Inverted Double Pendulum#
This environment is part of the Mujoco environments. Please read that page first for general information.
Action Space 
Box(1.0, 1.0, (1,), float32) 
Observation Shape 
(11,) 
Observation High 
[inf inf inf inf inf inf inf inf inf inf inf] 
Observation Low 
[inf inf inf inf inf inf inf inf inf inf inf] 
Import 

Description#
This environment originates from control theory and builds on the cartpole environment based on the work done by Barto, Sutton, and Anderson in “Neuronlike adaptive elements that can solve difficult learning control problems”, powered by the Mujoco physics simulator  allowing for more complex experiments (such as varying the effects of gravity or constraints). This environment involves a cart that can moved linearly, with a pole fixed on it and a second pole fixed on the other end of the first one (leaving the second pole as the only one with one free end). The cart can be pushed left or right, and the goal is to balance the second pole on top of the first pole, which is in turn on top of the cart, by applying continuous forces on the cart.
Action Space#
The agent take a 1element vector for actions.
The action space is a continuous (action)
in [1, 1]
, where action
represents the
numerical force applied to the cart (with magnitude representing the amount of force and
sign representing the direction)
Num 
Action 
Control Min 
Control Max 
Name (in corresponding XML file) 
Joint 
Unit 

0 
Force applied on the cart 
1 
1 
slider 
slide 
Force (N) 
Observation Space#
The state space consists of positional values of different body parts of the pendulum system, followed by the velocities of those individual parts (their derivatives) with all the positions ordered before all the velocities.
The observation is a ndarray
with shape (11,)
where the elements correspond to the following:
Num 
Observation 
Min 
Max 
Name (in corresponding XML file) 
Joint 
Unit 

0 
position of the cart along the linear surface 
Inf 
Inf 
slider 
slide 
position (m) 
1 
sine of the angle between the cart and the first pole 
Inf 
Inf 
sin(hinge) 
hinge 
unitless 
2 
sine of the angle between the two poles 
Inf 
Inf 
sin(hinge2) 
hinge 
unitless 
3 
cosine of the angle between the cart and the first pole 
Inf 
Inf 
cos(hinge) 
hinge 
unitless 
4 
cosine of the angle between the two poles 
Inf 
Inf 
cos(hinge2) 
hinge 
unitless 
5 
velocity of the cart 
Inf 
Inf 
slider 
slide 
velocity (m/s) 
6 
angular velocity of the angle between the cart and the first pole 
Inf 
Inf 
hinge 
hinge 
angular velocity (rad/s) 
7 
angular velocity of the angle between the two poles 
Inf 
Inf 
hinge2 
hinge 
angular velocity (rad/s) 
8 
constraint force  1 
Inf 
Inf 
Force (N) 

9 
constraint force  2 
Inf 
Inf 
Force (N) 

10 
constraint force  3 
Inf 
Inf 
Force (N) 
There is physical contact between the robots and their environment  and Mujoco attempts at getting realistic physics simulations for the possible physical contact dynamics by aiming for physical accuracy and computational efficiency.
There is one constraint force for contacts for each degree of freedom (3). The approach and handling of constraints by Mujoco is unique to the simulator and is based on their research. Once can find more information in their documentation or in their paper “Analyticallyinvertible dynamics with contacts and constraints: Theory and implementation in MuJoCo”.
Rewards#
The reward consists of two parts:
alive_bonus: The goal is to make the second inverted pendulum stand upright (within a certain angle limit) as long as possible  as such a reward of +10 is awarded for each timestep that the second pole is upright.
distance_penalty: This reward is a measure of how far the tip of the second pendulum (the only free end) moves, and it is calculated as 0.01 * x^{2} + (y  2)^{2}, where x is the xcoordinate of the tip and y is the ycoordinate of the tip of the second pole.
velocity_penalty: A negative reward for penalising the agent if it moves too fast 0.001 * v_{1}^{2} + 0.005 * v_{2} ^{2}
The total reward returned is reward = alive_bonus  distance_penalty  velocity_penalty
Starting State#
All observations start in state (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0) with a uniform noise in the range of [0.1, 0.1] added to the positional values (cart position and pole angles) and standard normal force with a standard deviation of 0.1 added to the velocity values for stochasticity.
Episode Termination#
The episode terminates when any of the following happens:
The episode duration reaches 1000 timesteps.
Any of the state space values is no longer finite.
The y_coordinate of the tip of the second pole is less than or equal to 1. The maximum standing height of the system is 1.196 m when all the parts are perpendicularly vertical on top of each other).
Arguments#
No additional arguments are currently supported (in v2 and lower), but modifications can be made to the XML file in the assets folder (or by changing the path to a modified XML file in another folder)..
env = gym.make('InvertedDoublePendulumv2')
There is no v3 for InvertedPendulum, unlike the robot environments where a v3 and beyond take gym.make kwargs such as xml_file, ctrl_cost_weight, reset_noise_scale etc.
There is a v4 version that uses the mujocobindings
env = gym.make('InvertedDoublePendulumv4')
Version History#
v4: all mujoco environments now use the mujoco bindings in mujoco>=2.1.3
v3: support for gym.make kwargs such as xml_file, ctrl_cost_weight, reset_noise_scale etc. rgb rendering comes from tracking camera (so agent does not run away from screen)
v2: All continuous control environments now use mujoco_py >= 1.50
v1: max_time_steps raised to 1000 for robot based tasks (including inverted pendulum)
v0: Initial versions release (1.0.0)