Symbolic dynamics (CasADi)

Optimization-based control methods like nonlinear MPC, trajectory optimization and moving-horizon estimation need symbolic dynamics: an expression graph the solver can differentiate through and evaluate analytically. Every dynamics in crazyflow.dynamics has a symbolic_dynamics function that returns CasADi MX expressions, validated to be numerically equivalent to the numeric dynamics implementation.

The return signature is always (X_dot, X, U, Y):

• X — state vector (CasADi MX column vector)
• U — input vector
• X_dot — state derivative, as a symbolic expression in X and U
• Y — output vector (position + attitude)

Physical parameters are bound with parametrize, exactly as for the numeric dynamics.

first_principles

import casadi as cs
from crazyflow.dynamics.first_principles import symbolic_dynamics
from crazyflow.dynamics.core import parametrize

symbolic_dynamics = parametrize(symbolic_dynamics, "cf2x_L250")

# include rotor velocity in the state vector
X_dot, X, U, Y = symbolic_dynamics(model_rotor_vel=True, model_dist_f=False, model_dist_t=False)

f = cs.Function("f", [X, U], [X_dot])

State vector layout with model_rotor_vel=True:

Indices	Variable	Units
0–2	pos	m
3–6	quat (xyzw)	—
7–9	vel	m/s
10–12	ang_vel	rad/s
13–16	rotor_vel	RPM

See crazyflow.dynamics.first_principles in the API reference for the full list of accepted parameters.

Fitted dynamics — quaternion form

symbolic_dynamics on the fitted dynamics converts them to quaternion + angular velocity state, matching the dynamics function signature. This makes it straightforward to swap between first_principles and a fitted dynamics in a solver without changing the state layout.

from crazyflow.dynamics.so_rpy_rotor_drag import symbolic_dynamics
from crazyflow.dynamics.core import parametrize

X_dot, X, U, Y = parametrize(symbolic_dynamics, "cf2x_L250")(model_rotor_vel=True)

Fitted dynamics — Euler form

The fitted dynamics also expose symbolic_dynamics_euler, which works directly in roll/pitch/yaw + RPY-rate state. This is the natural representation of these dynamics — they are fitted in Euler angles — and it avoids the trigonometric overhead of converting to and from quaternions inside the solver. For most NMPC applications on the fitted dynamics, this is the variant to use.

from crazyflow.dynamics.so_rpy_rotor_drag import symbolic_dynamics_euler
from crazyflow.dynamics.core import parametrize

symbolic_dynamics_euler = parametrize(symbolic_dynamics_euler, "cf2x_L250")
X_dot, X, U, Y = symbolic_dynamics_euler(model_rotor_vel=True)

State vector layout with model_rotor_vel=True:

Indices	Variable	Units
0–2	pos	m
3–5	rpy (roll/pitch/yaw)	rad
6–8	vel	m/s
9–11	drpy (RPY rates)	rad/s
12	thrust state	N

Wrapping for Acados / IPOPT

Both functions return raw CasADi expressions. Wrap them in a cs.Function to pass to any CasADi-based solver:

import casadi as cs
from crazyflow.dynamics.so_rpy_rotor_drag import symbolic_dynamics_euler
from crazyflow.dynamics.core import parametrize

symbolic_dynamics_euler = parametrize(symbolic_dynamics_euler, "cf2x_L250")
X_dot, X, U, Y = symbolic_dynamics_euler(model_rotor_vel=True)

f = cs.Function("f", [X, U], [X_dot])
# Pass f directly to Acados, IPOPT, or any CasADi-based solver

With disturbance states

Setting model_dist_f=True or model_dist_t=True appends the disturbance vectors to the state, which is useful for augmented-state estimators:

from crazyflow.dynamics.first_principles import symbolic_dynamics
from crazyflow.dynamics.core import parametrize

symbolic_dynamics = parametrize(symbolic_dynamics, "cf2x_L250")

# dist_f (3,) and dist_t (3,) appended to state
X_dot, X, U, Y = symbolic_dynamics(model_rotor_vel=True, model_dist_f=True, model_dist_t=True)
# X is now 17 + 3 + 3 = 23 elements long

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The dynamics covered so far all come with pre-fitted parameters for the supported Crazyflie platforms. For other drones, the next page explains how to extract parameters from your own flight data.