Mouse Constraint Derivation

You can read here how to compute the mouse constraint Jacobian, which I think is the simplest constraint along with a distance constraint. It can be helpfull when you’re selecting geometries in the scene using the mouse world space position (as a joint anchor point), since there is lack of controlability by directly applying forces at a particular point of a rigid body shape.

Errata: There is a minus sign before the stabilization term. It should be:

Lambda = (J * M * J^T)-1 * (-J * V – Beta / TimeStep * C)

Note: You can bound excessive impulses to avoid excessive velocity changes.

Thanks to Dirk Gregorius of Valve for pointing those things out via e-mail.

Practically, a single mouse constraint equation could be enforced as:

Vec3 P1 = GetWorldAnchorPosition(); // Could be anything static in R^3.
Vec3 P2 = GetWorldMousePosition();

Mat33 W2 = Diagonal( Body2->GetInverseMass() );
Mat33 I2 = Body2->GetWorldInverseInertia();

Vec3 COM2 = Body2->GetWorldCenterOfMass();
Vec3 Velocity2 = Body2->GetLinearVelocity();
Vec3 Omega2 = Body2->GetAngularVelocity();

Vec3 C = P2 - P1;
Vec3 Radius2 = P2 - COM2;
Vec3 Cdot = Velocity2 + Cross(Omega2, Radius2);

Mat33 R2 = Skew(Radius2);
Mat33 R2T = Transpose(R);
Mat33 K = W2 + (R2 * I2 * R2T);
Mat33 M = Inverse(K);

Vec3 VelocityBias = Beta / TimeStep * C;

Vec3 Impulse = M * (-Cdot + VelocityBias);

Body2->ApplyLinearImpulseAtPoint(Impulse, P2); 

And here is the result:

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