A key body of work in the field of complex, dynamic optimisation modelling is that of Anderson and Pandy. A single complex model they developed has provided numerous insights in multiple studies (Anderson & Pandy, 2001a, 2001b; Anderson & Pandy, 2003; Pandy, 2003).
The model in question was a three-dimensional, 23 DOF model with 54 active MTUs (Figure 2.6).It was made up of ten segments. The pelvis was a single rigid segment with six DOF, the head, arms and torso were modelled as a single rigid body (HAT) and the other eight segments were divided evenly between the two legs. The feet consisted of hindfoot and forefoot segments. The muscles were defined appropriately to best represent the anatomical structure. The HAT segment was controlled by six back and abdominal muscles and each leg had 24 muscles to control it. Certain muscles, such as the gluteus maximus and gluteus medius/minimus, had to be separated into two separate actuators due to the complex geometry at their pelvic origin. This assumption meant the model could better replicate their actions.
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Figure 2.6: Pandy and Anderson’s complex gait model (Anderson & Pandy, 2001b)
For any simulation it is important to prove its validity. For this particular model, practical testing was performed and the results of the comparison were analysed in different papers (Anderson & Pandy, 2001a; Anderson & Pandy, 2003; Pandy, 2003).
The experimental testing itself consisted of five healthy adult males, whose age, height and weight were all taken into consideration. Each participant was required to perform four laps of a 400m track as a warm up. On the third of these laps, the number of steps the participant took, and the time taken, were recorded. Due to the way in which the model was built, with the head, arms and trunk being modelled as a single segment, the participants performed all walking tasks with their arms folded across their chest. The participant then entered the gait laboratory and with the use of a metronome, they reproduced their natural outdoor walking rhythm along an 11m track, instrumented with force plates. During these indoor walking trials, the participant had passive reflective markers attached to them at specific locations. The motion of these markers and hence the particular body segments, was captured using specialist cameras. EMG recordings were also taken throughout the trials. Each participant performed five trials, all of which were video recorded as well. For each participant, anthropometric data was taken.
Using mean data collected from a gait lab study as the initial conditions, an optimisation problem was constructed. Assuming bilateral symmetry, half a gait cycle was simulated over a fixed time of 0.56s, which was the mean time taken for half a cycle in the gait lab testing. Constraints were applied so that joint angles and velocities, as well as muscle excitation and activation, at the end of the left side of gait were equal to those at the start of the right side of gait. The cost function of the optimisation was the total
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metabolic energy divided by the anterior-posterior displacement of the CM. The energy used by each muscle was the sum of basal metabolic heat, shortening heat, activation heat, maintenance heat and mechanical work. A penalty function was included in the cost function to avoid joint hyperextension. The authors were keen to highlight that their model was not given a ‘tracking’ problem; that is to say the kinematic motion was not strictly defined. Instead, only initial and final conditions were set.
One of its first uses was to investigate the differences between static and dynamic solutions and to justify the use of each for different scenarios (Anderson & Pandy, 2001b). Firstly, a dynamic simulation was performed. The cost function to be minimised was metabolic energy per unit of distance travelled with the constraint being that it had to produce a cyclic gait pattern. The activation profiles of the muscles were defined by first-order differential functions. There were two different static problems set up, relating to the way in which the muscles were modelled. In the first one, they behaved as ideal force generators; in the second they were constrained by their respective force-length- velocity profiles. In both cases, the joint moments produced by the forward dynamic solution were the inputs, the muscle activations were the variables and the sum of the squares of the muscle activations was the objective function to be minimised. The results showed a good agreement between all the models. This led the authors to conclude that, if the inverse dynamics problem can be solved accurately, the use of predictive dynamic optimisation over static is not justifiable. However in situations where accurate experimental data is unavailable or a time-dependent performance criterion is desired then it is very useful. The key conclusion the authors draw is that the two methods should complement one another.
The dynamic model was also compared to the gait lab data to see how well it was able to predict the basic kinematics and ground reaction forces (Anderson & Pandy, 2001a), as well as the individual muscle contributions to gait (Anderson & Pandy, 2003). Each muscle excitation history was defined by discretised ‘control nodes’. These were spread at equal time intervals across the excitation history and interpolated between. The values of all these nodes, as well as the initial values of each muscle excitation, were used as the control parameters in the dynamic optimisation. Once again, the cost function was minimising metabolic energy expenditure per unit distance travelled.
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The results of this model were relatively close to the experimental data. In contrast to Pandy’s double inverted pendulum model, which suggested that the first and second peaks in the vertical ground reaction force were caused by mechanisms at the knee and ankle respectively, the more complex model claimed it was hip and ankle mechanisms respectively (Pandy, 2003). This difference is explained by the increased complexity. Where the complex model is able to perform all six determinants of gait (Saunders et al., 1953), the double inverted pendulum can only reproduce three of them.
Upon closer inspection however there were some flaws. Although the predicted vertical component of GRF displayed the double peak shape, it contained a lot of spikes and was not a smooth curve like the empirical data. In addition to this, the contributions of
“inertial forces”, “centrifugal forces”, “muscle forces” etc. appeared to rise or drop instantaneously at milestones such as heel rise and contralateral heel strike.
A large kinematic anomaly was the excessive transverse pelvic rotation around heel strike. The explanation for this is the heel-strike force required to decelerate the swing leg. The participants did not exhibit this behaviour in the practical testing which suggests there is a more sophisticated method used by humans than the model is able to replicate. The explanation proffered for the spikes and discontinuities in ground reaction force components was due to the way in which the foot was modelled when in contact with the ground. A mass-spring-damper system was used. The model also predicted the metabolic energy consumption rate to be much greater than the results published elsewhere. This was explained by a lack of arm swing in the model and simplifications made in muscle modelling. For example, considerations of the difference between fast and slow twitch muscles were not made. They also state that, despite over 10,000 hours of processing time, it is possible that the solution hadn’t completely converged.
In spite of these drawbacks, most of the kinematic behaviour of the simulation appeared close to reality. This suggested that minimum metabolic energy expenditure per distance travelled may indeed be a valid criterion for walking. They were also able to postulate the individual contributions of the different muscles of the lower limbs at different times in the gait cycle and although the magnitudes may not have been perfect due to spikes and step changes, the general proportions make good references for future work, particularly when EMG data is unavailable.
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A slightly modified version of this model was used by Thelen and Anderson (2006) to investigate whether it could be solved within a more manageable time. The body was now modelled as an eight segment, 21 DOF structure, actuated by 92 MTUs. A number of extra considerations were made when solving the model. They state that due to measurement errors in practical data captured for comparison and modelling assumptions, kinematics and kinetics are often dynamically inconsistent. This means that models will predict extra, external forces known as residual forces. They hoped to produce a dynamically more consistent model by using a ‘residual elimination algorithm (REA)’ and taking into consideration time delays between muscle excitation and activation.
The solving method was unique too. They used a ‘computed muscle control (CMC)’
algorithm. This meant using the joint angle errors (when compared to those recorded from ten healthy male participants) at a given time to calculate the appropriate angular acceleration of the joint required, so as to match the joint angles at the next time instant. Muscle activation and contraction dynamics were integrated from the previous time step to work out the upper and lower bounds on the force that each muscle could produce at the current time step. A static optimisation was then used to calculate the appropriate muscle forces needed to achieve the necessary joint angular accelerations, by means of the equations of motion. This process was repeated for every time interval.
The results showed that the kinematic root mean-squared (RMS) errors were mostly less than 1° and the predicted muscle activation profiles, visually, appeared fairly consistent with the experimental data. It is important to highlight, however, that this method is a tracking problem, whereas the previous studies (Anderson & Pandy, 2001a; Anderson & Pandy, 2003) used a performance based dynamic optimisation which can produce novel motions.