Inverted Double Pendulum

This environment is part of the Mujoco environments which contains general information about the environment.

Action Space	Box(-1.0, 1.0, (1,), float32)
Observation Space	Box(-inf, inf, (11,), float64)
import	gymnasium.make("InvertedDoublePendulum-v4")

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 1-element 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 “Analytically-invertible 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 * x² + (y - 2)², where x is the x-coordinate of the tip and y is the y-coordinate of the tip of the second pole.

• velocity_penalty: A negative reward for penalising the agent if it moves too fast 0.001 * v₁² + 0.005 * v₂ 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 End

The episode ends when any of the following happens:

1.Truncation: The episode duration reaches 1000 timesteps. 2.Termination: Any of the state space values is no longer finite. 3.Termination: 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.

import gymnasium as gym
env = gym.make('InvertedDoublePendulum-v4')

There is no v3 for InvertedPendulum, unlike the robot environments where a v3 and beyond take gymnasium.make kwargs such as xml_file, ctrl_cost_weight, reset_noise_scale, etc.

import gymnasium as gym
env = gym.make('InvertedDoublePendulum-v2')

Version History

• v4: All MuJoCo environments now use the MuJoCo bindings in mujoco >= 2.1.3

• v3: Support for gymnasium.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)