Friction using delta time returns inconsistent result

Question

Slowing down a character using the following formula returns different results depending on the frame rate:

this.y += this.yVel * delta;
this.yVel *= Math.pow(0.99, delta);

Is there a better way to decelerate a player using delta time to allow for consistent movement across several framerates?

Answer 1

Normally, fixed delta time is used in physics simulation cause in this reason. On the other hand, frame base delta time is useful for rendering.

Of course, fixed delta base calculation result is make diff frame time object position. So you will need interpolation.

Answer 2

This line assumes that yVel is constant for the entirety of the duration delta:

this.y += this.yVel * delta;

But that's not true - we know by the end of duration delta that yVel has decreased to a fraction of \$0.99^\text{delta}\$ according to this line:

this.yVel *= Math.pow(0.99, delta);

So if we think of yVel as a function of time, \$v(t) = v_0 \cdot 0.99^t\$, then the total displacement in y over the course of the interval delta is the integral:

$$\begin{align} \Delta y &= \int_0^\text{delta}v_0 \cdot 0.99^t dt\\ &= v_0 \cdot \int_0^\text{delta} e^{\ln (0.99) \cdot t} dt\\ &= v_0 \cdot \left( \frac 1 {\ln(0.99)} e^{\ln(0.99) \cdot t}\right)\Bigg|^\text{delta}_0\\ &= \frac {v_0} {\ln(0.99)}\left(e^{\ln(0.99)\cdot \text{delta}} - 1 \right)\\ &=v_0 \cdot \frac {0.99^\text{delta} - 1} {\ln(0.99)} \end{align}$$

In code:

var frictionFactor = Math.Pow(0.99, delta);
this.y += this.yVel * (frictionFactor - 1)/Math.Log(0.99);
this.yVel *= frictionFactor;

This will be smoother, but can still vary from one run to another due to limited precision and accumulation of rounding errors. So as Shin Dongho mentioned, your best bet here is to fix your physics timestep at a known value, so you get consistent, reproducible results.