All UCM variables were computed at time states described previously (section 3.5.4). Briefly, the UCM was computed at five sequential 3% increments which
neighboured the lead-in, and lead-out, from the reference states of MX1 and MTC. The time slices used for computing the UCM are state-dependent and relevant to
hypothesised task goal events of MX1 and MTC.
The UCM is computed around the mean joint configuration from a given sample of trials. This mean configuration essentially represents the goal state of the effector system. The UCM does not depend upon a temporal sequencing order and therefore trials could be categorised as discrete occurrences based upon different trial ranking
148
criteria. Because the time slices are goal-dependent, it is assumed that the movement task goal at each time slice remains constant between trials. This is important in the computation of the UCM hypothesis. To ensure that this assumption has greater merit, the hypothesised task goals expressed by the effector trajectories within a swing trial were re-sampled into a discrete set of like trials. The criteria for re-ordering the swing trials were based upon initial conditions of the stance (or swing) end-effector at MX1 and the ‘final’ conditions at MTC. This ranking of trials was based upon hypotheses listed in Table 3.6.2.5.1 and illustrated in Figure 3.6.2.5.1.
Table 3.6.2.5.1.
Criteria for trial re-sampling into nine subgroups
Trials were excluded during the first wave of trial rankings if the vertical swing hip position was outside of 5-95% of the re-ordered ranked trials (Figure 3.6.2.5.1). Within these three sub-groups, a second wave of trial re-ordering of trials based upon rankings of a second parameter, in this example, those which had a vertical MTp position outside of 5-95%. So, the first wave included three subgroups according to hip height (Hhigh,
Have, Hlow) and these sub-groups were further sub-divided into three sub-subgroups
according to toe height position (Thigh, Tave, Tlow). The number of trials employed for
computing a UCM (n ≈ 50) is consistent with other UCM studies (e.g., Tseng et al. (2006)).
Re-sampling criteria 1: based upon initial condition
Re-sampling criteria 2: based upon the response made from
initial condition Stance
Effector
Vertical displacement of the stance effector at MX1
Vertical displacement of the stance effector between MX1 and MTC
Swing Effector
Vertical displacement of the swing effector at MX1
Vertical displacement of the swing effector between MX1 and MTC
149
The dilemma is the identification of which control parameter does the controller pay attention to at the MX1 initial condition and which control parameter does the controller pay attention to for the necessary response. There are multiple scenarios which could represent the state estimate of interest to the controller’s policy. The control parameter representing the required state at some initial condition is hypothesised to be associated with the effector state at MX1, and this could be the combination of the following: 1) the vertical displacement of the stance effector (external reference); 2) the length of the swing limb (internal reference); or 3) the toe- to-ground clearance height (i.e. combined effector - external reference). Because of the
HIGH 65-94 percentiles LOW 5-34 percentiles AVE 35-64 percentiles Figure 3.6.2.5.1
Subdivision of N swing trials into nine groups. From the full number of trials, the data was ranked according to a criteria listed in Table 3.8.2.5.1. Trials were
subsequently ranked in order of percentiles according to the criteria, i.e. 5-34%, 35- 64%, 65-94%. Data values located within lowest and highest five percentiles of the re-ordered trials were considered possible outliers to the group goal and thus they were excluded from analysis. Within each of these three groups, a second ranking criteria was performed based upon criteria 2. From this each group was further subdivided into three subsequent groups, also excluding extreme trials beyond 5- 95% of the re-ordered trials. For example, nine subgroups created from N=600 trials results in approximately n≈60 trials allocated to each of the nine re-sampled
groupings. Weak 5-34% Moderate 35-64% Strong 65-94% C1: C2: N=600 trials Weak 5-34% Moderate 35-64% Strong 65-94% Weak 5-34% Moderate 35-64% Strong 65-94%
150
difficulty obtaining an indication of the controller’s policy, the criteria was specific to the effector system and assumed that the effector was configuring its variance to control its effector endpoint.
The trials within each of the nine subgroups were used to calculate the Jacobian and null space based upon the mean of the segment angles which represent the mean ‘end-effector’ value.
3.6.2.5.1 Distribution of the re-sampled data
The three methods for sub-grouping the original data into nine sub categories involve data re-sampling. Following the first rank-order re-sampling criteria, each of the three sub-groups will contain a ‘biased’ sample of trials with distribution characteristics which are vastly different in terms of the skewed description of the data. This of course will most heavily affect those variables which are the basis of the re-sampling criteria. However, the other two variables will be also affected because of a co-dependent relationship (demonstrated by results of Section 7.3). For example, after the first criteria for re-sampling, the first and third distributions will not be normally distributed. This is because this ranking divides the distribution into three by partitioning the two ‘tail-ends’ from the middle section of the normally distributed. Hence, the two ‘tail- ends will be strongly negatively, and positively skewed. The second distribution will be less skewed, however, it will have a kurtosis due to a peaked nature. Given that there is a variable response to initial conditions by this second ‘grouping’ parameter, the new rank-ordered samples will have diluted, to some extent, the bias of the ‘parent’ distribution even though these ‘sibling’ samples are taken from a non-normal ‘parent’ distribution. This of course will be dependent upon the variant response of the
151