3.3 System Modelling using Energy
3.3.4 Modelling Methodology
[38]. Equating eqs. (3.30) and (3.29) allows for the representation of the J(q, ˙q) ˙q vector in terms of the mass matrix Mc(q), where
J(q, ˙q) ˙q +1 2M˙ c(q, ˙q) ˙q = ˙Mc(q, ˙q) ˙q− " ∂ ∂q 1 2˙q TM c(q) ˙q !#T ∴ J(q, ˙q) ˙q = 1 2M˙ c(q, ˙q) ˙q− " ∂ ∂q 1 2˙q TM c(q) ˙q !#T . (3.31)
The ˙q matrix is not invertible, but the individual entries may be represented with the knowledge that the J(q, ˙q) matrix is skew-symmetric (with the total number of individual elements in each J(q, ˙q) matrix = n(n−1)/2 for a nth order system).
Therefore, J(q, ˙q) = 0 J1(q, ˙q) J2(q, ˙q) · · · Jn−1(q, ˙q) −J1(q, ˙q) 0 Jn(q, ˙q) · · · J2(n−1)−1(q, ˙q) −J2(q, ˙q) −Jn(q, ˙q) . .. . .. ... .. . ... . .. . .. ... −Jn−1(q, ˙q) −J2(n−1)−1(q, ˙q) · · · 0 . (3.32)
It may be tedious to calculate each entry, but it is not required if one just wishes to characterise the behaviour of the system in terms of torques. Finding the J(q, ˙q) ˙q vector in this case would be sufficient.
3.3.4 Modelling Methodology
The method highlighted below presents an independent system modelling alterna- tive to the classical Lagrangian method as first presented in [93]. This method is modified here to include the explicit modelling of the energy that is shuffled within
Chapter 3. Introduction to System Modelling 30
the system (represented by the J(x, v)v vector). The procedure is outlined similarly to what is seen in [93]:
(1) Identify the relevant generalised coordinates of the system.
(2) Define Es, the mechanical energy of the system by determining the kinetic
and potential energies in terms of the generalised coordinates, and derive the mass matrix M(x) and the potential torque matrix K(x) from these energy expressions.
(3) Calculate the system’s net change in energy by taking d dtEs.
(4) Describe the factors that add and dissipate energy, namely the power-loss component −vTR(x, v) and the actuation matrix vTG(x)u. Equate these
factors to the change in energy described by d
dtEs. This is known as the system power equation.
(5) Calculate the entries of the energy-shuffling component J(x, v)v using eq. (3.31) that was derived in the last section.
(6) Substitute in all the relevant components into the system power equation and manipulate it into the generalised prototypical form, where the equations of motion may be solved for.
(7) Transform the equations of motion into the state-space representation using a relevant set of transformations.
This energy method is used for the modelling obligations of this investigation since the Lagrangian method is covered significantly in literature. The general modelling of the n-link pendulum system is demonstrated in section 7.3.1, but a more conceptual example of the application of the modelling methodology shown above can be seen in section 7.4, whereby the Acrobot is modelled for experimental purposes.
3.4
Conclusion
The foundations of two robust modelling techniques, namely the Classical La- grangian and the Energy modelling methods, were discussed in this chapter. Each technique was implemented on a model of the DIP (energy modelling example shown in later chapter) with the objective of developing equations of motion that describe the movements of each pendulum. The classical Lagrangian modelling method involves the implementation of the Euler-Lagrange equation to derive the equations of motion of a system, whereby calculus of variations is used to exploit the principle of least action. Whilst the method is effective and easy to implement, the procedural nature of this technique makes it difficult to identify underlying causes to the torques represented in the prototypical form. The energy method involves the
Chapter 3. Introduction to System Modelling 31
classification of the torques in the system according to the effect that each torque has on the energy of the system. The structure of each component in the proto- typical form is scrutinised to determine its influence on the system’s energy. This leads to the decomposition of the unclassified D(q, ˙q) matrix into conservative and non-conservative torques. The equations of motion are then determined through the evaluation of the power equation, which is found by evaluating the energy that is added or dissipated from the system due to friction and actuation. The technique highlighted by Naude has one major constraint when applied to com- pounded pendulum systems; the torques responsible for energy-shuffling within the system (represented by J(q, ˙q) ˙q) are unobservable in the energy domain (the torques contribute no energy to the system, and are also not responsible for any change of energy in the system). This chapter highlights a method of determining the entries of the internal shuffling vector J(q, ˙q) ˙q through the matrix evaluation of the Lagrangian. This is a modification to the method highlighted by Naude. Both methods allow for the satisfactory identification of the dynamics of a multi-body pendulum system. The energy method, however, has the added benefit of identify- ing the particular components of the dynamics and their effects on the total energy change in the system, and is therefore selected as the preferred modelling method in this investigation.
32
“Chaos is merely order waiting to be deciphered.” — José Saramago
Chapter 4
Stability Concepts
4.1
Chapter Overview
This chapter serves to introduce basic concepts of system stability that are relevant to this research project. A section in this chapter is dedicated to the formal definition of stability principles with regards to system equilibrium points, including the concepts of local and global stability, the varying forms of stability (marginal, exponential, asymptotic etc.) and the characteristics of an unstable system. Short discussions are included to provide fundamental knowledge about the implementation of pole-zero diagrams to graphically demonstrate the stability of a linear system, and the Routh- Hurwitz stability criterion, which is a technique that is integral to this research project. Whilst this chapter presents a superficial level of information on certain topics, sources are provided to the reader for the purpose of supplementation if required.