Half Cheetah

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

Action Space	Box(-1.0, 1.0, (6,), float32)
Observation Space	Box(-inf, inf, (17,), float64)
import	gymnasium.make("HalfCheetah-v5")

Description

This environment is based on the work of P. Wawrzyński in “A Cat-Like Robot Real-Time Learning to Run”. The HalfCheetah is a 2-dimensional robot consisting of 9 body parts and 8 joints connecting them (including two paws). The goal is to apply torque to the joints to make the cheetah run forward (right) as fast as possible, with a positive reward based on the distance moved forward and a negative reward for moving backward. The cheetah’s torso and head are fixed, and torque can only be applied to the other 6 joints over the front and back thighs (which connect to the torso), the shins (which connect to the thighs), and the feet (which connect to the shins).

Action Space

The action space is a Box(-1, 1, (6,), float32). An action represents the torques applied at the hinge joints.

Num	Action	Control Min	Control Max	Name (in corresponding XML file)	Joint	Type (Unit)
0	Torque applied on the back thigh rotor	-1	1	bthigh	hinge	torque (N m)
1	Torque applied on the back shin rotor	-1	1	bshin	hinge	torque (N m)
2	Torque applied on the back foot rotor	-1	1	bfoot	hinge	torque (N m)
3	Torque applied on the front thigh rotor	-1	1	fthigh	hinge	torque (N m)
4	Torque applied on the front shin rotor	-1	1	fshin	hinge	torque (N m)
5	Torque applied on the front foot rotor	-1	1	ffoot	hinge	torque (N m)

Observation Space

The observation space consists of the following parts (in order):

• qpos (8 elements by default): Position values of the robot’s body parts.

• qvel (9 elements): The velocities of these individual body parts (their derivatives).

By default, the observation does not include the robot’s x-coordinate (rootx). This can be included by passing exclude_current_positions_from_observation=False during construction. In this case, the observation space will be a Box(-Inf, Inf, (18,), float64), where the first observation element is the x-coordinate of the robot. Regardless of whether exclude_current_positions_from_observation is set to True or False, the x- and y-coordinates are returned in info with the keys "x_position" and "y_position", respectively.

By default, however, the observation space is a Box(-Inf, Inf, (17,), float64) where the elements are as follows:

Num	Observation	Min	Max	Name (in corresponding XML file)	Joint	Type (Unit)
0	z-coordinate of the front tip	-Inf	Inf	rootz	slide	position (m)
1	angle of the front tip	-Inf	Inf	rooty	hinge	angle (rad)
2	angle of the back thigh	-Inf	Inf	bthigh	hinge	angle (rad)
3	angle of the back shin	-Inf	Inf	bshin	hinge	angle (rad)
4	angle of the back foot	-Inf	Inf	bfoot	hinge	angle (rad)
5	angle of the front thigh	-Inf	Inf	fthigh	hinge	angle (rad)
6	angle of the front shin	-Inf	Inf	fshin	hinge	angle (rad)
7	angle of the front foot	-Inf	Inf	ffoot	hinge	angle (rad)
8	velocity of the x-coordinate of front tip	-Inf	Inf	rootx	slide	velocity (m/s)
9	velocity of the z-coordinate of front tip	-Inf	Inf	rootz	slide	velocity (m/s)
10	angular velocity of the front tip	-Inf	Inf	rooty	hinge	angular velocity (rad/s)
11	angular velocity of the back thigh	-Inf	Inf	bthigh	hinge	angular velocity (rad/s)
12	angular velocity of the back shin	-Inf	Inf	bshin	hinge	angular velocity (rad/s)
13	angular velocity of the back foot	-Inf	Inf	bfoot	hinge	angular velocity (rad/s)
14	angular velocity of the front thigh	-Inf	Inf	fthigh	hinge	angular velocity (rad/s)
15	angular velocity of the front shin	-Inf	Inf	fshin	hinge	angular velocity (rad/s)
16	angular velocity of the front foot	-Inf	Inf	ffoot	hinge	angular velocity (rad/s)
excluded	x-coordinate of the front tip	-Inf	Inf	rootx	slide	position (m)

Rewards

The total reward is: reward = forward_reward - ctrl_cost.

• forward_reward: A reward for moving forward, this reward would be positive if the Half Cheetah moves forward (in the positive \(x\) direction / in the right direction). \(w_{forward} \times \frac{dx}{dt}\), where \(dx\) is the displacement of the “tip” (\(x_{after-action} - x_{before-action}\)), \(dt\) is the time between actions, which depends on the frame_skip parameter (default is \(5\)), and frametime which is \(0.01\) - so the default is \(dt = 5 \times 0.01 = 0.05\), \(w_{forward}\) is the forward_reward_weight (default is \(1\)).

• ctrl_cost: A negative reward to penalize the Half Cheetah for taking actions that are too large. \(w_{control} \times \|action\|_2^2\), where \(w_{control}\) is ctrl_cost_weight (default is \(0.1\)).

info contains the individual reward terms.

Starting State

The initial position state is \(\mathcal{U}_{[-reset\_noise\_scale \times I_{9}, reset\_noise\_scale \times I_{9}]}\). The initial velocity state is \(\mathcal{N}(0_{9}, reset\_noise\_scale^2 \times I_{9})\).

where \(\mathcal{N}\) is the multivariate normal distribution and \(\mathcal{U}\) is the multivariate uniform continuous distribution.

Episode End

Termination

The Half Cheetah never terminates.

Truncation

The default duration of an episode is 1000 timesteps.

Arguments

HalfCheetah provides a range of parameters to modify the observation space, reward function, initial state, and termination condition. These parameters can be applied during gymnasium.make in the following way:

import gymnasium as gym
env = gym.make('HalfCheetah-v5', ctrl_cost_weight=0.1, ....)

Parameter	Type	Default	Description
xml_file	str	"half_cheetah.xml"	Path to a MuJoCo model
forward_reward_weight	float	1	Weight for forward_reward term (see Rewards section)
ctrl_cost_weight	float	0.1	Weight for ctrl_cost weight (see Rewards section)
reset_noise_scale	float	0.1	Scale of random perturbations of initial position and velocity (see Starting State section)
exclude_current_positions_from_observation	bool	True	Whether or not to omit the x-coordinate from observations. Excluding the position can serve as an inductive bias to induce position-agnostic behavior in policies (see Observation State section)

Version History

• v5:

  • Minimum mujoco version is now 2.3.3.

  • Added support for fully custom/third party mujoco models using the xml_file argument (previously only a few changes could be made to the existing models).

  • Added default_camera_config argument, a dictionary for setting the mj_camera properties, mainly useful for custom environments.

  • Added env.observation_structure, a dictionary for specifying the observation space compose (e.g. qpos, qvel), useful for building tooling and wrappers for the MuJoCo environments.

  • Return a non-empty info with reset(), previously an empty dictionary was returned, the new keys are the same state information as step().

  • Added frame_skip argument, used to configure the dt (duration of step()), default varies by environment check environment documentation pages.

  • Restored the xml_file argument (was removed in v4).

  • Renamed info["reward_run"] to info["reward_forward"] to be consistent with the other environments.

• 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. Added reward_threshold to environments.

• v0: Initial versions release.