A Computational Mechanism To Account For 
Averaged Modified Hand Trajectories 
Ealan A. Henis*and Tamar Flash 
Department of Applied Mathematics and Computer Science 
The Weizmann Institute of Science 
Rehovot 76100, Israel 
Abstract 
Using the double-step target displacement paradigm the mechanisms un- 
derlying arm trajectory modification were investigated. Using short (10- 
110 msec) inter-stimulus intervals the resulting hand motions were initially 
directed in between the first and second target locations. The kinematic 
features of the modified motions were accounted for by the superposition 
scheme, which involves the vectorial addition of two independent point-to- 
point motion units: one for moving the hand toward an internally specified 
location and a second one for moving between that location and the final 
target location. The similarity between the inferred internally specified lo- 
cations and previously reported measured end-points of the first saccades 
in double-step eye-movement studies may suggest similarities between per- 
ceived target locations in eye and hand motor control. 
1 INTRODUCTION 
The generation of reaching movements toward unexpectedly displaced targets in- 
volves more complicated planning and control problems than in reaching toward 
stationary ones, since the planning of the trajectory modification must be per- 
formed before the initial plan is entirely completed. One possible scheme to modify 
a trajectory plan is to abort the rest of the original motion plan, and replace it with 
a new one for moving toward the new target location. Another possible modifica- 
*Current address IRCS/GRASP, University of Pennsylvania. 
619 
620 Henis and Flash 
tion scheme is to superimpose a second plan with the initial one, without aborting 
it. Both schemes are discussed below. 
Earlier studies of reaching movements toward static targets have shown that point- 
to-point reaching hand motions follow a roughly straight path, having a typical bell- 
shaped velocity profile. The kinematic features of these movements were successfully 
accounted for (Figure 1, left) by the minimum-jerk model (Flash &: Hogan, 1985). In 
that model the X-components of hand motions (and analogously the Y-components) 
were represented by: 
X(t) = Xa + (Xr - Xa)(10T a - 157 q + 6T*), 
T = -- (1) 
t! 
A 
A 
Figure 1: The Minimum-jerk Model and The Non-averaged Superposition Scheme 
Computer 
c B 
ioo 
D msec 
Figure 2: The Experimental Setup and The Initial Movement Direction Vs. D 
A Computational Mechanism to Account for Averaged Modified Hand Trajectories 621 
where t! is the movement duration, and XB -XA is the X-component of movement 
amplitude. In a previous study (Henis & Flash, 1989; Flash & Henis, 1991) we have 
used the double-step target displacement paradigm (see below) with inter-stimulus 
intervals (ISis) of 50-400 msec. Many of the resulting motions were found to be 
initially directed toward the first target location (non-averaged) (for larger ISis a 
larger percentage of the motions were non-averaged). The kinematic features of 
these modified motions were successfully accounted for (Figure 1 right) by a super- 
position modification scheme that involves the vectorial addition of two time-shifted 
independent point-to-point motion units (Equation (1)) that have amplitudes that 
correspond to the two target displacements. 
In the present study shorter ISis of 10-110 msec were used, hence, all target displace- 
ments occurred before movement initiation. Most of the resulting hand motions 
were found to be initially directed in between the first and second target locations 
(averaged motions). For increasing values of D, where D = tiT - ISI (tiT is the 
reaction time), the initial motion direction gradually shifted from the direction of 
the first toward the direction of the second stimulus (Figure 2 right). The averaging 
phenomenon has been previously reported for hand (Van Sonderen et al., 1988) and 
eye (Aslin & Shea, 1987; Van Gisbergen et al., 1987) motions. In this work we 
wished to account for the kinematic features of averaged trajectories as well as for 
the dependence of their initial direction on D. 
It was observed (Van Sonderen et al., 1988) that when the first target displacement 
was toward the left and the second one was obliquely downwards and to the right 
most of the resulting motions were averaged. Averaged motions were also induced 
when the first target displacement was downwards and the second one was obliquely 
upwards and to the left. In this study we have used similar target displacements. 
Four naive subjects participated in the experiments. The motions were performed 
in the absence of visual feedback from the moving limb. In a typical trial, initially 
the hand was at rest at a starting position A (Figure 2 left). At t = 0 a visual target 
was presented at one of two equally probable positions B. It either remained lit 
(control condition, probability 0.4) or was shifted again, following an ISI, to one of 
two equally probable positions C (double-step condition, probability 0.3 each). In a 
block of trials one target configuration was used. Each block consisted of five groups 
of 56 trials, and within each group one ISI pair was used. The five ISI pairs were: 
10 and 80, 20 and 110, 30 and 150, 40 and 200, and 50 and 300 msec. The target 
presentation sequence was randomized, and included appropriate control trials. 
2 MODELING RATIONALE AND ANALYSIS 
2.1 THE SUPERPOSITION SCHEME 
The superposition scheme for averaged modified motions is based on the vectorial 
addition of two time-shifted independent elemental point-to-point hand motions 
that obey Equation (1). The first elemental trajectory plan is for moving between 
the initial hand location and an intermediate location Bi, internally specified. This 
plan continues unmodified until its intended completion. The second elemental 
trajectory plan is for moving between Bi and the final location of the target. The 
durations of the elemental motions may vary among trials, and are therefore a 
622 Henis and Flash 
priori unknown. With short ISis the elemental motion plans may be added (to give 
the modified plan) preceding movement initiation. Several possibilities for B i were 
examined: a) the first location of the stimulus, b) an a priori unknown position, c) 
same as (b) with Bi constrained to lie on the line connecting the first and second 
locations of the stimulus, and d) same as (b) with Bi constrained to lie on the 
line of initial movement direction. Version (a) is equivalent to the superposition 
scheme that successfully accounted for non-averaged modified trajectories (Flash & 
Henis, 1991). In versions (b), (c) and (d) it was assumed that due to the quick 
displacement of the target, the specification of the end-point for the first motion 
plan may differ from the actual first location of the target. The first motion unit 
was represented by: 
XX(t)--XA+(XB,--XA)(lOT a--15T 4+6Ts), where T-  (2) 
In (2), (XB --X) is the X-component of the first unit amplitude. The duration of 
this unit is denoted by T1, a priori unknown. The expression for Yl(t) was analogous 
to Equation (2). The X-component of the second motion unit was taken to be: 
X2(t)=(Xc--XB,)(iOT a--15T 4+6Ts), where T- t-t t-t (3) 
t! - is T 
In (3), (Xc - XB) is the X-component of the amplitude of the second trajectory 
unit. The start and end times of the second unit are denoted by t and t,, respec- 
tively. The duration of the second motion unit T. - t! -t is a priori unknown. The 
X-component of the entire modified motion (and similarly for the Y-component) 
was represented by: 
X() -- Xi() -'[- X.(t). (4) 
The unknown parameters Tx,T.,Bix and Bie that can best describe the entire 
measured trajectory were determined by using least-squares best-fit methods (Mar- 
quardt, 1963). This procedure minimized the sum of the position errors between 
the simulated and measured data points, taking into account (in versions (a), (c) 
and (d)) the assumed constraints on the location Bi. 
2.2 THE ABORT-REPLAN SCHEME 
In the abort-replan scheme it was assumed that initially a point-to-point trajectory 
plan is generated for moving toward an initial target (Equation (2)). The same four 
different possibilities for the end-point of the initial motion plan were examined. It 
was assumed that at some time-instant t the initial plan is aborted and replaced 
by a new plan for moving between the expected hand position at t = t and the 
final target location. The new motion plan was assumed to be represented by: 
5 
(5) 
i=0 
The coefficients aa, a4 and as were derived by using the the measured values of 
position, velocity and acceleration at t = t,. For versions (b), (c) and (d) the 
analysis was performed simultaneously for the X and Y components of the tra- 
jectory. Choosing a trial Bi and T1 the initial motion plan (Equation (2)) was 
A Computational Mechanism to Account for Averaged Modified Hand Trajectories 623 
calculated. Choosing a trial t,, the remaining three unknown coefficients a0, a and 
a. of Equation (5) were calculated using the continuity conditions of the initial and 
new position, velocity and acceleration at t = t, (method I). Alternatively, these 
three coefficients were calculated using the corresponding measured values at t = t, 
(method II). To determine the best choice of the unknown parameters Bix, Bi, T 
and t, the same least squares methods (Marquardt, 1963) were used as described 
above. For version (a), for each cartesian component, a point-to-point minimum- 
jerk trajectory AB was speed-scaled to coincide with the initial part of the measured 
velocity profile. The time t, of its deviation from the measured speed profile was 
extracted. From t on, the trajectory was represented by Equation (5). The values 
of a0, al and a. were derived by using the same least squares methods (method I). 
Alternatively, these values were determined by using the measured position, velocity 
and acceleration at t = t, (method II). 
3 RESULTS 
The motions recorded in the control trials were roughly straight with bell-shaped 
speed profiles. The mean reaction time in these control trials was 367.1 + 94.6 
msec (N = 120). The mean movement time was 574.1 q- 127.0 msec (N = 120). 
The change in target location elicited a graded movement toward an intermediate 
direction in between the two target locations, followed by a subsequent motion 
toward the final target (Figure 3, middle). Occasionally the hand went directly 
toward the final target location (Figure 3, right). For values of D less than 100 ms 
the movements were found to be initially directed toward the first target (Figure 3, 
left). As D increased, the initial direction of the motions gradually shifted (Figure 
2, right) from the direction of the first (non-averaged) toward the direction of the 
second (direct) target locations (The initial direction depended on D rather than 
on ISI). The mean reaction time to the first stimulus (RT1) was 350.4 q- 93.5 msec 
(N=192). The mean reaction time to the second stimulus (RT2) (inferred from the 
superposition version (b)) was 382.8 q- 119.9 msec (N=192). This value is much 
smaller than that predicted by successive processing of information: RT. = 2RT- 
ISI (Poulton, 1981), and might be indicative of the presence of parallel planning. 
The mean durations T and T. of the two trajectory units (of superposition version 
(b)) were: 373.0 q- 112.2 and 592.1 q- 98.1 msec (N = 192), respectively. 
3.1 MODIFICATION SCHEMES 
The most statistically successful model (Table-l) in accounting for the measured 
motions was the superposition version (b), which involves an internally specified 
location (a priori unknown) for the end-point of the first motion unit. In par- 
ticular, the averaged initial direction of the measured motions was accounted for. 
Superposition version (d) was equivalent to version (b). The velocities simulated on 
the basis the other tested schemes substantially deviated from the measured ones 
(Table I and Figure 4). It should be noted that in both the superposition and abort- 
replan versions (b), (c) and (d) there were 4, 3 and 3 unknown parameters. In the 
abort-replan versions (aI) there were 3 unknown parameters, compared to 2 in the 
superposition version (a). Hence the relative success of the superposition version 
(b) in accounting for the data was not due to a larger number of free parameters. 
624 Henis and Flash 
Table 1: Normalized Velocity Deviations and The t-score With SP(b)) 
SP(a) 
18.60 
 50.16 
(4.711) 
SP(b) SP(c) SP(d) AB(aI) AB(aII) AB(bI) AB(bII) AB(cI) AB(cII) AB(dI) AB(dII) 
0.035 0.126 0.042 0.083 0.084 0.081 0.078 0.084 0.083 0.082 0.085 
 0.036  0.132 + 0.045  0.093  0.088  0.101  0.102  0.108  0.096  0.097  0.101 
(0.000) (8.465) (1.546) (6.126) (6.559) (5.460) (5.050) (5.478) (5.959) (5.782) (5.935) 
Non- Averaged 
151 
D,80 
B A 
ISI * 150 
O' I0 
o 
Averaqed 
151, 40 
O* 250 
B A 
151. 
b  
Drecf 
C A 
151 
0,400 
+a 
B A 
151- 80 
D.450 
c 
Figure 3: Types of Modified Trajectories 
B+ .i..A 
SP(b) c 
8 
+ 
AB(blI) +c 
SP(b) 
loom 
+B 
AB(bII) 
+B 
Figure 4: Representative Simulated Vs. Measured Trajectories 
A Computational Mechanism to Account or Averaged Modified Hand Trajectories 625 
3.2 THE END-POINTS INFERRED FROM SUPERPOSITION (b) 
The mean locations Bi resulting from different trials performed by the same subject 
were computed by pooling together Bi of movements with the same D 4- 15 msec 
(Figure 5 left). For D _< 100 msec, the measured motions were non-averaged and 
the inferred Bi were in the vicinity of the first target. For increasing values of D, 
Bi gradually shifted from the first toward the second target location, following a 
typical path that curved toward the initial hand position. For D > 400 msec, Bi 
were in the vicinity of the second target location. Since initially the motions are 
directed toward Bi, this gradual shift of Bi as a function of D may account for 
the observed dependence of the initial direction of motion on D. The locations Bi 
obtained on the basis of the other tested schemes did not show any regular behavior 
as functions of D. 
4 DISCUSSION 
This paper presents explicit possible mechanisms to account for the kinematic fea- 
tures of averaged modified trajectories. the most statistically successful scheme in 
accounting for the measured movements involves the vectorial addition of two in- 
dependent point-to-point motion units: one for moving between the initial hand 
position and an internally specified location, and a second one for moving between 
that location and the final target location. Taken together with previous results for 
non-averaged modified trajectories (Flash &: Henis, 1991), it was shown that the 
same superposition principle may account for both modified trajectory types. The 
differences between the observed types stem from differences in the time available 
to modify the end-point of the first unit. Our simulations have enabled us to infer 
the locations of the intermediate target locations, which were found to be similar 
to previously reported (Ashn &: Shea, 1987) experimentally measured end-points of 
the first saccades, obtained by using the double-step paradigm (Figure 5 rightl). 
This result may suggests underlying similarities between internally perceived tar- 
get locations in eye and hand motor control and may suggest a common "where" 
command (Gielen et al., 1984; 1990) for both systems. 
c 
13 
B 
Figure 5: Inferred First Unit End-points and Measured Eye Positions 
Reprinted with permission from Vision Res., Vol. 7, No. 11, 1925-1942, As[in, R.N. 
and Shea S.L.: The Amplitude And Angle of Saccades to Double-Step Target Displace- 
ments, Copyright [1987], Pergamon Press plc. 
626 Henis and Flash 
Why is the internally specified location dependent on D, which is a parameter 
associated with both sensory information and motor execution? One possible ex- 
planation is that following the target displacement the effect of the first stimulus on 
the motion planning decays, and that of the second stimulus becomes larger. These 
changes may occur in the transformations from the visual to the motor system. A 
purely sensory change in the perceived target location was also proposed (Van Son- 
deren et al., 1988; Becker & Jurgens 1979). Another possibility is that the direction 
of hand motion is internally coded in the motor system (Georgopoulos et al., 1986), 
and it gradually rotates (within the motor system) from the direction of the first 
to the direction of the second target. It is not clear which of these possibilities 
provides a better explanation for the observations. 
In the superposition scheme there is no need to keep track of the actual or planned 
kinematic state of the hand. Hence, in contrast to the abort-replan scheme, an 
efference copy of the planned motion is not required. The successful use of motion 
plans expressed in extrapersonal coordinates provides support to the idea that arm 
movements are internally represented in terms of hand motion through external 
space. The construction of complex movements from simpler elementary building 
blocks is consistent with a hierarchical organization of the motor system. The 
independence of the elemental trajectories allows to plan them in parallel. 
Acknowledgement s 
This research was supported by a grant no. 8800141 from the United-States Israel 
Binational Science Foundation (BSF), Jerusalem, Israel. Tamar Flash is incumbent 
of the Corinne S. Koshland career development chair. 
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