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flightgear/src/FDM/YASim/Airplane.cpp

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#include "Atmosphere.hpp"
#include "ControlMap.hpp"
#include "Gear.hpp"
#include "Math.hpp"
#include "Glue.hpp"
#include "RigidBody.hpp"
#include "Surface.hpp"
#include "Thruster.hpp"
#include "Airplane.hpp"
namespace yasim {
// gadgets
inline float norm(float f) { return f<1 ? 1/f : f; }
inline float abs(float f) { return f<0 ? -f : f; }
Airplane::Airplane()
{
_emptyWeight = 0;
_pilotPos[0] = _pilotPos[1] = _pilotPos[2] = 0;
_wing = 0;
_tail = 0;
_ballast = 0;
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_cruiseP = 0;
_cruiseT = 0;
_cruiseSpeed = 0;
_cruiseWeight = 0;
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_approachP = 0;
_approachT = 0;
_approachSpeed = 0;
_approachAoA = 0;
_approachWeight = 0;
_dragFactor = 1;
_liftRatio = 1;
_cruiseAoA = 0;
_tailIncidence = 0;
}
Airplane::~Airplane()
{
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int i;
for(i=0; i<_fuselages.size(); i++)
delete (Fuselage*)_fuselages.get(i);
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for(i=0; i<_tanks.size(); i++)
delete (Tank*)_tanks.get(i);
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for(i=0; i<_thrusters.size(); i++)
delete (ThrustRec*)_thrusters.get(i);
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for(i=0; i<_gears.size(); i++)
delete (GearRec*)_gears.get(i);
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for(i=0; i<_surfs.size(); i++)
delete (Surface*)_surfs.get(i);
for(i=0; i<_contacts.size(); i++)
delete[] (float*)_contacts.get(i);
}
void Airplane::iterate(float dt)
{
// The gear might have moved. Change their aerodynamics.
updateGearState();
_model.iterate();
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// FIXME: Consume fuel
}
ControlMap* Airplane::getControlMap()
{
return &_controls;
}
Model* Airplane::getModel()
{
return &_model;
}
void Airplane::getPilotAccel(float* out)
{
State* s = _model.getState();
// Gravity
Glue::geodUp(s->pos, out);
Math::mul3(-9.8f, out, out);
// The regular acceleration
float tmp[3];
Math::mul3(-1, s->acc, tmp);
Math::add3(tmp, out, out);
// Convert to aircraft coordinates
Math::vmul33(s->orient, out, out);
// FIXME: rotational & centripetal acceleration needed
}
void Airplane::setPilotPos(float* pos)
{
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int i;
for(i=0; i<3; i++) _pilotPos[i] = pos[i];
}
void Airplane::getPilotPos(float* out)
{
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int i;
for(i=0; i<3; i++) out[i] = _pilotPos[i];
}
int Airplane::numGear()
{
return _gears.size();
}
Gear* Airplane::getGear(int g)
{
return ((GearRec*)_gears.get(g))->gear;
}
void Airplane::updateGearState()
{
for(int i=0; i<_gears.size(); i++) {
GearRec* gr = (GearRec*)_gears.get(i);
float ext = gr->gear->getExtension();
gr->surf->setXDrag(ext);
gr->surf->setYDrag(ext);
gr->surf->setZDrag(ext);
}
}
void Airplane::setApproach(float speed, float altitude)
{
// The zero AoA will become a calculated stall AoA in compile()
setApproach(speed, altitude, 0);
}
void Airplane::setApproach(float speed, float altitude, float aoa)
{
_approachSpeed = speed;
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_approachP = Atmosphere::getStdPressure(altitude);
_approachT = Atmosphere::getStdTemperature(altitude);
_approachAoA = aoa;
}
void Airplane::setCruise(float speed, float altitude)
{
_cruiseSpeed = speed;
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_cruiseP = Atmosphere::getStdPressure(altitude);
_cruiseT = Atmosphere::getStdTemperature(altitude);
_cruiseAoA = 0;
_tailIncidence = 0;
}
void Airplane::setElevatorControl(int control)
{
_approachElevator.control = control;
_approachElevator.val = 0;
_approachControls.add(&_approachElevator);
}
void Airplane::addApproachControl(int control, float val)
{
Control* c = new Control();
c->control = control;
c->val = val;
_approachControls.add(c);
}
void Airplane::addCruiseControl(int control, float val)
{
Control* c = new Control();
c->control = control;
c->val = val;
_cruiseControls.add(c);
}
int Airplane::numTanks()
{
return _tanks.size();
}
float Airplane::getFuel(int tank)
{
return ((Tank*)_tanks.get(tank))->fill;
}
float Airplane::getFuelDensity(int tank)
{
return ((Tank*)_tanks.get(tank))->density;
}
void Airplane::setWeight(float weight)
{
_emptyWeight = weight;
}
void Airplane::setWing(Wing* wing)
{
_wing = wing;
}
void Airplane::setTail(Wing* tail)
{
_tail = tail;
}
void Airplane::addVStab(Wing* vstab)
{
_vstabs.add(vstab);
}
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void Airplane::addFuselage(float* front, float* back, float width,
float taper, float mid)
{
Fuselage* f = new Fuselage();
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int i;
for(i=0; i<3; i++) {
f->front[i] = front[i];
f->back[i] = back[i];
}
f->width = width;
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f->taper = taper;
f->mid = mid;
_fuselages.add(f);
}
int Airplane::addTank(float* pos, float cap, float density)
{
Tank* t = new Tank();
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int i;
for(i=0; i<3; i++) t->pos[i] = pos[i];
t->cap = cap;
t->fill = cap;
t->density = density;
t->handle = 0xffffffff;
return _tanks.add(t);
}
void Airplane::addGear(Gear* gear)
{
GearRec* g = new GearRec();
g->gear = gear;
g->surf = 0;
_gears.add(g);
}
void Airplane::addThruster(Thruster* thruster, float mass, float* cg)
{
ThrustRec* t = new ThrustRec();
t->thruster = thruster;
t->mass = mass;
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int i;
for(i=0; i<3; i++) t->cg[i] = cg[i];
_thrusters.add(t);
}
void Airplane::addBallast(float* pos, float mass)
{
_model.getBody()->addMass(mass, pos);
_ballast += mass;
}
int Airplane::addWeight(float* pos, float size)
{
WeightRec* wr = new WeightRec();
wr->handle = _model.getBody()->addMass(0, pos);
wr->surf = new Surface();
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wr->surf->setPosition(pos);
wr->surf->setTotalDrag(size*size);
_model.addSurface(wr->surf);
_surfs.add(wr->surf);
return _weights.add(wr);
}
void Airplane::setWeight(int handle, float mass)
{
WeightRec* wr = (WeightRec*)_weights.get(handle);
_model.getBody()->setMass(wr->handle, mass);
// Kill the aerodynamic drag if the mass is exactly zero. This is
// how we simulate droppable stores.
if(mass == 0) {
wr->surf->setXDrag(0);
wr->surf->setYDrag(0);
wr->surf->setZDrag(0);
} else {
wr->surf->setXDrag(1);
wr->surf->setYDrag(1);
wr->surf->setZDrag(1);
}
}
void Airplane::setFuelFraction(float frac)
{
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int i;
for(i=0; i<_tanks.size(); i++) {
Tank* t = (Tank*)_tanks.get(i);
_model.getBody()->setMass(t->handle, t->cap * frac);
}
}
float Airplane::getDragCoefficient()
{
return _dragFactor;
}
float Airplane::getLiftRatio()
{
return _liftRatio;
}
float Airplane::getCruiseAoA()
{
return _cruiseAoA;
}
float Airplane::getTailIncidence()
{
return _tailIncidence;
}
char* Airplane::getFailureMsg()
{
return _failureMsg;
}
int Airplane::getSolutionIterations()
{
return _solutionIterations;
}
void Airplane::setupState(float aoa, float speed, State* s)
{
float cosAoA = Math::cos(aoa);
float sinAoA = Math::sin(aoa);
s->orient[0] = cosAoA; s->orient[1] = 0; s->orient[2] = sinAoA;
s->orient[3] = 0; s->orient[4] = 1; s->orient[5] = 0;
s->orient[6] = -sinAoA; s->orient[7] = 0; s->orient[8] = cosAoA;
s->v[0] = speed; s->v[1] = 0; s->v[2] = 0;
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int i;
for(i=0; i<3; i++)
s->pos[i] = s->rot[i] = s->acc[i] = s->racc[i] = 0;
// Put us 1m above the origin, or else the gravity computation in
// Model goes nuts
s->pos[2] = 1;
}
void Airplane::addContactPoint(float* pos)
{
float* cp = new float[3];
cp[0] = pos[0];
cp[1] = pos[1];
cp[2] = pos[2];
_contacts.add(cp);
}
float Airplane::compileWing(Wing* w)
{
// The tip of the wing is a contact point
float tip[3];
w->getTip(tip);
addContactPoint(tip);
if(w->isMirrored()) {
tip[1] *= -1;
addContactPoint(tip);
}
// Make sure it's initialized. The surfaces will pop out with
// total drag coefficients equal to their areas, which is what we
// want.
w->compile();
float wgt = 0;
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int i;
for(i=0; i<w->numSurfaces(); i++) {
Surface* s = (Surface*)w->getSurface(i);
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float td = s->getTotalDrag();
s->setTotalDrag(td);
_model.addSurface(s);
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float mass = w->getSurfaceWeight(i);
mass = mass * Math::sqrt(mass);
float pos[3];
s->getPosition(pos);
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_model.getBody()->addMass(mass, pos);
wgt += mass;
}
return wgt;
}
float Airplane::compileFuselage(Fuselage* f)
{
// The front and back are contact points
addContactPoint(f->front);
addContactPoint(f->back);
float wgt = 0;
float fwd[3];
Math::sub3(f->front, f->back, fwd);
float len = Math::mag3(fwd);
float wid = f->width;
int segs = (int)Math::ceil(len/wid);
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float segWgt = len*wid/segs;
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int j;
for(j=0; j<segs; j++) {
float frac = (j+0.5f) / segs;
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float scale = 1;
if(frac < f->mid)
scale = f->taper+(1-f->taper) * (frac / f->mid);
else
scale = f->taper+(1-f->taper) * (frac - f->mid) / (1 - f->mid);
// Where are we?
float pos[3];
Math::mul3(frac, fwd, pos);
Math::add3(f->back, pos, pos);
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// _Mass_ weighting goes as surface area^(3/2)
float mass = scale*segWgt * Math::sqrt(scale*segWgt);
_model.getBody()->addMass(mass, pos);
wgt += mass;
// Make a Surface too
Surface* s = new Surface();
s->setPosition(pos);
float sideDrag = len/wid;
s->setYDrag(sideDrag);
s->setZDrag(sideDrag);
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s->setTotalDrag(scale*segWgt);
// FIXME: fails for fuselages aligned along the Y axis
float o[9];
float *x=o, *y=o+3, *z=o+6; // nicknames for the axes
Math::unit3(fwd, x);
y[0] = 0; y[1] = 1; y[2] = 0;
Math::cross3(x, y, z);
Math::unit3(z, z);
Math::cross3(z, x, y);
s->setOrientation(o);
_model.addSurface(s);
_surfs.add(s);
}
return wgt;
}
// FIXME: should probably add a mass for the gear, too
void Airplane::compileGear(GearRec* gr)
{
Gear* g = gr->gear;
// Make a Surface object for the aerodynamic behavior
Surface* s = new Surface();
gr->surf = s;
// Put the surface at the half-way point on the gear strut, give
// it a drag coefficient equal to a square of the same dimension
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// (gear are really draggy) and make it symmetric. Assume that
// the "length" of the gear is 3x the compression distance
float pos[3], cmp[3];
g->getCompression(cmp);
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float length = 3 * Math::mag3(cmp);
g->getPosition(pos);
Math::mul3(0.5, cmp, cmp);
Math::add3(pos, cmp, pos);
s->setPosition(pos);
s->setTotalDrag(length*length);
_model.addGear(g);
_model.addSurface(s);
_surfs.add(s);
}
void Airplane::compileContactPoints()
{
// Figure it will compress by 20cm
float comp[3];
float DIST = 0.2f;
comp[0] = 0; comp[1] = 0; comp[2] = DIST;
// Give it a spring constant such that at full compression it will
// hold up 10 times the planes mass. That's about right. Yeah.
float mass = _model.getBody()->getTotalMass();
float spring = (1/DIST) * 9.8f * 10.0f * mass;
float damp = 2 * Math::sqrt(spring * mass);
int i;
for(i=0; i<_contacts.size(); i++) {
float *cp = (float*)_contacts.get(i);
Gear* g = new Gear();
g->setPosition(cp);
g->setCompression(comp);
g->setSpring(spring);
g->setDamping(damp);
g->setBrake(1);
// I made these up
g->setStaticFriction(0.6f);
g->setDynamicFriction(0.5f);
_model.addGear(g);
}
}
void Airplane::compile()
{
double ground[3];
ground[0] = 0; ground[1] = 0; ground[2] = 1;
_model.setGroundPlane(ground, -100000);
RigidBody* body = _model.getBody();
int firstMass = body->numMasses();
// Generate the point masses for the plane. Just use unitless
// numbers for a first pass, then go back through and rescale to
// make the weight right.
float aeroWgt = 0;
// The Wing objects
aeroWgt += compileWing(_wing);
aeroWgt += compileWing(_tail);
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int i;
for(i=0; i<_vstabs.size(); i++) {
aeroWgt += compileWing((Wing*)_vstabs.get(i));
}
// The fuselage(s)
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for(i=0; i<_fuselages.size(); i++) {
aeroWgt += compileFuselage((Fuselage*)_fuselages.get(i));
}
// Count up the absolute weight we have
float nonAeroWgt = _ballast;
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for(i=0; i<_thrusters.size(); i++)
nonAeroWgt += ((ThrustRec*)_thrusters.get(i))->mass;
// Rescale to the specified empty weight
float wscale = (_emptyWeight-nonAeroWgt)/aeroWgt;
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for(i=firstMass; i<body->numMasses(); i++)
body->setMass(i, body->getMass(i)*wscale);
// Add the thruster masses
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for(i=0; i<_thrusters.size(); i++) {
ThrustRec* t = (ThrustRec*)_thrusters.get(i);
body->addMass(t->mass, t->cg);
}
// Add the tanks, empty for now.
float totalFuel = 0;
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for(i=0; i<_tanks.size(); i++) {
Tank* t = (Tank*)_tanks.get(i);
t->handle = body->addMass(0, t->pos);
totalFuel += t->cap;
}
_cruiseWeight = _emptyWeight + totalFuel*0.5f;
_approachWeight = _emptyWeight + totalFuel*0.2f;
body->recalc();
// Add surfaces for the landing gear.
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for(i=0; i<_gears.size(); i++)
compileGear((GearRec*)_gears.get(i));
// The Thruster objects
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for(i=0; i<_thrusters.size(); i++) {
ThrustRec* tr = (ThrustRec*)_thrusters.get(i);
tr->handle = _model.addThruster(tr->thruster);
}
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// Ground effect
float gepos[3];
float gespan = _wing->getGroundEffect(gepos);
_model.setGroundEffect(gepos, gespan, 0.3f);
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solveGear();
solve();
// Do this after solveGear, because it creates "gear" objects that
// we don't want to affect.
compileContactPoints();
}
void Airplane::solveGear()
{
float cg[3], pos[3];
_model.getBody()->getCG(cg);
// Calculate spring constant weightings for the gear. Weight by
// the inverse of the distance to the c.g. in the XY plane, which
// should be correct for most gear arrangements. Add 50cm of
// "buffer" to keep things from blowing up with aircraft with a
// single gear very near the c.g. (AV-8, for example).
float total = 0;
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int i;
for(i=0; i<_gears.size(); i++) {
GearRec* gr = (GearRec*)_gears.get(i);
Gear* g = gr->gear;
g->getPosition(pos);
Math::sub3(cg, pos, pos);
gr->wgt = 1.0f/(0.5f+Math::sqrt(pos[0]*pos[0] + pos[1]*pos[1]));
total += gr->wgt;
}
// Renormalize so they sum to 1
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for(i=0; i<_gears.size(); i++)
((GearRec*)_gears.get(i))->wgt /= total;
// The force at max compression should be sufficient to stop a
// plane moving downwards at 3x the approach descent rate. Assume
// a 3 degree approach.
float descentRate = 3.0f*_approachSpeed/19.1f;
// Spread the kinetic energy according to the gear weights. This
// will results in an equal compression fraction (not distance) of
// each gear.
float energy = 0.5f*_approachWeight*descentRate*descentRate;
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for(i=0; i<_gears.size(); i++) {
GearRec* gr = (GearRec*)_gears.get(i);
float e = energy * gr->wgt;
float comp[3];
gr->gear->getCompression(comp);
float len = Math::mag3(comp);
// Energy in a spring: e = 0.5 * k * len^2
float k = 2 * e / (len*len);
gr->gear->setSpring(k);
// Critically damped (too damped, too!)
gr->gear->setDamping(2*Math::sqrt(k*_approachWeight*gr->wgt));
// These are pretty generic
gr->gear->setStaticFriction(0.8f);
gr->gear->setDynamicFriction(0.7f);
}
}
void Airplane::initEngines()
{
for(int i=0; i<_thrusters.size(); i++) {
ThrustRec* tr = (ThrustRec*)_thrusters.get(i);
tr->thruster->init();
}
}
void Airplane::stabilizeThrust()
{
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int i;
for(i=0; i<_thrusters.size(); i++)
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_model.getThruster(i)->stabilize();
}
void Airplane::runCruise()
{
setupState(_cruiseAoA, _cruiseSpeed, &_cruiseState);
_model.setState(&_cruiseState);
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_model.setAir(_cruiseP, _cruiseT);
// The control configuration
_controls.reset();
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int i;
for(i=0; i<_cruiseControls.size(); i++) {
Control* c = (Control*)_cruiseControls.get(i);
_controls.setInput(c->control, c->val);
}
_controls.applyControls(1000000); // Huge dt value
// The local wind
float wind[3];
Math::mul3(-1, _cruiseState.v, wind);
Math::vmul33(_cruiseState.orient, wind, wind);
// Cruise is by convention at 50% tank capacity
setFuelFraction(0.5);
// Set up the thruster parameters and iterate until the thrust
// stabilizes.
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for(i=0; i<_thrusters.size(); i++) {
Thruster* t = ((ThrustRec*)_thrusters.get(i))->thruster;
t->setWind(wind);
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t->setAir(_cruiseP, _cruiseT);
}
stabilizeThrust();
updateGearState();
// Precompute thrust in the model, and calculate aerodynamic forces
_model.getBody()->reset();
_model.initIteration();
_model.calcForces(&_cruiseState);
}
void Airplane::runApproach()
{
setupState(_approachAoA, _approachSpeed, &_approachState);
_model.setState(&_approachState);
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_model.setAir(_approachP, _approachT);
// The control configuration
_controls.reset();
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int i;
for(i=0; i<_approachControls.size(); i++) {
Control* c = (Control*)_approachControls.get(i);
_controls.setInput(c->control, c->val);
}
_controls.applyControls(1000000);
// The local wind
float wind[3];
Math::mul3(-1, _approachState.v, wind);
Math::vmul33(_approachState.orient, wind, wind);
// Approach is by convention at 20% tank capacity
setFuelFraction(0.2f);
// Run the thrusters until they get to a stable setting. FIXME:
// this is lots of wasted work.
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for(i=0; i<_thrusters.size(); i++) {
Thruster* t = ((ThrustRec*)_thrusters.get(i))->thruster;
t->setWind(wind);
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t->setAir(_approachP, _approachT);
}
stabilizeThrust();
updateGearState();
// Precompute thrust in the model, and calculate aerodynamic forces
_model.getBody()->reset();
_model.initIteration();
_model.calcForces(&_approachState);
}
void Airplane::applyDragFactor(float factor)
{
float applied = Math::sqrt(factor);
_dragFactor *= applied;
_wing->setDragScale(_wing->getDragScale() * applied);
_tail->setDragScale(_tail->getDragScale() * applied);
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int i;
for(i=0; i<_vstabs.size(); i++) {
Wing* w = (Wing*)_vstabs.get(i);
w->setDragScale(w->getDragScale() * applied);
}
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for(i=0; i<_surfs.size(); i++) {
Surface* s = (Surface*)_surfs.get(i);
s->setTotalDrag(s->getTotalDrag() * applied);
}
}
void Airplane::applyLiftRatio(float factor)
{
float applied = Math::sqrt(factor);
_liftRatio *= applied;
_wing->setLiftRatio(_wing->getLiftRatio() * applied);
_tail->setLiftRatio(_tail->getLiftRatio() * applied);
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int i;
for(i=0; i<_vstabs.size(); i++) {
Wing* w = (Wing*)_vstabs.get(i);
w->setLiftRatio(w->getLiftRatio() * applied);
}
}
float Airplane::clamp(float val, float min, float max)
{
if(val < min) return min;
if(val > max) return max;
return val;
}
float Airplane::normFactor(float f)
{
if(f < 0) f = -f;
if(f < 1) f = 1/f;
return f;
}
void Airplane::solve()
{
static const float ARCMIN = 0.0002909f;
float tmp[3];
_solutionIterations = 0;
_failureMsg = 0;
while(1) {
if(_solutionIterations++ > 10000) {
_failureMsg = "Solution failed to converge after 10000 iterations";
return;
}
// Run an iteration at cruise, and extract the needed numbers:
runCruise();
_model.getThrust(tmp);
float thrust = tmp[0];
_model.getBody()->getAccel(tmp);
float xforce = _cruiseWeight * tmp[0];
float clift0 = _cruiseWeight * tmp[2];
_model.getBody()->getAngularAccel(tmp);
float pitch0 = tmp[1];
// Run an approach iteration, and do likewise
runApproach();
_model.getBody()->getAngularAccel(tmp);
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float apitch0 = tmp[1];
_model.getBody()->getAccel(tmp);
float alift = _approachWeight * tmp[2];
// Modify the cruise AoA a bit to get a derivative
_cruiseAoA += ARCMIN;
runCruise();
_cruiseAoA -= ARCMIN;
_model.getBody()->getAccel(tmp);
float clift1 = _cruiseWeight * tmp[2];
// Do the same with the tail incidence
_tail->setIncidence(_tailIncidence + ARCMIN);
runCruise();
_tail->setIncidence(_tailIncidence);
_model.getBody()->getAngularAccel(tmp);
float pitch1 = tmp[1];
// Now calculate:
float awgt = 9.8f * _approachWeight;
float dragFactor = thrust / (thrust-xforce);
float liftFactor = awgt / (awgt+alift);
float aoaDelta = -clift0 * (ARCMIN/(clift1-clift0));
float tailDelta = -pitch0 * (ARCMIN/(pitch1-pitch0));
// Sanity:
if(dragFactor <= 0) {
_failureMsg = "Zero or negative drag adjustment.";
return;
} else if(liftFactor <= 0) {
_failureMsg = "Zero or negative lift adjustment.";
return;
}
// And the elevator control in the approach. This works just
// like the tail incidence computation (it's solving for the
// same thing -- pitching moment -- by diddling a different
// variable).
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const float ELEVDIDDLE = 0.0001f;
_approachElevator.val += ELEVDIDDLE;
runApproach();
_approachElevator.val -= ELEVDIDDLE;
_model.getBody()->getAngularAccel(tmp);
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float apitch1 = tmp[1];
float elevDelta = -apitch0 * (ELEVDIDDLE/(apitch1-apitch0));
// Now apply the values we just computed. Note that the
// "minor" variables are deferred until we get the lift/drag
// numbers in the right ballpark.
applyDragFactor(dragFactor);
applyLiftRatio(liftFactor);
// DON'T do the following until the above are sane
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if(normFactor(dragFactor) > 1.1
|| normFactor(liftFactor) > 1.1)
{
continue;
}
// OK, now we can adjust the minor variables:
_cruiseAoA += 0.5f*aoaDelta;
_tailIncidence += 0.5f*tailDelta;
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_approachElevator.val += 0.5f*elevDelta;
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_cruiseAoA = clamp(_cruiseAoA, -0.174f, 0.174f);
_tailIncidence = clamp(_tailIncidence, -0.174f, 0.174f);
_approachElevator.val = clamp(_approachElevator.val, -1.f, 1.f);
if(norm(dragFactor) < 1.00001 &&
norm(liftFactor) < 1.00001 &&
abs(aoaDelta) < .000017 &&
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abs(tailDelta) < .000017 &&
abs(elevDelta) < 0.00001)
{
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break;
}
}
if(_dragFactor < 1e-06 || _dragFactor > 1e6) {
_failureMsg = "Drag factor beyond reasonable bounds.";
return;
} else if(_liftRatio < 1e-04 || _liftRatio > 1e4) {
_failureMsg = "Lift ratio beyond reasonable bounds.";
return;
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} else if(Math::abs(_cruiseAoA) >= .174) {
_failureMsg = "Cruise AoA > 10 degrees";
return;
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} else if(Math::abs(_tailIncidence) >= .174) {
_failureMsg = "Tail incidence > 10 degrees";
return;
}
}
}; // namespace yasim