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* Copyright (c) 2013, Daniel Murphy
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package org.jbox2d.dynamics;
import org.jbox2d.callbacks.ContactImpulse;
import org.jbox2d.callbacks.ContactListener;
import org.jbox2d.common.MathUtils;
import org.jbox2d.common.Settings;
import org.jbox2d.common.Timer;
import org.jbox2d.common.Vec2;
import org.jbox2d.dynamics.contacts.Contact;
import org.jbox2d.dynamics.contacts.ContactSolver;
import org.jbox2d.dynamics.contacts.ContactSolver.ContactSolverDef;
import org.jbox2d.dynamics.contacts.ContactVelocityConstraint;
import org.jbox2d.dynamics.contacts.Position;
import org.jbox2d.dynamics.contacts.Velocity;
import org.jbox2d.dynamics.joints.Joint;
/*
Position Correction Notes
=========================
I tried the several algorithms for position correction of the 2D revolute joint.
I looked at these systems:
- simple pendulum (1m diameter sphere on massless 5m stick) with initial angular velocity of 100 rad/s.
- suspension bridge with 30 1m long planks of length 1m.
- multi-link chain with 30 1m long links.
Here are the algorithms:
Baumgarte - A fraction of the position error is added to the velocity error. There is no
separate position solver.
Pseudo Velocities - After the velocity solver and position integration,
the position error, Jacobian, and effective mass are recomputed. Then
the velocity constraints are solved with pseudo velocities and a fraction
of the position error is added to the pseudo velocity error. The pseudo
velocities are initialized to zero and there is no warm-starting. After
the position solver, the pseudo velocities are added to the positions.
This is also called the First Order World method or the Position LCP method.
Modified Nonlinear Gauss-Seidel (NGS) - Like Pseudo Velocities except the
position error is re-computed for each raint and the positions are updated
after the raint is solved. The radius vectors (aka Jacobians) are
re-computed too (otherwise the algorithm has horrible instability). The pseudo
velocity states are not needed because they are effectively zero at the beginning
of each iteration. Since we have the current position error, we allow the
iterations to terminate early if the error becomes smaller than Settings.linearSlop.
Full NGS or just NGS - Like Modified NGS except the effective mass are re-computed
each time a raint is solved.
Here are the results:
Baumgarte - this is the cheapest algorithm but it has some stability problems,
especially with the bridge. The chain links separate easily close to the root
and they jitter as they struggle to pull together. This is one of the most common
methods in the field. The big drawback is that the position correction artificially
affects the momentum, thus leading to instabilities and false bounce. I used a
bias factor of 0.2. A larger bias factor makes the bridge less stable, a smaller
factor makes joints and contacts more spongy.
Pseudo Velocities - the is more stable than the Baumgarte method. The bridge is
stable. However, joints still separate with large angular velocities. Drag the
simple pendulum in a circle quickly and the joint will separate. The chain separates
easily and does not recover. I used a bias factor of 0.2. A larger value lead to
the bridge collapsing when a heavy cube drops on it.
Modified NGS - this algorithm is better in some ways than Baumgarte and Pseudo
Velocities, but in other ways it is worse. The bridge and chain are much more
stable, but the simple pendulum goes unstable at high angular velocities.
Full NGS - stable in all tests. The joints display good stiffness. The bridge
still sags, but this is better than infinite forces.
Recommendations
Pseudo Velocities are not really worthwhile because the bridge and chain cannot
recover from joint separation. In other cases the benefit over Baumgarte is small.
Modified NGS is not a robust method for the revolute joint due to the violent
instability seen in the simple pendulum. Perhaps it is viable with other raint
types, especially scalar constraints where the effective mass is a scalar.
This leaves Baumgarte and Full NGS. Baumgarte has small, but manageable instabilities
and is very fast. I don't think we can escape Baumgarte, especially in highly
demanding cases where high raint fidelity is not needed.
Full NGS is robust and easy on the eyes. I recommend this as an option for
higher fidelity simulation and certainly for suspension bridges and long chains.
Full NGS might be a good choice for ragdolls, especially motorized ragdolls where
joint separation can be problematic. The number of NGS iterations can be reduced
for better performance without harming robustness much.
Each joint in a can be handled differently in the position solver. So I recommend
a system where the user can select the algorithm on a per joint basis. I would
probably default to the slower Full NGS and let the user select the faster
Baumgarte method in performance critical scenarios.
*/
/*
Cache Performance
The Box2D solvers are dominated by cache misses. Data structures are designed
to increase the number of cache hits. Much of misses are due to random access
to body data. The raint structures are iterated over linearly, which leads
to few cache misses.
The bodies are not accessed during iteration. Instead read only data, such as
the mass values are stored with the constraints. The mutable data are the raint
impulses and the bodies velocities/positions. The impulses are held inside the
raint structures. The body velocities/positions are held in compact, temporary
arrays to increase the number of cache hits. Linear and angular velocity are
stored in a single array since multiple arrays lead to multiple misses.
*/
/*
2D Rotation
R = [cos(theta) -sin(theta)]
[sin(theta) cos(theta) ]
thetaDot = omega
Let q1 = cos(theta), q2 = sin(theta).
R = [q1 -q2]
[q2 q1]
q1Dot = -thetaDot * q2
q2Dot = thetaDot * q1
q1_new = q1_old - dt * w * q2
q2_new = q2_old + dt * w * q1
then normalize.
This might be faster than computing sin+cos.
However, we can compute sin+cos of the same angle fast.
*/
/**
* This is an internal class.
*
* @author Daniel Murphy
*/
public class Island {
public ContactListener m_listener;
public Body[] m_bodies;
public Contact[] m_contacts;
public Joint[] m_joints;
public Position[] m_positions;
public Velocity[] m_velocities;
public int m_bodyCount;
public int m_jointCount;
public int m_contactCount;
public int m_bodyCapacity;
public int m_contactCapacity;
public int m_jointCapacity;
public Island() {
}
public void init(int bodyCapacity, int contactCapacity, int jointCapacity,
ContactListener listener) {
// System.out.println("Initializing Island");
m_bodyCapacity = bodyCapacity;
m_contactCapacity = contactCapacity;
m_jointCapacity = jointCapacity;
m_bodyCount = 0;
m_contactCount = 0;
m_jointCount = 0;
m_listener = listener;
if (m_bodies == null || m_bodyCapacity > m_bodies.length) {
m_bodies = new Body[m_bodyCapacity];
}
if (m_joints == null || m_jointCapacity > m_joints.length) {
m_joints = new Joint[m_jointCapacity];
}
if (m_contacts == null || m_contactCapacity > m_contacts.length) {
m_contacts = new Contact[m_contactCapacity];
}
// dynamic array
if (m_velocities == null || m_bodyCapacity > m_velocities.length) {
final Velocity[] old = m_velocities == null ? new Velocity[0] : m_velocities;
m_velocities = new Velocity[m_bodyCapacity];
System.arraycopy(old, 0, m_velocities, 0, old.length);
for (int i = old.length; i < m_velocities.length; i++) {
m_velocities[i] = new Velocity();
}
}
// dynamic array
if (m_positions == null || m_bodyCapacity > m_positions.length) {
final Position[] old = m_positions == null ? new Position[0] : m_positions;
m_positions = new Position[m_bodyCapacity];
System.arraycopy(old, 0, m_positions, 0, old.length);
for (int i = old.length; i < m_positions.length; i++) {
m_positions[i] = new Position();
}
}
}
public void clear() {
m_bodyCount = 0;
m_contactCount = 0;
m_jointCount = 0;
}
private final ContactSolver contactSolver = new ContactSolver();
private final Timer timer = new Timer();
private final SolverData solverData = new SolverData();
private final ContactSolverDef solverDef = new ContactSolverDef();
public void solve(Profile profile, TimeStep step, Vec2 gravity, boolean allowSleep) {
// System.out.println("Solving Island");
float h = step.dt;
// Integrate velocities and apply damping. Initialize the body state.
for (int i = 0; i < m_bodyCount; ++i) {
final Body b = m_bodies[i];
final Vec2 c = b.m_sweep.c;
float a = b.m_sweep.a;
final Vec2 v = b.m_linearVelocity;
float w = b.m_angularVelocity;
// Store positions for continuous collision.
b.m_sweep.c0.set(b.m_sweep.c);
b.m_sweep.a0 = b.m_sweep.a;
if (b.m_type == BodyType.DYNAMIC) {
// Integrate velocities.
// v += h * (b.m_gravityScale * gravity + b.m_invMass * b.m_force);
v.x += h * (b.m_gravityScale * gravity.x + b.m_invMass * b.m_force.x);
v.y += h * (b.m_gravityScale * gravity.y + b.m_invMass * b.m_force.y);
w += h * b.m_invI * b.m_torque;
// Apply damping.
// ODE: dv/dt + c * v = 0
// Solution: v(t) = v0 * exp(-c * t)
// Time step: v(t + dt) = v0 * exp(-c * (t + dt)) = v0 * exp(-c * t) * exp(-c * dt) = v *
// exp(-c * dt)
// v2 = exp(-c * dt) * v1
// Taylor expansion:
// v2 = (1.0f - c * dt) * v1
float a1 = MathUtils.clamp(1.0f - h * b.m_linearDamping, 0.0f, 1.0f);
v.x *= a1;
v.y *= a1;
w *= MathUtils.clamp(1.0f - h * b.m_angularDamping, 0.0f, 1.0f);
}
m_positions[i].c.x = c.x;
m_positions[i].c.y = c.y;
m_positions[i].a = a;
m_velocities[i].v.x = v.x;
m_velocities[i].v.y = v.y;
m_velocities[i].w = w;
}
timer.reset();
// Solver data
solverData.step = step;
solverData.positions = m_positions;
solverData.velocities = m_velocities;
// Initialize velocity constraints.
solverDef.step = step;
solverDef.contacts = m_contacts;
solverDef.count = m_contactCount;
solverDef.positions = m_positions;
solverDef.velocities = m_velocities;
contactSolver.init(solverDef);
// System.out.println("island init vel");
contactSolver.initializeVelocityConstraints();
if (step.warmStarting) {
// System.out.println("island warm start");
contactSolver.warmStart();
}
for (int i = 0; i < m_jointCount; ++i) {
m_joints[i].initVelocityConstraints(solverData);
}
profile.solveInit = timer.getMilliseconds();
// Solve velocity constraints
timer.reset();
// System.out.println("island solving velocities");
for (int i = 0; i < step.velocityIterations; ++i) {
for (int j = 0; j < m_jointCount; ++j) {
m_joints[j].solveVelocityConstraints(solverData);
}
contactSolver.solveVelocityConstraints();
}
// Store impulses for warm starting
contactSolver.storeImpulses();
profile.solveVelocity = timer.getMilliseconds();
// Integrate positions
for (int i = 0; i < m_bodyCount; ++i) {
final Vec2 c = m_positions[i].c;
float a = m_positions[i].a;
final Vec2 v = m_velocities[i].v;
float w = m_velocities[i].w;
// Check for large velocities
float translationx = v.x * h;
float translationy = v.y * h;
if (translationx * translationx + translationy * translationy > Settings.maxTranslationSquared) {
float ratio = Settings.maxTranslation
/ MathUtils.sqrt(translationx * translationx + translationy * translationy);
v.x *= ratio;
v.y *= ratio;
}
float rotation = h * w;
if (rotation * rotation > Settings.maxRotationSquared) {
float ratio = Settings.maxRotation / MathUtils.abs(rotation);
w *= ratio;
}
// Integrate
c.x += h * v.x;
c.y += h * v.y;
a += h * w;
m_positions[i].a = a;
m_velocities[i].w = w;
}
// Solve position constraints
timer.reset();
boolean positionSolved = false;
for (int i = 0; i < step.positionIterations; ++i) {
boolean contactsOkay = contactSolver.solvePositionConstraints();
boolean jointsOkay = true;
for (int j = 0; j < m_jointCount; ++j) {
boolean jointOkay = m_joints[j].solvePositionConstraints(solverData);
jointsOkay = jointsOkay && jointOkay;
}
if (contactsOkay && jointsOkay) {
// Exit early if the position errors are small.
positionSolved = true;
break;
}
}
// Copy state buffers back to the bodies
for (int i = 0; i < m_bodyCount; ++i) {
Body body = m_bodies[i];
body.m_sweep.c.x = m_positions[i].c.x;
body.m_sweep.c.y = m_positions[i].c.y;
body.m_sweep.a = m_positions[i].a;
body.m_linearVelocity.x = m_velocities[i].v.x;
body.m_linearVelocity.y = m_velocities[i].v.y;
body.m_angularVelocity = m_velocities[i].w;
body.synchronizeTransform();
}
profile.solvePosition = timer.getMilliseconds();
report(contactSolver.m_velocityConstraints);
if (allowSleep) {
float minSleepTime = Float.MAX_VALUE;
final float linTolSqr = Settings.linearSleepTolerance * Settings.linearSleepTolerance;
final float angTolSqr = Settings.angularSleepTolerance * Settings.angularSleepTolerance;
for (int i = 0; i < m_bodyCount; ++i) {
Body b = m_bodies[i];
if (b.getType() == BodyType.STATIC) {
continue;
}
if ((b.m_flags & Body.e_autoSleepFlag) == 0
|| b.m_angularVelocity * b.m_angularVelocity > angTolSqr
|| Vec2.dot(b.m_linearVelocity, b.m_linearVelocity) > linTolSqr) {
b.m_sleepTime = 0.0f;
minSleepTime = 0.0f;
} else {
b.m_sleepTime += h;
minSleepTime = MathUtils.min(minSleepTime, b.m_sleepTime);
}
}
if (minSleepTime >= Settings.timeToSleep && positionSolved) {
for (int i = 0; i < m_bodyCount; ++i) {
Body b = m_bodies[i];
b.setAwake(false);
}
}
}
}
private final ContactSolver toiContactSolver = new ContactSolver();
private final ContactSolverDef toiSolverDef = new ContactSolverDef();
public void solveTOI(TimeStep subStep, int toiIndexA, int toiIndexB) {
assert (toiIndexA < m_bodyCount);
assert (toiIndexB < m_bodyCount);
// Initialize the body state.
for (int i = 0; i < m_bodyCount; ++i) {
m_positions[i].c.x = m_bodies[i].m_sweep.c.x;
m_positions[i].c.y = m_bodies[i].m_sweep.c.y;
m_positions[i].a = m_bodies[i].m_sweep.a;
m_velocities[i].v.x = m_bodies[i].m_linearVelocity.x;
m_velocities[i].v.y = m_bodies[i].m_linearVelocity.y;
m_velocities[i].w = m_bodies[i].m_angularVelocity;
}
toiSolverDef.contacts = m_contacts;
toiSolverDef.count = m_contactCount;
toiSolverDef.step = subStep;
toiSolverDef.positions = m_positions;
toiSolverDef.velocities = m_velocities;
toiContactSolver.init(toiSolverDef);
// Solve position constraints.
for (int i = 0; i < subStep.positionIterations; ++i) {
boolean contactsOkay = toiContactSolver.solveTOIPositionConstraints(toiIndexA, toiIndexB);
if (contactsOkay) {
break;
}
}
// #if 0
// // Is the new position really safe?
// for (int i = 0; i < m_contactCount; ++i)
// {
// Contact* c = m_contacts[i];
// Fixture* fA = c.GetFixtureA();
// Fixture* fB = c.GetFixtureB();
//
// Body bA = fA.GetBody();
// Body bB = fB.GetBody();
//
// int indexA = c.GetChildIndexA();
// int indexB = c.GetChildIndexB();
//
// DistanceInput input;
// input.proxyA.Set(fA.GetShape(), indexA);
// input.proxyB.Set(fB.GetShape(), indexB);
// input.transformA = bA.GetTransform();
// input.transformB = bB.GetTransform();
// input.useRadii = false;
//
// DistanceOutput output;
// SimplexCache cache;
// cache.count = 0;
// Distance(&output, &cache, &input);
//
// if (output.distance == 0 || cache.count == 3)
// {
// cache.count += 0;
// }
// }
// #endif
// Leap of faith to new safe state.
m_bodies[toiIndexA].m_sweep.c0.x = m_positions[toiIndexA].c.x;
m_bodies[toiIndexA].m_sweep.c0.y = m_positions[toiIndexA].c.y;
m_bodies[toiIndexA].m_sweep.a0 = m_positions[toiIndexA].a;
m_bodies[toiIndexB].m_sweep.c0.set(m_positions[toiIndexB].c);
m_bodies[toiIndexB].m_sweep.a0 = m_positions[toiIndexB].a;
// No warm starting is needed for TOI events because warm
// starting impulses were applied in the discrete solver.
toiContactSolver.initializeVelocityConstraints();
// Solve velocity constraints.
for (int i = 0; i < subStep.velocityIterations; ++i) {
toiContactSolver.solveVelocityConstraints();
}
// Don't store the TOI contact forces for warm starting
// because they can be quite large.
float h = subStep.dt;
// Integrate positions
for (int i = 0; i < m_bodyCount; ++i) {
Vec2 c = m_positions[i].c;
float a = m_positions[i].a;
Vec2 v = m_velocities[i].v;
float w = m_velocities[i].w;
// Check for large velocities
float translationx = v.x * h;
float translationy = v.y * h;
if (translationx * translationx + translationy * translationy > Settings.maxTranslationSquared) {
float ratio = Settings.maxTranslation
/ MathUtils.sqrt(translationx * translationx + translationy * translationy);
v.mulLocal(ratio);
}
float rotation = h * w;
if (rotation * rotation > Settings.maxRotationSquared) {
float ratio = Settings.maxRotation / MathUtils.abs(rotation);
w *= ratio;
}
// Integrate
c.x += v.x * h;
c.y += v.y * h;
a += h * w;
m_positions[i].c.x = c.x;
m_positions[i].c.y = c.y;
m_positions[i].a = a;
m_velocities[i].v.x = v.x;
m_velocities[i].v.y = v.y;
m_velocities[i].w = w;
// Sync bodies
Body body = m_bodies[i];
body.m_sweep.c.x = c.x;
body.m_sweep.c.y = c.y;
body.m_sweep.a = a;
body.m_linearVelocity.x = v.x;
body.m_linearVelocity.y = v.y;
body.m_angularVelocity = w;
body.synchronizeTransform();
}
report(toiContactSolver.m_velocityConstraints);
}
public void add(Body body) {
assert (m_bodyCount < m_bodyCapacity);
body.m_islandIndex = m_bodyCount;
m_bodies[m_bodyCount] = body;
++m_bodyCount;
}
public void add(Contact contact) {
assert (m_contactCount < m_contactCapacity);
m_contacts[m_contactCount++] = contact;
}
public void add(Joint joint) {
assert (m_jointCount < m_jointCapacity);
m_joints[m_jointCount++] = joint;
}
private final ContactImpulse impulse = new ContactImpulse();
public void report(ContactVelocityConstraint[] constraints) {
if (m_listener == null) {
return;
}
for (int i = 0; i < m_contactCount; ++i) {
Contact c = m_contacts[i];
ContactVelocityConstraint vc = constraints[i];
impulse.count = vc.pointCount;
for (int j = 0; j < vc.pointCount; ++j) {
impulse.normalImpulses[j] = vc.points[j].normalImpulse;
impulse.tangentImpulses[j] = vc.points[j].tangentImpulse;
}
m_listener.postSolve(c, impulse);
}
}
}