/// Initialize the bodies, anchors, and reference angle using the world /// anchor. public void Initialize(Body b1, Body b2, Vector2 anchor) { body1 = b1; body2 = b2; localAnchor1 = body1.GetLocalPoint(anchor); localAnchor2 = body2.GetLocalPoint(anchor); referenceAngle = body2.GetAngle() - body1.GetAngle(); }
/// Initialize the bodies, anchors, axis, and reference angle using the world /// anchor and world axis. // Linear raint (point-to-line) // d = p2 - p1 = x2 + r2 - x1 - r1 // C = dot(perp, d) // Cdot = dot(d, cross(w1, perp)) + dot(perp, v2 + cross(w2, r2) - v1 - cross(w1, r1)) // = -dot(perp, v1) - dot(cross(d + r1, perp), w1) + dot(perp, v2) + dot(cross(r2, perp), v2) // J = [-perp, -cross(d + r1, perp), perp, cross(r2,perp)] // // Angular raint // C = a2 - a1 + a_initial // Cdot = w2 - w1 // J = [0 0 -1 0 0 1] // // K = J * invM * JT // // J = [-a -s1 a s2] // [0 -1 0 1] // a = perp // s1 = cross(d + r1, a) = cross(p2 - x1, a) // s2 = cross(r2, a) = cross(p2 - x2, a) // Motor/Limit linear raint // C = dot(ax1, d) // Cdot = = -dot(ax1, v1) - dot(cross(d + r1, ax1), w1) + dot(ax1, v2) + dot(cross(r2, ax1), v2) // J = [-ax1 -cross(d+r1,ax1) ax1 cross(r2,ax1)] // Block Solver // We develop a block solver that includes the joint limit. This makes the limit stiff (inelastic) even // when the mass has poor distribution (leading to large torques about the joint anchor points). // // The Jacobian has 3 rows: // J = [-uT -s1 uT s2] // linear // [0 -1 0 1] // angular // [-vT -a1 vT a2] // limit // // u = perp // v = axis // s1 = cross(d + r1, u), s2 = cross(r2, u) // a1 = cross(d + r1, v), a2 = cross(r2, v) // M * (v2 - v1) = JT * df // J * v2 = bias // // v2 = v1 + invM * JT * df // J * (v1 + invM * JT * df) = bias // K * df = bias - J * v1 = -Cdot // K = J * invM * JT // Cdot = J * v1 - bias // // Now solve for f2. // df = f2 - f1 // K * (f2 - f1) = -Cdot // f2 = invK * (-Cdot) + f1 // // Clamp accumulated limit impulse. // lower: f2(3) = max(f2(3), 0) // upper: f2(3) = min(f2(3), 0) // // Solve for correct f2(1:2) // K(1:2, 1:2) * f2(1:2) = -Cdot(1:2) - K(1:2,3) * f2(3) + K(1:2,1:3) * f1 // = -Cdot(1:2) - K(1:2,3) * f2(3) + K(1:2,1:2) * f1(1:2) + K(1:2,3) * f1(3) // K(1:2, 1:2) * f2(1:2) = -Cdot(1:2) - K(1:2,3) * (f2(3) - f1(3)) + K(1:2,1:2) * f1(1:2) // f2(1:2) = invK(1:2,1:2) * (-Cdot(1:2) - K(1:2,3) * (f2(3) - f1(3))) + f1(1:2) // // Now compute impulse to be applied: // df = f2 - f1 public void Initialize(Body b1, Body b2, Vector2 anchor, Vector2 axis) { body1 = b1; body2 = b2; localAnchor1 = body1.GetLocalPoint(anchor); localAnchor2 = body2.GetLocalPoint(anchor); localAxis1 = body1.GetLocalVector(axis); referenceAngle = body2.GetAngle() - body1.GetAngle(); }