reactphysics3d/src/engine/ConstraintSolver.h

/********************************************************************************
* ReactPhysics3D physics library, http://code.google.com/p/reactphysics3d/      *
* Copyright (c) 2010-2013 Daniel Chappuis                                       *
*********************************************************************************
*                                                                               *
* This software is provided 'as-is', without any express or implied warranty.   *
* In no event will the authors be held liable for any damages arising from the  *
* use of this software.                                                         *
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* Permission is granted to anyone to use this software for any purpose,         *
* including commercial applications, and to alter it and redistribute it        *
* freely, subject to the following restrictions:                                *
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* 1. The origin of this software must not be misrepresented; you must not claim *
*    that you wrote the original software. If you use this software in a        *
*    product, an acknowledgment in the product documentation would be           *
*    appreciated but is not required.                                           *
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* 2. Altered source versions must be plainly marked as such, and must not be    *
*    misrepresented as being the original software.                             *
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* 3. This notice may not be removed or altered from any source distribution.    *
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********************************************************************************/

#ifndef REACTPHYSICS3D_CONSTRAINT_SOLVER_H
#define REACTPHYSICS3D_CONSTRAINT_SOLVER_H

// Libraries

namespace reactphysics3d {

// Class ConstraintSolver
/**
 * This class represents the constraint solver that is used to solve constraints between
 * the rigid bodies. The constraint solver is based on the "Sequential Impulse" technique
 * described by Erin Catto in his GDC slides (http://code.google.com/p/box2d/downloads/list).
 *
 * A constraint between two bodies is represented by a function C(x) which is equal to zero
 * when the constraint is satisfied. The condition C(x)=0 describes a valid position and the
 * condition dC(x)/dt=0 describes a valid velocity. We have dC(x)/dt = Jv + b = 0 where J is
 * the Jacobian matrix of the constraint, v is a vector that contains the velocity of both
 * bodies and b is the constraint bias. We are looking for a force F_c that will act on the
 * bodies to keep the constraint satisfied. Note that from the virtual work principle, we have
 * F_c = J^t * lambda where J^t is the transpose of the Jacobian matrix and lambda is a
 * Lagrange multiplier. Therefore, finding the force F_c is equivalent to finding the Lagrange
 * multiplier lambda.

 * An impulse P = F * dt where F is a force and dt is the timestep. We can apply impulses a
 * body to change its velocity. The idea of the Sequential Impulse technique is to apply
 * impulses to bodies of each constraints in order to keep the constraint satisfied.
 *
 * --- Step 1 ---
 *
 * First, we integrate the applied force F_a acting of each rigid body (like gravity, ...) and
 * we obtain some new velocities v2' that tends to violate the constraints.
 *
 * v2' = v1 + dt * M^-1 * F_a
 *
 * where M is a matrix that contains mass and inertia tensor information.
 *
 * --- Step 2 ---
 *
 * During the second step, we iterate over all the constraints for a certain number of
 * iterations and for each constraint we compute the impulse to apply to the bodies needed
 * so that the new velocity of the bodies satisfy Jv + b = 0. From the Newton law, we know that
 * M * deltaV = P_c where M is the mass of the body, deltaV is the difference of velocity and
 * P_c is the constraint impulse to apply to the body. Therefore, we have
 * v2 = v2' + M^-1 * P_c. For each constraint, we can compute the Lagrange multiplier lambda
 * using : lambda = -m_c (Jv2' + b) where m_c = 1 / (J * M^-1 * J^t). Now that we have the
 * Lagrange multiplier lambda, we can compute the impulse P_c = J^t * lambda * dt to apply to
 * the bodies to satisfy the constraint.
 *
 * --- Step 3 ---
 *
 * In the third step, we integrate the new position x2 of the bodies using the new velocities
 * v2 computed in the second step with : x2 = x1 + dt * v2.
 *
 * Note that in the following code (as it is also explained in the slides from Erin Catto),
 * the value lambda is not only the lagrange multiplier but is the multiplication of the
 * Lagrange multiplier with the timestep dt. Therefore, in the following code, when we use
 * lambda, we mean (lambda * dt).
 *
 * We are using the accumulated impulse technique that is also described in the slides from
 * Erin Catto.
 *
 * We are also using warm starting. The idea is to warm start the solver at the beginning of
 * each step by applying the last impulstes for the constraints that we already existing at the
 * previous step. This allows the iterative solver to converge faster towards the solution.
 *
 * For contact constraints, we are also using split impulses so that the position correction
 * that uses Baumgarte stabilization does not change the momentum of the bodies.
 *
 * There are two ways to apply the friction constraints. Either the friction constraints are
 * applied at each contact point or they are applied only at the center of the contact manifold
 * between two bodies. If we solve the friction constraints at each contact point, we need
 * two constraints (two tangential friction directions) and if we solve the friction
 * constraints at the center of the contact manifold, we need two constraints for tangential
 * friction but also another twist friction constraint to prevent spin of the body around the
 * contact manifold center.
 */
class ConstraintSolver {

    private :

        // -------------------- Attributes -------------------- //

        /// Number of iterations of the contact solver
        uint mNbIterations;

        /// Current time step
        decimal mTimeStep;

    public :

        // -------------------- Methods -------------------- //

        /// Constructor
        ConstraintSolver();

        /// Destructor
        ~ConstraintSolver();

};

}

#endif