Talk:Feed forward (control)

Discussion 2004

The present form of the article describes the term "feed-forward" in a confusing way. At least as far as uses in aspects of "control theory" are concerned "feed-forward" and "open loop" have more in common than the corresponding current Wikipedia articles teach. It would be great if some experts voiced their thoughts and could share them with the public - thanks.

My first thought on seeing this page was that it would be best moved to Feed-forward neural network or some such, but looking at what links to it, it seems you are intending more of a page about the term "feed-forward" in general. My worry in that case is that you'll find yourself with not enough to write, and will be informed that Wikipedia is not a dictionary. I guess that depends what other fields use the term: perhaps it could end up as more of a disambiguation page - IMSoP 01:44, 4 Apr 2004 (UTC) (I'm not following you around, honest, I just fired up recentchanges and saw something I recognised)

Eh, physiology? Is this some new kind of "living perceptron"? Oh well, I guess I'll stop interrupting you and assume you know what you're doing. The reason I put "refers to" rather than "is", by the way, is because "feed-forward" isn't a noun - or I've never seen it used as one, anyway; so if you wanted to use is, you'd have to have a noun for the adjective feed-forward to modify: "...a feed-forward network is one which..." - IMSoP 02:05, 4 Apr 2004 (UTC)

Firstly, thanks. I like it when people review what I enter.

Secondly, I think you may be on the right track (I will move this page to "Feed-foreward regulatory network". Bensaccount 02:08, 4 Apr 2004 (UTC)

I guess I was trying to merge OneLook dictionary and textbook information in an attempt to figure out what "feed-forward" was. Bensaccount 02:19, 4 Apr 2004 (UTC)

Well, I can only comment on what I know, but in Artificial Intelligence, a feed-forward network would be an artificial neural network - such as a simple multi-layer perceptron - which had multiple layers connected in series, such that no neuron received input from any neuron to which its own output contributed. In other words, one in which there is no feedback from later processing to influence earlier layers. Such architectures allow more complex processing than single-layer networks, but are more easily analysable than more complex architectures such as fully-connected networks.

I won't copy that onto the page just yet, because I'm not entirely sure whether that was or wasn't the sense of the term you were originally aiming to explain, but I think you have become confused if you are associating perceptrons with physiology. - IMSoP 19:18, 4 Apr 2004 (UTC)

Feed-forward seems to be used in two contexts: All I know is that in the physiology context, feedforward is a type of neural regulatory system of the nervous system. I dont know the computing meaning. I seem to have mixed the two meaning up because I didn't realize they were different things. If you can fix it you should, I will be back later.Bensaccount 20:04, 4 Apr 2004 (UTC)

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Quoting from the entry:

When a hill is encountered the car slows down below the set speed. This speed error causes the engine throttle to be opened further, bringing the car back almost to its original speed. Almost its original speed because a feedback system needs some residual error that can be multiplied by loop gain to provide the necessary correction factor for the duration of the hill.

This is incorrect, but I'm not sure how to succinctly explain what is wrong. I'll go on about it here, instead.  :-)

What the article says is partly true: for a feedback control system which has only a proportional term, there will always be a residual error which the proportional term is unable to resolve.

In the early days of closed-loop control (in factories, say of a chemical process), people first discovered you could regulate the output rate of a process by measuring the deviation from a desired rate (set-point) and then altering the input flows by an amount proportional to that deviation.

The constant of proportionality was determined empirically. Immediately, folks noticed that the output of the system never quite reached the set point, but always ended up a bit below it. The operators quickly realized that they could measure this steady-state error and nudge the input controls up a bit by hand, bringing the system up to the exact desired set point value. As the system approaches the set point, the error drops to zero and so this so-called "integral term" in the feedback equation ceases to affect the systems behavior.

The operators were doing something akin to integrating the residual error over time, and adding that (times a constant of proportionality) back to the input rates. Soon, people realized that this too could be done automatically.

And the car's cruise control, being just such a system, does the same thing. It doesn't need any residual error to maintain the velocity; it simply compares the velocity to the set point and alters the throttle setting according to the control law of the system. See also "PID loop control".

Agreed, the page is incorrect, I was quite shocked to read something so wrong on the Wiki, I'll try to fudge it, but I don't really get what the point the person writing it was trying to make.

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Hello chaps. This is a tricky one. As a cybernetician a strict distinction between feedforward and feedback to me has always seemed elusive. Just because an envirionmemtal parameter is monitored doesn't necessarily imply feedforward. If the controlling model is seen as forecasting it still need not imply feedforward. Anyway this graphic may help. It's redrawn after Fig 1-1 in Mrosovsky's "Rheostasis" OUP 1990. It's defintely worth discussing!

Incidentally someone ought to discuss homeostasis or homeodynamics as Yates calls it, in one of these Wiki entries on Feedback. Stasis seems incorrect for an essentially dynamic process even though convergence to a fixed point is implied. Mrosovsky's fixed point moves hence his title but whither the much loved Rheostat of the electrical laboratory? This probably a bit off topic and should be raised in the Homeostasis entry but maybe it illuminates some of the difficulties.--Nick Green 17:08, 2 September 2006 (UTC)

Discussion 2009

The whole subject of feedback and open loop control seems to be badly understood everywhere, not just here on Wiki. I tried to get some help in improving these pages by reading Encyclopedia Brittanica but that was equally bad.

I will try to make it all a bit clearer, but it is going to be difficult since I am not an expert. OTOH, all the experts these days are so specialised it probably needs someone like me to provide a more general overview. A B McDonald (talk) 12:59, 21 April 2009 (UTC)

The new diagram is partly based on that by Nick Green on this page. I'll wait to see if there are any reactions to my changes before going on to correct the feedback articles. A B McDonald (talk) 15:30, 21 April 2009 (UTC)

Hi Folks

How annoying (but tantalising), to come across such a relatively common but poorly defined term. I have encountered it often in neuroscience. For its meaning here, can I suggest:

"In neuroscience, feedforward describes a neural system where the signal travels from input, through various synapses, to output, and where the input signal does not vary according to the effect of the environment or later elements of the system. Contrasted with feedback where the input signal is modified by the response of later parts of the system or the system's environment"?

This seems to be the sense in which it is used in neuroscience but I've never encountered a definition, only derived this from context - though it does seem to harmonise with Feedforward neural network... Anthony (talk) 07:08, 8 July 2009 (UTC)

-- Though I haven't witnessed or been involved in much of the past of this article, I suggest, for starters, that a split be considered. I'm an electrical engineer and having biological/psychological elements mixed in with control theory is a bit bothersome. It also slightly confuses me when I try to look for places to make additions. (A textbook I own gave me some good ideas about examples to use regarding control theory, such as how feed-forward techniques improve the tracking of a sinusoidal or parabolic signal.)

Just a thought, my two cents.

98.238.110.152 (talk) 16:21, 30 December 2009 (UTC) R. Smith

IMHO you should add new sections as needed under Applications. That is where these items appear in Feedback entry. I notice that on the Feedback page the applications have been ordered alphabetically, so if you are adding any it might be worthwhile reordering the ones here. If I were you I would go ahead and make the changes. I don't think that you will receive any criticism, unlike me when I try to insert my diagram into the Feedback pages :-( A B McDonald (talk) 20:56, 16 January 2010 (UTC)

Definition of the term

The lede reads: "Feed-forward is a term describing an element or pathway within a control system which passes a controlling signal from a source in the control system's external environment, often a command signal from an external operator, to a load elsewhere in its external environment."
I consider this formulation quite weak. I think that the principle of feed-forward should not be identified with elements or pathways. The article's scope clearly is control theory, thus it seems to be a good idea to rename it to "feed-forward control"—I think this would also simplify the definition: feed-forward control would be definable as a concept in control theory. Morton Shumway—talk 21:01, 7 September 2010 (UTC).

Suggested new material

Feedforward is a term that originated in the field of Robotic Controls. While it has been adopted by other fields and even popularized in articles and textbooks about management, its original meaning is made clear in the following brief citation:

“In feedforward control there is a coupling from the setpoint and/or from the disturbance directly to the control variable, that is, a coupling from an input signal to the control variable. The control variable adjustment is not error-based. In stead it is based on knowledge about the process in the form of a mathematical model of the process and knowledge about or measurements of the process disturbances.” From: Basic Dynamics and Control, Haugen, F, 2009, ISBN 978-82-91748-13-9

To suggest that feedforward control and open loop control are identical would be wrong just as it would be wrong to say that feedforward control could be based solely on operator input. Feedforward control requires a mathematical model of the process and/or machine being controlled and its mathematical relationship to any inputs or feedback the system might receive. Open loop control does not require the sophistication of a mathematical model of the physical system. Control based on operator input without integral processing and interpretation through a mathematical model of the system is a teleoperator system and is not feedforward controlled.

Feedforward control requires integration of the mathematical model into to the control algorithm such that it is used to determine the control actions based on what is known about the state of the system being controlled. In the case of control for a lightweight, flexible robotic arm, this could be as simple as compensating between when the robot arm is carrying a payload and when it is not. The target joint angles are adjusted to place the payload in the desired position based on knowing the deflections in the arm from the mathematical model's interpretation of the disturbance caused by the payload. Systems that plan actions and then pass the plan to a different system for execution do not satisfy the above definition of feedforward control. Unless the system includes a means to detect a disturbance or receive an input and process that input through its mathematical model do determine the required modification to the control action, it is not true feedforward control.

The term Feedforward Control was originally used by professors and graduate students at Georgia Tech and MIT. It is not typically hyphenated in scholarly works. Meckl and Seering of MIT and Book and Dickerson of Georgia Tech began the development of the concepts of Feedforward Control in the mid 1970s. The concept was clearly defined in many scholarly papers, articles and books by the mid 1980s.

• Oosting, K.W., Simulation of Control Strategies for a Two Degree-of-Freedom Lightweight Flexible Robotic Arm, Thesis, Georgia Institute of Technology, Dept. of Mechanical Engineering, 1987,
• Alberts, T.E., Augmenting the Control of A Flexible Manipulator with Passive Mechancal Damping, Phd. Thesis, Georgia Institute of Technology, Dept. of Mechanical Engineering, August 1986.
• Hastings, G.G., Controlling Flexible Manipulators, An Experimental Investigation, Ph.D. Dissertation, Dept. of Mech. Eng., Georgia Institute of Technology, August, 1986.
• Book, W.J. and Cetinkunt, S.,"Alear Optimum Control of Flexible Robot Arms OR Fixed Paths",IEEE Conference on Decision and Control. December 1985.
• Meckl, P.H. and Seering, W.P. /'Feedforward Control Techniques Achieve Fast Settling Time in Robots" Automatic Control Conference Procedings. 1986, pp 58-64.
• Sakawa, Y., Matsuno, F. and Fukushima, S., "Modeling and Feedback Control of a Flexible Arm", Journal of Robotic Systems. August 1985, pp 453-472.
• Truckenbrodt, A., "Modeling and Control of Flexible Manipulator Structures", 4th CISM-IFToMM Symp., Warszawa, 1981.
• Leu, M.C., Dukovski, V. and Wang, K.K., "An Analytical and Experimental Study of the Stiffness of Robot Manipulators with Parallel Mechanisms", 1985 ASME Winter Annual Meeting PRD-Vol. 15 Robotics and Manufacturing Automation, pp. 137-144
• Asada, H., Youcef-Toumi, K. and Ramirez, R.B., "Designing of the MIT Direct Drive Arm", Int. Symp. on Design and Synthesis, Japan, July 1984.
• Rameriz, R.B., Design of a High Speed Graphite Composite Robot Arm, M.S. Thesis, M.E. Dept., MIT, Feb. 1984.
• Balas, M.J., "Feedback Control of Flexible Systems", IEEE Trans. on Automatic Control, Vol.AC-23, No.4, Aug. 1978, pp.673-679.
• Balas, M.J., "Active Control of Flexible Systems", J. of Optim. Th. and App., Vol.25, No.3, July 1978,
• Book, W.J., Modeling, Design and Control of Flexible Manipulator Arms, PhD. Thesis, MIT, Dept. of Mech. Eng., April 1974.
• Maizza-Neto,, 0., Modal Analysis and Control of Flexible Manipulator Arms, PhD. Thesis-, MIT, Dept. of Mech. Eng., September 1974.
• Book, W.J., Maizzo Neto, 0. and Whitney, D.E., "Feedback Control of Two Beam, Two Joint Systems With Distributed Flexibility", Journal of Dynamic Systems, Measurement and Control, Vol.97, No.4, December 1975, pp.424-430.
• Book, W.J., "Analysis of Massless Elastic Chains With Servo Controlled Joints", Journal of Dynamic Systems, Measurement and Control, Vol.101, September 1979, pp.187-192.
• Book, W.J., "Recursive Lagrangian Dynamics of Flexible Manipulator Arms Via Transformation Matrices", Carnegie-Mellon University Robotics Institute Technical Report, CMU-RI-TR-8323, Dec. 1983.
• Hughes, P.C., "Dynamics of a Flexible Manipulator Arm for the Space Shuttle", AAS/AIAA Astrodynamics Conference, September 1977, Jackson Lake Lodge, Wyoming.
• Hughes, P.C., "Dynamics of a Chain of Flexible Bodies", Journal of Astronautical Sciences, 27,4, Oct.-Dec. 1979, pp.359-380.
• Meirovitch, L., "Modeling and control of Distributed Structures" Proc. of the Workshop on Application of Distributed System Theory to Large Space Structures, JPL/CIT, NTIS #N83- 36064, July 1,1983.
• Schmitz, E., "Experiments on the End-point Position Control of a Very Flexible One Link.Manipulator", Ph.D. Dissertation,-Stanford Univ., Dept. of Aero & Astro., June 1985.
• Martin, G.D., On the Control of Flexible Mechanical Systems, Ph.D. Dissertation, Stanford Univ., Dept. of E.E., May 1978.
• Zalucky, A. and Hardt, D.E., "Active Control of Robot Structure Deflections", J. of Dynamic Systems, Measurement and Control, Vol. 106, March 1984, pp. 63-69.
• Sangveraphunsiri, V., The Optimal Control and Design of a Flexible Manipulator Arm, Ph.D Dissertation, Dept. of Mech. Eng., Georgia Inst, of Tech., 1984.
• Alberts, T.E., Sangveraphunsiri, V. and Book, Wayne J., Optimal Control of a Flexible Manipulator Arm: Volume I, Dynamic Modeling, MHRC Technical Report, MHRC-TR-85-06, Georgia Inst, of Technology, 1985.
• Nemir, D. C , Koivo, A. J., and Kashyap, R. L., "Pseudolinks and the Self-Tuning Control of a Nonrigid Link Mechanism", Purdue University, Advance copy submitted for publication, 1987.
• Widmann, G. R. and Ahmad, S., "Control of Industrial Robots with Flexible Joints", Purdue University, Advance copy submitted for publication, 1987.
• Hollars, M. G., Uhlik, C. R., and Cannon, R. H., "Comparison of Decoupled and Exact Computed Torque Control for Robots with Elastic Joints", Advance copy submitted for publication, 1987.
• Cannon, R. H. and Schmitz, E., "Initial Experiments on the End- Point Control of a Flexible One Link Robot", International Journal of Robotics Research, November 1983.
• Oosting, K.W. and Dickerson, S.L. , “Low-Cost, High Speed Automated Inspection”, 1991, Industry Report
• Oosting, K.W. and Dickerson, S.L. , “Feed Forward Control for Stabilization”, 1987, ASME
• Oosting, K.W. and Dickerson, S.L. , “Control of a Lightweight Robot Arm”, 1986, IEEE International Conference on Industrial Automation
• Khatib and Oussama, SPRINGER HANDBOOK OF ROBOTICS, Springer Press, 2008.
• Oosting, K.W., “Actuated Feedforward Controlled Solar Tracking System,” 2009, Patent Pending
• Oosting, K.W., “Feedforward Control System for a Solar Tracker,” 2009, Patent Pending
• Oosting, K.W., “Smart Solar Tracking,” July, 2010, InterSolar NA Presentation

[This was posted by User:Trkwoo --Morton Shumway—talk 04:53, 20 March 2011 (UTC).] [tidy up Morton Shumway—talk 15:36, 21 March 2011 (UTC). ]

First, it would be helpful if you made the references you give into a list, for better readability. Morton Shumway—talk 04:55, 20 March 2011 (UTC).

—The concept is older than to be only found in the literature since the eighties, and it did not originate in robotic control. Rather it has been widely discussed in control theory, motor control etc. in an application to a number of systems, biological as well as artificial.

"Feedforward (MacKay 1966, Greene 1969) is the strategy whereby a controller monitors disturbances to a system directly and applies appropriate compensatory signals to the controlled system—rather than waiting for feedback on how the disturbances have affected the system before determining compensatory signals." (Arbib, Michael A. (1989): The Metaphorical Brain 2. Neural Networks and Beyond, New York: John Wiley & Sons, p. 99), the given references:

* MacKay, D. M. (1966): "Cerebral organization and the conscious control of action". In: J. C. Eccles (Ed.), Brain and conscious experience, Springer, pp. 422–440.

* Greene, P. H. (1969): "Seeking mathematical models of skilled actions". In: H. C. Muffley/D. Bootzin (Eds.), Biomechanics, Plenum, pp. 149–180.

See also this article (by Michael Arbib?): http://psychology.jrank.org/pages/1155/feedback-feedforward.html

—As to your emphasis on mathematical models, please see Miall (2003), p. 688: "Feedforward control schemes may be grouped as those based on direct control and those based on indirect control using internal models. Here, direct control means control without explicit knowledge of the behavior of the plant […]." (Miall, R. Christopher (2003): "Motor Control, Biological and Theoretical". In: Michael A. Arbib (Ed.), The Handbook of Brain Theory and Neural Networks, 2nd Edition, pp. 686–689.

Morton Shumway—talk 16:09, 21 March 2011 (UTC).

—Please see also my suggestion above. Morton Shumway—talk 00:26, 23 March 2011 (UTC).

Thanks for your comments and references. I have read virtually all of the material you reference. I was surprised to find that MacKay actually does use the term feedforward and with somewhat similar meaning as found in more current work in the field of Automatic Controls. However, MacKay does not speak of "Feedforward Control" and writes in the context of theory for how the brain controls the body. MacKay's work is brilliant, but is not really relevant to the discipline of "Feedforward Controls." As for the the requirement for a mathematical model, Miall goes on to say that the feedforward controller must be matched to the plant and speaks of the model that must be acquired by the controller if not explicitly programed into the controller. The model is mathematical in nature regardless of whether it is provided or learned. Perhaps the article on Feedforward should mention MacKay and other early writers as historically relevant to theories related to "feedforward systems" and how the brain works, but then direct those interested to a page on a more appropriate topic. In the same way, the article should direct those interested in Feedforward Controls to a new page with content something like:

“Feedforward Control” is a term that has specific meaning within the field of CPU based Automatic Controls. The discipline of “Feedforward Controls” as it relates to modern, CPU based automatic controls is widely discussed, but is seldom practiced due to the difficulty and expense of developing or providing for an adequate mathematical model required to facilitate this type of control. Open loop control and feedback control (often based on canned PID control algorithms; see PID Controller: http://en.wikipedia.org/wiki/PID_controller) are much more widely used[4, 32, 34]. While the term “feedforward control” has been adopted by other fields and even popularized in articles and textbooks about management, its meaning in modern controls is made clear in the following citation:

“In feedforward control there is a coupling from the set point and/or from the disturbance directly to the control variable, that is, a coupling from an input signal to the control variable. The control variable adjustment is not error-based. In stead it is based on knowledge about the process in the form of a mathematical model of the process and knowledge about or measurements of the process disturbances.” From: Basic Dynamics and Control, Haugen, F, 2009, ISBN 978-82-91748-13-9

Feedforward Control is distinctly different from open loop control and teleoperator systems. Feedforward control requires a mathematical model of the plant (process and/or machine being controlled) and the plant's relationship to any inputs or feedback the system might receive. Neither open loop control nor teleoperator systems require the sophistication of a mathematical model of the physical system or plant being controlled. Control based on operator input without integral processing and interpretation through a mathematical model of the system is a teleoperator system and is not considered feedforward control.

The mathematical model of the plant (machine, process or organism) used by the feedforward control system may be created and input by a controls engineer or it may be learned by the control system. Control systems capable of learning and/or adapting their mathematical model have become more practical as microprocessors speeds have increased. The discipline of modern Feedforward Controls was made possible by the invention of microprocessors.

Feedforward systems are also found in biological control by human and animal brains. One helpful article on this type of feedforward system can be found at: http://psychology.jrank.org/pages/1155/feedback-feedforward.html

Even in the case of biological feedforward systems, such as in the human brain, knowledge or a mental model of the plant (body) can be considered to be mathematical as the model is characterized by limits, rhythms, mechanics and patterns. [37, 38]

Historically, the use of the term “feedforward” is found in works by D. M. MacKay as early as 1956. While MacKay’s work is in the field of biological control theory, he speaks only of feedforward systems. MacKay does not mention “Feedforward Control” or allude to the discipline of “Feedforward Controls.” MacKay and other early writers who use the term “feedforward” are generally writing about theories of how human or animal brains work.[37] See article on Motor Skills: http://en.wikipedia.org/wiki/Motor_skill

Feedforward Control requires integration of the mathematical model into the control algorithm such that it is used to determine the control actions based on what is known about the state of the system being controlled. In the case of control for a lightweight, flexible robotic arm, this could be as simple as compensating between when the robot arm is carrying a payload and when it is not. The target joint angles are adjusted to place the payload in the desired position based on knowing the deflections in the arm from the mathematical model’s interpretation of the disturbance caused by the payload. Systems that plan actions and then pass the plan to a different system for execution do not satisfy the above definition of feedforward control. Unless the system includes a means to detect a disturbance or receive an input and process that input through the mathematical model to determine the required modification to the control action, it is not true feedforward control.

The discipline of “Feedforward Controls” was largely developed by professors and graduate students at Georgia Tech, MIT, Stanford and Canaigee Mellon. Feedforward is not typically hyphenated in scholarly publications. Meckl and Seering of MIT and Book and Dickerson of Georgia Tech began the development of the concepts of Feedforward Control in the mid 1970s. The discipline of Feedforward Controls was well defined in many scholarly papers, articles and books by the late 1980s. [1, 2, 31, 33, all]

Feedforward Control is also discussed in the field of Artificial Intelligence. See Feedforward neural network at: http://en.wikipedia.org/wiki/Feedforward_neural_network
[This was posted by User:Trkwoo Morton Shumway—talk 14:14, 13 April 2011 (UTC).]

If you like the article to be about a discipline called "feedforward control" or "feedforward controls" to cover a more specific use of the term than in the sense of 'the concept of feedforward in control theory', and to refer to a particular school, then you could do that in an according subsection. If any ambiguities arise due to a renaming of the article, it should stay "feedforward" as it now covers the more general aspects. Or you could add this material to an article on CPU based automatic controls.

As to the relevance of particular texts, much of the theory that is nowadays used all over the place has been developed in sometimes slightly different contexts. Still, many ideas remain the same throughout different disciplines. I am not sure about your idea of control theory, it's really not that important in the first place whether biological or artificial systems are analysed (or built).

Indeed Miall speaks of the however acquired model in covering the case of internal models. Of course you still might call the mechanism that switched on a heating when outside temperature drops below some value a 'mathematical model'.

Which way do you see your idea of the term applied in the notion of feedforward ANNs?

Morton Shumway—talk 12:17, 3 April 2011 (UTC).

Moved from article

Moved the note below from the article. Jim1138 (talk) 06:29, 31 October 2013 (UTC)

Note: This is a very poor presentation of the original work done by me. There are fatal errors in the approach and denigrate the value of this great concept. Indeed, this is the way beings in the nature adopt for their actions/inactions and is a virtual control system depicting life. What is done by others seems to be a piecemeal application without seeing the real nature of this work. I find that some of my recent ordinary comments about this approach has become somehow inserted in some recent reference paper work. I doubt very much that anyone can find the true approach without my help at all. There is none better than the original approach I developed from scratch, which is exactly what nature offers. The thesis was a very short and simplePeter Abbeywick (talk) 13:02, 21 January 2016 (UTC) version derived from a more complex concept (inserted by Peter Abbeywick - original thesis owner). — Preceding unsigned comment added by 110.175.124.231 (talk • contribs)

removed a link that pointed back to itself. Kvom01 (talk) 17:32, 7 February 2015 (UTC)

I am Peter Abbeywick again. It is interesting to find abuse of someone's research and making a mess of it. If this does not work it is because they do not have the basic knowledge. All the work done was based on my original application. When an application is used as theory, it leads to incorrect results. When such publications are made, Wikipedia becomes another worthless entity. Some claims that Feedforward Control existed before my work is the biggest joke. If this is the typical work you provide, this is not a trustworthy source of information. You may be wasting your time. When my words are repeated in articles about Feedforward Control without proper reference, I find it hilarious. I can assure that you will never get it right since it has an introduced error into it that was beyond my control. (My email: peter.abbeywick@gmail.com). — Preceding unsigned comment added by 110.175.124.231 (talk) 12:20, 21 January 2016 (UTC)