Abstract
A two-dimensional, three-joint, four-segment model was set to simulate
sit-to-stand (STS) movement. Redundancy of the system allowed us to compare the
model results of the foot-ground reaction forces to those measured and to set
an iteration algorithm to re-estimate joint positioning and anthropometric
properties, while solving for joint torques before and after seat-off. Model
applications for FES controlled systems are discussed.
1. Introduction
Sit-to-stand (STS) movement is a mechanically demanding task that requires coordinated operation of the muscles of the lower extremities. Spinal cord injured patients with upper motor neuron lesions or subjects with various motor impairments can regain the ability of standing up by means of voluntary movement of the upper body and FES-assisted activation of the knee extensors [1-3,5]. Recent studies employed inverse dynamics and semi-dynamic methods for the determination of joint torques in chair-rise of paraplegic subjects. All in all, these studies have shown that knowledge of the kinematics and joint torques of the entire body is essential for gaining quantitative evaluation of the underlying stimulation paradigm of the FES control system throughout the rising process [1-3].
It should be mentioned, though, that a simple dynamic linked segment model is not sufficient to provide a reliable solution to the inverse-dynamics problem. This is due to uncertainties related to joint positioning, segments’ anthropometry, deformability of the trunk during movement, synchronization errors between force plate and motion analysis system and digitization errors of the joint positions [4,6]. Nevertheless, if the system is over-determined, it may become possible to use the redundancies to obtain joint torques while being able to test the reliability of the results. Redundancy of the system allows us to
compare the model results of the foot-ground reaction forces to those measured and to set an iteration algorithm to re-estimate joint positioning and anthropometric properties, while solving for joint torques before and after seat-off. In the present study a model is presented to estimate lower-limb dynamics, based on kinematic data and forceplate measurements of the foot-ground and thigh-chair reaction forces. The results presented are used to examine the biomechanical factors underlying prediction of the knee extensor and hip and ankle flexor forces in FES-assisted standing up systems.
2. Model
A two-dimensional, three-joint, four-rigid-segment model was used. Anthropometric data were scaled using body height and mass. Kinematics was derived from angles qi (i = 1 … 3), expressing sagittal rotation of the ankle, knee and hip, respectively. Next, the Newton-Euler equations of motion were applied to each of the segments to compute the forces and moments in the joints:
dLj/dt = Fj-1 – Fj - mjg (1)
dHj/dt = |dj ´ Fj-1| – |pj ´ Fj| - tj-1-tj (2)
The left hand part in equation (1) and (2) represents the time rate change of the linear and angular momentum of the j’th segment center of mass, respectively. Fj and Fj-1 are the inter-segmental forces acting at the top-most (proximal) and the bottom-most (distal) ends (joints) of the j’th segment, respectively. Similarly, tj and tj-1 are the joint torques (scalars) acting on the same respective ends. pj and dj are the position vectors (with two components) connecting the center of mass of segment j to its proximal and distal joints, respectively. Equations (1) and (2) yield a set of 12 (= 3 x 4) scalar equations. There are potentially 15 unknowns: 9 (= 3 x 3) force components and torques acting at each of the joints; and 6 (= 3 x 2) unknowns at the body contact with the supports. However, since the chair-thigh and foot-ground reaction force and point of application were actually measured, the system was over-determined, allowing us to use the force plate measurements under the feet for comparison with the model results. Quantitative evaluation of the difference between the model estimated and the measured components of the force and center of pressure (COP) (platform A) was obtained by using the sum of the square errors (SSE):
SSE = Sk (qmodel[k] - qmeas[k])2 (3)
where qmodel[k] and qmeas[k] are the model and measured values at time point k, respectively. An iterative algorithm (Fig. 1) was used to minimize the SSE. The iteration process allowed for the readjustment of the positioning of the centers of the ankle and the knee from their predetermined initial positioning. Constraints on the anatomical boundaries of the feet and the ankle joint and seating boundaries were imposed during the iteration. At the end of the iterative process, the torques acting in the hip, knee and ankle joints were calculated.
The above method was tested with a group of six able-bodied subjects with no known history of injury or pathological disorders, which could affect normal posture or gait. Subjects were positioned with the trunk erect and with the arms folded over their chest on an armless, adjustable chair that was mounted on forceplate B. The height of the chair was such that the knees were flexed at 90o as the feet rested flat on forceplate A (both AMTI). Forceplate data from the last 5 of a total of 8 trials, were collected at 100 samples/s during 9s starting 3s prior to the signal. Kinematic data of the reflective markers were extracted from videotape recordings (at 50 frames/s, synchronized with the forceplate signals) using PEAK software. All data were digitally filtered with a zero-phase lag, bi-directional, fourth-order low-pass Butterworth filter at a cut-off frequency of 5 Hz.
3. Results and Discussion
Sensitivity of the model to perturbations in the segments anthropometry indicated that the torques obtained in the final iteration were only mildly affected. The effect of change in the location of the centers of mass (COM) of the trunk, shank and thigh (expressed in % distance from their respective joints) on the SSE values in the final iteration was as follows: ± 20% caused only minor variations in the vertical and AP forces but changed those of the COP by 20%. The effect on the predicted torque histories in the ankle, knee and hip joints was as follows: negligible for hip, 0.3% for knee extension, 0.02% for ankle plantarflexion and 10% for ankle dorsiflexion. These results showed that the iterative algorithm largely compensated for errors in the segment inertial parameters. We accordingly suggest that optimization on the locations of segments’ COM’s, in addition to the optimization on the joint centers of rotation, could be used to provide a reliable prediction for the torque history in the joints. Reducing errors in the dynamic solution underlying STS movements is necessary for appropriate design of feed-forward controllers in lower limb FES systems aimed to assist subjects with motor impairment to rise from a chair. For example, the estimated torque profiles can be taken to delineate the required stimulation parameters while using explicit recruitment curves [7].
Figure 2 shows the estimated torque curves in the hip, knee and ankle joints during STS movement of normal subject. The estimated torque profiles were normalized to body-mass times height. The initiation of the STS movement was accompanied by an increase in the hip flexion and ankle dorsiflexion torque at the ankle as the COP of the foot-ground force moved posteriorly, toward the heel. Peak hip flexion torque was reached soon after the onset of forward acceleration. The hip and the knee extension torques increased simultaneously during the forward acceleration phase and reached their peak values very close to the instant of seat-off (SO). Afterward, the extension torques at the hip and the knee joints started to descend, whereas the ankle torque reversed direction as the COP of the foot-ground forces moved anteriorly to the ankle joint at the end of the movement. These torque profiles served to illustrate the underlying FES sequences in the hip, knee and ankle actuators (Fig. 2, bottom).
