add gs

RCCService2018/content/LuaPackages/OtterImpl/spring.lua

@@ -0,0 +1,148 @@
+--[[
+	An analytical spring solution as a function of damping ratio and frequency.
+
+	Adapted from
+	https://gist.github.com/Fraktality/1033625223e13c01aa7144abe4aaf54d
+]]
+
+local assign = require(script.Parent.assign)
+
+local pi = math.pi
+local abs = math.abs
+local exp = math.exp
+local sin = math.sin
+local cos = math.cos
+local sqrt = math.sqrt
+
+local RESTING_VELOCITY_LIMIT = 1e-3
+local RESTING_POSITION_LIMIT = 1e-2
+
+local function step(self, state, dt)
+	-- Advance the spring simulation by dt seconds.
+	-- Take the damped harmonic oscillator ODE:
+	--    f^2*(X[t] - g) + 2*d*f*X'[t] + X''[t] = 0
+	-- Where X[t] is position at time t, g is desired position, f is angular frequency, and d is damping ratio.
+	-- Apply constant initial conditions:
+	--    X[0] = p0
+	--    X'[0] = v0
+	-- Solve the IVP to get analytic expressions for X[t] and X'[t].
+	-- The solution takes on one of three forms for d=1, d<1, and d>1
+
+	local d = self.__dampingRatio
+	local f = self.__frequency * 2 * pi -- Rad/s
+	local g = self.__goalPosition
+
+	local p0 = state.value
+	local v0 = state.velocity or 0
+
+	local offset = p0 - g
+	local decay = exp(-dt*d*f)
+
+	local p1, v1
+
+	if d == 1 then -- Critically damped
+		p1 = (v0*dt + offset*(f*dt + 1))*decay + g
+		v1 = (v0 - f*dt*(offset*f + v0))*decay
+
+	elseif d < 1 then -- Underdamped
+		local c = sqrt(1 - d*d)
+
+		local i = cos(f*c*dt)
+		local j = sin(f*c*dt)
+
+		-- Problem: Damping ratios close to 1 can cause numerical instability.
+		-- Solution: Rearrange to group terms involving j/c, then find an approximation z for j/c.
+		--    z = sin(dt*f*c)/c
+		-- Substitute a for dt*f
+		--    z = sin(a*c)/c
+		-- Take the 5th-order series expansion of z at c = 0
+		--    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 to mitigate precision issues
+		--    z ≈ a + ((a*a)*(c*c)*(c*c)/20 - c*c)*(a*a*a)/6
+
+		local z
+		if c > 1e-4 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
+
+		-- Repeat the process with a->dt and c->b=f*c for the f->0 case
+		local y
+		if f*c > 1e-4 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 - r1*offset)/(2*f*c)
+		local co1 = offset - co2
+
+		local e1 = co1*exp(r1*dt)
+		local e2 = co2*exp(r2*dt)
+
+		p1 = e1 + e2 + g
+		v1 = r1*e1 + r2*e2
+	end
+
+	local positionOffset = abs(p1 - self.__goalPosition)
+	local velocityOffset = abs(v1)
+
+	local complete = velocityOffset < RESTING_VELOCITY_LIMIT and positionOffset < RESTING_POSITION_LIMIT
+
+	if complete then
+		p1 = self.__goalPosition
+		v1 = 0
+	end
+
+	return {
+		value = p1,
+		velocity = v1,
+		complete = complete,
+	}
+end
+
+local function spring(goalPosition, inputOptions)
+	assert(typeof(goalPosition) == "number")
+
+	local options = {
+		dampingRatio = 1,
+		frequency = 1,
+	}
+
+	if inputOptions ~= nil then
+		assert(typeof(inputOptions) == "table")
+		assign(options, inputOptions)
+	end
+
+	local dampingRatio = options.dampingRatio
+	local frequency = options.frequency
+
+	assert(typeof(dampingRatio) == "number")
+	assert(typeof(frequency) == "number")
+
+	assert(dampingRatio * frequency >= 0, "Expected dampingRatio * frequency >= 0")
+
+	local self = {
+		__dampingRatio = dampingRatio,
+		__frequency = frequency, -- Hz
+		__goalPosition = goalPosition,
+		step = step,
+	}
+
+	return self
+end
+
+return spring