Introduction to EVE’s Motion Model

  These notes are intended for a broad readership in the EVE community, but they are most suitable for mathematically-minded players.  In this chapter, I analyze ship movement in detail.  I expect that the reader already has a reasonable amount of player–player combat experience.  If you do not, build up some empirical experience with small and medium ship combat against other players.  

All the data presented in this article was extracted from the Tranquility server, from the dates of December 20th, 2009 to March 18th, 2010 during the Dominion expansion [6].  I chose this expansion because changes to the velocity calculation were done in prior expansions (i.e. Quantum Rise).  I am aware that the motion constants have changed since the data was taken, as well as several bonuses for assault frigates.  Changes to tables and graphs may be coming in the future, but the conceptual results stand on their own and are still accurate and relevant in late 2015.

This chapter is organized around a first-order dynamical model of ship movement.  I use this framework to analyze a variety of maneuvers and even some tactics.  Note that I will assume that pilots have all skills at Level V.  If this seems somewhat unrealistic to you, consider this as a discussion of what is possible.  Indeed, if you are serious about flying a ship well, you will learn rather quickly that skill level is important.  The chapter following this one will discuss tracking and how it relates to the physical model of movement in EVE.  Before I launch into the details of object movement, it is worth noting a few things about objects themselves.

Introduction

A Quick Note on Objects

Marbles in Molasses — Equations of Motion

One-Dimensional Motion

A Note from Newton

Case 1 — 1-D Motion at Maximum Speed

Case 2 — Starting from a Stationary Position

Three-Dimensional Motion

Mass Matters — Ship Speed Calculation

Velocity Calculation Examples

Summary of Linear Motion Equations

Maneuverability Data for Selected Ships

Some Graphs

Copyright Notice

A Quick Note on Objects

It turns out that objects in New Eden are not point elements. They actually have a zero-meter-contour corresponding to an imaginary boundary within which you appear to be at zero meters from the object.  Note that I have not called this boundary a collision extent, because for many objects you can maneuver your ship closer even though you are, according to the overview, in contact with the object.  Depending on the type of object, the role of this nonzero distance is different.  Ships, cans, stations, and gates all have a nonzero radius inside which you can navigate.  Ships in EVE also have a spheroid, zero-meter boundary.  Figure I-1 shows an example for two objects in a 2-D projection of space.

The zero-meter contour ranges in size from tens of meters, for frigates and cans, to hundreds of meters for battleships.  I'm bringing this detail up now because the following notes require a reasonably precise notion of distance and proximity to objects.  It is helpful to keep the zero-meter distance in mind.  The game designers at CCP have indeed approximated many objects in their world as spherical cows, so to speak.

Note that this distance has nothing to do with the signature radius of the ship or object.  Signature radius is a parameter that is used for tracking and damage calculations and has nothing to do with object distances and is discussed in detail in following chapters. And now, on to a discussion of ship motion in EVE. . .

Marbles in Molasses — Equations of Motion

What we know from CCP's disclosures is that they have used the following model for ship movement in EVE: From a stationary position, when your ship is commanded to accelerate to a  velocity, , your velocity function in time is,

                                (1-1)

where, M is the mass of the ship in millions of kg, and  is the ship's "inertia modifier" (more on this later).

This curve is an exponentially asymptotic approach from 0 to VMAX, with time-constant, , which EVE calls the ship's agility  [2].  I have included an example time-response below showing an acceleration and deceleration event.

Anyone with a basic differential equations background will recognize Eqn. 1-1 as the solution to a first-order linear differential equation.  If having an equation were a substitute for having an understanding, we'd be done!  Unfortunately, it's not that simple.  How did they arrive at the first-order dynamics?  Is there any physical insight that pilots can gain from that model?  Are there any implications for successful piloting?  How does this inform ship maneuvering and collision events?  First, a discussion of the linear model.  Then I'll add some other details.  

One-Dimensional Motion

To derive the equations of motion in EVE, we have to go back to the basic physics model that gave rise to the equation in 1-1.  I do not have any special knowledge of how CCP designed their game; however, I believe that insight into ship movement in EVE is actually found in the equations for objects moving in a fluid drag medium.  Let's take a look at motion in a straight line first to see why this is the case. Later I will generalize the form to vector movement in three dimensions.

Objects in a linear drag situation can be accelerated by applying a force to them,  .  They also have a drag force opposing this motion which is linearly proportional to velocity.  The drag force on the object is proportional to how fast you are moving, and acts to oppose your current velocity, .  Figure I-3 shows an example situation and labels the forces at work on the mass, M.        Appreciable drag forces are not what you might expect in the vacuum of outer space, of course, but selecting a game rule is part of the designer’s creative oeuvre which can enrich the game, in this case, by being physically consistent for all objects and with intuition.

If you sum up all these forces on the object, you get a simple first-order linear differential equation.

                                        (1-2)

                                           (1-3)

That is the linear differential equation that I promised not to write down unless we had a sense of it's meaning so, I will try to explore its meaning in two ways.  First, by explaining a little about what the terms mean physically, and second, by filling in some base cases to show why this equation explains the behavior of ships in the EVE universe.  I have formatted these notes as an aside for readers who want more information about deriving the motion equations.  Readers may chose to skip this material if it suits them.

A Note from Newton

We have a mass that is being accelerated by forces.  Intuitively, we know that pushing on an object causes it to accelerate.  Newton captured the intuition for this in his Second Law, which states:

A body experiencing a force F experiences an acceleration a related to F by F = ma, where m is the mass of the body.

Equation 1-2 is exactly the same equation as F = ma, but for a ship in EVE.  There is a competition of forces at work on the ship, which is being captured by the subtraction of the accelerating force, Fa, and the drag force, Fd.  

Also notice that in these forms, I use the notation  to represent the acceleration, a, which just means 'the rate of change of velocity with time'.  I wrote it this way to keep the notation for velocity consistent for all the equations in this article.

Case 1 — 1-D Motion at Maximum Speed

First, consider the case where you are underway, moving in a straight line at maximum cruising speed.  The velocity is not changing in either magnitude or direction.  So,  is zero and,  is just the constant .  So, our equation of motion is now, .  Essentially, the drag force on the ship is equal to the applied acceleration force.  So, computing for our maximum 'thrust' from our ship parameters, , where  has units of [kg/s].

Case 2 — Starting from a Stationary Position

Next, consider the situation where you are sitting still and you start to accelerate.  The initial velocity is zero, , and you start to accelerate as quickly as possible in a straight line.  We learned from the maximum velocity case above, that maximum thrust force is .  So, our equation of motion in this case is just, .  This means the maximum acceleration you can exert on your ship, without getting a bump from a buddy, is .

If we test our differential equation (Eqn. 1-3) against what we are told our motion equations should be (Eqn. 1-1), we get some more clues as to what is going on. I have omitted the manipulations here; however, after a blizzard of cancellations the ship 'inertia factor' labeled, , turns out to be the reciprocal of our drag factor, .  This is to be expected, since the time constant was already known from (Eqn. 1-1).  What is important is that we now have motion parameters mapped to our ship parameters.  So, rewriting (Eqn. 1-3) with our 'inertia factor' term, and in vector form,

                                (1-4)

The form in (1-4), using the time-constant , is to my eye the easiest to understand.  Perhaps this is because, as an engineer, I am used to looking at equations with time-constants.  A scientist might use more Greek letters but it would be the same equation.  On the left-hand side we have terms that represent the ship's velocity, , and its change in velocity in time.  The right-hand side represents our desired motion commands captured in the vector, .  The unit-length vector, , is just the direction that you double-clicked in space.  With certain built-in motion programs like 'Orbit at R', the direction of this unit vector is determined for you automatically.

The form in Eqn. 1-4 has the benefit that it is in terms of velocity, which is our parameter of interest, and it captures the maneuverability of the ship in two scalar constants, VMAX, the maximum ship velocity, and , the acceleration time-constant.  So, in the frictionless vacuum of space, our snappy little ships are moving around like marbles in a bucket of molasses!  

From the perspective of masses moving in a lossy medium, EVE's name for the 'inertia factor' would be more correct if it were ‘drag constant’.  Inertia has units of [kg m/s], which would not suit how this term is used in the equations of motion whatsoever.  Calling it a drag factor is only slightly less misleading, of course, but it has the benefit that it corresponds to a physical model.

Three-Dimensional Motion

The vector notation can also be decomposed into three independent equations for motion in 3-space.  The total velocity vector magnitude would be limited to a magnitude of, , of course.  Or,

                                (1-5)

with the maximum velocity constraint on our command vector,

.

I haven't written out a polar coordinates form here, but I will use one in the next chapter.  Stay tuned.  I should probably also include a discussion on the independence of orthogonal motion programs, as well as linear superposition of motion programs.  Suffice it to say, (1) motion in each of these directions is independent, and; (2) contributing motion programs in the same dimension will result in ship motion that is the sum of the responses to those programs.  My development of orbit theory below relies on this property although I do not fully explore this concept here.

It is worth noting that EVE probably does not simulate this differential equation in its full-precision glory.  Instead, it likely uses a discrete-time difference equation to make the math go faster.  This should make no difference to the player except to note that time in the game is discrete in some way.  No doubt, the precise workings are a CCP secret.

Finally, some readers have noted that using a linear drag model may not be what CCP used to originate the equations of motion.  While this may be true, it does not make a great deal of difference, because the resulting behavior of ships in the game is well explained by this model, as you will see more of below.  There are, in fact, an unlimited number of natural systems that produce first-order motion dynamics.  The idea of using linear drag, or marbles in molasses, is just a simple one that is physically familiar to people.  This statement relies on the untestable assumption that the reader has left their computer desk at some point in their life.

Mass Matters — Ship Speed Calculation

Before wrapping up the discussion of the equations of motion in EVE-Online, it is worth discussing where my use of the term VMAX comes from.  In these notes, I use VMAX to signify the maximum speed that you can command your ship to move at, given the modules, active or otherwise, that your ship has online and active. There are several articles out there that discuss maximum speed calculations, specifically Farjung's article on the EVE-O boards [3].  I will summarize some of these calculations here for completeness.  Calculating the cruising speed of ships is straightforward; however, the effect of afterburners and micro-warp drives adds a counter-intuitive turn. some module names are clearly chosen for intuitive perspective and not necessarily consistent with a Force-Mass-Drag model.

I define Base Cruising Speed, VBASE, to be maximum ship speed without the aid of active modules such as micro-warp drives or afterburners.  This speed can be affected by passive ship modules, skills and implants.  Ship modules, such as overdrives, nanofibers and expanded cargo holds, change this base speed for a ship and can also modify the other motion parameters such as mass (in the case of armor plates) or ‘inertia’ (in the case of nanofibers).  From the perspective of game design and balance, the modules behave in ways that make sense and that is most important, giving ship fitting all three degrees of freedom -- changing the force, mass and ‘inertia’ separately.  

From the perspective of a Force-Mass-Drag model, however, some of these modules behave in ways that are counterintuitive.  For example, I showed above that the maximum cruising velocity of a ship arises when the force exerted by the ship is in balance with the drag force.  So, I can solve for the maximum velocity that a ship can produce as,

It is easy to imagine that the ‘Overdrive’ modules change the maximum force that a ship can deliver, but the Inertia Stabilizer modules change the ‘inertia’ (actually drag) factor, changing the time constant of the ship, but not changing the maximum velocity.  For velocity to be unchanged, but drag to be decreased, that would mean that Inertia Stabilizers are also increasing the force that EVE ships can exert?  This is a minor point because the change in ship motion performance can still be attributed to a physical mechanism and, although it is counterintuitive for a Force-Mass-Drag model, it is possible that another model was intended when the game was conceptualized.  

(Note that a previous version of these notes claimed that changing the mass of the ships by adding plates should have change the velocity of the ship.  This was incorrect.  Adding plates should not change ship velocity in the Force-Mass-Drag model).

If, for example, you add overdrives and plates to a ship, only the overdrives affect the base cruising speed.  This is nonintuitive because if EVE ships had a fixed thrust, their base speed would be affected by adding plates (mass) to the ships.  It seems that the design for ship movement includes a minimum base speed that is not degraded by increasing the base armor with plates and modules.

Adding plates does add to the mass of the ship.  For active propulsion modules, the mass of the plates and the mass changes from the afterburner both affect the new top speed.  So, the AB,MWD Speed , VAB , or VMWD, is dependent on mass.  The form seems to apply the assumption that afterburner and micro-warp drive modules apply a specific amount of force to the ship.  The equation for afterburner or micro-warp speed, sourced from [3], is,

,

where gAB,MWD is the speed boost factor for the module.

When the afterburner/micro-warp drive is active, two things happen.  First, it adds mass to your ship, , which affects your time-constant and agility.  Second, it multiplies the base speed of your ship by a factor that depends on a force, F, divided by your new mass.  The less you weigh before you activate the module, the faster you'll go, and the more agile you'll be on the way.  

Until 2015, propulsion modules had names that were not consistent with the approximate added force, however, these have been updated.  I may add a post to explain this.  As of end of 2015, the MWD classifications are as below.  

For readers who are still a little confused about this, I have included an example calculation in the block below.  Note mass units are in kilograms but I use Mkg to specify millions of kilograms to make it less cluttered.  This is basic stuff.  

Velocity Calculation Examples

Given a frigate with the following parameters:

Mass M = 1,000,000 kg = 1 Mkg

Inertia I = 2.32 s/Mkg

Base Cruising Speed = 300m/s

The time-constant for movement, without the modules active is,

These examples use the trivial case of a ship mass of 1,000,000 kg.  Otherwise, every bit of mass below this will only help improve the active ship speed you can achieve with active modules.

Now that I have elaborated a basic ship movement model, I'll summarize this first section and we can move on to orbital motion and other tactics.

Summary of Linear Motion Equations

Lets review what I’ve derived in this section:

1. When a command vector is entered into the eve interface, by double-clicking in space, aligning or approaching a stationary object, the ship accelerates in the direction of the command vector.
2. A ship's maximum acceleration is  or .  Afterburners and micro-warp drives add mass to your ship, dynamically changing the maneuvering time-constant.
3. A drag force acts in the opposite direction to the ship's velocity.   The magnitude of the drag force is proportional to the ship's velocity and a coefficient.  The drag coefficient is one-over the ships 'inertia factor', , and has units .  The 'inertia factor' term is affected by several different skills.
4. As the ship's velocity increases, the drag force increases until the ship reaches a terminal velocity, .  Base velocity is affected usually by one or more skills.  
5. Acceleration and deceleration exhibit symmetric dynamics (i.e. they have the same time-constant).  Unlike driving a car, the mechanism for both acceleration and deceleration motion in EVE are the same.
6. The time-to-warp is a linear multiple of the time-constant, .  Simply put,  is the time for velocity to reach , or about 62.3% of .  Time-to-warp is the time required to get to 75% of .
7. Base Cruising Speed is not affected by mass unless afterburners or micro-warp drives are active. This breaks somewhat with the force-mass linear drag model used for ship movement.  MWD/AB Speed is dependent on mass of the ship and module as well as the effect of armor plates.

The table below summarizes some ship maneuverability data from the era of the Dominion expansion [6].  This is included here as a simple comparison for pilots familiar with Tech. 1 hulls.  More ship maneuverability data is available in Appendix A.  

Maneuverability Data for Selected Ships

The first thing to notice is that for each successive ship class, the mass increases by an order of magnitude; however, the inertia, or drag factor, goes down less dramatically.  No doubt, these constants have been chosen so that the time-constants, and alignment times of subsequent ship classes, will be comparable.  After all, it would make little sense for a cruiser to take ten times longer to align than a frigate, and a battleship 100 times longer still.

Within each class there are some racial trends as well.  It is no surprise that the Minmatar ships have a significant base maneuvering advantage, primarily owing to their lower inertia modifier coefficient, not lower mass.  For other races of ships, there is often some mass or inertia advantage; however, because of high drag coefficient, other ships do not accelerate as quickly out-of-the-box.  The difference is substantial, with small and medium Minmatar ships being as much as 30% more maneuverable than other races. This is not a definitive statement on ship movement, of course, because it does not account for fitting and piloting, which are critical factors for determining the outcome of encounters.

Some Graphs

I have presented several different viewpoints on ship-motion graphs in this chapter, including AMAX versus VMAX, and τ versus VMAX two-dimensional charts. These plots are two-dimensional projections of what are actually three degrees of freedom (Inertia, Mass and VMAX) because it takes these three parameters (not including active module effects) to fully specify how a ship moves.  I would make three-dimensional projections of the parameters but its hard to move around them on a two-dimensional screen.  Choosing the axes for these projections depends on what you are trying to visualize, so there are other graphs that can be made and readers should keep that in mind.  I will include some other projections of the three-dimensional data in Appendix I.A.

Figure I-4 shows a comparison of small-ship motion parameters assuming no fitting or modules (i.e. without afterburner or micro-warp drives active).  I have arbitrarily graphed maximum base speed against maximum acceleration.  Some readers liked this because it contrasts the ability to accelerate, align and maneuver with the ability to catch targets.  This is only one viewpoint for comparing ship movement, and it should not be interpreted as a criticism of the parameters as they stand.  

Another version of this comparison for afterburner-active parameters reveals the importance of mass for active setups.  Figure I-5 shows a comparison of the velocity and acceleration parameters with an Afterburner II active.  Why you would fit an Afterburner on an Interdictor is not clear; however, I left it in for comparison while I have removed some of the other ships for clarity.  There are several notable changes between Figures I-4 and I-5 but I will not enter into a long discussion here.  After all, there are more interesting things to discuss.

So far, I have explained the basic mathematics of linear ship movement in space, where it may come from, and how it is parametrized. I've also summarized some example parameters for common ships. What we need to do now is to see how this applies to common maneuvers, such as automatic orbiting and, in the next chapter, tracking.

In Part II, I will show how these equations can be applied to model the behavior of several built-in motion commands that are commonly used by EVE pilots.  

Copyright Notice

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