Circular Motion

A few clarifying Q's...

Are we talking scripted motion or is it supposed to be physics based ie. caused by forces? Is this going to happen inside some physics / graphics / game engine or are you planning to write this bý hand?

cheers,
Mike

It's for a game that I'm developing with some friends! We are using C#, XNA and a private Engine!

Does that make things more clear?

Based on the figure, this is how you could achieve what you want:

1)Since you want your rigid body to rotate along a circle with constant speed, we do for each frame:

2)We can therefore compute the force to be applied to the vehicle. It is:

3)The rate of variation of the orientation is:


This won't give you a perfect circular motion( due to the re-orthonormalization of vehicle.orientation, and some other subtle things ). The faster you'll rotate the vehicle, the greater the error will be: so the faster you rotate the vehicle, the smaller the timestep.
As you've probably understood, vehicle.orientation is a 3x3 matrix. You could use an angle, but actually, I never use things like Vector2, or Matrix22 data structure, since, Vector3 and Matrix33, are sufficient for every type of problem in an euclidian space. Here, the vehicle.orientation will have vehicle.orientation(0,2)=vehicle.orientation(2,0)=
vehicle.orientation(1,2)=vehicle.orientation(2,1)=0

and vehicle.orientation(2,2)=1 ( after of course re-orthonormalization )

and all other vector3 variables will have their third component equal to zero, except the angular velocity who will verify the following property:
vehicle.angular_velocity(0)=vehicle.angular_velocity(1)=0
and vehicle.angular_velocity(2) != 0 will evaluate to true.

P.S: yes, this is C++ code...The C# implementation is left as an exercise for the reader.
edit
the equations I used are:

angular_velocity(t)=constant_vector_3//Vector3(0,0,constant_float)
acceleration(t)=skew_symmetric_matrix(angular_velocity(t))*velocity(t)
d(position(t))/dt = linear_velocity(t)
d(linear_velocity(t))/dt=acceleration(t)
d(orientation(t))/dt = skew_symmetric_matrix(angular_velocity(t))*orientation(t)

And I used the Integration Method of Euler, the simplest one, to compute position(t+dt), linear_velocity(t+dt) and orientation(t+dt). But you can use any other method(generally I use the fourth order Runge-Kutta method, RK4 pour les intimes.

[Edited by - johnstanp on March 10, 2010 12:10:25 PM]

A circumference in 2D is defined by the parametric equation
P(θ) = C + R*(cosθ, sinθ).
The angle θ is the angle between the line with direction (cosθ, sinθ) and the x-axis. r = R*(cosθ, sinθ) is the vector you have calculated in your code.

By basic calculus, the velocity of your object is the derivative of the position P(θ(t)) with respect to time t. It is quite easy to get the following formula:
v(t) = d/dt P(θ(t)) = d/dt (C + R*(cos(θ(t)), sin(θ(t)))) = R*(d/dt θ(t))*(-sin(θ(t)), cos(θ(t))).
ω(t) = (d/dt θ(t)) is the angular velocity.

There are two important considerations:
1. The vector r and the velocity v are perpendicular:
dot(r, v) = R*R*ω*(-cosθsinθ + sinθcosθ) = 0
2. The velocity can be calculated without any cross product. Indeed, if r = (rx, ry) the velocity is v = (-ω*ry, ω*rx)

The acceleration is
a(t) = d/dt v(t) = d/dt (R*ω(t)*(-sin(θ(t)), cos(θ(t)))) = R*(d/dt ω(t))*(-sin(θ(t)), cos(θ(t))) + R*ω(t)*(-cos(θ(t)), -sin(θ(t))).
(d/dt ω(t)) is the angular acceleration and, if the angular velocity is constant, it is zero and the acceleration is parallel to r and with opposite direction. So, if r = (rx, ry), a = (-ω*rx, -ω*ry).

In code it is something like (pseudocode):

You can then use the integration method you prefer.

Note that you probably have to 'renormalize' the radius to be sure the object is on the circumference and it is probably better to simply do a rotation around the point.