Wheel loaders and many other heavy vehicles steer by rotating the front and rear chassis sections around a central articulation joint. Unlike Ackermann steering, the wheels do not turn relative to their own chassis section. Instead, the two complete axle assemblies point in different directions.
This difference changes the kinematic model. The articulation joint is not the instantaneous center of curvature. During a turn, that center lies to the side of the vehicle at the intersection of the two axle lines.
A symmetric articulated vehicle. The front and rear axle lines meet at the instantaneous center of curvature.
Geometry and Notation
Let \(H\) be the articulation joint. The rear axle center \(R\) lies a distance \(l_r\) behind it, and the front axle center \(F\) lies a distance \(l_f\) ahead of it. The total wheelbase is
\[ L=l_r+l_f. \]
Write \(\theta_r\) for the rear chassis heading and \(\theta_f\) for the front chassis heading. The signed articulation angle is
\[ \phi=\theta_f-\theta_r. \]
With the unit direction
\[ \mathbf{e}(\theta)= \begin{pmatrix}\cos\theta\\\sin\theta\end{pmatrix}, \]
the two axle centers are
\[ R=H-l_r\mathbf{e}(\theta_r), \qquad F=H+l_f\mathbf{e}(\theta_f). \]
The No-Slip Constraints
In the planar kinematic model, each axle may move along its chassis direction but not sideways. Let \(v_r\) be the longitudinal speed at the rear axle center. Then
\[ \dot{R}=v_r\mathbf{e}(\theta_r). \]
The front axle has zero velocity in its lateral direction:
\[ \dot{F}\cdot\mathbf{e}_{\perp}(\theta_f)=0, \qquad \mathbf{e}_{\perp}(\theta)= \begin{pmatrix}-\sin\theta\\\cos\theta\end{pmatrix}. \]
Differentiating
\[ F=R+l_r\mathbf{e}(\theta_r)+l_f\mathbf{e}(\theta_f) \]
and using \(\theta_f=\theta_r+\phi\) gives
\[ \dot{F} =v_r\mathbf{e}(\theta_r) +l_r\dot{\theta}_r\mathbf{e}_{\perp}(\theta_r) +l_f(\dot{\theta}_r+\dot{\phi})\mathbf{e}_{\perp}(\theta_f). \]
Projecting this velocity onto \(\mathbf{e}_{\perp}(\theta_f)\) yields
\[ -v_r\sin\phi +l_r\dot{\theta}_r\cos\phi +l_f(\dot{\theta}_r+\dot{\phi})=0. \]
Solving for the rear heading rate gives the general articulated-vehicle relation
\[ \boxed{ \dot{\theta}_r =\frac{v_r\sin\phi-l_f\dot{\phi}} {l_f+l_r\cos\phi} }. \]
The term containing \(\dot{\phi}\) matters while the hydraulic joint is actively changing the articulation angle. Omitting it is a quasi-static steering approximation, not the general model.
Constant Articulation
When the articulation angle is held constant, \(\dot{\phi}=0\), so
\[ \dot{\theta}_r =\frac{v_r\sin\phi}{l_f+l_r\cos\phi}. \]
This is already enough to see why the ordinary bicycle-model expression \(v_r\tan\phi/L\) cannot simply be reused: articulated steering has two differently oriented rigid sections, not one body with a steered front wheel.
Symmetric Wheel Loader
For a symmetric vehicle,
\[ l_r=l_f=\frac{L}{2}. \]
Introduce the mean heading
\[ \theta=\frac{\theta_r+\theta_f}{2} =\theta_r+\frac{\phi}{2}. \]
At constant articulation, the joint velocity points along this mean heading. If \(v\) denotes the speed of the joint, its state equations are
\[ \boxed{ \begin{aligned} \dot{x} &=v\cos\theta,\\ \dot{y} &=v\sin\theta,\\ \dot{\theta} &=\frac{2v}{L}\sin\frac{\phi}{2}. \end{aligned} } \]
The signed curvature of the joint path is therefore
\[ \boxed{ \kappa=\frac{\dot{\theta}}{v} =\frac{2}{L}\sin\frac{\phi}{2} }. \]
For \(\phi\neq0\), the signed distance from the articulation joint to the ICC is
\[ \boxed{ \rho_H=\frac{1}{\kappa} =\frac{L}{2\sin(\phi/2)} }. \]
As \(\phi\to0\), the curvature tends to zero and the ICC moves to infinity, which is precisely straight-line motion.
Axle and Wheel Turning Radii
The articulation joint, axle centers, and individual wheels follow different concentric circles. For the symmetric model, the radius of either axle center is
\[ \rho_A =\sqrt{\rho_H^2-\left(\frac{L}{2}\right)^2} =\frac{L}{2}\left|\cot\frac{\phi}{2}\right|. \]
If the track width is \(w\), the ideal inner and outer wheel-path radii are
\[ \rho_{\mathrm{inner}}=\rho_A-\frac{w}{2}, \qquad \rho_{\mathrm{outer}}=\rho_A+\frac{w}{2}. \]
Thus \(w\) does not appear in the centerline curvature, but it does determine wheel-path clearance and the difference between inner and outer wheel speeds.
Small-Angle Approximation
For small articulation angles, \(\sin(\phi/2)\approx\phi/2\), so
\[ \kappa\approx\frac{\phi}{L}, \qquad \dot{\theta}\approx\frac{v}{L}\phi. \]
This explains why a linear steering law can look plausible near straight motion. The exact symmetric model uses \(2\sin(\phi/2)\), however, not \(\tan\phi\). At large articulation angles, the distinction is substantial.
Exact Integration over a Time Step
If \(v\) and \(\phi\) remain constant during a time step \(\Delta t\), then the curvature and angular velocity \(\omega=v\kappa\) are constant. The articulation joint follows an exact circular arc:
\[ \begin{aligned} x_{k+1} &=x_k+\frac{v}{\omega} \left[\sin(\theta_k+\omega\Delta t)-\sin\theta_k\right],\\ y_{k+1} &=y_k-\frac{v}{\omega} \left[\cos(\theta_k+\omega\Delta t)-\cos\theta_k\right],\\ \theta_{k+1} &=\theta_k+\omega\Delta t. \end{aligned} \]
Straight motion is the limit \(\omega\to0\). Handling that limit separately avoids division by a value close to zero.
Interactive Simulation
The vehicle below follows the symmetric joint equations derived above. The up and down arrow keys drive forward and backward. The left and right arrow keys change the articulation angle; releasing them holds the hydraulic joint at its current angle. The page follows the vehicle as it moves through the document.
Model Limits
This is a planar, no-slip kinematic model. It does not include tire deformation, lateral slip, axle oscillation, hydraulic dynamics, load transfer, terrain compliance, or the changing contact geometry of uneven ground. Those effects matter for dynamic control and simulation, but they do not change the geometric distinction between articulated steering and Ackermann steering.
The corresponding single-body steering geometry is derived in the Ackermann steering model.