Half Cheetah#
This environment is part of the Mujoco environments which contains general information about the environment.
Action Space 

Observation Space 

import 

Description#
This environment is based on the work of P. Wawrzyński in “A CatLike Robot RealTime Learning to Run”. The HalfCheetah is a 2dimensional 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 xcoordinate (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 xcoordinate of the robot.
Regardless of whether exclude_current_positions_from_observation
is set to True
or False
, the x and ycoordinates 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 
zcoordinate 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 xcoordinate of front tip 
Inf 
Inf 
rootx 
slide 
velocity (m/s) 
9 
velocity of the zcoordinate 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 
xcoordinate 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” (\(x_{afteraction}  x_{beforeaction}\)), \(dt\) is the time between actions, which depends on the
frame_skip
parameter (default is \(5\)), andframetime
which is \(0.01\)  so the default is \(dt = 5 \times 0.01 = 0.05\), \(w_{forward}\) is theforward_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('HalfCheetahv5', ctrl_cost_weight=0.1, ....)
Parameter 
Type 
Default 
Description 


str 

Path to a MuJoCo model 

float 

Weight for forward_reward term (see 

float 

Weight for ctrl_cost weight (see 

float 

Scale of random perturbations of initial position and velocity (see 

bool 

Whether or not to omit the xcoordinate from observations. Excluding the position can serve as an inductive bias to induce positionagnostic behavior in policies (see 
Version History#
v5:
Minimum
mujoco
version is now 2.3.3.Added support for fully custom/third party
mujoco
models using thexml_file
argument (previously only a few changes could be made to the existing models).Added
default_camera_config
argument, a dictionary for setting themj_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 nonempty
info
withreset()
, previously an empty dictionary was returned, the new keys are the same state information asstep()
.Added
frame_skip
argument, used to configure thedt
(duration ofstep()
), default varies by environment check environment documentation pages.Restored the
xml_file
argument (was removed inv4
).Renamed
info["reward_run"]
toinfo["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 asxml_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 mujocopy >= 1.50.
v1: max_time_steps raised to 1000 for robot based tasks. Added reward_threshold to environments.
v0: Initial versions release.