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

EOM 6DOF Simple Variable Mass Wind Angles

Robotics / Equations Of Motion

The six-degree-of-freedom equations of motion of a rigid body that burns fuel, carried in wind axes – the airspeed V with the incidence α and the sideslip β – with the attitude as the wind angles bank μ, flight-path angle γ and heading χ. 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. With a = F/m, the body rates ω = [p; q; r] and DCM_wb the body-to-wind rotation of (α, β):

• V' = a_x
• β' = p·sin α − r·cos α + a_y/V
• α' = q − (p·cos α + r·sin α)·tan β + a_z/(V·cos β)
• [p_w; q_w; r_w] = DCM_wb·[p − β'·sin α; q − α'; r + β'·cos α]
• μ' = p_w + (q_w·sin μ + r_w·cos μ)·tan γ, γ' = q_w·cos μ − r_w·sin μ, χ' = (q_w·sin μ + r_w·cos μ)/cos γ
• ω' = I^−T(M − dI/dt·ω − ω × (Iω))
• I(m) = I_e + (m − m_e)/(m_f − m_e)·(I_f − I_e), dI/dt = dm/dt·(I_f − I_e)/(m_f − m_e)
• m' = dm/dt, except zero while the tank is empty and draining or full and filling
• X_e' = DCM_we^T·[V; 0; 0]

Ports

• F – the applied force in wind 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_we^T·[V; 0; 0], [3,1].
• Xe – the position in the flat-earth frame, [3,1].
• Wind – the wind angles [μ; γ; χ] in radians, [3,1].
• DCMwe – the direction cosine matrix from the flat-earth frame to wind axes, [3,3].
• Vw – the velocity in wind axes, [V; 0; 0], [3,1].
• alpha_beta – [α; β] in radians, [2,1].
• alpha_beta_dot – [α'; β'] in rad/s, [2,1].
• wb – the body rates [p; q; r] in rad/s, [3,1].
• wb_dot – the angular accelerations [p'; q'; r'] in rad/s², [3,1].
• Ab – the acceleration in body axes, V_b × ω + DCM_wb^T·a with V_b = DCM_wb^T·[V; 0; 0], [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 [V alpha beta] – the initial airspeed, incidence and sideslip (radians), three values. Defaults to [0 0 0]; the airspeed divides, so a run needs a nonzero one.
• Initial Wind Angles [mu gamma chi] – the initial bank, flight-path and heading angles in radians, three values, with |γ| 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 2.
• 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 eleven outputs from them and the current inputs, 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 interpolated and 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 Wind (Wind Angles): Units → units (Metric (MKS) 1:1, English (velocity in ft/s) ↔ English (Velocity in ft/s)), Initial Position [Xe Ye Ze] → xme_0, Initial Velocity [V alpha beta] → Vm_0, Initial Wind Angles [mu gamma chi] → wind_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, rep = Wind Angles and vre_flag and abi_flag off: each of those moves that block's port list, and this block has one. The dialog's inertia is not read at this mass type (measured) and is not offered; the knots unit system is not offered either, because it converts velocities inside the integration, and an imported one is reported. "Sampling Time (s)" does not cross: the Simulink block is continuous and defines no SampleTime.

Notes

• Stateful, continuous and nonlinear: 13 continuous states. Three outputs – alpha_beta_dot, wb_dot and Ab – read the inputs, so the block has direct feedthrough, as the Simulink block does.
• At a limit the mass stops, but the inertia's rate does not: the Simulink block computes it from the mass flow rate on the wire, whatever the tank is doing, and this block reproduces that. 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.
• The angular acceleration uses the transpose of the inertia's inverse, I^−T, as the Simulink block does. For a symmetric inertia – any physical one – that is I⁻¹.
• V = 0, β = ±90° and γ = ±90° are singular, as on the fixed-mass block.