Car Physics : Pacejka Formula

Hi Folks, I am having a bit of trouble with the pacejka tyre formula. From the book I am reading the value at which most Forward force is generated is around 0.25-0.5. However when I run it through my formula I get the peak at a value of 8.1 longitudnal slip (which is incredibly high I think). The maximum value of the forward force is 7100N for a downward force (Fz) on the wheel of 4.2kN. This is a ratio of around 1.7, which according to the book is reasonable. Has anyone come across this before, perhaps Todd, Doug or Ruud? Thanks for the help (I am tearing my hair out!) Ashley PS. My email is spam-proofed, please remove ".THISREMOVE" to email me.

Hi Ash What are your pacejka constants? Chris Ashley McConnell wrote in message <7u246.29324$y5.97693@news2-hme0>... >Hi Folks, > >I am having a bit of trouble with the pacejka tyre formula. From the bookI >am reading the value at which most Forward force is generated is around >0.25-0.5. However when I run it through my formula I get the peak at a >value of 8.1 longitudnal slip (which is incredibly high I think). > >The maximum value of the forward force is 7100N for a downward force (Fz)on

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>the wheel of 4.2kN. This is a ratio of around 1.7, which according to the >book is reasonable. > >Has anyone come across this before, perhaps Todd, Doug or Ruud? > >Thanks for the help (I am tearing my hair out!) > >Ashley > >PS. My email is spam-proofed, please remove ".THISREMOVE" to email me. > >

Ashley McConnell wrote: > I am having a bit of trouble with the pacejka tyre formula. From the book I > am reading the value at which most Forward force is generated is around > 0.25-0.5. However when I run it through my formula I get the peak at a > value of 8.1 longitudnal slip (which is incredibly high I think). First, test your implementation of the modeling equations by running your model with coefficients from other sources. Make sure your program matches the published results. This gets the shape correct. (For example, choose one of the lateral models from Race Car Vehicle Dynamics Chap 14.) The input value for peak force can be calculated for this type of fit, but not in closed form (IIRC). (This is one reason why my company doesn't use it.) Once you know this value of the input, scale your input to put the peak where you want it in your fit. It sounds to me like you are attempting to adapt a fit originally built for lateral data to a longitudinal use. The lateral fit may have been in degrees rather than radians. This is actually a good thing. because it made it obvious that you have to rescale (renormalize) the data to your new applications input range. If the lateral fit had been in radians, the peak placement would have seemed somewhat reasonable and you wouldn't have found how to put it where you need/want it ;) (One problem with using other peoples data is that not everyone is careful enough to properly document details such as the units of various coefficients in their fits. This can make that work hard for other people to use.) Happy New Year, - Matt -- Matthew V. Jessick Motorsims, a division of Moto1 [email hidden] http://www.moto1.net Vehicle Dynamics Engineer (972)910-8866 Ext.125

Hi Chris, Thanks for the reply. Here are all the pacejka constants that I use: - 1.7990 \\a0 0.0 \\a1 1688.0 \\a2 4140.0 \\a3 6.026 \\a4 0.0 \\a5 -0.3589 \\a6 1.0 \\a7 0.0 \\a8 -0.006111 \\a9 -0.03224 \\a10 0.0 \\a111 0.0 \\a112 0.0 \\a12 0.0 \\a13 1.65 \\b0 0.0 \\b1 1688.0 \\b2 0.0 \\b3 229.0 \\b4 0.0 \\b5 0.0 \\b6 0.0 \\b7 -10.0 \\b8 0.0 \\b9 0.0 \\b10 2.0680 \\c0 -6.49 \\c1 -21.85 \\c2 0.416 \\c3 -21.31 \\c4 0.02942 \\c5 0.0 \\c6 -1.197 \\c7 5.228 \\c8 -14.84 \\c9 0.0 \\c10 0.0 \\c11 -0.0037360 \\c12 0.03891 \\c13 0.0 \\c14 0.0 \\c15 0.639 \\c16 1.693 \\c17 0.013 \\f0_for_rolling_resistance 0.0000065 \\K_for_rolling_resistance 0.315 \\Rolling_Radius 0.7 \\TyreInertia These are from the high performance rear wheel drive car that is in the appendix of G.Genta's book (Motor Vehicle Dynamics modelling and simulation). Thanks for the help Ash PS. If you have a set of pacejka contants that you KNOW are right they would be very appreciated :) Chris West <[email hidden]> wrote in message news:%s546.28447$[email hidden]...

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> Hi Ash > > What are your pacejka constants? > > Chris > > Ashley McConnell wrote in message <7u246.29324$y5.97693@news2-hme0>... > >Hi Folks, > > > >I am having a bit of trouble with the pacejka tyre formula. From the

book

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> I > >am reading the value at which most Forward force is generated is around > >0.25-0.5. However when I run it through my formula I get the peak at a > >value of 8.1 longitudnal slip (which is incredibly high I think). > > > >The maximum value of the forward force is 7100N for a downward force (Fz) > on > >the wheel of 4.2kN. This is a ratio of around 1.7, which according to

the

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> >book is reasonable. > > > >Has anyone come across this before, perhaps Todd, Doug or Ruud? > > > >Thanks for the help (I am tearing my hair out!) > > > >Ashley > > > >PS. My email is spam-proofed, please remove ".THISREMOVE" to email me. > > > > > >

Hi Matthew, Thanks for the reply - my reply is below:- Matthew V. Jessick <[email hidden]> wrote in message news:[email hidden]... > Ashley McConnell wrote: > > > I am having a bit of trouble with the pacejka tyre formula. From thebook I

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> > am reading the value at which most Forward force is generated is around > > 0.25-0.5. However when I run it through my formula I get the peak at a > > value of 8.1 longitudnal slip (which is incredibly high I think). > > First, test your implementation of the modeling > equations by running your model with > coefficients from other sources. > Make sure your program matches the published results. > This gets the shape correct. > (For example, choose one of the lateral models > from Race Car Vehicle Dynamics Chap 14.) >

I have simply ran a set of longitudnal slip ratio values (increasing by 0.01). These were not linked to any other process just put into the formula to get Fx. It DOES form the right shape and it does have a reasonable maximum value (7100N). Unfortunately I dont have the RCVD book, although I do have Giancarlo Gentas similar book (which is excellent btw). > The input value for peak force can be calculated > for this type of fit, but not in closed form (IIRC). > (This is one reason why my company doesn't use it.) > I am not sure what "closed form" is, my maths is letting me down a bit :(

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> Once you know this value of the input, scale your > input to put the peak where you want it in your fit. > > > It sounds to me like you are attempting to adapt a > fit originally built for lateral data to a longitudinal use. > The lateral fit may have been in degrees rather than radians. > This is actually a good thing. because it made it obvious > that you have to rescale (renormalize) the data to your > new applications input range. If the lateral fit had been > in radians, the peak placement would have seemed > somewhat reasonable and you wouldn't have found > how to put it where you need/want it ;)

I have coefficients for both lateral and longitudnal pacejka formula - the b0-bx set of coefficients (in my post above) are for longitudnal. I dont think its a case of trying to stick the lateral coefficents in a longitudnal formula, UNLESS the data is wrong (which wouldnt be the first time!) :) > > (One problem with using other peoples data is that > not everyone is careful enough to properly document > details such as the units of various coefficients in their fits. > This can make that work hard for other people to use.) Yeah, its hard to test code when you dont know for sure that the data is right :/ Thanks alot for the help, and Happy New Year Ash

Ashley McConnell wrote:

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> It DOES form the right shape and it does have a reasonable maximum value > (7100N). Unfortunately I dont have the RCVD book, although I do have > Giancarlo Gentas similar book (which is excellent btw). > > > The input value for peak force can be calculated > > for this type of fit, but not in closed form (IIRC). > > (This is one reason why my company doesn't use it.) > > > > I am not sure what "closed form" is, my maths is letting me down a bit :(

This is a bit of a red herring, sorry. I was assuming that you were developing some software to implement a tire fit and then would eventually want to generate new fits to give some adjusted tire characteristics. What I meant by "Closed Form" is that I don't believe you can develop a straightforward equation for the slip ratio for the peak force as a function of the coefficients. That is because of the inverse tangent functions used in the magic tire formulas. You would need to plot the functions (as you have done) or solve through iteration for the coefficient values to generate a particular desired value for the slip ratio to give the peak force. (In other words, if you eventually want your fits to have a particular peak position, it may be difficult to figure out how to cause that to happen.) > I have coefficients for both lateral and longitudnal pacejka formula - the > b0-bx set of coefficients (in my post above) are for longitudnal. I dont > think its a case of trying to stick the lateral coefficents in a longitudnal > formula, UNLESS the data is wrong (which wouldnt be the first time!) :) You are using the Pacejka et.al. tire models in a much more developed way than I ever have. I have only used single fits for particular conditions. Presumably, all those coefficients of yours (as you know of course ;) deal with how to systematically adjust the fits for various conditions of load, etc. I don't know how you can easily debug a problem with either the coefficients or your implementation of the equations without access to the experimental data the coefficient set is supposed to match. Hopefully, the source which gave the coefficient set contains sufficient plots to help you debug the system... Now that I understand more of where you are, I don't have many suggestions. You may be able to set several fixed values for normal force, camber, etc., (whatever inputs the coefficients exist to vary the curveswith) and then attempt to guess which of the coefficients or equations have the bug. Another idea is to plot the curves through the entire positive _and_ negative range of the slip ratio. That might indicate a sign problem. (Should the positive plot properly but the negative doesn't, etc. this might indicate a sign problem in a coefficient or equation.) Given what you have said about the curves looking correct but having a suspicious character for the longitudinal slip ratio for peak force, perhaps the problem is in the coefficients/code that handle the effect of normal force variations. I've seen data that indicates that more normal force can tend to stretch out the plots. (Move the peak toward higher slip ratios.) A bug may be causing you to get too much effect from normal force variations... or some other effect that tends to move the peak toward higher slip ratios. I'm doing some similar debugging with modelling that results in a "friction circle" for combined slip cases this weekend ;), so you have my sympathy! Luck, -- Matthew V. Jessick Motorsims, a division of Moto1 [email hidden] http://www.moto1.net Vehicle Dynamics Engineer (972)910-8866 Ext.125

Hi, Ashley, I'll echo part of Matt Jessick's comment. I've never seen the Pacejka's magic formula with so many constants before. Guess I've never seen the full version. Wish I could help you out on this one, but it sounds like you're ahead of me in this area. For the work I've been doing I'll probably be using the nondimensional technique described in Milliken's & Milliken's "Race Car Vehicle Dynamics", as it looks quite a bit simpler to work with. For now, I've been using simple approximations while developing my model. Saving the tires for last, I guess :0) BTW, how is that book you've got? Is it worth the $98 on Amazon.Com? Todd Wasson --- Performance Simulations Drag Racing and Top Speed Prediction Software http://PerformanceSimulations.Com

Matthew, Thanks for the detailed reply, Now I see were you are coming from, more of a manufacturing angle than simulation. You were concerned that it wouldnt be possible to create new tyres from results used in the pacejka formula. I am not using ALL the coefficients (in my reply above) in ONE pacejka formula. That list is the contents of my "tyre file". It has coefficients for the 3 different pacejka formulas for Longitudnal - Fx (all the b's), Lateral - Fy (all the a's) and Alligning Moment -Mz (all the c's). I am not concerned about combining the Fx and Fy values (as I know that a value of Fx will reduce the maximum value of Fy (and vice-versa). I will do this once I get a reasonable slip->Force function working correctly. I have tried what you suggested with -ve values of slip, but unfortunately it is a fairly good shape and semetrical, I have posted to the alt.binaries.simulators.autos newsgroup if you are interested, its only 5k. I am using kN for the force i,e. its 4.2kN per tyre. This is calculated using the mass of the car and its centre of gravity, using a method in G.Gentas book. (the mass of the car is 1480kg). Good luck with the friction circle stuff, that will be my next task once I get Fx and Fy working :) Thanks again for the help - much appreciated! Ash

Hi Todd, Thanks for the reply :) I am not using ALL the coefficients (in my reply above) in ONE pacejka formula. That list is the contents of my "tyre file". It has coefficients for the 3 different pacejka formulas for Longitudnal - Fx (all the b's), Lateral - Fy (all the a's) and Alligning Moment -Mz (all the c's). It is only the set of "b" constants that I am concerned with in the current longitudnal slip pacejka formula. The rest were just added for completeness The book is great, but I cannot compare to RCVD, because I didnt get it (Its alot dearer over here than G.Genta's). Chris recommended it to me, I think because it is slightly more slanted towards simulation. I would really like to get RCVD aswell sometime, it is always nice to get 2 versions of the story :) My book has some non-dimensional stuff for combining Fx and Fy (i..e acceleration is happening in both longitudnal and lateral directions at the same time). Not sure if it is the same as in RCVD. Thanks for the help - much appreciated. Ash J. Todd Wasson <[email hidden]> wrote in message news:[email hidden]... > Hi, Ashley, > > I'll echo part of Matt Jessick's comment. I've never seen the Pacejka's > magic formula with so many constants before. Guess I've never seen thefull > version. Wish I could help you out on this one, but it sounds like you're > ahead of me in this area. For the work I've been doing I'll probably beusing > the nondimensional technique described in Milliken's & Milliken's "RaceCar > Vehicle Dynamics", as it looks quite a bit simpler to work with. For now,I've > been using simple approximations while developing my model. Saving thetires

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> for last, I guess :0) > > BTW, how is that book you've got? Is it worth the $98 on Amazon.Com? > > > Todd Wasson > --- > Performance Simulations > Drag Racing and Top Speed Prediction > Software > http://PerformanceSimulations.Com

On Mon, 1 Jan 2001 16:39:24 -0000, "Ashley McConnell" <[email hidden]> wrote: >Hi Folks, > >I am having a bit of trouble with the pacejka tyre formula. From the book I >am reading the value at which most Forward force is generated is around >0.25-0.5. However when I run it through my formula I get the peak at a >value of 8.1 longitudnal slip (which is incredibly high I think). Seems way too high yes. I assume you mean the ratio? Anything above 1 is already more a slipping state than a driving state. >The maximum value of the forward force is 7100N for a downward force (Fz) on >the wheel of 4.2kN. This is a ratio of around 1.7, which according to the >book is reasonable. Yep, F1 tires seem to have values like that, but I guess the 1.7 drops down quickly when not in optimal conditions though. (F1-2000 starts at 2.4 for some, but before the loading factor is included probably, see a former thread some time ago). >Has anyone come across this before, perhaps Todd, Doug or Ruud? Haven't been near too much physics lately, sorry, been putting most energy into graphics, culling, ray-triangle intersection, sphere visibilities and such. Very interesting. When I get a track working (and rather, when I get a track that I can distribute), I'll probably move back to car physics, since I find the Ferrari P4 I've put up now doesn't act too real. >Thanks for the help (I am tearing my hair out!) Seems like the formula you use takes so many coefficients that it's a pain to debug. A little bit TOO much magic. But as the values seem quite ok, you must be very near. It's just that one '-' instead of a '+' can mean so much here. Ruud van Gaal, GPL Rank +53.25 Pencil art : http://www.marketgraph.nl/gallery/ Car simulation: http://www.marketgraph.nl/gallery/racer/

On Mon, 1 Jan 2001 16:39:24 -0000, "Ashley McConnell" >Hi Folks, >I am having a bit of trouble with the pacejka tyre formula. From the book I >am reading the value at which most Forward force is generated is around >0.25-0.5. However when I run it through my formula I get the peak at a >value of 8.1 longitudnal slip (which is incredibly high I think). Seems way too high yes. I assume you mean the ratio? Anything above 1 is already more a slipping state than a driving state. Yep, F1 tires seem to have values like that, but I guess the 1.7 drops down quickly when not in optimal conditions though. (F1-2000 starts at 2.4 for some, but before the loading factor is included probably, see a former thread some time ago). Haven't been near too much physics lately, sorry, been putting most energy into graphics, culling, ray-triangle intersection, sphere visibilities and such. Very interesting. When I get a track working (and rather, when I get a track that I can distribute), I'll probably move back to car physics, since I find the Ferrari P4 I've put up now doesn't act too real. Seems like the formula you use takes so many coefficients that it's a pain to debug. A little bit TOO much magic. But as the values seem quite ok, you must be very near. It's just that one '-' instead of a '+' can mean so much here. Ruud van Gaal, GPL Rank +53.25 Pencil art : http://www.marketgraph.nl/gallery/ Car simulation: http://www.marketgraph.nl/gallery/racer/

On Tue, 2 Jan 2001 14:23:00 -0000, "Ashley McConnell" <[email hidden]> wrote: ... >I have tried what you suggested with -ve values of slip, but unfortunately >it is a fairly good shape and semetrical, I have posted to the >alt.binaries.simulators.autos newsgroup if you are interested, its only 5k. Looks ok on sight, although in real life the negative side would be a braking curve, right? >I am using kN for the force i,e. its 4.2kN per tyre. This is calculated >using the mass of the car and its centre of gravity, using a method in >G.Gentas book. (the mass of the car is 1480kg). What force would that be? Note that F1 cars are more in the range of 700-800kg. Also, not many tires will reach a friction coefficient of 1.7 probably outside of F1 (and cart) racing. Or perhaps drag racers do. Ruud van Gaal, GPL Rank +53.25 Pencil art : http://www.marketgraph.nl/gallery/ Car simulation: http://www.marketgraph.nl/gallery/racer/

Hi Ash, >I am not using ALL the coefficients (in my reply above) in ONE pacejka >formula. That list is the contents of my "tyre file". It has coefficients >for the 3 different pacejka formulas for Longitudnal - Fx (all the b's), >Lateral - Fy (all the a's) and Alligning Moment -Mz (all the c's). Ah!! That makes much more sense now! I wrote a curve editor that lets you hit buttons to change some of the constants, but if I recall correctly, it only had four or five in the equation. Since most of the "b"s are 0, maybe the leftovers are the same thing I was toying with. It seems that one of the constants was for maximum grip/friction coefficient (the 1.65 you listed?), one defined the steepness of the initial slope (longitudinal stiffness, probably the 1688 number? I don't know! lol) While the other two controlled the curvature between the initial upwards slope and the final low value, after the peak. The -10 sounds familiar here. Actually, I think one of those two numbers controlled the value it dropped to after the peak, the other ends up working in conjunction with it to control how quickly it curves through the peak and drops down. If so, the 1688 or the 229 number might be screwy if you're getting wierd results that don't match up right. Could you post the equation your book shows for the longitudinal "b" section? Perhaps I can take a look through my little editor and see if we can discover something. If not, maybe we'll all learn something about Pacejka's brilliant formula anyway :0) Todd Wasson --- Performance Simulations Drag Racing and Top Speed Prediction Software http://PerformanceSimulations.Com

Hi Folks, Thanks everyone who helped! The problem was that the longitudnal slipratio needs to be in percentage (rather than a simple ratio). Thanks Chris! Now to solve the rest of the 132131919 problems :) Thanks again folks Ash

On 2 Jan 2001, J. Todd Wasson wrote: > I'll echo part of Matt Jessick's comment. I've never seen the Pacejka's > magic formula with so many constants before. Guess I've never seen the full There are several papers (SAE and others) on the Pacejka curve fitting techinque, but I don't have the references handy. It has always required large numbers of coefficients to describe a tire "completely" (over a range of loads, etc), and this work continues, I believe mostly at Delft University, Netherlands. I've not tested the data-set in Genta's book, but he has a web site at his university that you could check to see if there have been any corrections. The MRA-Radt Nondimensional technique only uses the Magic Formula curve fit because it produces a good fit to the normalized test data. The technique could use a different formula if it was desired -- see Chap 14 in RCVD for an introduction. > BTW, how is that book you've got? Is it worth the $98 on Amazon.Com? If you are referring to the Genta book, it's a lot cheaper in the USA if you order it from your local book store, I got it from Barnes & Noble; imo Amazon.com is often not the place to buy low-volume technical books. At B & N you can order it in and look at it, before you buy. While this may sound like a broken record (anyone remember what that means?<grin>), our book is available direct from the publisher, and they have a very reasonable shipping price for overseas if you pay by credit card. Last time I looked the link was: <http://www.sae.org/servlets/productDetail?PROD_TYP=BOOK&PROD_CD=R-146> Happy New Year, -- Doug Douglas Milliken <[email hidden]> Milliken Research Associates Inc.

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>There are several papers (SAE and others) on the Pacejka curve fitting >techinque, but I don't have the references handy. It has always required >large numbers of coefficients to describe a tire "completely" (over a range >of loads, etc), and this work continues, I believe mostly at Delft >University, Netherlands. I've not tested the data-set in Genta's book, >but he has a web site at his university that you could check to see if >there have been any corrections. > >

Great, thanks. I'll try to find this site. >The MRA-Radt Nondimensional technique only uses the Magic Formula curve fit >because it produces a good fit to the normalized test data. The technique >could use a different formula if it was desired -- see Chap 14 in RCVD for >an introduction. I've looked at this and made a small program that lets you play with these variables and see the results graphed at different loads and slip angles. However, I can't seem to get these to match any of the other real tire data shown in your book. I can get any particular curve to match well, but the resulting data for load variations on the same tire never matches up correctly. No load sensitivity in the equations? However, I was pleased to see the peaks of the force curves move along the slip angle direction with load changes, along with seeing some change in cornering stiffness as load changes. Perhaps I am doing something wrong here, but I haven't been able to figure out what it is yet. Thanks, Todd Wasson --- Performance Simulations Drag Racing and Top Speed Prediction Software http://PerformanceSimulations.Com

On 02 Jan 2001 07:06:08 GMT, [email hidden] (J. Todd Wasson) wrote: Hi Todd, > BTW, how is that book you've got? Is it worth the $98 on Amazon.Com? I've just ordered it (Giancarlo Genta's Motor Vehicle Dynamics), but on Barnes & Noble; it's US$59 there, which saves quite a lot wrt Amazon (probably too busy laying off people right now anyway) ;-) Just FYI. Ruud van Gaal, GPL Rank +53.25 Pencil art : http://www.marketgraph.nl/gallery/ Car simulation: http://www.marketgraph.nl/gallery/racer/

>I've just ordered it (Giancarlo Genta's Motor Vehicle Dynamics), but >on Barnes & Noble; it's US$59 there, which saves quite a lot wrt >Amazon (probably too busy laying off people right now anyway) ;-) > Thanks, Ruud :0) Todd Wasson --- Performance Simulations Drag Racing and Top Speed Prediction Software http://PerformanceSimulations.Com

J. Todd Wasson <[email hidden]> wrote: :>I've just ordered it (Giancarlo Genta's Motor Vehicle Dynamics), but :>on Barnes & Noble; it's US$59 there, which saves quite a lot wrt :>Amazon (probably too busy laying off people right now anyway) ;-) : : Thanks, Ruud :0) Not to mention you can use one of the following coupons for ordering $10 off (minimum purchase $30) BNREDF1 $10 off (minimum purchase 40) : SNAPBN1 ZZTDB7L HOLIDAY AMEXFEB (expires 02/28/01) ZZJXLSC (expires 02/28/01) DISCNOW (expires 03/31/01) FLEETBN (expires 04/30/01) to reduce the price to $49. TTT

On 1 Feb 2001 08:00:27 GMT, Tauno Taipaleenmaki <[email hidden]> wrote: > Not to mention you can use one of the following coupons for ordering > > $10 off (minimum purchase $30) BNREDF1 Yikes, couponz? Valid coupons, leftovers, where are these coming from? (and a little problem if suddenly 20 people start using them; I want the book more than the $10). Ruud van Gaal, GPL Rank +53.25 Pencil art : http://www.marketgraph.nl/gallery/ Car simulation: http://www.marketgraph.nl/gallery/racer/

Ruud van Gaal <[email hidden]> wrote: : Yikes, couponz? : Valid coupons, leftovers, where are these coming from? (and a little : problem if suddenly 20 people start using them; I want the book more : than the $10). Almost all the US Internet retailers have coupons and promotions. B&B spits out coupons quite often so I really don't think that they don't expect people to use them ? So no, it's not a problem if people use coupons because that's the reason they exists. And you wanting the book more than the $10 has nothing to do with it. TTT ps. most of the coupons listed earlier in this thread were actually for new customers (like myself, never used B&B before)

On 1 Feb 2001 18:02:00 GMT, Tauno Taipaleenmaki <[email hidden]> wrote: >Ruud van Gaal <[email hidden]> wrote: >: Yikes, couponz?...

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> Almost all the US Internet retailers have coupons and promotions. > B&B spits out coupons quite often so I really don't think that > they don't expect people to use them ? > > So no, it's not a problem if people use coupons because that's > the reason they exists. And you wanting the book more than the > $10 has nothing to do with it.

Sorry about the tone of my post. It just sounded a bit warez-y. :) Ruud van Gaal, GPL Rank +53.25 Pencil art : http://www.marketgraph.nl/gallery/ Car simulation: http://www.marketgraph.nl/gallery/racer/