Perceptual and Motor Skills, 1975, 40, 411-414. @ Perceptual and Motor Skills 1975
NOTE O N FITTS' LAW FOR MANIPULATIVE TEMPORAL MOTOR RESPONSES W I T H PATH CONSTRAINTS TAFULD 0.KVALSETH lnstitutt for Indurtliell @konomi og Organisa~jon Universitetet i Trondheiml
Summary.-This paper presents a first-order version of an earlier model by Kvilseth (1973) of manipulative motor responses involving serial hand movements with different types of movement-path constraints. This model represents a generalization of Fitts' law. Comparisons are also made between experimental movement times obtained and predetermined motion-rime systems predictions.
In an earlier study by this author (Kvblseth, 1973) of manipulative handcontrol subject to movement constraints, a second-order linear model was formulated between mean movement time ( Y ) and two indices (XI, X ? ) of task difficulty (ID). The independent variable XI was the ID or entropy measure originally defined by Fitts (1954) as
= lob (2A/W) 111 where A denotes the amplitude of the movement and W the accuracy or width of the target for terminating the movement. The variable X Z , which was defined as XI
Xa
= log,
(1/V)
PI
incorporated an additional constraint V in terms of ( a ) a center constraint as a slot or gate of width V, ( b ) a target height ( V ) constraint, or ( c ) a complete path constraint of constant width V throughout the movement path. For ( c ) , three different types of paths were used, i.e., a straight path, a sinusoidal path of one period and amplitude A/8, and a gaussian-appearing random continuous path. Each target pair and the corresponding movement path constraint were drawn on white paper and Ss were required to move a ball-point pen back and forth between the two targets as fast as possible while trying to keep the movements within the path constraints. All movements were made with S s hand while the arm remained fixed and rested on a table. The values used for A, W and V are given in Table 2 below. For all five constraint conditions considered, the second-order model for Y in XI and X 2 was found to provide an excellent fit to the experimental data that significantly exceeded the fit of the first-order model. Nevertheless, the first-order model with half as many unknown parameters did turn out to pro'The author is currently on leave from the School of Industrial and Systems Engineering, Georgia Institute of Technology, U.S.A. Present address: Norges Tekniske Hogskole, Trondheim, Norway.
412
T. 0. KVALSETH
duce quite a respectable and highly significant fit. It is the purpose of this note to present these results as well as those of a second-order polynomial model. Furthermore, some comparisons will be made between the experimental movement times obtained and predetermined motion-time systems predictions. Thus, the following two models were to be considered Y
= a,
= ao
+ a& + a x 2 + s + a , S + c; X = log,
( 2 A I W )(
131
~ I V ) 6~=, a d a l
and where the a4 and pi are unknown constant parameters and E is the error term. The formulation of the second-order model in Eq. 4 was prompted by the observation that plots of Y versus X appeared to indicate that Y was nonlinearly related to X according to the form given in Eq. 4 for some of the constraint conditions considered.
RESULTSAND DISCUSSION On the basis of the experimental data collected in an earlier study (Kvilseth, 1973), the least squares estimates obtained for the parameters in Eqs. 3 and 4 are given in Table 1. The coefficients of multiple determination Ra2 given in TABLE 1 PARAMETER ESTIMATES FOR ALTERNATIVECONSTRAINT CONDITIONS Parameter estimates*
Centef Constraint
Target height constraint
Complete path constraint Straight Sinusoidal Random
First-order model (Eq. 3 ) ao
-157.38
-139.38
-279.02
-474.03
-759.88
A
35.69
31.50
50.21
77.87
115.69
25.62 0.79 R.I3 0.77 0.53 R.2 Second-order model (Eq. 4 ) 6 0.7
25.83 0.94 0.94 0.87
52.58 0.76 0.76 0.67
106.79 0.92 0.90 0.92
163.87 0.84 0.80 0.84
0.8
1.0
1.4
1.4
a1 A
a?
Rm3
A $1 A
43.04 5.73
27.34 -19.96
2.37 3.94 0.95 0.79 0.0004 0.02 0.13 0.012 sec. as Y unit. "Estimates are based o n j3.
Re2 RBI' Raaa
824.81 -267.81 22.72 0.81 0.16 0.21
927.97
1594.38
-277.41
480.88
22.11 0.99 0.75 0.83
37.35 0.92 0.38 0.49
SERIAL MOVEMENT AND PATH CONSTRAINTS
413
Table 1 indicate that the first-order model of Eq. 3 produced a highly respectable fit to the experimental data for the various constraint conditions used. From 76 to 94% of the variation in movement time was explained by this model as compared to from 95 to 99.7 % obtained by Kvilseth ( 1973) for the secondorder model in XI and Xz. The partial determination coefficients R,I2 and Razz in Table 1 indicate that the variable XI contributed somewhat more to the movement time Y than did the variable X2 except for the cases of sinusoidal and random path constraints when the two variables were about equally "important." If the variables XI and X2 are interpreted as information-theory measures as Fitts (1954) originally proposed for XI, then the data given in Table 1 indicate that the maximum marginal information rate ax,/a? ranged from approximately 1 to 3 bits/sec. between the alternative constraint condiA
tions; Y denotes the mean movement time fitted by the model in Eq. 3. The marginal information rate ax&? varied between about 0.6 and 4 bits/sec. For the three types of complete path constraint considered, the inclusion of the second-order term X5esulted in a significant improvement in the model fit over the first-order relationship of Eq. 3. For the random path constraint, this improvement represented an almost 10% increase in the proportion of the variation in Y explained by the model. In terms of the partial determination coefficients RgI2 and RsZ2 associated with the regressors X and X2, respectively, the X2 was clearly determined to be the one that made the largest contribution to (the calculated value o f ) Y for all five constraint conditions as seen from the data in Table 1. The small values of RBI2 and RsZ2 given in this table for the center constraint case, in particular, were due to the multicollinearity or high correlation between the variables X and X2. A comparison between the mean movement times obtained in these experiments and c redetermined motion-time systems predictions showed generally large differences for all values used for A, W and V or for the ID variables XI, X2 and X. Such comparisons were made for three different predetermined motion-time systems, i.e., the Methods-Time Measurement system, the Basic Motion Timestudy system and the Work-Factor system. The movement times obtained in these experiments were converted into "normal times" by applying a rating factor of 150%. The percentages by which the predetermined motiontime systems predictions were found to differ from the experimental movement times obtained are given in Table 2 for those three of the five constraint conditions that would perhaps seem to be the most realistic ones in terms of actual work situations. It is apparent that all three predetermined motion-time systems are unsatisfactory for predicting movement times for manipulative hand movements subject to any of the types of movement constraints considered in these experi-
414
T. 0. KVALSETH
TABLE 2 PERCENTAGE DIFFERENCE BETWEENNORMALEXPERIMENTAL MOVEMENTTIMES AND PREDETERMINED MOTION-TIMESYSTEMSPREDICTIONS
A
W
V
1 1 1
1/2 1/4 1/8 1/16 1/2 1/4 1/8 1/16 1/2 1/4 1/8 1/16
1/16 1/8 1/4 1/2 1/16 1/8 1/4 1/2 1/16 1/8 1/4 1/2
1
2 2
2 2 4 4 4 4
Center constraint MTM BMT WF
Target - height constraint MTM BMT W F
Random path constraint MTM BMT W F
59.8 57.2 47.2 37.3 -7.7 23.5 -3.0 -18.2 67.2 59.0 38.5 33.3 16.7 -18.8 -6.7 -36.0 65.6 44.1 -3.2 39.1 1.0-77.1 6.3 -8.1 57.3 9.3 -28.0 59.0 12.8 -19.2 32.3 40.3 88.1 63.5 86.6 50.0 48.7 47.4 16.7 14.6 89.3 87.2 83.1 49.0 38.6 22.9 38.6 30.7 77.6 66.1 43.7 61.2 41.3 5.5 27.2 12.4 67.5 25.3 18.3 66.2 34.2 17.8 38.2 42.3 92.2 9 2 5 91.5 67.7 69.0 68.3 74.9 74.4 89.8 88 3 85.2 66.7 61.9 55.1 60.7 57.9 88.7 76 3 75.4 65.6 50.9 28.4 48.2 42.2 74.9 55.4 45.1 70.3 47.3 38.2 54.3 59.1 are based on [1.5Y - Y (PMTS)]100/1.5Y; 1.5 or 150% being the 3.0 14.7 42.3 68.2 18.8 49.0 51.9 81.3 73.9 65.6 63.7 74.3
Note.-Percentages rating factor employed. MTM = Methods-Time Measurement, BMT = Basic Motion Timestudy, W F = Work-Factor, and PMTS = Predetermined Motion-Time Systems. ments. Of the three systems, t h e Methods-Time Measurement system ~ r o d u c e d the largest deviations i n movement times from t h e experimental ones while, i n general, t h e most satisfactory movement time estimates evolved from t h e Work-Factor system.
REFERENCES F I ~ s P. , M. The information capacity of the human motor system in controlling the amplitude of movement. journal of Experimental Psychology, 1954, 47, 381-391. KVALSETH,T. 0. Fita' law for manipulative temporal motor res onses with and without path constraints. Perceptual and Motor Skdlr, 1973, 37, 827-431. Accepted December 2, 1974.
NOTE O N FITTS' LAW FOR MANIPULATIVE TEMPORAL MOTOR RESPONSES W I T H PATH CONSTRAINTS TAFULD 0.KVALSETH lnstitutt for Indurtliell @konomi og Organisa~jon Universitetet i Trondheiml
Summary.-This paper presents a first-order version of an earlier model by Kvilseth (1973) of manipulative motor responses involving serial hand movements with different types of movement-path constraints. This model represents a generalization of Fitts' law. Comparisons are also made between experimental movement times obtained and predetermined motion-rime systems predictions.
In an earlier study by this author (Kvblseth, 1973) of manipulative handcontrol subject to movement constraints, a second-order linear model was formulated between mean movement time ( Y ) and two indices (XI, X ? ) of task difficulty (ID). The independent variable XI was the ID or entropy measure originally defined by Fitts (1954) as
= lob (2A/W) 111 where A denotes the amplitude of the movement and W the accuracy or width of the target for terminating the movement. The variable X Z , which was defined as XI
Xa
= log,
(1/V)
PI
incorporated an additional constraint V in terms of ( a ) a center constraint as a slot or gate of width V, ( b ) a target height ( V ) constraint, or ( c ) a complete path constraint of constant width V throughout the movement path. For ( c ) , three different types of paths were used, i.e., a straight path, a sinusoidal path of one period and amplitude A/8, and a gaussian-appearing random continuous path. Each target pair and the corresponding movement path constraint were drawn on white paper and Ss were required to move a ball-point pen back and forth between the two targets as fast as possible while trying to keep the movements within the path constraints. All movements were made with S s hand while the arm remained fixed and rested on a table. The values used for A, W and V are given in Table 2 below. For all five constraint conditions considered, the second-order model for Y in XI and X 2 was found to provide an excellent fit to the experimental data that significantly exceeded the fit of the first-order model. Nevertheless, the first-order model with half as many unknown parameters did turn out to pro'The author is currently on leave from the School of Industrial and Systems Engineering, Georgia Institute of Technology, U.S.A. Present address: Norges Tekniske Hogskole, Trondheim, Norway.
412
T. 0. KVALSETH
duce quite a respectable and highly significant fit. It is the purpose of this note to present these results as well as those of a second-order polynomial model. Furthermore, some comparisons will be made between the experimental movement times obtained and predetermined motion-time systems predictions. Thus, the following two models were to be considered Y
= a,
= ao
+ a& + a x 2 + s + a , S + c; X = log,
( 2 A I W )(
131
~ I V ) 6~=, a d a l
and where the a4 and pi are unknown constant parameters and E is the error term. The formulation of the second-order model in Eq. 4 was prompted by the observation that plots of Y versus X appeared to indicate that Y was nonlinearly related to X according to the form given in Eq. 4 for some of the constraint conditions considered.
RESULTSAND DISCUSSION On the basis of the experimental data collected in an earlier study (Kvilseth, 1973), the least squares estimates obtained for the parameters in Eqs. 3 and 4 are given in Table 1. The coefficients of multiple determination Ra2 given in TABLE 1 PARAMETER ESTIMATES FOR ALTERNATIVECONSTRAINT CONDITIONS Parameter estimates*
Centef Constraint
Target height constraint
Complete path constraint Straight Sinusoidal Random
First-order model (Eq. 3 ) ao
-157.38
-139.38
-279.02
-474.03
-759.88
A
35.69
31.50
50.21
77.87
115.69
25.62 0.79 R.I3 0.77 0.53 R.2 Second-order model (Eq. 4 ) 6 0.7
25.83 0.94 0.94 0.87
52.58 0.76 0.76 0.67
106.79 0.92 0.90 0.92
163.87 0.84 0.80 0.84
0.8
1.0
1.4
1.4
a1 A
a?
Rm3
A $1 A
43.04 5.73
27.34 -19.96
2.37 3.94 0.95 0.79 0.0004 0.02 0.13 0.012 sec. as Y unit. "Estimates are based o n j3.
Re2 RBI' Raaa
824.81 -267.81 22.72 0.81 0.16 0.21
927.97
1594.38
-277.41
480.88
22.11 0.99 0.75 0.83
37.35 0.92 0.38 0.49
SERIAL MOVEMENT AND PATH CONSTRAINTS
413
Table 1 indicate that the first-order model of Eq. 3 produced a highly respectable fit to the experimental data for the various constraint conditions used. From 76 to 94% of the variation in movement time was explained by this model as compared to from 95 to 99.7 % obtained by Kvilseth ( 1973) for the secondorder model in XI and Xz. The partial determination coefficients R,I2 and Razz in Table 1 indicate that the variable XI contributed somewhat more to the movement time Y than did the variable X2 except for the cases of sinusoidal and random path constraints when the two variables were about equally "important." If the variables XI and X2 are interpreted as information-theory measures as Fitts (1954) originally proposed for XI, then the data given in Table 1 indicate that the maximum marginal information rate ax,/a? ranged from approximately 1 to 3 bits/sec. between the alternative constraint condiA
tions; Y denotes the mean movement time fitted by the model in Eq. 3. The marginal information rate ax&? varied between about 0.6 and 4 bits/sec. For the three types of complete path constraint considered, the inclusion of the second-order term X5esulted in a significant improvement in the model fit over the first-order relationship of Eq. 3. For the random path constraint, this improvement represented an almost 10% increase in the proportion of the variation in Y explained by the model. In terms of the partial determination coefficients RgI2 and RsZ2 associated with the regressors X and X2, respectively, the X2 was clearly determined to be the one that made the largest contribution to (the calculated value o f ) Y for all five constraint conditions as seen from the data in Table 1. The small values of RBI2 and RsZ2 given in this table for the center constraint case, in particular, were due to the multicollinearity or high correlation between the variables X and X2. A comparison between the mean movement times obtained in these experiments and c redetermined motion-time systems predictions showed generally large differences for all values used for A, W and V or for the ID variables XI, X2 and X. Such comparisons were made for three different predetermined motion-time systems, i.e., the Methods-Time Measurement system, the Basic Motion Timestudy system and the Work-Factor system. The movement times obtained in these experiments were converted into "normal times" by applying a rating factor of 150%. The percentages by which the predetermined motiontime systems predictions were found to differ from the experimental movement times obtained are given in Table 2 for those three of the five constraint conditions that would perhaps seem to be the most realistic ones in terms of actual work situations. It is apparent that all three predetermined motion-time systems are unsatisfactory for predicting movement times for manipulative hand movements subject to any of the types of movement constraints considered in these experi-
414
T. 0. KVALSETH
TABLE 2 PERCENTAGE DIFFERENCE BETWEENNORMALEXPERIMENTAL MOVEMENTTIMES AND PREDETERMINED MOTION-TIMESYSTEMSPREDICTIONS
A
W
V
1 1 1
1/2 1/4 1/8 1/16 1/2 1/4 1/8 1/16 1/2 1/4 1/8 1/16
1/16 1/8 1/4 1/2 1/16 1/8 1/4 1/2 1/16 1/8 1/4 1/2
1
2 2
2 2 4 4 4 4
Center constraint MTM BMT WF
Target - height constraint MTM BMT W F
Random path constraint MTM BMT W F
59.8 57.2 47.2 37.3 -7.7 23.5 -3.0 -18.2 67.2 59.0 38.5 33.3 16.7 -18.8 -6.7 -36.0 65.6 44.1 -3.2 39.1 1.0-77.1 6.3 -8.1 57.3 9.3 -28.0 59.0 12.8 -19.2 32.3 40.3 88.1 63.5 86.6 50.0 48.7 47.4 16.7 14.6 89.3 87.2 83.1 49.0 38.6 22.9 38.6 30.7 77.6 66.1 43.7 61.2 41.3 5.5 27.2 12.4 67.5 25.3 18.3 66.2 34.2 17.8 38.2 42.3 92.2 9 2 5 91.5 67.7 69.0 68.3 74.9 74.4 89.8 88 3 85.2 66.7 61.9 55.1 60.7 57.9 88.7 76 3 75.4 65.6 50.9 28.4 48.2 42.2 74.9 55.4 45.1 70.3 47.3 38.2 54.3 59.1 are based on [1.5Y - Y (PMTS)]100/1.5Y; 1.5 or 150% being the 3.0 14.7 42.3 68.2 18.8 49.0 51.9 81.3 73.9 65.6 63.7 74.3
Note.-Percentages rating factor employed. MTM = Methods-Time Measurement, BMT = Basic Motion Timestudy, W F = Work-Factor, and PMTS = Predetermined Motion-Time Systems. ments. Of the three systems, t h e Methods-Time Measurement system ~ r o d u c e d the largest deviations i n movement times from t h e experimental ones while, i n general, t h e most satisfactory movement time estimates evolved from t h e Work-Factor system.
REFERENCES F I ~ s P. , M. The information capacity of the human motor system in controlling the amplitude of movement. journal of Experimental Psychology, 1954, 47, 381-391. KVALSETH,T. 0. Fita' law for manipulative temporal motor res onses with and without path constraints. Perceptual and Motor Skdlr, 1973, 37, 827-431. Accepted December 2, 1974.
