- Open Access
Alignment of angular velocity sensors for a vestibular prosthesis
© DiGiovanna et al; licensee BioMed Central Ltd. 2012
- Received: 10 June 2011
- Accepted: 13 February 2012
- Published: 13 February 2012
Vestibular prosthetics transmit angular velocities to the nervous system via electrical stimulation. Head-fixed gyroscopes measure angular motion, but the gyroscope coordinate system will not be coincident with the sensory organs the prosthetic replaces. Here we show a simple calibration method to align gyroscope measurements with the anatomical coordinate system. We benchmarked the method with simulated movements and obtain proof-of-concept with one healthy subject. The method was robust to misalignment, required little data, and minimal processing.
- Vestibular prosthetics
- sensor alignment
Imagine waking up one morning, opening your eyes, and seeing your bedroom rotated 90 degrees. Suddenly, instead of the ceiling appearing above you, it is to your right. Would you be able to function? Could you adapt to such a sensory misalignment? In the long-term, the answer is yes. The brain is capable of resolving misalignment between normal and received sensory information through the mechanism of neural plasticity. Gonshor and Mevill Jones showed one dramatic example of such plasticity in their work using prism glasses . These glasses inverted subjects' view of the world (e.g. right is left), but over days subjects adjusted their vestibulo-ocular (VOR) reflexes; by 18 days the VOR had reversed to match visual information. However, such plasticity is neither immediate nor free. Subjects in preliminary trials reported "rapid and severe nausea"  and VOR following adaptation to vision reversal never fully mimicked normal responses.
Vestibular prosthetics transmit sensory information to the brain, but they also can induce sensory misalignment. These prosthetics should (partially) replace vestibular organs, which sense gravity, linear acceleration, and angular acceleration (rotation) of the head. Normally, information from these organs is transmitted to the brain where it is fused with visual and other sensory inputs to yield spatial orientation and on-going movements of the body [2, 3]. Vestibular prosthetics aim to address some symptoms of vestibular dysfunction including: spatial disorientation, postural instability, self-motion perception deficits, visual blurring during head motion ("oscillopsia") due to loss of VOR, and chronic disequilibrium. However, there can be misalignment between the information transmitted by the prosthetic and information formerly provided by the damaged sensory organs. Two existing approaches solve this problem: 1) align the prosthetic sensors to the user's anatomy during the implantation surgery; 2) allow brain plasticity to correct any misalignment of the implanted prosthetic. These approaches are not exclusive; in fact 2) will happen regardless of how or if 1) was completed. The required plasticity may not be as extreme as for prisms , but time and energy are still required.
We suggest that providing an initial condition that is close to natural semicircular canal SCC information will decrease learning time (effort) for the brain and also may increase mapping accuracy. This can be accomplished by approach 1) above, but it requires additional efforts by the surgical team and lengthens surgery time for the patient. We propose an alternative approach that will achieve the same ends as surgical prosthetic alignment but can be completed outside of the operating room. Our approach uses a simple sensor that is temporarily secured to the head via a bite-bar (a device commonly used to hold film during a dental x-ray). Measurements from this bite-bar are collected synchronously with the vestibular prosthetic sensor. The calibration method we propose finds the matrix necessary to align prosthetic sensor measurements to the bite-bar (also to the head by proxy); then using average anatomical positions (or medical imaging techniques) the sensor can be further aligned to the vestibular organs. Our calibration method is not restricted by time under anaesthesia and can take advantage of advanced imaging techniques. We will show that it is possible to replace the surgical alignment with our method; additionally, this will produce more accurate alignments of the vestibular prosthetic sensor to the damaged vestibular organs.
Aligning different sensor systems was an early problem for naval weapons systems that was addressed with strap-down gyro triads . Gyro digital outputs were processed by a PC using a least-squares optimization to find nine entries of the rotation matrix, which accounted for sensor misalignment. Vestibular prosthetics have a static displacement between the sensors (gyros and SCC); here we are only concerned with angular velocities. Thus finding a rotation matrix between the sensors will align them. This problem was anticipated and a practical solution patented in 2008 .
In this paper we will develop and test a simple optimization algorithm to find a rotation matrix for vestibular prosthetic sensor alignment. The algorithm transforms artificial sensor measurements (green coordinate system in Figure 1) into a physiologically appropriate coordinate system (e.g. blue coordinate system in Figure 1). Robust performance is demonstrated in simulations tailored to anticipated implementation issues, where simulated sensor readings were generated based on actual head measurements. Additionally, this method creates a more accurate signal than surgical alignment. We performed a proof-of-concept alignment of real sensor recordings (human data) to confirm simulations of independent sensor noise effects and test bite-bar sensor approximation of skull movements. Finally, based on the method's strengths and weakness, we proposed a clinical protocol and implementation to rapidly and accurately calibrate a vestibular prosthetic sensor for human patients.
A sequential quadratic programming algorithm (Matlab's fmincon function) was used to optimize the rotation angles (α, β, γ) given angular velocity measurements in the two coordinate systems while accounting for the presence of constraints (i.e. physiological limits and trigonometric functions in the rotation matrix). To decrease the probability of selecting a local minimum, we repeated the optimization from ten random, but widely dispersed (+/- 90°), initial conditions (angles).
Our approach has fewer parameters (3 vs. 9) - thus reducing the number of training samples - and it ensures orthogonality. However, the approach in Eqn. 3 has lower computational complexity (e.g., no trigonometric functions). Since our calculations can be performed off-line and do not require long time series, computational complexity was not a paramount concern.
The four coordinate systems to be used in this analysis are defined here. Figure 1 shows the Head-Mounted Motion Sensor (HMMS) coordinate system in green. The HMMS is aligned to the HEAD coordinate system (but the HMMS origin is translated from mid-skull). The origin of the HEAD coordinates is coincident with the origin of the REID coordinates . The semi-circular canal (CANAL) coordinates are shown in blue. The origin of the CANAL coordinates is coincident with origin of HEAD, but rotated (fixed-angle rotation) -19.9° about <Y> and +43.45° about <Z> to roughly approximate the average human CANAL orientation found in . (Of course, if desired, imaging or other methods could be used to find the canal orientation for each patient. This orientation would define patient-specific CANAL coordinates.) The final coordinate system is the Bite-Bar Sensor (BBS) coordinates that will be aligned with HEAD but the origin will not be coincident (BBS is not shown in Figure 1). Our main goal is to rotate HMMS measurements to align them with CANAL coordinates. The BBS and HEAD coordinates are crucial intermediaries to attain this goal.
Does HMMS alignment via the BBS provide any advantage over manual alignment of the HMMS and HEAD during surgery?
How does signal error change as a function of HMMS misalignment?
How does independent, additive sensor noise affect the alignment algorithm?
Can slow movements be used for training the alignment algorithm such that it can generalize to faster movements?
We assumed a BBS would be better aligned with HEAD because it can be done before the vestibular implant surgery via imaging technologies (e.g. x-ray, CT) and is not limited by surgery time or mastoid geometry. Specifically, the BBS would have normally distributed (zero-mean; 0.5° standard deviation) alignments errors (e i in the approximation (Eqn. 5) of R between HEAD and BBS). Though the BBS and surgical alignment errors are approximate, we have chosen values that are about right. Certainly, all the same technologies that might be available for a surgical alignment of the sensors are available for this post-surgical alignment - without the constraints present during surgery. Hence, for these simulations, what is important and justifiable is that the precision of the BBS alignment be assumed better than the precision of surgical HMMS alignment.
In Eqn. 10, the R is the optimized rotation matrix calculated using Eqn. 4 (where a = BBS and b = HMMS). Errors in this alignment method are quantified in the same fashion as surgical (see eqns. 8 and 9)
Multiple sensors were used for the proof-of-concept human experiments to compare angular velocity measurements in HEAD, BBS, and HMMS coordinate systems. An optic motion capture system (Vicon 460, Vicon Motion Systems, Oxford UK) was used precisely measure the position of the skull over time. Five cameras (Vicon M2 (Vicon Motion Systems, Oxford UK)) were placed around the subject with reflective markers attached to the skin over the skull providing spatially accurate (< 0.8 mm error) marker positions at 100 Hz resolution. Given these measurements, it was possible to calculate angular velocities in HEAD coordinates (details in next section). As shown in the prior section, HEAD measurements can be rotated into CANAL coordinates via the matrices in  or patient-specific coordinates using a CT scan.
While motion capture systems are often used in research laboratories, it is less common to find them in clinical settings. The systems are expensive; require a relatively large operating space; and trained personnel. Thus, we proposed a BBS as a replacement for both surgical alignment of the HMMS to HEAD and direct HEAD measurements. The BBS is a much smaller, easier to operate, and less expensive. Specifically, we mounted two 2D gyroscopes (LPR5150AL and LPY5150AL, STMicroelectronics) orthogonally to a common dental "bite-bar" (designed to hold film at a specific distance from the x-ray camera) to measure angular velocities of the head. These low-power, dual-axis, micromachined gyros are capable of measuring angular rate along pitch and roll (LPR5150AL) or along pitch and yaw (LPY5150AL) axes with a full scale of ±1500°/s and with a -3 dB bandwidth up to 140 Hz. Our assumption was that the molars are approximately perpendicular to the 'pitch' <Y> axis (interaural) of the head and the geometry of the bite bar created surfaces aligned to both the pitch and 'yaw' <Z> axes. By design, the orthogonal gyros mounted on these surfaces would be approximately parallel to the pitch, roll, and yaw axes of the head. This complete device is the BBS.
Finally, we needed a device to mimic the Head-Mounted Motion Sensors (HMMS) that would be present in vestibular prostheses. To address this, a commercially available inertial system (MTx by xSens) was rigidly attached to the subject's head. The MTx uses gyros, accelerometers, and magnetometers to determine angular rotations, orientation, and acceleration .
Calculating angular velocity
In the prior section we introduced three sensors (Vicon markers, bite-bar system, and MTx) that provide angular velocity in the HEAD, BBS, and HMMS coordinate systems respectively. The MTx system calculates angular velocity internally; thus HMMS measurements were read-out directly. Similarly, the BBS directly measures angular velocity and outputs a voltage for each axis. This voltage was converted to degrees per second using the manufacturer's data-sheet and a calibration voltage to find BBS measurements. On the other hand, Vicon measures marker positions; thus a transformation was required to get HEAD angular velocities. Markers were used to determine axes for the HEAD segment (specifically forehead and left and right ear markers). Given these axes, angular velocity was calculated:
1. Calculate HEAD origin as mean value of three markers [left_ear, right_ear, forehead]
2. Define <y> as (left_ear - right_ear)/norm (left_ear - right_ear)
3. Define <aux_x> as (forehead - origin)/norm (forehead - origin)
4. Define <z> as <aux_x> × <y> (where × is the cross product)
5. Define <x> as <y> × <z>
6. Create R = [<x> <y> <z>]
This technique was validated with artificial data (sinusoidal angular velocity at 50°/s). We found perfect correlation (R2 = 1) between the angular velocities applied to the data and those found using Eqns. 13 and 14. Additionally, we applied white noise equivalent to the Vicon camera calibration accuracy and then a low-pass filter (5th order Butterworth, fc = 10 Hz) similar to the post-processing in Vicon software (Workstation). This measurement noise reduces R2 to 0.9706. This should be considered the upper bound of performance for HEAD measurements.
Have subject sit in a chair in the middle of recording area.
- 2)Mount vestibular prosthetic sensors (HMMS) to subject.
Normally this would already be accomplished in surgery. Here we used a tight-fitting bike helmet with a rigidly attached MTx for healthy subjects.
- 3)Attach reflective markers to subject (for HEAD measurements)
Increasing the number of markers slightly improves calculation of rotation angles. However, exact placement at specific landmarks is not necessary.
Align bite-bar with interaural axis and have subject bite down firmly to keep it stable (for BBS measurements)
Begin recording Vicon, BBS, and MTx sensor measurements synchronously
- 6)Ask subject to move head for ~30 s completing the following movements:
Range of motion for each rotation axis slowly then quickly. (Referred to as Fast and Slow ROM.)
Normal (subject unrestricted) movements slowly then quickly. (Referred to as Fast and Slow Explore.)
Briefly (~2 s) rest between each movement type
To ensure we would have sufficient data for various analyses, we collected more data than necessary (in steps 3 and 6). To validate the robustness found in artificial data, we tried multiple sensor alignments (step 2). One healthy male (study author, age 29, height 183 cm) subject performed this recording protocol.
A common approach to minimize the effect of zero-mean noise sources is to use longer data segments because the optimizer finds the mean solution. In Figure 6 errors initially decrease with additional training samples for both metrics; however, this effect quickly saturates (i.e. above 1600 samples (16 s)). This saturation is reasonable because of the structure of the error. In Eqn. 12, the R component should improve with additional samples but the desired signal () remains corrupted by noise. Thus the overall room for improvement is limited.
Alignment of sensor pairs
Through the simulations above, we have shown the advantages of aligning the HMMS to the HEAD via a BBS rather than manual HMMS-HEAD alignment during surgery. These advantages form the primary purpose of this study. However, to demonstrate that the solution is practical, we also conducted a single proof-of-concept experiment with a human subject. In the experiment, we investigate if the BBS provided stable HEAD angular velocities measurements during actual movements. If there is no additional noise (e.g. bite-bar slip) during the experiment, the simulations have shown that it is possible to align the HMMS and BBS sensors with low error. For comparison, we used the Vicon system to directly measure head position and calculate HEAD angular velocities via Eqn. 13. Then we benchmarked HMMS-BBS against HMMS-HEAD alignment.
Before each successive trial (steps 4-6 of the recording protocol, see Methods) we introduced a progressively larger misalignment by rotating the MTx mounting plate (point of rigid attachment to the helmet) clockwise (average increments of 67 deg); this created a majority of misalignment in the HMMS yaw axis. For all trials, we used the optimization described in Methods-Optimization to solve for R using the cost defined in Eqn. 4. Separate optimizations were performed, one to align HMMS to HEAD; another to align HMMS to BBS. The HMMS was a common sensor, thus any HMMS measurement noise was common to both optimizations.
Average Sensor Alignment over Segments
Correlation (R 2 )
Correlation (R 2 )
RMS angular velocity
Range of velocities
The HMMS-BBS alignment was more accurate than HMMS-HEAD. This was surprising because the BBS was less rigidly attached to the skull and this sensor was a prototype composed of two approximately orthogonal 2D gyros, which could degrade measurement accuracy. However, HEAD measurements relied on markers that may "slip" (artificial motion of one or more markers during movement). This could strongly affect the head axis calculation used to determine angular velocity. It occurs more at higher speeds (see Figure 9) and could be reduced with additional facial markers, higher camera sampling rates, better marker trackingiii, improved camera calibration, and/or additional cameras.
However, there is no strong motivation to pursue such technical improvements for HEAD measurement when the BBS already performed better even as a prototype; with the important caveat that both measurement systems suffered from noise in this experiment. Additionally, the BBS requires little additional space to collect measurements (~10 cm clearance from the subject's mouth, compared to multiple cameras each at least 1 m away from the subject). To reduce the BBS measurement noise, dental impression material can be used to create a mould of the patient's teeth to increase stability of the bar, facilitate improved alignment with the skull (using the imaging technologies discussed in Methods), and significantly increase subject comfort during the calibration. (We suspect bite-bar slippage is partially responsible for HMMS-BBS alignment degradation at higher angular velocities, as the algorithm did not suffer such effects in simulations.)
In this paper we proposed a straightforward optimization and protocol to align angular velocity measurements from different sensors as would be necessary in a vestibular prosthetic. The implication of this work is that vestibular prosthetic sensor could replace "natural" SCC angular velocity sensing in the semi circular canals via a three-part alignment. The three components are: a) creating a BBS aligned with HEAD; b) finding the proper R between HMMS-BBS; c) rotating the HMMS measurements to the BBS and then to the SCC using the HEAD-CANAL alignment.
Encouragingly, this optimization can be completed accurately with only limited data and simple sensors. We thoroughly benchmarked the optimization in a simulation environment to understand both strengths and fundamental limitations. These simulations were carefully designed to mimic realistic constraints a vestibular prosthetic may face. The optimization was insensitive to dramatically larger (50×) errors in HMMS fixation to the skull - error metrics were significantly lower for the BBS optimization compared to our approximation of state of the art surgical alignment.
Additionally we compared two different sensors that could be used for acute measurements of head rotations during alignment of a vestibular prosthetic sensor. We found that the simple bite-bar sensor (BBS) could be more accurately aligned with the skull-fixed sensor (HMMS) compared to motion capture estimates of HEAD rotations. This suggests that the BBS had less measurement and/or movement noise than our attempt to directly measure HEAD because the other measurement noise source (HMMS) was common. Additionally, by testing multiple movement types and velocities we confirmed the simulation results, which suggested a series of slow (for a healthy subject) range of motion movements (total time < 10 s) was sufficient for accurate sensor alignment. (Specifically, sensor alignment did not improve by including either additional Slow ROM data or data from other movement types into the training set.)
Align (subject-specific) BBS with HEAD coordinate and have subject bite down
Begin recording BBS and HMMS measurements synchronously
Ask subject to move head slowly to create rotations in yaw, pitch, and roll (or allow the subject to remain still and mechanically rotate them)
Calculate BBS-HMMS and HEAD-CANALiv alignments
Download appropriate rotation matrix to prosthetic
Specifically, a patient might have a vestibular prosthetic implanted on day 0 and recover from the surgery and/or adapt to baseline electrical stimulation for N days . On day N+1, before using the gyroscope signals to modulate the stimulation rate, the patient returns to the clinic for calibration and performs the protocol above. The result is an optimized R matrix (based on Eqn. 4).
Multiply all nine rotation matrix (R) elements by 215
Round to the nearest integer
Download R integer elements into microcontroller (one time)
Perform alignment calculations (i.e. Eqn. 1)
Divide results by 215
Repeat steps 4 and 5 for each angular velocity sample
Through our simulations, we have addressed a natural criticism to this alignment method, "why not set a proper alignment during the prosthetic implant surgery?" We achieved more accurate alignment without any of the known risks. Specifically, surgical alignment will not always be possible. Second, when possible, it is not easy to rapidly assess alignment during surgery. Even if another method (e.g., high resolution imaging) were developed that assessed alignment during surgery, our alignment method would provide a simple way to verify the new method functionally and to implement the desired realignment. Additionally, assuming accurate sensor alignment could be achieved during surgery, why spend any surgical time/effort on this when better performance can be attained with a short (circa 10 min.) post-surgery procedure? Obviously, lengthening the surgery duration to perform an in situ calibration or to align sensors with some ideal orientation increases total surgery time, which could prove detrimental to patient recovery. Finally, the natural healing process may disrupt a surgical alignment, e.g. connective tissue growth could slightly move the sensor.
The alignment protocol investigated herein could be adapted (as suggested by  and recently shown in ) such that it directly transforms the sensor signals to yield desired VOR spatial characteristics or to yield some other functional behavioural outcome (i.e., balance or psychophysical performance). While the HMMS-CANAL coordinate transformation should only be performed once (assuming no change in sensors or implant), alignment using functional metrics could be updated periodically. This updating would create an asynchronous co-adaptation of the vestibular prosthetic and the user's brain. Such co-adaptation may yield both improved functional outcomes and more rapid prosthetic integration.
i This research was done in the context of the CLONS project, which aims to create a closed-loop prosthetic that interfaces directly to vestibular neurons - bypassing dysfunctional semi-circular canals (CLONS details in ). CLONS is an acronym for CLOsed-loop Neural prosthetics for vestibular disorderS.
ii The rotation order in Eqn. 2 can be directly related to Figure 1. The rotation involves initially rotating the frame about the Z-axis by yaw, then rotating about X' by roll, then rotating about Y'' by pitch. The ' and '' indicate that the first and second rotations affect the axis positions before the rotations occur.
iii The authors were not experts in motion capture. Although we operated the system correctly, a more skilled Vicon operator likely could reduce marker slip frequency.
iv The final alignment step is to translate the BBS approximation of HEAD angular velocity into CANAL coordinates. This could be achieved as we described in Methods-Coordinate Systems and Results-Simulations. Patient-specific HEAD-CANAL alignments could also be calculated with individual CT scans 
We thank Vito Monaco for invaluable assistance in validating angular velocity calculations based on marker data. We also thank Dr. Andrea Gesi for donating the bite-bar apparatus used in this study. Financial support for the authors was provided in part by the EU Future & Emerging Technologies Open Scheme, Project 225929 (CLONS).
- Gonshor A, Melvill Jones G: Extreme vestibulo-ocular adaptation induced by prolonged optical reversal of vision. Journal of Physiology 1976, 256: 381-414.PubMed CentralView ArticlePubMedGoogle Scholar
- Guyton A, Hall JE: Textbook of medical physiology. 10th edition. Philadelphia: W. B. Saunders Company; 2001.Google Scholar
- Merfeld DM: Spatial Orientation and the Vestibular System. In Sensation and Perception. Sunderland MA: Sinauer Associates, Inc; 2008.Google Scholar
- Fernandez C, Goldberg JM: Physiology of peripheral neurons innervating semicircular canals of the squirrel monkey. II. Response to sinusoidal stimulation and dynamics of peripheral vestibular system. J Neurophysiol 1971., 34:Google Scholar
- Gong W, Merfeld DM: Prototype neural semicircular canal prosthesis using patterned electrical stimulation. Ann Biomed Eng 2000, 28: 572-581.View ArticlePubMedGoogle Scholar
- Gong W, Merfeld DM: System design and performance of an unilateral semicircular canal prosthesis. IEEE Trans Biomed Eng 2002, 49: 175-181. 10.1109/10.979358View ArticlePubMedGoogle Scholar
- Merfeld DM, Gong W, Morrissey J, Saginaw M, Haburcakova C, Lewis RF: Acclimation to chronic constant-rate peripheral stimulation provided by a vestibular prosthesis. IEEE Trans Biomed Eng 2006, 53: 2362-2372. 10.1109/TBME.2006.883645View ArticlePubMedGoogle Scholar
- Merfeld DM, Haburcakova C, Gong W, Lewis RF: Chronic vestibulo-ocular reflexes evoked by a vestibular prosthesis. IEEE Trans Biomed Eng 2007, 54: 1005-1015.View ArticlePubMedGoogle Scholar
- Shkel AM, Zeng FG: An electronic prosthesis mimicking the dynamic vestibular function. Audiol Neurotol 2006, 113-122.Google Scholar
- Della Santina CC, Migliaccio AA, Patel AH: A multichannel semicircular canal neural prosthesis using electrical stimulation to restore 3-D vestibular sensation. IEEE Trans Biomed Eng 2007, 54: 1016-1030.PubMed CentralView ArticlePubMedGoogle Scholar
- Fridman GY, Davidovics NS, Dai C, Migliaccio AA, Della Santina CC: Vestibulo-Ocular Reflex Responses to a Multichannel Vestibular Prosthesis Incorporating a 3D Coordinate Transformation for Correction of Misalignment. J Assoc Res Otolaryngol 2010, 11: 367-381. 10.1007/s10162-010-0208-5PubMed CentralView ArticlePubMedGoogle Scholar
- Wall C, Merfeld DM, Rauch SD, Black FO: Vestibular prostheses: The engineering and biomedical issues. Journal of Vestibular Research 2003, 12: 95-113.Google Scholar
- Carta D, Lackowski D: Estimation of orthogonal transformations in strapdown inertial systems. Automatic Control, IEEE Transactions on 1972, 17: 97-100. 10.1109/TAC.1972.1099871View ArticleGoogle Scholar
- Merfeld DM: Sensor Signal Alignment. 2008.Google Scholar
- Craig JJ: Introduction to Robotics: Mechanics and Control. 2nd edition. Reading: Addison-Wesley Publishing Co., Inc; 1989.Google Scholar
- Della Santina CC, Potyagaylo V, Migliaccio AA, Minor LB, Carey JP: Orientation of human semicircular canals measured by three-dimensional multiplanar CT reconstruction. Journal of the Association for Research in Otolaryngology: JARO 2005, 6: 191-206. 10.1007/s10162-005-0003-xPubMed CentralView ArticlePubMedGoogle Scholar
- Roetenberg D, Luinge H, Slycke P: 6 DOF Motion Analysis Using Inertial Sensors. Measuring Behavior Conference; Maastricht, The Netherlands 2008.Google Scholar
- Guyot JP, Sigrist A, Pelizzone M, Kos MI: Adaptation to steady-state electrical stimulation of the vestibular system in the human. Annals of Otology, Rhinology and Laryngology 2011, 120: 143-149.View ArticleGoogle Scholar
- Merfeld DM: Sensor Signal Alignment. 2011.Google Scholar
- DiGiovanna J, Gong W, Haburcakova C, Kögler V, Carpaneto J, Genovese V, Merfeld DM, Demosthenous A, Guyot JP, Hoffmann KP, Berthoz A, Morari M, Micera S: Development of a closed-loop neural prosthesis for vestibular disorders. Journal of Automatic Control 2010, 20: 27-32. 10.2298/JAC1001027DView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.