A construction vehicle like a wheel loader typically has an articulated steering. The basic idea consists of folding the chassis in the steering direction.

With the track width $$w$$ (the lateral wheel separation), the wheel base $$l$$ (the longitudinal wheel separation), the steering angle $$\phi$$ and $$r$$ being the distance between the ICC (instantaneous center of curvature) and the center of the car.

## Forward Kinematic for wheel loader

A kinematic model for a wheel loader can be formulated similarly to the kinematic model of a ackerman vehicle. Considering the state of the vehicle is represented as a quadruple $$(x, y, \theta, \phi)$$, with $$\theta$$ being the heading and $$(x,y)$$ the position in the world. The most important thing about the steering of a wheel loader is that the rotational center is exactly in the middle of the vehicle where the inflexion point is. Given the speed $$s$$ of the vehicle we get

$\dot{x} = s\cos\theta\\\dot{y} = s\sin\theta\\\dot{\theta} = \frac{s}{l} \tan\phi \approx \frac{s}{l}\phi$

(Max taylor approximation error of the tangent simplification is about $$3^\circ$$ at $$30^\circ$$ steering lock)

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