Dynamic Optimization of the Sit-to-Stand Movement

Transcription

1 Journal of Applied Biomechanics, 011, 7, Human Kinetics, Inc. Dynamic Optimization of the Sit-to-Stand Movement Hiroshi R. Yamasaki, 1 Hiroyuki Kambara, and Yasuharu Koike 1 Showa University; Tokyo Institute of Technology The purpose of this study was to clarify criteria that can predict trajectories during the sit-to-stand movement. In particular, the minimum jerk and minimum torque-change models were examined. Three patterns of sit-to-stand movement from a chair, i.e., upright, natural, and leaning forward, were measured in five young participants using a 3-D motion analysis device (00 Hz. The trajectory of the center of mass and its smoothness were examined, and the optimal trajectories predicted by both models were evaluated. Trajectories of the center of mass predicted by the minimum torque-change model, rather than the minimum jerk model, resembled the measured movements in all rising movement patterns. The upright pattern required greater extension torque of the knee and ankle joints at the instant of seat-off. The leaning-forward pattern required greater extension hip torque and higher movement cost than the natural and upright patterns. These results indicate that the natural sit-to-stand movement might be a result of dynamic optimization. Keywords: dynamic optimization, center of mass trajectory, sit-to-stand movement The sit-to-stand movement is an important function required for performing activities of daily living. When rising from a chair, the sit-to-stand movement pattern varies depending on environmental parameters, such as chair height and foot position. However, if the seating position is constrained, a model of a specific sit-to-stand trajectory can be optimized (Pandy et al., 1995; Kuzelicki et al., 005. Since mass distribution of the body and position of the body s center of mass relative to the base of support are essential factors for maintaining balance while standing, it has been assumed that the nervous system generates forces to control motion of the center of mass (Shumway-Cook & Woollacott, 006. Therefore, we hypothesized that motion of the center of mass is optimized in a sit-to-stand movement. On the other hand, it has been suggested that arm movements, such as reaching movements, are optimized to minimize a cost, such as jerk (Flash & Hogan, 1985, torque change (Uno et al., 1989, or variance of the final position (Harris & Wolpert, The minimum torque-change model takes into account the dynamics of the body, whereas the minimum jerk model does not. Then, what criteria suitably predict the trajectory of the sit-to-stand movement? The extent to which previously proposed models can explain the whole-body, antigravity movement is still unclear. Hiroshi R. Yamasaki (Corresponding Author is with the Department of Physical Therapy, Showa University, Yokohama, Japan. Hiroyuki Kambara and Yasuharu Koike are with the Precision & Intelligence Laboratory, Tokyo Institute of Technology, Yokohama, Japan. The primary aim of the current study is to examine whether the minimum jerk and minimum torque-change models can predict trajectory, especially trajectory of the center of mass, during the sit-to-stand movement. Data Acquisition Methods Five university students, aged 0.4 ± 0.54 years (mean ± SD, without neurological or musculoskeletal disorders, volunteered to participate in this study. All participants gave informed consent. All procedures were approved by the ethics committee of Showa University. They were asked to perform three patterns of sit-to-stand movement with their arms crossed over their chests: upright, natural, and leaning forward from a given seated position. The initial posture was set at 0 dorsiflexion of the ankle, 90 flexion of the knee, and 70 flexion of the hip (Figure 1. For the natural pattern, we instructed participants to stand up at natural speed. For the leaning-forward pattern, we told participants to stand by leaning the trunk forward deeply at the beginning of the movement. For the upright pattern, we told participants to stand without leaning the trunk forward. Participants repeated each of the three patterns 15 times. The measurement was started by recording the natural sit-to-stand movement pattern for each participant. The order of recording measurements for the upright and leaning-forward patterns was randomly assigned for each participant. We confirmed that participants performed the different sit-to-stand movements according to instructions provided during the experiment. Position data of markers placed on the 306

2 The Sit-to-Stand Movement 307 term. Mass and inertia of the segments were estimated from the weight and height of each participant (Winter, 004. We estimated the position of the body s center of mass using the rigid body model (Figure 1. The model assumed that the ankle joint was fixed on the ground. Measured jerk cost of movement at the center of mass was obtained by integrating the Cartesian jerk for the center of mass over the rising time. The normalized jerk cost (NJC was computed as follows: NJC = Jc / (D /T 5 ( where Jc denotes the measured jerk cost, D denotes the distance between the initial and final position of the center of mass, and T indicates movement time (Nagasaki, The torque-change cost (TC was obtained by integrating the time derivatives of the torques of three joints. The normalized torque-change cost (NTC was obtained as follows: NTC = TCi / (dτi / T (3 Figure 1 Definition of joint angles in the sit-to-stand movement. θ 1 : dorsiflexion of the ankle, θ : flexion of the knee, θ 3 : flexion of the hip. A positive value indicates the angle in the counter-clockwise direction. Dotted line: initial posture. Thin line: final posture. ankle, knee, hip, and shoulder joints were recorded using a 3-D motion analysis device (Vicon MX at a sampling frequency of 00 Hz and low-pass filtered at a cutoff frequency of 3 Hz to yield the angular data. The vertical force of the seat was measured using a force plate (Kistler 987B at a sampling frequency of 1 khz to monitor the instant of seat-off. Data Analysis Movement Time. Movement time was determined as the time during which the linear velocity of the center of mass exceeded 5% of its peak. Rising time was defined as the time between seat-off (i.e., the moment when the vertical force of the seat decreased to zero and termination of movement. Calculation of Joint Torque, Center of Mass, and Measured Costs. We used the dynamic equation for a three-linked rigid body model composed of the shin, thigh, and trunk with the head and arms (HAT to calculate the joint torque τ: (, = R( + H(, + G( (1 where R indicates the inertial term, H includes the Coriolis and centripetal terms, and G denotes the gravitational where TCi and dτi indicate the torque-change cost and magnitude of absolute change in torque from seat-off to the termination of movement at the i-th joint, respectively, and T is movement time. Linearity of the Center of Mass Trajectory. Linearity of the center of mass trajectory during the rising time in Cartesian coordinates was quantified by the following equation: Linearity = d / L (4 where L is the length of a line connecting the positions of the center of mass at the instant of seat-off and at the termination of movement, and d is the shortest distance from the instantaneous position of the center of mass to the line. The trajectory convex upward was defined as positive. Maximum and minimum values of linearity during movement were obtained for each trial. Variability of the Center of Mass Position. Spatial variability of the center of mass during movement was evaluated by the following equation: Variability= 1 n 1 n ( xtj xt ytj yt n + ( n (5 j= 1 j= 1 where n is the total number of trials (n = 15 in each of the three patterns of movement, x tj and y tj are the horizontal and vertical positions of the center of mass at time t in the j-th trial. In addition, x t and y t are the averaged horizontal and vertical positions of the center of mass at time t, respectively. Since movement times were different from trial to trial, movement time was normalized to 1, and values of variability were calculated for 100 points that equally divide the normalized movement time.

3 308 Yamasaki, Kambara, and Koike Statistical Analyses. To examine the relationship between initial, maximum, and final variabilities of the center of mass, we computed Pearson s correlation coefficients for the ensemble data of the combined three patterns of sit-to-stand movement. One-way repeatedmeasures ANOVA and the post hoc Tukey test were used to examine effects of the movement pattern on the normalized jerk cost and normalized torque-change cost using ensemble data from all participants. Statistics software (SPSS 15.0 was used to perform the analyses. Simulation. For each participant and movement pattern, the average center of mass position, velocity, and acceleration data at seat-off and termination of movement were given as the initial and final boundary conditions for computing the minimum Cartesian jerk and minimum torque-change trajectories. Minimum Jerk Trajectory of the Center of Mass. The theoretical minimum jerk trajectory of the center of mass was obtained by minimizing the following cost function: tf C j = 1 ( x + y dt (6 t= t0 where x and y denote the Cartesian coordinates of the center of mass position. Applying variational calculus methods, we can obtain an optimal trajectory of the center of mass analytically as, xt ( = x + ( x x ( 10k 15k + 6k 0 f 0 + Tv ( k 10k + 15k 6k x 0 + Tv ( v ( 4k + 7k 3k xf x 0 T + a 0 k 3 k + k 4 x ( T + ( a xf a x 0( k k + k yt ( = y + ( y y ( 10k 15k + 6k 0 f 0 + Tv ( k 10k + 15k 6k y0 + Tv ( v ( 4k + 7k 3k yf y0 T + a 0 k 3 k + k 4 y ( T + ( a yf a y0( k k + k (7 (8 where x 0 and y 0 are the position, v x0 and v y0 are the velocity, and a x0 and a y0 are the acceleration of the center of mass at seat-off in the horizontal and vertical directions, respectively, and x f, y f, v xf, v yf, a xf, and a yf are those at the termination of movement. The term k denotes time sequence normalized with the rising time T = t f t 0, that is, the time from seat-off to termination of movement. It should be noted that the minimum jerk model cannot predict the angle and torques at the joints, whereas the minimum torque-change model can predict the angular data as described below. Minimum Torque-Change Trajectory. The theoretical objective cost function C T was defined as follows: 3 1 tf dt i C (9 T = dt t= t0 i 1 dt = where i is the torque at the i-th joint out of three joints. A three-link rigid body model with link parameters corresponding to the anthropometric size of each participant was adopted to calculate the minimum torquechange trajectory. Since the viscous coefficients of the lower extremities during movement were unknown, we assumed the link had zero viscous parameters in this study. Using dynamic programming and variational calculus methods, the optimization problem results in a two-point boundary value problem of the set of nonlinear ordinary differential equations (Uno et al., 1989: dx / dt = f( Xu, = [ T, T T, ] T d / dt = ( f / X u = T = z (10 where X is the state vector containing the angle, angular velocity, and torque, and z denotes the n-dimensional part corresponding to z in the 3n-dimensional Lagrangemultiplier vector. The equations were solved using the bvp6c function embedded in MATLAB to yield the optimal angular and torque trajectories. Quantities for the Model Evaluation Trajectories predicted by the minimum jerk and minimum torque-change models were evaluated using kinematic parameters of the center of mass, namely, the linearity index, peak velocity, and time to peak velocity. Results The average movement times of the five participants for the upright, natural, and leaning-forward patterns were 1.07 ± 0.03 s, 1.07 ± 0.0 s, and 1.1 ± 0.0 s, respectively. On average, the movement time for the leaningforward pattern was longer than for the upright and natural patterns, which had comparable movement times. Figure shows the temporal trajectories of joint angle, angular velocity, torque, and torque change for a typical participant performing the upright pattern. Trajectories from all 15 trials made by the participant are superimposed with the prediction of the minimum torque-change model. As mentioned earlier, the minimum jerk model cannot predict joint kinematics and dynamics. The minimum torque-change model was also successful in reproducing the measured joint kinematics in the natural and leaning-forward patterns (data not shown.

4 The Sit-to-Stand Movement 309 Average angles, angular velocities, and torques at seat-off and termination of movement across the five participants are shown in Table 1. The flexion angle of the hip joint at seat-off was greatest in the leaning-forward pattern and least in the upright pattern of sit-to-stand movement. The extension angular velocity of the hip at seat-off was largest in the upright pattern and smallest in the leaning-forward pattern. Comparing the average torque magnitudes at seatoff revealed that extension torques at the ankle and knee joints in the upright pattern were greater than those in the natural and leaning-forward patterns, whereas the extension torque of the hip joint was greatest in the leaning-forward pattern. These results demonstrate that extension torques in the lower joints lifting the body are related to the hip flexion angle at seat-off. Figure Profiles of angular and torque data from a study participant performing the upright sit-to-stand movement (UP. Dotted lines: experimental data. Thick line: prediction of the minimum torque-change model. Table 1 Average values at seat-off and termination of the sit-to-stand movement (n = 5 Seat-Off Termination Ankle Knee Hip Ankle Knee Hip Angle UP 1.9 ( ( ( ( ( (0.105 (rad NT 1.33 ( ( ( ( ( (0.090 LF 1.33 ( ( ( ( ( (0.104 Angular UP 0.54 ( ( ( ( ( (0.074 velocity NT 0.45 ( ( ( ( ( (0.11 (rad/s LF 0.45 ( ( ( ( ( (0.171 Torque UP ( ( ( ( ( (6.16 (N m NT ( ( ( ( ( (.44 LF 0.35 ( ( ( ( ( (3.61 Note. Mean (SD, UP: upright, NT: natural, and LF: leaning forward.

5 310 Yamasaki, Kambara, and Koike All participants stood steadily at the termination of movement. There were no significant differences in the final posture between the three patterns of movement except for the angular velocities of the hip joint. Figure 3 shows the spatial variability in the center of mass at the initial position, the maximum variability, and the variability at the final position for all participants. Initial and final variabilities (less than 5 mm were smaller than the maximum variability (more than 30 mm. Initial variability in the natural pattern tended to be slightly greater than that of the upright and leaningforward patterns for three participants. Final variability was greater in the natural pattern than in the upright and leaning-forward patterns in all but one participant. There was a weak correlation between initial and final variabilities (r =.57, p =.0, but no consistent relationship between the initial and maximum variabilities (r =.14, p =.60, or between the maximum and final variabilities (r =.30, p =.6. The average maximum and minimum linearity, and their timing across participants are summarized in Table, along with the linearity and timings predicted by the models. Timing values indicate the percent of time from seat-off to termination of movement. The center of mass had a convex downward trajectory, and the timing of occurrence of the maximum and minimum was delayed in the leaning-forward pattern compared with the upright and natural patterns. These trends in linearity and timing for the different movement patterns were predicted by both the minimum jerk and minimum torque-change models. Figure 3 Initial, maximum, and final variabilities of the center of mass for all participants. UP: upright, NT: natural, LF: leaning forward. Table Maximum and minimum linearity and their timing Linearity Timing (% Measured MTC MJ Measured MTC MJ Max UP ( ( ( ( ( (9.4 NT ( ( ( ( ( (9.8 LF ( ( ( ( ( (13.8 Min UP ( ( ( ( ( (6.7 NT ( ( ( ( ( (6.3 LF ( ( ( ( ( (6.7 Note. Mean (SD. MTC: minimum torque-change prediction, MJ: minimum jerk prediction, UP: upright, NT: natural, and LF: leaning forward.

6 The Sit-to-Stand Movement 311 Examining the values in detail showed that minimum torque-change predictions were closer to the measured values than minimum jerk predictions. On average, the maximum linearity indexes predicted by the minimum jerk model were smaller than measured; notably, the prediction for the leaning-forward pattern (0.008 was about one-quarter of the measured linearity index (0.031, indicating that the minimum jerk model predicted a relatively straight trajectory of the center of mass during movement. The maximum linearity timings predicted by the minimum jerk model were delayed compared with the measured timings. In the natural pattern, for instance, the minimum jerk model predicted the occurrence at 61.6% of the rising time, whereas the measured timing and minimum torque-change prediction were 50.9% and 51.1% of the rising time, respectively. Minimum torque-change predictions were closer to the measured timings than minimum jerk predictions for all three patterns of sit-to-stand movement. Figure 4 shows the position, velocity, and acceleration profiles for the center of mass after seat-off for one participant performing the natural movement pattern. Trajectories of the 15 trials and predictions of the minimum jerk and minimum torque-change models are superimposed. Since horizontal peak velocity was observed before seat-off in most of the trials, we only analyzed the magnitude of vertical peak velocity and timing. We observed the minimum jerk prediction of vertical peak velocity to be smaller than the minimum torque-change prediction and measured data. Table 3 summarizes the average peak velocities and timings for all participants. On average, the minimum torque-change prediction of Figure 4 Trajectory of the center of mass from a study participant performing the natural sit-to-stand movement. Dotted lines: experimental data. Black thick line: prediction of the minimum torque-change model. Gray thick line: prediction of the minimum jerk model. Table 3 Average vertical peak velocity (PV and timing of peak velocity (TPV (n = 5 PV (mm/s TPV (% Measured MTC MJ Measured MTC MJ UP ( ( ( ( ( (3.6 NT 4.7 ( ( ( ( ( (4. LF 418. ( ( ( ( ( (9.5 Note. Mean (SD. MTC: minimum torque-change prediction, MJ: minimum jerk prediction, UP: upright, NT: natural, and LF: leaning forward.

7 31 Yamasaki, Kambara, and Koike peak velocity was comparable with the measured peak velocity. The timing of vertical peak velocity (TPV of the center of mass predicted by the minimum torquechange model was comparable with the measured TPV. In addition, both models predicted that the TPV in the leaning-forward pattern was delayed compared with the natural and upright patterns. In sum, predictions of the minimum torque-change model more closely resembled the measured features of the center of mass trajectories than the minimum jerk model. Figure 5 shows the normalized jerk cost (NJC and the normalized torque-change cost (NTC for the three movement patterns. Both NJC and NTC were significantly higher in the leaning-forward pattern than in the natural and upright patterns (F, = 3.03, p <.01 for NJC, and F, = 3.03, p <.01 for the NTC. There were no significant differences in either cost between the natural and upright patterns. Discussion This study examined whether the minimum jerk and minimum torque-change models can predict the trajectory of the center of mass during the sit-to-stand movement. Participants were able to perform different patterns of the sit-to-stand movement from a sitting posture. Trajectories after seat-off in all three patterns were well reproduced by the minimum torque-change model. This result suggests that the minimum torque-change model is applicable to whole body movements as well as arm movements. In this study, both the minimum jerk and minimum torque-change models were able to predict peak vertical velocities and peak velocity timing of the body s center of mass after seat-off. In addition, the time shift of TPV across different patterns was also predicted by both models. Our results indicate that predictions of the minimum torque-change model resembled the measured vertical peak velocity and timing for all three movement patterns (Table 3 as well as the joint kinematics. Differences in the trajectories predicted by the minimum jerk and minimum torque-change models suggest that the effect of joint dynamics during the rising movement that is, interaction torques and gravity are significant. Although the average initial and final boundary conditions were distinctive for each participant, the same conditions were applied to the simulation of both the minimum jerk and minimum torque-change trajectories. Therefore, differences found in the curvature and the timings of peak velocity of the optimal trajectories come from the difference in the optimization criterion itself, namely, the effect of dynamics among the segments. The results suggest that the trajectories during the sit-to-stand movement are the result of optimizations that account for dynamic properties of the body. Results of this study also indicate that there are criteria underlying selection of the natural sit-to-stand movement in young participants. The NJC and NTC for the leaning-forward pattern were greater than for the natural and upright patterns. Differences in movement time are related to the normalized costs. Considering that the upright pattern requires greater muscle torque at the knee and ankle joints to lift the body than the natural and leaning-forward patterns, the natural pattern may be selected as the less torque-demanding, smoother rising pattern among the possible repertoire of rising movements. The strategy of the rising movement would depend on factors such as the individual s muscle strength and standing balance, as well as seat height. Extension torque at the hip joint was greatest in the leaning-forward pattern. Horak and Nashner (1986 proposed the hip strategy, in which the hip joint is used dominantly, rather than the Figure 5 Average movement costs for all trials. NJC: normalized jerk cost, NTC: normalized torque-change cost. Horizontal bar denotes significant differences between the groups (p <.01. UP: upright, NT: natural, LF: leaning forward.

8 The Sit-to-Stand Movement 313 ankle joint, to recover balance during standing. It has been suggested that elderly persons or individuals with neurological disorders may select the hip strategy to maintain standing balance against external disturbances. It is possible that participants in this study might select the leaning-forward pattern when muscles in their lower extremities become fatigued. The effect of various seating conditions on the selection of rising strategy remains to be examined. It should be noted that our results regarding variability of the center of mass are not consistent with predictions of the minimum variance model (Harris & Wolpert, The model predicts augmented variability in the final position. Furthermore, the model may predict that the smallest variability would be in the natural pattern. Although the final position of the center of mass might be controlled within the boundary of the base of support, the variability-minimization principle may not be selected by the central nervous system to execute the natural sitto-stand movement. The limitation of the current study is that the simulation was based on a model in which the ankle was fixed on the ground. In fact, a large dorsiflexion torque was predicted in the simulation before seat-off. In reality, such torque would induce the unsticking of the toe from the ground. It remains to be explored whether the movement before seat-off can be predicted by the optimization models. Acknowledgments This work was partially supported by the Grant-in-Aid for Scientific Research on Priority Areas Emergence of Adaptive Motor Function through Interaction between Body, Brain and Environment and the Strategic Promotion Program for Brain Science Research from MEXT. References Flash, T., & Hogan, N. (1985. The coordination of arm movements: An experimentally confirmed mathematical model. The Journal of Neuroscience, 5, Harris, C.M., & Wolpert, D.M. (1998. Signal-dependent noise determines motor planning. Nature, 394, Horak, F.B., & Nashner, L.M. (1986. Central programming of postural movements: adaptation to altered supportsurface configurations. Journal of Neurophysiology, 55, Kuzelicki, J., Zefran, M., Burger, H., & Bajd, T. (005. Synthesis of standing-up trajectories using dynamic optimization. Gait & Posture, 1, Nagasaki, H. (1991. Asymmetric trajectory formation in cyclic forearm movements in man. Experimental Brain Research, 87, Pandy, M.G., Garner, B.A., & Anderson, F.C. (1995. Optimal control of non-ballistic muscular movements: A constraintbased performance criterion for rising from a chair. Journal of Biomechanical Engineering, 117, Shumway-Cook, A., & Woollacott, M.H. (006. Normal postural control. In Motor Control: Translating Research into Clinical Practice (3rd ed.. Philadelphia: Lippincott/ Williams & Wilkins. Uno, Y., Kawato, M., & Suzuki, R. (1989. Formation and control of optimal trajectories in human multijoint arm movements: Minimum-torque change model. Biological Cybernetics, 61, Winter, D.A. (004. Anthropometry. In Biomechanics and Motor Control of Human Movement (3rd ed.. New York: John Wiley & Sons.