Pendulum with moving pivot

Question

I'm making a game which you can see here, if you are on Windows or Linux: http://insertnamehere.org/birdsofprey/

If you click and hold your mouse on a bird, you can see I'm just swinging the bird back and forth in pendulum motion. I would like to, instead, implement a more realistic motion, where the movement of your mouse affects the swinging of the bird like a pendulum with a moving pivot.

I found a document on this topic but the equations rely on knowing the pivot's acceleration (X'' and Y''), which I do not; I am only repeatedly translating the bird graphic to the current mouse position.

I have the bird's angle (-180 to 180 degrees), angular velocity and acceleration. I will need to alter these three variables each time the mouse is moved, so I will also have the last (x,y) and the new mouse (x,y).

Is this enough to make a good simulation of a pendulum with moving pivot?

Answer 1

The actual solution may not be what you want, since it will come with perks like the birds swinging almost a full $-180^\circ$ to $180^\circ$ everytime when pulled hard and then immediately held still. Also you will have to solve a differential equation in real time.

May I offer an alternative instead which will make it look real while being easy to implement.

Since you have a simple oscillation, I assume you simply vary the angle as $\alpha(t)=r\cos(2\pi f t)+3\pi/2$ for some range $r$ and frequency $f$ in your code.

To add a touch of reality, you can vary the 'down' angle based on the horizontal velocity of the mouse. If the velocity is $v$, then calculate $\beta(v)=3\pi/2-\tan^{-1}(Av)$. This will give some pseudo-direction for the adjusted direction of pull.

Also decrease the range of the oscillation as the velocity increases: $r(v)=1/(1+(Bv)^2)$.

Finally use the following function to get the angle: $\alpha(t)=r(v)\cos(2\pi f t)+\beta(v)$.

You will have to choose fractions $A$ and $B$ till you hit upon the right motion that pleases you. $A=B=0$ will give your original motion back. Larger $A$ will make the change in angle more sensitive to speed, and larger $B$ will dampen the oscillations more at a given velocity.

Answer 2

The mouse velocity is [(current mouse position) - (previous mouse position)]/timestep (presumably you're using a constant timestep of 10ms or something?) and then the acceleration is similarly [(current mouse velocity) - (previous mouse velocity)]/timestep. (At least, approximately.)

Once you have the acceleration, I guess you can use the formulas you already have.

Answer 3

In case you really need to simulate accurate physics, you can use the solution given below.

In the picture above, vector $A = g-a$ where $g$ is acceleration due to gravity, and $a$ is the acceleration of mouse which you need to compute in real time. You will have to translate $g=9.8m/s^2$ into $pixel/s^2$ and store as a constant. For $A$, you need to calculate $A_x$ and $A_y$ based on the last two frames.

We need to find $\alpha(t)$.

We have $\beta=tan^{-1}\frac{A_x}{A_y}$ and $A_1=|A|\cos(\alpha-\beta)$. Angular acceleration will be $-A_1/h$ ($h$ will vary from bird to bird).

You will need to solve the differential equation $\frac{d^2\alpha}{dt^2}=A_1/h$ in real time to make the bird swing realistically.

The obvious way to solve the differential equation numerically is given below -

Store angular velocity $\omega=\frac{d}{dt}\alpha$. At each frame of the animation, assuming $\Delta t$ time has passed since last frame, do

1. $\omega\leftarrow\omega+\frac{A_1}{h}\Delta t$
2. $\alpha\leftarrow\alpha+\omega\Delta t$