split SkeletonModel and Ragdoll classes apart

This commit is contained in:
Andrew Meadows 2014-08-14 14:29:03 -07:00
parent ca4a23e532
commit 46c91052c9
7 changed files with 263 additions and 162 deletions

View file

@ -83,10 +83,11 @@ MyAvatar::MyAvatar() :
for (int i = 0; i < MAX_DRIVE_KEYS; i++) {
_driveKeys[i] = 0.0f;
}
_skeletonModel.setEnableShapes(true);
// The skeleton is both a PhysicsEntity and Ragdoll, so we add it to the simulation once for each type.
_physicsSimulation.setEntity(&_skeletonModel);
_physicsSimulation.setRagdoll(&_skeletonModel);
_skeletonModel.setEnableShapes(true);
Ragdoll* ragdoll = _skeletonModel.buildRagdoll();
_physicsSimulation.setRagdoll(ragdoll);
}
MyAvatar::~MyAvatar() {
@ -1604,10 +1605,10 @@ void MyAvatar::updateCollisionWithAvatars(float deltaTime) {
if (simulation != &(_physicsSimulation)) {
skeleton->setEnableShapes(true);
_physicsSimulation.addEntity(skeleton);
_physicsSimulation.addRagdoll(skeleton);
_physicsSimulation.addRagdoll(skeleton->getRagdoll());
}
} else if (simulation == &(_physicsSimulation)) {
_physicsSimulation.removeRagdoll(skeleton);
_physicsSimulation.removeRagdoll(skeleton->getRagdoll());
_physicsSimulation.removeEntity(skeleton);
skeleton->setEnableShapes(false);
}

View file

@ -23,13 +23,21 @@
#include "Menu.h"
#include "MuscleConstraint.h"
#include "SkeletonModel.h"
#include "SkeletonRagdoll.h"
SkeletonModel::SkeletonModel(Avatar* owningAvatar, QObject* parent) :
Model(parent),
Ragdoll(),
_owningAvatar(owningAvatar),
_boundingShape(),
_boundingShapeLocalOffset(0.0f) {
_boundingShapeLocalOffset(0.0f),
_ragdoll(NULL) {
}
SkeletonModel::~SkeletonModel() {
if (_ragdoll) {
delete _ragdoll;
_ragdoll = NULL;
}
}
void SkeletonModel::setJointStates(QVector<JointState> states) {
@ -155,9 +163,6 @@ void SkeletonModel::getBodyShapes(QVector<const Shape*>& shapes) const {
void SkeletonModel::renderIKConstraints() {
renderJointConstraints(getRightHandJointIndex());
renderJointConstraints(getLeftHandJointIndex());
//if (isActive() && _owningAvatar->isMyAvatar()) {
// renderRagdoll();
//}
}
class IndexValue {
@ -489,21 +494,25 @@ bool SkeletonModel::getEyePositions(glm::vec3& firstEyePosition, glm::vec3& seco
}
void SkeletonModel::renderRagdoll() {
if (!_ragdoll) {
return;
}
const QVector<VerletPoint>& points = _ragdoll->getRagdollPoints();
const int BALL_SUBDIVISIONS = 6;
glDisable(GL_DEPTH_TEST);
glDisable(GL_LIGHTING);
glPushMatrix();
Application::getInstance()->loadTranslatedViewMatrix(_translation);
int numPoints = _ragdollPoints.size();
int numPoints = points.size();
float alpha = 0.3f;
float radius1 = 0.008f;
float radius2 = 0.01f;
glm::vec3 simulationTranslation = getTranslationInSimulationFrame();
glm::vec3 simulationTranslation = _ragdoll->getTranslationInSimulationFrame();
for (int i = 0; i < numPoints; ++i) {
glPushMatrix();
// NOTE: ragdollPoints are in simulation-frame but we want them to be model-relative
glm::vec3 position = _ragdollPoints[i]._position - simulationTranslation;
glm::vec3 position = points[i]._position - simulationTranslation;
glTranslatef(position.x, position.y, position.z);
// draw each point as a yellow hexagon with black border
glColor4f(0.0f, 0.0f, 0.0f, alpha);
@ -517,109 +526,18 @@ void SkeletonModel::renderRagdoll() {
glEnable(GL_LIGHTING);
}
// virtual
void SkeletonModel::initRagdollPoints() {
clearRagdollConstraintsAndPoints();
_muscleConstraints.clear();
initRagdollTransform();
// one point for each joint
int numStates = _jointStates.size();
_ragdollPoints.fill(VerletPoint(), numStates);
for (int i = 0; i < numStates; ++i) {
const JointState& state = _jointStates.at(i);
// _ragdollPoints start in model-frame
_ragdollPoints[i].initPosition(state.getPosition());
}
}
void SkeletonModel::buildRagdollConstraints() {
// NOTE: the length of DistanceConstraints is computed and locked in at this time
// so make sure the ragdoll positions are in a normal configuration before here.
const int numPoints = _ragdollPoints.size();
assert(numPoints == _jointStates.size());
float minBone = FLT_MAX;
float maxBone = -FLT_MAX;
QMultiMap<int, int> families;
for (int i = 0; i < numPoints; ++i) {
const JointState& state = _jointStates.at(i);
int parentIndex = state.getParentIndex();
if (parentIndex == -1) {
FixedConstraint* anchor = new FixedConstraint(&_translationInSimulationFrame, &(_ragdollPoints[i]));
_fixedConstraints.push_back(anchor);
} else {
DistanceConstraint* bone = new DistanceConstraint(&(_ragdollPoints[i]), &(_ragdollPoints[parentIndex]));
bone->setDistance(state.getDistanceToParent());
_boneConstraints.push_back(bone);
families.insert(parentIndex, i);
}
float boneLength = glm::length(state.getPositionInParentFrame());
if (boneLength > maxBone) {
maxBone = boneLength;
} else if (boneLength < minBone) {
minBone = boneLength;
}
}
// Joints that have multiple children effectively have rigid constraints between the children
// in the parent frame, so we add DistanceConstraints between children in the same family.
QMultiMap<int, int>::iterator itr = families.begin();
while (itr != families.end()) {
QList<int> children = families.values(itr.key());
int numChildren = children.size();
if (numChildren > 1) {
for (int i = 1; i < numChildren; ++i) {
DistanceConstraint* bone = new DistanceConstraint(&(_ragdollPoints[children[i-1]]), &(_ragdollPoints[children[i]]));
_boneConstraints.push_back(bone);
}
if (numChildren > 2) {
DistanceConstraint* bone = new DistanceConstraint(&(_ragdollPoints[children[numChildren-1]]), &(_ragdollPoints[children[0]]));
_boneConstraints.push_back(bone);
}
}
++itr;
}
float MAX_STRENGTH = 0.6f;
float MIN_STRENGTH = 0.05f;
// each joint gets a MuscleConstraint to its parent
for (int i = 1; i < numPoints; ++i) {
const JointState& state = _jointStates.at(i);
int p = state.getParentIndex();
if (p == -1) {
continue;
}
MuscleConstraint* constraint = new MuscleConstraint(&(_ragdollPoints[p]), &(_ragdollPoints[i]));
_muscleConstraints.push_back(constraint);
// Short joints are more susceptible to wiggle so we modulate the strength based on the joint's length:
// long = weak and short = strong.
constraint->setIndices(p, i);
float boneLength = glm::length(state.getPositionInParentFrame());
float strength = MIN_STRENGTH + (MAX_STRENGTH - MIN_STRENGTH) * (maxBone - boneLength) / (maxBone - minBone);
if (!families.contains(i)) {
// Although muscles only pull on the children not parents, nevertheless those joints that have
// parents AND children are more stable than joints at the end such as fingers. For such joints we
// bestow maximum strength which helps reduce wiggle.
strength = MAX_MUSCLE_STRENGTH;
}
constraint->setStrength(strength);
}
}
void SkeletonModel::updateVisibleJointStates() {
if (_showTrueJointTransforms) {
if (_showTrueJointTransforms || !_ragdoll) {
// no need to update visible transforms
return;
}
const QVector<VerletPoint>& ragdollPoints = _ragdoll->getRagdollPoints();
QVector<glm::vec3> points;
points.reserve(_jointStates.size());
glm::quat invRotation = glm::inverse(_rotation);
for (int i = 0; i < _jointStates.size(); i++) {
JointState& state = _jointStates[i];
points.push_back(_ragdollPoints[i]._position);
points.push_back(ragdollPoints[i]._position);
// get the parent state (this is the state that we want to rotate)
int parentIndex = state.getParentIndex();
@ -653,15 +571,14 @@ void SkeletonModel::updateVisibleJointStates() {
}
}
// virtual
void SkeletonModel::stepRagdollForward(float deltaTime) {
setRagdollTransform(_translation, _rotation);
Ragdoll::stepRagdollForward(deltaTime);
updateMuscles();
int numConstraints = _muscleConstraints.size();
for (int i = 0; i < numConstraints; ++i) {
_muscleConstraints[i]->enforce();
SkeletonRagdoll* SkeletonModel::buildRagdoll() {
if (!_ragdoll) {
_ragdoll = new SkeletonRagdoll(this);
if (_enableShapes) {
buildShapes();
}
}
return _ragdoll;
}
float DENSITY_OF_WATER = 1000.0f; // kg/m^3
@ -679,8 +596,13 @@ void SkeletonModel::buildShapes() {
return;
}
initRagdollPoints();
float massScale = getMassScale();
if (!_ragdoll) {
_ragdoll = new SkeletonRagdoll(this);
}
_ragdoll->initRagdollPoints();
QVector<VerletPoint>& points = _ragdoll->getRagdollPoints();
float massScale = _ragdoll->getMassScale();
float uniformScale = extractUniformScale(_scale);
const int numStates = _jointStates.size();
@ -700,14 +622,14 @@ void SkeletonModel::buildShapes() {
Shape* shape = NULL;
int parentIndex = joint.parentIndex;
if (type == Shape::SPHERE_SHAPE) {
shape = new VerletSphereShape(radius, &(_ragdollPoints[i]));
shape = new VerletSphereShape(radius, &(points[i]));
shape->setEntity(this);
_ragdollPoints[i].setMass(massScale * glm::max(MIN_JOINT_MASS, DENSITY_OF_WATER * shape->getVolume()));
points[i].setMass(massScale * glm::max(MIN_JOINT_MASS, DENSITY_OF_WATER * shape->getVolume()));
} else if (type == Shape::CAPSULE_SHAPE) {
assert(parentIndex != -1);
shape = new VerletCapsuleShape(radius, &(_ragdollPoints[parentIndex]), &(_ragdollPoints[i]));
shape = new VerletCapsuleShape(radius, &(points[parentIndex]), &(points[i]));
shape->setEntity(this);
_ragdollPoints[i].setMass(massScale * glm::max(MIN_JOINT_MASS, DENSITY_OF_WATER * shape->getVolume()));
points[i].setMass(massScale * glm::max(MIN_JOINT_MASS, DENSITY_OF_WATER * shape->getVolume()));
}
if (parentIndex != -1) {
// always disable collisions between joint and its parent
@ -717,7 +639,7 @@ void SkeletonModel::buildShapes() {
} else {
// give the base joint a very large mass since it doesn't actually move
// in the local-frame simulation (it defines the origin)
_ragdollPoints[i].setMass(VERY_BIG_MASS);
points[i].setMass(VERY_BIG_MASS);
}
_shapes.push_back(shape);
}
@ -729,17 +651,11 @@ void SkeletonModel::buildShapes() {
// joints that are currently colliding.
disableCurrentSelfCollisions();
buildRagdollConstraints();
_ragdoll->buildRagdollConstraints();
// ... then move shapes back to current joint positions
if (_ragdollPoints.size() == numStates) {
int numStates = _jointStates.size();
for (int i = 0; i < numStates; ++i) {
// ragdollPoints start in model-frame
_ragdollPoints[i].initPosition(_jointStates.at(i).getPosition());
}
}
enforceRagdollConstraints();
_ragdoll->slamPointPositions();
_ragdoll->enforceRagdollConstraints();
}
void SkeletonModel::moveShapesTowardJoints(float deltaTime) {
@ -747,8 +663,9 @@ void SkeletonModel::moveShapesTowardJoints(float deltaTime) {
// unravel a skelton that has become tangled in its constraints. So let's keep this
// around for a while just in case.
const int numStates = _jointStates.size();
assert(_jointStates.size() == _ragdollPoints.size());
if (_ragdollPoints.size() != numStates) {
QVector<VerletPoint>& ragdollPoints = _ragdoll->getRagdollPoints();
assert(_jointStates.size() == ragdollPoints.size());
if (ragdollPoints.size() != numStates) {
return;
}
@ -757,32 +674,22 @@ void SkeletonModel::moveShapesTowardJoints(float deltaTime) {
float fraction = glm::clamp(deltaTime / RAGDOLL_FOLLOWS_JOINTS_TIMESCALE, 0.0f, 1.0f);
float oneMinusFraction = 1.0f - fraction;
glm::vec3 simulationTranslation = getTranslationInSimulationFrame();
glm::vec3 simulationTranslation = _ragdoll->getTranslationInSimulationFrame();
for (int i = 0; i < numStates; ++i) {
// ragdollPoints are in simulation-frame but jointStates are in model-frame
_ragdollPoints[i].initPosition(oneMinusFraction * _ragdollPoints[i]._position +
ragdollPoints[i].initPosition(oneMinusFraction * ragdollPoints[i]._position +
fraction * (simulationTranslation + _rotation * (_jointStates.at(i).getPosition())));
}
}
void SkeletonModel::updateMuscles() {
int numConstraints = _muscleConstraints.size();
for (int i = 0; i < numConstraints; ++i) {
MuscleConstraint* constraint = _muscleConstraints[i];
int j = constraint->getParentIndex();
int k = constraint->getChildIndex();
assert(j != -1 && k != -1);
// ragdollPoints are in simulation-frame but jointStates are in model-frame
constraint->setChildOffset(_rotation * (_jointStates.at(k).getPosition() - _jointStates.at(j).getPosition()));
}
}
void SkeletonModel::computeBoundingShape(const FBXGeometry& geometry) {
// compute default joint transforms
int numStates = _jointStates.size();
QVector<glm::mat4> transforms;
transforms.fill(glm::mat4(), numStates);
QVector<VerletPoint>& ragdollPoints = _ragdoll->getRagdollPoints();
// compute the default transforms and slam the ragdoll positions accordingly
// (which puts the shapes where we want them)
for (int i = 0; i < numStates; i++) {
@ -791,7 +698,7 @@ void SkeletonModel::computeBoundingShape(const FBXGeometry& geometry) {
int parentIndex = joint.parentIndex;
if (parentIndex == -1) {
transforms[i] = _jointStates[i].getTransform();
_ragdollPoints[i].initPosition(extractTranslation(transforms[i]));
ragdollPoints[i].initPosition(extractTranslation(transforms[i]));
continue;
}
@ -799,7 +706,7 @@ void SkeletonModel::computeBoundingShape(const FBXGeometry& geometry) {
transforms[i] = transforms[parentIndex] * glm::translate(joint.translation)
* joint.preTransform * glm::mat4_cast(modifiedRotation) * joint.postTransform;
// setting the ragdollPoints here slams the VerletShapes into their default positions
_ragdollPoints[i].initPosition(extractTranslation(transforms[i]));
ragdollPoints[i].initPosition(extractTranslation(transforms[i]));
}
// compute bounding box that encloses all shapes
@ -918,9 +825,12 @@ const int BALL_SUBDIVISIONS = 10;
// virtual
void SkeletonModel::renderJointCollisionShapes(float alpha) {
if (!_ragdoll) {
return;
}
glPushMatrix();
Application::getInstance()->loadTranslatedViewMatrix(_translation);
glm::vec3 simulationTranslation = getTranslationInSimulationFrame();
glm::vec3 simulationTranslation = _ragdoll->getTranslationInSimulationFrame();
for (int i = 0; i < _shapes.size(); i++) {
Shape* shape = _shapes[i];
if (!shape) {

View file

@ -15,18 +15,20 @@
#include "renderer/Model.h"
#include <CapsuleShape.h>
#include <Ragdoll.h>
#include "SkeletonRagdoll.h"
class Avatar;
class MuscleConstraint;
class SkeletonRagdoll;
/// A skeleton loaded from a model.
class SkeletonModel : public Model, public Ragdoll {
class SkeletonModel : public Model {
Q_OBJECT
public:
SkeletonModel(Avatar* owningAvatar, QObject* parent = NULL);
~SkeletonModel();
void setJointStates(QVector<JointState> states);
@ -96,12 +98,11 @@ public:
bool getEyePositions(glm::vec3& firstEyePosition, glm::vec3& secondEyePosition) const;
virtual void updateVisibleJointStates();
// virtual overrride from Ragdoll
virtual void stepRagdollForward(float deltaTime);
SkeletonRagdoll* buildRagdoll();
SkeletonRagdoll* getRagdoll() { return _ragdoll; }
void moveShapesTowardJoints(float fraction);
void updateMuscles();
void computeBoundingShape(const FBXGeometry& geometry);
void renderBoundingCollisionShapes(float alpha);
@ -115,10 +116,6 @@ public:
protected:
// virtual overrrides from Ragdoll
void initRagdollPoints();
void buildRagdollConstraints();
void buildShapes();
/// \param jointIndex index of joint in model
@ -147,7 +144,7 @@ private:
CapsuleShape _boundingShape;
glm::vec3 _boundingShapeLocalOffset;
QVector<MuscleConstraint*> _muscleConstraints;
SkeletonRagdoll* _ragdoll;
};
#endif // hifi_SkeletonModel_h

View file

@ -0,0 +1,148 @@
//
// SkeletonRagdoll.cpp
// interface/src/avatar
//
// Created by Andrew Meadows 2014.08.14
// Copyright 2014 High Fidelity, Inc.
//
// Distributed under the Apache License, Version 2.0.
// See the accompanying file LICENSE or http://www.apache.org/licenses/LICENSE-2.0.html
//
#include <DistanceConstraint.h>
#include <FixedConstraint.h>
#include "SkeletonRagdoll.h"
#include "MuscleConstraint.h"
#include "../renderer/Model.h"
SkeletonRagdoll::SkeletonRagdoll(Model* model) : Ragdoll(), _model(model) {
assert(_model);
}
SkeletonRagdoll::~SkeletonRagdoll() {
}
// virtual
void SkeletonRagdoll::stepRagdollForward(float deltaTime) {
setRagdollTransform(_model->getTranslation(), _model->getRotation());
Ragdoll::stepRagdollForward(deltaTime);
updateMuscles();
int numConstraints = _muscleConstraints.size();
for (int i = 0; i < numConstraints; ++i) {
_muscleConstraints[i]->enforce();
}
}
void SkeletonRagdoll::slamPointPositions() {
QVector<JointState>& jointStates = _model->getJointStates();
int numStates = jointStates.size();
for (int i = 0; i < numStates; ++i) {
_ragdollPoints[i].initPosition(jointStates.at(i).getPosition());
}
}
// virtual
void SkeletonRagdoll::initRagdollPoints() {
clearRagdollConstraintsAndPoints();
_muscleConstraints.clear();
initRagdollTransform();
// one point for each joint
QVector<JointState>& jointStates = _model->getJointStates();
int numStates = jointStates.size();
_ragdollPoints.fill(VerletPoint(), numStates);
slamPointPositions();
}
// virtual
void SkeletonRagdoll::buildRagdollConstraints() {
QVector<JointState>& jointStates = _model->getJointStates();
// NOTE: the length of DistanceConstraints is computed and locked in at this time
// so make sure the ragdoll positions are in a normal configuration before here.
const int numPoints = _ragdollPoints.size();
assert(numPoints == jointStates.size());
float minBone = FLT_MAX;
float maxBone = -FLT_MAX;
QMultiMap<int, int> families;
for (int i = 0; i < numPoints; ++i) {
const JointState& state = jointStates.at(i);
int parentIndex = state.getParentIndex();
if (parentIndex == -1) {
FixedConstraint* anchor = new FixedConstraint(&_translationInSimulationFrame, &(_ragdollPoints[i]));
_fixedConstraints.push_back(anchor);
} else {
DistanceConstraint* bone = new DistanceConstraint(&(_ragdollPoints[i]), &(_ragdollPoints[parentIndex]));
bone->setDistance(state.getDistanceToParent());
_boneConstraints.push_back(bone);
families.insert(parentIndex, i);
}
float boneLength = glm::length(state.getPositionInParentFrame());
if (boneLength > maxBone) {
maxBone = boneLength;
} else if (boneLength < minBone) {
minBone = boneLength;
}
}
// Joints that have multiple children effectively have rigid constraints between the children
// in the parent frame, so we add DistanceConstraints between children in the same family.
QMultiMap<int, int>::iterator itr = families.begin();
while (itr != families.end()) {
QList<int> children = families.values(itr.key());
int numChildren = children.size();
if (numChildren > 1) {
for (int i = 1; i < numChildren; ++i) {
DistanceConstraint* bone = new DistanceConstraint(&(_ragdollPoints[children[i-1]]), &(_ragdollPoints[children[i]]));
_boneConstraints.push_back(bone);
}
if (numChildren > 2) {
DistanceConstraint* bone = new DistanceConstraint(&(_ragdollPoints[children[numChildren-1]]), &(_ragdollPoints[children[0]]));
_boneConstraints.push_back(bone);
}
}
++itr;
}
float MAX_STRENGTH = 0.6f;
float MIN_STRENGTH = 0.05f;
// each joint gets a MuscleConstraint to its parent
for (int i = 1; i < numPoints; ++i) {
const JointState& state = jointStates.at(i);
int p = state.getParentIndex();
if (p == -1) {
continue;
}
MuscleConstraint* constraint = new MuscleConstraint(&(_ragdollPoints[p]), &(_ragdollPoints[i]));
_muscleConstraints.push_back(constraint);
// Short joints are more susceptible to wiggle so we modulate the strength based on the joint's length:
// long = weak and short = strong.
constraint->setIndices(p, i);
float boneLength = glm::length(state.getPositionInParentFrame());
float strength = MIN_STRENGTH + (MAX_STRENGTH - MIN_STRENGTH) * (maxBone - boneLength) / (maxBone - minBone);
if (!families.contains(i)) {
// Although muscles only pull on the children not parents, nevertheless those joints that have
// parents AND children are more stable than joints at the end such as fingers. For such joints we
// bestow maximum strength which helps reduce wiggle.
strength = MAX_MUSCLE_STRENGTH;
}
constraint->setStrength(strength);
}
}
void SkeletonRagdoll::updateMuscles() {
QVector<JointState>& jointStates = _model->getJointStates();
int numConstraints = _muscleConstraints.size();
glm::quat rotation = _model->getRotation();
for (int i = 0; i < numConstraints; ++i) {
MuscleConstraint* constraint = _muscleConstraints[i];
int j = constraint->getParentIndex();
int k = constraint->getChildIndex();
assert(j != -1 && k != -1);
// ragdollPoints are in simulation-frame but jointStates are in model-frame
constraint->setChildOffset(rotation * (jointStates.at(k).getPosition() - jointStates.at(j).getPosition()));
}
}

View file

@ -0,0 +1,42 @@
//
// SkeletonkRagdoll.h
// interface/src/avatar
//
// Created by Andrew Meadows 2014.08.14
// Copyright 2014 High Fidelity, Inc.
//
// Distributed under the Apache License, Version 2.0.
// See the accompanying file LICENSE or http://www.apache.org/licenses/LICENSE-2.0.html
//
#ifndef hifi_SkeletonRagdoll_h
#define hifi_SkeletonRagdoll_h
#include <QVector>
#include <Ragdoll.h>
#include "../renderer/JointState.h"
class MuscleConstraint;
class Model;
class SkeletonRagdoll : public Ragdoll {
public:
SkeletonRagdoll(Model* model);
virtual ~SkeletonRagdoll();
void slamPointPositions();
virtual void stepRagdollForward(float deltaTime);
virtual void initRagdollPoints();
virtual void buildRagdollConstraints();
void updateMuscles();
private:
Model* _model;
QVector<MuscleConstraint*> _muscleConstraints;
};
#endif // hifi_SkeletonRagdoll_h

View file

@ -164,6 +164,9 @@ public:
const QVector<LocalLight>& getLocalLights() const { return _localLights; }
void setShowTrueJointTransforms(bool show) { _showTrueJointTransforms = show; }
QVector<JointState>& getJointStates() { return _jointStates; }
const QVector<JointState>& getJointStates() const { return _jointStates; }
protected:
QSharedPointer<NetworkGeometry> _geometry;

View file

@ -52,11 +52,11 @@ public:
void setMassScale(float scale);
float getMassScale() const { return _massScale; }
protected:
void clearRagdollConstraintsAndPoints();
virtual void initRagdollPoints() = 0;
virtual void buildRagdollConstraints() = 0;
protected:
float _massScale;
glm::vec3 _ragdollTranslation; // world-frame
glm::quat _ragdollRotation; // world-frame