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Symbolic rigid body models

A symbolic rigid-body model states the variables, coordinate frames, forces, constraints, and balance laws before choosing numerical values. This exposes incompatible assumptions early and supplies exact special cases for later calculations.

Parameters and state

For one rigid body, constant parameters can include mass \(M\), body-frame center of mass, body-frame inertia tensor \(\mathbf I_b\), geometry, and material or contact coefficients. A common world-frame state is

\[\mathcal S(t)=\{\mathbf R,\mathbf p,\mathbf Q,\mathbf L\}\]

where \(\mathbf R\) is the center position, \(\mathbf p\) is total linear momentum, \(\mathbf Q\) maps body coordinates to world coordinates, and \(\mathbf L\) is measured about the center of mass.

Velocities can be derived from momenta. Avoid storing both as unrelated authoritative fields.

import sympy as sp

I1, I2, I3 = sp.symbols("I1 I2 I3", positive=True)
w1, w2, w3 = sp.symbols("w1 w2 w3", real=True)
inertia = sp.diag(I1, I2, I3)
omega = sp.Matrix([w1, w2, w3])
angular_momentum = inertia * omega
print("Body-frame angular momentum:", angular_momentum)

Translational equations

The center-of-mass equations are

\[\dot{\mathbf R}=\frac{\mathbf p}{M},\qquad \dot{\mathbf p}=\mathbf F_{\mathrm{ext}}\]

The first equation converts momentum to center velocity. The second is Newton's law for the complete body. Internal forces maintaining rigidity cancel from the total; external contact, gravity, and applied forces remain.

Dimensions provide an immediate check. Position rate has length per time, momentum divided by mass also has length per time, and force has the same units as momentum rate.

Torque from applied forces

If force \(\mathbf F_k\) acts at world point \(\mathbf x_k\), torque about the center is

\[\boldsymbol\tau_{\mathrm{cm}}=\sum_k(\mathbf x_k-\mathbf R)\times\mathbf F_k+\sum_j\boldsymbol\tau_j^{\mathrm{couple}}\]

A pure applied couple contributes torque without net force. Gravity in a uniform field acts effectively through the center of mass and therefore produces no torque about the center, although it accelerates the whole body.

Angular momentum and frame conversion

The world angular-momentum balance is

\[\dot{\mathbf L}_w=\boldsymbol\tau_w\]

The body-frame inertia tensor is constant for a rigid body. To obtain angular velocity, transform angular momentum into the body frame:

\[\mathbf L_b=\mathbf Q^{\mathsf T}\mathbf L_w,\qquad \boldsymbol\omega_b=\mathbf I_b^{-1}\mathbf L_b\]

It can then be transformed back with

\[\boldsymbol\omega_w=\mathbf Q\boldsymbol\omega_b\]

This sequence prevents the common error of multiplying a world vector by a body-frame tensor. Equivalent world-inertia formulas are valid if every quantity is transformed consistently.

theta = sp.Symbol("theta", real=True)
rotation = sp.Matrix(
    [
        [sp.cos(theta), -sp.sin(theta), 0],
        [sp.sin(theta), sp.cos(theta), 0],
        [0, 0, 1],
    ]
)
world_l = rotation * angular_momentum
recovered_omega = sp.simplify(inertia.inv() * rotation.T * world_l)
assert recovered_omega == omega
print("Recovered body angular velocity:", recovered_omega)

Orientation kinematics

For a body-to-world rotation matrix, one convention gives

\[\dot{\mathbf Q}=[\boldsymbol\omega_w]_\times\mathbf Q\]

where the skew-symmetric matrix

\([\boldsymbol\omega]_\times\)

acts so that \([\boldsymbol\omega]_\times\mathbf a=\boldsymbol\omega\times\mathbf a\). If angular velocity is expressed in the body frame, multiplication appears on the other side. The chosen orientation direction and frame must be documented together.

A unit-quaternion equation is an alternative, but its factor, multiplication order, and component layout depend on convention. Copying a formula from a different convention can produce a normalized yet incorrect orientation.

Body-frame Euler equations

In principal body axes, the rotational dynamics can be written

\[I_1\dot\omega_1+(I_3-I_2)\omega_2\omega_3=\tau_1\]

with two equations obtained by cyclic permutation. The coupling terms explain why the scalar formula \(\tau=I\alpha\) is not valid component by component for arbitrary three-dimensional motion.

For a spherical inertia tensor where \(I_1=I_2=I_3\), the coupling terms vanish. For rotation exactly about a principal axis, the other angular-velocity components vanish and the familiar scalar equation is recovered.

Energy and invariants

Rigid-body kinetic energy is

\[K=\frac{\mathbf p\cdot\mathbf p}{2M}+\frac12\boldsymbol\omega_b^{\mathsf T}\mathbf I_b\boldsymbol\omega_b\]

For a torque-free, force-free body, linear momentum, world angular momentum, and kinetic energy are constant. Orientation still changes. A valid rotation matrix also satisfies

\[\mathbf Q^{\mathsf T}\mathbf Q=\mathbf I,\qquad \det\mathbf Q=1\]

These provide separate checks: preserving matrix orthogonality does not prove physical energy conservation, and conserving angular momentum does not prove the orientation update is accurate.

Constraints and contacts

A hinge, rolling contact, or fixed point adds constraint equations and unknown reaction forces. Reactions should be solved so allowed acceleration or velocity constraints hold. They should not be omitted from force and torque balances merely because their values are unknown.

For ideal constraints, reaction forces may do no virtual work in allowed directions. Frictional and driven constraints can transfer energy. State which case applies before using energy conservation.

Symbolic reductions as tests

Reduce the general equations before assigning numbers:

Each reduction is an exact fixture for signs, frames, and components. If a later calculation cannot reproduce it as the relevant parameters approach the limit, the model or implementation is inconsistent.

From equations to an update

The symbolic equations define derivatives; a numerical method decides how to approximate their integral. A staged method must evaluate force and torque at the states prescribed by that method. Updating orientation, then calculating torque from the new orientation while translation still uses the old state mixes stages.

# Euler's torque-free rigid-body equations in principal axes.
angular_acceleration = -inertia.inv() * omega.cross(angular_momentum)
energy = (omega.dot(angular_momentum)) / 2
energy_rate = sum(sp.diff(energy, w) * a for w, a in zip(omega, angular_acceleration))
assert sp.simplify(energy_rate) == 0
print("Angular acceleration:", angular_acceleration)

Reduce the step and compare at fixed physical times. Renormalizing orientation may be a useful constraint correction, but record it and still test phase, energy, and angular momentum.

Assumptions before algebra

The symbols become meaningful only after the mechanical idealization is declared. State whether the body is rigid, whether mass and body-frame inertia remain constant, whether the center used in the equations is the center of mass, and whether the reference frame is inertial. Specify whether gravity is uniform, whether contacts are persistent or impulsive, and whether applied forces depend on pose or velocity. These decisions determine which terms belong in force, torque, and energy equations.

Also state the system boundary. A motor torque can be external to the rotor but internal to a rotor-plus-motor system. Bearing reactions may exert zero work in an ideal fixed constraint while still exerting force and torque. Friction changes the mechanical energy ledger, and a moving support can transfer both energy and momentum. Symbolic cancellation is valid only when the paired terms actually belong to the selected system.

Finally, distinguish coordinates from physical vectors. Components change under a basis rotation even when the vector does not. A body-fixed basis rotates with the object; a world basis does not. Writing frame labels on intermediate quantities may look verbose, but it prevents a silent mismatch from surviving into compact matrix notation. Check a rotated description of the same state: scalar energy should be unchanged, while vector predictions should rotate with the basis.

Reliability checks

A symbolic rigid-body model is a compact contract connecting geometry, balances, and frames. Once that contract is correct, numerical methods can be judged against exact limits instead of against a merely plausible moving shape.

A worked Python notebook

Use principal moments 1, 2, and 3 with the first two angular velocity components fixed at one. The question for this calculation is precise: how does rotational kinetic energy change when we change third angular velocity component, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

The inertia tensor is diagonal only in its principal body axes. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from -3 to 3 rad/s. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for third angular velocity component, expressed in rad/s. The output is rotational kinetic energy, expressed in J. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = \frac{3 q^{2}}{2} + \frac{5}{2}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 1 rad/s, the expression gives 4 J. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("3*q**2/2+5/2", locals={"q": q})
reference_input = 1.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "rad/s")
print("Predicted rotational kinetic energy:", reference_output, "J")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 16 J. The midpoint, at 0 rad/s, gives 2.5 J. The upper endpoint gives 16 J. Read these together with the assumptions: they describe different choices of third angular velocity component within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([-3.0, 0.0, 3.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} rad/s -> {output_value:.6g} J")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows third angular velocity component; the vertical axis shows rotational kinetic energy. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(-3.0, 3.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Third angular velocity component (rad/s)")
ax.set_ylabel("Rotational kinetic energy (J)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does rotational kinetic energy change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1 rad/s is 3 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 3 q\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 1.03
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "J")
print("Full expression:", full_prediction, "J")
print("Approximation difference:", full_prediction - linear_prediction, "J")

Here the changed input is 1.03 rad/s. The tangent estimate is 4.09 J, while the full expression gives 4.09135 J. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 6.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of third angular velocity component inside the stated interval, predict whether rotational kinetic energy will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: The inertia tensor is diagonal only in its principal body axes. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of J: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(3.0) - evaluate(-3.0))
for points in [101, 201, 401]:
    grid = np.linspace(-3.0, 3.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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