# State estimation and experimental verification of a robotic vehicle with six in-wheel drives using Kalman filter

## Abstract

In this research we present an algorithm for a six-wheeled robotic vehicle with articulated suspension (RVAS) to estimate the vehicle velocity and acceleration states, slip ratio and the tire forces. The estimation algorithm consists of six parts. In the first part, a wheel state estimator estimates the wheel rotational speed and its angular acceleration using Kalman filter, which is used to estimate the longitudinal tire force distribution in the second part. The third part is to estimate respective longitudinal, lateral, and vertical speeds of the vehicle and wheels. Based on these speeds, the slip ratio and slip angle are estimated in the fourth part. In the fifth part, the vertical tire force is then estimated. In the sixth part, the lateral tire force is then estimated. For a simulation test environment, the RVAS dynamic model is developed using Matlab and Simulink. The estimation algorithm is then verified in simulation using the vehicle test data and different test scenarios. It is found from simulation results that the proposed estimation algorithm can estimate the vehicle states, longitudinal tire forces efficiently. Moreover, a small prototype of the robotic vehicle is fabricated for experimental verification of the estimation algorithm. Various experiments are executed in pavement and off-road driving to estimate the wheel angular position, velocity and acceleration states and finally the slip ratio is estimated in these situations.

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## Abbreviations

g:

Gravity acceleration

$$c_{d}$$ :

Rotational damping coefficient of the arm rod

$${I}_{x} {, I}_{y} {, I}_{z}$$ :

Moments of inertia of the vehicle about the roll, pitch and yaw axes respectively

$$I_{arm\_x}{, I}_{arm\_y}{, I}_{arm\_z}$$ :

Moments of inertia of the arm rod about its x, y, z axes respectively

$$I_{w\_x}$$ :

Moment of inertia of the wheel about its x axis

$$I_{w\_z}$$ :

Moment of inertia of the wheel about its z axis

$$J_{w}$$ :

Moment of inertia of the wheel about its rotational axis

$$k_{s}$$ :

Rotational spring stiffness of the arm rod

$$l_{arm}$$ :

Length of the arm rod

$$l_{f}$$ :

Distance from the vehicle c.g to the front arm axle

$$l_{m}$$ :

Distance from the vehicle c.g. to the middle arm axle

$$l_{r}$$ :

Distance from the vehicle c.g. to the rear arm axle

$$m_{arm} {, m}_{s} {, m}_{w}$$ :

Masses of the arm rod, sprung mass of the vehicle and the in-wheel motor respectively

$$M_{sxi}$$ :

Internal moment acting on the sprung mass about the roll axis at the rotational center of the ith arm rod

$$M_{szi}$$ :

Internal moment acting on the sprung mass about the yaw axis at the rotational center of the ith arm rod

$$M_{z\_des}$$ :

Yaw moment input

$$t_{w}$$ :

Half of the vehicle width

$$\omega_{arm\_i}$$ :

Angular velocity vector of the ith arm rod

$${\omega }_{{{\text{cg}}}}$$ :

Vehicle angular velocity vector

$${\omega }_{i}$$ :

Angular speed at the ith wheel

$$\hat{\omega }_{i}$$ :

Estimated angular speed at the ith wheel

$$\theta_{arm\_i }(0)$$ :

Initial arm angle at time 0

R:

Tire radius of the ith wheel

$${\uplambda }_{{\text{i}}}$$ :

Slip ratio at the ith wheel

$$F_{sxi}$$, $${ }F_{syi}$$, $${ }F_{szi}$$ :

Longitudinal, lateral, and vertical internal forces acting on a sprung mass at the rotational center of the ith arm rod respectively

$$F_{txi}$$ , $${ }F_{tyi}$$ , $${ }F_{tzi}$$ :

Longitudinal, lateral, and vertical tire forces at the ith wheel respectively

$$\dot{r}_{cg},$$ $$\, \ddot{r}_{cg}$$ :

Vehicle velocity, and acceleration vectors

$$\ddot{r}_{{\text{i}}}$$ :

Translational acceleration vector of the rotational center of the ith arm rod

$$\ddot{r}_{{{\text{wi}}}}$$ :

Translational acceleration vector of the ith wheel

$$T_{i}$$ :

Measured wheel torque of the ith wheel

$$T_{self\_i}$$ :

Self-aligning torque at the ith wheel

$$T_{S\& D\_i}$$ :

Sum of spring and damping torques of the ith arm rod

$${\text{v}}_{x} \text{, }{\text{v}}_{{\text{y}}}\text{, }{\text{v}}_{{\text{z}}}$$ :

Vehicle longitudinal, lateral and vertical velocities

$$\alpha_{i}$$ :

Slip angle at the ith wheel

$$\alpha_{cg}$$ :

Vehicle angular acceleration vector

$$\alpha_{arm\_i}$$ :

Angular acceleration vector of the ith arm rod

$$\dot{\phi }\text{, }\dot{\theta }\text{, }\dot{\varphi }$$ :

Roll rate, pitch rate, yaw rate

$$\theta_{arm\_i} \text{, } \theta_{arm\_static}$$ :

Arm angle at ith arm rod, and static arm

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## Acknowledgements

This research was supported by a grant from Defense Acquisition Program Administration and Agency for Defense Development under contract UD180045RD.

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Correspondence to Youngshik Kim.

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Ali, H.F.M., Oh, SW. & Kim, Y. State estimation and experimental verification of a robotic vehicle with six in-wheel drives using Kalman filter. Microsyst Technol 27, 2419–2432 (2021). https://doi.org/10.1007/s00542-020-05148-2