The Different Models of the Biped Walking Robots.
Contents
 1 INTRODUCTION
 2 AIMS AND OBJECTIIVES
 3 CHAPTER 3.LITERATURE REVIEW
 4 3.1 WL10RD, BIPER3 and 4
 5 3.2 WHL11
 6 3.3 WL12RIII
 7 3.4 WASUBOT and Manny
 8 3.5 P1, P2 and P3
 9 3.6 Main point according Fumio Kanehiro
 10 Year
 11 Researcher
 12 Area of Development
 13 1850
 14 1872
 15 1893
 16 1961
 17 1968
 18 1977
 19 1980
 20 1981
 21 1985
 22 1992
 23 1996
 24 1998
 25 2000
 26 2002
 27 2004
 28 2006
 29 2007
 30 CHAPTER 4 BIPEDAL WALKING
 31 4.1 Why Study Legged Robots?
 32 4.2 Walking Gait
 33 4.3 Why pointed feet?
 34 4.4 Terminology
 35 4.5Passive Dynamic Walking
 36 4.6Robot Walking Hypothesis
 37 4.7 Dynamics Model
 38 4.7.1 Lagrangian mechanics
 39 4.7.2 Ground Contact
 40 4.7.3 Model Equations
 41 4.7.4 Knee Angle limiter
 42 4.8Control Model
 43 4.8.1 PD Controller
 44 4.8.2 Reference Signals
 45 CHAPTER 6 IMPLEMENTATION
 46 Field
 47 Description
 48 Units
 49 6.1 Biped block
 50 6.1.1 Dynamic model
 51 6.1.2 Ground Contact Model
 52 6.1.3 Knee Angle Limiter block
 53 6.2 PD Control Block
 54 CHAPTER 7 BIPED WALKING RESULTS
 55 BIPEDAL RUNNING
 56 8.1 Introduction
 57 8.2 Running Cycle
 58 8.3 Robot Running Hypothesis
 59 8.4 Trajectory with telescopic legs
 60 8.5 Running with segmented legs
 61 8.5.1 Reference and effective stiffness
 62 Implementation
 63 METHODOLOGY OF BIPEDAL RUNNING
 64 FUTURE WORK
 65 CONCLUSION
INTRODUCTION
AIMS AND OBJECTIIVES
CHAPTER 3.LITERATURE REVIEW
This portion would discuss the background research in detail, the methodologies and other useful aspects involved in designing the earlier models of the bipeds and pros and cons of the different models of the biped walking robots.
The first biped walking robot was established in 1893 by a native Canadian; Prof. George Moore (Mechanical Man, 1893) reported he build a robot which was a figure of a man, constructed of iron and fitted with internal mechanism, used steam for motion was intended to move similar to the walking gait of a human being. I t appeared like an oldfashioned knight. The walking speed for this model was around four to five miles per hour and stood six feet tall in height. (Roshiem, 1994) stated that the steam man was powered by a gas fired boiler with a power of 0.5 h.p. Swing arm provided it with stability as it guided him in circles.
(Machado, Silva, 2005) told that computer controlled biped systems has been a much focused area at Waseda University, Japan, since the end of 1960’s. At the Humanoid Research Laboratory a biped robot was transpired by Ichiro Kato on 1969 which was called WAP1. The robot mainly consisted of artificial rubber muscles for its actuation. Playback of priory taught movement was used for the biped locomotion. The main restraint in WAP1 was its low speed. It was followed by WAP2 and WAP3. (Thomas Isaac, 2004) stated that in 1971 WAP3 was developed which could not only move on flat surfaces but could also move up and down on the stair case by moving its center of gravity on the frontal plane. WAP3 was the first in the world to achieve the threedimensional walking and turning. It was directed by a control based memory.
3.1 WL10RD, BIPER3 and 4
As stated (Elliot Nicholls, 1998) in 1985 another robot used the quasidynamic walking which was named WL10RD and it was developed by the same research team as above. Since this time the development in this research has been rather drippy. BIPER3 was developed in 1984 by Miura and Shimuyana which completely flake out the static balance entirely. This was modeled after the human walking on stilts, showed true active balance. This robot contained three actuators in which one is used to change the angle separating the legs towards the motion direction and the other two which lifted the legs to the side in the sidelong plane. This robot is termed as three degree of freedom robot. Later another robot was developed named BIPER4 and this was extended to seven degree of freedom robot.
Another robot was developed by Raibert using the methodology that the robot used uncluttered driven leg for the leaping motion and was attached to a chain which restricted pitch motion, vertical and horizontal translation around a radial path inscribed by the chain. The current progress of the leaping motion of the robot was tracked using the state machine activated by the sensor feedback. The state machine was then used to modify the control algorithm which modified three parameters of leaping stride forward speed, foot placement and body attitude.
Hodgins, Koechling and Raibert developed a dynamic running robot which extended the earlier study of onelegged hopping robot into two and three dimensional. The two dimensional robot used the same control methodology as previously used in the hopping robot in two dimension controlling the three aspects of the running stride which are body height, foot placement and body attitude using the state machine. The robot was controlled differently using the different sates of the sate machine.
3.2 WHL11
(Karsten Burns, 2010) describes that in 1985, biped walking robot was developed by Waseda University’s Humanoid research laboratory in partnership with Hitachi Ltd. The robot could walk on flat path at the rate of 13 seconds a foot and it was achieved by putting an onboard computer and a hydraulic pump in addition to that of earlier WL series robot. It was seen that this robot could walk up to 64 km. The main cons in this robot were that it was unable to walk on inclined surfaces. So since the work is being done to stabilize the movement to develop a control system that can adjust gaits.
3.3 WL12RIII
Another biped walking robot was established in 1989 as described by (Agrosy 2010) named WL12RIII and it used the principle of stabilizing its walking on different paths using the trunk motion. An algorithm was developed used to calculate trajectory of the trunk and was done by introducing a “virtual surface” which is derived from the geometry of the path and the trajectory of the feet. When experimented on a stair with a height of 10cm and the time it took to go up and down was 2.6 seconds and when experimented on a inclination of 10 degrees, it took 1.6 seconds to walk down the path. Kato came up with a control methodology where the path of the robot is unknown as well as the external forces acting on it by using combinational motion of the trunk and lower portion of the biped. So finally the step size of the biped was reduced to 0.64 of a second.
3.4 WASUBOT and Manny
(Agrosy, 2010) has explained in his site that WASUBOT is another biped robot. Its basic principle is the same as that of the WABOT2. The change was rather a clarification in maintenance of the robot and WAM 8’s as the arms. In 1989 at Battelle’s Pacific Northwest Laboratories in Richand, a full manlike behavior robot named Manny was developed. It took three years and $2 million to develop this robot by the work of 12 researchers. It was delivered to the US Army’s Dugway proving ground in Utah. Manny had 42 degree of freedom.
3.5 P1, P2 and P3
According to (Daniel Ichbiah 2005) with the aim of producing bipedal walking robots in 1993 the first series of P prototypes came into existence. First developed was P1 which was quite big and heavy like a metal monster which was 6 feet and 2 inches in height and was 175 kilograms in weight. Instead of a face it had a big screen of rectangular shape. In challenge of making robots more or less like the humans they came up with others models like P2 and P3. P2 came into being officially on 20th of September 1996 and cost around $ 105.3 million. It was shorter in height but weighed more like 208 kilos. September marked a critical moment in the history of Biped robots by the coming of P3. It was made of purified white steel and plastic and resembles to an astronaut. It was based on the same technological model and had same technological capabilities but its height was 5 feet and 4 inches and weighed 130 kilos quite short and light weighted as compared to the earlier models. P3 walked at the same speed as the human. Coming of P3 was a big step towards the modelling of Biped robots quite similar to humans.
Honda was keen to accept this challenge of reducing some more weight. In 2003 an upgraded version of Asimo was introduced it was a great machine which had some social ethics of greeting. It could also pass information like the weather forecast.
According to (Daniel Ichbiah, 2005), in the summer of 2003 HPR2 came in to existence by the production team of AIST. It was a prototype which could get down and stand up all by itself. It worked using the limb coordination software and Kawada industries were able to design its body with flexible joints but the main problem was it was heavier and taller than Asimo.
(Kieth Kleiner, 2009) told that Toyota in 2009 developed a humanoid robot which could run upto 7 km/h overcoming Honda’s Asimo which could run to 6 km/h. Hondo Asimo robot is capable of ascending and descending on a staircase and handling varying circumstances unlike Toyota which can only on flat surfaces.
(Kim Tae Gyu, 2009) Humanoid robot RX is a running robot which was developed by South Korea which after Japan is the second in the world which was put forward by a local avocation working with not much of a budget
3.6 Main point according Fumio Kanehiro
According to (Daniel Ichbiah 2005) Fumio Kanehiro who is the member ASIT said “One of the main problems with humanoid robots is that they easily fall over. When a biped robot stands on its two feet, only a very small area in contact with the ground supports it, while its centre of gravity is at waist level, which is relatively high.”
Year 
Researcher 
Area of Development 
1850 
Chebyshev 
Design Linkage used in early walking mechanism 
1872 
Muybridge 
Uses stop motion photography to document running animals 
1893 
Rygg 

1961 
Space General 
Eight legged kinematic machine walks in outdoor terrain 
1968 
Frank and McGhee 
Simple digital logic controls walking of Phony Pony 
1977 
McMahon and Greene 
Human runner with speed record on tuned track at Harvard 
1980 
Kato 
Hydraulic biped walks with quasidynamic gait 
1981 
Miura and Shimoyama 
Walking biped balances actively in threedimensional space 
1985 
A. Takanishi et al 
Realization of dynamicbipedwalking stabilized with trunk motion under known external force 
1992 
Kajita et al 
Dynamic Walking Control of aBiped RobotAlong a Potential Energy Conserving Orbit 
1996 
Kun, A., Miller II 
daptive dynamic balance of abipedusing neural networks 
1998 
Park, JH and Rhee 
ZMP trajectory generation for reduced trunk motions ofbiped robots 
2000 
elexistence technology was adapted in a new type of cockpit system to control a humanoid biped robot 

2002 
S. Meyret, A. Muller 
Adaptative neurofuzzy control of the rabbitbiped robot 
2004 
J. Gutman, M. Fukuchi 
Detection of stair dimensions for the path planning ofbiped robot 
2006 
A. Sutherland 
Torso Driven Walking Algorithm for Dynamically Balanced Variable SpeedBiped Robots 
2007 
F. Asano and Z.W. Luo 
Asymptotically stable gait generation forbiped robotbased on mechanical energy balance 
Table. Development in the field of legged robots
CHAPTER 4 BIPEDAL WALKING
4.1 Why Study Legged Robots?
(M.H Raibert 1986) mentioned there is a contemplative reason for researching robots with legs, apart from the outright adventure of developing robots that can actually run. Reason being the mobility and they provide an exertive suspension from the gait of the feet. Moving through difficult terrain, where other cannot go is one of the main reasons for legged robots. Legged robot perform actively while moving on a rough terrain unlike the traditional wheeled robots which can only move on flatter surfaces and in result this limits the wheeled robots to move on half the earth’s landmass. Legged robots use isolated footholds that maximise support and friction but wheeled robots need a constant path of support. Another advantage is, despite pronounced variations in the terrain, the payload is free to travel stably. Legged robots can also tip through the hurdled obstacles.
(Yang and Kim, 1998;; Spenneberg, et al., 2004) have investigated a further lately, worries malfunction tolerance during immobile stable locomotion. The effect of a failure in one of the wheels of a wheeled vehicle is a stern lost of mobility, since all wheels of these kinds of vehicles should be in permanent contact with the ground during locomotion. However, legged vehicles may present a superfluous number of legs and, therefore, can maintain static balance and continue its locomotion even with one or more of its legs damaged.
(M.H Raibert 1986) explains another main reason for studying legged robots is to boost a greater knowledge of human and animal movement. This point could further be explained by actually analyzing the athletes during the instant replays. We could analyze the complexities and the various procedures involved in the various postures and position of legs while they are performing different tasks and can study the movements while they swing, throw, drift, maintaining balance and speed as they go or otherwise drive their body through space. This performance can not only be seen in the athletes performing on Television we can have our sight set at the local playground where one’s own child in coming forward from a phase of crawling on four sets to walking on two legs and then running, jumping, climbing and performing various other exercises.
4.2 Walking Gait
Studying and analysing (M.W.Whittle, 2003) normal gait has quiet an importance. It is also important to know the nomenclature used to describe the gait. This section gives a detailed overview of the gait cycle before formally developing a detailed mathematical and software model.
It is challenging (Joel, DeLisa, 1998) to come up with a formal definition of the walking gait without sounding pompous as it looks quite a simple task. An informal definition of the walking can be put forward as a method of locomotion involving the use of two legs, to provide both support and propulsion. Clinical study of the gait is the most commonly used technique from the various techniques. A single gait cycle can be described as the sequence or interval between a foot strikes to foot strike of the same leg.
There have been (Lamm R.D., 1995) two main classifications of a complete gait cycle phase: stance phase and the swing phase. The phases are also subdivided as shown in the figure below.
right left left right right left
initial pre initial pre initial pre
contact swing contact swing contact swing
Time percent of cycle
Double R.Single Support Double L.Single SupportDouble
Support Support Support
0% 15% 45% 60% 100%
0% 40% 55% 85% 100%
The interval of the stance phase is taken as 60% of the total interval. Double support phase and single support phase are the two subdivisions of the stance phase. Double support phase can be encompassed as when both the legs are in contact with the ground. It has been analysed that at average walking the double support is 10% of the total gait interval, but as the speed increases, the double support interval decreases. The remaining interval is the single support phase. To avoid buckling of the support foot in the stance phase, the muscles like tibialis anterior, the quadriceps, the hamstring, the hip abductors come into function
Energy conservation during a walking cycle can be categorized into three main events. The events include controlling the forward movement during the deceleration towards the end of the swing phase, jolt absorption while the foot land on the ground and momentum during pushoff when the centre of gravity is pushed up and forward. A human’s centre of mass is a located at the hip joint. Centre of mass does not deviate up or down, when a body is moving in a straight line, at his moment not much energy is required. This straight line is only possible wheels are placed instead of the feet, but this is not the case in humans so it deviates in a vertical and lateral sinusoidal displacement. At midpoint centre of mass is at the highest position and centre of mass during human locomotion goes in rhythmic flow of upwards and downwards motion. At time of the double support phase the centre of mass is at the lowest point
4.3 Why pointed feet?
(Westervelt et al, 2007) degree of actuation of the system is an essential source of complication or in a more absolute domain, the degree underactuation. Underactuation is used to describe devices that have lower number of actuators then degree of freedom. Inverted pendulum is a classic example of under actuated system. The model taken in this report assumes that at stance leg there is no possibility of actuation and legs are terminated in points. There could be a major concern over modelling feet pointed because “real robots have feet”. Against the mechanical bipedal walking which is to be contrasted, if one takes a person’s walking as the defected benchmark, then the flatfooted walking accomplished by current robots needs to be improved? In particular, toe roll toward the finish of the single support phase needs to be certified as part of the gait plan. Currently, this is not legitimate specifically because it leads to underactuation, which cannot be indulged with the control design philosophy based on trajectory tracking and a quasistatic stability principle, such as the zero moment point (ZMP).
A model of an anthropomorphic walking gait should at minimum regarded as a fully actuated phase where the stance foot is level on the ground, pursued by an underactuated phase where the stance heel rises from the ground and the stance foot rotates with reference to the toe, and a double support phase where leg switch over takes place, optionally, heel strike and heel roll could also be incorporated, which would yield a second underactuated phase in the gait. In either case, a model of walking with a point contact is an essential element of a general model of walking that is more anthropomorphic in nature than the existing flatfooted walking paradigm. Because of the fact that the model with point feet is quite simple as compared to a more complete anthropomorphic gait model, it makes possible the development of new feedback designs and dynamic stability analysis methods that are suitable for moving past quasistatic walking.
4.4 Terminology
Some basic nomenclature would be put forward before going towards the formal mathematical modelling of the biped robot. This nomenclature would permit a casual explanation of the essential fundamentals of a dynamic model of a bipedal robot to be given which, in turn, will allow some demanding characteristics of the control problem to be elevated. A biped is referred to as an open kinematic chain which consists of two subchains which could be referred to as legs and torso. These are all connected at a common point called a hip. During the walking cycle of the biped, either one foot or both foot are on the ground. These are referred to as single support phase and double support phase. According to the model taken in this report, single support phase is defined as the phase of the walking cycle when only one foot is in contact with the ground. Stance leg is the contacting leg and the other is called the swing leg. When both feet are in contact with the ground, this is referred to as the double support phase
It is required that the movement of the robot’s centre of mass is strictly monotonic. Nonslipping nature of the feet is assumed when in contact with the ground. A planar biped is a biped whose motions only take place in the sagittal plane. Sagittal plane is the longitudinal planes that divide the body into right and left sections. Three dimensional bipeds have locomotion both in sagittal and frontal plane.
A statically stable gait is a rhythmic movement in which the centre of the mass of the biped lies in the support polygon. All the point of contact on the ground forms a convex hull which is referred to as support polygon. A dynamically stable gait is a rhythmic movement where the centre of pressure of the biped is on the boundary of the support polygon for at least part of the cycle.
4.5Passive Dynamic Walking
Passive dynamic walking is purely stationed on the recovery of the biped dynamic structure. Passive walkers are capable of maintaining a stable, rhythmic walking motion but they do not need external energy. The passive dynamic walker uses one leg to freely swing using its own weight and the other one supporting the swing. The second leg turns to swing while weight moves foot to another.
The subject has been studied widely. (T. McGeer, 1990) showed that a biped robot, suitably constructed to walk without external support. Very little energy is required when the knee joints are used that is why people use walking to benefit the body and legs. (M. Coleman ja, 1998) purely passive walker’s biggest limitation is their ability to walk only downhill. Increasing the system’s lowpower control can be achieved walker who keeps a stable walking motion ina flat or slight uphill, but whose energy is close to the minimum.
(Ted McGeer,1990) the practical enthusiasm for working on passive walking is, first, that it makes for mechanical simplicity and relatively high efficiency. Second, control of speed and direction is simplified when one doesn’t have to worry about the details of generating the gait. Moreover, the simplicity of the machine promotes understanding. Consider an analogy with the development of powered flight: The Wrights put their initial efforts into studying gliders, as did their predecessors Cayley and Lilienthal. Once they had a reasonable grasp of dynamics and control, adding a power plant was a relatively minor modification. (In fact their engine wasn’t very good for its day, but their other strengths led them to outstanding success.) As I’ll explain, adding power to a passive walker involves a comparably minor modification.
4.6Robot Walking Hypothesis
(Westervelt et al, 2007) a biped walking robot would be modelled based on the properties listed below. In order to ensure the robot satisfies these properties, a controller would be imposed.
 The robot consist of periodic phase of single support and double support
 Throughout the contact, the stance leg end acts as an ideal pivot, during the single support phase. The ratio of the horizontal component to the vertical component does not excel the coefficient of static friction. The vertical component of the ground reaction device is nonnegative.
 There is no slip or rebounding of the swing leg, while the previous stance leg discharges without correspondence with the ground.
 With respect to the two legs, the motion is symmetric in steady state.
 In each step, the swing leg begins from strictly behind the stance leg and at impact is positioned in front of the stance leg.
 Walking takes place on level surface from left to the right
4.7 Dynamics Model
The biped under consideration is a simple two foot robot to ensure the possibility of mathematical simulation. The model simulates the complete walking system. (Olli Haavisto et al, 2007) biped robot is a twodimensional system with five links including a torso and two identical legs with knees. This model has been used by many researchers
(Hardt et al, 1999; Juang, 2000) as it explains the walking motion of a biped robot quiet well.
mo
(xo, yo)
ro
α lo
r1
m1 l1
βL βR ML1 r2
γR MR2
y m2 l2
L ML2
x FLx FRx
FLy FRy
The biped robot under consideration needs seven variables to describe the position in twodimensional coordinate system which means that the robot has seven degrees of freedom. The coordinate (xo, yo) describe the position of the centre of the mass of the torso in a fixed position and α being the angle of the torso with respect to the normal. βL is the angle that the left thigh joint makes with respect to the torso and βR is the joint angle of the right thigh with respect to the torso. γL and γR are the left and right knee joint angle with respect to the left and right thigh joint respectively.
q = [xo, yo, α, βL, βR, γL, γR]T
(l0, l1, l2) denotes the link length of the torso, thigh and the shin respectively and (m0, m1, m2) denotes the masses. The link’s centre of the mass is located at a distance (r0, r1, r2) from the corresponding joint. External forces are used to model the walking plane that involves both the legs. The forces are in effect, when the leg touches the ground, to support the leg. The force is zeroed when the leg is not in contact with the ground or when the leg rises.
F = [FLx, FRx, FLy, FRy]T
The actual control signal for the model is the four moments, two of them actuated between torso and both the thighs and two are actuated at the knee joints.
M = [ML1, MR1, ML2, MR2]T
The two foot walking model used here has been studied in detail. (C. Chevallereau, 2003) developed a robot named RABBIT in which it was assumed that the when the swing foot hits the ground the other foot immediately rises into the air. All the walking stages have been described in this model but thelegwill always causea a steppedchange inthe systemmode and is calculatedseparately.
4.7.1 Lagrangian mechanics
(David McMahon, 2008) by taking the difference between the kinetic and the potential energies, a function lagrangian can be constructed. It can be used to derive the equations of the motion and is an equivalent to the Newtonian method. The lagrangian is a fundamental concept which captures all the dynamics of the system and allows us to determine many useful properties such as averages and dynamic behaviours.
(R.D. Gregg, 2011) a mechanical system with ndegree of freedoms with Q as the configuration space is described by elements (q, q) of tangent bundle (the space of configurations and their tangential velocities) TQ and Lagrangian function L : TQ → R given in coordinates by
Lq, q= Kq, q V(q) = 12qTM(q)q V(q)
Kq, q = kinetic energy
Vq = potential energy
M(q) = generalized mass/inertia matrix
By the least action principle, system integral curve necessarily satisfy the Euler Lagrange (EL) equations
ddt∇qL ∇qL= τ
where ndimensions vector τ contains the external joint torques. This system of secondorder ordinary differential equations gives the dynamics for the actuated mechanism in phase space TQ. These equations have the special structure
Mqq+ Cq, q+ Nq= Bu
where n x n matrix C(q, q)contains the Coriolis/centrifugal terms, vector Nq= ∇qVq contains the partial torques and n x n matrix B maps actuator input vector u ε Rm to joint torques τ = Bu ε Rn.
4.7.2 Ground Contact
(Heikki, 2004) using a set of x, y points, ground surface can be modelled which are linked through straight lines. In order to make the next ground point as origin point to negative x direction of the leg tip new coordinate system is defined i.e.(x’, y’). Below figure shows that the position of the axis x’ is tangential to the ground whereas y’ equals the surface normal direction.
y’
x’
y (x’0, 0)
Ft (x’G, y’G)
x
Fn
Figure: The leg tip touches the ground in point (x’0, 0)(grey) and penetrates it.The current position of the leg tip is (x’G, y’G)(black).
Normal and tangential forces are applied to the leg tip when it touches the ground at the point(x’, 0). In order that the situation is analogous to spring damper system, the output of the PD controller is used to calculate the normal force.
Fn=kyy’Gbyy’G
y’G = current (negative) leg tip y’ coordinate
ky = ground normal elastic constant
by = normal damping ratio
In order to prevent the leg sticking to the ground, only positive values would be taken for the normal force. In the tangential direction the force Ft acting is due to friction. Similar to the normal force calculation, PD controller is used to determine static friction force. The nominal value is the initial touching ground x’0
Ft=kxx’Gx’0bxx’G
kx and bx = ground tangential properties
If there is an exceeding required force than the maximum static friction force
Ft,max=μsFn
μs = static friction coefficient
If the leg starts to slide than the tangential force is
Ft=μkFn
μk = kinetic friction coefficient
The accumulated value of x’0 is constantly set to the corresponding leg tip x’G position throughout sliding. In order to attain the dynamic model input forces, the normal and tangential forces are needed to be projected to the original coordinate system (x, y) after these forces are computed.
4.7.3 Model Equations
(Olli Haavisto, 2004) system status is determined by the generalised coordinates and their time derivative, coordinates with each consequent to a generalized power
Fq = [Fxo, Fyo, Fα, FβL, FβR, FγL, FγR]T
The generalised coordinated of the centre of mass of the thigh of the left and the right leg is denoted by (xL1, yL1) and (xR1, yR1) respectively. The coordinates of the centre of the mass of the left and right shin link is represented by (xL2, yL2) and (xR2, yL2) respectively. The leg tip position of the left and right leg is denoted by (xLG, yLG) and (xRG, yRG) respectively. These notations can be represented in equations as:
xL1= x0 r0sinα r1sin(αβL)
yL1= y0r0cosαr1cos(αβL)
xL2= x0 r0sinα l1sin(αβL)r2sin(αβL+γL)
yL2= y0 r0cosα l1cos(αβL)r2cos(αβL+γL)
xLG= x0 r0sinα l1sin(αβL)l2sin(αβL+γL)
yLG= y0 r0cosα l1cos(αβL)l2cos(αβL+γL)
The kinetic energy can be easily expressed in terms of the Cartesian coordinates taking in to account all the links. The total kinetic energy can be written as
T=12(m0x02+y02+m1xL12+yL12+yR12+yR12+m2xL22+yL22+yR22+yR22
The generalized forces in terms of generalized components can be written as
Fx0=FLx+FRx
Fy0=m0+2m1+2m2g+FLy+FRy
Fα=∂yL1∂αm1+∂yL2∂αm2+∂yR1∂αm1+∂yR2∂αm2g+∂yLG∂αFLy+∂yRG∂αFRy+∂xLG∂αFLx+∂xRG∂αFRx
FβL=∂yL1∂βLm1+∂yL2∂βLm2g+∂yLG∂βLFLy+∂xLG∂βLFLx+ML1
FγL=∂yL2∂γLm2g+∂yLG∂γLFLy+∂xLG∂γLFLx+ML2
For the right leg the coordinates will be replaced by the coordinates of the left leg and (FLx, FRx, FLy, FRy) as mentioned earlier are the external forces of both legs.
Using the equation of the generalized coordinates, kinetic energy in terms of generalized coordinates and generalized forces are placed in the Lagrange equation and the dynamic equation can be written as
Aqq= b(q, q, M, F)
Using mathematica software this could be simplified,
Aq ε R7x7 is the inertia matrix and b(q, q, M, F) ε R7x1 is the vector containing the right hand sides of the seven partial differential equations. Appendix B contains both A and b in closed form formulas.
4.7.4 Knee Angle limiter
(Heikki, 2004), the knee angle limiter is set according to the ground surface. This is mainly done as same as the calculation for the ground contact normal force. A PD controller is used to control the maximum and the minimum limit of the knee angle. In order to prevent the joint rotating over or under the limit the controller output is added to the corresponding knee joint moment. It is desirable that the knee shouldn’t rotate over or under the zero angle of γL or γR which is bending towards wrong direction.
4.8Control Model
(R.K.Mittal et al, 2003) the control needs the data of the mathematical model and some kind of intelligence to proceed on the model. (Bijoy K. Ghosh, 1999) There are three functional abilities that a control design can be categorized to in which firstly being that at the task level the robotic system should be controlled directly i.e. without planning type decomposition to joint level commands, it should take task level commands directly. Secondly, rather than for a specific task, the robot control system should be designed for a large scale of tasks so the system becomes task independent. In the end robot should have the capability of handling some unexpected or uncertain events. A traditional control system is described in the figure shown below
yd(s) e(s) y(t)
s
4.8.1 PD Controller
In modelling control for robots there is no compromise on the response time and the overshoot of a system because of the stability of the system. (Franklin et al, 2002) Proportional action provides an instantaneous response to the control error. This is useful for improving the response of a stable system but cannot control an unstable system by itself. Derivative action acts on the rate of change of the control error; this provides a fast response as opposed to the integral action. (G.C. Goodwin et al, 2001) proportional derivative control is essential for fast response self. Derivative controllers do not need a steady state error of zero. Proportional controllers are fast, derivatives controllers are also fast so combination makes very fast controller.
(O. Haavisto, 2004) a discrete PD controller was developed in order to test the biped and get the system walking. The gait pattern formation is controlled by the walker with four separate discrete PD controllers, which are fed in order to change the reference signals. The left knee angle γL and the right knee angle γR provide their own separate controllers. The other controller controls the corner of the biped thighs i.e. difference of thigh angles ∆β= βRβL. The fourth PD controller controls the torso angle α, keeping the position of the torso upright.
Knee angles controller provides control signals directly for the moments ML2 and MR2. The angle between the thighs ∆β= βRβL, control signals have a positive effect on the right thigh torque MR1 and negative effect on the left thigh torque ML1. Both feet on the ground, influence the control signal evenly on the both thighs. The torso angle control was detached from the actual gait control so that it is feasible to decide the used angle separately. The control signal, however, needs to be added to the thigh moments. During the double support phase, the control signal is influencing both thigh moments, but during the single support phase only to the stance leg moment. System is used as the discrete PD controllers, whose transfer function has the form
ukh= Pekh+Dh∆e(kh)
k = sample number, h = sample interval
e(kh) is calculated by subtracting the adjustable parameter reference value. The expression ∆e(kh) can be directly obtained from current and previous signal value
∆ekh=ekhe(k1h)
By a suitable choice of the proportional controller gain P, the steady state error requirements can be met.
4.8.2 Reference Signals
Signals are formed step by step adding, subtracting or keep references to constant values between the sampling systems status. References are constant at the beginning of the double support phase but when the momentum is shifted forward then the knee reference is increased the angle between the thighs is reduced when the foot is raised and system moves to change the phase. Swing leg is transferred front ways by reducing the angle between the thighs to a constant value, so that the foot does not hit the base too early. When the leg is swung forward the knee is straightened before the touchdown as it remains far from the base. The creation of the reference signals was planned so that the left leg is for all time the swinging leg. When a new double support phase begins, the left and right leg signals are in the state input and reference output switched. Below shows the table of the parameters used in the controller
P 
D 

Double Support Phase: 

The angle between the thighs Δβ 
60 
1 
Knee angle γL, γR 
40 
0,5 
Torso angle α 
40 
2 
Phase Change: 

The angle between the thighs Δβ 
70 
6 
Support leg knee angle 
30 
2 
Swing leg knee angle 
10 
0,1 
Table. PDcontrollerused forwalkingdownthe parameters.
CHAPTER 6 IMPLEMENTATION
The biped walking model is implemented using Similink. The biped model in simulink consists of three blocks implementing dynamic equation of the biped model, calculating the ground support forces and limiting the knee angle. This model simulates complete walking model of the biped. The parameter definition of the biped model is shown in the table below.
Field 
Description 
Units 

Robot dimensions: 

l 
Robot link length [l0, l1, l2] 
m 

r 
Centre of mass distances [r0, r1, r2] 
m 

m 
Link masses [m0, m1, m2] 
kg 

Robot initial state: 

coordinates 
[x0, y0, α, βL, βR, γL,γR] 
m, (rad) 

speeds 
[x0, y0, α, βL, βR, γL, γR] 
m/s, (rad/s) 

Knee angle limiter parameters: 

kk 
elastic constant 
Nm/(rad) 

bk 
damping ratio 
Nms/(rad) 

min 
minimum angle for γL and γR 
(rad) 

max 
maximum angle for γL and γR 
(rad) 

Ground Properties: 

ground 
surface points x1, x2, …y1, y2, … 
m 

ky 
normal elastic constant 
kg/s2 

by 
normal damping ratio 
kg/s 

kx 
tangential elastic constant 
kg/s2 

bx 
tangential damping ratio 
kg/s 

mus 
static friction coefficient μs 

muk 
kinetic friction coefficient μk 

Additional parameters: 

acceleration of gravity 
m/s2 

sample time 
s 
These would be explained one by one in detail.
6.1 Biped block
6.1.1 Dynamic model
This block output is a 14 x 1 matrix which calculates the position of the seven variables in which two are the position of the centre of the torso and five angles q = (x0, y0, α, βL, βR, γL, γR) as described earlier and the rate of change of these variables q = (x0, y0, α, βL, βR, γL, γR) after solving the equation
q=bq, q, M, F*A1(q)
Inertia matrix A(q) is a 7 x 7 matrix and the complete formulas are shown in Appendix B. Matrix A depends on position of the seven variables. The matrix b is a 7 x 1 matrix and depends on 22 variables. The seven position variables q of the biped robot, their corresponding first derivatives q, the moments calculated from the PD controller and contact forces (two are used at a time because of one leg being in contact with the ground).
6.1.2 Ground Contact Model
Ground support forces and the sensor values are the main output of this block. Initially the leg tip cartesian coordinates are determined using the formulae described earlier and the speed formulae are determined by taking the first derivative of the cartesian coordinates. The two identical control forces block handles the leg separately.
This block calculates the tangential and normal forces by projecting the leg tip coordinate to the local (x’, y’) coordinate system and an in the end projects them to support forces.
Sensor values tell whether the foot is in contact with the ground or not. When the leg is contact with the ground the sensor value is set to 1 and 0 otherwise. The senor value of the leg is determined by the relational operator as shown below in the figure. If the position of the leg tip is smaller or equal to the ground level that the relational will give a 1 on the output.
6.1.3 Knee Angle Limiter block
If the angle of the joint does not remain in the given limits this block adds a moment value to the knee joint moments. The selector selects the position of the knee angles γL and γR and compares if the knee angle has gone beyond the maximum or minimum limit or not. The moments are added to the right and left knee moment and nothing is added to the other two moments.
In this block if the knee angle goes over or under the limit than PD controller calculates the moment value and adds it to the previous values.
6.2 PD Control Block
PDcontrolwas implementedin Simulinkblock,which istofeedthe systemstate, updatesiton the basis ofreference signalandcalculates thedifference variableon the basis of appropriatemoments.Figurebelowshowsblockstructurewhere thereferencesignals and theactualformation ofthe PDcontrolisseparatedinto separate subsectors create references and controller
Figure. PD controller block with create reference and controller sub block
Figure below showstheCreatereferencesblockcontent.Reference signalis calculated assumingthat theswinglegisalwaysleftandrightleg.Therefore at the second stepthe signal of the biped must be changed.
The reference creation is created using three main steps, as the there are two main phases in walking i.e. double support phase and the single support phase so the reference are created at these phases. There is another block in which the legs are switched as the controller is designed only for the left foot so when the leg strikes the ground the same process is switched to the other leg. If in the double support phase when the leg strikes the ground, difference between the horizontal distances i.e. xR – x0 is more than 0.118 then the references block is enabled and references are calculated. The calculated step length error is subtracted from the horizontal distance of the robot. When the single support phase is enabled the references are created for the four controllers as described before.
The Controller block initially determines the differences of the controlled variables and reference values. The constraints for the PD controllers are chosen in accordance with the phase of the step. After the control signals are calculated, they are changed to the moments.
The four similar PD controllers take the difference signals and the parameters (P, D) as inputs and give the resultant control signals as outputs. The only special feature in the discrete PD controllers is the speed term zeroing. When the D parameter is transformed the speed term is zeroed for one time step in order to evade peaks in the output signal. Because all of the controlled variables cannot be affected straight through the inputs of the biped system, a conversion of the control values is needed. The Δβcontrol signal is separated to both thigh moments, and the torso angle control signal is added to the thigh moment of the leg that is touching the ground. If both legs are in contact with the ground, the control is divided equally between the two moments.
CHAPTER 7 BIPED WALKING RESULTS
In this section complete results of the walking of the bipedal would be discussed. As described earlier there are seven variables that mostly influence the biped robots x0, y0, α, βL, βR, γL, γR. The position of these variables will be shown graphically and finally a complete walking cycle of the bipedal walking robot would be shown.
A complete walking cycle was simulated and result were taken through the GUI
The above figure shows a complete walking step of the biped robot, the dotted foot is taken as the left foot and the plain blue as the right leg. The left leg in this case is the swing leg and the right leg is the stance leg. The left foot take off from the ground and then takes a complete forward step and then gets back to the original position to become the stance leg and other one right becomes the swing leg. The horizontal coordinate of the centre of the mass of the torso will increase almost linearly with respect to time as shown in the graph. The yaxis shows the horizontal mass centre x0 w.r.t to xaxis and time on the xaxis
The vertical movement of the centre of the would be similar to a projectile motion as the biped moves forward with time the centre of the torso will go upwards and then back down to the original position. The initial value for y0 is given in the mfile as 1.3749. There is a slight movement at the beginning that is because initially for some time there is a double support phase before the swinging of leg. In the figure shown below yaxis denotes the vertical position of the torso’s centre and time on the xaxis.
The angle which torso makes with respect to the normal as said earlier is denoted by α. Angle α starts from the negative angle w.r.t. normal and then increases for some time and then come back to the original position which can be seen in the walking cycle figure.
Below figure shows the angle of the left and right thigh which it makes w.r.t. the torso. When either of the thigh angles β goes to zero it means the thigh is straightened w.r.t the torso and is done when the leg strikes the ground which is shown in the walking cycle . So below graph shows the right and left thigh angle βL and βR.
When the leg is raised from the ground, the knee is bended then again coming in line with the thigh just before again coming in contact with the ground. The figure shown below shows the knee angles γL and γR.
Below shows the sensor values of the left and the right leg. When the leg is contact with the ground the value of the sensor is one and zero if it is not in contact with the ground.
If one of the legs is taken the leg tip position in the x direction moves linearly and then zero for time until the other leg finishes the step and same is with the vertical movement the leg tip goes up and then comes back to ground and then remain zero until other leg comes back to ground.
BIPEDAL RUNNING
8.1 Introduction
(J.R.Ridgley, 2001) walking has been a focal point for a very large body of researchers, as it is considered the most basic form of legged locomotion with running as an enhancement. There is a popular phrase “one must learn to walk before one can run”. Less power and stress on the component of a system is required for walking. There is a significant practical advantage of the legged machines which can run over those limited to walking motion, but not much of a research has been done into the design of practical running machines. This is because there are a number of significant problems while constructing practical running machines. There are concerns which have been adequately resolved by modern technology. For example, mechanical structures and mechanisms can simply be developed which will consistently maintain the heaps of running, at least for devices developed on a reasonable extent of size for devices with masses on the order of 0.01kg to 100kg. At smaller extent than this, conventional machining methods are hard to use, and on larger scales the power requirements located on mechanical components may become unfeasible. Embedded control computers are extensively accessible which can easily process data rapidly enough to control the sensors and actuators which are expected to be used to control a running machine. Sensors which can calculate forces, accelerations, distances, angles, and velocities are easily accessible and only modestly expensive in contrast to other components of a typical robot. Actuators with adequate power density, for example electromagnetic and pneumatic actuators, are common.
8.2 Running Cycle
Stance phase takeoff flight phase impact
phase phase
Figure. Running cycle
(M.R.Anglin, 2011) running cycle can be described in two main categories flight phase and the stance phase. Each of thesephasesdefines the position of the foot at a certain portion of therunningcycle. Researchers who study a person’s walking andrunninglocomotion can use the information obtained to find and solverunningproblems.
The swing phase of arunningcycle occurs when none of the foot is in contact with the ground. The stance phase occurs between the time the leg tip is in contact with the ground and continues until the leg tip leaves the ground. The stance phase of arunningcycle can additionally be categorised into three sub phases: contact, midstance, and propulsion. During the contact sub phase, the leg tip comes in contact with the ground, and the sub phase carries on until the complete foot comes in contact with the ground. This contact can put a lot of pressure, occasionally equivalent to three times a person’s body weight or more, onto the foot. After the contact sub phase comes the midstance sub phase. During this phase, the body bends forward and shifts over the foot to get ready for the next sub phase, the propulsion sub phase.
A runner in the propulsion sub phase of therunninggait will see himself propelled forward. Straight away following this sub phase is the float phase. During the propulsion sub phase, the leg tip will leave the ground and the person is shifted forward. The sub phase ends with the leg tip takingoff the ground. Therunningcycle will continue until the foot touches the ground again, completing the cycle.
8.3 Robot Running Hypothesis
(Westervelt et al, 2007) in order to ensure that the robot’s consequent motion satisfies the below properties consistent with the notion of a simple running gait, conditions on the controller will be imposed.
 Running can be categorised into three main phases’ single support, flight and impact.
 Throughout the contact, the stance leg end acts as an ideal pivot, during the single support phase. The ratio of the horizontal component to the vertical component does not excel the coefficient of static friction. The vertical component of the ground reaction device is nonnegative.
 During the flight phase, a nonzero horizontal distance is covered by the centre of the mass.
 When the former swing leg end come in contact with the ground the flight phase ends.
 There is no slipping or rebounding of leg at impact.
 There is a symmetric motion of successive single support and flight phase with respect to two legs in the steady state.
 Running would take place on level surface from left to right.
8.4 Trajectory with telescopic legs
(Raibert et al, 1986) and (T.DMcGeer, 1990) have deeply studied the running in the robots. (Gieger etal., 2001) are also working on the realization of a biped running robot. They have projected a running controller based on feedback linearization. (Shuuji Kajita, 2002) established a technique for running pattern generation using the dynamics of a simple inverted pendulum. In order to conduct the simulation a very simple has been taken with a point mass of m and a mass less telescopic leg.
g
l
l y y
Fz > 0 Fz = 0
(a) (b)
Figure. (a) the telescopic leg when in contact with the ground (b) takeoff phase
The length of the leg in controlled as
l=l0+αsinωt
l0 = neutral leg length
α = the amplitude of the vibration
ω = frequency
Fy is the reaction force and is zero when the foot is in the air but when in contact with the ground is given by
Fy=ml+g=m(αω2sinωt+g)
g = gravitational acceleration
In a certain phase Fy is negative when αω2 > g. This happens when the foot of the leg is firmly in contact with the ground. When Fy approaches zero it means that the foot is leaving the contact with the ground which means robot jumps into the air and then the robot takes a free fall trajectory. The liftoff timing φA and the moment of takeoff yA can be written as
φA=1ωsin1gαω2
yA=αωcosωφA=(αω)2(gω)2
Now two legs are considered in the running model which is shown in the figure below.
m
lR lL
y
0
Figure. Biped Model
The length of the right leg lR can be represented as
lR=l0+αsinωt
The time when both the foot are of the ground or the flight phase time Tflight can be written as
Tflight=2yAg
In order to achieve smooth landing the left leg lL length can be written as
lL=l0+αsinωtφdelay
Where φdelay can be formulated as
φdelay=Tflight2π2ωφA
The velocity and acceleration of the mass change is continuous at the moment of landing. At the support phase the biped is supported by one of the legs so the duration of the support phase can be written as
Tsupport=πω+2φA
Now the formulation for the horizontal locomotion is done in the sagittal plane (x,y) and for ideal condition it is assumed that the torque around the contact point equal to zero (τy = 0). This results the constraint between floor reaction forces and mass location as
FxFy=xy
Fx = horizontal floor reaction force
x = horizontal position
The horizontal can be calculated as
x=1mFx
= αω2sinωt+gl0+αsinωtx
The above equation is for the support phase where πωφA<t< 2πω+φA
8.5 Running with segmented legs
The above model only had one degree of freedom in order to design a humanoid robot a two segment leg robot model would be discussed in detail. (J. Rummel, 2008) for this action a stance leg would be taken which is represented by a linear spring of rest length l0 and leg stiffness k. For this model only forces directed from a fixed contact point at the ground to the centre of the mass would be generated and the amount the amount of the force depends on the leg compression∆l=l0l(t). A constant leg stiffness k is assumed for a linear relationship between leg compression and leg force.
Two mass less segments of length l1 and l2 defines the segmented leg. These two segments are connected by inter segmental leg joint with an inner angle β. In order to produce springlike forces in a segmented leg a torsional spring of stiffness c at inter segmental joint with joint torque, is introduced.
τ∆β=c∆β
∆β=β0β =denotes the amount flexion in the joint
β0 = rest angle
β denotes the instantaneous joint angle and is defined as the function of leg length l and it can be written as
βl=cos1(l12+l22l2)2l1l2
Rest angle β0 which corresponds to a rest length of the leg is needed in order to calculate the amount of joint flexion ∆β
l0β0=l12+l222l1l1cos(β0)
In consequence, any amount of joint flexion ∆β interprets into an equivalent amount of leg compression ∆l depending on the selected rest angle β0. The leg force could be written as
Flegτ=ll1l2τsinβ
8.5.1 Reference and effective stiffness
Implementation
A complete model of running on segmented legs would be modelled using simulink. The complete model is shown below
Figure. Simulink model for running with segmented legs
Firstly flight phase block would be described. In the flight control block the position of the leg x and y which are calculated before are subtracted from the position of the leg at touchdown which are written as
yL=l0sin(α0)
xL=l0cos(α0)
When x – xL is zero that means the leg is contact with the ground with the angle α0 and then the contact phase is triggered and then and the position are send to the next block for the calculation of contact force, the angle between the segments and the acceleration. The formulae of these parameters are already shown.
The l during the flight changes according to simple Pythagoras theorem can be formulated as
l=x2+y2
When the l0 = l that means it is contact with the ground and then the F, β and a are calculated
In the integration sub block the acceleration calculated is integrated to first velocity vx and vy and then again integrated to position x and y
As yapex is the maximum height he leg goes and when the leg does not reach that height that means the leg has fallen and simulation is stopped and when the vertical velocity becomes zero that means one step is completed and then the step count is incremented.