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Psychophysical determination of coordinate representation of human arm orientation
Neuroscience Vol. 13, No. 2, Printed in Great Britain
pp.
S-604,
0306-4522/84 $3.00 + 0.00 Pergamon Press Ltd (CJ 1984 IBRO
1984
PSYCHOPHYSICAL DETERMINATION OF COORDINATE REPRESENTATION OF HUMAN ARM ORIENTATION J. F. Laboratory
of Neurophysiology,
SOECHTING
and B. Ross
University of Minnesota Medical School, Minneapolis, U.S.A.
MN 55455,
Ah&act-The coordinate representation of the sense of limb orientation was investigated psychophysically by asking subjects to match the orientation of the arm or of the forearm in several different coordinate representations. Movement of all degrees of freedom of one arm was permitted while
movement of the other limb was restricted to the degree of freedom investigated in that particular experiment. Performance on the tasks was assessed by calculating the standard deviation of the difference in the angles of the two limbs. According to this criterion, we suggest that limb orientation is represented by the angular elevation of the limb and by the yaw angle, referred to a spatial reference frame.
The organization and control of limb movement can be interpreted from the point of view that there exist a series of hierarchical operations which involve coordinate transformations. As an example, one can consider the movement of the arm to a target, a task which has recently been investigated in a number of laboratories.s~g,“,‘5*20*25 It seems reasonable to assume that the location of such a target is described in a coordinate representation extrinsic to the individual, for example Cartesian or polar coordinates. For the limb to move accurately to the target, its location must also be specified in terms of the joint angles of the limb segments involved. In addition, there is experimental evidence that the spatial trajectory of the limb is also specified, 5*2’although the question of whether this specification is in terms of extrinsicI or intrinsic (joint angle) coordinates25 is unresolved. Finally, joint torques adequate to produce this trajectory must be determined’ and the appropriate set of muscles activated. There is also experimental evidence that these latter two operations are distinct, since the same joint torque is produced by different patterns of activation of different muscles participating in a given movement.” Each of these sets of variables (joint angles, joint torques and muscle forces) can be viewed as defining a coordinate system. Going from one set of variables to another then involves a coordinate transformation. Recently, Pellioniasz and Llinb’8~‘g have provided a theoretical framework within which such coordinate transformations can be studied. In general, they noted, the axes of such coordinate systems will not be orthogonal to each other, and that sensory quantities will be represented by their projections onto these coordinate axes (covariant components), while the components of motor quantities add vectorially (con-
travariant components) to produce the desired resultant. They concluded that this endows the central nervous system with tensorial properties which are invariant of the particular frame of reference in which these quantities are represented. Nevertheless, in any instance, such transformations from covariant to contravariant representations must be implemented in a particular coordinate system(s). Robinson2’ has recently presented an analysis of the vestibulo-ocular reflex from this point of view, the sensory coordinate system being specified by the planes of the semicircular canals and that of the motor output by the lines of action of the extraocular muscles. Simpson et al.” have shown that visual information about movement is also represented in the coordinate system of the semicircular canals. For this particular system, the choice of coordinate system appears obvious from anatomy. Such is not the case, however, for limb movements in three-dimensional space. For example, the shoulder joint can be idealized as having three degrees of freedom. As will be shown in detail in the next section, the three joint angles corresponding to these degrees of freedom can be measured in a number of different ways. Similarly, the choice of a coordinate system on the motor side is not obvious because a large number of muscles act at the shoulder. Furthermore, for the reasons mentioned above, the transformation from joint angles to muscle force may involve an intermediate step, namely the specification of joint torques, whose coordinate system also needs to be determined. In this paper, we address the problem of the coordinate representation of joint angles of the human arm. One common means of assessing position sense psychophysically has been to use a matching
Address for correspondence: J. F. Soechting, Department of Physiology, 6-255 Millar Hall, 435 Delaware St. S.E., Minneapolis, MN 55455, U.S.A.
task: a given joint angle in the reference limb is achieved and the subject is then asked to match the joint angle with the other limb.*,‘0,‘2,‘7 In these experiments, movement was always restricted to one degree
N.S.C. 1312-N
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J. F. Soechting and R. Ross
of freedom in each limb and these experiments therefore do not discriminate among different coordinate representations.*’ The experiments to be described in this paper were designed to overcome this limitation. Movement in
all degrees of freedom was permitted in the reference limb, while movement of the other limb (the matching arm) was always limited to the degree of freedom investigated. Our hypothesis is that the “natural” coordinate representation of joint angles would be the one in which the standard deviation in the difference between joint angles of the two limbs would be least. Using this approach under more restricted conditions (location of the arm in the sagittal plane), we found that subjects were better able to match the orientation of their forearms in space than they were able to match the joint angle of the elbow.24 EXPERIMENTAL
PROCEDURES
Angular coordinates of the shoulder joint
The shoulder joint can be viewed as having three degrees of freedom, if translation due to scapular motion is neglected. At least two angles are required to define the orientation of the arm (or the location of the elbow), the third degree of freedom in such a definition corresponding to a rotation about the long axis of the humerus and not affecting the location of the elbow. Customarily, however three angles are used to define the location of the elbow relative to the shoulder: the angle of forward flexion (pitch), the angle of abduction (roll) and the angle of horizontal rotation (yaw). These angles are depicted in Fig. 1. as are alternate ways of defining the orientation of the upper arm. For purposes of illustration, coordinate representations of the right arm are depicted. The shoulder (S) is defined to be at the origin of the Cartesian coordinates X, Y and Z and the point E represents the position of the elbow. The axes X, Y and Z define the principal planes, i.e. XZ is the sagittal plane, YZ the frontal plane and XY the horizontal plane. The angles q,. rt2 and n, in Fig. l(A)-(C) are defined in one of the principal planes. In each instance, the point a is the projection of the elbow onto one of the principal planes and the angles I,. r12and rt3 are the apical angles of the shaded triangles. The angle r), represents the angle of horizontal rotation (yaw), n2 that of forward flexion (pitch) and n3 that of abduction (roll). There are other ways of defining the angular orientation of the arm however, Imagine three orthogonal axes fixed to the humerus such that the humerus is initially vertical along the Z-axis (Fig. I). Any arbitrary position of the elbow can then be achieved by two successive rotations about two of the axes fixed to the humerus. (These rotations, which detine Euler angles, are not commutative that is, the angles depend on the order in which the rotations are performed.) Thus, in Fig. I(A) the angles rh and 0, are defined by an initial rotation about the axis of the humerus (Z-axis), which gives PI,, and then about an axis perpendicular to the plane of the upper arm and forearm (an axis which coincides with the Y-axis when rt, = 0) to define 0,. The angle 8, will be called the angular elevation of the arm. Similarly, in Fig. l(B), there is a rotation first about the Y-axis (to give q2) and then about an X-axis fixed to the humerus (to give 0,). in other words forward flexion followed by abduction. The order is reversed in Fig. l(C); abduction (rb) is followed by _-...____-____. *We have adopted the convention that whenever a&s are expressed without a subscript, the subscript I is implicit (e.g. e = 8, ).
forward
flexion (Ox). As was mentroned a~ the beginnmg ot this section. the shoulder joint has at least three degrees of freedom of motion. Two of them are defined In Fig. I(A)_(C): in each instance. the two rotations described above arc followed by a third about the axis of the humerus. Note that this last rotation does not affect the posttton o) the elbow but will influence the position of the wrist in
space. If we define (.y, .v. I) to be the position of the elbou m the XYZ coordinate system. then
e, = (.P + .+z
tan q, = j’;s
tan
tan rtr 2 .y r
tan 0, _ J./(,x: + ; ‘)1
tan q, = j,,‘;
tan O,=s~(~-‘+r~)~.
(1)
Equations relating the angles to the components of the position of the elbow (equation I) can be derived from the rotation matrix’.’ which relates the coordinates of a point detined in the XYZ coordinate system tixed in space to the coordinates of the same point in an !7%! coordinate system fixed to the arm. When the rotations are defined as in Fig. I (A)* the rotation matrix is given by x
=
cos 0 cos ‘I
)’
=
cos 0 sin q
H 2
= [
-sin0
-sin7
sin e cos q
2
cos r) sin 0 sin n 0
cos U
B 1-H
f (2)
where (x. y. z) and (8,7, L?)represent the position of a point defined in the fixed and moving coordinate systems, respectively. For example, the location of the elbow is given by I,(O, 0, I) in barred coordinates (!, being the length of the arm). In fixed coordinates, according to equation (2), its location is I, (sin e cos q, sin 0 sin rj, cm 0). It may be seen that the detinitions of 7, and 0, given in equation (I) follow directly. Task
In a given experiment, subjects were asked to match one ofthesixangles(rl,,42.9,;B,.8~, 0,)showninFig.lwith their right and kft arms. They were presented a target, generally in the right, anterior quadrant (x > 0, y > 0), asked to close their eyes and then to point to the target with their right arm. In two experiments, the right arm was positioned by the experimenter. The left arm was constrained to move in one of the three principal planes (sagittal, frontal or horizontal). Note that when the location of the arm is restricted to the sag&al plane, 8,, q2ad 8, are identical, as are 0,, fJ and n, for arm locations in the frontal plane. Therefore, the angles described by the left arm used to match the orientation of the right arm are independent of the coordinate description. The subjects were asked to match the specihed angle formed by their right arm with a planar movement of their left arm. Thus, the angles 0,. q2and 0, were matched by a forward flexion of the kft arm, 0,. O2and ‘1, by abduction and 9,. by a movement of the left arm in the horizontal plane. In any given experimental session, subjects were asked to match two of the angles. Their performance on the tasks was quantitated by calculating the coefficient of correlation @) between the ass&a of the two arms and the standard deviation of the dil%rence. Before the experiment bqaa, the subject13wem shown a small threed&nsIoaal se& model (timBar to Fig. I) which deIined the an&a tbay were to matds. brey were also told that matcbhrg B, (ms&u elevation) was equivalent to matching the height z of the right and kft elbow, and matching 0, and 0, equivalent to matching the lateral ( y) and anterior (x) projections of the elbows, respectively. Matching r~r(forward flexion) and ‘I, (abduction) is quivaht to aaching the ulapenof the two amm. They were tild to imagine a sandwich board or an A-frame attncbal to their trunk and that nr or n, would be matebed if the two
Coordinate representation arms were parallel to the planes of these structures. The subjects were given a few practice trials with eyes open to ensure that they understood the tasks. Data recording and analysis The orientation of the right arm in space was measured by means of a pair of ultrasound emitters and a set of three orthogonal linear microphones which were aligned with the Cartesian coordinate axes of Fig. l.25 The two emitters were strapped to the arm, one at the shoulder level and the other at elbow level, on a line parallel to the arm. Using this system, the position of these two points in space was obtained and the value of the angles ‘1,- q, and 0, - 0r were calculated according to equation (1). The left arm was strapped to a manipulandum which pivoted about the axis of the shoulder and which was constrained to rotate about either the Y-axis (forward flexion) or about the X-axis (abduction). Angular position of the left arm was measured by means of a potentiometer. Angular coordinates of the elbow joint The elbow joint can be viewed as having two degrees of freedom (flexion-extension and pronation-supination). Only the former of these affects the orientation of the arm, while the latter influences the orientation of the hand. (The question of the coordinate representation of the hand is beyond the scope of this paper.) The third degree of freedom at the shoulder joint (axial rotation), however, does influence forearm rotation. Thus also for the forearm two angles are required to specify its orientation. The possible coordinate representations of forearm orientation which were investigated were influenced by our findings on the coordinate representation of the arm and by previous psychophysical studies regarding position sense at the elbow.24 As we show in results, the subjects’ performance was best when the task required them to match the joint angles shown in Fig. lA), that is, the angular elevation of the arm (0,) and the yaw angle (q,). Furthermore, previous work” showed that subjects were able to match the elevation of their forearm (angle of the forearm measured relative to the vertical) better than they were able to match the angle of forearm extension (angle subtended by the forearm and the arm). In those experiments, the arm was restricted to lie in the sagittal plane. Thus in this limiting case, forearm orientation is also described by its angular elevation. We hypothesized two possible coordinate representations of the forearm which are consistent with this limiting case. The first of these is shown in Fig. 3(A) and is analogous to the coordinate representation of the arm. The angle b, represents the elevation of the forearm and is measured in the vertical plane aEW, E denoting the elbow and W the wrist. The yaw angle a, is the angle between this vertical plane and the sagittal plane X-Z. An alternative coordinate representation is shown in Fig. 4(A). It represents an Eulerian description of the three degrees of freedom at the shoulder. Two angles are given by r~and-0 (as in Fig. lA), corresponding to rotations about Z and Y. The third angle, z2, denotes the amount of axial rotation of the arm (about the axis of the humerus). The reference value, a2 = 0 corresponds to the instance in which the plane of the arm (SEW) is vertical. The angle 1, represents the degree of freedom at the elbow and is equal to the angular elevation of the forearm when a2 = 0. It is unchanged by axial rotation. Note that b, and & are identical when a, is zero, that is when the arm is in the vertical plane. Furthermore, as noted above, when the arm is in the sagittal plane (a2 = 0 and ‘1,= 0) subjects are able to match this angle better than they are able to match the angular extension at the elbow (4). The angles a, and b, can be calculated from tan a, = -y/x
tan/I, = -(x2+y2)$r
(3)
597
of human arm orientation
where x, y, z denote the position of the wrist measured relative to the elbow in the XYZ coordinate system (Fig.
3A). The angle jr is given by b2 = 4 - 0,
(4)
where 0, is defined by equation (1) and Q is the angle of elbow extension. Finally, a2 is given by sin a2 sin 8, = n,
(5)
where n, is the Z-component of the normal to the plane formed by the arm and the forearm. The experimental design used to evaluate position sense of the forearm was similar to that described for the shoulder. Subjects were asked to match the angles b, and /I2 with their two arms. The left forearm was again strapped to the manipulandum which pivoted about the axis of the elbow and was constrained to move in the sag&al plane. The left arm was close to the vertical. Thus, for the left arm, the angles /I, and fir were identical. The position of the wrist and elbow of the right arm was determined by means of the ultrasound emitters described above. From these measures, the angles a, and /3, were calculated according to equation (3). The elbow angle C$was measured electrogoniometrically.2 At the onset of each experiment, the center of rotation of the right shoulder was determined. The subject was asked to move the arm in an arc with ultrasound emitters attached at the elbow and below the shoulder. The best estimate of the centre of rotation was determined graphically from these data. From the location of the shoulder, elbow and the wrist, the notmal vector to the plane of the arm was calculated by standard trigonometric methods and the angle 0, according to equation (I). Finally, the angles a, and fi2 were determined according to equations (4) and (5). The data reported in this paper summarize the results of 33 experiments involving 18 subjects. RESULTS
The results of experiments dealing with the coordinate representation of the orientation of the arm, are summarized in Table 1. In each experimental session, a paired comparison of a subject’s performance on two matching tasks was obtained. Thus, in the first four experiments, the subjects attempted to match either the angular elevation (0,) Fig. 1A) or the angle of forward flexion (qt, Fig. 1B) with both arms. Movement of the right arm was unrestricted, while that of the left arm was confined to the sagittal plane (forward flexion, FF). n Represents the number of trials in each task, while the values reported in the next two columns are the standard deviation (SD) in degrees of the difference between the angles of the right and left arms and the coefficient of correlation @) of the linear regression between them. As may be seen, all four subjects performed better when the task required them to match 19,(the angular elevation of the arm), in the sense that the standard deviation was smaller and p was larger on this task than when the subjects attempted to match the angle of forward flexion (or). This is also evident in Fig. 2(A) and (B), which show the performance of one subject on the two tasks. Each data point presents the results of one trial. A 45” line has been drawn for reference; it represents perfect performance on the task.
59s
J. F. Soechting and B. Ross
Table 1. Performance in matching orientation of the arm 0, FF
n
SD
p
n
40 30 41 40
7.0+* 6.5,’ 6.2+ 6.S’
0.953 0.962 0.952 0.972
41 37 41 41
6.5+*
O.Y60
Av.
;; q? Q
SD 12.9 15.6 8.8 12.8
P 0.939 0.850 0.908 0.850
12.5
0.887
FF
41 40
8.0** 7.4**
Passive 0.939 q* 0.965 q2
39 40
13.2 11.8
0.915 0.932
FF
36 40 38 41
%I** 6.3.. 6.5+* 8.1**
0.960 0.947 0.983 0.964
35 40 39 40
11.4 14.1 13.0 16.6
0.763 0.589 0.889 0.823
6.5..
0.960
13.8
0.766
6.8*+ 5.1 5.6’+ 5.8’.
0909 0.959 0.963 0.957
11.0 5.1 9.6 12.7
0.792 0.939 0.872 0.776
5.8
0.947
9.6
0.844
4.8” 4.7*
0.963 0.966
15.3 6.5
0.738 0.925
Av. ABD
41 41 40 40
Av. ABD
40 40
6), 8, OI 8,
‘I, ?I> & q,
0, e*
40 41 40 40
40 40
Each row presents standard deviation (SD) and coefficient of correlation @) for one sub&t; IeA columns: matching angular elevation of the arm by forward flexion (FF) or by abduction (ABD) and right columns: matching the angle indicated. l*p < 0.005. *P < 0.05.
A statistical analysis of the variance of each subject’s performance on the two tasks was performed. For all four subjects, the variance on the task which required matching t12 was sign.i&antly greater, as indicated by the symbols in Table l(**P < 0.005, *P < 0.05). In addition, a statistical analysis of the pooled data was performed (paired Student’s f-test on the logarithmic ratio of the variance). Also ac-
cording to this criterion. the subjects pert’ormed significantly better in matching the elevation of the arm ((I,) than they did in matching the angle of forward flexion (q2). Similar results were also obtained when the right arm was passively positioned. Also in this case, the two subjects were able to match angular elevation 8, better than they were able to match the angle of forward flexion. The subjects’ performance in matching 0, was also significantly better than their ability to match 0, (Fig. IC). Results obtained for one subject on these two tasks are shown in Fig. 2(C) and (D), and the data for four subjects are summarized in Table I. Thus the subjects’ performance in matching forearm elevation was superior to their performance on the two alternative tasks, suggesting that the coordinate system illustrated in Fig. I (A) is the one which is used preferentially to represent the angular orientation of the arm. At the conclusion of each experiment, each subject was interviewed. All but one expressed the opinion that matching forearm elevation was easier. At the beginning of the experiment, as the tasks were explained, opinions were mixed, a number of subjects volunteering the opinion that one of the two other tasks would be easier. If our hypothesis is correct that the coordinate system shown in Fig. l(A) is the preferred coordinate representation, then the subjects should also be better able to match forearm elevation (0,) than the angle of abduction measured in one of two possible ways (&, Fig. I B; q3, Fig. IC). This prediction was verified, as shown in Table 1. In these last two sets of experiments, the left arm was constrained to move in the frontal plane (abduction, ABD) and the subjects were required to match 0, and q, or &. Five of the six subjects performed significantly better when asked to match the elevation of the arm. Our considerations of possible coordinate systems
Vertical Fig. 1. Possible coordinate rqxcacntations of the spatial orientation of the arm. Each part of this t@re shows dilTefent angular masmos wtdoh may be used to describe the orientation of the arm. The pmnt E dmotea the location of the dbow and S the location of the shoulder, w&h is al the origin of the cartcsian~~notesy~xyz.~&~~rxms~aritbtba~~Qfthebody_x, anteriorY latarPlagdZ,v~.Ib:~~imliate~ptojcctioaoftheannontooncof thethre;~rmdpalpbesaadrk~~,,~~~q,ue~-in~~~.IhepnolG8,is
measured in the vertical plane of the arm relative to the Z-axis and the angle9 0, and 0, are mcasurod in the plane of the triangle GE, where II is the projection of E onto one of the principal planes.
Coordinate representation
of human arm orientation
80° -
00
40°
0,
80°
O0
40°
0,
80°
00
400
72
80°
40.
e,
80.
Fig. 2. Results of matching arm orientation in different coordinate representations. The results presented in (A) and (B) were obtained from one subject, those in (C) and (D) from another subject. In (A) and (C), the subjects matched the angle 0, (Fig. 1A) of their right arm by means of forward flexion of their left arm. Each data point represents the results from one trial. The 45” line, which is given for reference, indicates perfect performance on this task. In (B) and (D) the same subjects matched the angle g, (Fig. 1B) and G3(Fig. 1C) of their right arm by means of forward flexion of their left arm. Note that in both
instances, the subjects’ performance was better on the task which required them to match 0,. Details of the experimental design are given in the text.
which might be used to represent the orientation
of the forearm was limited by the findings on position sense at the shoulder described above and also by some preliminary observations reported elsewhere.24 Arguing by analogy with the results described above, which indicate that yaw angle (q, ) and elevation (0,) are most appropriate to describe the orientation of the arm, one might expect a similar coordinate description to hold for forearm orientation. This possibility is shown in Fig. 3(A), where a, represents the yaw angle measured relative to the frontal axis (X) and /?, the angle of forearm elevation measured in the vertical plane of the forearm. An alternative possibility is shown in Fig. 4(A). The actual position of the right limb is indicated by the heavy lines, S, E and W denoting the location of the shoulder, elbow and wrist. The angle fir denotes the elevation the forearm would have if the arm were to lie in a vertical plane. The angle a2 represents the amount of angular rotation, about the axis of the humerus, required to
bring the forearm from its actual position to the virtual position in the vertical plane. (Note that the amount of elbow extension does not change during this virtual rotation, i.e. it is performed with the elbow locked.) A matched comparison of the performance of seven subjects on these two tasks was made. Representative results from two subjects are shown in Figs 3(B) and 4(B), while the data from all subjects are summarized in Table 2. (Note that no data on f12are presented for one subject. During the experiment, it was obvious that matching error on this task often exceeded 20” and the experiment was terminated.) On average, the standard deviation on the task involving matching /?, was slightly less (8.2” vs 9.8’) and the coefficient of correlation slightly greater. These differences are not statistically significant, however. During these experiments, it appeared that the subjects made the largest errors in matching forearm elevation (/?,) when the position of the right hand was
J.F. Soechting and B. Ross
600
S
2oo /
80 °
/: o
40 °
X
•
|
0o
40 °
I
i
/31
l
,
80 °
120 °
Fig. 3. One possible coordinate representation of the orientation of the forearm. (A) schematicallydepicts the arm, S, E and W representing the position of the shoulder, elbow and wrist. The angle ~ and 0 are identical to r/I and 0= defined in Fig. I(A). The line ab represents the projection of the forearm onto the horixontal plane (XY), and the angle =l (yaw) is the angle between that line and the anterior (X) direction. The angle/3, (elevation) is measured in the vertical plane (aEWb) relative to the vertical (Z) axis. (13)shows the performance of one subject in matching the angular elevation of the two forearms. The filled circles indicate trials in which the distance d between the hands was greater than 35 cm, the open cirles those in which d was less than 35 c~. close to the midline of the body, that is when the right and left hands were near each other. In fact, in many of these instances, subjects would report that they knew that the attempted match was in error, but that nevertheless "it felt right". Furthermore, the errors were generally consistent in that subjects tended to underestimate the angle /~, with their left forearm, that is, they tended to flex the forearm more than required.
The values of the standard deviation and correlation coefficient for the trials in which forearm elevation was matched, excluding those trials in which the distance between the hands was less than 35 cm, are reported in the second column in Table 2. Note that when these trials are excluded, the standard deviation decreased for all subjects, by 22~ on average, and the coefficient of correlation increased. Excluding trials in which inter-hand distance was less
B 80 o
x
s
Z
4o0
i. 0*
I
,
40 °
~=
I
80 e
Fig. 4. Another possible coo~inate representation of the orientation of the forearm. Note that the perspective of the arm in (A) is different from that used in Fig. 3(A). The point a represents the projection of the location of the elbow onto the horizontal plane. The plane SaE is vertical. W represents the actual location of the wrist and the dashed fine represents the orientation the forearm would have if it were to lie in the vertical plane SaE. The angular rotation required to bring the forearm from the actual to the virtual orientation is given by =2; it is about the axis of the humerus SE. This rotation is performed with the elbow locked, i.e. the elbow angle ~ does not change. The angle ~2 is measured in the vertical plane SaE and describes the angle of virtual forearm orientation relative to the vertical axis. (B) shows the performance of one subject on this task.
Coordinate representation
Table 2. Performance in matching orientation of the forearm B,: . all trials n
SD
P
41 42 45 39 43 45 46
7.2 10.6 8.6 5.0 7.2 9.5 9.6
0.948 0.900 0.949 0.939 0.931 0.908 0.827
Av.
8.2
0.915
fit : d > 35 cm -i SD p
n
31 33 38 28 28 40 34
82 SD
p
3.5 6.8* 7.5 4.1** 6.1** 1.5** 9.5
0.963 0.953 0.958 0.949 0.954 0.931 0.840
38 9.0 46 8.1 37 8.4 41 9.5 46 12.0 44 11.7
0.883 0.948 0.891 0.870 0.910 0.840
6.4*
0.936
9.8
0.890
The left column summarizes data for the task of matching angular elevation of the forearm (B, ), the middle column for the same task with trials in which the two hands were within 35 cm excluded, and the third column for the task of matching &. **p < 0.005. *P < 0.05.
than 35 cm did not improve the measures of performance for the task of matching fir, the standard deviation decreasing on average by less than 1% (and no more than 4% in any subject) while p actually decreased 2%. When trials in which the two hands are close together are excluded, four of six subjects performed significantly better matching forearm elevation than they did matching /$. The average performance of all six subjects was also significantly better.
20
Sn
-20
(cm)
S,
Fig. 5. Orientation of the two arms during one trial in which the distance between the two hands was less than 35 cm. In this trial, the subject attempted to match the angular elevation of the forearm (8,). The points S, E and Wdenote the location of the shoulder, elbow and wrist; the subscripts R and L indicating the right and left limb. The points n and 6 are the projection of Ea and R’s on the horizontal plane. The dashed line for the right limb indicates the estimated position of the right hand, the location of the wrist and elbow having been measured. The left limb was in the sagittal (XZ) plane and the solid line and the points EL and W, denote the orientation of the limb and the position of the left elbow and wrist. The dashed line indicates the angular elevation of the left arm which would have matched that of the right forearm, as specified by the task. Note that the actual position of the left forearm more closely matched the height of the two hands.
of human arm orientation
601
There are two possible reasons why trials in which the right hand was close to the mid-sagittal plane should lead to large errors in matching forearm elevation. First, in these instances, the axial rotation (az) of the arm was appreciable. If the subjects were in fact matching /?* rather than fl,, they would tend to underestimate /?, (J& always being no larger than /&). This possibility was investigated by asking subjects to position their arm obliquely (axial rotation a2 ranging from 30” to 90’) with the hand far from the mid-sagittal plane. Such orientations of the arm did not, however, lead to errors in matching /?, significantly larger than average. The second possibility is that, when the right hand
was close to the midline, the subjects actually tended to match the position (primarily the height) of the two hands. This is illustrated in Fig. 5, which depicts schematically the orientations of the two arms for one trial in which the error in matching /$ was large. The symbols S, E and W denote the position of the shoulder, elbow and wrist, the subscripts R and L refer to the right and left limb, respectively. The planes S,aE, and ub W, are vertical. The dashed line denotes the orientation of the left forearm which would match the angle of elevation of the right forearm. Note that the actual position of the left forearm instead matches more closely the height (Z-axis) of the two hands, as hypothesized. Therefore, it seems safe to conclude that one of the measures used to specify the orientation of the arm is the angle of elevation, as defined by 0, for the shoulder (Fig. 1A) and by 8, for the elbow (Fig. 3A). In each instance, a second angular measure is needed to fully determine the limb’s orientation in space, the most logical candidate being the yaw angle (q, or a,). In a series of experiments, we also investigated the ability of subjects to match these two angles with their arms. The results of these experiments are summarized in Table 3 and representative data are shown in Fig. 6. In the experiments in which subjects were asked to match the yaw angle at the shoulder (q,), the left arm was constrained to move in a horizontal plane and the matching was performed by an anti-symmetric rotation of the two arms. That is, a rightward rotation of the right arm (corresponding to a positive value of q, ) was matched by a leftward rotation of the
Table 3. Performance in matching yaw angle of the arm (q, ) and of the forearm (a,) ‘11 n
SD
p
n
aI SD
p
51 56 40 41
10.3 11.2 8.5 11.1
0.856 0.897 0.869 0.839
45 44 47 43
10.6 9.4 10.4 14.0
0.953 0.944 0.942 0.867
Av.
10.3 0.865
Av.
11.1 0.926
J. F. Soechting and B. Rosh
A
,.6 .
.
.
.
.
. .C,
.
4’ . . /. . . 80” . .’
I
/ .
1
&T.,
O0
?
400
80°
I
-80°
-400
a
I
I
I
00
4o”
80”
Fig. 6. Performance of two subjects in matching yaw angles of the arm and forearm. (A) shows data points of trials in which the subject attempted to match the yaw angle r) (Fig. 1A) of the arm and (8) results of an experiment in which a subject attempted to match the yaw angle a of the forearm (Fig. 3A).
left arm. Similarly, in the experiments in which the task was to match the yaw angle of the forearm, the task was accomplished by an anti-symmetric rotation of the two forearms, the left forearm being horizontal, the left arm vertical and the axis of rotation about the humerus. The performance of the subjects in matching yaw angles was distinctly inferior to their performance in matching the angle of elevation. One possibility which might account for this finding is that the matching paradigm chosen was not the appropriate one, that is, that matching yaw angles would be produced by a symmetric rotation of both limbs. This possibility was explored in two experiments in which subjects were asked to match yaw angle at the shoulder. The errors in matching joint angles under this condition were even greater than those reported in Table 3. DISCUSSION
Our choice of a natural, or preferred, coordinate representation of the arm is based on the hypothesis that the task of matching joint angles with the two arms would be performed best in this natural coordinate representation, that is, that the standard deviation of the error would be least in that particular coordinate representation. According to this criterion, the data presented in this paper indicate that most probably the sense of limb orientation is expressed psychophysically in the coordinate system illustrated in Fig. 3(A), namely that the orientation of each limb segment is described by the yaw angle (r), or I, ) and the angular elevation (0, or /l, ). Note that these at&es describe the orientation of the limb segments relative to an absolute frame of reference, referred to the sagittal plane and to the vertical,
rather than a relative orientation, such as for example, the joint angle of the forearm relative to the arm. The performance of our subjects was not uniformly good in matching these four angles. The angular elevation of the arm (6,) was matched quite well, with an average SD of 6. I’ when the right arm was actively positioned by the subject. When the right arm was passively displaced by the experimenter, the average error was slightly larger, in agreement with findings that the error in matching joint angles increases when one limb is displaced passively.” In 15 of 16 experiments, the error in matching limb elevation was significantly less than that for the other angular measures considered for the shoulder joint. Subjects did not perform as well in matching the anguIar elevation of the forearm (SD = 8.2”). We have suggested that the error was larger for this angular measure because subjects tended to match the position of the two hands (their vertical height) when the hands were close together. In fact, when trials in which the two hands were within 35 cm of each other were eliminated, the error in matching forearm elevation (SD = 6.4”) was comparable to that in matching the elevation of the arm. The errors in matching yaw angles were considerably larger than those for the angular elevation of the limbs. Nevertheless, we can see no obvious alternative angular measures which, in conjunction with the angular elevation, would specify the oriertttttion of the limb. In our experiments, yaw angle was measured relative to a spatially fixed coordinate frame, namely the sagittal plane. It is poibk that subjects, instead, were matching yaw angks relative to a perceptual sagittal plqne (paocptual straightahead) which could vary from trial to trid, and that UtistIWy aowunt for the laager error. If it is accepted that the coordinate system we have
Coordinate representation of human arm orientation proposed is a preferred one, the question arises: which of the parameters relevant to the organization and control of limb movements are described in this coordinate system? One likely possibility is that this representation of sensory information is used to orient the arm relative to objects in space. Stevens26 in a recent psychophysical study has proposed that the orientation of surfaces is perceived in angular coordinates (slant and tilt) similar to those proposed here. The major difference is that slant and tilt are measured relative to the line of sight rather than relative to the vertical as in our case. However, Gibson6 has suggested that surface orientation is perceived relative to both line of sight and the gravitational vertical. If surface orientation relative to this latter axis is also perceived by similar angular measures, the slant angle would be identical to the angular elevation and the tilt angle equivalent to the yaw angle in our angular description. The question of whether or not movement trajectories are also planned in this coordinate systems and if this coordinate representation is also an appropriate one for describing joint torques remains open.
603
Regarding the neural correlates of the psychophysical variables, only speculations can be offered at this time. The assumption that joint afferents contribute exclusively to limb position sense has been questioned recently’3T’4and it has been suggested that tactile and spindle receptors also contribute. Our finding that forelimb orientation is expressed in a frame of reference relative to the trunk (or some spatial frame of reference) rather than relative to the upper arm implies that there must also be a convergence of sensory inputs from different limb segments. Furthermore, it seem plausible that vestibular and visual information are also utilized. Neurons responding to movements of several joints have been found in primary somatosensory cortex3 and posterior parietal cortex.‘4*‘6*22The precise information encoded by such neurons remains to be determined, however. Acknowledgements-This work was supported by USPHS Grant NS-15018 and by NSF Grant BNS-8117625. The authors thank Dr. A. P. Georgopoulos for his valuable advice concerning statistical analysis of the data and Dr. R. E. Poppele for his critical reading of the manuscript.
REFERENCES 1. Benati M., Gaglio S., Morass0 P., Tagliasco V. and Zaccaria R. (1980) Anthropomorphic robotics. I. Representing mechanical complexity. Biol. Cybern. 38, 125-140. 2. Chao E. Y., An K. N., Askew L. J. and Morrey B. F. (1980) Electrogoniometer for the measurement of human elbow joint rotation. J. biomech. Engng 102, 301-310. 3. Costanzo R. M. and Gardner E. P. (1981) Multi-joint neurons in somatosensory cortex of awake monkeys. Bruin Res. 214, 321-334. 4. Duffy F. H. and Burchfiel J. L. (1971) Somatosensory system: organizational hierarchy from single units in monkey area 5. Science, N.Y. 172, 273-275. 5. Georgopoulos A. P., Kalaska J. F. and Massey J. T. (1981) Spatial trajectories and reaction times of aimed movements: effects of practice, uncertainty and change in target location. J. Neurophys. 46, 725-743. 6. Gibson J. J. (1950) The perception of visual surfaces. Am. J. Psychol. 63, 367-384. 7. Goldstein H. (1950) Classical Mechanics, pp. 93-109. Addison-Wesley, Reading, Massachusetts. 8. Goodwin G. M., McCloskey D. I. and Matthews P. B. C. (1972) The contribution of muscle afferents to kinaesthesia shown by vibration induced illusion of movements and by the effects of paralysing joint afferents. Brain 95, 705-748. 9. Hollerbach J. M. and Flash T. (1982) Dynamic interactions between limb segments during planar arm movement. Biol. Cybern. 44, 67-77.
10. Horch K. W., Clark F. J. and Burgess P. R. (1975) Awareness of knee joint angle under static conditions. J. Neurophysiol. 38, 1436-1447.
11. Lacquaniti F. and Soechting J. F. (1982) Coordination of arm and wrist motion during a reaching task. J. Neurosci. 2, 399408. 12. Laszlo J. I. (1967) Kinaesthetic and exteroceptive information in the performance of motor skills. Physiol. Behav. 2, 359-365. 13. Matthews P. B. C. (1982) Where does Sherrington’s muscular sense originate? Muscles, joints, corollary discharge? A. Rev. Neurosci. 5, 189-218. 14. McCloskey D. I. (1978) Kinesthetic sensibility. Physiol. Reu. 58, 763-820. 15. Morass0 P. (1981) Spatial control of arm movements. Expi Brain Res. 42, 223-237. 16. Mountcastle V. B., Lynch J. C., Georgopoulos A., Sakata H. and Acuna C. (1975) Posterior parietal association cortex of the monkey: command functions for operations within extrapersonal space. J. Neurophysiol. 38, 871-908. 17. Paillard J. (1973) Muscular proprioception and position sense. Archs itul. Biol. 111, 451-461.
18. Pellionisz A. and Llinas R. (1980) Tensorial approach to the geometry of brain function: cerebellar coordination via a metric tensor. Neuroscience 5, 11251136. 19. Pellionisz A. and Llinas R. (1982) Space-time representation in the brain. The cerebellum as a predictive space-time metric tensor. Neuroscience 7, 2949-2970. 20. Prablanc C., Echallier J. F., Komilis E. and Jeannerod M. (1979) Optimal response of eye and hand motor systems in pointing at a visual target. I. Spatio-temporal characteristics of eye and hand movements and their relationships when varying the amount of visual information. Biol. Cybern. 35, 113-124. 21. Robinson D. A. (1982) The use of matrices in analyzing the three-dimensional behavior of the vestibulo-ocular reflex. Biol. Cybern. 46, 53-66. 22. Sakata H., Takaoka Y., Kawarasaki A. and Shibutani H. (1973) Somatosensory parietal cortex (area 5) of the rhesus monkey. Brain Res. 64, 85102.
properties of neurons in the superior
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23. Simpson J. I., Graf W. and Leonard C. (1981) The coordinate system of visual climbing fibers to the flocculus. In Proaress in Oculornotor Research kds) Fuchs A. F. and Becker W.. vv. 415-484. Elsevier, New York. 24. So&hting J. F. (1982) Does posit&n s&se at the elbow reflect a se&-if elbow joint angle or one of limb orientation? Brain Res. 248, 392-395. 25. Soechting J. F. and Lacquaniti F. (1981) Invariant characteristics
of a pointing movement in man. J. Neurosci. I, 710-720. 26. Stevens K. A. (1983) Surface tilt (the direction of slant): a neglected psychophysical variable. Percept. Psychophys. 33, 241-250. (Accepted 21 March 1984)
pp.
S-604,
0306-4522/84 $3.00 + 0.00 Pergamon Press Ltd (CJ 1984 IBRO
1984
PSYCHOPHYSICAL DETERMINATION OF COORDINATE REPRESENTATION OF HUMAN ARM ORIENTATION J. F. Laboratory
of Neurophysiology,
SOECHTING
and B. Ross
University of Minnesota Medical School, Minneapolis, U.S.A.
MN 55455,
Ah&act-The coordinate representation of the sense of limb orientation was investigated psychophysically by asking subjects to match the orientation of the arm or of the forearm in several different coordinate representations. Movement of all degrees of freedom of one arm was permitted while
movement of the other limb was restricted to the degree of freedom investigated in that particular experiment. Performance on the tasks was assessed by calculating the standard deviation of the difference in the angles of the two limbs. According to this criterion, we suggest that limb orientation is represented by the angular elevation of the limb and by the yaw angle, referred to a spatial reference frame.
The organization and control of limb movement can be interpreted from the point of view that there exist a series of hierarchical operations which involve coordinate transformations. As an example, one can consider the movement of the arm to a target, a task which has recently been investigated in a number of laboratories.s~g,“,‘5*20*25 It seems reasonable to assume that the location of such a target is described in a coordinate representation extrinsic to the individual, for example Cartesian or polar coordinates. For the limb to move accurately to the target, its location must also be specified in terms of the joint angles of the limb segments involved. In addition, there is experimental evidence that the spatial trajectory of the limb is also specified, 5*2’although the question of whether this specification is in terms of extrinsicI or intrinsic (joint angle) coordinates25 is unresolved. Finally, joint torques adequate to produce this trajectory must be determined’ and the appropriate set of muscles activated. There is also experimental evidence that these latter two operations are distinct, since the same joint torque is produced by different patterns of activation of different muscles participating in a given movement.” Each of these sets of variables (joint angles, joint torques and muscle forces) can be viewed as defining a coordinate system. Going from one set of variables to another then involves a coordinate transformation. Recently, Pellioniasz and Llinb’8~‘g have provided a theoretical framework within which such coordinate transformations can be studied. In general, they noted, the axes of such coordinate systems will not be orthogonal to each other, and that sensory quantities will be represented by their projections onto these coordinate axes (covariant components), while the components of motor quantities add vectorially (con-
travariant components) to produce the desired resultant. They concluded that this endows the central nervous system with tensorial properties which are invariant of the particular frame of reference in which these quantities are represented. Nevertheless, in any instance, such transformations from covariant to contravariant representations must be implemented in a particular coordinate system(s). Robinson2’ has recently presented an analysis of the vestibulo-ocular reflex from this point of view, the sensory coordinate system being specified by the planes of the semicircular canals and that of the motor output by the lines of action of the extraocular muscles. Simpson et al.” have shown that visual information about movement is also represented in the coordinate system of the semicircular canals. For this particular system, the choice of coordinate system appears obvious from anatomy. Such is not the case, however, for limb movements in three-dimensional space. For example, the shoulder joint can be idealized as having three degrees of freedom. As will be shown in detail in the next section, the three joint angles corresponding to these degrees of freedom can be measured in a number of different ways. Similarly, the choice of a coordinate system on the motor side is not obvious because a large number of muscles act at the shoulder. Furthermore, for the reasons mentioned above, the transformation from joint angles to muscle force may involve an intermediate step, namely the specification of joint torques, whose coordinate system also needs to be determined. In this paper, we address the problem of the coordinate representation of joint angles of the human arm. One common means of assessing position sense psychophysically has been to use a matching
Address for correspondence: J. F. Soechting, Department of Physiology, 6-255 Millar Hall, 435 Delaware St. S.E., Minneapolis, MN 55455, U.S.A.
task: a given joint angle in the reference limb is achieved and the subject is then asked to match the joint angle with the other limb.*,‘0,‘2,‘7 In these experiments, movement was always restricted to one degree
N.S.C. 1312-N
595
596
J. F. Soechting and R. Ross
of freedom in each limb and these experiments therefore do not discriminate among different coordinate representations.*’ The experiments to be described in this paper were designed to overcome this limitation. Movement in
all degrees of freedom was permitted in the reference limb, while movement of the other limb (the matching arm) was always limited to the degree of freedom investigated. Our hypothesis is that the “natural” coordinate representation of joint angles would be the one in which the standard deviation in the difference between joint angles of the two limbs would be least. Using this approach under more restricted conditions (location of the arm in the sagittal plane), we found that subjects were better able to match the orientation of their forearms in space than they were able to match the joint angle of the elbow.24 EXPERIMENTAL
PROCEDURES
Angular coordinates of the shoulder joint
The shoulder joint can be viewed as having three degrees of freedom, if translation due to scapular motion is neglected. At least two angles are required to define the orientation of the arm (or the location of the elbow), the third degree of freedom in such a definition corresponding to a rotation about the long axis of the humerus and not affecting the location of the elbow. Customarily, however three angles are used to define the location of the elbow relative to the shoulder: the angle of forward flexion (pitch), the angle of abduction (roll) and the angle of horizontal rotation (yaw). These angles are depicted in Fig. 1. as are alternate ways of defining the orientation of the upper arm. For purposes of illustration, coordinate representations of the right arm are depicted. The shoulder (S) is defined to be at the origin of the Cartesian coordinates X, Y and Z and the point E represents the position of the elbow. The axes X, Y and Z define the principal planes, i.e. XZ is the sagittal plane, YZ the frontal plane and XY the horizontal plane. The angles q,. rt2 and n, in Fig. l(A)-(C) are defined in one of the principal planes. In each instance, the point a is the projection of the elbow onto one of the principal planes and the angles I,. r12and rt3 are the apical angles of the shaded triangles. The angle r), represents the angle of horizontal rotation (yaw), n2 that of forward flexion (pitch) and n3 that of abduction (roll). There are other ways of defining the angular orientation of the arm however, Imagine three orthogonal axes fixed to the humerus such that the humerus is initially vertical along the Z-axis (Fig. I). Any arbitrary position of the elbow can then be achieved by two successive rotations about two of the axes fixed to the humerus. (These rotations, which detine Euler angles, are not commutative that is, the angles depend on the order in which the rotations are performed.) Thus, in Fig. I(A) the angles rh and 0, are defined by an initial rotation about the axis of the humerus (Z-axis), which gives PI,, and then about an axis perpendicular to the plane of the upper arm and forearm (an axis which coincides with the Y-axis when rt, = 0) to define 0,. The angle 8, will be called the angular elevation of the arm. Similarly, in Fig. l(B), there is a rotation first about the Y-axis (to give q2) and then about an X-axis fixed to the humerus (to give 0,). in other words forward flexion followed by abduction. The order is reversed in Fig. l(C); abduction (rb) is followed by _-...____-____. *We have adopted the convention that whenever a&s are expressed without a subscript, the subscript I is implicit (e.g. e = 8, ).
forward
flexion (Ox). As was mentroned a~ the beginnmg ot this section. the shoulder joint has at least three degrees of freedom of motion. Two of them are defined In Fig. I(A)_(C): in each instance. the two rotations described above arc followed by a third about the axis of the humerus. Note that this last rotation does not affect the posttton o) the elbow but will influence the position of the wrist in
space. If we define (.y, .v. I) to be the position of the elbou m the XYZ coordinate system. then
e, = (.P + .+z
tan q, = j’;s
tan
tan rtr 2 .y r
tan 0, _ J./(,x: + ; ‘)1
tan q, = j,,‘;
tan O,=s~(~-‘+r~)~.
(1)
Equations relating the angles to the components of the position of the elbow (equation I) can be derived from the rotation matrix’.’ which relates the coordinates of a point detined in the XYZ coordinate system tixed in space to the coordinates of the same point in an !7%! coordinate system fixed to the arm. When the rotations are defined as in Fig. I (A)* the rotation matrix is given by x
=
cos 0 cos ‘I
)’
=
cos 0 sin q
H 2
= [
-sin0
-sin7
sin e cos q
2
cos r) sin 0 sin n 0
cos U
B 1-H
f (2)
where (x. y. z) and (8,7, L?)represent the position of a point defined in the fixed and moving coordinate systems, respectively. For example, the location of the elbow is given by I,(O, 0, I) in barred coordinates (!, being the length of the arm). In fixed coordinates, according to equation (2), its location is I, (sin e cos q, sin 0 sin rj, cm 0). It may be seen that the detinitions of 7, and 0, given in equation (I) follow directly. Task
In a given experiment, subjects were asked to match one ofthesixangles(rl,,42.9,;B,.8~, 0,)showninFig.lwith their right and kft arms. They were presented a target, generally in the right, anterior quadrant (x > 0, y > 0), asked to close their eyes and then to point to the target with their right arm. In two experiments, the right arm was positioned by the experimenter. The left arm was constrained to move in one of the three principal planes (sagittal, frontal or horizontal). Note that when the location of the arm is restricted to the sag&al plane, 8,, q2ad 8, are identical, as are 0,, fJ and n, for arm locations in the frontal plane. Therefore, the angles described by the left arm used to match the orientation of the right arm are independent of the coordinate description. The subjects were asked to match the specihed angle formed by their right arm with a planar movement of their left arm. Thus, the angles 0,. q2and 0, were matched by a forward flexion of the kft arm, 0,. O2and ‘1, by abduction and 9,. by a movement of the left arm in the horizontal plane. In any given experimental session, subjects were asked to match two of the angles. Their performance on the tasks was quantitated by calculating the coefficient of correlation @) between the ass&a of the two arms and the standard deviation of the dil%rence. Before the experiment bqaa, the subject13wem shown a small threed&nsIoaal se& model (timBar to Fig. I) which deIined the an&a tbay were to matds. brey were also told that matcbhrg B, (ms&u elevation) was equivalent to matching the height z of the right and kft elbow, and matching 0, and 0, equivalent to matching the lateral ( y) and anterior (x) projections of the elbows, respectively. Matching r~r(forward flexion) and ‘I, (abduction) is quivaht to aaching the ulapenof the two amm. They were tild to imagine a sandwich board or an A-frame attncbal to their trunk and that nr or n, would be matebed if the two
Coordinate representation arms were parallel to the planes of these structures. The subjects were given a few practice trials with eyes open to ensure that they understood the tasks. Data recording and analysis The orientation of the right arm in space was measured by means of a pair of ultrasound emitters and a set of three orthogonal linear microphones which were aligned with the Cartesian coordinate axes of Fig. l.25 The two emitters were strapped to the arm, one at the shoulder level and the other at elbow level, on a line parallel to the arm. Using this system, the position of these two points in space was obtained and the value of the angles ‘1,- q, and 0, - 0r were calculated according to equation (1). The left arm was strapped to a manipulandum which pivoted about the axis of the shoulder and which was constrained to rotate about either the Y-axis (forward flexion) or about the X-axis (abduction). Angular position of the left arm was measured by means of a potentiometer. Angular coordinates of the elbow joint The elbow joint can be viewed as having two degrees of freedom (flexion-extension and pronation-supination). Only the former of these affects the orientation of the arm, while the latter influences the orientation of the hand. (The question of the coordinate representation of the hand is beyond the scope of this paper.) The third degree of freedom at the shoulder joint (axial rotation), however, does influence forearm rotation. Thus also for the forearm two angles are required to specify its orientation. The possible coordinate representations of forearm orientation which were investigated were influenced by our findings on the coordinate representation of the arm and by previous psychophysical studies regarding position sense at the elbow.24 As we show in results, the subjects’ performance was best when the task required them to match the joint angles shown in Fig. lA), that is, the angular elevation of the arm (0,) and the yaw angle (q,). Furthermore, previous work” showed that subjects were able to match the elevation of their forearm (angle of the forearm measured relative to the vertical) better than they were able to match the angle of forearm extension (angle subtended by the forearm and the arm). In those experiments, the arm was restricted to lie in the sagittal plane. Thus in this limiting case, forearm orientation is also described by its angular elevation. We hypothesized two possible coordinate representations of the forearm which are consistent with this limiting case. The first of these is shown in Fig. 3(A) and is analogous to the coordinate representation of the arm. The angle b, represents the elevation of the forearm and is measured in the vertical plane aEW, E denoting the elbow and W the wrist. The yaw angle a, is the angle between this vertical plane and the sagittal plane X-Z. An alternative coordinate representation is shown in Fig. 4(A). It represents an Eulerian description of the three degrees of freedom at the shoulder. Two angles are given by r~and-0 (as in Fig. lA), corresponding to rotations about Z and Y. The third angle, z2, denotes the amount of axial rotation of the arm (about the axis of the humerus). The reference value, a2 = 0 corresponds to the instance in which the plane of the arm (SEW) is vertical. The angle 1, represents the degree of freedom at the elbow and is equal to the angular elevation of the forearm when a2 = 0. It is unchanged by axial rotation. Note that b, and & are identical when a, is zero, that is when the arm is in the vertical plane. Furthermore, as noted above, when the arm is in the sagittal plane (a2 = 0 and ‘1,= 0) subjects are able to match this angle better than they are able to match the angular extension at the elbow (4). The angles a, and b, can be calculated from tan a, = -y/x
tan/I, = -(x2+y2)$r
(3)
597
of human arm orientation
where x, y, z denote the position of the wrist measured relative to the elbow in the XYZ coordinate system (Fig.
3A). The angle jr is given by b2 = 4 - 0,
(4)
where 0, is defined by equation (1) and Q is the angle of elbow extension. Finally, a2 is given by sin a2 sin 8, = n,
(5)
where n, is the Z-component of the normal to the plane formed by the arm and the forearm. The experimental design used to evaluate position sense of the forearm was similar to that described for the shoulder. Subjects were asked to match the angles b, and /I2 with their two arms. The left forearm was again strapped to the manipulandum which pivoted about the axis of the elbow and was constrained to move in the sag&al plane. The left arm was close to the vertical. Thus, for the left arm, the angles /I, and fir were identical. The position of the wrist and elbow of the right arm was determined by means of the ultrasound emitters described above. From these measures, the angles a, and /3, were calculated according to equation (3). The elbow angle C$was measured electrogoniometrically.2 At the onset of each experiment, the center of rotation of the right shoulder was determined. The subject was asked to move the arm in an arc with ultrasound emitters attached at the elbow and below the shoulder. The best estimate of the centre of rotation was determined graphically from these data. From the location of the shoulder, elbow and the wrist, the notmal vector to the plane of the arm was calculated by standard trigonometric methods and the angle 0, according to equation (I). Finally, the angles a, and fi2 were determined according to equations (4) and (5). The data reported in this paper summarize the results of 33 experiments involving 18 subjects. RESULTS
The results of experiments dealing with the coordinate representation of the orientation of the arm, are summarized in Table 1. In each experimental session, a paired comparison of a subject’s performance on two matching tasks was obtained. Thus, in the first four experiments, the subjects attempted to match either the angular elevation (0,) Fig. 1A) or the angle of forward flexion (qt, Fig. 1B) with both arms. Movement of the right arm was unrestricted, while that of the left arm was confined to the sagittal plane (forward flexion, FF). n Represents the number of trials in each task, while the values reported in the next two columns are the standard deviation (SD) in degrees of the difference between the angles of the right and left arms and the coefficient of correlation @) of the linear regression between them. As may be seen, all four subjects performed better when the task required them to match 19,(the angular elevation of the arm), in the sense that the standard deviation was smaller and p was larger on this task than when the subjects attempted to match the angle of forward flexion (or). This is also evident in Fig. 2(A) and (B), which show the performance of one subject on the two tasks. Each data point presents the results of one trial. A 45” line has been drawn for reference; it represents perfect performance on the task.
59s
J. F. Soechting and B. Ross
Table 1. Performance in matching orientation of the arm 0, FF
n
SD
p
n
40 30 41 40
7.0+* 6.5,’ 6.2+ 6.S’
0.953 0.962 0.952 0.972
41 37 41 41
6.5+*
O.Y60
Av.
;; q? Q
SD 12.9 15.6 8.8 12.8
P 0.939 0.850 0.908 0.850
12.5
0.887
FF
41 40
8.0** 7.4**
Passive 0.939 q* 0.965 q2
39 40
13.2 11.8
0.915 0.932
FF
36 40 38 41
%I** 6.3.. 6.5+* 8.1**
0.960 0.947 0.983 0.964
35 40 39 40
11.4 14.1 13.0 16.6
0.763 0.589 0.889 0.823
6.5..
0.960
13.8
0.766
6.8*+ 5.1 5.6’+ 5.8’.
0909 0.959 0.963 0.957
11.0 5.1 9.6 12.7
0.792 0.939 0.872 0.776
5.8
0.947
9.6
0.844
4.8” 4.7*
0.963 0.966
15.3 6.5
0.738 0.925
Av. ABD
41 41 40 40
Av. ABD
40 40
6), 8, OI 8,
‘I, ?I> & q,
0, e*
40 41 40 40
40 40
Each row presents standard deviation (SD) and coefficient of correlation @) for one sub&t; IeA columns: matching angular elevation of the arm by forward flexion (FF) or by abduction (ABD) and right columns: matching the angle indicated. l*p < 0.005. *P < 0.05.
A statistical analysis of the variance of each subject’s performance on the two tasks was performed. For all four subjects, the variance on the task which required matching t12 was sign.i&antly greater, as indicated by the symbols in Table l(**P < 0.005, *P < 0.05). In addition, a statistical analysis of the pooled data was performed (paired Student’s f-test on the logarithmic ratio of the variance). Also ac-
cording to this criterion. the subjects pert’ormed significantly better in matching the elevation of the arm ((I,) than they did in matching the angle of forward flexion (q2). Similar results were also obtained when the right arm was passively positioned. Also in this case, the two subjects were able to match angular elevation 8, better than they were able to match the angle of forward flexion. The subjects’ performance in matching 0, was also significantly better than their ability to match 0, (Fig. IC). Results obtained for one subject on these two tasks are shown in Fig. 2(C) and (D), and the data for four subjects are summarized in Table I. Thus the subjects’ performance in matching forearm elevation was superior to their performance on the two alternative tasks, suggesting that the coordinate system illustrated in Fig. I (A) is the one which is used preferentially to represent the angular orientation of the arm. At the conclusion of each experiment, each subject was interviewed. All but one expressed the opinion that matching forearm elevation was easier. At the beginning of the experiment, as the tasks were explained, opinions were mixed, a number of subjects volunteering the opinion that one of the two other tasks would be easier. If our hypothesis is correct that the coordinate system shown in Fig. l(A) is the preferred coordinate representation, then the subjects should also be better able to match forearm elevation (0,) than the angle of abduction measured in one of two possible ways (&, Fig. I B; q3, Fig. IC). This prediction was verified, as shown in Table 1. In these last two sets of experiments, the left arm was constrained to move in the frontal plane (abduction, ABD) and the subjects were required to match 0, and q, or &. Five of the six subjects performed significantly better when asked to match the elevation of the arm. Our considerations of possible coordinate systems
Vertical Fig. 1. Possible coordinate rqxcacntations of the spatial orientation of the arm. Each part of this t@re shows dilTefent angular masmos wtdoh may be used to describe the orientation of the arm. The pmnt E dmotea the location of the dbow and S the location of the shoulder, w&h is al the origin of the cartcsian~~notesy~xyz.~&~~rxms~aritbtba~~Qfthebody_x, anteriorY latarPlagdZ,v~.Ib:~~imliate~ptojcctioaoftheannontooncof thethre;~rmdpalpbesaadrk~~,,~~~q,ue~-in~~~.IhepnolG8,is
measured in the vertical plane of the arm relative to the Z-axis and the angle9 0, and 0, are mcasurod in the plane of the triangle GE, where II is the projection of E onto one of the principal planes.
Coordinate representation
of human arm orientation
80° -
00
40°
0,
80°
O0
40°
0,
80°
00
400
72
80°
40.
e,
80.
Fig. 2. Results of matching arm orientation in different coordinate representations. The results presented in (A) and (B) were obtained from one subject, those in (C) and (D) from another subject. In (A) and (C), the subjects matched the angle 0, (Fig. 1A) of their right arm by means of forward flexion of their left arm. Each data point represents the results from one trial. The 45” line, which is given for reference, indicates perfect performance on this task. In (B) and (D) the same subjects matched the angle g, (Fig. 1B) and G3(Fig. 1C) of their right arm by means of forward flexion of their left arm. Note that in both
instances, the subjects’ performance was better on the task which required them to match 0,. Details of the experimental design are given in the text.
which might be used to represent the orientation
of the forearm was limited by the findings on position sense at the shoulder described above and also by some preliminary observations reported elsewhere.24 Arguing by analogy with the results described above, which indicate that yaw angle (q, ) and elevation (0,) are most appropriate to describe the orientation of the arm, one might expect a similar coordinate description to hold for forearm orientation. This possibility is shown in Fig. 3(A), where a, represents the yaw angle measured relative to the frontal axis (X) and /?, the angle of forearm elevation measured in the vertical plane of the forearm. An alternative possibility is shown in Fig. 4(A). The actual position of the right limb is indicated by the heavy lines, S, E and W denoting the location of the shoulder, elbow and wrist. The angle fir denotes the elevation the forearm would have if the arm were to lie in a vertical plane. The angle a2 represents the amount of angular rotation, about the axis of the humerus, required to
bring the forearm from its actual position to the virtual position in the vertical plane. (Note that the amount of elbow extension does not change during this virtual rotation, i.e. it is performed with the elbow locked.) A matched comparison of the performance of seven subjects on these two tasks was made. Representative results from two subjects are shown in Figs 3(B) and 4(B), while the data from all subjects are summarized in Table 2. (Note that no data on f12are presented for one subject. During the experiment, it was obvious that matching error on this task often exceeded 20” and the experiment was terminated.) On average, the standard deviation on the task involving matching /?, was slightly less (8.2” vs 9.8’) and the coefficient of correlation slightly greater. These differences are not statistically significant, however. During these experiments, it appeared that the subjects made the largest errors in matching forearm elevation (/?,) when the position of the right hand was
J.F. Soechting and B. Ross
600
S
2oo /
80 °
/: o
40 °
X
•
|
0o
40 °
I
i
/31
l
,
80 °
120 °
Fig. 3. One possible coordinate representation of the orientation of the forearm. (A) schematicallydepicts the arm, S, E and W representing the position of the shoulder, elbow and wrist. The angle ~ and 0 are identical to r/I and 0= defined in Fig. I(A). The line ab represents the projection of the forearm onto the horixontal plane (XY), and the angle =l (yaw) is the angle between that line and the anterior (X) direction. The angle/3, (elevation) is measured in the vertical plane (aEWb) relative to the vertical (Z) axis. (13)shows the performance of one subject in matching the angular elevation of the two forearms. The filled circles indicate trials in which the distance d between the hands was greater than 35 cm, the open cirles those in which d was less than 35 c~. close to the midline of the body, that is when the right and left hands were near each other. In fact, in many of these instances, subjects would report that they knew that the attempted match was in error, but that nevertheless "it felt right". Furthermore, the errors were generally consistent in that subjects tended to underestimate the angle /~, with their left forearm, that is, they tended to flex the forearm more than required.
The values of the standard deviation and correlation coefficient for the trials in which forearm elevation was matched, excluding those trials in which the distance between the hands was less than 35 cm, are reported in the second column in Table 2. Note that when these trials are excluded, the standard deviation decreased for all subjects, by 22~ on average, and the coefficient of correlation increased. Excluding trials in which inter-hand distance was less
B 80 o
x
s
Z
4o0
i. 0*
I
,
40 °
~=
I
80 e
Fig. 4. Another possible coo~inate representation of the orientation of the forearm. Note that the perspective of the arm in (A) is different from that used in Fig. 3(A). The point a represents the projection of the location of the elbow onto the horizontal plane. The plane SaE is vertical. W represents the actual location of the wrist and the dashed fine represents the orientation the forearm would have if it were to lie in the vertical plane SaE. The angular rotation required to bring the forearm from the actual to the virtual orientation is given by =2; it is about the axis of the humerus SE. This rotation is performed with the elbow locked, i.e. the elbow angle ~ does not change. The angle ~2 is measured in the vertical plane SaE and describes the angle of virtual forearm orientation relative to the vertical axis. (B) shows the performance of one subject on this task.
Coordinate representation
Table 2. Performance in matching orientation of the forearm B,: . all trials n
SD
P
41 42 45 39 43 45 46
7.2 10.6 8.6 5.0 7.2 9.5 9.6
0.948 0.900 0.949 0.939 0.931 0.908 0.827
Av.
8.2
0.915
fit : d > 35 cm -i SD p
n
31 33 38 28 28 40 34
82 SD
p
3.5 6.8* 7.5 4.1** 6.1** 1.5** 9.5
0.963 0.953 0.958 0.949 0.954 0.931 0.840
38 9.0 46 8.1 37 8.4 41 9.5 46 12.0 44 11.7
0.883 0.948 0.891 0.870 0.910 0.840
6.4*
0.936
9.8
0.890
The left column summarizes data for the task of matching angular elevation of the forearm (B, ), the middle column for the same task with trials in which the two hands were within 35 cm excluded, and the third column for the task of matching &. **p < 0.005. *P < 0.05.
than 35 cm did not improve the measures of performance for the task of matching fir, the standard deviation decreasing on average by less than 1% (and no more than 4% in any subject) while p actually decreased 2%. When trials in which the two hands are close together are excluded, four of six subjects performed significantly better matching forearm elevation than they did matching /$. The average performance of all six subjects was also significantly better.
20
Sn
-20
(cm)
S,
Fig. 5. Orientation of the two arms during one trial in which the distance between the two hands was less than 35 cm. In this trial, the subject attempted to match the angular elevation of the forearm (8,). The points S, E and Wdenote the location of the shoulder, elbow and wrist; the subscripts R and L indicating the right and left limb. The points n and 6 are the projection of Ea and R’s on the horizontal plane. The dashed line for the right limb indicates the estimated position of the right hand, the location of the wrist and elbow having been measured. The left limb was in the sagittal (XZ) plane and the solid line and the points EL and W, denote the orientation of the limb and the position of the left elbow and wrist. The dashed line indicates the angular elevation of the left arm which would have matched that of the right forearm, as specified by the task. Note that the actual position of the left forearm more closely matched the height of the two hands.
of human arm orientation
601
There are two possible reasons why trials in which the right hand was close to the mid-sagittal plane should lead to large errors in matching forearm elevation. First, in these instances, the axial rotation (az) of the arm was appreciable. If the subjects were in fact matching /?* rather than fl,, they would tend to underestimate /?, (J& always being no larger than /&). This possibility was investigated by asking subjects to position their arm obliquely (axial rotation a2 ranging from 30” to 90’) with the hand far from the mid-sagittal plane. Such orientations of the arm did not, however, lead to errors in matching /?, significantly larger than average. The second possibility is that, when the right hand
was close to the midline, the subjects actually tended to match the position (primarily the height) of the two hands. This is illustrated in Fig. 5, which depicts schematically the orientations of the two arms for one trial in which the error in matching /$ was large. The symbols S, E and W denote the position of the shoulder, elbow and wrist, the subscripts R and L refer to the right and left limb, respectively. The planes S,aE, and ub W, are vertical. The dashed line denotes the orientation of the left forearm which would match the angle of elevation of the right forearm. Note that the actual position of the left forearm instead matches more closely the height (Z-axis) of the two hands, as hypothesized. Therefore, it seems safe to conclude that one of the measures used to specify the orientation of the arm is the angle of elevation, as defined by 0, for the shoulder (Fig. 1A) and by 8, for the elbow (Fig. 3A). In each instance, a second angular measure is needed to fully determine the limb’s orientation in space, the most logical candidate being the yaw angle (q, or a,). In a series of experiments, we also investigated the ability of subjects to match these two angles with their arms. The results of these experiments are summarized in Table 3 and representative data are shown in Fig. 6. In the experiments in which subjects were asked to match the yaw angle at the shoulder (q,), the left arm was constrained to move in a horizontal plane and the matching was performed by an anti-symmetric rotation of the two arms. That is, a rightward rotation of the right arm (corresponding to a positive value of q, ) was matched by a leftward rotation of the
Table 3. Performance in matching yaw angle of the arm (q, ) and of the forearm (a,) ‘11 n
SD
p
n
aI SD
p
51 56 40 41
10.3 11.2 8.5 11.1
0.856 0.897 0.869 0.839
45 44 47 43
10.6 9.4 10.4 14.0
0.953 0.944 0.942 0.867
Av.
10.3 0.865
Av.
11.1 0.926
J. F. Soechting and B. Rosh
A
,.6 .
.
.
.
.
. .C,
.
4’ . . /. . . 80” . .’
I
/ .
1
&T.,
O0
?
400
80°
I
-80°
-400
a
I
I
I
00
4o”
80”
Fig. 6. Performance of two subjects in matching yaw angles of the arm and forearm. (A) shows data points of trials in which the subject attempted to match the yaw angle r) (Fig. 1A) of the arm and (8) results of an experiment in which a subject attempted to match the yaw angle a of the forearm (Fig. 3A).
left arm. Similarly, in the experiments in which the task was to match the yaw angle of the forearm, the task was accomplished by an anti-symmetric rotation of the two forearms, the left forearm being horizontal, the left arm vertical and the axis of rotation about the humerus. The performance of the subjects in matching yaw angles was distinctly inferior to their performance in matching the angle of elevation. One possibility which might account for this finding is that the matching paradigm chosen was not the appropriate one, that is, that matching yaw angles would be produced by a symmetric rotation of both limbs. This possibility was explored in two experiments in which subjects were asked to match yaw angle at the shoulder. The errors in matching joint angles under this condition were even greater than those reported in Table 3. DISCUSSION
Our choice of a natural, or preferred, coordinate representation of the arm is based on the hypothesis that the task of matching joint angles with the two arms would be performed best in this natural coordinate representation, that is, that the standard deviation of the error would be least in that particular coordinate representation. According to this criterion, the data presented in this paper indicate that most probably the sense of limb orientation is expressed psychophysically in the coordinate system illustrated in Fig. 3(A), namely that the orientation of each limb segment is described by the yaw angle (r), or I, ) and the angular elevation (0, or /l, ). Note that these at&es describe the orientation of the limb segments relative to an absolute frame of reference, referred to the sagittal plane and to the vertical,
rather than a relative orientation, such as for example, the joint angle of the forearm relative to the arm. The performance of our subjects was not uniformly good in matching these four angles. The angular elevation of the arm (6,) was matched quite well, with an average SD of 6. I’ when the right arm was actively positioned by the subject. When the right arm was passively displaced by the experimenter, the average error was slightly larger, in agreement with findings that the error in matching joint angles increases when one limb is displaced passively.” In 15 of 16 experiments, the error in matching limb elevation was significantly less than that for the other angular measures considered for the shoulder joint. Subjects did not perform as well in matching the anguIar elevation of the forearm (SD = 8.2”). We have suggested that the error was larger for this angular measure because subjects tended to match the position of the two hands (their vertical height) when the hands were close together. In fact, when trials in which the two hands were within 35 cm of each other were eliminated, the error in matching forearm elevation (SD = 6.4”) was comparable to that in matching the elevation of the arm. The errors in matching yaw angles were considerably larger than those for the angular elevation of the limbs. Nevertheless, we can see no obvious alternative angular measures which, in conjunction with the angular elevation, would specify the oriertttttion of the limb. In our experiments, yaw angle was measured relative to a spatially fixed coordinate frame, namely the sagittal plane. It is poibk that subjects, instead, were matching yaw angks relative to a perceptual sagittal plqne (paocptual straightahead) which could vary from trial to trid, and that UtistIWy aowunt for the laager error. If it is accepted that the coordinate system we have
Coordinate representation of human arm orientation proposed is a preferred one, the question arises: which of the parameters relevant to the organization and control of limb movements are described in this coordinate system? One likely possibility is that this representation of sensory information is used to orient the arm relative to objects in space. Stevens26 in a recent psychophysical study has proposed that the orientation of surfaces is perceived in angular coordinates (slant and tilt) similar to those proposed here. The major difference is that slant and tilt are measured relative to the line of sight rather than relative to the vertical as in our case. However, Gibson6 has suggested that surface orientation is perceived relative to both line of sight and the gravitational vertical. If surface orientation relative to this latter axis is also perceived by similar angular measures, the slant angle would be identical to the angular elevation and the tilt angle equivalent to the yaw angle in our angular description. The question of whether or not movement trajectories are also planned in this coordinate systems and if this coordinate representation is also an appropriate one for describing joint torques remains open.
603
Regarding the neural correlates of the psychophysical variables, only speculations can be offered at this time. The assumption that joint afferents contribute exclusively to limb position sense has been questioned recently’3T’4and it has been suggested that tactile and spindle receptors also contribute. Our finding that forelimb orientation is expressed in a frame of reference relative to the trunk (or some spatial frame of reference) rather than relative to the upper arm implies that there must also be a convergence of sensory inputs from different limb segments. Furthermore, it seem plausible that vestibular and visual information are also utilized. Neurons responding to movements of several joints have been found in primary somatosensory cortex3 and posterior parietal cortex.‘4*‘6*22The precise information encoded by such neurons remains to be determined, however. Acknowledgements-This work was supported by USPHS Grant NS-15018 and by NSF Grant BNS-8117625. The authors thank Dr. A. P. Georgopoulos for his valuable advice concerning statistical analysis of the data and Dr. R. E. Poppele for his critical reading of the manuscript.
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