Dynamic Models

Gym-Khana provides four vehicle dynamics models of increasing complexity. All use numba JIT compilation for real-time performance.

Source: gymkhana/envs/dynamic_models/

KS — Kinematic Single Track

gymkhana/envs/dynamic_models/kinematic.py

Simplest model using pure kinematics without tire forces. Suitable for low-speed scenarios where tire slip is negligible.

ST — Single Track

gymkhana/envs/dynamic_models/single_track.py

Single-track dynamics model with basic tire modeling. Models lateral dynamics but without an explicit tire force model.

MB — Multi-Body

gymkhana/envs/dynamic_models/multi_body/

Most detailed model with full tire modeling and multi-body dynamics. Provides the highest fidelity but parameters are only available for a full-scale vehicle (not 1/10 scale).

STD — Single Track Drift

gymkhana/envs/dynamic_models/single_track_drift/

Single-track dynamics with PAC2002 (Pacejka Magic Formula) tire model. Recommended for RL drift training. This model captures realistic tire force saturation and slip behavior needed for drifting.

Use with:

config = {
    'model': 'std',
    'control_input': ['accl', 'steering_angle'],
    'params': GKEnv.f1tenth_std_vehicle_params(),
}

Vehicle parameters

The dynamic model’s physical parameters include:

  • mu: surface friction coefficient

  • C_Sf / C_Sr: cornering stiffness coefficient, front/rear [1/rad]

  • lf / lr: distance from CG to front/rear axle [m]

  • h: height of center of gravity [m]

  • m: total vehicle mass [kg]

  • I: moment of inertia about z axis [kg m^2]

  • s_min / s_max: steering angle constraints [rad]

  • sv_min / sv_max: steering velocity constraints [rad/s]

  • v_switch: velocity at which acceleration can no longer create wheel spin [m/s]

  • a_max: maximum longitudinal acceleration [m/s^2]

  • v_min / v_max: longitudinal velocity bounds [m/s]

  • width / length: vehicle dimensions [m]

Parameters can be passed via the params key in the config dict, or updated at runtime with env.update_params().

API reference

Kinematic Single Track (KS) vehicle dynamics model.

gymkhana.envs.dynamic_models.kinematic.vehicle_dynamics_ks(x: ndarray, u_init: ndarray, params: dict)

Compute Kinematic Single Track vehicle dynamics.

Reference: CommonRoad vehicle models, section 5.

Parameters:

  • x – State vector of shape (5,): [x_pos, y_pos, steering_angle, velocity, yaw_angle].

  • u_init – Control input [steering_velocity, acceleration].

  • params – Vehicle parameters dict. Uses lf, lr, s_min, s_max, sv_min, sv_max, v_switch, a_max, v_min, v_max. See gymkhana.envs.dynamic_models for full parameter descriptions.

Returns:

Time derivatives of the state vector, shape (5,).

gymkhana.envs.dynamic_models.kinematic.vehicle_dynamics_ks_cog(x: ndarray, u_init: ndarray, params: dict)

Compute Kinematic Single Track dynamics referenced at the centre of gravity.

Unlike vehicle_dynamics_ks() (which references the rear axle), this variant computes position derivatives at the vehicle’s centre of gravity by incorporating the kinematic slip angle beta. Used internally by the STD model for the low-speed kinematic blending regime.

Reference: CommonRoad vehicle models, section 5.

Parameters:

  • x – State vector of shape (5,): [x_pos, y_pos, steering_angle, velocity, yaw_angle].

  • u_init – Control input [steering_velocity, acceleration].

  • params – Vehicle parameters dict. Uses lf, lr, s_min, s_max, sv_min, sv_max, v_switch, a_max, v_min, v_max. See gymkhana.envs.dynamic_models for full parameter descriptions.

Returns:

Time derivatives of the state vector, shape (5,).

gymkhana.envs.dynamic_models.kinematic.get_standardized_state_ks(x: ndarray) → dict

Extract standardized state dict from KS model state vector.

Parameters:

x – KS state vector [x_pos, y_pos, steering_angle, velocity, yaw_angle].

Returns:

x, y, delta, v_x, v_y, yaw, yaw_rate, slip. v_y, yaw_rate, and slip are zero (kinematic model).

Return type:

Dict with keys

Single Track (ST) vehicle dynamics model.

gymkhana.envs.dynamic_models.single_track.vehicle_dynamics_st(x: ndarray, u_init: ndarray, params: dict)

Compute Single Track vehicle dynamics.

Reference: CommonRoad vehicle models, section 7.

Parameters:

  • x – State vector of shape (7,): [x_pos, y_pos, steering_angle, velocity, yaw_angle, yaw_rate, slip_angle].

  • u_init – Control input [steering_velocity, acceleration].

  • params – Vehicle parameters dict. Uses mu, C_Sf, C_Sr, lf, lr, h, m, I, s_min, s_max, sv_min, sv_max, v_switch, a_max, v_min, v_max. See gymkhana.envs.dynamic_models for full parameter descriptions.

Returns:

Time derivatives of the state vector, shape (7,).

gymkhana.envs.dynamic_models.single_track.get_standardized_state_st(x: ndarray) → dict

Extract standardized state dict from ST model state vector.

Parameters:

x – ST state vector [x_pos, y_pos, steering_angle, velocity, yaw_angle, yaw_rate, slip_angle].

Returns:

x, y, delta, v_x, v_y, yaw, yaw_rate, slip.

Return type:

Dict with keys

Multi-Body (MB) vehicle dynamics model.

gymkhana.envs.dynamic_models.multi_body.multi_body.vehicle_dynamics_mb(x: ndarray, u_init: ndarray, params: dict)

Compute Multi-Body vehicle dynamics.

DoT (Department of Transportation) based multi-body model. Reference point is the center of mass. State vector is 29-dimensional; use DynamicModel.get_initial_state() to initialize correctly.

Parameters:

  • x – State vector of shape (29,) (see source for component index map).

  • u_init – Control input [steering_velocity, acceleration].

  • params – Vehicle parameters dict (see gymkhana.envs.dynamic_models).

Returns:

Time derivatives of the state vector, shape (29,).

gymkhana.envs.dynamic_models.multi_body.multi_body.get_standardized_state_mb(x: ndarray) → dict

[X,Y,DELTA,V_X, V_Y,YAW,YAW_RATE,SLIP]

Single Track Drift (STD) vehicle dynamics model with PAC2002 tire model.

gymkhana.envs.dynamic_models.single_track_drift.single_track_drift.vehicle_dynamics_std(x: ndarray, u_init: ndarray, params: dict)

Compute Single Track Drift vehicle dynamics.

Extends the ST model with a PAC2002 (Pacejka Magic Formula) tire model to capture tire force saturation and slip needed for drifting.

Reference: CommonRoad STD model (vehicle_dynamics_std.py).

Author: Teodor Ilie (18-October-2025).

Parameters:

  • x – State vector of shape (9,): [x_pos, y_pos, steering_angle, velocity, yaw_angle, yaw_rate, slip_angle, front_wheel_omega, rear_wheel_omega].

  • u_init – Control input [steering_velocity, acceleration].

  • params – Vehicle parameters dict. Uses base params plus STD-specific keys h_s, R_w, I_z, I_y_w, T_sb, T_se, and PAC2002 tyre coefficients. See gymkhana.envs.dynamic_models for full parameter descriptions.

Returns:

Time derivatives of the state vector, shape (9,).

gymkhana.envs.dynamic_models.single_track_drift.single_track_drift.get_standardized_state_std(x: ndarray) → dict

Extract standardized state dict from STD model state vector.

Parameters:

x – STD state vector [x_pos, y_pos, steering_angle, velocity, yaw_angle, yaw_rate, slip_angle, omega_front, omega_rear].

Returns:

x, y, delta, v_x, v_y, yaw, yaw_rate, slip.

Return type:

Dict with keys