1. ZMP-BASED LOCOMOTION
Robotics Course
Lesson 22

2. 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.

3. 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.

4. DYNAMIC WALKING SCHEMA
You can look at the figure

5. 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 scenario 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.

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

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

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

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

10. MECHANICAL STRUCTURE
The biped robot is configured of two legs, each having 4 degrees of freedom (DOF).
The trunk is used to stabilize the robot during the walking gait.
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, used to stabilize the robot,
is moved to the angle calculated by the controller, the centre of mass (COM) position will change to a point were the robot's structure is stable.

11. 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.

12. 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.

13. 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 achieve desired accelera-tions to counteract existing undesired accelerations.

14. LEGGED ROBOT LOCOMOTION
Three main approaches:
trajectory based methods,
heuristic control methods, and
cpg-and-reflex based methods

15. TRAJECTORY PLANNING Question: how to generate good trajectories?
(Very) simplest options:
Provide a few desired positions (angles) over time (controller with step functions)
Provide smooth hand-tuned trajectories (e.g. with spline fitting)
These are open-loop solutions, i.e. no feedback to the trajectory planner

16. ZERO MOMENT POINT APPROACH
Method for proving that a trajectory is stable
Zero Moment Point (ZMP): point on the ground about at which the net moment of the inertial forces and the gravity forces has no component along the horizontal planes (a.k.a. center of pressure, CoP).
ZMP is different from the projection of the center of mass (CoM) on the ground
ZMP ~ projection on the ground of the point around which the robot is rotating

17. TRAJECTORY-BASED METHODS
Main idea: design walking kinematic trajectories, and use the dynamic equations to test and prove that locomotion is stable
Trajectories are designed by trial-and-error, or from human recordings
Most successful approach: Zero Moment Point (ZMP) method (Vukobratovic 1990)

18. ZMP APPROACH Foot print polygon
Locomotion is stable if the ZMP remain within the foot-print polygons

19. ZMP APPROACH

20. ZMP APPROACH Most used method: Example of online stabilization:
Human motion capture for getting trajectories,
Modify trajectories such that locomotion is stable according to the ZMP criterion
Add online stabilization to deal with perturbations
Example of online stabilization:
Use of hip actuators to manipulate the ZMP

21. ZMP STABILIZATION DIAGRAM

22. ZMP APPROACH CONCLUSIONS
Pros:
Well-defined methodology for proving stability
Well-suited for expensive robots that should never fall

23. ZMP APPROACH CONCLUSIONS
Cons:
Requires additional online control to deal with perturbations
Transitions from online control back to desired trajectories can be tricky
Define good trajectories can be time- consuming
Requires a perfect knowledge of the robot’s dynamics and environment

24. MINIMALIST CONTROL DIAGRAM

25. PID CONTROLLER
Stands for Proportional Integrative Derivative Controller
A feedback controller extensively used in many devices, e.g. thermostat, robots, ...

26. PID CONTROLLER

27. PID CONTROLLER

28. PID CONTROLLER

29. PROPORTIONAL CONTROL

30. PROPORTIONAL CONTROL

31. PROPORTIONAL CONTROL

32. STEADY-STATE ERROR

33. PROPORTIONAL INTEGRATIVE

34. FULL-PID CONTROL

35. PID CONTROL SUMMARY

36. MINIMALIST CONTROL DIAGRAM

37. CONTROLLER WITH STEP FUNCTION
Simply provide a new desired angle every so often:
Problems:
very saccadic motion,
depends strongly on PID gains

38. CONTROLLER WITH HAND-TUNED TRAJECTORIES
Provide a trajectory as a vector:

39. CONTROLLER WITH HAND-TUNED TRAJECTORIES
Hand-tuned trajectory: only give a few points and use of spline-fitting
Problems:
How to be sure that the locomotion is stable?
Need for mathematical tools to prove stability, such ZMP is.

40. OPEN-LOOP SINUS-BASED CONTROLLER
Desired angle for each DOF
Problem: finding, for each DOF i suitable

41. OPEN-LOOP SINUS-BASED CONTROLLER
Parameters can be found by hand, or using a genetic algorithm
For instance, genetic algorithm for optimizing the speed of walking

42. HEURISTIC CONTROL: VIRTUAL MODEL CONTROL
Most successful example: Virtual Model Control (G.Pratt)
Idea: create virtual elements to keep the robot upright and have it move forward
Then compute the necessary torques such that the robot motors replicate the effect of those virtual elements

43. HEURISTIC CONTROL: VIRTUAL MODEL CONTROL
For each virtual element producing a force F, the joint torque needed to produce that virtual force can be computed with:
J is the Jacobian relating the reference frame of the virtual element to the robot

44. VIRTUAL MODEL CONTROL
Only some motors should be activated at particular phases in the locomotor cycle
Finite state machine (set of if-then rules) for cycling through different actuation phases

45. CONTROL DIAGRAM: VIRTUAL MODEL CONTROL

46. VIRTUAL MODEL CONTROL: SUMMARY
Pros:
Intuitive way of designing a controller
Does not need an accurate model of the environment
Robust against perturbations
Cons:
Requires a perfect knowledge of the robot’s dynamics
Need to make sure that the virtual forces can actually be generated by the robot’s motors
Finite-state machine for cycling through the different phases is a somewhat rigid mechanism