TransitionDynamics::dynamicsShape definitions

The user guide should contain more explanation on how calculate values for TransitionDynamics. Here I have summarized (after derivation) some formulas to be used. The "step" shape is not considered since it is an instantaneous change.

When dynamicsDimension == “time” OR dynamicsDimension == “distance”:

Independent dimension -> X(time or distance)

Dependent dimension -> Y

Initial state -> i

Final state -> f

value -> V = X_f - X_i

Y_i = Y(X_i)

Y_f = Y(X_f)

Rate of change at initial state -> (dY/dX)_i = initRate (Propose to use "initRate" attribute when dynamicsShape == "cubic")

Rate of change at final state -> (dY/dX)_f = 0

1. Case: dynamicsShape == “linear”

   Y = A(X - X_i) + B

   Where

   B = Y_i

   A=((Y_f - Y_i))/V

2. Case: dynamicsShape == “sinusoidal”

   V = X_f - X_i = value

   Y = A.sin⁡(B(X - X_i)) + C

   dY/dX = A.B.cos⁡(B(X - X_i))

   Where

   C = Y_i

   B = ((2n-1)π)/2V ∀nϵZ^+

   A = (Y_f - Y_i)/sin⁡(((2n-1)π)/2)

   It should be noted that these values also define the initial rate of change of Y.

   (dY/dX)_i = ((Y_f - Y_i)/sin⁡(((2n-1)π)/2) )(((2n-1)π)/2V).cos⁡((((2n-1)π)/2V)(X - X_i))

3. Case: dynamicsShape == ”cubic”

   Y = A(X - X_i)³ + B(X - X_i)² + C(X - X_i) + D

   dY/dX = 3A(X - X_i)² + 2B(X - X_i) + C

   Where

   D = Y_i

   C = (dY/dX)_i

   B = (Y_f - Y_i - (2V/3).(dY/dX)_i)/(V²/3)

   A = (Y_f - Y_i - (V/2).(dY/dX)_i)/(-V³/2)

   As it can be seen, cubic needs an additional parameter (initial rate of change (dY/dX)_i) to be completely defined.

When dynamicsDimension == “rate”:

The rate implies the rate of change w.r.t to time (t).

value -> V = dY/dt

Independent dimension -> t

Dependent dimension -> Y

Initial state -> i

Final state -> f

Since only rate is given, time at final state can be calculated for each dynamicsShape.

1. Case: dynamicsShape == “linear”

   The rate is constant with time.

   V = value = (dY/dt)

   Y = Y_i + V(t - t_i)

2. Case: dynamicsShape == “sinusoidal”

   The rate is at the point of inflexion. Initial and final rates are assumed to be zero.

   Y = A.sin⁡(B(t - t_i)) + C

   dY/dt = A.B.cos⁡(B(t - t_i))

   where

   t_f = t_i + π(Y_f - Y_i)/(V.2.sqrt(2))

   C = Y_i

   B = π/(2(t_f - t_i))

   A = Y_f - Y_i

3. Case: dynamicsShape == “cubic”

   The rate is at the point of inflexion. Initial and final rates are assumed to be zero.

   Y = A(t - t_i)³ + B(t - t_i)² + C(t - t_i) + D

   dY/dt = 3A(t - t_i)² + 2B(t - t_i) + C

   Where

   t_f = t_i + (Y_f - Y_i)/(2V/3)

   D = Y_i

   C = (dY/dt)_i = 0

   B = 2V/(t_f - t_i)

   A = -4V/3(t_f - t_i)²

These equations could be prone to error since they were derived manually, hence, re-checking is necessary.