Part II: Analysis of Basic Ship Maneuvers

In Part I, I showed how the ship motion is based on the physics of objects in a lossy medium, and developed the equations of motion that can be used for individual motion of any ship.  In this section, I show how these equations can be applied to model the behavior of any ship motion commands that the pilot can make.  In Part III, I will show how ship motion commands are affected by the motion of other ships or by collisions with them.

Part II: Analysis of Basic Ship Maneuvers

Changing Directions & Turning

Minimum Velocity Point

Orbital Motion

Automatic Orbits

Automatic Orbit Data

But wait. . .

Closed-form Orbit Equations

Comparison of Orbit Data with Closed-Form Equations

Manual Orbits

Manual Orbit Data

Copyright Notice

Changing Directions & Turning

Using the model from Equation 1-5, I can compute the velocity characteristics for ships as they make turns in space.  For starters, it is simplest to compute ship behavior for a fixed command vector that does not change in time.  If a ships is already moving at VMAX in the x direction, I want to know how a ship behaves when the pilot enters a new direction for movement an angle θ.  

Consider an initial 2-D problem setup with initial motion vector of,

Now at time, t = 0, we issue a command to the ship model as a maximum-velocity vector at an angle θ.  The command vector is,

To solve for the motion of the ship, I must write the solution to the differential equation in each dimension.  First, I'll do the case in the x direction which is complicated slightly because there is a nonzero initial condition.  Using superposition of the homogeneous and particular solutions I can write the x direction response, and simplifying,

          .

The y direction is considerably simpler requiring only acceleration from zero,

          .

Finding the magnitude of the ship's velocity in time is just the length of the velocity vector,

Figure I-9 shows the x and y component velocities and total velocities versus time, normalized to the time-constant, τ.  This normalization is possible because in the velocity equation above, the time variable is always in a ratio with the time-constant of your ship.  Thus, turning feels similar between ships, with the same shape of curve scaled only in time, and the minimum velocity ratio scaled by your maximum velocity and depending only on the angle of the turn. The applicability of this theory might be for the execution of turns while taking missiles fire -- your minimum velocity during such maneuvers can be important to mitigating damage.

If the derivation above looks complicated to you, do not be intimidated.  Linear differential equations of this type really are very simple.  So simple, in fact, that you can write the solutions based solely on inspection.  Let’s press on...

Minimum Velocity Point

Solving for the minimum velocity point in the turning transient, I find that it occurs independent of the command vector angle, occurring at time,

This time is independent of the velocity or the new motion angle.  This is because the geometric sum (vector length) of the two transients always has the same final magnitude; so the minimum velocity is not the same, but the time at which that minimum occurs is always the same.  To compute that minimum velocity, I can just write vtotal(t) at this time tmin.  

This relationship is useful for getting a sense of the velocity penalties you experience given the turn angles you choose.  When facing missile spam this performance is something to keep in mind.  Note that this form is independent of τ meaning that all ships experience these minimum velocity conditions during turning transients, scaled by their own maximum velocity, of course.  

Figure I-10 shows this function for angles from zero to π, or zero to 180o.  In this figure I also show a set of data points taken on Tranquility.  I used a frigate with no-module and afterburner cases, which correspond to the two sets of data points.  I took this data using the 'Tactical Overlay' which allowed me to create 45o, 90o, and 135o angles with reasonable accuracy.

There are two points that I find easy to remember when piloting.  These are the 90o point, where the fraction of maximum velocity is  or 71%.  Also the 120o point is easy to remember at just 50% of the maximum velocity.  Finally, I have included Figure I-11 that shows the velocity profiles for several angles to show that total velocity minima occur at the same time independent of other parameters.  I would like to take time-series data but without video sequences it is considerably easier to just find the minima as in Figure I-10.

This analysis is a simple example of how formal methods can be used to get a sense of how piloting decisions are linked with piloting performance.  In subsequent piloting sections I will try to continue in a similar manner.

Orbital Motion

With all this math on motion mechanics we should also be able to explain the behavior of ships when moving in a circular path, right?  Circular motion, or orbit motion, is important in EVE as it has many uses in combat situations.  Depending on how an orbit is used it can be used to counter turret tracking, or as a way to simply maintain a distance relationship without staying still.  This section evaluates circular motion as accomplished with the 'Orbit at R' command in the EVE interface, and evaluates the limitations of these 'Automatic' orbits.  A more detailed discussion of orbits and tracking tactics will be presented in Chapter II.

Automatic Orbits

First, let's be clear about what the 'Orbit At R' command appears to do.  All it does is command your ship to continually accelerate towards a point that is a distance R away from the selected object.  This is somewhat different than an ideal orbit because all of the acceleration in an ideal orbit situation is entirely directed towards the center-point of the circle defining the orbit.  Given that your ship accelerates towards an off-center point, not all of the acceleration that a ship can produce is exerted towards the center of the orbit.  This bears some analysis to determine how much this compromises your ship's ability to achieve a minimum orbit distance.  

Keep in mind that if you are closer to the target object than the distance, R, then this command pilots directly towards the closest point on the R-sphere around the target.  Simply put, activating the 'orbit at R' command in this circumstance means your ship pilots away from the target in the most direct line possible.  When the distance R is reached, the behavior changes to the normal orbit behavior discussed in the first paragraph.

So, if the ship accelerates toward a point that is R1 away from the target, at what distance does it end up orbiting at?  And how is the speed and distance determined from ship parameters?  Figure I-6 shows a ship at A orbiting a stationary point B at radius R2, having been commanded to 'orbit at R1'.  Also, note that a right triangle has been inscribed in the R2 circle with the opposite side of length R1 and the hypotenuse of length R2.  The angle formed at the piloted ship I'll call, .

EVE happens in 3-space, but normally orbits occur in a plane, so I have composed the motion equations in two-dimensions, the x-y plane.  In this arrangement, the ship's orbit velocity is vy.

with the limitation,

.

Let's narrow things down with some boundary conditions.  First, note  for the ship at the position as shown.  Now I will add  the effect of centripetal force on the ship.  This is a potentially dangerous step because nothing of what I have done so far seems to indicate that this force exists.  Lets just call it an intuition air-strike.  The orbit acceleration force,, has to balance with the centripetal force on the ship.  Rewriting the x-y form of the differential equation including the centripetal force balance projected in each of the axes,

                                                (1-6)

In this form, I am projecting the force applied in acceleration of the ship along the x and y axes separately.  I can rewrite these projections in terms of radii from the right triangle in Figure I-6, with some help from trigonometric identities.

                                (1-7)

                                           (1-8)

We now have two equations for the forward velocity of the ship, and we want to solve for the orbit distance, R2, and the orbit speed, vy.  Solving explicitly for either of these variables is extremely messy.  A form for R2,

                                  (1-9)

Or, eliminating to isolate a vy form,

                                (1-10)

Both of these have closed-form solutions, and I present these in the following section.  

Automatic Orbit Data

There are two things I can do with solutions to Eqn. 1-9/1-10, messy or not.  First, I can check to make sure that it explains the behavior of ships in the game.  Second, I can come up with some quick guidelines for what we can learn from these more complex forms in terms of fitting and improving our orbital movement.

Fitting my Enyo with an afterburner and a micro-warp drive and orbiting a small stationary target at several ranges (500m, 1000m, 1750m, 2500m, 5000m, 7500m), I obtain the graph below.  I also graphed the solutions to Eqn. 1-9/1-10 in green to show how the theory for ship motion in EVE agrees in Figure I-7.

Enyo Parameters

I have repeated this sort of experiment with cruisers, destroyers and battlecruiser ships, although I won’t take up more space to present all of those plots here.  Noting the agreement in the graph, the model has captured, in part, the quantitative character of ship behavior, namely the orbit distance and velocity curves for ships.  

But wait. . .

Looking at the figure above, it looks as though I've got it all figured out.  I could just leave it here and proceed with a discussion of closed-form expressions for angular velocity and no one would be the wiser.  Unfortunately, that would be a lie.  The truth is that I have computed a solution based on my rationalization of what may be happening in EVE — a first-order linear drag model with centripetal forces.  This model can not be dismissed outright, after all it correctly models the relationship between orbit radius and velocity, and many other behaviors you may wish to consider.  What is not consistent with the in-game behavior, is the exact solution for points on this vy curve.

The origin of the discrepancy is, in part, due to the fact that I am using a set of continuous equations to solve for the equilibrium of forces on the ship.  The in-game computation, however, is simulating a discrete time-series of movements and forces which have been designed to improve computation workload, memory usage, client synchronization or other proprietary goals.

The way to make the equations agree fully with the in-game data is to notice that a finite non-zero distance has been added to the target orbit distance command, R1.  This is not intuitive, but it makes sense that the game designers wanting to have another knob to adjust automatic orbit performance.   I have developed a model that give the exact behavior of the ship in orbit situations, based on Eqn. 1-6.  I am showing the most general closed-form below, but be warned that the closed form conveys little additional insight into the motion process.  The valuable contribution from this orbit section is to show that our physical model has correctly captured the ship behavior.  As a side benefit, we also have a useful tool to predict ship behavior in orbit situations where the orbited object is stationary or nearly stationary.  

Closed-form Orbit Equations

Predicting your orbit speed and distance can be of importance for speed tanking.  This is particularly true if you are designing kiting cruiser fits whose final orbit speed is critical to avoiding damage.  In the previous section, I described a closed-form relationships for orbit velocity and orbit distance.  In this section I present the cartesian form of this equation which could be of use to tool developers for fitting parameterization.  I am also adding a dataset for a 100MN afterburner cruiser case.

Note that in the equations below, I am assuming that the orbit is around an object with a zero-meter radius.  To correct for the fact that most objects in EvE have a non-zero radius, which I refer to as R0.  It is necessary to add the radius of the orbited ship or object to the orbit radius command to get the effective orbit  command radius, R1.  The figure below illustrates how I have labeled the variables for the ‘Orbit at Rc’ command by adding R0 to the command distance, R1 = R0 + Rc.  

Based on equation (1-9) I solve for orbit radius, R2, and equation (1-10) allows me to solve for the orbit velocity, vy, or vorbit, as I will now be referring to it.  Note that both of the closed forms for these equations have four possible solutions, depending on the result of the sign of higher-order roots.  Choosing the correct form is trivial, based on comparing the numerical results for both equations.  The closed-form equations for the orbit radius, R2, given that you command an orbit radius of R1,

        (1-11)

The orbit velocity can be derived from equations (1-10),

        (1-12)

Now that I have an elaborate equation like this, it is always helpful to check boundary conditions to see if they agree with intuition for the system we are studying.  Consider the case when the ship is piloted in such a way that the acceleration of the ship is towards the center of the target.  That is, if you piloted manually, and accelerated towards the center of the orbit circle (R1 = 0), instead of an edge that was offset by R1, what would be the result for R2 and vorbit?  Would this case agree with our intuition from the centripetal force model in Eqn. 1-6?  

If we apply a condition of R1 → 0 to both equations (1-11) and (1-12) above, it is immediately obvious that both quantities will approach zero.  This may trouble you until you note that the only way to orbit at an arbitrarily close distance, say five meters, would be to have an infinite force available to counteract the centripetal force.  Ships in EVE, however, have only limited acceleration available to them, so orbits for R1 → 0 would approach zero velocity.

Another way to look at this boundary condition is to consider the acceleration of the ship.  The idea that made orbit analysis possible was the claim that centripetal forces were balanced with the ship’s acceleration force as the ship is turning.  If I write the centripetal acceleration force as a function of the commanded orbit radius, R1, then I find that as R1 approaches zero, the centripetal force is,

It should be possible to reach the maximum possible acceleration that the ship is capable of, or,

Sure enough, when I plot the equations above for R1 → 0 below, I find that the acceleration towards the center of the orbit circle is limited to the maximum acceleration possible for the ship, VMAX / τ.  This confirms that the equations capture the limitations of the ship’s acceleration for circular motion of all types, not just for ‘Orbit at R’ programs.  Once again, our ship motion model in EVE shows itself to be consistent with a physical analog that gives us insight into how ships will behave outside the built-in orbit functionality.  

What insight do these closed forms provide us?  Be careful.  This is a trick question.  

At first, my answer was rather contrite — “none!”  The value of this chapter is the consistency and completeness of the vector differential equation model.  My thinking was that any of the tools that derive from the model are useful, but in themselves they are not the insightful part of this work.  Given my training, I should have known that being dismissive of data—theory mismatches was a mistake. If you have captured a theoretical model, and compared it with careful experiments, discrepancies almost always reveal details in the model that you may not have previously considered or that you considered incorrectly.  So, the key to my trick question is that insight is not in the closed form itself but in the ways, if any, that this theory disagrees with data.  I think you will see below that there is still more to learn about orbits.

All that said, when deriving a solution or closed form it is helpful to look for insight into what the terms represent from the perspective of the operative or physical elements in the system you are studying.  If any reader can parse the meaning of the terms in these equations it would be a helpful contribution to this work.  I have prepared some examples to show the correctness of this model for a broad set of circumstances including cruisers, frigates and battleships.

Comparison of Orbit Data with Closed-Form Equations

The challenge with accurately predicting the properties of an automatic orbit lies partly in understanding the effective diameter of the object you are orbiting.  Each of the three figures below includes two predicted orbit result curves, one assumes that the can I was orbiting has zero effective radius, R0 = 0m, and another assumes the can has an effective radius of R0 > 0m.  You can see that including the zero-meter contour is important in improving the prediction for close-in orbits.  

What you will see below does not qualify as a proof, and does not include any claims about the statistical fitness of data and theory.  In other motion model examples in this chapter, you will notice that the fits are close enough that statistical analysis would add little value.  For closed-form orbits, however, there is still some fitting to be done, which I believe can be explained by adding a ‘fudge’ factor, but this assertion still needs to be proven.

Cruiser Parameters with Oversized 100MN AB

Cruiser Parameters with 10MN MWD

Cruiser Parameters for 10MN AB

As you can see from all of these examples, there is a problem with the agreement when I don’t use a orbit distance correction factor, R0.  There may be several reasons why this is showing up in the data.  It is possible that the way I am taking the data is creating a systematic error.  It is alternatively possible the measured orbit data is correct, but that my equations are derived incorrectly creating a false expectation for the orbit distance and angular velocity.  Finally, it is possible that CCP has added an orbit distance ‘fudge factor’ which depends on whether the ship is fit with an afterburner or a micro-warp drive.  

I am skeptical of the second explanation, because I believe the boundary conditions prove that we are on the right track as R1 approaches zero.  It is possible that using a jettisoned can as an orbit target has some strange effects, but the offset distance should be the same regardless of what type of module is fitted.  To address the question of whether the effect is module dependent, I devised an experiment where the mass, velocity and drag (inertia) are all equal, even though the base speed and module choice are different in each case:

Frigate Parameters (Raptor)

If CCP is adding a ‘fudge factor’ which depends on the module, the orbit performance data will diverge for these three seemingly identical arrangements.

Taking a look at the figures, I see clearly that the ship motion of each of these ship configurations is almost identical.  Second, the model (Eqn. (1-11) and (1-12)) does not agree with the R2 distance of the orbit.  This tells me that the the model is consistent for a given set of parameters, and also that the discrepancy between my closed form equations and the orbit behavior is not dependent on the motion parameters or on the presence of an active module!

If the discrepancy is not built in based on modules, then perhaps it is based on the ship type?  The right experiment is to equalize the motion parameters between two different hull types and compare the orbit performance.  

*** work ongoing ***

In this section I have developed closed forms for orbit velocity and orbit distance, Equations (1-11) and (1-12), derived from Equations (1-9) and (1-10).  Measurements across ship class and propulsion module show that these equations compute orbit radius and orbit velocity accurately from known ship parameters.  These equations are a useful tool in determining angular velocity when orbiting stationary or slow-moving targets.

Manual Orbits

Recall that in combat situations, motion is important for mitigating damage from missiles, drones, and turrets.  While missile damage depends on (among other things) your absolute speed, turret hit probabilities depend on (among other things) the ratio of your transverse velocity to your range, i.e. target angular velocity.  At this point it is possible to have a discussion about transverse and angular velocity.  I will, however, delay the urge to get into this in detail at this point because it is better served in a chapter dedicated to tracking mechanics.  From the solution to Eqn. 1-9/1-10, I could write a messy closed form for a ship's maximum angular velocity on a stationary target, .  Instead, I will provide some reasoning about the limits to angular velocity and how to overcome them.

By using the 'Orbit at R' command that is built into the EVE interface, you are executing a motion program that has built-in limitations — the interface does not permit pilots to program an orbit any closer than 500m.  If, however, you pilot manually by accelerating towards the center of your orbit, or slightly off-center orbit, you can achieve much tighter orbits, and higher angular velocities.  It’s not hard to do this in a frigate when attacking battlecruiser- and battleship-sized targets as they are almost stationary relative to your movement.  I will admit that I have not found this technique widely useful; however, it has helped me mitigate damage in some circumstances.

From the analytical perspective, we can revisit Eqn. 1-7 with new conditions to compute the limits to angular velocity.  When I analyzed the forces at work on the ship I projected the acceleration in the x-axis and wrote a balance with the centripetal force.  When the pilot can apply all of their acceleration towards the center of the orbit, then no projection is necessary.  The first equation from Eqn. 1-6 simplifies to,

.

Using this to write an angular velocity limit for ideal manual piloting, and considering that we can orbit at some distance R < 500m,

.

Let’s see how this hypothetical limit compares with the data taken for the Enyo using automatic orbits.  Recall that our 'Orbit at R' real velocities are constrained by solutions to Eqn. 1-7 and 1-8 and the relevant extensions for solutions on the vy-R2 curve.

Manual Orbits

From the Enyo data shown above, I have graphed the angular velocities achieved under three cases: No active module, afterburner II active, and micro-warp drive active.  I have also practiced manual orbits myself, and I can say that, as an example, for the afterburner II case, I can achieve  at a target distance of about 800m.  While this is not in itself an impressive result — some frigates accomplish this under automatic conditions — it does exceed what is normally possible for this ship hull.

Note that the manual limit curves in Figure I-8 represent the theoretical maximum that can be achieved consistently in circular orbit conditions [Clarification to this section is posted on EVEisMath.blogspot.com].  They are not curves that are intended to agree with the automatic data points.  In fact, you can see from the no-module case that orbits of this type are not able to approach high-.  This is in part because, unlike the close-in cases like afterburner or micro-warp drive, the speed and mass of the no-module frigate are low, meaning it follows the artificial program of the 'Orbit at R' command.  So, if you want to get the most out of your orbit, use an afterburner, and learn to pilot manually.  

*** add section on \tau periodic ***

In this section, I have tried to show how orbits can be understood as part of the physical model for EVE ship movement, and how these maneuvers can be improved with some additional effort on behalf of the pilot.  By overcoming the geometric limitation of the 'Orbit at R' command through manual piloting, ship performance in EVE can be improved. When and where to apply these improvements is another story altogether.  Practice is, as always, the best guide.

I mentioned above that with the model that I had developed I would also outline some guidelines for how modules and fittings affect basic movement parameters.  In the Module Effects on Motion Section, I summarize how modules, skills and projected effects contribute to some of the summary performance numbers in this section.

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