Biol. Cybern. 91, 104–114 (2004) DOI 10.1007/s00422-004-0501-7 © Springer-Verlag 2004
Balance control during an arm raising movement in bipedal stance: which biomechanical factor is controlled? Myriam Ferry1 , Luc Martin1 , Nicolas Termoz1,2 , Julie Cˆot´e2,4 , Franc¸ois Prince2,3 1
Laboratoire Sport et Performance Motrice EA 597, U.F.R.A.P.S. Universit´e Joseph Fourier, BP 53, 38041 Grenoble cedex 09, France 2 ˆ Laboratoire de posture et locomotion, Centre de R´eadaptation Marie-Enfant, Hopital Sainte Justine, Montr´eal, QC, Canada, H1T 1C9 3 D´epartement de kin´esiologie/chirurgie de Montr´eal, Montr´eal, QC, Canada, H3C 3J7 4 Department of Kinesiology and Physical Education, McGill University, 475 Pine Avenue West, Montreal, QC, Canada, H2W 1S4 Received: 5 June 2003 / Accepted: 15 June 2004 / Published online: 28 August 2004
Abstract. In order to obtain new insight into the control of balance during arm raising movements in bipedal stance, we performed a biomechanical analysis of kinematics and dynamical aspects of arm raising movements by combining experimental work, large-scale models of the body, and techniques simulating human behavior. A comparison between experimental and simulated joint kinematics showed that the minimum torque change model yielded realistic trajectories. We then performed an analysis based on computer simulations. Since keeping the center of pressure (CoP) and the projection of the center of mass (CoM) inside the support area is essential for equilibrium, we modeled an arm raising movement where displacement of one or the other variable is limited. For this optimization model, the effects of adding equilibrium constraints on movement trajectories were investigated. The results show that: (a) the choice of the regulated variable influences the strategy adopted by the system and (b) the system was not able to regulate the CoM for very fast movements without compromising its balance. Consequently, we suggest that the system is able to maintain balance while raising the arm by only controlling the CoP. This may be done mainly by using hip mechanisms and controlling net ankle torque. Keywords: Coordination – Dynamic optimization – Minimum torque change criterion – Center of mass – Center of pressure
1 Introduction Disturbances of postural equilibrium can result either from unexpected external forces or from one’s own voluntary movement. Specifically, when raising the arm, the erect posture is affected both by reactive forces and changes in body joint configurations. Consequently, a classical approach to studying the coordination between posture and movement is to analyze equilibrium processes Correspondence to: M. Ferry (e-mail:
[email protected], Tel.: +33-4-76635088, Fax: +33-4-76514469)
related to arm movements (Pozzo et al. 2001; Hodges et al. 1999; Patla et al. 2002). The state of equilibrium can be reached in diverse ways since a dynamical system has redundant degrees of freedom (DOF). For example, when a subject is asked to reach a stationary target, there is an infinite number of possible trajectories that his limb can follow to successfully perform the task. Thus, maintenance of dynamic equilibrium requires appropriate control strategies in order to master various DOFs and to achieve whole-body stabilization. Given the high position of its center of mass (CoM) relative to its base of support (BoS), the body has a high potential energy, which leads to the prioritization of equilibrium control during almost all motor tasks, including quiet standing. A large number of studies on voluntary movement from a standing posture (Patla et al. 2002; Massion 1992; Mouchnino et al. 1996) have analyzed the coordination between movement and posture only from the perspective of the CoM position and displacement. However, in addition to the control of the whole-body CoM, the coordination between movement and posture can be investigated by analyzing the shifts in center of pressure (CoP) associated with arm raising (Hay and Redon 2001). With computational resources available today, largescale models of the body and techniques simulating human behavior (Flash and Hogan 1985; Ramos and Stark 1990; Chang et al. 2001) can provide data to support hypotheses on how neuromuscular and musculoskeletal systems interact to produce movements. Then, the problem of indeterminacy can be solved by defining an objective function and using dynamic optimization (Chang et al. 2001; Klein Breteler et al. 2001); in fact, one way to select a unique trajectory is to introduce additional constraints on the task characteristics (for example, defining the initial and final positions of the end effector), thereby further constraining the body’s effective DOFs. Until now, two main cost functions have been used: minimum jerk (Flash 1987) and minimum torque change (Uno et al. 1989). Some studies (Klein Breteler et al. 2001; Dornay et al. 1996) have shown that the minimum torque change trajectories are more similar to the observed trajectories than the trajectories simulated with the minimum jerk model.
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Using an optimization approach to generate the predicted arm raising motion patterns can lead to solutions by adding constraint functions. In an optimization study, Chang et al. (2001) imposed a constraint on the maintenance of balance while lifting a weight. However, they considered only the anterior–posterior (A/P) position of the CoM. Other studies (Alexandrov et al. 2001a,b) have considered the CoM as well as the CoP but imposed values that were kept constant during the whole-trunk bending task to model “ideal” movements. Thus, no study has considered the A/P displacements of the CoP as an equilibrium constraint. In the present study, we propose a biomechanical analysis of kinematics as well as dynamical aspects of an arm raising movement to better understand the mechanisms of coordination between movement and posture. The first step of this study is to formulate the optimization problem to obtain the best fit to the experimental trajectories. We will therefore discuss the relevance of the minimum torque change model as an adequate tool to represent movement trajectories by comparing the characteristics of motion patterns of actual (experimental) and predicted (generated by an optimization program) data. Specifically, for a given task with determined initial and final joint angles, we are interested in finding joint motion trajectories that stay within the boundaries of the predetermined constraints. In a second step, we propose an analysis based on computer simulations. Since keeping the CoP and the projection of the CoM inside the BoS is essential for equilibrium, we model an arm raising movement where the displacement of either of these two variables is limited. Subsequently, for this trajectory-formation model, the effects of adding equilibrium constraints on movement trajectories will be investigated. The final purpose of this study is to formulate a new approach to the equilibrium constraint control using an optimization procedure. Solving this optimization problem can help to determine which strategy of equilibrium control a system uses during an arm raising movement: does it prefer to regulate a dynamical parameter (CoP) or a position parameter (CoM)? 2 Methods 2.1 Experimental design Experimental data were collected from six subjects (two males and four females; age: 21–25 years, M = 23.30, SD = 1.69). Their mean height was 1.69 m (SD = 0.051) and their mean weight was 62.5 kg (SD = 11.3). Subjects gave their informed consent before participation. Each subject voluntarily perturbed his/her own balance by rapidly raising the fully extended arms from a resting position (along the body) to shoulder level in the forward direction. This task was chosen because it is the one most often studied to investigate the coordination between movement and posture (Pozzo et al. 2001; Patla et al. 2002; Hay and Redon 2001). A diode was placed 1 m ahead of the subject and served as a target to reach. Participants were instructed to perform fluent movements, not move their feet, and maintain full elbow and knee extensions
during movement. All movements were self-initiated and assumed to be left–right symmetrical for modeling purposes. Each participant performed five rapid arm movements. Four infrared-light-emitting diode markers (sampling frequency 200 Hz) were placed on the subjects’ left side at the following anatomical landmarks: lateral maleolus (ankle joint), great trochanter (hip joint), acromion (shoulder joint), and styloid process (wrist joint). Their coordinates were recorded using an opto-electronic camera (Selspot) placed 3 m from the subjects’ sagittal axis. A/P and vertical ground reaction forces and torques were obtained from an AMTI force platform (Advanced Mechanical Technology Inc.) with a sampling frequency of 500 Hz. Force plate measurement and displacement data of each marker were filtered using a second-order (zero-lag) Butterworth filter with a cutoff frequency of 6 Hz. Intersegmental angles were calculated at ankle (θ1 ), hip (θ2 ), and shoulder (θ3 ) joints according to the convention in Fig. 1. 2.2 Multibody model The general configuration of the 2D model used is presented in Fig. 1. The model had n + 1 segments labeled from 0 (feet) to n (topmost segment, the arm), linked by n joints, where in the present configuration of the model n = 3. The feet remained at rest, in static contact with the force plate. Each segment used in the model was represented by its mass m, length l, and distance r between the segment’s distal end and its CoM. Many documented sources of error affect variables calculated with the inverse dynamics method, in particular anthropometric error. The “normality” represents ±2 standard deviations compared to documented anthropometric parameters. For this reason, we optimized anthropometric values computing specific coefficients for each subject (Vaughan et al. 1982). We considered that the margin of error compared to normality was 20%. Thus, minimizing deviations between CoP values calculated with optimized anthropometric parameters and experimental CoP values, we calculated optimized anthropometric coefficients for each subject. In the optimization process of anthropometric parameters, we sought to compute the following coefficients: m1 /weight, r1 /l1 , I1 , m2 /weight, r2 /l2 , I2 , m3 /weight, r3 /l3 , I3 , m0 /weight, and r0 /0.2. Some anthropometric parameters were not optimized: l1 , l2 , and l3 . The values of these coefficients are presented in Appendix A1 . Each segment’s mass and CoM location were calculated using optimized anthropometric parameters. Simulated joint torques were calculated using the inverse dynamics method and Lagrangian formalism. Note that the configuration of this inverse model is similar to that of the minimum torque change model used in this study, with the same parameters. The torque vector Ti was expressed by the following equation: Ti = [A(θ )]θ¨i + [C(θ )]θ˙i2 + [B(θ )]θ˙i θ˙j + [Q(θ )] ,
(1)
T where Ti is the moment applied at T 1 , . . . , θn is joint i, θ=θ the angular position vector, θ¨i = θ¨1 , . . . , θ¨n is the angular
106
O3
where M is the whole-body mass, mi is the mass of segment i, and XCoMi is the A/P CoM coordinate of the ith segment. The displacement of the CoP in the A/P direction, XCoP , is defined by ankle joint torque T1 and support forces R:
θ3
T1 + CoP ∧ R = 0.
O4 θ2
Then: XCoP =
li
(4)
2.3 Minimum torque change model
Gi
2.3.1 Cost function. Based on the idea that movement optimization must be related to movement dynamics (Dornay et al. 1996), as the performance criterion (objective function) CT we minimized the sum of squares of the rate of torque integrated over the duration of the entire movement:
ri RI
θ1 O1 Go →
y
1 CT = 2
l0
X x→
r0 Fig. 1. Configuration of the 2D body segment model consisting of (n + 1) segments linked by n joints. For each segment i ≥ 0 : li is the length, Gi the center of mass, ri the distance between Gi , and joint Oi r0 the horizontal distance between G0 and O1 , and Ii, the moment of inertia with respect to the axis (Gi , z). For i ≥ 1, θi gives the orientation of segment i − 1 (with respect to the vertical direction for segment i = 1) whereas for segment i = 2, 3 the orientation of segment i − 1 is relative to the joint angle)
acceleration vector, θ˙i2 and θ˙i θ˙j are velocity vectors such T T that θ˙i2 = θ12 ,. . . ,θn2 and θ˙i θ˙j = θ˙1 θ˙2 , θ˙1 θ˙3 , . . . , θ˙n−1 θ˙n and where A is the inertial matrix, B is the Coriolis matrix, C is the centrifugal matrix and Q is the gravity force vector. The torque vector is detailed in Appendix A2 . In these calculations, the CoM is a weighted average of the CoMs of each ith segment. Consequently, the A/P CoM position XCoM is given by: n 1 mi × XCoMi , M i=1
T1 − lo Rx + m0 r0 g , Ry
where T1 is the torque actuated at the ankle joint, l0 , m0 , and r0 are respectively the height, the mass of the foot, and the distance between G0 and O1 (Fig. 1). ¨ CoM and Ry = M(g + Y ¨ CoM ) are related to the Rx = M X rate of change of the horizontal and vertical linear moments of the whole system (Cahou¨et et al. 2002), g is the ¨ CoM gravity acceleration, M is the whole-body mass, and X ¨ CoM are respectively A/P and vertical acceleration of and Y the CoM.
O2
XCoM =
(3)
(2)
tf n t=0 i=1
dT 2i dt, dt
(5)
where t is the movement duration of an arm raising task starting at time zero t = 0 and ending at time tf , n is the number of the joint angle (n = 3), and Ti is the torque at the ith joint angle. 2.3.2 Constraints allowing comparison between simulated and experimental data. First, to compare experimental and simulated trajectories, constraints are required in the optimization search; to produce appropriate results, we specified initial and final experimental angle values (6–8) in order to allow the model to adopt the same initial and final posture. Additional constraints were imposed as follows. We considered that the initial (at t0 ) and final (at tf ) velocity and acceleration at each joint were set at zero (9). Then, the optimization problem can be mathematically formulated as follows: Find θ1 , θ2 , θ3
that minimize (5) θ1 (t0 ) and θ1 (tf ) θ2 (t0 ) and θ2 (tf ) θ3 (t0 ) and θ3 (tf ) subject to θ ˙i (t0 ) = θ¨i (t0 ) = θ˙i (tf ) = θ¨i (tf ) = 0; 0; 0T
(6) (7) (8) (9)
Experimental angles values
107
where θ1 , θ2 , θ3 are respectively ankle, hip, and shoulder joint angles: the initial and final angles are constrained to the observed. These values are presented in Appendix B for each subject. 2.3.3 Constraints used in the simulation process. Second, to investigate the effects of adding equilibrium constraints, two conditions were tested. “CoM condition” represents the condition in which the equilibrium constraint was related to the A/P displacements of the CoM, and “CoP condition” represents the condition in which the equilibrium constraint was related to the A/P displacements of the CoP. We then compared movements simulated over a wide range of speeds. Movement durations that were tested were 1, 0.8, 0.5, 0.35 s. The measures used to characterize postural stability were joint kinematics (θi ), joint kinetics (Ti ), and A/P displacements of the CoM and the CoP. We considered first that the initial (at t = 0) and final (at tf ) velocity and acceleration at each joint were set at zero (9). Moreover, other constraints in the joint-angle domain were taken into account to produce realistic results respecting the physiological joint limits of the subject. We determined each joint’s lower and upper angle values (12) during the performance of the movement. Therefore, given the initial and final end-effector positions (10, 11), instead of experimental angle values, we needed to find a continuous motion trajectory that would lead a particularly, articulated figure from the initial posture to the target posture without violating the subject’s balance. These conditions can be expressed by constraints in the optimization search. Another constraint imposed in the optimization search is the maintenance of balance during arm raising. To keep the subject’s CoM projection XCoM (t) or his center of pressure XCoP (t) within the BoS, we limited the A/P motion of these two variables (13). These constraints are imposed in order to prevent toe- or heel-off in the model. The minimization problem was expressed by: Find θ1 , θ2 , θ3 that minimize (5) ˙ (t ) = θ¨i (t0 ) = θ˙i (tf ) = θ¨i (tf ) = 0; 0; 0T θ i 0 cos(θ ) + l2 cos(θ1 + θ2 ) l 1 1 −l3 cos(θ1 + θ2 + θ3 ) = Xw subject to l1 sin(θ1 ) + l2 sin(θ1 + θ2 ) −l3 sin(θ1 + θ2 + θ3 ) = Yw θ < θi < θu l Xa < XEQ (t) < Xb
(9) (10) (11) (12) (13)
where θl and θu are respectively the lower and upper bounds; these values are presented in Appendix C; XEQ is the A/P displacements of the equilibrium variable (CoM or CoP) and Xa (0.10 m) and Xb (−0.01 m) are respectively the boundary positions allowed for the CoM or the CoP in backward and forward directions with respect to the ankle joint; li is the length of the ith segment, and Xw and Yw are respectively the A/P and vertical positions of the end effector. The A/P and vertical initial (Xwi = 0.27 m and Ywi = 0.76 m) and final (Xwf = 0.77 m and Ywf = 1.56 m) positions of the end effector were similar across both velocity conditions.
2.3.4 Time discretization and angular trajectory model. Discretizing the time (ti ) of angular trajectories, the infinite dimension of the minimization problem (5) was converted into a finite one:
k dT32 dT12 dT22 (ti ) + (ti ) + (ti ) t , (14) Ck = dt dt dt i=1
where k is the number of discrete time intervals ti and t Tc . is the time increment such that t = (k−1) Defining a state vector q at each discrete time ti , such that qT = (θ1T , θ2T , θ3T ), the problem was to find q such that (14) was minimal, with constraints (6–9), for the comparison between experimental and simulated data, and (9–13) for the simulation process, respected at each discrete time ti . To solve this optimization problem, joint angular velocities and accelerations have been traditionally expressed as a function of state vector q using a finite difference method (Chang et al. 2001). This method, however convenient, led to unrealistic predictions, since it did not ensure that angular kinematics were continuously differentiable functions. One alternative was to use polynomial approximations (Wada et al. 2001) for angular displacements. Considering that angular displacements were time functions, we chose a sixth-order polynomial since it was the order that gave us the best fit to experimental trajectories. Since the relationship between joint torques, joint angles, and their time derivatives is nonlinear, the optimization problem was reformulated: θi (t) =
6
xj t j ,
(15)
j =0
where θi (t) is the postural angle at time t, xj is the coefficient of the sixth-order polynomial (j = 0, 1, . . . , 6). Then, using (15), the objective function was expressed in terms of a function of xj : ... CT(t, θi , θ˙i , θ¨i θ i ) = Ck(t, xj ) .
(16)
The problem consisted of determining solutions x that could minimize the nonlinear objective function, starting at an initial estimate x0 , and bounded by constraints (6–9), for the comparison between experimental and simulated data, and by constraints (9–13) for the simulation process. The constraint optimization problem was solved with a sequential quadratic programming (SQP) method (Boggs and Tolle 1996) using Matlab (Math Works Natick, MA, USA). 2.4 Data analysis The five arm movements performed by each participant were averaged in order to have a mean trial for each subject. Then we analyzed a mean experimental joint angle trace and mean A/P displacements of the CoM and CoP for each participant. To compare minimum torque change trajectories with the experimental ones, the two sets of
108
joint angles and CoM and CoP trajectories were normalized relative to their time duration. Deviations between simulated and experimental trajectories were quantified using the root mean square error (RMS): tf (s(t) − r(t))2 dt, (17) RMS = 1/tf 0
where s(t) represents the experimental curve and r(t) the simulated one. Equation (17) is applied to the joint angles and CoM and CoP trajectories. The RMS was calculated for the six subjects and by means of a Mann and Whitney test (MW) difference between the RMS value and 0 for all participants (p < 0.05). For each subject, the linear relationship between these two data sets was quantified using the coefficient of determination r 2 . We also used a statistical test to infer if the correlation coefficient was significant (tr).
CoP position. During simulated arm raising, the CoP was displaced in a similar forward direction (Fig. 3). Indeed greater differences were found in the A/P displacements of the CoP between the two conditions (RMS = 0.018 and r 2 = 0.65). To prove the relevance of the minimum torque change criterion, we also compared experimental CoP trajectory with CoP trajectory calculated with inverse dynamics method (RMS = 0.029 and r 2 = 0.063). The RMS and r 2 values showed that, despite the differences between experimental and simulated CoP values, the difference between experimental and inverse dynamics data justified the sensibility of the CoP to kinematic and anthropometric errors. The comparison between anthropometric parameters extracted from the tables and those optimized showed that, for all subjects, r1 / l1 , rp /0.2, and m1 /weight became greater when they were optimized; r2 / l2 , m2 /weight, I2 , r3 / l3 , m3 /weight, I3 were less important when they were optimized; and only I1 showed differExperimental
3 Results Ankle angle (Rad)
1,51 1,5 1,49 1,48 1,47
(b)
0,15 Hip angle (Rad)
We randomly extracted one mean sample arm raising data from one subject from the experimental database (Fig. 2). The computer optimization program generated the predicted arm raising motion pattern for this sample. At first glance, the minimum torque change model was able to reproduce the spatial characteristics of the measured trajectories. Qualitative and quantitative similarities and differences between simulated and experimental joint trajectories stand out (Fig. 2). The stereotypical pattern for each joint was the following: a plantar flexion at the ankle joint (θ1 ) (Fig. 2a), a flexion at the hip joint (θ2 ) (Fig. 2b), and an extension at the shoulder joint (θ3 ) (Fig. 2c). To compare those trajectories in detail, we computed the mean RMS value and their standard deviations (SD) for θ1 , θ2 , θ3 , and CoM and CoP A/P motion. The RMS and r 2 values for all subjects and mean RMS and r 2 values and their SDs are presented in Table 1. There was no significant deviation across subjects between simulated and mean observed trajectories for the three joint angles (MW test, p > 0.05). The RMS values between experimental and observed trajectories were not significantly different from zero. There was a significant correlation between the overall simulated and experimental data sets (tr; p < 0.05). A quantitative comparison between the simulated and observed postural angle trajectories indicated that the minimum torque change model showed that the angles reached larger values earlier during the movement compared to experimental data. Arm raising produced a forward CoM displacement (Fig. 3). More interestingly, the postural perturbation induced in the experimental condition a CoM shift quite similar to that of the simulation [quantitatively (RMS = 0.002 and r 2 =0.98)]. The analysis of experimental CoP displacements showed one main detectable mechanical event, which was a large forward displacement, after which there was a small adjustment before reaching final
(a)
0,13 0,11 0,09 0,07 0,05
(c)
Shoulder angle (Rad)
3.1 Comparison between experimental and simulated data
Simulated
2 1,5 1 0,5 0
0.64
Time (s)
Fig. 2. We randomly extracted one mean sample arm raising data set from the experimental database. A comparison between the measured and simulated postural angle trajectories of each joint (a ankle, b hip, c shoulder). Each curve represents data for simulated minimum torque change trajectories and mean measured trajectories, for one subject. The movement duration for each condition is 0.64 s. We consider that the movement is performed at rapid speed
109 Table 1. Root mean square error (RMS) and coefficient of determination r 2 calculated for each subject, for each joint angle, and A/P shifts of the CoM and the CoP
Subject
θ1 θ2 θ3 CoM CoP (1)* CoP (2)*
RMS r2 RMS r2 RMS r2 RMS r2 RMS r2 RMS r2
1
2
3
4
5
6
0.049 0.850 0.079 0.990 0.130 0.970 0.002 0.995 0.014 0.759 0.035 0.165
0.026 0.730 0.024 0.730 0.180 0.940 0.002 0.974 0.022 0.854 0.021 0.000
0.032 0.560 0.027 0.670 0.200 0.950 0.001 0.990 0.015 0.495 0.029 0.002
0.042 0.790 0.006 0.980 0.180 0.950 0.001 0.993 0.005 0.941 0.024 0.003
0.036 0.920 0.010 0.940 0.150 0.960 0.001 0.972 0.036 0.451 0.034 0.141
0.003 0.760 0.019 0.880 0.170 0.950 0.002 0.977 0.015 0.433 0.032 0.067
Mean RMS (SD)
Mean r 2 (SD)
0.003 (0.000)
0.770 (0.110)
0.028 (0.023)
0.870 (0.120)
0.170 (0.022)
0.950 (0.008)
0.002 (0.000)
0.980 (0.009)
0.018 (0.009)
0.650 (0.203)
0.029 (0.005)
0.005 (0.067)
*CoP (1): comparison between experimental CoP and simulated CoP trajectories, *CoP (2): comparison between experimental CoP and “inverse dynamic CoP” trajectories. Area (minimum and maximum : ± 20 %) of the CoP values CoM experimental
CoPexperimental
CoMsimulated
CoPsimulated
CoP calculated by inverse dynamics methods 0,13
A/P positions (m)
0,12 0,11 0,1 0,09 0,08 0,07 0,06 0,05
0
Time (s)
0.64
Fig. 3. Comparison between the measured and simulated CoM and a comparison between experimental, simulated, and “inverse dynamic” CoP trajectories. An area is plotted with different values of anthropometric parameters: it corresponded to CoP values as a result of parameter changes of + and −20%
ent variations across subjects. Then, smaller anthropometric coefficients of the trunk and the arm and greater coefficients of the lower limb (except inertia) resulted in smaller effects of arm movements on the CoP. 3.2 “CoM condition” vs. “CoP condition”: influence of movement velocity For both conditions, the increase in movement velocity induced a slight increase in amplitude of hip and ankle dis-
placements (Fig. 4). However, for the movement duration of 0.35 s, the greater movement velocity produced smaller amplitudes of hip and ankle displacements only in the “CoP condition”. The shoulder joint was the least affected by equilibrium constraints and movement velocity. Therefore, for movement durations of 1 and 0.8 s, there were no qualitative and quantitative differences between data computed with the “CoM condition” and those computed with the “CoP condition”, although quantitative differences were observed at movement durations of 0.5 and 0.35 s. This was measured by comparing the mean postural angle values and amplitudes of joint displacements in the “CoM condition” with those in the “CoP condition” (Fig. 4). Figure 5 shows the effect of movement velocity on joint torques: the increase in movement velocity induced greater joint torque values, regardless of the equilibrium constraint. However, for a movement duration of 0.35 s, both equilibrium constraints induced different variations: in the “CoM condition”, the amplitude of ankle joint torque was larger (139 Nm) than in the “CoP condition” (75.1 Nm) (Fig. 5a), whereas in the “CoP condition”, the amplitude of hip joint torque variation (89.7 Nm) was greater than in the “CoM condition” (44.3 Nm) (Fig. 5b). Therefore, there was no difference in shoulder joint torque between the “CoM condition” and the “CoP condition”, at any movement duration (Fig. 5c). Figure 6 shows the opposite effect of velocity on CoP and CoM displacements in both constraint conditions. The amplitudes of CoM shifts decreased with increasing arm raising speed, although at very rapid speed they slightly increased in the “CoP condition” (+0.005 m) with respect to a movement performed within 0.5 s. The average amplitude of CoM displacements was 0.017 m for the “CoM condition” and 0.018 m for the “CoP condition” (Fig. 6a). Moreover, an increase in the movement velocity induced greater amplitudes of CoP displacements, which were enhanced by a factor of about 6.6 in the “CoM condition” and about 3.7 in the “CoP condition” when executed
110
“CoM condition”
“CoM condition”
“CoP condition
“CoP condition
(a)
Ank le joint tor que (N.m)
(a)
1,55
Ank le angle (Rad)
1,54 1,53 1,52
106 86 66 46
1,51
26
1,5
6 -14
1,49
Hip joint tor que (N.m)
(b)
(b)
0,3
Hip angle ( Rad)
126
0,24
27 12 -3 -18 -33 -48
0,18 -63
Shoulder joint tor que ( N.m)
(c)
0,12 (c)
1,6 Shoulder angle (Rad)
1,4 1,2 1
155 120 85 50 15 -20 -55 -90
0,8
-125
0,6 1
0,4
0,8
0,5
0,35
Movement time (s)
0,2 1
0,8
0,5
0,35
Movement time (s)
Fig. 4. Comparison between mean postural angle values and range of joint motions between the “CoM condition” and the “CoP condition”: a ankle, b hip, c shoulder; different movement durations are simulated: 1 s (slow movement), 0.8 s (“natural” movement), 0.5 s (rapid movement), and 0.35 s (very rapid movement)
at very rapid speed. Consequently, for a movement duration of 0.35 s, the amplitude of the CoP shift in the “CoM condition” was about 0.205 m and the CoP overcame the anterior limit of the BoS (Fig. 6b).
4 Discussion The first results of this study allowed us to examine the relevance of the minimum torque change criterion. This trajectory-formation model was selected because some studies (Klein Breteler et al. 2001; Dornay
Fig. 5. Comparison between mean joint torque values and range of joint torques between the “CoM condition” and the “CoP condition” (a ankle, b hip, c shoulder)
et al. 1996) state that the predictions of the minimum torque change model represent actual trajectories quite well in comparison with other trajectory-formation models. In fact, the spatial characteristics of observed angle trajectories are consistent with optimization models that claim that when subjects perform basic pointing movements, they try to minimize certain movement costs. However, some quantitative differences are observed between experimental and simulated CoM and CoP trajectories. A comparison between experimental and simulated CoM trajectories shows smaller differences because the CoM position is roughly a function of three angles. By comparison, there were greater differences when computing CoP trajectories because this parameter depends not only on angular positions but also on angular velocities and accelerations. CoP motion estimations are obtained by
111
“CoM condition”
“CoP condition
(a)
CoM displacements (m)
0,04
0,035
0,03
0,025
0,02
(b)
CoP displacements (m)
0,18
0,13
0,08
0,03
-0,02
1
0,8
0,5
0,35
Movement time (s)
Fig. 6. Comparison between mean CoP and CoM values and range of A/P CoP and CoM displacements between “CoM condition” and “CoP condition”
solving the inverse dynamics problem using numerical double-differentiation position measurements. Then, discrepancies observed between experimental and “inverse dynamics” CoP displacements are due to the greater sensitivity of the CoP estimation to inaccuracies in numerically determined accelerations (Cahou¨et et al. 2002). This inverse dynamics approach requires only kinematic measurements, but it may generate inaccurate CoP motion estimations because the accuracy of the solution strongly depends on accuracy of input data (Hatze 2000); simulated CoP motion obtained by solving optimization problems fits better with experimental CoP values than CoP motion estimations obtained by solving inverse dynamics methods. Moreover, adjusting some anthropometric parameters can help to improve the predictions. Smaller parameter values of trunk and arm and greater parameter values of lower limb allow to reduce the variations between the CoP position compared to the experimental one. The optimization of these last parameters is valid for each participant tested (both for male and female subjects). One can generalize the adjustment of these anthropometric parameters to the whole of the subjects, whatever their sex, size, and weight. We can conclude that anthropometric error can be corrected partly by adjusting some of the anthropometric parameters, then improving the predictions of the model.
These latter results showed the relevance of the minimum torque change criterion; since there are smaller differences between observed and simulated angle trajectories, larger discrepancies are observed on CoP displacements compared to those of the CoM. Despite few differences between modeled and observed motor responses, the data from this experiment are consistent with the general hypothesis of the model, i.e., that body kinematics are manipulated to control posture and raise the arms. However, the above discussion suggests that the optimization approach does not represent a general law of human movement control; indeed the obtained results are specific to the task imposed on the system. The models (e.g., minimum jerk, minimum torque change), which are based on the optimization method, may be valid only in a restricted domain (Okadome and Honda 1999). The results showed that the proposed method generated the “optimal” raising arm motion with reduced objective function values. There are still questions that need to be addressed. It appears that even for simple arm raising, it may be that more than one performance criterion is required to predict or explain this behavior. Differences in the joints and CoM and CoP kinematics between the experimental and the predicted pattern were observed in most trials. The addition of objective functions based on joint kinematics might provide a closer simulation of relative segment motion and CoP shifts. In addition, different criteria may need to be adopted at different periods of the arm raising task. The use of different or more “appropriate” performance criteria during different stages of the arm raising task is worthy of future study. The second section of this investigation is based on the analysis of computer simulations in which we have modeled the equilibrium constraints with the A/P displacements of either the CoM or the CoP. The experimental analysis of body kinematics and kinetics produces parameters that may illustrate some control strategies accounted for in the balance maintenance model. These strategies likely refer to the joint about which most of the movement occurs and correspond to the manner in which the system may restore balance. For movement durations of 1 and 0.8 s, similar motion patterns were observed between the “CoM condition” and the “CoP condition”: the influence of these two movement velocities on postural disturbance is identical. This similarity can be considered in the choice of the minimization criterion. Indeed, as we use the same cost function (14) and as equilibrium constraints are not active (13), it is not surprising that angular trajectories, joint torque profiles, and CoM and CoP trajectories are similar in these two velocity conditions. The system will predominantly use ankle mechanisms because, for one, the maximum value of net ankle torque (73.88 N m) is more important than that of the hip joint (12.95 N m) and also the amplitude of ankle joint displacements increases slightly with increased movement velocity, whereas the amplitude of hip joint displacements does not change. Thus, subjects may control their balance mainly by recruiting the ankle joint with a slight contribution of the hip (Winter et al. 1996; Gatev et al. 1999).
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Consistent with the kinematic pattern changes, net joint torque patterns also change at movement durations of 0.5 and 0.35 s. The hip and ankle joint displacement amplitudes associated with increasing movement velocities differ, and in particular the changes associated with movement duration of 0.35 s are not similar between the “CoM condition” and the “CoP condition”. Not only kinematic but also kinetic analysis suggests that the choice of the equilibrium constraint will influence the strategy adopted by the system, especially at 0.35 s movement time. In the “CoM condition”, the system develops an important net ankle torque (125.2 N m) so that the physiological limits of the subject are reached (Martin et al. 1993). If the equilibrium constraint is related to the A/P shift of the CoP, then the strategy adopted by the system is modified. The system produces smaller ankle joint torque (71.45 Nm) and greater hip joint torque (27 Nm) than in the “CoM condition” (125.6 and 16.44 Nm, respectively). These torque patterns are consistent with experimental results of Eng et al. (1992). In light of these findings, one can speculate that the system modifies its strategy using hip mechanisms to maintain balance during the performance of arm raising while modifying the ankle torque to control the CoP. In fact, studies suggest that the CoP displacements are directly influenced by neuromuscular activity acting at joint levels (Morasso and Shieppati 1999). Thus, maintaining dynamic equilibrium will result in the regulation of the A/P displacement of the CoP through the control of the net ankle torque with concomitant hip mechanisms. These results suggest that using an equilibrium constraint related to the A/P displacements of the CoM will not produce realistic results in terms of kinetics (joint torque) for movement duration of 0.5 and 0.35 s. For the “CoP condition”, in both velocity conditions, neither the CoM nor the CoP went out of the stability limits, but in the “CoM condition”, for a movement duration of 0.35 s, the CoP overcame the anterior edge of the BoS (0.20 m), thereby compromising its balance. However, the overcoming of the CoP over the anterior edge of the BoS, i.e., the biomechanical system falling forward, can be used as supplementary information, but these values might be different for reallife conditions because the model does not comprise knee and spine. Moreover, increasing velocity has opposite effects on the CoM and CoP displacements: as reported by Pozzo et al. (2002), the amplitudes of CoM displacements decrease with an increase in movement velocity whereas the amplitude of the CoP displacements increases. Consequently, previous results suggest that equilibrium constraints can only be related to the A/P displacement of the CoP in this optimization procedure. In this case, one postulates that the biomechanical variable regulated by the system will be a dynamical parameter. In fact, it is more plausible that the system controls the CoP instead of the CoM: changes in the CoP reflect the system’s response to movements involving the whole-body CoM; the CoP describes the forces that must be produced to return to a balance position. With changes in body position associated with arm raising, the distance between the CoM’s projec-
tion and the CoP increases, making the subject inherently less stable and necessitating muscular actions to return the CoM to a stable position within the support area. It follows that this active postural control will be achieved by controlling a variable other than the CoM and the system will preferably choose the CoP. Moreover, since pressure receptors are located under the feet, proprioceptive information and, consequently, the CoP position information can be directly available to the system. In comparison, it is more difficult for the system to have insight about CoM positions because, first, the CoM is a global, whole-body parameter, implying that obtaining information about this parameter may necessitate complex computations. Consequently, the idea that the body’s CoM position is the reference value (Mouchnino et al. 1996; Hay and Redon 2001) and that it is the variable controlled by the CNS is challenged. In this study, the combination of experimental and modeling work has provided new insight into the control of balance during voluntary movements such as arm raising in bipedal stance. In light of these results, we suggest that in such optimization procedures, the biomechanical control parameter that is used to model the system should be a dynamical parameter rather than a global position one. Further studies should investigate the influence of varying the duration of the movement and the BoS’s size on inferred postural control mechanisms.
Appendix A1 In the optimization process of anthropometric parameters, we sought to compute the following coefficients: m1 /weight m2 /weight m3 /weight m0 /weight
r1 / l1 r 2 / l2 r3 / l3 r0 /0.2
I1 I2 I3
where: some anthropometric parameters were not optimized: l1 , l2 and l3 ; the calculation of these coefficients for each subject revealed that some coefficients did not change across subjects; they are as follows: m1 /weight m2 /weight m3 /weight m0 /weight r1 / l1 r2 / l2
0.3516 0.5394 0.0800 0.0290 0.6960 0.4469
Other coefficients varied from one subject to another: I1 I2 I3
r0 /0.2 r3 / l 3
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Appendix A2
c11 = c22 = c33 = 0, c12 = −d1 sin θ2 − d2 sin θ2 + d3 sin(θ2 + θ3 )
The model of the human body consists of four linked segments (shanks, thigh, trunk, and arm). The segments are connected by one-degree-of-freedom joints. The model is bidimensional and can only be used for A/P movements of the body. The equations of joint torques acting on this system can be computed using the Lagrangian method. The equation can be written in compact form:
c13 = d3 sin(θ2 + θ3 ) + d4 sin θ3 , c23 = d4 sin θ3 .
Ti = [A(θ)]θ¨i + [C(θ)]θ˙i2 + [B(θ)]θ˙i θ˙j + [Q(θ)] ,
(1)
T where θ = θ1 , . . . , θn is the angular position vector, ¨θ = θ¨1 , . . . , θ¨n T is the angular acceleration vector, θ˙ 2 and T θ˙ θ˙ are velocity vectors such that θ˙ 2 = θ12 , . . . , θn2 and T θ˙ θ˙ = θ˙1 θ˙2 , θ˙1 θ˙3 , . . . , θ˙n−1 θ˙n and where A is the inertial matrix, B is the Coriolis matrix, C is the centrifugal matrix, and Q is the gravity forces vector. For the calculation of the matrices below, we used the following coefficients:
d1 d2 d3 d4
= m 2 r 2 l1 , = m 3 l1 l2 , = m 3 r 3 l1 , = m 3 r 3 l2 ,
where mi , li and ri are respectively the mass, length, and distance (between the segment’s distal end and its center of mass) of each segment i. The inertial matrix A consists of the following elements: a12 a13 a23 a11
a12 a13 a22 a23 a33
= a21 , = a31 , = a32 , = I1 + I2 + I3 + m1 r12 + m2 r22 + m2 l12 + m3 l22 + m3 l12 +m3 r32 + 2d1 cos θ2 + 2d2 cos θ2 − 2d3 cos(θ2 + θ3 ) −2d4 cos θ3 , = I2 + I3 + m2 r22 + m3 l22 + m3 r32 + d1 cos θ2 + d2 cos θ2 −d3 cos(θ2 + θ3 ) − 2d4 cos θ3 , = I3 + m3 r32 − d3 cos(θ2 + θ3 ) − d4 cos θ3 , = I2 + I3 + m2 r22 + m3 l22 + m3 r32 − 2d4 cos θ3 , = I3 + m3 r32 − d4 cos θ3 , = I3 + m3 r32 .
The Coriolis matrix B consists of the following elements: b223 b312 b112 b113 b123 b213 b313
= b213 , = −b213 , = 2(− sin θ2 (d1 + d2 ) + d3 sin(θ2 + θ3 )), = 2(d4 sin θ2 + d3 sin(θ2 + θ3 )), = 2(d4 sin θ3 − d3 sin(θ2 + θ3 )), = 2d4 sin θ3 , = b323 = b212 = 0.
The centrifugal matrix C consists of the following elements: c21 = −c12 , c31 = −c13 , c32 = −c23 ,
The gravity forces vector Q consists of the following elements: q1 = g(cos θ1 (m1 r1 + m2 l1 + m3 l1 ) + cos(θ1 + θ2 ) × (m2 r2 + m3 l2 ) − cos(θ1 + θ2 + θ3 )m3 r3 ), q2 = g(cos(θ1 + θ2 )(m2 r2 + m3 l2 ) − cos(θ1 + θ2 + θ3 )m3 r3 ), q3 = −g cos(θ1 + θ2 + θ3 )m3 r3 . Appendix B To compare experimental and simulated trajectories, and to produce appropriate results, we specified initial and final experimental angle values [constraints (6–8)] for each subject. These values are expressed in radians. Subject Subject Subject Subject Subject Subject 1 2 3 4 5 6 θ1 (t0 ) θ2 (t0 ) θ3 (t0 ) θ1 (tf ) θ2 (tf ) θ3 (tf )
1.503 1.526 0.222 0.0962 0.193 1.515
1.481 1.472 0.253 0.325 0.160 1.494
1.501 1.506 0.136 0.111 0.186 1.675
1.479 1.494 0.163 0.117 0.215 1.651
1.473 1.500 0.137 0.0846 0.238 1.543
1.523 1.536 0.189 0.134 0.200 1.531
Appendix C To produce realistic results respecting physiological limits of the subjects, each joint had lower (θl ) and upper (θu ) values [constraint (12)] during the performance of the movement: θ1 θ2 θ3
θl −θ1 + π/4 −θ2 − π/4 −θ3 − π/4
θu θ1 − 3π /4 θ2 + π /4 θ3 − 3π /4
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