Need guidance on what math is needed to understand constrained rigid body motion

This can go as deep as you like (and I'm not familiar with Box2D) but could I suggest:

• All of algebra 2 (excluding complex numbers and series)
• All of linear algebra
• All of AP calculus AB

Depending on your existing skills, you may need to pick up some of the prerequisite skills for the above.

Cheers,

Chris

(ex-engineer with programming experience, now teaching maths and physics)

@Chris,

Well okay, I guess the lesson learned here is that if you ask a very general question you get a very general answer. To be as concrete as possible, this ~100 line code example from Box2D_lite is what I am trying to understand. I have several years of experience with programming and physics simulations, a firm grasp of vectors, and a basic understanding of matrices. I understand everything going on here except the matrix math around line 44-64. Could you please help me do a breakdown of what is going on here. Thanks in advance!

Cheers mike

/*
* Copyright (c) 2006-2007 Erin Catto http://www.gphysics.com
*
* Permission to use, copy, modify, distribute and sell this software
* and its documentation for any purpose is hereby granted without fee,
* provided that the above copyright notice appear in all copies.
* Erin Catto makes no representations about the suitability
* of this software for any purpose.
* It is provided "as is" without express or implied warranty.
*/

#include "Joint.h"
#include "Body.h"
#include "World.h"

void Joint::Set(Body* b1, Body* b2, const Vec2& anchor)
{
body1 = b1;
body2 = b2;

Mat22 Rot1(body1->rotation);
Mat22 Rot2(body2->rotation);
Mat22 Rot1T = Rot1.Transpose();
Mat22 Rot2T = Rot2.Transpose();

localAnchor1 = Rot1T * (anchor - body1->position);
localAnchor2 = Rot2T * (anchor - body2->position);

P.Set(0.0f, 0.0f);

softness = 0.0f;
biasFactor = 0.2f;
}

void Joint::PreStep(float inv_dt)
{
// Pre-compute anchors, mass matrix, and bias.
Mat22 Rot1(body1->rotation);
Mat22 Rot2(body2->rotation);

r1 = Rot1 * localAnchor1;
r2 = Rot2 * localAnchor2;

// deltaV = deltaV0 + K * impulse
// invM = [(1/m1 + 1/m2) * eye(2) - skew(r1) * invI1 * skew(r1) - skew(r2) * invI2 * skew(r2)]
// = [1/m1+1/m2 0 ] + invI1 * [r1.y*r1.y -r1.x*r1.y] + invI2 * [r1.y*r1.y -r1.x*r1.y]
// [ 0 1/m1+1/m2] [-r1.x*r1.y r1.x*r1.x] [-r1.x*r1.y r1.x*r1.x]
Mat22 K1;
K1.col1.x = body1->invMass + body2->invMass; K1.col2.x = 0.0f;
K1.col1.y = 0.0f; K1.col2.y = body1->invMass + body2->invMass;

Mat22 K2;
K2.col1.x = body1->invI * r1.y * r1.y; K2.col2.x = -body1->invI * r1.x * r1.y;
K2.col1.y = -body1->invI * r1.x * r1.y; K2.col2.y = body1->invI * r1.x * r1.x;

Mat22 K3;
K3.col1.x = body2->invI * r2.y * r2.y; K3.col2.x = -body2->invI * r2.x * r2.y;
K3.col1.y = -body2->invI * r2.x * r2.y; K3.col2.y = body2->invI * r2.x * r2.x;

Mat22 K = K1 + K2 + K3;
K.col1.x += softness;
K.col2.y += softness;

M = K.Invert();

Vec2 p1 = body1->position + r1;
Vec2 p2 = body2->position + r2;
Vec2 dp = p2 - p1;

if (World::positionCorrection)
{
bias = -biasFactor * inv_dt * dp;
}
else
{
bias.Set(0.0f, 0.0f);
}

if (World::warmStarting)
{
// Apply accumulated impulse.
body1->velocity -= body1->invMass * P;
body1->angularVelocity -= body1->invI * Cross(r1, P);

body2->velocity += body2->invMass * P;
body2->angularVelocity += body2->invI * Cross(r2, P);
}
else
{
P.Set(0.0f, 0.0f);
}
}

void Joint::ApplyImpulse()
{
Vec2 dv = body2->velocity + Cross(body2->angularVelocity, r2) - body1->velocity - Cross(body1->angularVelocity, r1);

Vec2 impulse;

impulse = M * (bias - dv - softness * P);

body1->velocity -= body1->invMass * impulse;
body1->angularVelocity -= body1->invI * Cross(r1, impulse);

body2->velocity += body2->invMass * impulse;
body2->angularVelocity += body2->invI * Cross(r2, impulse);

P += impulse;
}