{"id":46,"date":"2011-12-21T20:55:45","date_gmt":"2011-12-21T19:55:45","guid":{"rendered":"http:\/\/www.edy.es\/dev\/?p=46"},"modified":"2020-04-30T21:30:08","modified_gmt":"2020-04-30T19:30:08","slug":"facts-and-myths-on-the-pacejka-curves","status":"publish","type":"post","link":"https:\/\/www.edy.es\/dev\/2011\/12\/facts-and-myths-on-the-pacejka-curves\/","title":{"rendered":"Facts and myths on the Pacejka curves"},"content":{"rendered":"

\"\"<\/a>When developing vehicle physics the tire is the single component that has the most influence in the entire system. When designing a tire simulation the Pacejka Curves arise sooner or later.<\/p>\n

What’s “Pacejka” exactly?<\/h4>\n

First of all, the Pacejka Magic formula, also called simply Magic Formula “MF” or “MF-tyre”, is a function\u00a0y=f(x)<\/em> which is used to predict and simulate the forces developed\u00a0by a tire. Given an input x<\/em> and a list of coefficients (b0..b12<\/em>\u00a0in the picture), the function calculates the force y<\/em>\u00a0developed by the tire. Three different functions are used, one for longitudinal forces, other for lateral forces, and the last one for the self-aligning torque (the feedback force you experiment on your car’s steering wheel). Each function has its own set of coefficients.<\/p>\n

The coefficients affect the final shape of the curve in a wide variety of ways (see the pic). The goal is\u00a0fitting the curve with the results obtained in the empiric experiments<\/strong>\u00a0on the specific tire the curve is being calculated for. Once the proper coefficients for the curve are found, the behavior of that tire can be easy and realistically predicted without having to actually use the real tire.\u00a0One can imagine using a tire testing facility, testing a specific tire in a wide variety of conditions and measuring the results. Then, based on those results, one could start toying with the coefficients of the Pacekja curves until getting a curve\u00a0that acceptably fits the experimental results.<\/p>\n

What about the coefficient sets for Pacejka curves?<\/h4>\n

Pacejka and other models are developed by and for the automotive industry. Real coefficient sets are scarce and hard to find, as they are heavily protected intellectual data. Tire manufacturers sell the coefficient sets to interested customers who can then simulate how a specific tire would behave in a vehicle (the MSC Adams<\/a> software is an industry-standard for modelling this kind of simulations). A Pacejka coefficient set for a single tire model can easily cost around $1000-1500.<\/p>\n

You may find many Pacejka sets over there (for instance as part of the car add-ons for the Racer.nl<\/a> simulator). But most of them are not real<\/em> Pacejka sets as they haven’t been calculated based on measures on the real tires. Anyone can use a Pacejka editor<\/a> and get an arbitrary set of coefficients. For that, the Pacejka formula is a realistic and flexible method for simulating an imaginary<\/em> tire based on the behavior expected in a simulated vehicle.<\/p>\n

I’ve written a document which describes how each Pacejka coefficient affects the resulting curve<\/a>. This is useful for designing tire friction curves matching a specific behavior in vehicle simulation games.<\/p>\n

Is Pacejka a necessary or best method for simulating tires?<\/h4>\n

Surprisingly, the curves themselves are not the most important part of a tire simulation. Remember, a Pacejka curve is simply a function y=f(x)<\/em>. Any function with shape similar to Pacejka (peak – asymptote) will work, no matter how it’s calculated. It could result more or less realistic, but even just\u00a0 a flat, clamped slope will do a decent work.<\/p>\n

The critical\u00a0part\u00a0for simulating a tire is how<\/em> are you calculating the parameters you will feed the curves with. The “x”<\/em> in y=f(x)<\/em>. And here, my friend, is where things get serious.<\/p>\n

What are the problems with the Pacejka-based methods?<\/h4>\n

Pacejka and other models are based on the amount of slip<\/em> of the tire over the ground. The longitudinal version uses the difference between the tire’s angular velocity and the actual velocity over the ground (slip ratio<\/em><\/strong>). The lateral version is based on the angle between the wheel’s heading direction and the true movement direction (slip angle<\/em><\/strong>).<\/p>\n

However the\u00a0slip ratio<\/em> is a tricky concept. Having slip doesn’t mean that the tire is actually sliding over the ground. The slip ratio reflects the fact that the drive wheels compress the tire at the front of the contact patch where the radius of the wheel is smaller, so it effectively travels faster than the ground underneath. A portion of the contact patch is adherent to the surface while the other portion is slipping. This slip ratio<\/em> generates the force that moves the car. In a car cruising at constant speed the drive wheels will be spinning a bit faster than the non-drive wheels.<\/em><\/p>\n

There are several different definitions for slip ratio and slip angle available. These ones are a commonly used definition borrowed from the great book from Brian Beckman\u00a0The Physics Of Racing<\/a><\/em>:<\/p>\n

\"\"<\/p>\n

\u03c9 \u00a0 \u00a0angular velocity
\nRe<\/sub>\u00a0 effective radius
\nV \u00a0 \u00a0lineal velocity
\nVy<\/sub> \u00a0lateral velocity
\nVx<\/sub> \u00a0longitudinal velocity<\/p>\n

The slip ratio \u03c3<\/em><\/strong> and slip angle \u03b1<\/em><\/strong> are the input for the Pacejka formulas, which yield the longitudinal and lateral forces exerted by the tire. Although this kind of methods is known to provide the state-of-the-art tire simulation, they still exhibit severe problems:<\/p>\n