Write the body-axis equations before the model

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Course: Read the forces that steer the car

Module: Model the car's reaction to sudden inputs

Estimated duration: 55 minutes

The skill in this lesson is simple to say and easy to skip: before you choose a tire law, before you linearize, before you build a step-steer response, and before you ask software for an answer, you write the planar rigid-body equations in body axes.

That first page is the frame of the whole model. It tells you what is force balance, what is moment balance, what is kinematics, and what is only a later constitutive assumption. If you start by pasting in a tire formula, a state-space matrix, or a simulation block, you can still get numbers. What you lose is the ability to audit the numbers. The body-axis Newton-Euler equations are the audit trail.

The vehicle-dynamics reason for using body axes is practical. The ground frame G is fixed to the road. The body frame B is attached to the vehicle at its mass center C. In the planar rigid-vehicle model, the vertical ground axis Z and the vehicle vertical axis z are parallel, and the vehicle heading psi is the angle between the vehicle x-axis and the ground X-axis. The velocity of the mass center is expressed in B as a forward component vx and a lateral component vy. The yaw rate is r, the time derivative of heading. The applied force components are Fx and Fy in the vehicle axes, and the applied yaw moment is N about the z-axis.

Those definitions are not bookkeeping. They are what make the equations physically readable. Traction, braking, and lateral tire forces are generated by the vehicle and are most naturally expressed along the vehicle axes. A braking force acts along the car's longitudinal axis. A lateral tire force acts laterally through the contact patches. A yaw moment acts about the vehicle vertical axis. If you keep those quantities in the moving body frame, the force inputs stay simple. The price you pay is that acceleration in a rotating frame is not just the derivative of vx and vy. The rotating-frame term must be kept.

The vector form is the first thing you write:

BF = m * BGaB = m * Bv_dot + m * BGomegaB x BvB

BL = BI * BGomega_dotB + BGomegaB x (BI * BGomegaB)

For the planar yaw-only vehicle, use:

BvB = [vx, vy, 0]

BGomegaB = [0, 0, r]

BF = [Fx, Fy, 0]

BM = [0, 0, N]

The cross product omega x v becomes [-r * vy, r * vx, 0]. So the planar body-axis force equations are:

Fx = m * (vx_dot - r * vy)

Fy = m * (vy_dot + r * vx)

The simplest planar yaw moment equation is:

N = Iz * r_dot

That yaw equation assumes you have kept only yaw rotation about the body z-axis and are using the relevant yaw mass moment Iz about the mass center. If the model includes roll, pitch, product of inertia effects, or a full three-dimensional angular velocity vector, you do not collapse the rotational Newton-Euler equation to N = Iz * r_dot. You keep the full vector moment equation until the assumptions justify the scalar reduction.

The ground-path kinematics are separate from the force balances. Body velocities tell you what the car is doing relative to its own axes. The ground frame tells you where the car goes on the road. For planar motion, the usual kinematic map is:

X_dot = vx * cos(psi) - vy * sin(psi)

Y_dot = vx * sin(psi) + vy * cos(psi)

psi_dot = r

This is the first major separation you must protect. The equations for Fx, Fy, and N describe how forces and moments change the body-axis velocity components and yaw rate. The equations for X, Y, and psi describe how those body-axis velocities carry the vehicle through the ground frame. Mixing those two jobs is one of the fastest ways to create a model that looks plausible and cannot be debugged.

The principle: write the rigid-body shell before you write the behavior inside it

A vehicle-dynamics model has layers. The first layer is rigid-body motion: translational force balance, rotational moment balance, and coordinate-frame kinematics. The second layer is force generation: tire forces, drive force, braking force, aerodynamic force, road grade, or any other applied load. The third layer is the analysis you perform on that model: linearization, step response, frequency response, numerical integration, or controller design.

This lesson is about the first layer. You are not yet deciding whether the tire is linear, nonlinear, saturated, temperature-dependent, or load-sensitive. You are deciding where force enters the rigid body and what acceleration it must produce. You are making the dynamic skeleton strong enough that later approximations have something honest to approximate.

The body-axis equations answer three questions.

First, what external force is required for the mass center acceleration? That is the translational Newton equation in the moving frame. It gives Fx = m * (vx_dot - r * vy) and Fy = m * (vy_dot + r * vx) for the planar case.

Second, what external moment is required for angular acceleration? That is the rotational Newton-Euler equation. In the yaw-only planar case it reduces to N = Iz * r_dot, but the full body-frame equation reminds you that angular momentum is a vector and that rotating bodies can carry coupling terms when the motion is not a one-axis simplification.

Third, how do the body velocities update ground position and heading? That is not a force law. It is coordinate kinematics. You need it to draw a path, compare with GPS, or integrate a lap, but it does not replace the body-axis force balances.

The mechanism: why the rv terms are not optional

The most important part of the planar body-axis equations is the pair of terms that beginners are tempted to delete:

-r * vy in the longitudinal equation

+r * vx in the lateral equation

These terms appear because B rotates with the vehicle. If a vector is expressed in a rotating coordinate frame, its inertial derivative has two parts: the derivative of the components seen in the rotating frame, plus omega x v. In a car, that means a constant body-axis velocity vector can still be changing direction in the ground frame because the car is yawing.

You can see it in a steady corner. Suppose the car has constant vx, near-zero vy, and a nonzero yaw rate r. In body axes, vx_dot can be zero and vy_dot can be zero. If you incorrectly treat acceleration as only [vx_dot, vy_dot], you conclude that the lateral force is zero. That is absurd for a cornering vehicle. The correct lateral equation gives Fy = m * r * vx when vy_dot is zero and vy is zero. The car can have no change in body-axis speed components and still need lateral force because the velocity vector is turning in the ground frame.

That is the whole reason to write the equation first. The rotating-frame term protects the physics from your intuition. It shows that yaw rate turns longitudinal speed into lateral acceleration demand. It also shows that lateral velocity contributes to longitudinal acceleration demand through -r * vy. These are not tire-model details. They are rigid-body kinematics inside Newton's law.

Sub-skill 1: choose the frame before choosing states

Start every derivation by naming the frames. Write G for the ground frame and B for the body frame. Attach B at the mass center C. In the planar model, keep x forward, y lateral, and z vertical. State the heading angle psi from the ground X-axis to the body x-axis. Then write the velocity of C in body components:

BvC = [vx, vy, 0]

Only after that should you name the states. A clean minimum state set for planar rigid-body motion is vx, vy, r, psi, X, and Y. Some analyses use sideslip beta instead of vy, or speed V plus beta instead of vx and vy. That is fine later. For the first derivation, vx and vy are the least ambiguous because they match the body-axis Newton-Euler expansion directly.

If the model is for transient response rather than just a path trace, include r as a dynamic state, not only psi. Heading tells where the car points. Yaw rate tells how fast the body frame is rotating. The force equations need r because the body frame rotation creates the omega x v terms. The moment equation needs r_dot because yaw moment changes yaw rate.

Your frame choice also defines signs. If y is left-positive, then leftward lateral force is positive, leftward lateral velocity is positive, and positive yaw is the rotation from x toward y. If you choose another convention, the equations must be changed consistently. The equations are not magic symbols; they are compact sign contracts.

Sub-skill 2: write vector Newton-Euler before scalar expansion

Do not begin with Fx = something. Begin with the vector equation. The vector form forces you to account for the moving frame:

BF = m * Bv_dot + m * BGomegaB x BvB

BL = BI * BGomega_dotB + BGomegaB x (BI * BGomegaB)

Then insert the planar definitions. With omega = [0, 0, r] and v = [vx, vy, 0], compute omega x v. The result is [-r * vy, r * vx, 0]. Now the force equations are not memorized formulas. You know where each term came from.

This matters when the problem changes. If you later add roll and yaw, the same Newton-Euler structure still applies, but omega is no longer only [0, 0, r]. If you move to a multibody model, you still need all kinematic quantities in a common frame before summing and comparing them. If you use a ground-frame formulation, you still must transform forces and mass properties consistently. The vector equation is the stable object. The scalar equations are a special case.

Sub-skill 3: keep force generation out of the first line

At this stage, Fx and Fy are resultant applied force components at the rigid-body level. They may later be sums of tire forces, drive force, braking force, aerodynamic drag, and other loads. But do not substitute those pieces until the rigid-body balance is written.

A clean first page might say:

Fx = sum of body-axis longitudinal external forces

Fy = sum of body-axis lateral external forces

N = sum of external yaw moments about C

Then, and only then, you may expand the sums. For a bicycle model, Fy might become front lateral force plus rear lateral force, with moment N built from axle distances and lateral forces. For a four-wheel model, Fx and Fy might be sums over all four contact patches after each tire force is transformed into body axes. For a braking transient, Fx might include braking force and drag. The rigid-body equation does not care how you generated the force. It only cares about the resultant force and moment applied to the body.

This separation keeps your model testable. If the car accelerates incorrectly, you can ask whether the rigid-body equation is wrong or the force law feeding it is wrong. If you combine them too early, every error hides inside one large expression.

Sub-skill 4: separate dynamics from kinematics

The body-axis force equations give vx_dot and vy_dot once Fx, Fy, vx, vy, and r are known:

vx_dot = Fx / m + r * vy

vy_dot = Fy / m - r * vx

The yaw moment equation gives:

r_dot = N / Iz

These are dynamic equations. They tell how the state changes because forces and moments act.

The ground-position equations are kinematic equations:

X_dot = vx * cos(psi) - vy * sin(psi)

Y_dot = vx * sin(psi) + vy * cos(psi)

psi_dot = r

Do not use X_dot and Y_dot as substitutes for vx and vy unless you explicitly transform between frames. Do not feed a ground-frame lateral acceleration into a body-frame lateral equation without rotating it. Do not compare a body-axis velocity component to a ground-axis velocity component and call the difference an error. If the coordinate frame changes, the component meaning changes.

This is why coordinate frames are not a formal nuisance. They are tools for defining motion. A point position in three-dimensional space requires a frame and coordinate components. A velocity component also requires a frame. So does an acceleration component, a force component, and a moment component. The frame label is part of the quantity.

Worked example 1: the planar rigid vehicle under traction and lateral force

Use the rigid-vehicle situation from the corpus: a planar vehicle has body frame B attached at mass center C, a ground frame G fixed to the ground, heading psi between the two x-axes, and forward and lateral force components Fx and Fy. You want the equations before you know any tire details.

Write the body velocity and angular velocity:

BvC = [vx, vy, 0]

BGomegaB = [0, 0, r]

Now expand the translational equation:

BF = m * Bv_dot + m * BGomegaB x BvB

omega x v = [-r * vy, r * vx, 0]

So:

Fx = m * (vx_dot - r * vy)

Fy = m * (vy_dot + r * vx)

Now solve for the state derivatives:

vx_dot = Fx / m + r * vy

vy_dot = Fy / m - r * vx

This version is ready for a simulator or a hand calculation because the unknowns are explicit. It is also ready for inspection. If the vehicle is not yawing, r = 0, the equations reduce to Fx = m * vx_dot and Fy = m * vy_dot in the body frame. If the vehicle is yawing while moving forward, the lateral equation includes the r * vx term even if vy_dot is zero. Both checks are useful.

For yaw, write the scalar planar equation only after naming the assumption:

Assumption: yaw-only angular velocity about the body z-axis, mass center frame, yaw inertia Iz.

Then:

N = Iz * r_dot

If you cannot honestly state that assumption, keep the full rotational equation. The full form is not advanced decoration. It is the reminder that angular momentum is frame-dependent and that rotating bodies with more than one angular-velocity component can have coupling terms.

Worked example 2: steady cornering exposes the missing term

Now use a simple steady-cornering check. The car is in planar motion with constant vx, nearly zero vy, and constant yaw rate r. This is not a tire model. It is only a rigid-body consistency test.

Set:

vx_dot = 0

vy_dot = 0

vy = 0

r is not zero

The longitudinal equation gives:

Fx = 0

The lateral equation gives:

Fy = m * r * vx

That result is the point of the exercise. The car needs lateral force even though the body-axis lateral velocity component is not changing. The acceleration is present because the body frame is rotating and the forward velocity vector is turning with the vehicle. If your first derivation gives Fy = 0 in this situation, you have treated body-axis components as if they lived in a non-rotating frame.

This check is one of the best calibration cues for the lesson. You do not need a track map, a tire curve, or a data system to use it. You only need to ask whether the equations can represent a vehicle traveling around a constant-radius path. If they cannot, the rigid-body foundation is wrong.

Worked example 3: adding roll changes the moment equation before it changes the tire model

The corpus also gives a vehicle with roll and yaw rotations, with heading psi and roll angle phi describing the orientation of B relative to G. That situation is a useful boundary marker for the planar lesson.

If you include roll, do not pretend the yaw-only scalar moment equation is still the whole rotational dynamics. The translational body-axis idea remains useful, and the Newton-Euler structure remains the same, but the angular velocity vector and mass moment matrix are no longer the one-number yaw story. The rotational equation must be kept as:

BL = BI * BGomega_dotB + BGomegaB x (BI * BGomegaB)

Only after defining the angular velocity components, inertia matrix, and assumptions may you reduce it. This is where many vehicle models quietly become inconsistent. The author adds roll angle because roll matters, then keeps a yaw-only rotational equation because it is familiar. The body-axis equation protects you from that mismatch. It asks whether the angular momentum model matches the rotations you allowed.

You do not need to solve the roll model in this lesson. You only need to know when the planar model has been exceeded. The planar lesson is valid when the intended model is a rigid vehicle in horizontal planar motion with yaw as the rotational degree of freedom. Once roll is a state, the rotational side of Newton-Euler must be revisited.

Calibration cues: how you know the first page is right

A correct body-axis first page has several signatures.

The frame labels are visible. You can point to G, B, C, x, y, z, X, Y, Z, and psi. You can say which quantities are expressed in B and which describe ground-frame position. If the page has vx and X_dot but no transformation between them, it is not done.

The translational equation contains the rotating-frame terms. In planar form, the longitudinal equation has -r * vy inside the parentheses, and the lateral equation has +r * vx inside the parentheses. If those terms are missing, the body-axis acceleration has been treated as an inertial-frame derivative.

The r = 0 limit is sensible. With no yaw rate, the equations reduce to ordinary component-wise force balance in the body frame. That does not prove the model is complete, but it catches sign errors and frame confusion.

The steady-cornering check is sensible. With vx constant, vy near zero, and r nonzero, the lateral force is not zero. That catches the most common missing omega x v error.

The yaw equation is matched to the rotational assumptions. If the model is yaw-only, N = Iz * r_dot is a reasonable scalar reduction. If the model includes roll and yaw, the full rotational Newton-Euler form stays on the page until the additional assumptions are made explicit.

The force law is still replaceable. You should be able to erase the tire-force section and leave the rigid-body equations intact. That means you have not hidden the dynamics inside a tire model.

The later linearization point is clear. The nonlinear body-axis equations are the object you linearize around an operating condition. You do not linearize a vague diagram. You linearize equations. When the next lesson studies stability or a step-steer first answer, it should be operating on this foundation.

Common mistakes and what good looks like

Mistake 1: starting with the tire model.

The bad version begins with cornering stiffness, slip angle, or a front and rear lateral-force formula before the rigid body has been written. It feels efficient because it looks closer to the final bicycle model. It is fragile because you have not yet defined what Fy and N must accomplish.

Good looks like this: first write BF = m * Bv_dot + m * omega x Bv and BL = BI * omega_dot + omega x BI omega. Then expand the planar equations. Then define the tire forces that feed Fx, Fy, and N.

Mistake 2: deleting omega x v.

The bad version writes Fx = m * vx_dot and Fy = m * vy_dot while still claiming vx and vy are body-axis components. It will pass a straight-line test and fail a cornering test.

Good looks like this: Fx = m * (vx_dot - r * vy) and Fy = m * (vy_dot + r * vx). You should be able to explain that the extra terms appear because the body frame is rotating.

Mistake 3: mixing ground-frame and body-frame components.

The bad version compares X_dot to vx or uses a ground-frame acceleration directly in the body-axis force balance. The numbers may be close at small heading angles, which makes the mistake hard to notice early.

Good looks like this: body-axis velocity components are transformed to ground-frame position rates through psi. If you need ground-frame acceleration, transform consistently. If you need body-frame force balance, keep the components in B.

Mistake 4: treating yaw rate as heading.

The bad version uses psi where r belongs, or assumes that heading angle itself produces the rotating-frame acceleration terms. Heading is orientation. Yaw rate is angular velocity. The cross product uses angular velocity.

Good looks like this: psi_dot = r is the kinematic relation, and r appears in the dynamic force equations. The moment equation gives r_dot from yaw moment.

Mistake 5: using N = Iz * r_dot after adding roll.

The bad version adds roll angle phi to the orientation but leaves the rotational dynamics as if yaw is the only angular motion. That can make the model internally inconsistent before any tire or suspension detail is added.

Good looks like this: if roll is included, return to the vector moment equation. Define the angular velocity vector, inertia matrix, and assumptions. Reduce only after the reduction is justified.

Mistake 6: hiding sign conventions.

The bad version copies formulas without saying which direction is positive for y, yaw, force, and moment. The model may work until someone changes a tire-force sign or imports data from a different convention.

Good looks like this: the first page states the axes and sign convention. Every force and moment is assigned to that convention before it is summed.

Drill: the three-pass blank-sheet derivation

Use this drill before your next modeling session or before reading a worked solution. It takes about 35 minutes and requires only a blank page.

Pass 1, five minutes: draw the frames and name the quantities. Draw G fixed to the ground and B attached at the mass center C. Mark x forward, y lateral, z vertical, heading psi, body velocities vx and vy, yaw rate r, forces Fx and Fy, and yaw moment N. Do not write any tire equations.

Pass 2, ten minutes: write the vector equations and expand them. Start with BF = m * Bv_dot + m * omega x Bv and BL = BI * omega_dot + omega x BI omega. Substitute Bv = [vx, vy, 0] and omega = [0, 0, r]. Compute omega x v and write Fx = m * (vx_dot - r * vy), Fy = m * (vy_dot + r * vx), and the yaw-only moment equation N = Iz * r_dot under its assumption.

Pass 3, ten minutes: run three checks. First set r = 0 and confirm ordinary component force balance. Second set vx_dot = 0, vy_dot = 0, vy = 0, and r nonzero; confirm that Fy = m * r * vx. Third write the ground kinematics X_dot, Y_dot, and psi_dot without mixing them into the force balance.

Pass 4, ten minutes: rewrite the equations as state derivatives ready for integration: vx_dot, vy_dot, r_dot, X_dot, Y_dot, and psi_dot. Leave Fx, Fy, and N as named inputs or resultant forces. The success criterion is that another person can replace the force model without rewriting your rigid-body equations.

Do the drill three times on different days. The goal is not memorization of symbols. The goal is that the order becomes automatic: frames, vector Newton-Euler, planar expansion, kinematic map, checks, then force law.

How this lesson connects to the neighboring lessons

The next stability lesson should begin after this one. Linearization is not a replacement for writing the nonlinear equations. It is a local approximation of them. The corpus example on rotational stability perturbs a rigid body's angular velocity and ignores nonlinear terms to examine stability. That is the same kind of move you will make in vehicle handling: first define the nonlinear rigid-body equations, then perturb around an operating condition, then keep the terms that belong in the linear model.

The step-steer lesson also depends on this page. A step steer is an input to the force-generation layer. The first answer of the car is the response of the rigid-body states after those forces and moments enter the body-axis equations. If the body-axis equations are wrong, the step response may still have a curve, but the curve will be explaining the wrong system.

The frequency-response lesson depends on the same discipline. Frequency response is meaningful only after you know which input perturbs which state through which equations. Steering frequency response should not be a floating graph divorced from the force and moment balances underneath it.

When this principle breaks down

The instruction to write planar body-axis equations first does not mean every vehicle problem is planar. It means every model needs its rigid-body equations before its force law and analysis method. If the problem includes roll and yaw, use the body-axis Newton-Euler form but do not force it into the yaw-only scalar equation. If the problem is a multibody vehicle model, transform kinematic characteristics into a common coordinate frame before combining them. If the problem is a full three-dimensional motion problem, the planar equations become a special case, not the governing model.

It also does not mean body axes are always the only possible frame. Some problems are easier in a global frame, and multibody calculations may choose a common frame for summing momentum or mass properties. The point is not loyalty to a symbol. The point is consistency. In the planar vehicle transient problem, body axes are usually the practical choice because the important applied forces are naturally described relative to the vehicle.

The final test

Before moving on, look at your derivation and ask five questions.

Can I identify every coordinate frame used on the page?

Can I point to the omega x v term and explain why it exists?

Can I recover steady-cornering lateral force with vx constant and r nonzero?

Can I keep the tire model separate from the rigid-body equations?

Can I say exactly which assumption allowed N = Iz * r_dot?

If the answer is yes, you have the foundation needed for transient response. If the answer is no, do not linearize yet. Do not tune a tire law yet. Do not ask for a frequency response yet. Go back to the body-axis Newton-Euler equations and make the first page honest.

Worked example: planar rigid vehicle under traction and lateral force

The corpus situation is a rigid vehicle in planar motion with a ground frame fixed to the road and a body frame attached at the mass center. The body velocity is resolved into vx and vy, and the yaw rate is r. Starting from the body-frame Newton-Euler equation, the cross product omega x v expands to [-r * vy, r * vx, 0], so the force balances become Fx = m * (vx_dot - r * vy) and Fy = m * (vy_dot + r * vx). The useful teaching move is to solve those for vx_dot and vy_dot while leaving Fx and Fy as resultant forces. That keeps the rigid-body model independent of whatever tire, braking, or traction law you later attach.

Worked example: steady cornering as a missing-term check

A steady-cornering check exposes the most common error. Set vx_dot = 0, vy_dot = 0, vy = 0, and keep r nonzero. A model that has dropped the rotating-frame term predicts no lateral force. The correct body-axis equation gives Fy = m * r * vx. The car needs lateral force because the forward velocity vector is changing direction in the ground frame even when its body-axis components are steady.

Worked example: roll and yaw mark the edge of the planar reduction

The corpus also includes a vehicle with roll and yaw rotations, described by heading psi and roll angle phi. That example is a boundary warning. Once roll is included, the yaw-only scalar equation N = Iz * r_dot is no longer automatically the full rotational dynamics. The body-frame moment equation should remain in vector form until the angular velocity vector, inertia matrix, and assumptions have been stated. The planar lesson is still useful, but its scalar yaw reduction must not be carried into a roll-yaw model without review.

Common mistakes

The recurring mistakes are starting with tire forces before writing the rigid-body balance, deleting omega x v, mixing ground-frame and body-frame components, confusing heading psi with yaw rate r, using a yaw-only moment equation after adding roll, and hiding sign conventions. Good work has the opposite signatures: visible frame labels, explicit rotating-frame terms, a separate ground-kinematic map, psi_dot = r, a moment equation matched to the modeled rotations, and stated positive directions for forces and moments.

Drill: three-pass blank-sheet derivation

The drill is a repeated blank-sheet derivation. First draw G, B, C, psi, vx, vy, r, Fx, Fy, and N. Second write the vector Newton-Euler equations and expand the planar cross product. Third run the checks r = 0 and steady cornering with r nonzero. Fourth rewrite the result as state derivatives for vx, vy, r, X, Y, and psi. Do the drill three times on separate days. The success criterion is that you can derive the equations without dropping the rv terms and without inserting a tire model into the rigid-body balance.

Cross-reference: linearization comes after the body-axis equations

The sibling lessons on stability, step-steer response, and steering frequency response all depend on this first page. Linearization is a local approximation of nonlinear equations, not a substitute for writing them. Step response and frequency response are meaningful only after the force and moment inputs act through a defined rigid-body model.

Author Review

No quiz questions are attached to this lesson.

Sources

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