body under consideration is part of a body segment, then external forces to be shown include those forces that the rest of the body apply on the body segment under consideration. Although such forces are treated as internal forces in the overall motion of the entire body, they need to be considered as external forces when the movement of an individual body segment is studied.
Some of the external forces acting on an object (part of an object) may be known in both direction and magnitude e.g. gravitational force. In other cases, we might know something about the direction of an external force. When the friction between objects can be neglected, direction of contact force becomes perpendicular to the surface of contact. Frictional force is always in the direction that opposes relative sliding on the surface of contact. There are also propulsive forces that actuate movement and motion. For example, in the case of a bicycle, the propulsive force is in the direction of motion. In speed skating, the propulsive force lies in a plane that is at right angles to the gliding direction. [9]
1.2.3 Impulse and Momentum
The impact forces affect the movement and motion of human body. The analysis of impact forces is based on the mathematical relationship between impulse and momentum. A large force acting on a body during a very brief period of time may instantaneously alter the velocity of the body. Such forces often occur when two bodies collide. [9] Impulse ζ of a force F during a time interval tf - ti is defined as the time integral of the force over tf - ti.
= dt
t f t i
F (1-11)
The impulse acting on an object is directly related to the change in linear momentum. The linear momentum L of an object at time t is defined as:
L = m v c (1-12) where m is the mass of the object v c^ is the velocity of its centre of mass at time t According to Newton‟s second law, the equation of motion of the centre of mass of an object is:
d L /dt = F (1-13)
where ∑ F is the resulting force acting on the body Integrating the equation of motion for the centre of mass between time t = ti and t = tf , we obtain the following relationship:
m v c^ f - m v ci= (1-14)
where m is the mass of the body
