Integrate function - c++

Question

I have this function to reach a certain 1 dimensional value accelerated and damped with overshoot. That is: given an inital value, a velocity and a acceleration (force/mass), the target value is attained by accelerating to it and gets increasingly damped while getting closer to the target value.
This all works fine, howver If i want to know what the TotalAngle is after time 't' I have to run this function say N steps with a 'small' dt to find the 'limit'.
I was wondering If i can (and how) to intergrate over dt so that the TotalAngle can be determined given a time 't' initially.
Regards, Tanks for any help.
dt = delta time step per frame
input = 1
TotalAngle = 0 at t=0
Velocity = 0 at t=0
void FAccelDampedWithOvershoot::Update(float dt, float input, float& Velocity, float& TotalAngle)
{
const float Force = 500000.f;
const float DampForce = 5000.f;
const float MaxAngle = 45.f;
const float InvMass = 1.f / 162400.f;
float target = MaxAngle * input;
float ratio = (target - TotalAngle) / MaxAngle;
float fMove = Force * ratio;
float fDamp = -Velocity * DampForce;
Velocity += (fMove + fDamp) * invMass * dt;
TotalAngle += Velocity * dt;
}

Answer 1

Updated with fixed bugs in math
Originally I've lost mass and MaxAngle a few times. This is why you should first solve it on a paper and then enter to the SO rather than trying to solve it in the text editor.
Anyway, I've fixed the math and now it seems to work reasonably well. I put fixed solution just over previous one.
Well, this looks like a Newtonian mechanics which means differential equations. Let's try to solve them.
SO is not very friendly to math formulas and I'm a bit bored to type characters so here is what I use:
F = Force
Fd = DampForce
MA = MaxAngle
A= TotalAngle
v = Velocity
m = 1 / InvMass
' for derivative i.e. something' is 1-st derivative of something by t and something'' is 2-nd derivative
if I divide you last two lines of code by dt and merge in all the other lines I can get (I also assume that input = 1 as other case is obviously symmetrical)
v' = ([F * (1 - A / MA)] - v * Fd) / m
and applying A' = v we get
m * A'' = F(1 - A/MA) - Fd * A'
or moving to one side we get a simple 2-nd order differential equation
m * A'' + Fd * A' + F/MA * A = F
IIRC, the way to solve it is to first solve characteristic equation which here is
m * x^2 + Fd * x + F/MA = 0
x[1,2] = (-Fd +/- sqrt(Fd^2 - 4*F*m/MA))/ (2*m)
I expect that part under sqrt i.e. (Fd^2 - 4*F*m/MA) is negative thus solution should be of the following form. Let
Dm = Fd/(2*m)
K = sqrt(F/MA/m - Dm^2)
(note the negated value under sqrt so it works now) then
A(t) = e^(-Dm*t) * [P * sin(K*t) + Q * cos(K*t)] + C
where P, Q and C are some constants.
The solution is easier to find as a sum of two solutions: some specific solution for
m * A'' + Fd * A' + F/MA * A = F
and a general solution for homogeneou
m * A'' + Fd * A' + F/MA * A = 0
that makes original conditions fit. Obviously specific solution A(t) = MA works and thus C = MA. So now we need to fit P and Q of general solution to match starting conditions. To find them we need
A(0) = - MA
A'(0) = V(0) = 0
Given that e^0 = 1, sin(0) = 0 and cos(0) = 1 you get something like
Q = -MA
P = 0
or
P = 0
Q = - MA
C = MA
thus
A(t) = MA * [1 - e^(-Dm*t) * cos(K*t)]
where
Dm = Fd/(2*m)
K = sqrt(F/MA/m - Dm^2)
which kind of makes sense given your task.
Note also that this equation assumes that everything happens in radians rather than degrees (i.e. derivative of [sin(t)]' is just cos(t)) so you should transform all your constants accordingly or transform the solution.
const float Force = 500000.f * M_PI / 180;
const float DampForce = 5000.f * M_PI / 180;
const float MaxAngle = M_PI_4;
which on my machine produces
Dm = 0.000268677541
K = 0.261568546
This seems to be similar to original funcion is I step with dt = 0.01f and the main obstacle seems to be precision loss because of float
Hope this helps!

Answer 2

This is not a full answer and I am sure someone else can work it out, but there is no room in the comments and it may help you find a better solution.
The image below shows the velocity (blue) as your function integrates at time steps 1. The red shows the function below that calculates the value for time t
The function F(t)
F(t) = sin((t / f) * pi * 2) * (1 / (((t / f) + a) ^ c)) * b
With f = 23.7, a = 1.4, c = 2, and b= 50 that give the red plot in the image above
All the values are just approximation.
f determines the frequency and is close to a match,
a,b,c control the falloff in amplitude and are a by eye guestimate.
If it does not matter that you have a perfect match then this will work for you. totalAngle uses the same function but t has 0.25 added to it. Unfortunately I did not get any values for a,b,c for totalAngle and I did notice that it was offset so you will have to add the offset value d (I normalised everything so have no idea what the range of totalAngle was)
Function F(t) for totalAngle
F(t) = sin(((t+0.25) / f) * pi * 2) * (1 / ((((t+0.25) / f) + a) ^ c)) * b + d
Sorry only have f = 23.7, c= 2, a~1.4 nothing for b=? d=?