Updated Calculation Order (#1019)

* Updated order Calculation

Updated order of `x` and `theta` calculations so that they are no longer one timestamp behind `x_dot` and `theta_dot`.

#1018

* Added semi-implicit euler option

* Got implicit and standard euler mixed up

* switched default option

gym/envs/classic_control/cartpole.py

@@ -62,6 +62,7 @@ def __init__(self):
         self.polemass_length = (self.masspole * self.length)
         self.force_mag = 10.0
         self.tau = 0.02  # seconds between state updates
+        self.kinematics_integrator = 'euler'
 
         # Angle at which to fail the episode
         self.theta_threshold_radians = 12 * 2 * math.pi / 360
@@ -97,10 +98,16 @@ def step(self, action):
         temp = (force + self.polemass_length * theta_dot * theta_dot * sintheta) / self.total_mass
         thetaacc = (self.gravity * sintheta - costheta* temp) / (self.length * (4.0/3.0 - self.masspole * costheta * costheta / self.total_mass))
         xacc  = temp - self.polemass_length * thetaacc * costheta / self.total_mass
-        x  = x + self.tau * x_dot
-        x_dot = x_dot + self.tau * xacc
-        theta = theta + self.tau * theta_dot
-        theta_dot = theta_dot + self.tau * thetaacc
+        if self.kinematics_integrator == 'euler':
+            x  = x + self.tau * x_dot
+            x_dot = x_dot + self.tau * xacc
+            theta = theta + self.tau * theta_dot
+            theta_dot = theta_dot + self.tau * thetaacc
+        else: # semi-implicit euler
+            x_dot = x_dot + self.tau * xacc
+            x  = x + self.tau * x_dot
+            theta_dot = theta_dot + self.tau * thetaacc
+            theta = theta + self.tau * theta_dot
         self.state = (x,x_dot,theta,theta_dot)
         done =  x < -self.x_threshold \
                 or x > self.x_threshold \