Mercurial > hg > nsaunier > traffic-intelligence

annotate trafficintelligence/traffic_engineering.py @ 1306:4bc0651d91f9 default tip

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bug corrected generating last velocity twice and saving it (not saved, duplicated at loading time
author Nicolas Saunier <nicolas.saunier@polymtl.ca>
date Mon, 30 Mar 2026 15:31:34 -0400
parents d828f3f76273
children
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
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4bd7cc69b6cd added traffic engineering utilities with first version of LaneGroups
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1 #! /usr/bin/env python
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2 ''' Traffic Engineering Tools and Examples'''
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3
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4 from math import ceil
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5 from numpy import e, log, arange
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6 from numpy.random import triangular
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7 from scipy import stats
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8
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9 from matplotlib.pyplot import figure, plot, xlabel, ylabel, xlim, ylim
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10
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11 from trafficintelligence import prediction
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12
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13 #########################
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14 # Simulation
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15 #########################
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16
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17 def generateTimeHeadways(meanTimeHeadway, simulationTime):
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18 '''Generates the time headways between arrivals
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19 given the meanTimeHeadway and the negative exponential distribution
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20 over a time interval of length simulationTime (assumed to be in same time unit as headway'''
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21 from random import expovariate
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22 headways = []
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23 totalTime = 0
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24 flow = 1/meanTimeHeadway
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25 while totalTime < simulationTime:
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26 h = expovariate(flow)
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27 headways.append(h)
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28 totalTime += h
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29 return headways
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30
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31 class RoadUser(object):
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32 '''Simple example of inheritance to plot different road users (no history)'''
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33 def __init__(self, position, velocity):
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34 'Both fields are 2D numpy arrays'
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35 self.position = position.astype(float)
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36 self.velocity = velocity.astype(float)
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37
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38 def move(self, deltaT):
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39 self.position += deltaT*self.velocity
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40
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41 def draw(self, init = False):
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42 if init:
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43 self.plotLine = plot(self.position[0], self.position[1], self.getDescriptor())[0]
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44 else:
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45 self.plotLine.set_data(self.position[0], self.position[1])
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46
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47 class PassengerVehicle(RoadUser):
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48 def getDescriptor(self):
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49 return 'dr'
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50
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51 class Pedestrian(RoadUser):
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52 def getDescriptor(self):
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53 return 'xb'
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54
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55 class Cyclist(RoadUser):
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56 def getDescriptor(self):
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57 return 'og'
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58
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59 class SLUser(object):
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60 '''Class for single lane road users
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61
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62 Warning: does not work with decimal time, time must be integer based'''
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63 def __init__(self, t0, x0, v0, sigma):
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64 'sigma is the width of the triangular distribution around 1 for speed noise in ]0,0.1]'
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65 self.t = [t0]
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66 self.x = [x0]
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67 self.v = [v0]
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68 self.sigma = max(0.001,min(0.1,sigma))
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69 self.left = 1.-self.sigma
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70 self.right = 1.+self.sigma
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71
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72 def getT(self):
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73 return self.t
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74
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75 def existsAt(self, t):
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76 return t in self.t
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77
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78 def getX(self):
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79 return self.x
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80
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81 def getXAt(self, t):
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82 if self.existsAt(t):
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83 i = self.t.index(t)
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84 return self.x[i]
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85 else:
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86 return None
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87
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88 def getV(self):
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89 return self.v
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90
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91 def getVAt(self, t):
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92 if self.existsAt(t):
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93 i = self.t.index(t)
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94 return self.v[i]
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95 else:
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96 return None
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97
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98 def update(self, a, deltat):
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99 self.t.append(self.t[-1]+1)
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100 self.v.append(max(0,self.v[-1]+a*deltat*triangular(self.left,1., self.right)))
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101 self.x.append(self.x[-1]+self.v[-1]*deltat)
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102
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103 def plotX(self, deltat):
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104 plot([t*deltat for t in self.t], self.x)
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105
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106 def plotV(self, deltat):
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107 plot([t*deltat for t in self.t], self.v)
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108
790
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109 #########################
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110 # queueing models
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111 #########################
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112
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113 class CapacityReduction(object):
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114 def __init__(self, beta, reductionDuration, demandCapacityRatio = None, demand = None, capacity = None):
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115 '''reduction duration should be positive
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116 demandCapacityRatio is demand/capacity (q/s)'''
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117 if demandCapacityRatio is None and demand is None and capacity is None:
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118 print('Missing too much information (demand, capacity and ratio)')
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119 import sys
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120 sys.exit()
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121 if 0 <= beta < 1:
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122 self.beta = beta
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123 self.reductionDuration = reductionDuration
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124
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125 if demandCapacityRatio is not None:
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126 self.demandCapacityRatio = demandCapacityRatio
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127 if demand is not None:
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128 self.demand = demand
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129 if capacity is not None:
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130 self.capacity = capacity
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131 if capacity is not None and demand is not None:
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132 self.demandCapacityRatio = float(self.demand)/self.capacity
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133 if demand <= beta*capacity:
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134 print('There is no queueing as the demand {} is inferior to the reduced capacity {}'.format(demand, beta*capacity))
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135 else:
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136 print('reduction coefficient (beta={}) is not in [0, 1['.format(beta))
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137
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138 def queueingDuration(self):
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139 return self.reductionDuration*(1-self.beta)/(1-self.demandCapacityRatio)
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140
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141 def nArrived(self, t):
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142 '''since the beginning of the capacity reduction'''
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143 if self.demand is None or t<0:
790
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144 print('Missing demand field')
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145 return None
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146 return self.demand*t
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147
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148 def nServed(self, t):
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149 '''since the beginning of the capacity reduction'''
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150 if self.capacity is None or t<0:
790
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151 print('Missing capacity field')
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152 return None
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153 if 0<=t<=self.reductionDuration:
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154 return self.beta*self.capacity*t
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155 elif self.reductionDuration < t:
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156 qDuration = self.queueingDuration()
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157 if t <= qDuration:
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158 return self.beta*self.capacity*self.reductionDuration+self.capacity*(t-self.reductionDuration)
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159 else:
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160 return self.beta*self.capacity*self.reductionDuration+self.capacity*(qDuration-self.reductionDuration)+self.demand*(t-qDuration)
790
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161
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162 def nQueued(self, t):
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163 return self.nArrived(t)-self.nServed(t)
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164
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165 def maxNQueued(self):
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166 return self.nQueued(self.reductionDuration)
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167
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168 def totalDelay(self):
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169 if self.capacity is None:
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parents: 789
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170 print('Missing capacity field')
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171 return None
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172 return self.capacity*self.reductionDuration**2*(1-self.beta)*(self.demandCapacityRatio-self.beta)/(2*(1-self.demandCapacityRatio))
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173
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174 def averageDelay(self):
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175 return self.reductionDuration*(self.demandCapacityRatio-self.beta)/(2*self.demandCapacityRatio)
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176
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177 def averageNQueued(self):
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178 return self.totalDelay()/self.queueingDuration()
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179
198
c91c8fd8bf1e simple vehicle model with constant acceleration
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180
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181 #########################
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182 # fundamental diagrams
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183 #########################
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184
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15e244d2a1b5 corrected bug with circular import for VideoFilenameAddable, moved to base module
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185 class FundamentalDiagram(object):
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186 ''' '''
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187 def __init__(self, name):
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188 self.name = name
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189 self.kj = None
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190 self.kc = None
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191 self.vf = None
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192 self.qmax = None
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193
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194 def getJamDensity(self):
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195 return self.kj
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196
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197 def getCriticalDensity(self):
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198 return self.kc
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199
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200 def getCapacity(self):
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201 return self.qmax
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202
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203 def getFreeFlowSpeed(self):
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204 return self.vf
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205
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206 def q(self, k):
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diff changeset
207 return k*self.v(k)
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208
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209 @staticmethod
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210 def meanHeadway(k):
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diff changeset
211 return 1/k
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212
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213 @staticmethod
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214 def meanSpacing(q):
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215 return 1/q
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216
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217 def plotVK(self, language='fr', units={}):
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diff changeset
218 densities = [k for k in arange(1, self.kj+1)]
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diff changeset
219 figure()
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diff changeset
220 plot(densities, [self.v(k) for k in densities])
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diff changeset
221 xlim(xmin=0)
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diff changeset
222 ylim(ymin=0)
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diff changeset
223 xlabel('Densite (veh/km)') # todo other languages and adapt to units
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diff changeset
224 ylabel('Vitesse (km/h)')
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225
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diff changeset
226 def plotQK(self, language='fr', units={}):
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diff changeset
227 densities = [k for k in arange(1, self.kj+1)]
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diff changeset
228 figure()
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diff changeset
229 plot(densities, [self.q(k) for k in densities])
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diff changeset
230 xlim(xmin=0)
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diff changeset
231 ylim(ymin=0)
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diff changeset
232 xlabel('Densite (veh/km)') # todo other languages and adapt to units
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diff changeset
233 ylabel('Debit (km/h)')
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diff changeset
234
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235 class GreenshieldsFD(FundamentalDiagram):
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236 '''Speed is a linear function of density'''
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237 def __init__(self, vf, kj):
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238 FundamentalDiagram.__init__(self,'Greenshields')
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239 self.vf=vf
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240 self.kj=kj
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241 self.kc=kj/2
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242 self.qmax=vf*kj/4
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243
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244 def v(self,k):
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diff changeset
245 from numpy import log
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fa07a78b29f6 minor modif, change to makefile for repo opencv installation
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diff changeset
246 return self.vf*(1-k/self.kj)
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247
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248
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diff changeset
249 class GreenbergFD(FundamentalDiagram):
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diff changeset
250 '''Speed is the logarithm of density'''
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diff changeset
251 def __init__(self, vc, kj):
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diff changeset
252 FundamentalDiagram.__init__(self,'Greenberg')
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diff changeset
253 self.vc=vc
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diff changeset
254 self.kj=kj
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255 self.qmax = self.kc*self.vc
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diff changeset
256 self.kc = self.kj/e
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257
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diff changeset
258 def v(self,k):
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diff changeset
259 return self.vc*log(self.kj/k)
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260
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261 class TriangularFD(FundamentalDiagram):
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262 def __init__(self, vf = None, kc = None, kj = None, qmax = None, w = None):
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diff changeset
263 FundamentalDiagram.__init__(self,'Triangular')
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diff changeset
264 if vf is not None and kj is not None:
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diff changeset
265 self.vf=vf
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diff changeset
266 self.kj = kj
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267 if qmax is not None:
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268 self.qmax = qmax
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269 self.kc = qmax/vf
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270 self.w = qmax/(self.kc-kj)
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271 elif kc is not None:
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272 self.kc = kc
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273 self.qmax = vf*kc
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274 self.w = self.qmax/(self.kc-kj)
73
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275
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276 def v(self, k):
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277 if k<self.kc:
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278 return self.vf
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279 else:
1303
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280 return self.qmax/(self.kj-self.kc)*(1-self.kj/k)
1206
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281
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282 def generateDensities(n, maxDensity):
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283 return stats.uniform.rvs(size=n)*maxDensity
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284
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285 def generateSpeedVolumes(fd, n, maxDensity, maxHGVProportion = 0, etrucks = 2.5):
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286 densities = generateDensities(n, maxDensity)
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287 speeds = [fd.v(k) for k in densities]
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288 volumes = [fd.q(k) for k in densities]
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289 if maxHGVProportion > 0:
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290 hgvProportions = stats.uniform.rvs(size=n)*maxHGVProportion # en pourcent
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291 volumes = [v/(1+(etrucks-1)*p/100) for v,p in zip(volumes, hgvProportions)]
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292 else:
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293 hgvProportions = None
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294 return speeds, volumes, hgvProportions
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295
1262
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296 higwayMaxDensityLOS = {'A':7, 'B':11, 'C':16, 'D':22, 'E': 28}
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297
1206
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298 def highwayLOS(k):
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299 'returns the highway level of service for density k in veh/km'
1262
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300 for los, kmax in higwayMaxDensityLOS.items():
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301 if k<kmax: return los
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302 return 'F'
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303
33
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304 #########################
116
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305 # intersection
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306 #########################
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307
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15e244d2a1b5 corrected bug with circular import for VideoFilenameAddable, moved to base module
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diff changeset
308 class FourWayIntersection(object):
116
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diff changeset
309 '''Simple class for simple intersection outline'''
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310 def __init__(self, dimension, coordX, coordY):
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diff changeset
311 self.dimension = dimension
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312 self.coordX = coordX
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diff changeset
313 self.coordY = coordY
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314
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315 def plot(self, options = 'k'):
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diff changeset
316 from matplotlib.pyplot import plot, axis
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317
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diff changeset
318 minX = min(self.dimension[0])
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diff changeset
319 maxX = max(self.dimension[0])
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diff changeset
320 minY = min(self.dimension[1])
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diff changeset
321 maxY = max(self.dimension[1])
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diff changeset
322
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diff changeset
323 plot([minX, self.coordX[0], self.coordX[0]], [self.coordY[0], self.coordY[0], minY],options)
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diff changeset
324 plot([self.coordX[1], self.coordX[1], maxX], [minY, self.coordY[0], self.coordY[0]],options)
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diff changeset
325 plot([minX, self.coordX[0], self.coordX[0]], [self.coordY[1], self.coordY[1], maxY],options)
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diff changeset
326 plot([self.coordX[1], self.coordX[1], maxX], [maxY, self.coordY[1], self.coordY[1]],options)
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diff changeset
327 axis('equal')
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diff changeset
328
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diff changeset
329 #########################
33
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parents:
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330 # traffic signals
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parents:
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331 #########################
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parents:
diff changeset
332
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15e244d2a1b5 corrected bug with circular import for VideoFilenameAddable, moved to base module
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diff changeset
333 class Volume(object):
33
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parents:
diff changeset
334 '''Class to represent volumes with varied vehicule types '''
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parents:
diff changeset
335 def __init__(self, volume, types = ['pc'], proportions = [1], equivalents = [1], nLanes = 1):
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parents:
diff changeset
336 '''mvtEquivalent is the equivalent if the movement is right of left turn'''
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parents:
diff changeset
337
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parents:
diff changeset
338 # check the sizes of the lists
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parents:
diff changeset
339 if sum(proportions) == 1:
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diff changeset
340 self.volume = volume
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diff changeset
341 self.types = types
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parents:
diff changeset
342 self.proportions = proportions
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parents:
diff changeset
343 self.equivalents = equivalents
314
539e2b4cfaa3 modified for 4740 tps
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parents: 302
diff changeset
344 self.nLanes = nLanes
33
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diff changeset
345 else:
302
9d88a4d97473 corrected bug in compute-homography
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parents: 300
diff changeset
346 print('Proportions do not sum to 1')
33
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parents:
diff changeset
347 pass
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diff changeset
348
314
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parents: 302
diff changeset
349 def checkProtected(self, opposedThroughMvt):
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parents: 302
diff changeset
350 '''Checks if this left movement should be protected,
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parents: 302
diff changeset
351 ie if one of the main two conditions on left turn is verified'''
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parents: 302
diff changeset
352 return self.volume >= 200 or self.volume*opposedThroughMvt.volume/opposedThroughMvt.nLanes > 50000
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parents: 302
diff changeset
353
205
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diff changeset
354 def getPCUVolume(self):
33
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parents:
diff changeset
355 '''Returns the passenger-car equivalent for the input volume'''
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parents:
diff changeset
356 v = 0
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parents:
diff changeset
357 for p, e in zip(self.proportions, self.equivalents):
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parents:
diff changeset
358 v += p*e
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parents:
diff changeset
359 return v*self.volume
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parents:
diff changeset
360
665
15e244d2a1b5 corrected bug with circular import for VideoFilenameAddable, moved to base module
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 652
diff changeset
361 class IntersectionMovement(object):
33
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parents:
diff changeset
362 '''Represents an intersection movement
4bd7cc69b6cd added traffic engineering utilities with first version of LaneGroups
Nicolas Saunier <nico@confins.net>
parents:
diff changeset
363 with a volume, a type (through, left or right)
4bd7cc69b6cd added traffic engineering utilities with first version of LaneGroups
Nicolas Saunier <nico@confins.net>
parents:
diff changeset
364 and an equivalent for movement type'''
205
aeaaf5579b46 minor changes to traffic engineering
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parents: 198
diff changeset
365 def __init__(self, volume, mvtEquivalent = 1):
33
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parents:
diff changeset
366 self.volume = volume
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parents:
diff changeset
367 self.mvtEquivalent = mvtEquivalent
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parents:
diff changeset
368
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parents:
diff changeset
369 def getTVUVolume(self):
205
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parents: 198
diff changeset
370 return self.mvtEquivalent*self.volume.getPCUVolume()
33
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parents:
diff changeset
371
665
15e244d2a1b5 corrected bug with circular import for VideoFilenameAddable, moved to base module
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diff changeset
372 class LaneGroup(object):
33
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parents:
diff changeset
373 '''Class that represents a group of mouvements'''
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parents:
diff changeset
374
34
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parents: 33
diff changeset
375 def __init__(self, movements, nLanes):
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diff changeset
376 self.movements = movements
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diff changeset
377 self.nLanes = nLanes
33
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parents:
diff changeset
378
34
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diff changeset
379 def getTVUVolume(self):
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diff changeset
380 return sum([mvt.getTVUVolume() for mvt in self.movements])
33
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parents:
diff changeset
381
206
82b4101d9a2f re-arranged and commnted signal cycle calculations
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parents: 205
diff changeset
382 def getCharge(self, saturationVolume):
82b4101d9a2f re-arranged and commnted signal cycle calculations
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parents: 205
diff changeset
383 return self.getTVUVolume()/(self.nLanes*saturationVolume)
82b4101d9a2f re-arranged and commnted signal cycle calculations
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parents: 205
diff changeset
384
82b4101d9a2f re-arranged and commnted signal cycle calculations
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parents: 205
diff changeset
385 def optimalCycle(lostTime, criticalCharge):
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parents: 205
diff changeset
386 return (1.5*lostTime+5)/(1-criticalCharge)
82b4101d9a2f re-arranged and commnted signal cycle calculations
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parents: 205
diff changeset
387
82b4101d9a2f re-arranged and commnted signal cycle calculations
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parents: 205
diff changeset
388 def minimumCycle(lostTime, criticalCharge, degreeSaturation=1.):
82b4101d9a2f re-arranged and commnted signal cycle calculations
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parents: 205
diff changeset
389 'degree of saturation can be used as the peak hour factor too'
82b4101d9a2f re-arranged and commnted signal cycle calculations
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 205
diff changeset
390 return lostTime/(1-criticalCharge/degreeSaturation)
34
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parents: 33
diff changeset
391
665
15e244d2a1b5 corrected bug with circular import for VideoFilenameAddable, moved to base module
Nicolas Saunier <nicolas.saunier@polymtl.ca>
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diff changeset
392 class Cycle(object):
34
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parents: 33
diff changeset
393 '''Class to compute optimal cycle and the split of effective green times'''
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parents: 33
diff changeset
394 def __init__(self, phases, lostTime, saturationVolume):
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parents: 33
diff changeset
395 '''phases is a list of phases
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parents: 33
diff changeset
396 a phase is a list of lanegroups'''
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parents: 33
diff changeset
397 self.phases = phases
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parents: 33
diff changeset
398 self.lostTime = lostTime
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parents: 33
diff changeset
399 self.saturationVolume = saturationVolume
33
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parents:
diff changeset
400
206
82b4101d9a2f re-arranged and commnted signal cycle calculations
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parents: 205
diff changeset
401 def computeCriticalCharges(self):
314
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402 self.criticalCharges = [max([lg.getCharge(self.saturationVolume) for lg in phase]) for phase in self.phases]
34
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parents: 33
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403 self.criticalCharge = sum(self.criticalCharges)
206
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parents: 205
diff changeset
404
82b4101d9a2f re-arranged and commnted signal cycle calculations
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 205
diff changeset
405 def computeOptimalCycle(self):
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406 self.computeCriticalCharges()
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Nicolas Saunier <nicolas.saunier@polymtl.ca>
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diff changeset
407 self.C = optimalCycle(self.lostTime, self.criticalCharge)
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parents: 205
diff changeset
408 return self.C
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Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 205
diff changeset
409
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Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 205
diff changeset
410 def computeMinimumCycle(self, degreeSaturation=1.):
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Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 205
diff changeset
411 self.computeCriticalCharges()
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parents: 205
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412 self.C = minimumCycle(self.lostTime, self.criticalCharge, degreeSaturation)
82b4101d9a2f re-arranged and commnted signal cycle calculations
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parents: 205
diff changeset
413 return self.C
34
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parents: 33
diff changeset
414
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parents: 33
diff changeset
415 def computeEffectiveGreen(self):
314
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parents: 302
diff changeset
416 #from numpy import round
206
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parents: 205
diff changeset
417 #self.computeCycle() # in case it was not done before
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diff changeset
418 effectiveGreenTime = self.C-self.lostTime
34
388a5a25fe92 Code for lab5 works
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parents: 33
diff changeset
419 self.effectiveGreens = [round(c*effectiveGreenTime/self.criticalCharge,1) for c in self.criticalCharges]
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parents: 33
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420 return self.effectiveGreens
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parents: 33
diff changeset
421
33
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parents:
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422
4bd7cc69b6cd added traffic engineering utilities with first version of LaneGroups
Nicolas Saunier <nico@confins.net>
parents:
diff changeset
423 def computeInterGreen(perceptionReactionTime, initialSpeed, intersectionLength, vehicleAverageLength = 6, deceleration = 3):
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parents:
diff changeset
424 '''Computes the intergreen time (yellow/amber plus all red time)
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parents:
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425 Deceleration is positive
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parents:
diff changeset
426 All variables should be in the same units'''
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parents:
diff changeset
427 if deceleration > 0:
36
571b11304ec9 corrected bug
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parents: 34
diff changeset
428 return [perceptionReactionTime+float(initialSpeed)/(2*deceleration), float(intersectionLength+vehicleAverageLength)/initialSpeed]
33
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parents:
diff changeset
429 else:
992
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parents: 796
diff changeset
430 print('Issue deceleration should be strictly positive')
33
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parents:
diff changeset
431 return None
37
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parents: 36
diff changeset
432
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parents: 36
diff changeset
433 def uniformDelay(cycleLength, effectiveGreen, saturationDegree):
911b52744ceb added uniform delay function
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parents: 36
diff changeset
434 '''Computes the uniform delay'''
796
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 790
diff changeset
435 return 0.5*cycleLength*(1-float(effectiveGreen)/cycleLength)**2/(1-float(effectiveGreen*saturationDegree)/cycleLength)
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 790
diff changeset
436
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 790
diff changeset
437 def randomDelay(volume, saturationDegree):
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 790
diff changeset
438 '''Computes the random delay = queueing time for M/D/1'''
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 790
diff changeset
439 return saturationDegree**2/(2*volume*(1-saturationDegree))
87
f234154207d4 distance and time to change speed
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parents: 73
diff changeset
440
790
944949c8ef3e minor name change
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parents: 789
diff changeset
441 def incrementalDelay(T, X, c, k=0.5, I=1):
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parents: 789
diff changeset
442 '''Computes the incremental delay (HCM)
493
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Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 479
diff changeset
443 T in hours
850ed17c7b2f added some computation of delay
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 479
diff changeset
444 c capacity of the lane group
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Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 479
diff changeset
445 k default for fixed time signal
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Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 479
diff changeset
446 I=1 for isolated intersection (Poisson arrival)'''
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Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 479
diff changeset
447 from math import sqrt
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Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 479
diff changeset
448 return 900*T*(X - 1 + sqrt((X - 1)**2 + 8*k*I*X/(c*T)))
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parents: 479
diff changeset
449
87
f234154207d4 distance and time to change speed
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diff changeset
450 #########################
f234154207d4 distance and time to change speed
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diff changeset
451 # misc
f234154207d4 distance and time to change speed
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diff changeset
452 #########################
f234154207d4 distance and time to change speed
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 73
diff changeset
453
f234154207d4 distance and time to change speed
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 73
diff changeset
454 def timeChangingSpeed(v0, vf, a, TPR):
789
3666342dabe2 minor changes
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 665
diff changeset
455 'for decelerations, a < 0'
3666342dabe2 minor changes
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 665
diff changeset
456 return TPR-(vf-v0)/a
87
f234154207d4 distance and time to change speed
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parents: 73
diff changeset
457
f234154207d4 distance and time to change speed
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 73
diff changeset
458 def distanceChangingSpeed(v0, vf, a, TPR):
789
3666342dabe2 minor changes
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 665
diff changeset
459 'for decelerations, a < 0'
1165
c5863e04d302 corrected bug
Nicolas Saunier <nicolas.saunier@polymtl.ca>
parents: 1029
diff changeset
460 return TPR*v0+(vf**2-v0**2)/(2*a)