The dynamics of sensorimotor calibration in reaching-to-grasp ...

J Neurophysiol 110: 2857–2862, 2013. First published September 25, 2013; doi:10.1152/jn.00112.2013.

The dynamics of sensorimotor calibration in reaching-to-grasp movements Geoffrey P. Bingham1 and Mark A. Mon-Williams2 1

Department of Psychological and Brain Sciences, Indiana University, Bloomington, Indiana; and 2Institute of Psychological Sciences, University of Leeds, Leeds, United Kingdom

Submitted 12 February 2013; accepted in final form 21 September 2013

Bingham GP, Mon-Williams MA. The dynamics of sensorimotor calibration in reaching-to-grasp movements. J Neurophysiol 110: 2857–2862, 2013. First published September 25, 2013; doi:10.1152/jn.00112.2013.— Reach-to-grasp movements require information about the distance and size of target objects. Calibration of this information could be achieved via feedback information (visual and/or haptic) regarding terminal accuracy when target objects are grasped. A number of reports suggest that the nervous system alters reach-to-grasp behavior following either a visual or haptic error signal indicating inaccurate reaching. Nevertheless, the reported modification is generally partial (reaching is changed less than predicted by the feedback error), a finding that has been ascribed to slow adaptation rates. It is possible, however, that the modified reaching reflects the system’s weighting of the visual and haptic information in the presence of noise rather than calibration per se. We modeled the dynamics of calibration and showed that the discrepancy between reaching behavior and the feedback error results from an incomplete calibration process. Our results provide evidence for calibration being an intrinsic feature of reach-to-grasp behavior. calibration; sensorimotor adaptation; dynamics; cue combination; feedback control CALIBRATION IS CENTRAL TO skilled human behavior because it allows previously acquired skills to operate in new environments or in the presence of neurophysiological change (caused by factors such as muscular fatigue). The role of calibration has been well established in the reflexive behaviors found in mammals (see Tresilian 2012 for a review). For example, the vestibular-ocular reflex is known to calibrate to an error signal provided by retinal slip so head rotations are counteracted by rotational eye movements in the opposite direction (Tresilian 2012). Calibration also appears to be fundamental to learned motor skills as well as reflexive behavior. A number of studies have provided evidence for the calibration of targeted actions, including visually guided locomotion (e.g., Bruggeman and Warren 2010; Rieser et al. 1995), throwing (e.g., Martin et al. 1996; Rieser et al. 1995), and reaching-to-grasp (e.g., Coats et al. 2008; Mon-Williams and Bingham 2007). The modification of reach-to-grasp behavior requires terminal feedback (an error signal regarding reach distance and grasp size and orientation) that can be provided by vision (e.g., Bingham et al. 2000; Martin et al. 1996; Rieser et al. 1995) or haptics (e.g., Coats et al. 2008). Mon-Williams and Bingham (2007) explored the alteration of reach distance by gradually distorting the haptic feedback obtained when participants grasped visible target objects. The results indicated that the relationship between visually specified distance and reach distance could be captured by a straight-line mapping function. Mon-Williams and Bingham showed that the bias and slope of this function could

Address for reprint requests and other correspondence: G. P. Bingham, Dept. of Psychological and Brain Sciences, Indiana Univ., 1101 E. 10th St., Bloomington, IN 47405 (e-mail: [email protected]). www.jn.org

be changed in response to distorted haptic feedback and suggested that this provided evidence for calibration in reach-tograsp behavior. It is notable, however, that Mon-Williams and Bingham (2007) found that behavior was modified to an extent less than the error signal provided by the haptic feedback. In fact, all calibration studies employing distorted haptic feedback have resulted in responses that show incomplete compensation for the error signal. Theoretically, reach-to-grasp calibration should result in reaches that yield responses to visually specified distances that accurately reflect the distorted haptic feedback. For instance, if the distorted haptic feedback indicates that targets are 6 cm closer then reaches should alter by this amount (and direction) when reaching for visually specified targets. But the results of existing studies have not yielded this full effect of distorted haptic feedback (for example, reaches have only exhibited ⬇4 cm change with distorted haptic feedback equal to 6 cm). An alternative explanation regarding the incomplete compensation for the signaled error is that the changes reflect the weighting of the visual and haptic information in the presence of noise rather than the calibration of reaching behavior. Modified weak fusion (Landy et al. 1995) is a model of cue combination wherein the combination is a weighted average related to the inverse of a cue’s variance. Weak fusion was originally applied to the combination of different sources of visual information. Subsequently, it has also been applied to the combination of information coming from different modalities, for instance, vision with audition (Heron et al. 2004) and vision with haptics (van Beers et al. 1999). Weak fusion (or weighted averages) is thus applied as a model of sensory integration (Ernst and Bulthoff 2004; Ghahramani et al. 1997). Van Beers et al. (2002) applied modified weak fusion to the combination of visual and kinesthetic information regarding the spatial location of the hand. Van Beers et al. argued that the specification of reachable locations varying in the coronal plane would be better provided (lower variance) visually than kinesthetically. In contrast, they suggested that locations varying along the midsagittal plane (that is, in depth) would be better specified by kinesthesis than vision. In the latter case, they predicted that kinesthetic information should be weighted more heavily than visual information. In their study, a location was specified kinesthetically by the participant’s right hand pointing above a table but underneath a mirror that reflected a computer graphics display sitting above. The location of the right hand (occluded by the mirror) was specified visually in the display by a spot that was projected onto the table top by the mirror. When the kinesthetic and visually specified locations were separated in depth, the participants pointed below the table with their occluded left hand to a location

0022-3077/13 Copyright © 2013 the American Physiological Society

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about two thirds of the way closer to the kinesthetic location than the visual. The results of van Beers et al. 1999 and van Beers et al. 2002 suggest that modifications in reach distance following distorted haptic feedback might reflect the weighting attached to the visual and haptic information rather than calibration per se. The modifications would result because the increased noise associated with the haptic feedback would cause the nervous system to adjust the weight attached to the visual and haptic information (Burns and Blohm 2010; Burns et al. 2011; Sober and Sabes 2005, 2003). In line with this suggestion, Smeets et al. (2006) have argued that the observed drift in the perceived location of an unseen hand can be predicted on the basis of optimal sensory combination and the assumption that the visual estimate of hand position becomes gradually less precise with each movement made by the unseen hand. We wished to test whether the incomplete compensation observed in reach-tograsp behavior following distorted haptic feedback reflected the dynamics of calibration or the weighting attached to the visual and haptic information. To test between these alternative hypotheses, we used a model of the dynamics involved in calibration developed by Burge et al. (2008). Burge et al. used less than unity gain (K ⬍ 1), so their model was similar to feedback control models from engineering wherein models of real-time feedback control use less than unity gain to keep the dynamics stable in the face of visual feedback delays (DiStefano et al. 1990; Franklin et al. 2009; Jagacinski and Flach 2003). However, the constraint in the case of human calibration is not real-time delays but the effect of perceptual or motor noise (i.e., variable error in contrast to any constant error or bias). Strong corrections of a constant error or bias would destabilize the performance dynamics in the face of variable errors, because the correction would also be applied to the variable error. Burge et al. (2008) used a Kalman filter to transform levels of variable error to determine the value of the feedback gain. They showed that the model yields an exponential response to error feedback as follows: ␥t ⫹ ␣, C ⫽ ␤⫺ e

Before participating in the experiment, all participants read and signed consent forms approved by the Indiana University IRB. Apparatus. Participants sat near the corner of an L-shaped table so that one surface of the table lay in their sagittal plane and the other arm of the table was in the coronal plane to their left (Fig. 1; Mon-Williams and Bingham 2007). A semisilvered mirror (40% light transmission) 33.7 cm ⫻ 24.3 cm extended across the corner of the L so that it was 45° to the line of sight. The mirror was placed so that the top edge was at average eye height. The table allowed participants to reach comfortably behind the mirror. The mirror allowed us to create the illusion that an object was behind the mirror by using an image of an object physically located in front of the mirror. Our design allowed us to produce a visually perfect “virtual object” by removing the physical object behind the mirror. It also allowed us to produce an environment in which the visual and physical properties of an object were in perfect agreement or an environment where there was a mismatch. We adjusted each participant’s eye height to the mirror by changing the seat height. A wooden surface 38 cm ⫻ 80 cm was placed over each arm of the L at 40 cm below eye height. Each of these surfaces was cut diagonally to fit against the bottom of the mirror. Targets were placed on this surface extending to the left of the participant (i.e., in front of the mirror). These targets were viewed in the mirror as if located on the surface behind the mirror extending away from the participant. Cylindrical target objects could be placed on the two surfaces so that they appeared coincident and so that they gave the impression that there was a single object. The illusion was absolutely convincing. Target cylinders were hardwood, 7 cm in height, 7 cm in diameter, and painted matte black. The target placed in front of the mirror was covered with bright dots 1 cm in diameter, whereas the target placed behind the mirror was only black. A black panel was fixed to the back of the mirror by a screw at the upper left corner of the mirror so that the panel also rested on a nail on the bottom right corner of the mirror. A string attached to the right upper corner of the

A

Target and feedback objects

L-shaped table

Semisilvered mirror, angled at 45°

(1)

where C is the correction (or rescaling) on a given trial in response to the feedback, t is time, ␤ is a scaling constant reflecting the size of the error, ␣ is the asymptote of the change (e.g., visual distance minus change indicated by distorted feedback), and ␥ is a time constant. The time constant, ␥, is directly proportional to the feedback gain.1 Burge et al. tested this model with distorted visual feedback from reaches to visual targets. We applied this model to study the response of the nervous system to distorted haptic feedback provided over different times.

Occluder

B

Target object

Semisilvered mirror, angled at 45° L-shaped table

METHODS

Participants. Eighteen participants (8 female, 10 male) took part, with nine (4 female, 5 male) participating in the 20-block condition and nine (4 female, 5 male) in the 30-block condition. Participants were remunerated at a rate of $7/h for their time. All participants had normal or corrected-to-normal vision and were free of motor deficits. 1

This equation has been used previously to model calibration or “adaptation” of reaches (e.g., Bingham and Romack 1999).

Feedback object Occluder not shown Fig. 1. Illustration of the apparatus used in these experiments. See the text for description. A shows the apparatus from above and B shows a perspective view.

J Neurophysiol • doi:10.1152/jn.00112.2013 • www.jn.org

CALIBRATION DYNAMICS

Fig. 2. Results for the 20-block group. Mean reach distances plotted as a function of block order together with fitted model functions. Horizontal lines show the visually specified distances of targets at 15, 21, and 26 cm. Œ, Mean reach distances. Error bars are standard error.
, Exponential model curve fitted to 20 blocks of trials including the Snapback data. }, Exponential model curve fitted to 14 blocks of trials excluding the Snapback data.

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understood that they would be reaching to targets at different distances and that they would never contact an object at the medium and far distances, but would be contacting an object during feedback trials at the near distance. Visible targets were placed at one of three distances from the hand: near ⫽ 15 cm, medium ⫽ 21 cm, and far ⫽ 26 cm. Reaches were tested in blocks of trials with distance randomized. A block consisted of three reaches, one reach to each distance. Distorted feedback was only provided for one target with the other targets allowing us to probe transfer to the other target distances. In the first phase (“changing feedback”), eight blocks of feedback trials were tested with progressively distorted haptic feedback provided only for the near object. Every two blocks, the haptic feedback object at the near distance was moved 1 cm closer to the participant. In the second phase (“steady-state feedback”), feedback trials were tested in which constant (3 cm closer) distorted haptic feedback was provided only for the near object. Vision was occluded between trials. The 20-block group received 6 blocks in this condition. The 30-block group received 16 blocks in this condition. In the third phase (“snapback”), no feedback was provided for 6 blocks of trials; that is, only virtual objects were grasped at all distances. In summary, both groups were exposed to three phases (changing feedback, steady state, snap back) with the only difference being the number of steady-state blocks administered to the two groups. A technical problem meant that the data of one participant in the 20-block group was lost so we report results for 8 participants in the 20-block group and 9 in the 30-block group. RESULTS

panel extended up over a pulley so that when the experimenter, who was standing on the inside of the L, pulled the string, the panel rotated around the screw up and away from the mirror, giving the participant a simultaneous view of the image in the mirror and the coincident scene behind the mirror. With the panel up, the participant could see his or her hand behind the mirror grasping a virtual object. This method is how visual feedback could be provided. (In the present experiments, this was only used by the experimenters to calibrate the apparatus to ensure that visible and grasped target locations were as intended.) With the panel in the down position, the scene behind the mirror was occluded. Reach kinematics were measured using a three-marker Ascension Mini-bird (Ascension Technology, Burlington, VT) electromagnetic measurement system. Movements were sampled at 60 Hz. We calibrated the measurement volume, checking loci every 2 cm in a three-dimensional grid over the reach space. Measurements were reliable and accurate to within 1 mm. Markers (1.1 cm ⫻ 0.8 cm ⫻ 0.8 cm) were placed on the nail of the index finger, the thumbnail, and the wrist of the right hand with double-sided tape. (The mean of the index and thumb marker locations was used to evaluate reach distance.) The wires were gathered around the forearm with Velcro. The Mini-bird control boxes and computers were outside the drapes. Participants performed reaches using their right arm. Quasi-Newton estimation in Systat 5.2 was used for model fitting. Procedure. Each reach started with the thumb and index finger of the hand placed together and resting on a raised start location behind the mirror at the edge of the table. Participants practiced reaching-tograsp a purely virtual target and also a physical object. Participants

Participants in the 20-block group exhibited the often observed pattern of response to the progressively distorted haptic feedback (Fig. 2). In feedback trials, reaches to the near target simply reflected the distance of the haptic feedback object whereas reaches to the other two distances exhibited change but with a characteristic lag (i.e., the reaches were modified but to a smaller extent than reaches to the target with distorted feedback). Once the feedback was removed in the snapback condition, the reaches to the near target exhibited the typical snapback relative to the visual targets (i.e., a similar modification to that shown by reaches to the medium and far targets). Thus reach distances in snapback exhibited two-thirds of the difference between the distorted haptic feedback and the visual targets. We fitted the model to the mean trajectories separately for each distance twice, once fitting to the full 20 blocks of data and again fitting only to the first 14 blocks, excluding the snapback blocks. If the behavior had settled to the steady-state level of the distorted haptic feedback, then the asymptotic ␣ parameter values should have been the same in the two cases. The results are shown in Table 1, where it is readily apparent that these values were different. For the fits in Table 1, we first transformed the data by subtracting visual target distance minus the 3-cm haptic feedback level (i.e., far target data minus 23 cm, medium minus 18 cm, and near minus 12 cm).

Table 1. Fitted parameter and r2 values for model fits to the mean data 20 Block

14 Block

Target Distance

␤

␥

␣

r

␤

␥

␣

r2

Near Medium Far

4.646 3.986 3.997

0.369 0.141 0.143

0.320 1.028 1.028

0.79 0.97 0.98

5.225 7.802 8.606

0.181 0.044 0.039

⫺0.958 ⫺3.238 ⫺3.949

0.96 0.98 0.99

2

The model was fitted separately for each of the 3 target distances to 20 blocks of data and 14 blocks of data. J Neurophysiol • doi:10.1152/jn.00112.2013 • www.jn.org

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Fig. 4. Mean standard deviations for each block plotted as a function of block. Œ, 30-block group;
, 20-block group.

Fig. 3. Results for the 30-block group. Mean reach distances plotted as a function of block order together with fitted model functions. Horizontal lines show the visually specified distances of targets at 15, 21, and 26 cm. Œ, Mean reach distances. Error bars are standard error.
, Exponential model curve fitted to 30 blocks of trials including the Snapback data. }, Exponential model curve fitted to 24 blocks of trials excluding the Snapback data.

Asymptotes (␣ values) thus should be zero if the reach distances had settled to the distance specified by the haptic feedback. We fitted the model to the data for each participant at each target distance using both 20 and 14 blocks to test difference from zero and group differences between the values of the ␣ parameter. A repeated-measures ANOVA was performed on the ␣ parameter values with Block (20, 14) and Distance (near, medium, far) as factors. Block was significant [F(1, 7) ⫽ 15.8, P ⬍ 0.01], as was Distance [F(2, 14) ⫽ 4.0, P ⬍ 0.05]. The interaction was not (P ⬎ 0.3). The difference in distances was produced by the fact that reach distances equaled the distorted haptic feedback distance during feedback blocks at the near distance only. We performed an independent (or group) t-test (two tailed) to test difference of fitted ␣ parameter values from zero, only for the 20-block fits. The result was significant [t(23) ⫽ 5.2, P ⬍ 0.001]. Using the mean ␣ parameter value computed over the medium and far target distances for the 20-block fits, we found that reach distances had moved to 67.72% of the difference between visual target distances and the steady-state distorted haptic feedback, toward the haptic feedback. Next, we turned to results for the 30-block group. As shown in Fig. 3, the results were much the same as for the 20-block group with an essential difference. By the end of the feedback blocks in snapback condition, reach distances had settled to the

distance specified by the distorted haptic feedback. Furthermore, reach distances remained at this distance after haptic feedback was removed and only visual information about target distances was available. As found in previous studies using these methods, the overall random error progressively increased during snapback blocks (that is, the standard deviations for each block at the near target increased progressively), but the constant error exhibited no trend and remained stable at the distance specified by the haptic feedback. There was no tendency to return to the visually specified distance [in agreement with Mon-Williams and Bingham (2007) who found similar stability after feedback was removed]. We fitted the model to the mean trajectories, fitting separately to 30 blocks and 24 blocks of data. (The latter excluded snapback blocks.) The resulting parameter values are shown in Table 2, where it can be seen that the values for the asymptotes (that is, ␣ parameter values) were not different from one another or from zero. A repeated-measures ANOVA on ␣ values with Block and Distance as factors yielded no significant effects or interactions (all P ⬎ 0.6). The overall mean was ⫺0.23, and the 30- and 24-block fit means were ⫺0.26 and ⫺0.20, respectively. We performed an independent t-test (two tailed) to test difference from zero. The result failed to reach significance (P ⬎ 0.4). This time, we also performed this test for the fits to 24 blocks and it also was not significant (P ⬎ 0.4). The mean ␣ parameter value computed for medium and far target distances showed that reach distances covered 100% of the distance from visually specified target distances (103.54%). Finally, we compared ␣ parameter values for the 30-block and 20-block groups when fitted with the full number of blocks in each case. A repeated-measures ANOVA with Block (20, 30) and Distance (near, medium, far) only yielded a significant main effect of Block [F(1, 8) ⫽ 6.0, P ⬍ 0.04]. Neither Distance nor the interaction was significant.

Table 2. Fitted parameter and r2 values for model fits to the mean data 30 Block Target Distance Near Medium Far

24 Block

␤

␥

␣

r

␤

␥

␣

r2

5.162 4.130 4.124

0.326 0.157 0.132

0.072 ⫺0.168 ⫺0.044

0.89 0.89 0.91

5.157 4.210 4.401

0.286 0.134 0.101

⫺0.121 ⫺0.382 ⫺0.508

0.95 0.92 0.93

2

The model was fitted separately for each of the 3 target distances to 30 blocks of data and 24 blocks of data. J Neurophysiol • doi:10.1152/jn.00112.2013 • www.jn.org

CALIBRATION DYNAMICS DISCUSSION

We tested a model of calibration dynamics (Burge et al. 2008), to explore the use of haptic feedback in the calibration of visual information about target distances used to guide reach-to-grasp behavior.2 We tested whether previous findings of incomplete calibration merely reflected the number of trials with distorted feedback given the underlying dynamics of calibration or whether they indicated the weighting of visual and haptic information in the presence of noise [see van Beers (2009) for a model that minimizes movement variability via modifications of central planning signals that exclude execution noise but include the effects of planning]. Calibration dynamics determine how long it takes for responses to settle to the distance specified by feedback information. In the condition where distorted feedback was provided over a relatively short period (20 blocks), we found incomplete compensation for the distorted haptic feedback. This finding suggests the possibility that the modification in behavior reflects the combined weighting of the sensory information rather than calibration. In this context, our results were similar to those of van Beers et al. (2002) such that reach distances in posttest (snapback) conditions exhibited two-thirds of the distance specified by the haptic feedback relative to the visually specified distances. Critically, however, when we tested this again with a different group of participants who received a larger number of feedback trials over a longer duration, we found that posttest reach distances exhibited 100% of the distance specified by the haptic feedback. This is exactly what is required for calibration to yield accurate targeted reaching-to-grasp performance. Thus partial calibration was achieved in previous studies because the duration of the feedback was insufficient to allow performance to settle. This finding is consistent with the model proposed by Smith et al. (2006), who argued that motor learning is underpinned by longer and shorter acting processes that interact. The results of the present experiment did not reflect a kinesthetic bias. On the one hand, we found that there was no mean trend in the snapback condition. As found in previous studies (e.g., Mon-Williams and Bingham 2007), linear regressions performed on reach distances as a function of blocks in the snapback condition were not significant with the slopes and correlation coefficients both close to zero, indicating that the calibration was stable on average. On the other hand, the variable error began to increase as soon as feedback was removed. We computed a standard deviation at each distance for each block and computed mean standard deviations for each block within each of the two groups. These are shown in Fig. 4, where it can be readily seen that the variability began to increase as soon as feedback was removed and continued to increase throughout the snapback condition. Bingham et al. (2007) found that accurate haptic feedback yielded accurate reaching, but that feedback was required every fourth reach or so to preserve accurate performance, because otherwise reach distances began to drift. Such random drift is represented in Fig. 4, another characteristic of calibration dynamics. 2 We did not attempt to test the use of perceptual resolution and motor variance to predict the feedback gain, only the exponential form of the equation. However, we found values for the feedback gain of ⬇0.30. These were similar to those found by Burge et al. (2008) for visual feedback with a resolution of ⬇1 cm in reach space. Standard deviations computed across participants in each block were ⬇1.25 cm (Fig. 4). Thus, although the latter also include motor noise, the values were in a comparable range.

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Random drift is one reason that calibration is required to preserve accurate reaching. In developing their model of calibration dynamics, Burge et al. (2008) showed that the spatial resolution of information contributes to the calibration dynamics but the temporal stability of information must also be an important factor. Stability is a central phenomenon in dynamics and was explicitly central to the formulation of the model by Burge et al. The modification of reach-to-grasp behavior in response to distorted terminal feedback can be seen to have the requisite flexibility needed for sensorimotor adaptation but sufficient time constants to ensure the stable responses required for skilled behavior in the face of temporary variability within the environment or the neurophysiology of the system. The extent to which our findings generalize to other visuomotor behaviors remains an empirical question. Studies regarding the perception of hand location have shown systematic, stable, but idiosyncratic errors when participants report the location of an unseen hand (Rincon-Gonzalez et al. 2011). Thus the present findings may reflect the nature of the task and the presence of haptic feedback [it has been argued that adaptation level depends on the type of feedback; van der Kooij et al. (2013)]. It is not known what brain regions are responsible for calibration; this requires further exploration, though it seems reasonable to assume that calibration processes will be found to involve both the parietal and premotor cortex (Graziano 1999). GRANTS This work was supported by National Eye Institute Grant R01-EY011741-06 (Principle Investigator: G. P. Bingham; Consultant: M. A. MonWilliams). DISCLOSURES No conflicts of interest, financial or otherwise, are declared by the author(s). AUTHOR CONTRIBUTIONS G.P.B. conception and design of research; G.P.B. performed experiments; G.P.B. analyzed data; G.P.B. and M.A.M.-W. interpreted results of experiments; G.P.B. prepared figures; G.P.B. and M.A.M.-W. drafted manuscript; G.P.B. and M.A.M.-W. edited and revised manuscript; G.P.B. and M.A.M.-W. approved final version of manuscript. REFERENCES Bingham G, Coats R, Mon-Williams M. Natural prehension in trials without haptic feedback but only when calibration is allowed. Neuropsychologia 45: 288 –294, 2007. Bingham GP, Zaal F, Robin D, Shull JA. Distortions in definite distance and shape perception as measured by reaching without and with haptic feedback. J Exp Psychol Hum Percept Perform 26: 1436 –1460, 2000. Bingham GP, Romack JL. The rate of adaptation to displacement prisms remains constant despite acquisition of rapid calibration. J Exp Psychol Hum Percept Perform 25: 1331–1346, 1999. Bruggeman H, Warren WH. The direction of walking— but not throwing or kicking—is adapted by optic flow. Psychol Sci 21: 1006 –1013, 2010. Burge J, Ernst MO, Banks MS. The statistical determinants of adaptation rate in human reaching. J Vis 8: 20.1–19, 2008. Burns JK, Blohm G. Multi-sensory weights depend on contextual noise in reference frame transformations. Front Hum Neurosci 4: 221, 2010. Burns JK, Nashed JY, Blohm G. Head roll influences perceived hand position. J Vis 11: 3, 2011. Coats R, Bingham GP, Mon-Williams M. Calibrating grasp size and reach distance: interactions reveal integral organization of reaching-to-grasp movements. Exp Brain Res 189: 211–220, 2008. DiStefano JJ, Stubberud AR, Williams IJ. Feedback and Control Systems. New York: McGraw-Hill, 1990.

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