1. Walking Robots Lecture 9 - Week 5
Advanced Programming for 3D Applications CE

2. Types of Locomotion in Nature

3. Real Robots Sneak (Epson, Japan)
Rollerwalker (University of Tokyo, Japan)
U-BOT (University of Massachusetts, USA)

4. Real Robots (cont.) The Self Deploying Microglider (EPFL, France) Aiko
(SINTEF Applied Cybernetics, Japan)

5. Battlefield Extraction-assist Robot (Vecna Technologies, USA)
Real Robots (cont.)
Battlefield Extraction-assist Robot
(Vecna Technologies, USA)
Asimo
(Honda, Japan)

6. Why Legs? Potentially less weight Better handling of rough terrains
Only about a half of the world’s land mass is accessible by current man-built vehicles
Do less damage to terrains (environmentally conscious)
More energy-efficient
More maneuverability
Use of isolated footholds that optimize support and traction
(i.e. ladder)
Active suspension
Decouples the path of body from the path of feet

7. Why Legs? (cont.) Aren’t wheels and caterpillars good enough?
Wheels and caterpillars always need “continuous” support from the ground. Legs can enable a robot to make use of “discreet” footholds.

8. Why Bipeds? Why 2 legs? 4 or 6 legs give more stability, don’t they?
A biped robot body can be made shorter along the walking direction and can turn around in small areas
Light weight
More efficient due to less number of actuators needed
Everything around us is built to be comfortable for use by human form
Social interaction with robots and our perception (HRI perspective)
Form will become as important as functionality in the future
Our instinctive desire to create a replica of ourselves (maybe?)

9. Joints in a Leg
At least 2 DOF (degrees of freedom) needed to move a leg
A lift motion + a swing motion
A human leg has 30 DOF
Hip joint = 3 DOF
Knee joint = 1 ~ 2 DOF (almost a hinge)
Ankle joint = 1 DOF (hinge)
24 DOF for the foot!
In many cases, a robot leg has 3 DOF
Control becomes increasingly complex with added DOF
With 4 DOF, ankle joint can be added
Reasonably walking biped robots have been built with as few as 4 DOF

10. Joints in a Leg (cont.)
Picture of a joint model

11. Stability
Stability means the capability to maintain the body posture given the control patterns
Statically stable walking implies that the posture can be achieved even if the legs are frozen / the motion is stopped at any time, without loss of stability
Dynamic stability implies that stability can only be achieved through active control of the leg motion
Statically stable systems can be controlled using kinematic models
Dynamic walking requires use of dynamical models

12. Gaits Gaits determine the sequence of configurations of the legs
A sequence of lift and release events of individual legs
Gaits can be divided into 2 main classes
Periodic gaits  repeat the same sequence of movements
Non-periodic or free gaits  no periodicity in the control and could be controlled by the layout of environment
The number of possible events N for a walking machine with k legs is:
N = (2k – 1)!
For a biped robot (k = 2), there are 3! = 6 possible events
Lift left leg, lift right leg, release left leg, release right leg, lift both legs, release both legs

13. Gaits (cont.)
An example of a static gait with 6 legs

14. Gaits and Stability
People, and humanoid robots, are not statically stable
Standing up and walking appear effortless to us, but we are actually using active control of our balance
We use muscles and tendons
Robots use motors
In order to remain stable, the robot’s Center of Gravity must fall under its polygon of support
The polygon is basically the projection between all of its support points onto the surface
In a biped robot, the polygon is really a line
The center of gravity cannot be aligned in a stable way with a point on that line to keep the robot upright

15. Gaits and Stability (cont.)
Each vertex = support foot Dot = center of gravity
Quadruped Robot – Gait Motion (

16. Control of a Walking Robot
3 things that control must consider for walking:
Gait: the sequence of leg movements
Foot placement
Body movement for supporting legs
Leg control patterns
Legs have 2 major states:
Stance: On the ground
Fly: In the air moving to a new position
Fly state has 3 major components:
Lift phase: leaving the ground
Transfer: moving to a new position
Landing: smooth placement on the ground
More DOF for the legs means
Smoother movement, but
Increasingly complex controls

17. Walking vs Running
Motion of a legged system is called walking if in all instances at least one leg is supporting the body
- Honda Asimo walking (
- First Asimo version is E0 in It took seconds for 1 complete step
If there are instances where no legs are on the ground, it is called running
- Honda Asimo running (
- Honda Asimo running (close-up) (
Walking can be statically or dynamically stable
- With 2 legs, almost always dynamically stable
Running is always dynamically stable

18. Biped Walking = Rolling
Rolling is quite efficient
Biped walking is similar to rolling a polygon
Polygon side length = step length
As step length gets shorter, more like rolling a circle

19. Walking State Methodology
Walking algorithm for biped robots often derived from classical control theory
Uses a reference trajectory for the robot to follow
Reference trajectories can rarely be defined to work in the real world
Irregular terrains and encountering different obstacles, etc.
Uses static balance poses to define points of tending to balance during a gait
The point that a biped robot tends to balance is called a state
The walking states are chosen as the maximum and minimum tending to balance stance equilibrium positions where little or no torque needs to be applied to maintain the state

20. Walking State Methodology (cont.)
Marching gait example
5 states where the robots tends to either balance or tend to topple
The center of gravity tends to shift as shown by the cube on top of the robot

21. Walking State Methodology (cont.)
While advancing to new states during the actual walking locomotion, an autonomous robot’s software should ideally extrapolate the gait from balanced state to the next.

22. Walking State Methodology (cont.)
In states 2 and 4, we can interpret the robot as tending to an out of balance point. If the leg that is bent continues in the same direction, then the robot will topple.
The control algorithm should not counter the tending to topple position by bending the other knee on the other leg or shifting the original leg back to its initial position.
The control algorithm should continue with the balance control state, expecting that to prevent a fall, the robot has to counter balance by shifting the center of gravity to either the neutral position or to the next tending to out of balance point on the opposite side.

23. Walking State Methodology (cont.)
The velocity and acceleration of the balance control state is determined by the weight and dynamics of the robot.
All the specific movements pre-determined (hard coded) for each state
Example (Clyon, Florida International University) (

24. Passive Walking
An approach to robotics movement control based on utilizing the gravity and the momentum of swinging limbs for greater efficiency.
Conserves momentum
Less number of actuators
Natural (anthropormorphic)
In a purely passive dynamic walking, gravity and natural dynamics alone generate the walking cycle
Active input is necessary only to modify the cycle, as in turning or changing speed
Examples
3 legs (
2 legs (

25. Zero Moment Point (ZMP)
Introduced in 1968 by Miomir Vukobratovic
Specifies the point with respect to which dynamic reaction force at the contact of the foot with the ground does not produce any moment (i.e. the point where total inertia force equals 0)
Assumes the contact area is planar and has sufficiently high friction to keep the feet from sliding (no sliding assumption)
The trajectory is planned using the angular momentum equation to ensure that the generated joint trajectories guarantee the dynamical postural stability of the robot, which usually is quantified by the distance of the zero moment point in the boundaries of a predefined stability region.

26. Zero Moment Point (ZMP) (cont.)
Ground reaction force and ZMP are generally measured with a series of sensors embedded in the feet
Pressure sensitive transducers, foot switches, strain gage based sensors, force sensitive resistors, and novel force-torque transducers

27. Zero Moment Point (ZMP) (cont.)
Center of pressure (CoP) is a ground reference point where the resultant of all ground reaction forces acts
At this point, it is assumed that all of the forces that act between the body and the ground through the foot can be simplified to a single ground reaction force vector and a free torque vector
If the horizontal forces between the feet and the ground can be neglected, then the CoP can be defined as the centroid of the vertical force distribution

28. Zero Moment Point (ZMP) (cont.)

29. Zero Moment Point (ZMP) (cont.)
For flat horizontal ground surfaces, ZMP == CoP
At any point P under the robot, the reaction can be represented by a force and a moment Mgrf

30. Zero Moment Point (ZMP) (cont.)
Around the ZMP (localized at rzmp ) the moment around the horizontal axis are zero and there is only a component of moment around the vertical axis
The resulting moment of force exerted from the ground on the body about the ZMP is always around the vertical axis
At the ZMP is a reference point at the ground in which the net moment due to inertial and gravitational forces has no component along the (horizontal) axes (parallel to the ground)
The trajectory that the ZMP follows is utilized and planned such that they are within the supporting polygon defined by the location and shape of the foot print

31. Zero Moment Point (ZMP) (cont.)
Anyways, in a very brief summary…

32. Zero Moment Point (ZMP) (cont.)
Anyways, in a very brief summary…

33. Zero Moment Point (ZMP) (cont.)
Anyways, in a very brief summary…

34. Zero Moment Point (ZMP) (cont.)
Honda’s Asimo
( )
AIST’s HRP-2
( )
AIST’s HRP-3
( )

35. Zero Moment Point (ZMP) (cont.)
Formulas from wikipedia

36. Zero Moment Point (ZMP) (cont.)

37. Zero Moment Point (ZMP) (cont.)

38. Sources (cited within this presentation)
Robot Locomotion by Henrik Christensen ( )
Walking Robots and Especially Hexapods by Marek Perkowski ( )
Estimation of ground reaction force and zero moment point on a powered ankle-foot prosthesis by Martinez Villalpando and Ernesto Carlos ( )
Design of a Biped Robot by Andre Senior and Sabri Tosunoglu
Overview of ZMP-based Biped Walking by Shuuji Kajita ( )
(on ZMP)