1. BIPEDAL LOCOMOTION
Antonio D'Angelo

2. BIPEDAL WALKING
In recent years the interest to study the bipedal walking has been growing.
Also the demand for build bipedal robots has been increasing
The design for the bipedal robot is rather different from conventional robots: there are limits in the amount on actuators size and weight.
To understand the mechanical bipedal robots mechanics design, is necessary first to understand the bipedal walking process or bipedal locomotion.

3. STATIC WALKING
If the static walking is used then the control architecture has to make sure that the projection of the center of gravity on the ground is always inside the foot support area.
Within this approach only slow walking speeds can be achieved, and only on flat surfaces.

4. DYNAMIC WALKING
Within dynamics walking the center of mass can be outside of the support area, but the zero momentum point (ZMP), which is the point where the total angular momentum is zero, cannot.
Dynamic walkers can achieve
faster walking speeds,
running ,
stair climbing,
execution of successive flips,
and even walking with no actuators.

5. STATICALLY STABLE
Static walking assumes that the robot is statically stable.
This mean that, at any time, if all motion is stopped the robot will stay indefinitely in a stable position.
It is necessary that the projection of the center of mass of the robot on the ground must be contained within the foot support area.

6. SUPPORT PHASES
The support area is either the foot surface in case of one supporting leg or the minimum convex area containing both foot surfaces in case both feet are on the ground. These are referred to as single and double support phases, respectively.

7. WALKING SPEED
Also, walking speed must be low so that inertial forces are negligible. This type of walking requires
large feet,
strong ankle joints
and can achieve only slow walking speeds.

8. SIMPLE MODEL OF WALKING
Inverted Pendulum Model
Influence of the Dynamics
Center of Mass
Center of Pressure

9. CENTER OF MASS - I
A bipedal robot gait is said to be statically stable and a humanoid posture is said to be balanced if the ground projection of its center of mass (COM), falls within the convex hull of the foot support area (the support polygon).

10. CENTER OF MASS - II
The center of mass is calculated according to its distance-weighted average location of the individual mass particles in the robot
where Pmi is the location of the mass particle i, and Mi is the mass of particle i

11. CENTER OF PRESSURE - I
The center of pressure (COP) is the pivot point of the human foot, the center point of the convex hull of the foot where it supports the most pressure.

12. CENTER OF PRESSURE - II
The center of pressure is calculated according to its distance-weighted average location of the individual pressures on the foot
where Ppi is the location of the pressure particle i, and Pi is the pressure of particle i

13. INVERTED PENDULUM MODEL- I
The human walking motion shows some similarities with the inverted pendulum mechanics.
The pendulum pivot point is placed approximately at the center of pressure on the foot. The pendulum mass is placed approximately at the center of mass.

14. INVERTED PENDULUM MODEL- II
A simple pendulum model of bipedal walking: m represents the center of gravity; l is the length of the leg and  represents the stance leg angle

15. EQUILIBRIUM POINT - I
A pendulum has an equilibrium point in the straight up position and will accelerate in the direction of whichever side it is on.
The further the mass is from the vertical, the faster it will accelerate.
Note that there is no torque at the pivot point.

16. EQUILIBRIUM POINT - II
Suppose the mass is traveling from left to right.
If the mass is on the left-hand side, it will slow down towards the vertical.
If the mass has passed the vertical, it will accelerate to its right.
At this point the system is converting kinetic energy into gravitational energy when it travels from left to vertical and convert it back into kinetic energy from the vertical to the right.

17. PENDULUM DYNAMICS The initial kinetic energy is:
The change to potential energy is:
By setting the change of potential energy equal kinetic energy:
For small angle approximation, we get

18. LINEAR ACTUATOR PENDULUM MODEL
Let’s add a linear actuator along the length of the pendulum.
The force on the point mass lies along the length of the leg.
The acceleration of the mass in the radial direction depends on the actuator force F, the gravitational force, and the fictitious centrifugal force due to the rotation of the pendulum.

19. LINEAR ACTUATOR DYNAMICS
Assume the mass of the pendulum is travelling from left to right.
With the actuator pulling the mass, the rotation motion will accelerate.
Extending the mass will decelerate the speed.
This is the same as sitting on a spinning chair where the pivot point is underneath the chair, when opening the arms during spinning, the rotation will slow down, while closing up the arms will increase the rotational speed.

20. MULTI JOINT SIMILARITIES
Multi joint model with the parameters mass point m, length l of the leg.
The Force Gain from the Actuator

21. MULTI JOINT MODEL - I
This model shares some characteristics with the linear actuator model, but it is implemented with a different mechanism.
In order to transform a linear actuator model into a multi joint pendulum model, we have to show that both dynamics are identical.
We illustrate the similarities, by using a two degrees of freedom (DOF) multi joint model with the parameters mass point m, length l of the leg.

22. MULTI JOINT MODEL - II
With a different mechanical design, to obtain the same parameters we have to use inverse kinematics to calculate the angles for each individual joint.
The only difference between the two models is the force gain from the actuator, for linear actuator model it is a linear force.
This model has a torque generated by the knee servo but it has a minor influence.

23. ANKLE PENDULUM MODEL - I
Adding a foot and an ankle to the multi joints leg.
Balancing the mass to shift it left and right from the vertical above the pivot point.

24. ANKLE PENDULUM MODEL - II
Controlling the speed of the rotation motion of the inverted pendulum by adjusting the length of the leg is not a well-balanced system.
Adding a foot and an ankle brings many benefits:
it yields to a larger supporting area (convex hull of the feet) for the center of mass to stay below when traveling.
it also helps to control the speed of the pendulum and leads to a more stable system.

25. CONTROLLING THE JOINTS TORQUE
Balancing the mass is made by shifting it left and right from the vertical above the pivot point.
The pivot point of a human is the center of pressure (COP), thus
if the COP is left of the COM then the mass point will accelerate to the right;
if the COP is right of the COM then the mass point will accelerate to the left.
By controlling the joints torque, we can arbitrarily control the location of the center of pressure.

26. WALKING ALGORITHM Inverse Kinematics Bezier Curve
Software Design Scheme
Implementation

27. INVERSE KINEMATICS - I
Inverse Kinematics computes the joint parameters necessary for a given destination point
where P represents the center of mass, L2 the hip joint link and L1 the knee joint link

28. INVERSE KINEMATICS - II
It is an important aspect since the most difficult problem in robot walking is stability. After working out the center of mass, we need to keep the COM within the supporting area.
The starting point comes from the relations

29. INVERSE KINEMATICS - III
By controlling the joint angles, we can control the stability during the motions:
knee angle
hip angle

30. BEZIER CURVE - I
Bezier curve in its most common form is a simple cubic equation. It was developed by Pierre Bezier in the 1970's for CAD/CAM operations. It can be used for drawing model and also animation.
The cubic equation requires four point parameters, the first point is the starting point (original endpoint), second and third are the control points used to adjust the curve, the fourth point is the end point (destination endpoint).

31. BEZIER CURVE - II
It is evaluated on arbitrary real values t between 0 and 1. If t increases it will return a number close to the destination endpoint, if t is close to 0 it will return a number close to the original endpoint.

32. SOFTWARE DESIGN SCHEME
Implementation overview of walking gait generation

33. WALKING GAIT OVERVIEW The main module controls the walking style.
The pattern generation system is invoked by
configuring the servos,
setting up the control curve (Bezier Curve) and
splitting the walking gait into six different phases.
The vision system is devoted to
capture images from the camera sensor,
set up the obstacle parameters,
detect obstacles the humanoids need to avoid or move toward.

34. WALKING BEHAVIOR
Main module works as an higher-level path planner. It is used to determine the behavior of the robot by controlling
the walking speed,
the number of steps for walking or turning, a left or right turn,
how robot deals with obstacle; walk passes it while turn runs away from it,
the step size (the distance between front foot to rear foot).
Main also controls the overall flow of the gait starting at position A and move to destination B.

35. PATTERN GENERATION SYSTEM
The pattern generation system is brought about three different modules
Pattern
Leg servo
Bezier Curve
The main contributions to stability are the simple pendulum model, the inverted kinematics and the walking phases.

36. PATTERN GENERATION OVERVIEW
How the walking gait is controlled

37. LEG SERVO
The legservo module is an interface between all the servos and it is used to adjust and maintain the range of the servos, by imitation of humans as close as possible.
For example, it is only possible for humans to bend the knee until the heel almost touches the hip, and extend it until the lower leg and higher one are lined up together.
Servo ranges can take one byte only so they are valued from 0 to 255. Selecting a value results in an angle for each joint such that it succeds in turning the servo into the proper location to generate the locomotion.

38. BEZIER MODULE
The Bezier Curve module implements a simple cubic equation, given a set of control points to generate a curve.
Applying an arbitrary value t between 0 to 1 into Bezier function, we can obtain the servo angle for different time instance.
The Bezier Curve module is an interface for setting up the control points and retrieving the servo value in a particular time instance. Different settings can achieve different gaits for different situations.

39. PATTERN MODULE - I
The pattern module is the heart of the PGS and it works as a listener for the main module which distinguishes commands like speed, patterns or phases. At least four different patterns need to implemented: walk, turn, kick and bow.
Six different phases that contribute for a walking cycle are also considered: three symmetrical phases for each leg and both.

40. PATTERN MODULE - II The pattern module will initialize
the predefined control points for the Bezier Curve module,
the joints constraints for each servo in the legservo module.
Then, according to the input command, whatever pattern, phase to start and to end, and whatever speed, the pattern module will adjust the control curve and the joints constraints respectively to start walking.

41. PATTERN GENERATION - I
The Pattern Generation System supports the main dynamics for the walking gait
to maintain the stability of the robot during the walking cycle, and
to ensure the stability of the robot during the transition between patterns (walk, kick, turn and bow).
The system can also provides tools for implementing technology support such as, reinforcement learning, optimal path planning ...etc.

42. PATTERN GENERATION - II
Simple pendulum model will help to understand and exploit the dynamics during the walking cycle in the following ways.
the pivot point is placed approximately in the center of pressure.
the pendulum mass is placed around the center of mass, and further the mass is from vertical (ground projection from the pivot point), faster the pendulum will accelerate.
rotation motion will accelerate when the actuator is pulling the mass and rotation motion will decelerate when the actuator extends.

43. PATTERN GENERATION - III
By controlling the joints torque, we can arbitrarily control the location of the center of pressure, and when COP is left of the COM then the mass point will accelerate to the right.
These models help us to establish the walking motion, and understand the ways to achieve stability by controlling the servos.

44. WALKING PATTERN

45. PATTERN GENERATION - IV
Inverted Kinematics is used to compute the joint parameter values necessary for a given destination point (COM):
in order to keep the walking balance we had to ensure the COM to lie underneath the supporting region during the trajectory;
this also limits the joint range and, with the help of Bezier Curve, we can setup a proper walking pattern to enforce it.
the six phases break the walking cycle into three different states per foot and maintaining the stab- ility becomes more feasible and easier to handle.

46. PATTERN GENERATION - V
PGS initiates a walking cycle action by receiving a high- level command from the main module.
The basic command instructions include walking pattern (walk, kick, turn, and bow), starting phase, ending phase and speed of the walk cycle.
After the command has been issued the phase handler turns the pattern flag on/off with respect to the input pattern, initializes Bezier Curve by setting up control points, then forwards the phase info to generate an angle process.
This process then initializes the leg joints position into ready mode, gets the servo values for each joint and triggers the walking cycle.

47. WALKING CYCLE - I
Walking Cycle has six phases, three phases for each leg. Angle generation process will activate the cycle/phases, according to the given start and end phase.
It will select either one leg stand phase, or ready for landing phase, or two legs stand phase. The number in the phases corresponds to the timing or state of the cycle between left leg being the supporting leg and right leg being the swinging leg or right leg being the supporting and left leg being the swinging leg.

48. WALKING CYCLE - II
Data flow diagram illustrating the role of Pattern Generation System

49. WALK PATTERN - I
Figure shows a walk pattern of the walking cycle.

50. WALK PATTERN - II
During this pattern the robot will start at two standing legs, that is, one leg stand in front and the other stand rear.
The reason is to move the COM forward to the supporting area of the front leg and ready for the rear leg to lift over the ground.
Since the robot is in static mode, the pivot point COP stays under the COM.
Then next step is to transit forward into one leg stand phase.
During the transition the torque created by the turning of ankle servo will slide the COP towards the inner area of the support foot for a while.

51. WALK PATTERN - III
When the COM is not found vertically above COP, the robot will land on one side and lift the rear leg above the ground.
When the ankle servo reaches the desire angle, the torque will disappear and then the COP will move back under the COM and this is the one leg stand phase.
The final step is moving the rear leg forward, turn the joint angles to a landing position and this is the ready to land phase.

52. WALK PATTERN - IV
The switching back to two standing legs can be simply obtained by turning the ankle servo of the supporting leg in the other way.
In this case the COP will slide towards to the outer side of the foot due to the torque generated by the servo.
When COM is not lined up vertically with COP, it will fall into the free leg direction and land back into two standing legs phase.

53. KICK PATTERN - I
Data flow diagram, illustrating the kick pattern of the walking cycle.

54. KICK PATTERN - II
Similar to the walk pattern, it has the same transition between two standing legs phase to one standing leg, and ready to land phase to two standing legs except it will jump to another states that are outside the cycle.
The phase handler will switch the kick motion flag on when a high-level input command issues a kick pattern request.
From one standing leg phase onward, it will check if the kick flag is on because if it is not then it will carry on with the walk pattern.
Otherwise, it will jump to a process that will switch the walk parameters to kick parameters.

55. KICK PATTERN - III
This process will also change the robot configu- ration by landing the torso forward and move the COM to the front edge of the supporting foot.
This is important because the next process will start kicking. When pulling the free backward leg and it is ready to initiate a kick, it will move the COM back into the middle of the supporting leg.
Kicking is the next process, swinging the free leg forward and kick the ball.

56. KICK PATTERN - IV
The traslation of the COM will end up at the front edge of the supporting leg again.
The final process will change the kicking parameters back to walking parameters, switching the turn flag off and carrying on with the ready to land phase.

57. TURN PATTERN - I
Data flow diagram, illustrating the turn pattern of the walking cycle.

58. TURN PATTERN - II
For this pattern, there is no parameter change because it shares the same parameters as walking. The phase handler will turn the turn flag on and this flag consists of two states: right turn and left turn.
From two standing legs phase, it will verify the turn flag, and if the turn flag is off, it will switch to the normal walking cycle.
Differently, if the flag is set to right turn, then it will disable the left ankle servo, otherwise it will disable the right ankle servo.

59. TURN PATTERN - III
The module will also perform a verification for the timing of the phases; if it is not the right phase for the given leg then it will results in a normal routine for transition between two standing legs to one standing leg phase.
Repeated applications until the correct phase for the correct turn is reached.
When the verification for the timing of the phases had passed, it will jump to the turn process.

60. TURN PATTERN - IV Within the turn process,
because the front leg ankle servo is already disabled,
by pulling the front leg back and pushing the rear leg forward,
it will twist the body to the desire direction,
and switch the front leg to rear and rear leg to front.
The final step is to enable the blocked ankle servo and the dynamics will be ready for the next walking cycle.

61. DYNAMICAL WALKING
Biped dynamic walking allows the center of gravity to be outside the support region for limited amounts of time.
There is no absolute criterion that determines whether the dynamic walking is stable or not.
Indeed a walker can be designed to recover from different kinds of instabilities.
However, if the robot has active ankle joints and always keeps at least one foot flat on the ground then the Zero Momentum Point (ZMP) can be used as a stability criterion.

62. ZMP DEFINITION
The ZMP is the point where the robot's total moment at the ground is zero.
As long as the ZMP is inside the support region the walking is considered dynamically stable because is the only case where the foot can control the robot's posture.
It is clear that for robots that do not continuously keep at least one foot on the ground or that do not have active ankle joints (walking on stilts), the notion of support area does not exist, therefore the ZMP criterion cannot applied.

63. DYNAMIC WALKING SCHEMA
You can look at the figure

64. COMPUTING ZMP - I
The position of the ZMP is computed by finding the point (X, Y, Z ) where the total torque is zero.
Since we are only interested in the ground plane we assume that Z = 0.
To avoid confusion, torque and moment mean in this work the same thing.
The robot has n links; each link is subject to a
total force Fi applied at a point determined by the
vector Ri relative to the center of gravity of the link.
Ti determines the total motor torque applied to the link.
Rz is the ZMP vector and T is the robots total torque.

65. COMPUTING ZMP - II
An example of the forces applied to a link is represented in the figure

66. COMPUTING ZMP - III
The force, torque and position vectors have the following coordinates:

67. COMPUTING ZMP - IV Then the total torque is computed as:
where x represents the cross product. The equation is then expanded as:

68. ZMP POSITION
Making Z=0 and solving these equations for X and Y we obtain the ZMP coordinates:

69. MECHANICAL STRUCTURE
The biped robot is configured of two legs, each having 4 degrees of freedom (DOF).
Three of these are rotational on the pitch axis at the hip, knee and ankle.
The fourth is also rotational an located at the hip on the yaw axis.
The trunk (an inverted pendulum) has 1 DOF.
The trunk is used to stabilize the robot during the walking gait.
As the trunk is moved to the angle calculated by the controller, the centers of mass (COM) position will change to a point were the robot's structure is stable.

70. STABILITY - I
In traditional legged robots, stability is maintained by having at least three contact points with the ground surface at all time.
With biped machines, only two points are in contact with the ground surface; for that reason algorithms to achieve balance must be implemented.
To solve the biped robot stability at walking, a simplified model of feet force sensors feedback can be used as a controlling input which tries to maintain stability at walking.
Even so the mechanical design goal is to ensure that the robot at walking will achieve dynamic balance.

71. STABILITY - II
Dynamic balance is partially provided by the control algorithm; however the mechanics design plays a very important role for the robot's ability to do the correct movements.
A way to reach it, is finding a correct mass distribution for the robot, thus the robot will be able to achieve the stability at walking.
These movements can therefore be made rapidly without generating large moments which would further destabilize the robot.

72. STABILITY - III
To achieve this, the COM should be placed in a location low enough to stabilize the robot inertially, but high enough so that it can be moved only small amounts to correct for undesired behavior.
The correct placing for the COM is the lower trunk, similar to humans. This provides for stability and allows the trunk to be moved, shifting the COM to archive desired accelera-tions to counteract existing undesired accelerations.