Evaluation of balance control ability would become important in the rehabilitation training. In this paper, in order to make clear usefulness and limitation of a traditional simple inverted pendulum model in balance prediction in sit-to-stand movements, the traditional simple model was compared to an inertia (rotational radius) variable inverted pendulum model including multiple-joint influence in the balance predictions. The predictions were tested upon experimentation with six healthy subjects. The evaluation showed that the multiple-joint influence model is more accurate in predicting balance under demanding sit-to-stand conditions. On the other hand, the evaluation also showed that the traditionally used simple inverted pendulum model is still reliable in predicting balance during sit-to-stand movement under non-demanding (normal) condition. Especially, the simple model was shown to be effective for sit-to-stand movements with low center of mass velocity at the seat-off. Moreover, almost all trajectories under the normal condition seemed to follow the same control strategy, in which the subjects used extra energy than the minimum one necessary for standing up. This suggests that the safety considerations come first than the energy efficiency considerations during a sit to stand, since the most energy efficient trajectory is close to the backward fall boundary.
Lower limb motor functions are important for the activities of daily living (ADL), participating in social activities, and preventing bedridden state. Therefore, rehabilitation training of sit-to-stand movement is considered to be the first step to prevent motor-disabled patients and elderly people from being bedridden. In the rehabilitation, joint angle trajectories and/or joint torques are commonly measured for evaluation of motor function. During lower limb movement, however, balance control is also required for developing the movement safely. Since sit-to-stand movement requires control of stability in addition to muscular strength [
Bipedal balance has been studied since it is expected to have important applications in preventing health problems associated with falls [
In order to calculate these BOS-CM conditions for predicting the ability to control gait balance, those previous studies modeled human bipedal gait as a single joint, simple inverted pendulum [
In this paper, in order to make clear usefulness and limitation of the traditional simple inverted pendulum model in balance prediction in sit-to-stand movements, the balance prediction obtained by the method of using the traditional simple model was compared to the prediction of a complex model that included multiple-joint influence. Here, since a telescopic pendulum model has been shown to be no less informative than more demanding multisegment models [
In the sit-to-stand movement, the center of mass (CM) velocity-position is in a given initial state when the subject leaves the chair. Then, the task of sit-to-stand consists of stopping the CM somewhere over the base of support (BOS) while satisfying the restrictions imposed by the friction coefficient, the foot geometry, maximum and minimum physiological ankle torque, and the condition that the foot segment should not move.
The sit-to-stand balance control feasibility can be calculated by finding all the CM velocity-position conditions when leaving the chair that allow the CM to arrive and stop over the base of support using the inverted pendulum model shown in Figure
The inverted pendulum used for modeling sit-to-stand movement in the sagittal plane. Note that the length
An example of the dynamic balance condition obtained through a simple inverted pendulum. The horizontal axis represents the posterior position of the CM, measured with respect to the toe and normalized to the subject’s foot length (thus the 0 and 1 represent the toe and heel position, resp.). The vertical axis represents the anterior velocity of the CM normalized to the subject’s height. The solid lines enclose the CM velocity-position conditions that allow recovering static balance (gray-shaded area), that is, the dynamic balance conditions. The broken line represents the most energy-efficient trajectory (zero external torque).
The inverted pendulum model with a static support segment (Figure
In previous studies, a simple inverted pendulum model with constant pendulum length
That is,
In this paper, the methodology presented by Pai and Patton [
It is important to mention that the inclusion of the inertia (rotational radius) variation affects not only the rotational movement equation (
Six healthy male subjects (
Experimental data were recorded using 15 reflective markers with an 8-camera, 3D motion analysis system at a data sampling rate of 120 Hz (Vicon, Oxford Metrics, UK). Force plates were used for finding the timing of seat-off. All the theoretical balance predictions were calculated using MATLAB (Math Works, Inc., USA).
First, every subject was asked to perform two self-selected most natural sit-to-stands, to measure their sit-to-stand inertia function. The rotational inertia function was estimated by assigning the measured CM rotational radius (
Next, the subjects were asked to stand up at different initial conditions of the CM position which in turn will lead to different initial conditions of the CM velocity-position when the subject leaves the chair. In order to create different velocity-position CM conditions at the seat-off, the horizontal position of the feet was varied while sitting. The feet were shifted −0.2~1.2 foot lengths forward from the foot position where the ankle was at 90 degree, since those distances were showed to include successful and not successful stand ups. Every subject was asked to perform a total of 23 to 25 sit-to-stands. It is important to note that the actual position of the feet is rather unimportant since the real important information will be the horizontal BOS-CM distance that is precisely known from the markers position measurements.
The BOS-CM horizontal position was gradually increased to demanding standing up conditions for every subject. A threshold of the BOS-CM horizontal distance of the first unsuccessful sit-to-stand was set to divide the data into two groups. The first group would be the conditions where every subject was able to make a successful sit-to-stand (normal condition). The second group would be all the data after the threshold (demanding condition), which was used for evaluating the theoretical balance predictions since it would include unstable sit-to-stand movements. From the results, the sitting BOS-CM horizontal distance threshold was 2.48 foot length. A total of 146 sit-to-stand measurements were performed, but due to markers disappearances or subjects’ mistakes (BOS movement), 127 valid sit-to-stands were analyzed, in which 80 sit-to-stands were classified as the normal condition and 47 sit-to-stands as the demanding one.
Figure
An example of balance control prediction for normal sit-to-stand conditions (a) and demanding sit-to-stand conditions (b) (subject C). The solid lines show the boundaries for sit-to-stand balance control calculated with the inertia variable model, while the broken lines show the boundaries calculated with a simple inverted pendulum model. The open circles represent the successful sit-to-stand data while the closed circles represent the unsuccessful sit-to-stand data. In order to consider a data to be stable, the whole error bar on the point should be inside the boundaries.
Normal sit-to-stand data
Demanding sit-to-stand data
From the 80 normal sit-to-stands measured, it was found that 72 (90%) of them had extra kinetic energy compared to the (calculated) zero torque trajectory, while the other 8 data were almost on their zero torque trajectories crossing the zero torque condition after leaving chair. Figure
Measured CM sit-to-stand trajectories after leaving chair. Here, the 72 trajectories that showed a more energetic (faster) CM trajectory than the most efficient zero torque trajectory (in broken lines) are shown.
An example of a nonstandard control strategy performed by subject A. The CM position at the seat-off was 1.39 foot length. It is possible to see that at the moment of the seat-off, the CM is in a backward position near the most efficient zero torque compared to the trajectories of the same subject in Figure
The results from the demanding sit-to-stand conditions showed that the inertia variable pendulum model is better than a simple inverted pendulum model for evaluating the stability of a sit-to-stand movement. The specificity for the demanding sit-to-stand was improved from 41% to 82% by using the model including multiple-joint influences. On the other hand, both of the maps correctly classified all the normal (natural) sit-to-stand data as seen in Figure
An example of validity threshold for the simple inverted pendulum model. The plot on the map shows average velocity at the seat-off for a natural sit-to-stand.
The simple inverted pendulum model is also considered to be reliable for rehabilitation assessment and balance analysis in sit-to-stand movements of motor-disabled subjects and elderly persons. For sit-to stand movements that have low CM velocity at the seat-off, there was no large difference in the map between the simple inverted pendulum model and the inertia variable pendulum model as shown in Figure
As seen in Figure
It seems that the natural gait control strategy first tries to satisfy the stability requirements rather than optimizing the energy usage. However, energy considerations are often used when analyzing gait and designing assistive and rehabilitation technology [
As for the data that crossed the zero torque line as shown in Figure
The balance prediction in sit-to-stand movements obtained using a traditional simple inverted pendulum was compared to the prediction of an inertia variable inverted pendulum model including multiple-joint influence. The results showed that the multiple-joint influence model is more accurate in predicting balance during sit-to-stand movements under demanding conditions and also that the traditionally used simple inverted pendulum is still reliable in predicting balance during normal or nondemanding sit-to-stand movements. Especially, the simple inverted pendulum model could be effective for sit-to-stand movements with low CM velocity at the seat-off. In addition, almost all CM trajectories during normal sit-to-stands seemed to follow the same control strategy, in which the subjects used extra energy than the minimum one necessary for standing up. This suggests that the safety considerations come first before the energy-efficiency considerations during a sit-to-stand since the most energy efficient trajectory is close to the backward fall boundary.
Equations (
For the CM,
From the constraint of the ground reaction force (
Equation (
For the maximum posterior friction condition, (
Equation (
For the COP after the toe condition, (
The authors confirm that there is no conflict of interests in relation to this work.
This work was supported in part by the Ministry of Education, Culture, Sports, Science and Technology of Japan under a grant-in-aid for challenging exploratory research.