# 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

7

angular velocity of the angle between the two poles

-Inf

Inf

hinge2

hinge

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)