Robot System Identification: Learning the Physics

Why Does This Exist?

A simulator contains an inertia value, but where did that value come from? A CAD estimate may omit a tool, a gearbox or a newly attached load. System identification estimates a system's model from measured inputs and responses. For a structured mechanical model, that can mean fitting inertia, friction or actuator parameters.

A state describes the current condition, such as joint position and velocity. A parameter describes the model, such as the inertia relating torque to acceleration. State estimation tracks changing state; identification asks which model parameters explain its changes. Some methods estimate both, but the questions remain distinct.

This lesson starts from robot dynamics with a known equation form. It shows why collecting varied data can matter more than collecting many repeated samples. Learning an unrestricted visual world model is a broader modelling problem, covered in world models.

Think of It Like This

How It Actually Works

State a model the measurements can test

Use a synthetic horizontal rotary joint with no gravity torque or contact. Let $\omega$ be angular velocity in rad/s, $a=\dot\omega$ angular acceleration in rad/s², and $\tau$ delivered joint torque in N m. Assume constant inertia $I>0$ in kg m² and linear viscous friction coefficient $b\geq0$ in N m s/rad:

The resisting friction torque is $-b\omega$, so $b\omega$ appears on the required-effort side. We assume calibrated, synchronized torque and motion values and omit motor dynamics, static friction, backlash and temperature drift. These are modelling assumptions, not properties guaranteed by a real joint.

For observation $i$, write the row $\phi_i=[a_i,\omega_i]$ and parameter vector $\theta=[I,b]^T$. Stack rows into data matrix $\Phi$ and torques into vector $y$. Then $y\approx\Phi\theta$. The superscript $T$ means transpose. Least squares chooses $\hat\theta$ minimizing $\|\Phi\theta-y\|_2^2$, the sum of squared torque residuals. MIT's system-identification chapter develops structured mechanical parameter fitting and experiment design.

Which observations separate inertia from friction?

Take these independently stipulated, noiseless snapshots:

The first two equations are $I+b=0.7$ and $-I+b=-0.3$. Adding gives $2b=0.4$, so $b=0.2$ N m s/rad. Subtracting gives $2I=1.0$, so $I=0.5$ kg m². The third row checks $0(0.5)+(-1)(0.2)=-0.2$ N m. The coefficient of inertia changes sign between the first two observations while the coefficient of friction stays fixed; that lets us distinguish their effects.

Each solid line in the diagram is a set of parameter pairs that explains one observation. One says $b=0.7-I$; the other says $b=I-0.3$. They meet at the mint ring, $(I,b)=(0.5,0.2)$. The horizontal axis is inertia; the vertical axis is viscous coefficient in N m s/rad. The lines describe possible models, not a joint's physical path through space.

For a held-out snapshot $a=2$ rad/s² and $\omega=-0.5$ rad/s, the fitted model predicts $\tau=0.5(2)+0.2(-0.5)=0.9$ N m. This is a synthetic consistency check, not experimental evidence that the model describes hardware.

Why doesn't more steady-speed data solve it?

At steady speed, $a=0$, so inertia disappears from the equation. A row with $\omega=1$ rad/s and $\tau=0.2$ N m identifies $b=0.2$, but allows every positive $I$. This is the dotted horizontal line in parameter space. Repeating the row a thousand times can reduce some noise in the friction estimate; it supplies no inertia information. Setting $a=2\omega$ in every row would also fail: then only the combination $2I+b$ is identifiable.

The algebraic requirement is full column rank of $\Phi$ for the parameters being estimated. Here the two columns must be independent. Near dependence produces a poorly conditioned problem: small measurement errors can cause large parameter changes. Persistent excitation is the related requirement that an input and its resulting motions keep supplying independent information over time. A finite full-rank dataset illustrates the principle without proving every online estimator's excitation assumptions.

On multi-link robots, even varied data may identify only combinations called base parameters, rather than every link parameter separately. Also scale the columns before interpreting a raw condition number: acceleration and velocity have different units and magnitudes. QR or singular-value decomposition is preferable to explicitly forming and inverting normal equations in a numerical implementation.

How do you tell a useful fit from a convenient fit?

Keep validation motions separate from fitting motions, including the operating range that matters to the controller. A small torque residual on training data is not a test of long-horizon motion. Roll out the fitted dynamics with recorded input and initial condition, then compare position and velocity traces. One-step and rollout errors answer different questions; robot simulation adds numerical solver and integration choices that can also affect the comparison.

Inspect residuals against velocity, acceleration, configuration and time. A residual changing sign with speed may suggest a missing friction term; a systematic time shift may suggest misalignment. Neither pattern alone diagnoses a unique cause. Filtered encoder derivatives are particularly delicate: noise is amplified by differentiation, while filtering can shift phase. Delivered torque derived from current also depends on torque constants, transmission effects and calibration.

Ordinary least squares assumes a suitable noise model. Noise in the regressors creates an errors-in-variables problem; feedback can correlate inputs with disturbances. Integral formulations, instrumental variables or prediction-error methods may be needed. Positive inertia constraints prevent an obviously impossible fit but don't make an uninformative dataset informative.

Use structured identification when known physics and a modest parameter set explain the behavior of interest. Use a richer model when residual structure persists under good measurements and excitation. Collecting that richer data and validating the model costs more than replacing a scalar fit with a larger optimizer. Plan excitation within the robot's motion and effort constraints, rather than treating aggressive motion as a substitute for informative experiment design.

Code

The tiny two-parameter solver below exposes the rank check and the scalar arithmetic. It forms normal-equation sums for transparency; use a scaled QR/SVD solver for noisy, larger or ill-conditioned problems.

1def fit_joint(rows: list[tuple[float, float, float]]) -> tuple[float, float]:2    aa = sum(a * a for a, w, torque in rows)3    ww = sum(w * w for a, w, torque in rows)4    aw = sum(a * w for a, w, torque in rows)5    ay = sum(a * torque for a, w, torque in rows)6    wy = sum(w * torque for a, w, torque in rows)7    determinant = aa * ww - aw * aw8    if aa == 0 or ww == 0 or determinant <= 1e-10 * aa * ww:9        raise ValueError("data do not separate inertia and friction")10    return ((ay * ww - wy * aw) / determinant,11            (wy * aa - ay * aw) / determinant)12
13observations = [(1.0, 1.0, 0.7), (-1.0, 1.0, -0.3), (0.0, -1.0, -0.2)]14inertia, friction = fit_joint(observations)15prediction = inertia * 2 + friction * (-0.5)16assert abs(inertia - 0.5) < 1e-1217assert abs(friction - 0.2) < 1e-1218assert abs(prediction - 0.9) < 1e-1219print(f"I={inertia:.3f}, b={friction:.3f}, held-out torque={prediction:.3f}")20try:21    fit_joint([(0.0, 1.0, 0.2)] * 100)22except ValueError as error:23    print(error)24else:25    raise AssertionError("steady-speed data cannot identify inertia")

Output: I=0.500, b=0.200, held-out torque=0.900 and data do not separate inertia and friction. Rejecting this dataset is the useful result, rather than silently returning one arbitrary inertia.

Watch Out For

The Quick Version

• System identification estimates a model from applied inputs and measured responses.
• Independent parameter effects must appear in the data; steady-speed motion alone cannot identify inertia.
• A low training residual doesn't establish identifiable parameters or accurate rollouts.
• Calibration, synchronization, measurement noise and controller feedback affect the estimator's validity.
• Validate the fitted model on separate motions over the operating conditions the controller will use.

What to Read Next

• Robot Dynamics: Forces, Inertia and Torque
• Robot State Estimation and Sensor Fusion
• Robot Simulation and Physics
• World Models for Physical Prediction
• Model Predictive Control for Robots