Racing Game Custom Physics Simulation (Part 2)

Preface

In part 1 we have shown how to make a car(truck) glide on a track made out of 3D triangles.

We were able to steer the car and it was “glued” to the track to make it seem like there are physics involved.

In this part we are going to have an actual physics simulation(to some degree).

The Models

The model of the track remains the same as in part 1(a set of 3D triangles).

The car model is made of 5 points: 4 for the wheels and 1 for the bottom center. Just like in part 1.

However, we will also maintain variables to keep the orientation of the car and the gravity velocity of each of the 4 wheels(in addition to the car’s gravity velocity).

Steering

Since we now maintain the orientation of the car from the previous frame we can’t just assume the steering vector is calculated aligned to the xz plane.

To calculate the new steering vector we rotate the Look and Right vectors by a delta of an angle on the Look x Right plane:

				Look = LastRight*sin(DeltaAngle)+ LastLook*cos(DeltaAngle);
				Right = LastRight*cos(DeltaAngle)- LastLook*sin(DeltaAngle);

The delta of the angle is the angular speed multiplied by the frame’s time delta.

The forward velocity is calculated the same as in part 1 but notice the Look vector might not be on the xz plane this time.

Gravity

This time we are also going to take gravity into account.

Our gravity is the following vector (0, -9.8, 0).

It is a vector pointing downwards with an acceleration of 9.8 meters per square second.

We compose the car’s velocity from two components. The steering velocity from above and the Gravity velocity.

The reason we separate the two is because it would be easier to set the gravity velocity to 0 whenever a point in the model hits the ground or track.

We also need to maintain a separate gravity velocity for each of the 5 points in the car model to make the physics simulation of the car work as if we had an object with volume.

The Simulation

After we calculated the Look and Right orientation vectors and the steering velocity vector we need to apply gravity.

For each of the 5 gravity velocity vectors we add the Gravity vector multiplied by the frame’s time delta.

				CurrentGravityVelocity=CurrentGravityVelocity+Gravity*t;
				for (unsigned int i=0; i<WheelsGravityVelocity.size(); i++)
					WheelsGravityVelocity[i]=WheelsGravityVelocity[i]+Gravity*t;

The next step is to calculate the current position of the wheels and the position of the car center.

For the car center position, we add the current steering velocity vector and the current gravity velocity vector multiplied by the frame’s time delta.

Like so:

				Pos = Pos+(Look*CurrentFlatSpeed+CurrentGravityVelocity)*t;

For each of the 4 wheels we calculate the wheel’s position relative to the car’s center using the wheels base dimensions and the Look and Right orientation vectors.

We then add the car’s center position to each of the 4 wheels positions.

The last step is to add the steering velocity and the gravity velocity multiplied by the time delta. The steering velocity is the same as it was for the car’s center but notice the gravity velocity might be unique for each wheel point(We saved them in their own variables).

				for (unsigned int i=0; i<Wheels.size(); i++)
					Wheels[i] = Right*WheelsDelta[i].x+Look*WheelsDelta[i].z+Pos+(Look*CurrentFlatSpeed+WheelsGravityVelocity[i])*t;

In a similar fashion to part 1 we now check which triangles each one of the 5 points of the model intersect with.

The difference is that instead of setting the points to the intersection height of the respective triangles, we only set them to the intersection height in case their own height is lower.

In addition we set to zero the Gravity velocity of each point only if  it’s height was adjusted by a triangle.

Now we have 5 points of the car displaced into new heights. The car’s center point will be used for the new position, but like in part 1, we need to calculate the new orientation from the 4 wheel points.

The current car model simulates a rigid body. However while adjusting the wheels’ heights the car’s wheel base is now deformed.

We will restore the original form of the wheel base by treating the 4 wheels as if they have springs among themselves(a total of 6 springs).

This will make the 4 wheel points simulate the wheels base as a if it was a rigid body.

In order to restore the original form of the wheels base we go over all the 6 springs and adjust them to be closer to their original length.

We iterate over this process for ten times and at the end we would get something closer to the original form.

				for (unsigned int k=0; k<10; k++)
				{
					for (unsigned int i=0; i<Wheels.size(); i++)
						for (unsigned int j=i+1; j<Wheels.size(); j++)
						{
							Graphics2D::Position v = Wheels[i]-Wheels[j];
							Graphics2D::Position center = (Wheels[i]+Wheels[j])*0.5;
							double l = (WheelsDelta[i]-WheelsDelta[j]).Length();
							double radius = (0.1*l+0.9*v.Length())/2.0;
							Wheels[i] = (Wheels[i]-center).Normalize()*radius+center;
							Wheels[j] = (Wheels[j]-center).Normalize()*radius+center;
						}
				}

Now that we have the wheel points placed on the wheels base frame we can calculate the new Look and Right orientation vectors in a similar fashion we did in part 1.

				Look = (Wheels[0]-Wheels[3]).Normalize();
				Right = (Wheels[1]-Wheels[0]).Normalize();

Conclusion

We now have a more physically based simulation that also support falling off from edges.

The result of this simulation can be seen in this video:

For the sake of completion I am adding the entire code for the update function.

 

			void Update (double t)
			{
				double WheelFactor = 0;
				if (Input.GetLeft())
					WheelFactor = -1;
				else if (Input.GetRight())
					WheelFactor = 1;
				if (Input.GetThrust())
					CurrentFlatSpeed += MaxFlatSpeed*t/AccelLatency;
				CurrentFlatSpeed = std::max(std::min(CurrentFlatSpeed, MaxFlatSpeed), 0.0);

				double DeltaAngle = WheelFactor*t;
				Graphics2D::Position Look = LastRight*sin(DeltaAngle)+ LastLook*cos(DeltaAngle);
				Graphics2D::Position Right = LastRight*cos(DeltaAngle)- LastLook*sin(DeltaAngle);
				CurrentGravityVelocity=CurrentGravityVelocity+Gravity*t;
				for (unsigned int i=0; i<WheelsGravityVelocity.size(); i++)
					WheelsGravityVelocity[i]=WheelsGravityVelocity[i]+Gravity*t;
				CurrentVelocity = Look*CurrentFlatSpeed+CurrentGravityVelocity;
				CarParms->SetLook (Look, Graphics2D::Position(0, 1, 0));
				std::vector<Graphics2D::Position> Wheels;
				Wheels.resize(WheelsDelta.size());
				for (unsigned int i=0; i<Wheels.size(); i++)
					Wheels[i] = Right*WheelsDelta[i].x+Look*WheelsDelta[i].z+Pos+(Look*CurrentFlatSpeed+WheelsGravityVelocity[i])*t;

				Pos = Pos+CurrentVelocity*t;
				if (Pos.y<0.0)
					CurrentGravityVelocity.y = std::max(0., CurrentGravityVelocity.y);
				Pos.y = std::max(0.0, Pos.y);

				unsigned int StartX = std::min((unsigned int)(std::max((Pos.x-Min.x)/(Max.x-Min.x), 0.0)), TrackGrid[0].size()-1);
				unsigned int StartZ = std::min((unsigned int)(std::max((Pos.z-Min.z)/(Max.z-Min.z), 0.0)), TrackGrid.size()-1);

				std::list<unsigned int>::iterator q;
				for (q = TrackGrid[StartZ][StartX].begin(); q != TrackGrid[StartZ][StartX].end(); q++)
				{
					const math::Ray r(float3(Pos.x, 100.0, Pos.z), float3(0, -1, 0));
					float d = 0;
					math::float3 Point;
					if (TrackGeometry[*q].Intersects(r, &d, &Point))
					{
						if (r.pos.y-d>=Pos.y)
						{
							CurrentGravityVelocity.y = std::max(0., CurrentGravityVelocity.y);
							Pos.y = r.pos.y-d;
						}
					}
				}
				CarParms->SetPosition (Pos);
//				std::vector<bool> IsWheelContact;
//				IsWheelContact.resize(4, false);
				for (unsigned int i=0; i<Wheels.size(); i++)
				{
					Graphics2D::Position p = Wheels[i];
					unsigned int StartX = std::min((unsigned int)(std::max((p.x-Min.x)/(Max.x-Min.x), 0.0)), TrackGrid[0].size()-1);
					unsigned int StartZ = std::min((unsigned int)(std::max((p.z-Min.z)/(Max.z-Min.z), 0.0)), TrackGrid.size()-1);

					std::list<unsigned int>::iterator q;
					for (q = TrackGrid[StartZ][StartX].begin(); q != TrackGrid[StartZ][StartX].end(); q++)
					{
						const math::Ray r(float3(p.x, 100.0, p.z), float3(0, -1, 0));
						float d = 0;
						math::float3 Point;
						if (TrackGeometry[*q].Intersects(r, &d, &Point))
						{
							if (r.pos.y-d>=Wheels[i].y)
							{
								WheelsGravityVelocity[i].y = std::max(0., WheelsGravityVelocity[i].y);
//								IsWheelContact[i] = true;
								Wheels[i].y = r.pos.y-d;
							}
						}
					}
					if (FloorHeight>=Wheels[i].y)
					{
						WheelsGravityVelocity[i].y = std::max(0., WheelsGravityVelocity[i].y);
//						IsWheelContact[i] = true;
						Wheels[i].y = FloorHeight;
					}
				}
				for (unsigned int k=0; k<10; k++)
				{
					for (unsigned int i=0; i<Wheels.size(); i++)
						for (unsigned int j=i+1; j<Wheels.size(); j++)
						{
							Graphics2D::Position v = Wheels[i]-Wheels[j];
							Graphics2D::Position center = (Wheels[i]+Wheels[j])*0.5;
							double l = (WheelsDelta[i]-WheelsDelta[j]).Length();
							double radius = (0.1*l+0.9*v.Length())/2.0;
							Wheels[i] = (Wheels[i]-center).Normalize()*radius+center;
							Wheels[j] = (Wheels[j]-center).Normalize()*radius+center;
						}
				}
				Look = (Wheels[0]-Wheels[3]).Normalize();
				Right = (Wheels[1]-Wheels[0]).Normalize();
				LastLook = Look;
				LastRight = Right;
				Graphics2D::Position Up = Look.Cross(Right);
				CarParms->SetLook(Look, Up);
			}

Simple Truck Racing Physics(Part 1)

Preface

I am working on a new 3D racing game.

For this racing game I need a track with mounds, hills and ramps.

I am going to cover my progress in making this racing game’s physics simulation.

The Models

At this point the track is made of a series of 3D triangles.

The 3D triangles might be constructed in a way that they form a road with mounds, turns or slopes but they don’t have to.

At this point the track geometry is used for both rendering and representing the terrain geometry in the simulation.

The track dimensions I used for testing are 110×110 square meters.

We also have the truck which has a 3D model representing it visually.

Inside the simulation the truck is made out of 5 points. The center bottom of the truck and 4 more points representing the wheels.

The truck’s size is a 2x2x5 cubic meters box.

The wheels base is 1.8×4.4 square meters.

Steering

For the steering of the truck I am saving the truck’s absolute direction inside a single angle.

I calculate the Look vector from the angle like this:

				Look = Graphics2D::Position(sin(CarAngle), 0, cos(CarAngle));

When I want the truck to rotate I add an angular speed multiplied by the frame’s time to the angle I mentioned above.

I then recalculate the Look vector every new frame.

In order for the truck to move forward we need to add the movement vector to the truck’s current position.

The truck’s movement vector is calculated like this:

				Move = Look*CurrentFlatSpeed*t;
				Pos = Pos+Move;

We don’t want the truck’s speed to accelerate instantaneously so we add the maximum speed multiplied by the frame’s time step but divided by the latency we want it to take to reach maximum speed.

				if (Input.GetThrust())
					CurrentFlatSpeed += MaxFlatSpeed*t/AccelLatency;
				CurrentFlatSpeed = std::max(std::min(CurrentFlatSpeed, MaxFlatSpeed), 0.0);

Terrain checks

At this point we can drive and steer the truck but we are completely ignoring the track(or terrain).

In order for the truck to “glide” on the terrain we will go over every triangle in our track mesh and test to see if the (x, z) part of the center bottom of the truck is inside the projection of the triangle on the xz plane.

(The center bottom of the truck is actually it’s position).

In order to test that we use a ray to triangle intersection test while the ray is from (truck position X, 1000, truck position Z) to (truck position X, 0, truck position Z).

If the ray intersects the triangle then the truck’s center is inside the projection of the triangle. We can then extract the height of the intersection between the ray and the triangle and use that as the new height(y axis value) of our truck.

(For the ray/triangle intersection we use MathGeoLib by clb).

This will make our truck go over the track’s topology but the truck will remain aligned as if it was on a flat surface.

In order to recalculate the truck’s alignment we do the same test we did with the truck’s center but with the 4 wheels instead.

Before we do that we calculate the absolute position of the 4 truck wheels from the truck’s wheels base rotated by the truck’s steering angle and added to the truck’s center bottom. Like so:

 

				Look = Graphics2D::Position(sin(CarAngle), 0, cos(CarAngle));
				Right = Graphics2D::Position(0, 1, 0).Cross(Look);
				for (unsigned int i=0; i<4; i++)
					WheelPos[i] = Right*WheelBase[i].x+Look*WheelBase[i].z+Pos;

We now do the same calculation over all the triangles and calculate the new height for each of the 4 wheels.

We then calculate the new Look and Right vectors of the truck from two vectors.

The Look vector will be the vector pointing from the rear left wheel to the front left wheel and the Right vector will be the vector pointing from the front left wheel to the front right wheel.

Don’t forget we want the normalized vectors.

				Look = (WheelPos[0]-WheelPos[3]).Normalize();
				Right = (WheelPos[1]-WheelPos[0]).Normalize();

That’s it. This will give us the following simulation result.

Optimizations

You probably noticed that we went through all the triangles in the track for each of the 5 points in the truck model.

This might be problematic to the performance and most of the triangles won’t intersect with the truck model.

In order to optimize this we prepare a 2D array where each array cell contains a linked list.

The 2D array represents a grid on the xz plane. The grid divides the plane into squares.

Each cell of the 2D array contains a list of all the triangles that their xz plane Axis Aligned Bounding Square intersects with the square in the grid that the cell represents.

This way every square in the grid will have a list that will contain all the triangles that intersect with the square(and maybe a little bit more that don’t).

So every time we want to test a point in the truck model against the track’s triangles we only need to test it against the triangles in the list of the square the point is at.

For the sake of completion here is the code to calculate a 10 by 10 triangle test optimization grid:

 

				std::vector<math::Triangle> TrackGeometry;
				std::vector<std::vector<std::list<unsigned int> > > TrackGrid;

				std::vector<Graphics2D::Position> & Positions = TrackMesh->GetPosition(0);
				std::vector<unsigned int> & Indices = TrackMesh->GetIndex(0);
				TrackGrid.resize(10);
				for (unsigned int i=0; i<TrackGrid.size(); i++)
					TrackGrid[i].resize(10);
				TrackGeometry.resize (Indices.size()/3);
				Min = Positions[0];
				Max = Positions[0];
				for (unsigned int i=0; i<Positions.size(); i++)
				{
					Min.x = std::min(Min.x, Positions[i].x);
					Min.y = std::min(Min.y, Positions[i].y);
					Min.z = std::min(Min.z, Positions[i].z);
					Max.x = std::max(Max.x, Positions[i].x);
					Max.y = std::max(Max.y, Positions[i].y);
					Max.z = std::max(Max.z, Positions[i].z);
				}
				for (unsigned int i=0; i<TrackGeometry.size(); i++)
				{
					Graphics2D::Position LocalMin, LocalMax;
					LocalMin = Positions[Indices[i*3]];
					LocalMax = Positions[Indices[i*3]];
					for (unsigned int k=1; k<3; k++)
					{
						LocalMin.x = std::min(LocalMin.x, Positions[Indices[i*3+k]].x);
						LocalMin.z = std::min(LocalMin.z, Positions[Indices[i*3+k]].z);
						LocalMax.x = std::max(LocalMax.x, Positions[Indices[i*3+k]].x);
						LocalMax.z = std::max(LocalMax.z, Positions[Indices[i*3+k]].z);
					}
					TrackGeometry[i].a = float3(Positions[Indices[i*3]].x, Positions[Indices[i*3]].y, Positions[Indices[i*3]].z);
					TrackGeometry[i].b = float3(Positions[Indices[i*3+1]].x, Positions[Indices[i*3+1]].y, Positions[Indices[i*3+1]].z);
					TrackGeometry[i].c = float3(Positions[Indices[i*3+2]].x, Positions[Indices[i*3+2]].y, Positions[Indices[i*3+2]].z);
					unsigned int StartX = std::min((unsigned int)(std::max((LocalMin.x-Min.x)/(Max.x-Min.x), 0.0)), TrackGrid[0].size()-1);
					unsigned int StartZ = std::min((unsigned int)(std::max((LocalMin.z-Min.z)/(Max.z-Min.z), 0.0)), TrackGrid.size()-1);
					unsigned int EndX = std::min((unsigned int)(std::max((LocalMax.x-Min.x)/(Max.x-Min.x), 0.0)), TrackGrid[0].size()-1);
					unsigned int EndZ = std::min((unsigned int)(std::max((LocalMax.z-Min.z)/(Max.z-Min.z), 0.0)), TrackGrid.size()-1);
					for (unsigned int z1=StartZ; z1<=EndZ; z1++)
						for (unsigned int x1=StartX; x1<=EndX; x1++)
							TrackGrid[z1][x1].push_front(i);
				}
				WheelsDelta.resize(4);
				WheelsDelta[0] = Graphics2D::Position(-0.9, 0, 2.2);
				WheelsDelta[1] = Graphics2D::Position(0.9, 0, 2.2);
				WheelsDelta[2] = Graphics2D::Position(0.9, 0, -2.2);
				WheelsDelta[3] = Graphics2D::Position(-0.9, 0, -2.2);

 

What’s next?

Our current simulation doesn’t have much of a physics feel to it.

The car is basically glued to the terrain. We also don’t deal with ledges.

In part 2 the simulation will get more interesting.