コード例 #1
0
ファイル: interpolation.py プロジェクト: EdsterG/openrave
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
コード例 #2
0
def _ImposeVelocityLimit(curve, vm):
    """_ImposeVelocityLimit imposes the given velocity limit to the ParabolicCurve. In case the velocity
    limit cannot be satisfied, this function will return an empty ParabolicCurve.

    """
    # Check types
    if type(vm) is not mp.mpf:
        vm = mp.mpf("{:.15e}".format(vm))

    # Check inputs
    assert (vm > zero)
    assert (len(curve) == 2)
    assert (Add(curve[0].a, curve[1].a) == zero)

    if Sub(Abs(curve[0].v0), vm) > epsilon:
        # Initial velocity violates the constraint
        return ParabolicCurve()

    if Sub(Abs(curve[1].v1), vm) > epsilon:
        # Final velocity violates the constraint
        return ParabolicCurve()

    vp = curve[1].v0
    if Abs(vp) <= vm:
        # Velocity limit is not violated
        return curve

    # h = Sub(Abs(vp), vm)
    # t = mp.fdiv(h, Abs(curve[0].a))

    ramp0, ramp1 = curve
    h = Sub(Abs(vp), vm)
    t = mp.fdiv(h, Abs(ramp0.a))

    # import IPython; IPython.embed()

    ramps = []
    if IsEqual(Abs(ramp0.v0), vm) and (mp.sign(ramp0.v0) == mp.sign(vp)):
        assert (IsEqual(ramp0.duration, t))  # check soundness
    else:
        newRamp0 = Ramp(ramp0.v0, ramp0.a, Sub(ramp0.duration, t), ramp0.x0)
        ramps.append(newRamp0)

    nom = h**2
    denom = Mul(Abs(curve[0].a), vm)
    newRamp1 = Ramp(Mul(mp.sign(vp), vm), zero,
                    Sum([t, t, mp.fdiv(nom, denom)]), curve.x0)
    ramps.append(newRamp1)

    if IsEqual(Abs(ramp1.v1), vm) and (mp.sign(ramp1.v1) == mp.sign(vp)):
        assert (IsEqual(ramp1.duration, t))  # check soundness
    else:
        newRamp2 = Ramp(Mul(mp.sign(vp), vm), ramp1.a, Sub(ramp1.duration, t))
        ramps.append(newRamp2)

    return ParabolicCurve(ramps)
コード例 #3
0
ファイル: interpolation.py プロジェクト: EdsterG/openrave
def _ImposeVelocityLimit(curve, vm):
    """_ImposeVelocityLimit imposes the given velocity limit to the ParabolicCurve. In case the velocity
    limit cannot be satisfied, this function will return an empty ParabolicCurve.

    """
    # Check types
    if type(vm) is not mp.mpf:
        vm = mp.mpf("{:.15e}".format(vm))

    # Check inputs
    assert(vm > zero)
    assert(len(curve) == 2)
    assert(Add(curve[0].a, curve[1].a) == zero)

    if Sub(Abs(curve[0].v0), vm) > epsilon:
        # Initial velocity violates the constraint
        return ParabolicCurve()

    if Sub(Abs(curve[1].v1), vm) > epsilon:
        # Final velocity violates the constraint
        return ParabolicCurve()
    
    vp = curve[1].v0
    if Abs(vp) <= vm:
        # Velocity limit is not violated
        return curve

    # h = Sub(Abs(vp), vm)
    # t = mp.fdiv(h, Abs(curve[0].a))
    
    ramp0, ramp1 = curve
    h = Sub(Abs(vp), vm)
    t = mp.fdiv(h, Abs(ramp0.a))

    # import IPython; IPython.embed()

    ramps = []
    if IsEqual(Abs(ramp0.v0), vm) and (mp.sign(ramp0.v0) == mp.sign(vp)):
        assert(IsEqual(ramp0.duration, t)) # check soundness
    else:
        newRamp0 = Ramp(ramp0.v0, ramp0.a, Sub(ramp0.duration, t), ramp0.x0)
        ramps.append(newRamp0)
        
    nom = h**2
    denom = Mul(Abs(curve[0].a), vm)
    newRamp1 = Ramp(Mul(mp.sign(vp), vm), zero, Sum([t, t, mp.fdiv(nom, denom)]), curve.x0)
    ramps.append(newRamp1)

    if IsEqual(Abs(ramp1.v1), vm) and (mp.sign(ramp1.v1) == mp.sign(vp)):
        assert(IsEqual(ramp1.duration, t)) # check soundness
    else:
        newRamp2 = Ramp(Mul(mp.sign(vp), vm), ramp1.a, Sub(ramp1.duration, t))
        ramps.append(newRamp2)

    return ParabolicCurve(ramps)
コード例 #4
0
ファイル: interpolation.py プロジェクト: labimage/openrave
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
コード例 #5
0
ファイル: interpolation.py プロジェクト: EdsterG/openrave
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):
        log.info("duration = {0} is negative".format(newDuration))
        return ParabolicCurve()
    if (newDuration <= epsilon):
        # Check if this is a stationary trajectory
        if (FuzzyEquals(x0, x1, epsilon) and FuzzyEquals(v0, v1, epsilon)):
            log.info("stationary trajectory")
            ramp0 = Ramp(v0, 0, 0, x0)
            newCurve = ParabolicCurve(ramp0)
            return newCurve
        else:
            log.info("newDuration is too short for any movement to be made")
            return ParabolicCurve()

    # Correct small discrepancies if any
    if (v0 > vm):
        if FuzzyEquals(v0, vm, epsilon):
            v0 = vm
        else:
            log.info("v0 > vm: {0} > {1}".format(v0, vm))
            return ParabolicCurve()
    elif (v0 < -vm):
        if FuzzyEquals(v0, -vm, epsilon):
            v0 = -vm
        else:
            log.info("v0 < -vm: {0} < {1}".format(v0, -vm))
            return ParabolicCurve()
    if (v1 > vm):
        if FuzzyEquals(v1, vm, epsilon):
            v1 = vm
        else:
            log.info("v1 > vm: {0} > {1}".format(v1, vm))
            return ParabolicCurve()
    elif (v1 < -vm):
        if FuzzyEquals(v1, -vm, epsilon):
            v1 = -vm
        else:
            log.info("v1 < -vm: {0} < {1}".format(v1, -vm))
            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", end=' ')
        # mp.nprint(D2, n=_prec)
        # print("vmnew", end=' ')
        # 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
コード例 #6
0
ファイル: rys_aux.py プロジェクト: thegodone/OpenMolcas
def gauss(alpha, beta, eps, nloop=30):
  n = len(alpha)
  if (len(beta) != n):
    raise

  # initialize
  root = alpha[:]
  weight = [one]+[zero for i in range(n-1)]
  aux = list(map(mp.sqrt, beta[1:]))
  aux.append(zero)

  for l in range(n):
    for j in range(nloop):
      for m in range(l, n):
        if (m == n-1):
          break
        if (abs(aux[m]) <= eps*(abs(root[m])+abs(root[m+1]))):
          break
      dp = root[l]
      if (m == l):
        break
      dg = (root[l+1]-dp)/(two*aux[l])
      dr = mp.sqrt(dg*dg+one)
      if (mp.sign(dg) < 0):
        dr *= -1
      dg = root[m]-dp+aux[l]/(dg+dr)
      ds = one
      dc = one
      dp = zero
      mml = m-l
      for ii in range(1, mml+1):
        i = m-ii
        df = ds*aux[i]
        db = dc*aux[i]
        if (abs(df) < abs(dg)):
          ds = df/dg
          dr = mp.sqrt(ds*ds+one)
          aux[i+1] = dg*dr
          dc = one/dr
          ds = ds*dc
        else:
          dc = dg/df
          dr = mp.sqrt(dc*dc+one)
          aux[i+1] = df*dr
          ds = one/dr
          dc = dc*ds
        dg = root[i+1]-dp
        dr = (root[i]-dg)*ds+two*dc*db
        dp = ds*dr
        root[i+1] = dg+dp
        dg = dc*dr-db
        df = weight[i+1]
        weight[i+1] = ds*weight[i]+dc*df
        weight[i] = dc*weight[i]-ds*df
      root[l] = root[l]-dp
      aux[l] = dg
      aux[m] = zero
    if (j >= nloop-1):
      raise

  # scale, sort and return
  return (list(t) for t in zip(*sorted(zip(root, [beta[0]*w*w for w in weight]))))
コード例 #7
0
ファイル: interpolation.py プロジェクト: AbuShaqra/openrave
def _Stretch1D(curve, newDuration, vm, am):

    log.debug("\nx0 = {0}; x1 = {1}; v0 = {2}; v1 = {3}; vm = {4}; am = {5}; prevDuration = {6}; newDuration = {7}".\
              format(mp.nstr(curve.x0, n=_prec), mp.nstr(curve.EvalPos(curve.duration), n=_prec),
                     mp.nstr(curve.v0, n=_prec), mp.nstr(curve.EvalVel(curve.duration), n=_prec),
                     mp.nstr(vm, n=_prec), mp.nstr(am, n=_prec), mp.nstr(curve.duration, n=_prec),
                     mp.nstr(newDuration, n=_prec)))
    
    # Check types
    if type(newDuration) is not mp.mpf:
        newDuration = mp.mpf("{:.15e}".format(newDuration))
    if type(vm) is not mp.mpf:
        vm = mp.mpf("{:.15e}".format(vm))
    if type(am) is not mp.mpf:
        am = mp.mpf("{:.15e}".format(am))

    # Check inputs
    # assert(newDuration > curve.duration)
    assert(vm > zero)
    assert(am > zero)

    if (newDuration < -epsilon):
        return ParabolicCurve()
    if (newDuration <= epsilon):
        # Check if this is a stationary trajectory
        if (FuzzyEquals(curve.x0, curve.EvalPos(x1), epsilon) and FuzzyEquals(curve.v0, curve.v1, epsilon)):
            ramp0 = Ramp(curve.v0, 0, 0, curve.x0)
            newCurve = ParabolicCurve(ramp0)
            return newCurve
        else:
            # newDuration is too short to any movement to be made
            return ParabolicCurve()

    v0 = curve[0].v0
    v1 = curve[-1].v1
    d = curve.d

    # 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)))

    assert(not (sum1 > zero))
    assert(not (sum2 < zero))
    
    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:
        # 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:
        # 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:
        # 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)):
            a0new = Mul(mp.sign(a0new), am)

        if (Abs(a0new) < epsilon):
            a1new = mp.fdiv(B2, C2)
            if (Abs(a1new) > Add(am, epsilon)):
                log.warn("abs(a1new) > am; cannot fix this case")
                # Cannot fix this case
                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, curve.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, curve.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)
            assert(t0new > 0)
            t1new = mp.fdiv(Sub(v1, vmnew), a1new)
            assert(t1new > 0)
            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()
                t0new = mp.fdiv(Sub(Sub(vmnew, v0), B), A)
                t1new = Sub(newDuration, t0new)
                assert(t1new > zero)
                a0new = Add(A, Mul(mp.fdiv(one, t0new), B))
                a1new = Add(A, Mul(mp.fdiv(one, Neg(t1new)), B))
                ramp1 = Ramp(v0, a0new, t0new, curve.x0)
                ramp2 = Ramp(ramp1.v1, a1new, t1new)
                newCurve = ParabolicCurve([ramp1, ramp2])

            else:
                # t0new = mp.fdiv(Sub(vmnew, v0), a0new)
                # assert(t0new > 0)
                ramp1 = Ramp(v0, a0new, t0new, curve.x0)
                
                # t1new = mp.fdiv(Sub(v1, vmnew), a1new)
                # assert(t1new > 0)
                ramp3 = Ramp(ramp1.v1, a1new, t1new)
                # print "a1",
                # mp.nprint(a1, n=_prec)
                # print "a1new",
                # mp.nprint(a1new, n=_prec)
                
                # print "t0trimmed",
                # mp.nprint(t0trimmed, n=_prec)
                # print "t0new",
                # mp.nprint(t0new, n=_prec)
                # print "t1trimmed", 
                # mp.nprint(t1trimmed, n=_prec)
                # print "t1new", 
                # mp.nprint(t1new, n=_prec)
                # print "T", 
                # mp.nprint(newDuration, n=_prec)
                
                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, curve.x0)
        ramp2 = Ramp(ramp1.v1, a1, t1)
        newCurve = ParabolicCurve([ramp1, ramp2])
        return newCurve