EOM 6DOF Simple Variable Mass Euler Angles — Robotics/Equations Of Motion

EOM 6DOF Simple Variable Mass Euler Angles

Robotics / Equations Of Motion

The six-degree-of-freedom equations of motion of a rigid body that burns fuel, carried in body axes with the attitude as Euler angles (the 3-2-1 sequence). The mass is a state driven by the mass flow rate and held between the empty and full masses; the inertia tensor follows it linearly between its empty and full values:

• V_b' = F/m − ω × V_b
• ω' = I⁻¹(M − ω × (Iω) − dI/dt·ω)
• I(m) = I_e + (m − m_e)/(m_f − m_e)·(I_f − I_e), dI/dt = m'·(I_f − I_e)/(m_f − m_e)
• m' = dm/dt, except zero while the tank is empty and draining or full and filling
• the Euler-angle kinematics of EOM 6DOF Euler Angles, singular at θ = ±90°
• X_e' = DCM_be^T V_b

Ports

• F – the applied force in body axes, [3,1].
• M – the applied moment about the centre of gravity in body axes, [3,1].
• m_dot – the mass flow rate dm/dt (negative while burning), a scalar [1,1].
• Ve – the velocity in the flat-earth frame, DCM_be^T V_b, [3,1].
• Xe – the position in the flat-earth frame, [3,1].
• Euler – the Euler angles [φ; θ; ψ] in radians, [3,1].
• DCMbe – the direction cosine matrix from the flat-earth frame to body axes, [3,3].
• Vb – the velocity in body axes [u; v; w], [3,1].
• wb – the body rates [p; q; r] in rad/s, [3,1].
• Fuel – the fuel tank status: 1 full (m ≥ m_f), −1 empty (m ≤ m_e), 0 otherwise, [1,1].

Parameters

• Units – Metric (MKS) (the default) or English (velocity in ft/s). The two are the same arithmetic, so the choice only says which units the numbers are in.
• Initial Position [Xe Ye Ze] – three values. Defaults to [0 0 0].
• Initial Velocity [U V W] – the initial body velocity, three values. Defaults to [0 0 0].
• Initial Euler Angles [phi theta psi] – in radians, three values, with |θ| below 90°. Defaults to [0 0 0].
• Initial Body Rates [p q r] – in rad/s, three values. Defaults to [0 0 0].
• Initial Mass – m₀, from the empty to the full mass. Defaults to 1.
• Empty Mass – m_e, the lower limit. Defaults to 0.5.
• Full Mass – m_f, the upper limit, greater than the empty mass. Defaults to 3.
• Empty Inertia – I_e, the invertible [3,3] tensor at the empty mass. Defaults to eye(3).
• Full Inertia – I_f, the invertible [3,3] tensor at the full mass. Defaults to 2*eye(3).
• Integration Substeps – M, how many fourth-order Runge-Kutta steps a sample is integrated with on the discrete solver and in exported code, a whole number of 1 or more. Defaults to 100. No Simulink counterpart.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. A core holds the 13 states, publishes the outputs from them, and integrates one sample with the inputs held and M Runge-Kutta substeps – with M = 100, the same computation as Simulink's fixed-step ode4 at one hundredth of the sample time, the fuel limit applied at every stage and the mass clamped after every substep exactly as the Simulink block does it. The inertia is inverted in closed form at every derivative evaluation.

The three HDL targets are simulation-only real arithmetic, quantized at the port: a sine of a state has no Q16.16 form. The cores simulate correctly and are not offered as synthesizable.

Simulink bridge

Import and export, mapped to Aerospace Blockset's aerolib6dof2/Simple Variable Mass 6DOF (Euler Angles): Units → units (the two offered values 1:1), Initial Position [Xe Ye Ze] → xme_0, Initial Velocity [U V W] → Vm_0, Initial Euler Angles [phi theta psi] → eul_0, Initial Body Rates [p q r] → pm_0, Initial Mass → mass_0, Empty Mass → mass_e, Full Mass → mass_f, Empty Inertia → inertia_e and Full Inertia → inertia_f. Always emitted with mtype = Simple Variable and vre_flag and mass_flag off: each of those moves that block's port list, and this block has one. The knots unit system is not offered – it converts velocities inside the integration – and an imported one is reported. The dialog's inertia is not read at this mass type (measured) and is not offered. "Sampling Time (s)" does not cross: the Simulink block is continuous and defines no SampleTime.

Notes

• Stateful, continuous and nonlinear: 13 continuous states. The outputs are states or functions of them, so the block has no direct feedthrough and a loop through it is not an algebraic loop.
• At a limit the mass stops and the inertia rate with it. On the continuous solver the mass is held by the zero rate alone and can overshoot a limit by one solver step; the discrete solver and every exported core also clamp it, as Simulink does.
• θ = ±90° is the Euler kinematics' singularity, as in the fixed-mass block.