There are two issues in balancing a stick pivoting on a finger tip (or mechanically on a moving cart): maintaining the stick angle near to vertical and maintaining the horizontal position within the bounds of reach or cart track. The (linearised) dynamics of the angle are second order (although driven by pivot acceleration), and so, as in human standing, control of the angle is not, by itself very difficult. However, once the angle is under control, the position dynamics are, in general, fourth order. This makes control quite difficult for humans (and even an engineering control system requires careful design). Recently, three of the authors have experimentally demonstrated that humans control the stick angle in a special way: the closed-loop inverted pendulum behaves as a non-inverted pendulum with a virtual pivot somewhere between the stick centre and tip and with increased gravity. Moreover, they suggest that the virtual pivot lies at the radius of gyration (about the mass centre) above the mass centre. This paper gives a continuous-time control-theoretical interpretation of the virtual-pendulum approach. In particular, by using a novel cascade control structure, it is shown that the horizontal control of the virtual pivot becomes a second-order problem which is much easier to solve than the generic fourth-order problem. Hence, the use of the virtual pivot approach allows the control problem to be perceived by the subject as two separate second-order problems rather than a single fourth-order problem, and the control problem is therefore simplified. The theoretical predictions are verified using the data previously presented by three of the authors and analysed using a standard parameter estimation method. The experimental data indicate that although all subjects adopt the virtual pivot approach, the less expert subjects exhibit larger amplitude angular motion and poorly controlled translational motion. It is known that human control systems are delayed and intermittent, and therefore, the continuous-time strategy cannot be correct. However, the model of intermittent control used in this paper is based on the virtual pivot continuous-time control scheme, handles time delays and moreover masquerades as the underlying continuous-time controller. In addition, the event-driven properties of intermittent control can explain experimentally observed variability.
When characterising typical human movement profiles, the optimal number of trials analysed for each participant should ensure a stable mean. Sequential analysis is one method able to establish the number of trials to stability by assessing a moving point mean against a set bandwidth. As the total trial number determining this bandwidth is selected arbitrarily, the effect of applying different total trial numbers on the results of sequential analysis was investigated. Twenty participants performed 30 trials of overarm throwing, and sequential analyses were applied to three dimensional (3-D) kinematic data over 10, 20 and 30 trial numbers. We found a total of 20 to be the preferred trial number for sequential analyses. Erroneous results were produced consistently by 10 trial number groups, while moving point means were statistically unchanged after the 10th trial. Subsequently, sequential analyses were applied to 20 trials to establish trials to stability in discrete and time series elements of the 3-D kinematic data. The results suggest that a trial size between 13 and 17 provides stable means for overarm throwing kinematics.
We investigated the acquisition of skill in balancing a stick (52 cm, 34 g) on the fingertip in nine participants using three-dimensional motion analysis. After 3.5 h of practice over 6 wk, the participants could more consistently balance the stick for longer durations with greatly reduced magnitude and speed of stick and finger movements. Irrespective of level of skill, the balanced stick behaved like a normal noninverted pendulum oscillating under greater-than-gravity torque with simple harmonic motion about a virtual pivot located at the radius of gyration above the center of mass. The control input parameter was the magnitude ratio between the torque applied on the stick by the participant and the torque due to gravity. The participants utilized only a narrow range of this parameter, which did not change with practice, to rotate the stick like a linear mass-spring system. With increased skill, the stick therefore maintained the same period of oscillation but showed marked reductions in magnitude of both oscillation and horizontal translation. Better balancing was associated with 1) more accurate visual localization of the stick and proprioceptive localization of the finger and 2) reduced cross-coupling errors between finger and stick movements in orthogonal directions; i.e., finger movements in the anteroposterior plane became less coupled with stick tip movements in the mediolateral plane, and vice versa. Development of this fine motor skill therefore depended on perceptual and motor learning to provide improved estimation of sensorimotor state and precision of motor commands to an unchanging internal model of the rotational dynamics.
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