Going the other way

Introduction

Problem statement and conventions

• Ball speed V_ball
• Vertical and horizontal launch angles LAy and LAx
• Spin, either as:
• Total spin S and spin axis tilt σ, or
• Backspin Sy and sidespin Sx
• Coefficient of restitution e
• Clubhead and ball masses M and m respectively

• Clubhead speed V_clubhead
• Spin loft L (what TrackMan used to call spin loft, the vertical component)
• Angle of attack K
• Face-to-path angle A
• Clubhead path angle P

1. Please bear in mind that this is not a new model of clubhead-ball impact, nor is it a test to validate such a model. It is much more modest than that. In 2007, I came up with the model described on page 1. There are plenty of other models of impact before and since, and some are undoubtedly more accurate; but this is the model we concern ourselves with here. All we do here is to reverse the direction of the model, solve what the inputs to that model must have been given the outputs. That is an exercise in algebra and, in the case of two equations, numerical analysis. Physics is not involved any more (that was taken care of in the forward model). Not only is physics not involved, the only involvement of golf is to define reasonable ranges of the parameters.
2. In the work that follows, we will refer to steps from page 1 as the source of formulas we use. For instance, we might introduce the formula S  = 160 * V_clubhead * sin(Φ) with no more justification than "from Step 6". That means it was derived in detail in the impact-to-launch process and computed in step #6, so you can refer back to that if you want more depth.
3. It is assumed that impact is center-strike. That is, there is no gear effect spin, nor loss of smash factor due to strike quality.
4. Some of the parameters may need to be estimated from "typical" values. For instance:
   • A driver head typically weighs 200 grams and has a coefficient of restitution close to the legal maximum of 0.83.
   • A five-iron head may weigh 256 grams with a COR of about 0.77 or 0.78.
   • A golf ball typically weighs within a half gram of the legal maximum of 46 grams.
   These parameters will have some effect on the result. Some testing shows that over a reasonable range of values the differences, while detectable, are not large. For the critical values of loft and face-to-path angle, the difference will seldom exceed one degree.
5. Some of the equations involved may not have closed-form solutions. For those, we will find numerical solutions, which may be of the form:
• A readily computable approximation, close enough to give results comparable to the precision of the instrumentation typically used.
• An iterative solution, with the intent of rapid convergence to a result comparable to the precision of the instrumentation typically used.

Straightforward algebra

Spin

Obliqueness

S = 160 * V_clubhead * sin(Φ)

Vball  =  Vclubhead

1 + e 1 + m/M

cos(Φ)

The tough ones

Finishing off

LAy  =  f*L + K
LAx  =  f*A + P

K  =  LAy - f*L
P  =  LAx - f*A

Numerical solutions

Overly simplified solution For small values of x, it is well known that tan(x)≅x. The diagram is a graph of tan(x) and x, along with the percentage error we incur by using x instead of tan(x). As we see from the diagram, those values are limited to less than 20 degrees, and probably only around 10 degrees, before the error gets to be too large to trust. But that is enough for drivers. What can we do if we can substitute x for tan(x)? Well, the first equation would become: Sx Sy  = A L So we now know the ratio between A and L. Let's call that ratio r, and we now know we can approximate it as Sx/Sy. That means that A  = Sx Sy L So we could plug this into the second equation and have only one equation and one unknown (L) to solve. cos(Φ)  =   cos( Sx Sy L) cos(L) This actually works pretty well for drivers, because the loft is seldom more than 15°. Let's see if we can find a way to make this approach work for higher lofted clubs as well.

Approximating the first equation Let's start out by deciding that we aren't interested in a reading if A (the face-to-path angle) is more than 10°. There are so many things wrong, or at least very non-standard, with such a swing that accuracy really isn't an issue. So we can still use A as an approximation for tan(A), even if we can't do that with L. What we want to find is the ratio A/L, in terms of quantities we know. We will call that ratio r. What we know is: Sx Sy  = tan(A) tan(L)   ≅   A tan(L) What we want to find is: r  =   A L We can rewrite A/L as r  =   A L   =   A tan(L) tan(L) L We already know that A/tan(L) is well approximated by Sx/Sy, so we have r  =   Sx Sy tan(L) L So we are left needing tan(L)/L, but we don't know L. However, we do know that L≅Φ for L>>A. So let's think about using Φ instead of L in the formula for r. That would give us: r  =   Sx Sy tan(Φ) Φ This turns out to work really well. We already know Φ. As noted, for L considerably larger than A, Φ is a good proxy for L. But what about for a small L, something in the same order as A. Well, A is small in all interesting cases, less than 10°. And we know that for small Φ, tan(Φ)/Φ is very close to 1. So we are left with the successful approximation we used for drivers, r=Sx/Sy. So we have found a good, useful value for r, one that is within a percent or less of its actual value. We can plug this r into the second equation, which becomes: cos(Φ)  =   cos(L) cos(r L) Let's look at how we can solve this equation.

Iterating the second equation This is simpler than we had before. At least it is just one unknown, the spin loft L. Let's look at two numerical solutions to the equation: An iterative solution, where we take an initial guess at L, evaluate the function, and use the result to give a better value of L. Repeat (iterate) this process until L isn't changing any more; that is the solution. We do that in this section. Use a polynomial expansion to approximate cos(L) and solve the resulting polynomial equation. We will do that in the next section. Iterative solutions typically require writing the equation in the form f(x)=0. For our problem, we have a function in L. For an iterative solution, it would be written: f(L)  =  cos(L) cos(r L)  -  cos(Φ)  =  0 The simplest fast iterative solution is Newton-Raphson iteration. The pattern of iteration is: Li+1  =  Li - f(Li) df(Li)/dL  =  Li  - f(Li) f ' (Li) It requires that we have or can obtain a first derivative for the function. We can differentiate our function the way we would in any first-semester course in calculus.. f ' (L)  =  -r cos(L) sin(r L)  - cos(r L) sin (L)   So our iteration formula becomes: Li+1  =  Li + cos(Li) cos(r Li)  -  cos(Φ) r cos(Li) sin(r Li) + cos(r Li) sin (Li) Iterative methods usually rely on a decent initial value for Li. The the closer the initial value is to the actual L, the faster the iteration will converge. But we already know that, for most cases of interest, the A/L ratio r is small and therefore Φ is rather close to L. So let us start with L0=Φ . This should promote very fast convergence. I implemented the convergent procedure on an Excel spreadsheet pretty easily. With Φ, the starting value, being so close to L, the solution, convergence was very fast. For most of my testing, it only took one step to get to less than a tenth of a degree of the final value. (In reality, it would take a second step to recognize this fact by comparing the last two values.)

Approximating the second equation as a polynomial Let's review the problem again. The equation we need to solve is: cos(Φ)  =   cos(L) cos(r L) Since we know both Φ and r, this is just an equation in L. Suppose we expand the cosine function as a power series. The MacLaurin series for cos(x) is: cos(x)  =  1 - x2 2!  + x4 4!  - x6 6!  + . . . (This is on many available lists of series expansions, and plenty of derivations around. For our purposes here we'll just accept it as given.) How many terms of the series will we need to maintain enough accuracy in our calculations. We will be using lofts up to perhaps 50°, that of a typical gap wedge. Let's look at how errors build up for the first two terms and three terms of the series. The graph shows us that the percentage error of a second-order series is above 1% before the loft reaches 40°, and is rising rapidly at that point. So we will clearly want more terms than two. The fourth-order series does a lot better; the percentage error never gets above 0.1% in our range of interest, and is small enough to be ignored completely for most of the range. Let's use the fourth-order series: cos(x)  ≅  1 - x2 2  +  x4 24 If we plug it into the equation we want to solve, we come up with: cos(Φ)  =  (1 -  L2 2  + L4 24 ) (1 - (r L)2 2  + (r L)4 24 ) Just looking at this, we can guess it will be pretty difficult. Multiplying it out leaves us with an eighth-order equation to solve. (Yes, I know there is a transformation that can make it a fourth-order equation, but that is still difficult.) But wait! One of the trig identities that didn't look too appetizing before might help a lot once we decide to use this MacLaurin series expansion. 2 cos(x) cos(y)  =  cos(x-y) + cos(x+y) Plugging this into our original equation gives us: 2 cos(Φ)   =  cos (L - rL) + cos (L + rL)   =  cos (L (1 - r)) + cos (L (1 + r)) Now we plug the MacLaurin series expansion into this new formulation, and we get: 2 cos(Φ)  =  (1 -  L2 (1-r)2 2  + L4 (1-r)4 24 ) + (1 - L2 (1+r)2 2  + L4 (1+r)4 24 ) Let's factor this so it looks like the sort of polynomial equation that the handbooks tell us how to solve. L4 (1-r)4 + (1+r)4 24   -  L2 (1-r)2 + (1+r)2 2   +  2 (1 - cos(Φ))  =  0  This is a fourth-order (quartic) equation in L. But note that there are no odd-order terms in it. That tells us we can use the simple transformation y=L2 and be left with a quadratic equation in y. y2 (1-r)4 + (1+r)4 24   -  y (1-r)2 + (1+r)2 2   +  2 (1 - cos(Φ))  =  0 We learned in high school how to solve quadratic equations. We take the coefficient of y2 and call it 'a', the coefficient of y and call it 'b', and the rest of the equation is 'c'. The solution for y is simply: y  =   - b ± sqrt (b2- 4ac) 2a In the case of our golf problem, the negative part of ± is used; it gives a loft in the range we are looking, 0-50° (roughly 0-1 radians). At that point, all we need to do is take the square root of y to get loft L.

Sensitivity to unknown parameters

1. Nominal COR and mass. "Nominal" depends on the type of club. Clubhead mass does not vary much for a given class of club (say, a 5-iron), seldom more than ±10 grams. Here are the high and low limits of the range for COR.
• Rigid faces can be as low as 0.77. This might be typical of a "blade" iron.
• Flex faces are limited by rule to be no more than 0.83. Most drivers are close to this number.Some irons, and many hybrids and fairway woods try to reach 0.83, but few get that far.
2. Low COR and nominal mass.
3. High COR and nominal mass.
4. Nominal COR and 10 grams under nominal mass.
5. Nominal COR and 10 grams over nominal mass.

• If your launch monitor shows clubhead speed as well as ball speed, you might be able to tease out of that some info about e and M. See my article on smash factor for the math. Unfortunately, it may not be possible to tease them apart. Smash factor depends on them both, and with the same algebraic expression for them that the reverse model uses in its second step.
• Refer to the catalogs of independent component manufacturers that try to "clone" the features of brand name clubheads. They all list the mass of the clubhead they sell, and I would guess it is usually similar to the brand name they are cloning. But I'm guessing here; I'm not sure. Here are two online catalogs to start you off: Diamond Tour Golf and Monark Golf.
• Don't expect much help from the brand name web sites. They give the specifications for the clubs, not the components. The closest you will get to finding M in their specifications is the swingweight -- which isn't really helpful for this purpose. If you have a friend in the engineering department of the brand name club you are working with, you might try asking; but don't hold your breath.

Summary

K  =  LAy - f*L
P  =  LAx - f*A

Test results