Acceleration in Unity

I am trying to emulate acceleration and deceleration in Unity.

I have written to code to generate a track in Unity and place an object at a specific location on the track based on time. The result looks a little like this. The issue I currently have is that each section of the spline is a different length and the cube moves across each section at a different, but uniform, speed. This causes there to be sudden jumps in the change of the speed of the cube when transitioning between sections.

In order to try and fix this issue, I attempted to use Robert Penner's easing equations on the GetTime(Vector3 p0, Vector3 p1, float alpha) method. However, whilst this did help somewhat, it was not sufficient. There were still jumps in speed in between transitions.

Does anyone have any ideas on how I could dynamically ease the position of the cube to make it look like it was accelerating and decelerating, without large jumps in speed between segments of the track?

I have written a script that shows a simple implementation of my code. It can be attached to any game object. To make it easy to see what is happening when the code runs, attach to something like a cube or sphere.

using System.Collections.Generic;
using UnityEngine;
#if UNITY_EDITOR
using UnityEditor;
#endif

public class InterpolationExample : MonoBehaviour {
[SerializeField]
private float currentTime;
private float lastTime = 0;
[SerializeField]
private float timeModifier = 1;
[SerializeField]
private bool running = true;
private bool runningBuffer = true;

[SerializeField]
[Range(0, 1)]
private float catmullRomAlpha = 0.5f;
[SerializeField]
private List<SimpleWayPoint> wayPoints = new List<SimpleWayPoint>
{
new SimpleWayPoint() {pos = new Vector3(-4.07f, 0, 6.5f), time = 0},
new SimpleWayPoint() {pos = new Vector3(-2.13f, 3.18f, 6.39f), time = 1},
new SimpleWayPoint() {pos = new Vector3(-1.14f, 0, 4.55f), time = 6},
new SimpleWayPoint() {pos = new Vector3(0.07f, -1.45f, 6.5f), time = 7},
new SimpleWayPoint() {pos = new Vector3(1.55f, 0, 3.86f), time = 7.2f},
new SimpleWayPoint() {pos = new Vector3(4.94f, 2.03f, 6.5f), time = 10}
};

[SerializeField]
private bool debugWayPoints = true;
[SerializeField]
private WayPointDebugType debugWayPointType = WayPointDebugType.SOLID;
[SerializeField]
private float debugWayPointSize = 0.2f;
[SerializeField]
private Color debugWayPointColour = Color.green;
[SerializeField]
private bool debugTrack = true;
[SerializeField]
[Range(0, 1)]
private float debugTrackResolution = 0.04f;
[SerializeField]
private Color debugTrackColour = Color.red;

[System.Serializable]
private class SimpleWayPoint
{
public Vector3 pos;
public float time;
}

[System.Serializable]
private enum WayPointDebugType
{
SOLID,
WIRE
}

private void Start()
{
wayPoints.Sort((x, y) => x.time.CompareTo(y.time));
wayPoints.Insert(0, wayPoints);
}

private void LateUpdate()
{
//This means that if currentTime is paused, then resumed, there is not a big jump in time
if(runningBuffer != running)
{
runningBuffer = running;
lastTime = Time.time;
}

if(running)
{
currentTime += (Time.time - lastTime) * timeModifier;
lastTime = Time.time;
if(currentTime > wayPoints[wayPoints.Count - 1].time)
{
currentTime = 0;
}
}
transform.position = GetPosition(currentTime);
}

#region Catmull-Rom Math
public Vector3 GetPosition(float time)
{
//Check if before first waypoint
if(time <= wayPoints.time)
{
return wayPoints.pos;
}
//Check if after last waypoint
else if(time >= wayPoints[wayPoints.Count - 1].time)
{
return wayPoints[wayPoints.Count - 1].pos;
}

//Check time boundaries - Find the nearest WayPoint your object has passed
float minTime = -1;
float maxTime = -1;
int minIndex = -1;
for(int i = 1; i < wayPoints.Count; i++)
{
if(time > wayPoints[i - 1].time && time <= wayPoints[i].time)
{
maxTime = wayPoints[i].time;
int index = i - 1;
minTime = wayPoints[index].time;
minIndex = index;
}
}

float timeDiff = maxTime - minTime;
float percentageThroughSegment = 1 - ((maxTime - time) / timeDiff);

//Define the 4 points required to make a Catmull-Rom spline
Vector3 p0 = wayPoints[ClampListPos(minIndex - 1)].pos;
Vector3 p1 = wayPoints[minIndex].pos;
Vector3 p2 = wayPoints[ClampListPos(minIndex + 1)].pos;
Vector3 p3 = wayPoints[ClampListPos(minIndex + 2)].pos;

return GetCatmullRomPosition(percentageThroughSegment, p0, p1, p2, p3, catmullRomAlpha);
}

//Prevent Index Out of Array Bounds
private int ClampListPos(int pos)
{
if(pos < 0)
{
pos = wayPoints.Count - 1;
}

if(pos > wayPoints.Count)
{
pos = 1;
}
else if(pos > wayPoints.Count - 1)
{
pos = 0;
}

return pos;
}

//Math behind the Catmull-Rom curve. See here for a good explanation of how it works. https://stackoverflow.com/a/23980479/4601149
private Vector3 GetCatmullRomPosition(float t, Vector3 p0, Vector3 p1, Vector3 p2, Vector3 p3, float alpha)
{
float dt0 = GetTime(p0, p1, alpha);
float dt1 = GetTime(p1, p2, alpha);
float dt2 = GetTime(p2, p3, alpha);

Vector3 t1 = ((p1 - p0) / dt0) - ((p2 - p0) / (dt0 + dt1)) + ((p2 - p1) / dt1);
Vector3 t2 = ((p2 - p1) / dt1) - ((p3 - p1) / (dt1 + dt2)) + ((p3 - p2) / dt2);

t1 *= dt1;
t2 *= dt1;

Vector3 c0 = p1;
Vector3 c1 = t1;
Vector3 c2 = (3 * p2) - (3 * p1) - (2 * t1) - t2;
Vector3 c3 = (2 * p1) - (2 * p2) + t1 + t2;
Vector3 pos = CalculatePosition(t, c0, c1, c2, c3);

return pos;
}

private float GetTime(Vector3 p0, Vector3 p1, float alpha)
{
if(p0 == p1)
return 1;
return Mathf.Pow((p1 - p0).sqrMagnitude, 0.5f * alpha);
}

private Vector3 CalculatePosition(float t, Vector3 c0, Vector3 c1, Vector3 c2, Vector3 c3)
{
float t2 = t * t;
float t3 = t2 * t;
return c0 + c1 * t + c2 * t2 + c3 * t3;
}

//Utility method for drawing the track
private void DisplayCatmullRomSpline(int pos, float resolution)
{
Vector3 p0 = wayPoints[ClampListPos(pos - 1)].pos;
Vector3 p1 = wayPoints[pos].pos;
Vector3 p2 = wayPoints[ClampListPos(pos + 1)].pos;
Vector3 p3 = wayPoints[ClampListPos(pos + 2)].pos;

Vector3 lastPos = p1;
int maxLoopCount = Mathf.FloorToInt(1f / resolution);

for(int i = 1; i <= maxLoopCount; i++)
{
float t = i * resolution;
Vector3 newPos = GetCatmullRomPosition(t, p0, p1, p2, p3, catmullRomAlpha);
Gizmos.DrawLine(lastPos, newPos);
lastPos = newPos;
}
}
#endregion

private void OnDrawGizmos()
{
#if UNITY_EDITOR
if(EditorApplication.isPlaying)
{
if(debugWayPoints)
{
Gizmos.color = debugWayPointColour;
foreach(SimpleWayPoint s in wayPoints)
{
if(debugWayPointType == WayPointDebugType.SOLID)
{
Gizmos.DrawSphere(s.pos, debugWayPointSize);
}
else if(debugWayPointType == WayPointDebugType.WIRE)
{
Gizmos.DrawWireSphere(s.pos, debugWayPointSize);
}
}
}

if(debugTrack)
{
Gizmos.color = debugTrackColour;
if(wayPoints.Count >= 2)
{
for(int i = 0; i < wayPoints.Count; i++)
{
if(i == 0 || i == wayPoints.Count - 2 || i == wayPoints.Count - 1)
{
continue;
}

DisplayCatmullRomSpline(i, debugTrackResolution);
}
}
}
}
#endif
}
}

• Is your ultimate goal more uniform movement along the curve? – wondra Jun 6 '18 at 20:53
• @wondra I am not looking to have it move at a constant speed over the whole track if that is what you mean. I would be more looking for smooth transitions in between the segments of the track instead of sudden speed jumps like there are currently if you try that script – Dan Jun 6 '18 at 21:40
• @wondra I am happy for the different segments to have different speeds as I need the object to be at the defined points at the defined times. I just want to make it look smoother than it currently is – Dan Jun 6 '18 at 21:45
• You could do some lerping, e.g. along the lines of transform.position = Vector3.Lerp(transform.position, GetPosition(currentTime), Time.deltaTime); – ElDuderino Jun 7 '18 at 14:57
• @ElDuderino Thanks for the comment, whilst that definitely made it more smooth and transition nicely, the object deviated from the track quite significantly which unfortunately means I can not use that approach – Dan Jun 7 '18 at 16:26