Spring.lua

-- Simulates the motion of a critically damped spring
-- @author fractality

--[[
MIT License

Copyright (c) 2020 Parker Stebbins

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
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furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
--]]
------------------------------------------------------------------------
-- API:
--   Spring Spring.new(double damp, double freq, vector pos)
--   void Spring:SetGoal(vector goal)
--   void Spring:SetFrequency(double freq)
--   void Spring:SetDampingRatio(double damp)
--   vector Spring:GetPosition()
--   vector Spring:GetVelocity()
--   vector Spring:Update(double dt)
--   void Spring:Reset(vector state)
--
-- Notes:
--   The state vector type must implement the following metamethods:
--     vector __mul(vector, double)
--     vector __add(vector, vector)
--     vector __sub(vector, vector)
------------------------------------------------------------------------

local Spring = {}
Spring.__index = Spring

local pi = math.pi
local exp = math.exp
local sin = math.sin
local cos = math.cos
local sqrt = math.sqrt

local EPS = 1e-4

function Spring.new(dampingRatio, frequency, position)
	assert(type(dampingRatio) == "number")
	assert(type(frequency) == "number")
	assert(dampingRatio * frequency >= 0, "Spring does not converge")

	return setmetatable(
		{
			d = dampingRatio,
			f = frequency,
			g = position,
			p = position,
			v = position * 0 -- Match the original vector type
		},
		Spring
	)
end

function Spring:Reset(position)
	self.p = position
	self.v = position * 0
end

function Spring:SetGoal(newGoal)
	self.g = newGoal
end

function Spring:SetFrequency(newFreq)
	self.f = newFreq
end

function Spring:SetDampingRatio(newDamp)
	self.d = newDamp
end

function Spring:GetGoal()
	return self.g
end

function Spring:GetPosition()
	return self.p
end

function Spring:GetVelocity()
	return self.v
end

function Spring:Update(dt)
	local d = self.d
	local f = self.f * 2 * pi
	local g = self.g
	local p0 = self.p
	local v0 = self.v

	local offset = p0 - g
	local decay = exp(-d * f * dt)

	local p1, v1

	if d == 1 then -- Critically damped
		p1 = (offset * (1 + f * dt) + v0 * dt) * decay + g
		v1 = (v0 * (1 - f * dt) - offset * (f * f * dt)) * decay
	elseif d < 1 then -- Underdamped
		local c = sqrt(1 - d * d)

		local i = cos(f * c * dt)
		local j = sin(f * c * dt)

		-- Damping ratios approaching 1 can cause division by small numbers.
		-- To fix that, group terms around z=j/c and find an approximation for z.
		-- Start with the definition of z:
		--    z = sin(dt*f*c)/c
		-- Substitute a=dt*f:
		--    z = sin(a*c)/c
		-- Take the Maclaurin expansion of z with respect to c:
		--    z = a - (a^3*c^2)/6 + (a^5*c^4)/120 + O(c^6)
		--    z ≈ a - (a^3*c^2)/6 + (a^5*c^4)/120
		-- Rewrite in Horner form:
		--    z ≈ a + ((a*a)*(c*c)*(c*c)/20 - c*c)*(a*a*a)/6

		local z
		if c > EPS then
			z = j / c
		else
			local a = dt * f
			z = a + ((a * a) * (c * c) * (c * c) / 20 - c * c) * (a * a * a) / 6
		end

		-- Frequencies approaching 0 present a similar problem.
		-- We want an approximation for y as f approaches 0, where:
		--    y = sin(dt*f*c)/(f*c)
		-- Substitute b=dt*c:
		--    y = sin(b*c)/b
		-- Now reapply the process from z.

		local y
		if f * c > EPS then
			y = j / (f * c)
		else
			local b = f * c
			y = dt + ((dt * dt) * (b * b) * (b * b) / 20 - b * b) * (dt * dt * dt) / 6
		end

		p1 = (offset * (i + d * z) + v0 * y) * decay + g
		v1 = (v0 * (i - z * d) - offset * (z * f)) * decay
	else -- Overdamped
		local c = sqrt(d * d - 1)

		local r1 = -f * (d - c)
		local r2 = -f * (d + c)

		local co2 = (v0 - offset * r1) / (2 * f * c)
		local co1 = offset - co2

		local e1 = co1 * exp(r1 * dt)
		local e2 = co2 * exp(r2 * dt)

		p1 = e1 + e2 + g
		v1 = e1 * r1 + e2 * r2
	end

	self.p = p1
	self.v = v1

	return p1
end

return Spring