Two Mass Drivetrain — Robotics/Actuators And Drivetrain

Two Mass Drivetrain

Robotics / Actuators And Drivetrain

A motor inertia and a load inertia coupled by a shaft that is not rigid – the flexible-joint plant, and the reason a position loop that is perfectly stable on the bench oscillates once the load is bolted on.

Four states, x = [θ_m, ω_m, θ_l, ω_l]:

• J_mω_m' = τ_m − k(θ_m−θ_l) − c(ω_m−ω_l) − b_mω_m
• J_lω_l' = k(θ_m−θ_l) + c(ω_m−ω_l) − b_lω_l − τ_load

The resonance sits at ω_n = √(k(1/J_m + 1/J_l)) with damping ratio ζ = (c/2)· √((1/J_m + 1/J_l)/k) – a frequency neither inertia carries on its own, which is exactly why two separate integrators cannot stand in for this block.

Ports

• tau_m – the motor torque applied to the motor inertia (N·m), a scalar [1,1]. This is one drivetrain, so use one block per axis.
• tau_load – the external load torque (N·m), a scalar. It opposes motion: a positive value decelerates the load, matching the sign convention DC Motor uses for its own load port.
• w_m – motor speed (rad/s). This is what the motor encoder reads, and what a naive loop closes on.
• w_l – load speed (rad/s). This is what the machine actually does.
• tau_shaft – the torque the coupling transmits, k(θ_m−θ_l) + c(ω_m−ω_l) (N·m). Exposed as a port because it is the quantity that breaks couplings, and because it is the one output that reads all four states.

Parameters

• Motor Inertia (kg.m^2) – J_m, positive. Rotor plus everything rigidly attached to it.
• Load Inertia (kg.m^2) – J_l, positive.
• Shaft Stiffness (N.m/rad) – k, zero or more. At exactly zero the two inertias are uncoupled and each simply integrates its own torque – a legal degenerate case, and a useful sanity check.
• Shaft Damping (N.m.s/rad) – c, zero or more. This is the internal damping of the coupling, acting on the speed difference; it is what limits the resonant peak.
• Motor Damping (N.m.s) – b_m, zero or more, acting on motor speed against ground (bearings, windage).
• Load Damping (N.m.s) – b_l, zero or more, the same against the load.
• Initial Shaft Twist (rad) – the initial θ_m−θ_l, i.e. a pre-wound shaft. Nonzero here is what makes the plant ring from the first sample.
• Initial Motor Speed (rad/s) / Initial Load Speed (rad/s) – the two speed states at t = 0.
• 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. The block exports the discretized state space, by whichever method the model selects (ZOH, FOH, Tustin, Matched, either Euler) – the coefficients are baked in at export time and are not tunable on the generated core, so re-export after changing a drivetrain.

Every row is emitted as a multiply-accumulate over the four states and two inputs, including the entries that are zero, so all ten backends sum the terms in the same order. Floating-point addition is not associative, and that uniformity is what keeps the software targets bit-identical to the C++ reference.

The three HDL targets are synthesizable Q16.16 at ordinary drivetrain parameters – no division and nothing transcendental survives into the body. That is a property of the discretized coefficients rather than of the block: a very small inertia drives 1/J, and with it B_d, up by orders of magnitude, and a shaft stiff enough to put the resonance near the sample rate does the same to A_d. Check the numbers if either applies.

Simulink bridge

No equivalent (Support::None). Drivetrain blocks ship in Simscape Driveline, which is not installed on this machine (measured – isfolder([matlabroot '/toolbox/simscape']) is false), and a physical-network block would not map onto a signal-flow port list in any case. A model that must cross can carry the same plant as a State-Space block: A, B, C, D exactly as printed at the top of this block's source, with states (θ_m, ω_m, θ_l, ω_l).

Notes

• Linear and stateful, and it carries a proper continuous state space – so model reduction can absorb it into a neighbouring plant in series, parallel or feedback, and the discretization follows the model's own method rather than a hardcoded Euler step.
• The plant is marginally stable, deliberately. Only the shaft difference enters the equations, so the rigid-body mode (both shafts turning together) has no restoring term and the two absolute positions random-walk under a zero-mean torque. That is the machine, not a defect – an unloaded drivetrain does drift. It is also why the initial condition is a twist rather than two absolute angles: only the difference is observable.
• Damping is split three ways on purpose. c acts on the speed difference (inside the coupling) while b_m and b_l act against ground. They are not interchangeable: raising c damps the resonance, raising b_m or b_l mostly just wastes torque.
• No backlash, no friction, no torque limit. Put Backlash or Stribeck Friction on the shaft and Saturation on the torque if the mechanism needs them – keeping them out is what leaves this block linear, and with it everything in the first note.
• Pairs with DC Motor upstream: its w output is the motor side, its tau_load port takes the reaction. Together they are the plant a cascade loop is actually tuned against.