Integrated kinetic and kinematic control

Both kinetic and kinematic control methods share the same joint space configuration, and they

can be integrated into a more advanced control scheme. Section 3.6.5.1 on page 67 shows how the

CoM position of the overall body can be controlled, while Section 3.6.5.2 provides a minimisation

approach to the supporting torque to expend the least effort while maintaining a posture, which

helps in the power efficiency of the robot.

It is straight forward to integrate the CoM and torque minimisation as secondary demands to

the standard IK problem, by setting the joint velocities

˙

q0 = J_G†x˙G− (I − J_G†JG)∇H (3.72)

substituting Eq. 3.72 for ˙q0 in Eq. 3.35 on page 55, results in the following joint velocities

˙

q = J†x + (I − J˙ †J )(J_G†x˙G+ (I − J_G†JG)∇H (3.73)

Here, the supporting torque minimisation behaviour is introduced as a third priority task,

through its projection into the null space of the secondary, CoM control task. It could have

been added as a secondary task by directly sharing the kinematic null space with the CoM task.

Moreover, the priority levels between the CoM and end effectors can be inverted favouring the CoM

demand, as is the situation in balancing the robot, as follow:

˙

q = J_G†x˙G+ (I − JG†JG)(J†x + (I − J˙ †J )∇H (3.74)

Equation 3.74 prioritises the CoM demand over the other end effectors. This is achieved by

projecting the other constraint into the null space of the CoM task Jacobian. Normally, in a

balancing situation, the support foot, and the CoM will have higher priority than the rest of the

end effectors. This way, the joint configuration ensures that the CoM and the support foot demands

are satisfied before solving for the other constraints in the system, so that the resulting pose will

3.7.Conclusions 71

3.7

Conclusions

This chapter shows simple models that approximate the humanoids motion while maintaining

standing balance in the face of large disturbances and achieving stable locomotion. These models

build upon prior work in the fields of robotics and Bio-mechanics.

In Chapter 4 , these simple models will be used to define regions of stability for the different

balancing strategies. These bounds can then be applied to complex humanoids or human subjects

to decide on the best balancing strategy or in predicting a fall. Furthermore, the main novelty of

this work is built upon these models, namely the Spherical Inverted Pendulum model and balancing

frame work in Chapters 5 and 6.

Also in this chapter, a formulation of the inverse kinematic problem to convert the end effector

and CoM constraint from Cartesian space to the joint space where the robot is controllable was

presented. It was also shown that closed formed solutions to the IK problem for redundant complex

robot systems such as humanoid robots, are either very difficult to find or do not exist. Furthermore,

the multi-task approach for kinematic redundancy makes a closed form analytical solution not

possible, due to the possibility of conflicts between the different demands.

In order to solve the problems arising due to the redundancy of the biped robot, the differential

kinematics and the Jacobian matrix were introduced. This method solves the inverse kinematics

problem by finding the joint velocities needed to satisfy the differential velocity in the end effectors.

This chapter discussed four different methods to solve the differential kinematics available in the

literature. These methods where compared in terms of their tracking performance, singularity

robustness and computational complexity. Redundancy is resolved through different approaches

such as joint limits and augmented task space constraint. Finally, the CoM Jacobian calculations

were presented and provided a method to minimise the joint gravity torques.

Out of the four numerical inverse kinematic methods, the feedback inverse kinematic (FIK)

method was found to be the most suitable for the application in this thesis. This is due to its linear

complexity in the number of degrees of freedom in the joint space and the end constraint degrees

of freedom. This FIK solver is implemented in C++ and used to validate the Spherical Inverted

Pendulum model introduced in Chapter 6. The FIK and SIP are validated on the Nao robot in

Chapter 7, where the robot is tasked with executing different demands that manipulate the hands

Chapter 4

Balance Strategies & Stability

Regions

This chapter builds on the humanoid models used to describe the balancing and walking dynamics

analysed in the previous chapters to define the stability bounds of the biped. In chapter 3, three

different methods for modelling the balance of a bipedal robot are described along with their

control inputs, as well as defining the three balance strategies. They were exploited by humans

in maintaining balance in the face of large disturbances as found by the biomechanical research

community. However, there was no discussion of the ankle torque limits, or whether the system

will be stabilized using one strategy or another.

This chapter will develop the stability regions defined by the state-space location of the CoM for

the different balancing strategies. These regions and their boundaries will be applied in determining

the appropriate strategy for maintaining the humanoid balance in later chapters of the thesis.

4.1

Standing Balance

Section 3.1 introduces the different standing balance strategies defined by the biomechanical re-

search community [Winter, 1995]. These are namely the following three strategies:

1. Ankle Strategy / CoP Balance

2. Hips Strategy / CMP Balance

3. Step Strategy

In this section, each of these strategies will be analysed within the prospect of humanoid balance

control. Also the stability region bounds for each one will be defined. These bounds will help

in predicting a fall and choosing the correct strategy to execute on the robot in the case of a

disturbance.