MTQ

class MTQ : public Actuator

#include <MTQ.h>

Magnetorquer (MTQ) actuator class.

Represents a magnetic torque coil that produces torque through interaction with Earth’s magnetic field. The torque is:

\[ \boldsymbol{\tau} = \mathbf{m} \times \mathbf{B} \]

where \(\mathbf{m} = u \cdot \text{axis}\) is the magnetic dipole moment (A·m²) and \(\mathbf{B}\) is the magnetic field in body frame.

Unlike reaction wheels, MTQs have no internal momentum state. The torque is always reaction-free (momentum is transferred to the Earth’s field).

See also

Actuator for base class interface

Public Types

using Vec3 = Actuator::Vec3

using BaseState = Actuator::BaseState

using Mat13 = Actuator::Mat13

using Mat73 = Actuator::Mat73

using T113 = Actuator::T113

using T173 = Actuator::T173

using T773 = Actuator::T773

Public Functions

MTQ(const Vec3 &axis, double max_dipole)

Construct a magnetorquer with specified parameters.

Parameters:

• axis – Unit vector specifying the coil direction in body frame.

• max_dipole – Maximum magnetic dipole moment magnitude (A·m²).

Vec3 torque(double u, const BaseState &x, const Vec3 &B_body) const

Compute the magnetic torque produced by this MTQ.

The torque is the cross product of magnetic dipole moment and field:

\[ \boldsymbol{\tau} = (u \cdot \text{axis}) \times \mathbf{B}_{\text{body}} \]

Parameters:

• u – Control input (dipole moment magnitude, A·m²).

• x – Base state (contains quaternion for ECI→body transformation).

• B_body – Magnetic field vector in body frame (Tesla).

Returns:

3D torque vector (Newton-meters).

Mat13 dtorq_du(double u, const BaseState &x, const Vec3 &B_body) const

Jacobian of torque with respect to control input.

Parameters:

• u – Control input.

• x – Base state.

• B_body – Magnetic field in body frame.

Returns:

1×3 Jacobian matrix.

Mat73 dtorq_dbasestate(double u, const BaseState &x, const Vec3 &B_body, const Eigen::Matrix<double, 4, 3> &dB_dq) const

Jacobian of torque with respect to base state.

Account for the fact that magnetic field depends on attitude (quaternion).

Parameters:

• u – Control input.

• x – Base state.

• B_body – Magnetic field in body frame.

• dB_dq – Jacobian of magnetic field with respect to quaternion (4×3).

Returns:

7×3 Jacobian matrix.

T113 ddtorq_dudu(double u, const BaseState &x, const Vec3 &B_body) const

Hessian of torque control input (second derivatives).

Parameters:

• u – Control input.

• x – Base state.

• B_body – Magnetic field in body frame.

Returns:

Tensor3 with 3 slices of 1×1 matrices.

T173 ddtorq_dudbasestate(double u, const BaseState &x, const Vec3 &B_body, const Eigen::Matrix<double, 4, 3> &dB_dq) const

Hessian of mixed torque derivatives (control-state).

Parameters:

• u – Control input.

• x – Base state.

• B_body – Magnetic field in body frame.

• dB_dq – Jacobian of magnetic field with respect to quaternion.

Returns:

Tensor3 with 3 slices of 1×7 matrices.

T773 ddtorq_dbasestatedbasestate(double u, const BaseState &x, const Eigen::Matrix<double, 4, 3> &dB_dq, const std::array<Eigen::Matrix<double, 4, 4>, 3> &d2B_dq2) const

Hessian of state-state torque derivatives.

Parameters:

• u – Control input.

• x – Base state.

• dB_dq – Jacobian of magnetic field with respect to quaternion.

• d2B_dq2 – Array of three 4×4 Hessian matrices for magnetic field.

Returns:

Tensor3 with 3 slices of 7×7 matrices.

virtual Vec3 torque(double u, const BaseState &x) const

Compute the torque vector produced by this actuator.

Returns the torque vector as a function of control input and spacecraft state.

Parameters:

• u – Control input (typically in [-u_max, u_max]).

• x – Base state vector (7D: [av; q; …]).

Returns:

3D torque vector in body frame (Newton-meters).

virtual Mat13 dtorq_du(double u, const BaseState &x) const

Jacobian of torque with respect to control input.

Returns \(\frac{\partial \boldsymbol{\tau}}{\partial u}\) as a 1×3 row vector.

Parameters:

• u – Control input.

• x – Base state.

Returns:

1×3 Jacobian matrix.

virtual Mat73 dtorq_dbasestate(double u, const BaseState &x) const

Jacobian of torque with respect to base state.

Returns \(\frac{\partial \boldsymbol{\tau}}{\partial \mathbf{x}}\) as a 7×3 matrix.

Parameters:

• u – Control input.

• x – Base state (7D).

Returns:

7×3 Jacobian matrix.

virtual T113 ddtorq_dudu(double u, const BaseState &x) const

Hessian of torque with respect to control input (second derivatives).

Returns a 3-slice tensor of \(\frac{\partial^2 \tau_i}{\partial u^2}\).

Parameters:

• u – Control input.

• x – Base state.

Returns:

Tensor3 with 3 slices of 1×1 matrices (one per torque component).

virtual T173 ddtorq_dudbasestate(double u, const BaseState &x) const

Hessian of torque with respect to control input and base state.

Returns mixed partial derivatives \(\frac{\partial^2 \tau_i}{\partial u \partial x_j}\).

Parameters:

• u – Control input.

• x – Base state.

Returns:

Tensor3 with 3 slices of 1×7 matrices.

virtual T773 ddtorq_dbasestatedbasestate(double u, const BaseState &x) const

Hessian of torque with respect to base state (full second derivatives).

Returns \(\frac{\partial^2 \tau_i}{\partial x_j \partial x_k}\) for each component i.

Parameters:

• u – Control input.

• x – Base state.

Returns:

Tensor3 with 3 slices of 7×7 matrices.