[InverseDynamics] Support for Jacobians & derivatives

This change adds support for calculating Jacobians and
dot(Jacobian)*u terms, along with the required support for
the 3xN matrices in the standalone math library.
It also adds functions to compute kinematics only (position, velocity, accel).
To facilitate tests, the Cl also adds a RandomTreeCreator to create
randomized multibody trees.
Thanks to Thomas Buschmann for this contribution!

src/BulletInverseDynamics/details/IDLinearMathInterface.hpp

@@ -8,13 +8,13 @@
 #include "../../LinearMath/btMatrix3x3.h"
 #include "../../LinearMath/btVector3.h"
 #include "../../LinearMath/btMatrixX.h"
+#define BT_ID_HAVE_MAT3X
 
 namespace btInverseDynamics {
 class vec3;
 class vecx;
 class mat33;
 typedef btMatrixX<idScalar> matxx;
-typedef btVectorX<idScalar> vecxx;
 
 class vec3 : public btVector3 {
 public:
@@ -47,11 +47,11 @@ inline mat33 operator/(const mat33& a, const idScalar& s) { return a * (1.0 / s)
 
 inline mat33 operator*(const idScalar& s, const mat33& a) { return a * s; }
 
-class vecx : public vecxx {
+class vecx : public btVectorX<idScalar> {
 public:
-	vecx(int size) : vecxx(size) {}
-	const vecx& operator=(const vecxx& rhs) {
-		*static_cast<vecxx*>(this) = rhs;
+	vecx(int size) : btVectorX(size) {}
+	const vecx& operator=(const btVectorX<idScalar>& rhs) {
+		*static_cast<btVectorX*>(this) = rhs;
 		return *this;
 	}
 
@@ -108,6 +108,65 @@ inline vecx operator/(const vecx& a, const idScalar& s) {
 
 	return result;
 }
+
+// use btMatrixX to implement 3xX matrix
+class mat3x : public matxx {
+public:
+    mat3x(){}
+    mat3x(const mat3x&rhs) {
+        matxx::resize(rhs.rows(), rhs.cols());
+        *this = rhs;
+    }
+    mat3x(int rows, int cols): matxx(3,cols) {
+    }
+    void operator=(const mat3x& rhs) {
+	if (m_cols != rhs.m_cols) {
+            error_message("size missmatch, cols= %d but rhs.cols= %d\n", cols(), rhs.cols());
+            abort();
+	}
+        for(int i=0;i<rows();i++) {
+            for(int k=0;k<cols();k++) {
+                setElem(i,k,rhs(i,k));
+            }
+        }
+    }
+    void setZero() {
+        matxx::setZero();
+    }
+};
+
+
+inline vec3 operator*(const mat3x& a, const vecx& b) {
+    vec3 result;
+    if (a.cols() != b.size()) {
+        error_message("size missmatch. a.cols()= %d, b.size()= %d\n", a.cols(), b.size());
+        abort();
+    }
+    result(0)=0.0;
+    result(1)=0.0;
+    result(2)=0.0;
+    for(int i=0;i<b.size();i++) {
+        for(int k=0;k<3;k++) {
+            result(k)+=a(k,i)*b(i);
+        }
+    }
+    return result;
+}
+
+
+inline void resize(mat3x &m, idArrayIdx size) {
+    m.resize(3, size);
+    m.setZero();
+}
+
+inline void setMatxxElem(const idArrayIdx row, const idArrayIdx col, const idScalar val, matxx*m){
+    m->setElem(row, col, val);
+}
+
+inline void setMat3xElem(const idArrayIdx row, const idArrayIdx col, const idScalar val, mat3x*m){
+    m->setElem(row, col, val);
+}
+
 }
 
 #endif  // IDLINEARMATHINTERFACE_HPP_