def _SolveForT0(A, B, t, tInterval):
"""There are four solutions to the equation (including complex ones). Here we use x instead of t0
for convenience.
"""
"""
SolveQuartic(2*A, -4*A*T + 2*B, 3*A*T**2 - 3*B*T, -A*T**3 + 3*B*T**2, -B*T**3)
"""
if (Abs(A) < epsilon):
def f(x):
return Mul(number('2'), B) * x * x * x - Prod(
[number('3'), B, t]) * x * x + Prod(
[number('3'), B, t, t]) * x - Mul(B, mp.power(t, 3))
sols = [mp.findroot(f, x0=0.5 * t)]
else:
sols = SolveQuartic(
Add(A, A), Add(Prod([number('-4'), A, t]), Mul(number('2'), B)),
Sub(Prod([number('3'), A, t, t]), Prod([number('3'), B, t])),
Sub(Prod([number('3'), B, t, t]), Mul(A, mp.power(t, 3))),
Neg(Mul(B, mp.power(t, 3))))
realSols = [
sol for sol in sols if type(sol) is mp.mpf and sol in tInterval
]
if len(realSols) > 1:
# I think this should not happen. We should either have one or no solution.
raise NotImplementedError
elif len(realSols) == 0:
return None
else:
return realSols[0]
def _CalculateLeastUpperBoundInoperativeInterval(x0, x1, v0, v1, vm, am):
# All input must already be of mp.mpf type
d = x1 - x0
temp1 = Prod([
number('2'),
Neg(Sqr(am)),
Sub(Prod([number('2'), am, d]), Add(Sqr(v0), Sqr(v1)))
])
if temp1 < zero:
T0 = number('-1')
T1 = number('-1')
else:
term1 = mp.fdiv(Add(v0, v1), am)
term2 = mp.fdiv(mp.sqrt(temp1), Sqr(am))
T0 = Add(term1, term2)
T1 = Sub(term1, term2)
temp2 = Prod([
number('2'),
Sqr(am),
Add(Prod([number('2'), am, d]), Add(Sqr(v0), Sqr(v1)))
])
if temp2 < zero:
T2 = number('-1')
T3 = number('-1')
else:
term1 = Neg(mp.fdiv(Add(v0, v1), am))
term2 = mp.fdiv(mp.sqrt(temp2), Sqr(am))
T2 = Add(term1, term2)
T3 = Sub(term1, term2)
newDuration = max(max(T0, T1), max(T2, T3))
if newDuration > zero:
dStraight = Prod([pointfive, Add(v0, v1), newDuration])
if Sub(d, dStraight) > 0:
amNew = am
vmNew = vm
else:
amNew = -am
vmNew = -vm
# import IPython; IPython.embed()
vp = Mul(pointfive, Sum([Mul(newDuration, amNew), v0,
v1])) # the peak velocity
if (Abs(vp) > vm):
dExcess = mp.fdiv(Sqr(Sub(vp, vmNew)), am)
assert (dExcess > 0)
deltaTime = mp.fdiv(dExcess, vm)
newDuration = Add(newDuration, deltaTime)
newDuration = Mul(newDuration, number('1.01')) # add 1% safety bound
return newDuration
else:
log.debug('Unable to calculate the least upper bound: T0 = {0}; T1 = {1}; T2 = {2}; T3 = {3}'.\
format(mp.nstr(T0, n=_prec), mp.nstr(T1, n=_prec), mp.nstr(T2, n=_prec), mp.nstr(T3, n=_prec)))
return number('-1')
def _Interpolate1DNoVelocityLimit(x0, x1, v0, v1, am):
# Check types
if type(x0) is not mp.mpf:
x0 = mp.mpf("{:.15e}".format(x0))
if type(x1) is not mp.mpf:
x1 = mp.mpf("{:.15e}".format(x1))
if type(v0) is not mp.mpf:
v0 = mp.mpf("{:.15e}".format(v0))
if type(v1) is not mp.mpf:
v1 = mp.mpf("{:.15e}".format(v1))
if type(am) is not mp.mpf:
am = mp.mpf("{:.15e}".format(am))
# Check inputs
assert (am > zero)
# Check for an appropriate acceleration direction of the first ramp
d = Sub(x1, x0)
dv = Sub(v1, v0)
difVSqr = Sub(v1**2, v0**2)
if Abs(dv) < epsilon:
if Abs(d) < epsilon:
# Stationary ramp
ramp0 = Ramp(zero, zero, zero, x0)
return ParabolicCurve([ramp0])
else:
dStraight = zero
else:
dStraight = mp.fdiv(difVSqr, Prod([2, mp.sign(dv), am]))
if IsEqual(d, dStraight):
# With the given distance, v0 and v1 can be directly connected using max/min
# acceleration. Here the resulting profile has only one ramp.
a0 = mp.sign(dv) * am
ramp0 = Ramp(v0, a0, mp.fdiv(dv, a0), x0)
return ParabolicCurve([ramp0])
sumVSqr = Add(v0**2, v1**2)
sigma = mp.sign(Sub(d, dStraight))
a0 = sigma * am # acceleration of the first ramp
vp = sigma * mp.sqrt(Add(Mul(pointfive, sumVSqr), Mul(a0, d)))
t0 = mp.fdiv(Sub(vp, v0), a0)
t1 = mp.fdiv(Sub(vp, v1), a0)
ramp0 = Ramp(v0, a0, t0, x0)
assert (IsEqual(ramp0.v1, vp)) # check soundness
ramp1 = Ramp(vp, Neg(a0), t1)
curve = ParabolicCurve([ramp0, ramp1])
assert (IsEqual(curve.d, d)) # check soundness
return curve
def Interpolate1DFixedDuration(x0, x1, v0, v1, newDuration, vm, am):
x0 = ConvertFloatToMPF(x0)
x1 = ConvertFloatToMPF(x1)
v0 = ConvertFloatToMPF(v0)
v1 = ConvertFloatToMPF(v1)
vm = ConvertFloatToMPF(vm)
am = ConvertFloatToMPF(am)
newDuration = ConvertFloatToMPF(newDuration)
log.debug("\nx0 = {0}; x1 = {1}; v0 = {2}; v1 = {3}; vm = {4}; am = {5}; newDuration = {6}".\
format(mp.nstr(x0, n=_prec), mp.nstr(x1, n=_prec), mp.nstr(v0, n=_prec), mp.nstr(v1, n=_prec),
mp.nstr(vm, n=_prec), mp.nstr(am, n=_prec), mp.nstr(newDuration, n=_prec)))
# Check inputs
assert (vm > zero)
assert (am > zero)
if (newDuration < -epsilon):
return ParabolicCurve()
if (newDuration <= epsilon):
# Check if this is a stationary trajectory
if (FuzzyEquals(x0, x1, epsilon) and FuzzyEquals(v0, v1, epsilon)):
ramp0 = Ramp(v0, 0, 0, x0)
newCurve = ParabolicCurve(ramp0)
return newCurve
else:
# newDuration is too short to any movement to be made
return ParabolicCurve()
d = Sub(x1, x0)
# First assume no velocity bound -> re-interpolated trajectory will have only two ramps.
# Solve for a0 and a1 (the acceleration of the first and the last ramps).
# a0 = A + B/t0
# a1 = A + B/(t - t0)
# where t is the (new) total duration, t0 is the (new) duration of the first ramp, and
# A = (v1 - v0)/t
# B = (2d/t) - (v0 + v1).
newDurInverse = mp.fdiv(one, newDuration)
A = Mul(Sub(v1, v0), newDurInverse)
B = Sub(Prod([mp.mpf('2'), d, newDurInverse]), Add(v0, v1))
interval0 = iv.mpf([zero, newDuration]) # initial interval for t0
# Now consider the interval(s) computed from a0's constraints
sum1 = Neg(Add(am, A))
sum2 = Sub(am, A)
C = mp.fdiv(B, sum1)
D = mp.fdiv(B, sum2)
log.debug("\nA = {0}; \nB = {1}; \nC = {2}; \nD = {3}; \nsum1 = {4}; \nsum2 = {5};".\
format(mp.nstr(A, n=_prec), mp.nstr(B, n=_prec), mp.nstr(C, n=_prec), mp.nstr(D, n=_prec),
mp.nstr(sum1, n=_prec), mp.nstr(sum2, n=_prec)))
if (sum1 > zero):
# This implied that the duration is too short
log.debug("the given duration ({0}) is too short.".format(newDuration))
return ParabolicCurve()
if (sum2 < zero):
# This implied that the duration is too short
log.debug("the given duration ({0}) is too short.".format(newDuration))
return ParabolicCurve()
if IsEqual(sum1, zero):
raise NotImplementedError # not yet considered
elif sum1 > epsilon:
log.debug("sum1 > 0. This implies that newDuration is too short.")
return ParabolicCurve()
else:
interval1 = iv.mpf([C, inf])
if IsEqual(sum2, zero):
raise NotImplementedError # not yet considered
elif sum2 > epsilon:
interval2 = iv.mpf([D, inf])
else:
log.debug("sum2 < 0. This implies that newDuration is too short.")
return ParabolicCurve()
if Sub(interval2.a, interval1.b) > epsilon or Sub(interval1.a,
interval2.b) > epsilon:
# interval1 and interval2 do not intersect each other
return ParabolicCurve()
# interval3 = interval1 \cap interval2 : valid interval for t0 computed from a0's constraints
interval3 = iv.mpf(
[max(interval1.a, interval2.a),
min(interval1.b, interval2.b)])
# Now consider the interval(s) computed from a1's constraints
if IsEqual(sum1, zero):
raise NotImplementedError # not yet considered
elif sum1 > epsilon:
log.debug("sum1 > 0. This implies that newDuration is too short.")
return ParabolicCurve()
else:
interval4 = iv.mpf([Neg(inf), Add(C, newDuration)])
if IsEqual(sum2, zero):
raise NotImplementedError # not yet considered
elif sum2 > epsilon:
interval5 = iv.mpf([Neg(inf), Add(D, newDuration)])
else:
log.debug("sum2 < 0. This implies that newDuration is too short.")
return ParabolicCurve()
if Sub(interval5.a, interval4.b) > epsilon or Sub(interval4.a,
interval5.b) > epsilon:
log.debug("interval4 and interval5 do not intersect each other")
return ParabolicCurve()
# interval6 = interval4 \cap interval5 : valid interval for t0 computed from a1's constraints
interval6 = iv.mpf(
[max(interval4.a, interval5.a),
min(interval4.b, interval5.b)])
# import IPython; IPython.embed()
if Sub(interval3.a, interval6.b) > epsilon or Sub(interval6.a,
interval3.b) > epsilon:
log.debug("interval3 and interval6 do not intersect each other")
return ParabolicCurve()
# interval7 = interval3 \cap interval6
interval7 = iv.mpf(
[max(interval3.a, interval6.a),
min(interval3.b, interval6.b)])
if Sub(interval0.a, interval7.b) > epsilon or Sub(interval7.a,
interval0.b) > epsilon:
log.debug("interval0 and interval7 do not intersect each other")
return ParabolicCurve()
# interval8 = interval0 \cap interval7 : valid interval of t0 when considering all constraints (from a0 and a1)
interval8 = iv.mpf(
[max(interval0.a, interval7.a),
min(interval0.b, interval7.b)])
# import IPython; IPython.embed()
# We choose the value t0 (the duration of the first ramp) by selecting the mid point of the
# valid interval of t0.
t0 = _SolveForT0(A, B, newDuration, interval8)
if t0 is None:
# The fancy procedure fails. Now consider no optimization whatsoever.
# TODO: Figure out why solving fails.
t0 = mp.convert(interval8.mid) # select the midpoint
# return ParabolicCurve()
t1 = Sub(newDuration, t0)
a0 = Add(A, Mul(mp.fdiv(one, t0), B))
if (Abs(t1) < epsilon):
a1 = zero
else:
a1 = Add(A, Mul(mp.fdiv(one, Neg(t1)), B))
assert (Sub(Abs(a0), am) < epsilon
) # check if a0 is really below the bound
assert (Sub(Abs(a1), am) < epsilon
) # check if a1 is really below the bound
# import IPython; IPython.embed()
# Check if the velocity bound is violated
vp = Add(v0, Mul(a0, t0))
if Abs(vp) > vm:
vmnew = Mul(mp.sign(vp), vm)
D2 = Prod([
pointfive,
Sqr(Sub(vp, vmnew)),
Sub(mp.fdiv(one, a0), mp.fdiv(one, a1))
])
# print "D2",
# mp.nprint(D2, n=_prec)
# print "vmnew",
# mp.nprint(vmnew, n=_prec)
A2 = Sqr(Sub(vmnew, v0))
B2 = Neg(Sqr(Sub(vmnew, v1)))
t0trimmed = mp.fdiv(Sub(vmnew, v0), a0)
t1trimmed = mp.fdiv(Sub(v1, vmnew), a1)
C2 = Sum([
Mul(t0trimmed, Sub(vmnew, v0)),
Mul(t1trimmed, Sub(vmnew, v1)),
Mul(mp.mpf('-2'), D2)
])
log.debug("\nA2 = {0}; \nB2 = {1}; \nC2 = {2}; \nD2 = {3};".format(
mp.nstr(A2, n=_prec), mp.nstr(B2, n=_prec), mp.nstr(C2, n=_prec),
mp.nstr(D2, n=_prec)))
temp = Prod([A2, B2, B2])
initguess = mp.sign(temp) * (Abs(temp)**(1. / 3.))
root = mp.findroot(lambda x: Sub(Prod([x, x, x]), temp), x0=initguess)
# import IPython; IPython.embed()
log.debug("root = {0}".format(mp.nstr(root, n=_prec)))
a0new = mp.fdiv(Add(A2, root), C2)
if (Abs(a0new) > Add(am, epsilon)):
if FuzzyZero(Sub(Mul(C2, a0new), A2), epsilon):
# The computed a0new is exceeding the bound and its corresponding a1new is
# zero. Therefore, there is no other way to fix this. This is probably because the
# given newDuration is less than the minimum duration (x0, x1, v0, v1, vm, am) can
# get.
log.debug(
"abs(a0new) > am and a1new = 0; Cannot fix this case. This happens probably because the given newDuration is too short."
)
return ParabolicCurve()
a0new = Mul(mp.sign(a0new), am)
if (Abs(a0new) < epsilon):
a1new = mp.fdiv(B2, C2)
if (Abs(a1new) > Add(am, epsilon)):
# Similar to the case above
log.debug(
"a0new = 0 and abs(a1new) > am; Cannot fix this case. This happens probably because the given newDuration is too short."
)
return ParabolicCurve()
else:
if FuzzyZero(Sub(Mul(C2, a0new), A2), epsilon):
# import IPython; IPython.embed()
a1new = 0
else:
a1new = Mul(mp.fdiv(B2, C2),
Add(one, mp.fdiv(A2, Sub(Mul(C2, a0new), A2))))
if (Abs(a1new) > Add(am, epsilon)):
a1new = Mul(mp.sign(a1new), am)
a0new = Mul(mp.fdiv(A2, C2),
Add(one, mp.fdiv(B2, Sub(Mul(C2, a1new), B2))))
if (Abs(a0new) > Add(am, epsilon)) or (Abs(a1new) > Add(am, epsilon)):
log.warn("Cannot fix acceleration bounds violation")
return ParabolicCurve()
log.debug(
"\na0 = {0}; \na0new = {1}; \na1 = {2}; \na1new = {3};".format(
mp.nstr(a0, n=_prec), mp.nstr(a0new, n=_prec),
mp.nstr(a1, n=_prec), mp.nstr(a1new, n=_prec)))
if (Abs(a0new) < epsilon) and (Abs(a1new) < epsilon):
log.warn("Both accelerations are zero. Should we allow this case?")
return ParabolicCurve()
if (Abs(a0new) < epsilon):
# This is likely because v0 is at the velocity bound
t1new = mp.fdiv(Sub(v1, vmnew), a1new)
assert (t1new > 0)
ramp2 = Ramp(v0, a1new, t1new)
t0new = Sub(newDuration, t1new)
assert (t0new > 0)
ramp1 = Ramp(v0, zero, t0new, x0)
newCurve = ParabolicCurve([ramp1, ramp2])
return newCurve
elif (Abs(a1new) < epsilon):
t0new = mp.fdiv(Sub(vmnew, v0), a0new)
assert (t0new > 0)
ramp1 = Ramp(v0, a0new, t0new, x0)
t1new = Sub(newDuration, t0new)
assert (t1new > 0)
ramp2 = Ramp(ramp1.v1, zero, t1new)
newCurve = ParabolicCurve([ramp1, ramp2])
return newCurve
else:
# No problem with those new accelerations
# import IPython; IPython.embed()
t0new = mp.fdiv(Sub(vmnew, v0), a0new)
if (t0new < 0):
log.debug(
"t0new < 0. The given newDuration not achievable with the given bounds"
)
return ParabolicCurve()
t1new = mp.fdiv(Sub(v1, vmnew), a1new)
if (t1new < 0):
log.debug(
"t1new < 0. The given newDuration not achievable with the given bounds"
)
return ParabolicCurve()
if (Add(t0new, t1new) > newDuration):
# Final fix. Since we give more weight to acceleration bounds, we make the velocity
# bound saturated. Therefore, we set vp to vmnew.
# import IPython; IPython.embed()
if FuzzyZero(A, epsilon):
log.warn(
"(final fix) A is zero. Don't know how to fix this case"
)
return ParabolicCurve()
t0new = mp.fdiv(Sub(Sub(vmnew, v0), B), A)
if (t0new < 0):
log.debug("(final fix) t0new is negative")
return ParabolicCurve()
t1new = Sub(newDuration, t0new)
a0new = Add(A, Mul(mp.fdiv(one, t0new), B))
a1new = Add(A, Mul(mp.fdiv(one, Neg(t1new)), B))
ramp1 = Ramp(v0, a0new, t0new, x0)
ramp2 = Ramp(ramp1.v1, a1new, t1new)
newCurve = ParabolicCurve([ramp1, ramp2])
else:
ramp1 = Ramp(v0, a0new, t0new, x0)
ramp3 = Ramp(ramp1.v1, a1new, t1new)
ramp2 = Ramp(ramp1.v1, zero, Sub(newDuration,
Add(t0new, t1new)))
newCurve = ParabolicCurve([ramp1, ramp2, ramp3])
# import IPython; IPython.embed()
return newCurve
else:
ramp1 = Ramp(v0, a0, t0, x0)
ramp2 = Ramp(ramp1.v1, a1, t1)
newCurve = ParabolicCurve([ramp1, ramp2])
return newCurve
def SolveQuartic(a, b, c, d, e):
"""
SolveQuartic solves a quartic (fouth order) equation of the form
ax^4 + bx^3 + cx^2 + dx + e = 0.
For the detail of formulae presented here, see https://en.wikipedia.org/wiki/Quartic_function
"""
# Check types
if type(a) is not mp.mpf:
a = mp.mpf("{:.15e}".format(a))
if type(b) is not mp.mpf:
b = mp.mpf("{:.15e}".format(b))
if type(c) is not mp.mpf:
c = mp.mpf("{:.15e}".format(c))
if type(d) is not mp.mpf:
d = mp.mpf("{:.15e}".format(d))
if type(e) is not mp.mpf:
e = mp.mpf("{:.15e}".format(e))
"""
# Working code (more readable but probably less precise)
p = (8*a*c - 3*b*b)/(8*a*a)
q = (b**3 - 4*a*b*c + 8*a*a*d)/(8*a*a*a)
delta0 = c*c - 3*b*d + 12*a*e
delta1 = 2*(c**3) - 9*b*c*d + 27*b*b*e + 27*a*d*d - 72*a*c*e
Q = mp.nthroot(pointfive*(delta1 + mp.sqrt(delta1*delta1 - 4*mp.power(delta0, 3))), 3)
S = pointfive*mp.sqrt(-mp.fdiv(mp.mpf('2'), mp.mpf('3'))*p + (one/(3*a))*(Q + delta0/Q))
x1 = -b/(4*a) - S + pointfive*mp.sqrt(-4*S*S - 2*p + q/S)
x2 = -b/(4*a) - S - pointfive*mp.sqrt(-4*S*S - 2*p + q/S)
x3 = -b/(4*a) + S + pointfive*mp.sqrt(-4*S*S - 2*p - q/S)
x4 = -b/(4*a) + S - pointfive*mp.sqrt(-4*S*S - 2*p - q/S)
"""
p = mp.fdiv(
Sub(Prod([number('8'), a, c]), Mul(number('3'), mp.power(b, 2))),
Mul(number('8'), mp.power(a, 2)))
q = mp.fdiv(
Sum([
mp.power(b, 3),
Prod([number('-4'), a, b, c]),
Prod([number('8'), mp.power(a, 2), d])
]), Mul(8, mp.power(a, 3)))
delta0 = Sum([
mp.power(c, 2),
Prod([number('-3'), b, d]),
Prod([number('12'), a, e])
])
delta1 = Sum([
Mul(2, mp.power(c, 3)),
Prod([number('-9'), b, c, d]),
Prod([number('27'), mp.power(b, 2), e]),
Prod([number('27'), a, mp.power(d, 2)]),
Prod([number('-72'), a, c, e])
])
Q = mp.nthroot(
Mul(
pointfive,
Add(
delta1,
mp.sqrt(
Add(mp.power(delta1, 2),
Mul(number('-4'), mp.power(delta0, 3)))))), 3)
S = Mul(
pointfive,
mp.sqrt(
Mul(mp.fdiv(mp.mpf('-2'), mp.mpf('3')), p) +
Mul(mp.fdiv(one, Mul(number('3'), a)), Add(Q, mp.fdiv(delta0, Q))))
)
# log.debug("p = {0}".format(mp.nstr(p, n=_prec)))
# log.debug("q = {0}".format(mp.nstr(q, n=_prec)))
# log.debug("delta0 = {0}".format(mp.nstr(delta0, n=_prec)))
# log.debug("delta1 = {0}".format(mp.nstr(delta1, n=_prec)))
# log.debug("Q = {0}".format(mp.nstr(Q, n=_prec)))
# log.debug("S = {0}".format(mp.nstr(S, n=_prec)))
x1 = Sum([
mp.fdiv(b, Mul(number('-4'), a)),
Neg(S),
Mul(
pointfive,
mp.sqrt(
Sum([
Mul(number('-4'), mp.power(S, 2)),
Mul(number('-2'), p),
mp.fdiv(q, S)
])))
])
x2 = Sum([
mp.fdiv(b, Mul(number('-4'), a)),
Neg(S),
Neg(
Mul(
pointfive,
mp.sqrt(
Sum([
Mul(number('-4'), mp.power(S, 2)),
Mul(number('-2'), p),
mp.fdiv(q, S)
]))))
])
x3 = Sum([
mp.fdiv(b, Mul(number('-4'), a)), S,
Mul(
pointfive,
mp.sqrt(
Sum([
Mul(number('-4'), mp.power(S, 2)),
Mul(number('-2'), p),
Neg(mp.fdiv(q, S))
])))
])
x4 = Sum([
mp.fdiv(b, Mul(number('-4'), a)), S,
Neg(
Mul(
pointfive,
mp.sqrt(
Sum([
Mul(number('-4'), mp.power(S, 2)),
Mul(number('-2'), p),
Neg(mp.fdiv(q, S))
]))))
])
return [x1, x2, x3, x4]
def f(x):
return Mul(number('2'), B) * x * x * x - Prod(
[number('3'), B, t]) * x * x + Prod(
[number('3'), B, t, t]) * x - Mul(B, mp.power(t, 3))