Diseño del Programa de formación acción

FIGUREF.2: Obtained FRFs for different amplitudes.

allows for a cleaner FRF response. The coherence reduces at high frequency where mea- surement noise has greater impact. Therefore, we can conclude that our FRF estimation was good enough andcoherent.

F.3

Estimating the parameters of the FRF

To get the transfer functionHmic, the parameters of the curve of the identified FRF must

be identified. This can be done through a least-squares routine in Matlab. This can more easily be done through thetfest Matlab function or through the system identification app in Matlab. The result obtained forHmicafter parameter identification is the following:

Hmic =

3.43s+60.65

s2_{+4.828s+61.92}·e

68 Appendix F. Prosthesis Dynamics Identification details

FIGUREF.3: Frequency response of identified Hmicand experimental FRF

of amplitude = 20%.

69

Appendix G

Stability Analysis

Coupling an admittance controlled device with external environment dynamics (Ze) cre-

ates a negative force feedback loop (Fig. G.1). The stability of the coupled system de- pends of the dynamics of apparent dynamicsYa andZe (see Fig. F.1). Because there is

no knowledge ofZe, one way of ensuring stability in our system is to ensure it is energy

passive, as explained in Chapter 2.

FIGUREG.1: Schematic of coupled dynamics of prosthesis and environ- ment dynamics

However, passive behaviour of a controlled device cannot always be achieved due to weak dynamic performance of the controlled device. Then,Yacan be stable but nonpas-

sive.Yais composed of virtual dynamicsYvand Michelangelo prosthesis dynamicsHmic. Yvis designed to be passive, but,is Hmicpassive?

A system is passive ifand only if it is positive real, that is,Hmicwill be passive iff:

1. Hmic(s)has no poles in<(s)>0

2. the phase ofHmic(s)lies between−90◦and 90◦.

The phase plot of the identified dynamics of the Michelangelo, Hmic, shows an abso-

lute phase lag greater than 90◦, therefore, it isnotpassive (see Fig. F.3). Then, the apparent dynamicsYa will not be passive either, asYa is just the multiplication, in frequency do-

main, ofYvwithHmic. IfYais not passive,how do we check if the coupled system can actually be stable? To answer this question, we can assess whatrange of environment dynamics

Ze could complementarily stabilize Ya for a stable prosthesis-environment interaction by

following theNyquist stability criterionfor closed-loop systems.

The closed-loop transfer function of the coupled system in Fig. G.1 isYa/(1+YaZe).

From the Nyquist criterion we know that any interconnection of stable systems is always (marginally) stable if the absolute phase of the loop-gain (L) is smaller than 180◦. In our case, the loop-gain of the closed-loop transfer function is L = YaZe. Then, we can

assess if the closed-loop system is stable by looking at the phase of the loop-gain. More specifically, an easy check for stability is to look at the phase margin: if it is positive,

|∠L| ≤180◦and therefore, stable; if negative, we confirm that|∠L|180◦, and therefore

70 Appendix G. Stability Analysis Because real objects consist of inertias, damping and stiffness and these are passive elements, we can modelZeasZe = ke+be/s, where ke andbe will be a range of values

of environment stiffness and damping, respectively. The range of values ofkeandbethat

complementarily stabiliseYais calledez-width.

The computation of the ez-width in this work is done in the program ’getEZwidth.m’. This program loads the identified transfer function Hmic, computes the loop-gain and

assesses stability based on phase margin for a range of values of ke and be. Finally, it

builds the plots for the ez-width.

The sources for the explanations on this appendix are [33], [34].

G.1

A note for coherence

Note that the identified dynamics of the Michelangelo Hmic are identified for position

input-output. That means that in our scheme (Fig. F.1) we should integrateωd intoθd,

and use desired (angular) position as input for the prosthetic controller. For coherence with identifiedHmic, the output should be angular positionθ. Therefore, we should dif-

ferentiate outputθ intoω, which will be the input for Ze. However, this does not affect

the loop-gain of the closed-loop transfer function and the ez-width analysis and results would be unaltered.

Furthermore, note that the input forZeis linear velocity and that the range of values

for ke and be is given in linear units and not rotational (see Chapter 2). To transform

angular velocityωinto linear velocityv, one should multiply by the radius of the index