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This paper presents a vehicle dynamics prediction system, which consists of a sensor fusion system and a vehicle parameter identification system. This sensor fusion system can obtain the six degree-of-freedom vehicle dynamics and two road angles without using a vehicle model. The vehicle parameter identification system uses the vehicle dynamics from the sensor fusion system to identify ten vehicle parameters in real time, including vehicle mass, moment of inertial, and road friction coefficients. With above two systems, the future vehicle dynamics is predicted by using a vehicle dynamics model, obtained from the parameter identification system, to propagate with time the current vehicle state values, obtained from the sensor fusion system. Comparing with most existing literatures in this field, the proposed approach improves the prediction accuracy both by incorporating more vehicle dynamics to the prediction system and by on-line identification to minimize the vehicle modeling errors. Simulation results show that the proposed method successfully predicts the vehicle dynamics in a left-hand turn event and a rollover event. The prediction inaccuracy is 0.51% in a left-hand turn event and 27.3% in a rollover event.

In recent years, many vehicle control research propose using the future vehicle dynamics information to assist drivers' maneuvers. For example, the vehicle path predictions can provide the future position error for the vehicle guidance controls. Compared with the conventional “look-down” sensing system that provides current position error, the path prediction not only provides the information that are easier perceived by human drivers, but also provides additional information of road conditions, weather conditions,

In general, the future vehicle dynamics is predicted by using a vehicle mathematics model to numerically propagate current state values with time. Therefore, a vehicle dynamics prediction system needs a mathematic model and current vehicle state values [

The second concern is the parameter uncertainty in the vehicle model. Most dynamics prediction methods use presumed vehicle parameters in the vehicle mathematics model [

As mentioned earlier, another key factor of the vehicle dynamics prediction is the sensor system for obtaining current vehicle state values. Since the vehicle system is highly nonlinear and many of its dynamics cannot be measured directly, the vehicle dynamics are often obtained by two ways: one is the observer-based sensor fusion system; the other is the kinematics-based sensor fusion system. The observer-based method needs a vehicle model and less number of sensors. On the contrary, the kinematics-based method does not require a vehicle model but needs more sensors. Since the state estimation accuracy of the observer-based method greatly relies on the incorporated model accuracy, the observer-based method is less preferred as compared with the kinematics-based method. Lots of kinematics-based sensor fusion systems employ a GPS and an IMU (three-axis accelerometer and three-axis gyroscope) to measure 6 degree-of-freedom (DOF) motions of an object. This sensor fusion system has been widely used in many applications, such as aircraft systems [

From pervious discussion, it can be concluded that a precise vehicle model is important for the accuracy of the dynamics prediction. How precise this vehicle model can be is determined by the accompanied sensor fusion system that provides current vehicle state values both for the state propagations and for the real time identification of vehicle parameters. In our previous work [

Three coordinate systems are introduced to describe a vehicle moving on a sloped road (see,

Three sets of Euler angles are used to describe the relationships between any two out of three coordinate systems. The first set of Euler angles (_{g}, θ_{g}, ϕ_{g}

The second set of Euler angles (_{r}, ϕ_{r}, ψ_{r}_{r}

The third set of Euler angles (_{v}, θ_{v}, ϕ_{v}

Since two sets of Euler angles are enough to describe the relationships between three coordinate systems, complying with the above angle definitions, the following relationship can be established for these angles:

An additional auxiliary frame (aux-frame) is obtained by rotating the z-axis of the road frame until the x-axis of the road frame is aligned with the x-axis of the body frame. The aux-frame is used because it describes vehicle translational motions in an intuitive manner while preserving the information of other vehicle dynamics relative to the road level. In the following vehicle modeling, vehicle translational motions are described in the aux-frame, and the rotational motions are described by angles _{v}, θ_{v}, ϕ_{v}

Since lots of the vehicle dynamics cannot be directly measured by individual sensors, a sensor fusion system is constructed to obtain the vehicle dynamics on a sloped road. The proposed sensor fusion system consists of a group of sensors, a kinematic model related to those sensor outputs, and a state estimation algorithm. They are discussed in the following.

Different from a conventional GPS system, a three-antenna GPS is used here because it not only provides absolute position measurements

Four suspension displacement sensors are installed at four corners of a vehicle. The suspension deflection can be related to the vehicle attitude and vertical displacement of the vehicle CG, both relative to the road frame:
_{r}_{i}_{f} and l_{r} are the distances from CG to the front and rear axis, respectively; t_{f} and t_{r} are one half of the distances of the front and rear track, respectively.

An IMU sensor is installed at the center of gravity of the vehicle to measure the 6 DOF movements. They are used here to improve the estimation accuracy of the vehicle dynamics.

As discussed in this paper, three sets of Euler angles (nine angles in total) parameterize this vehicle attitude determination system. The relationships stated in _{g}, θ_{g}, ψ_{g}_{r}

In addition to the angle measurements above, the vehicle rotational dynamics are also present in the IMU measurements, GPS position measurements, and suspension displacement measurements. In order to improve the robustness and accuracy of the angle determination, all the sensor measurements should be used. Thus, the estimation of vehicle dynamics is done for the rotational dynamics and translational dynamics simultaneously. In that case, since _{g}, θ_{g}, ψ_{g}, ϕ_{r}, θ9_{r}, ψ_{v}

Thus, a kinematic model that can coordinate the outputs of IMU, GPS and suspension displacement sensors is:
_{g}, y_{g}, z_{g}_{g}, ẏ_{g}, ż_{g}_{ω}

For a dynamics model shown in

The system output equation h(x) in

_{r}, θ_{r}, ψ_{v}_{r}, θ_{r}_{r}, θ_{r}_{g}, y_{g}, z_{g}

It should be emphasized that the

Since the outputs of the GPS, IMU and suspension displacement sensors are unsynchronized and contaminated by different noise characteristics (see

The above state estimation process can provide noiseless information for the displacement, velocity, and attitude of the vehicle, but not for the angular velocity and accelerations. Without this information, the subsequent vehicle parameter identification can be much complicated. Potentially, the angular velocity and accelerations can also be obtained from Kalman filtering by including those two states as system states in

The alpha-beta filter is a steady-state filter for noisy signals. Its algorithm is shown as follows:
_{α}_{α}_{s}_{α}_{x}, ω̇_{x}^{T}; the corresponding

As mentioned earlier, a dynamics prediction system needs a precise vehicle model. Furthermore, some of the parameters in that vehicle model should be identified in real time. To meet both requirements, we propose the following vehicle model for the dynamics prediction, which consists of 6 DOF vehicle dynamics, road angles, tire-road friction, nonlinear suspension,

where _{a}, y_{a}, z_{a}_{x,tire}, _{y,tire}, _{z,spring}) are the translational forces generated by tires and suspension systems; (_{x}, M_{y}, M_{z}_{x,tire}, _{y,tire}, _{z,spring}), vehicle attitude (_{v}, θ_{v}, ϕ_{v}_{x}, I_{y}, I_{z}_{x}, ω_{y}, ω_{z}_{i}_{a,tire},_{i}_{i}_{i}_{ω}

The suspension system is modeled as a nonlinear spring-mass-damper system. Thus, the translational force generated by the suspension can be described as follows [_{s,i}_{1}, _{2}, _{3} parameterize the stiffness; _{s,i}_{u,i}

The adhesive force generated by tire is a highly nonlinear function of variables including slip ratios, slip angles, vertical loads, _{a,tire}, _{b,tire}) for simplicity.

_{λ,i}_{α,i}_{i}_{i}_{x,tire}, _{y,tire}) can be obtained as follows:

Noted that two rear wheel angles (_{3}, _{4}) are zeros for a front-steer vehicle, and two front wheel angles (_{1}, _{2}) are known values because they can be obtained by the steering wheel angle and the Ackerman principle [

In this approach, the parameters shown in the vehicle model in

After feeding the vehicle dynamics,

In order to apply the recursive-least-square (RLS) algorithm to identify vehicle parameters, the vehicle parameters and the corresponding measured dynamics are rearranged into the following format:
_{rls}

The following vehicle parameters are identified using the above RLS algorithms: vehicle mass (_{tot}_{x}, I_{y}, I_{z}_{λ,1∼4}), and the cornering stiffness (_{α,f}, C_{α,r}_{s}_{s}_{u}_{ω}

In this case, it is possible to manipulate the signal processing steps and formulate four independent RLS problems for identifying the above ten parameters, which can greatly reduce the computation loads and efforts of searching the optimal

The translational dynamics in z direction in the vehicle model in _{11} and _{11} are the elements in weighting and scaling matrices, respectively.

The wheel dynamics in the vehicle model in

Noted that the matrix _{rls,2} is singular when one of the slip ratios is zero. It can be understood that the tracking stiffness cannot be identified when there is no traction force. Moreover, the angular rates of four tires are directly measured by tachometers (see

The longitudinal and lateral dynamics in the vehicle model

Noted that the matrix _{rls,3} is singular when the summation of the front (or rear) two slip angles is zero or the summation of the front steering wheel angles is zero. Again, it can be understood that the cornering stiffness cannot be identified when there is no lateral force. Moreover, the slip angles are calculated using the measurements from the sensor fusion system and the steering wheel angle.

The rotational dynamics in the vehicle model in

From a system observability viewpoint [

Numerical simulations are used to demonstrate the feasibility of the proposed dynamics prediction method. In these simulations, a vehicle moves at a longitudinal speed of 90 km/h. The steering wheel angles and the generated tire torques are both varying with time at the frequency of 1 Hz (see,

The simulation results of the proposed sensor fusion system are shown in

The vehicle dynamics and sensor fusion outputs presented in the global frame (show in _{tot}, I̅_{x}, I̅_{y}, I̅_{z}^{−3}%, 0.12%, 5.05%, 4.37%). The estimation accuracy is good mainly because the incorporated suspension displacement sensors are relatively accurate. On the other hand, as shown in

Continuing from previous simulations, the driver is assumed to hold still the steering wheel and the gas/brake pedal at the time instant 10.25 s to do a left-hand turn on the road. The prediction system is turned on at the 10.25 s to predict the vehicle dynamics for the next 4.75 s, using the vehicle dynamics from the sensor fusion system at the 10.25 s and the vehicle model with the parameters identified from _{a}, y_{a}, z_{a}_{v}, θ_{v}, ϕ_{v}

In another example, the driver holds the steering wheel still but generate 1,000 _{2} = _{3} = 1,000 _{a}, y_{a}, z_{a}_{v}, θ_{v}, ϕ_{v}

Although the vehicle rollover accident can be foreseen in this case, the prediction is a bit inaccurate. According to the parameter identification results shown in

The tire forces of two prediction cases are also shown in

From above discussion, one may propose using nonlinear tire models, such as Pacejka's magic formula [

This vehicle parameter identification is challenging mainly because the system has a low degree-of-observability [_{x}, I̅_{y}, I̅_{z}

One alternative to improve this parameter identification is to increase the degree-of-observability by choosing proper weighting and scaling matrices shown in _{4}) changes the minimum eigenvalue of the estimation matrix

A vehicle dynamics prediction system, consisting of a kinematics-based sensor fusion system and a vehicle parameter identification system, is proposed and verified by simulation results. The sensor fusion system can obtain the 6 DOF vehicle dynamics and the two road angles accurately. The estimation error for each vehicle dynamics is shown in

The prediction accuracy of the rollover event is worse than that of the left-hand turn event. It is mainly because the identified linear tire model cannot accurately describe the nonlinear tire adhesive force in the rollover event. Using a nonlinear tire model for the dynamics prediction is possible but not practical in this case, because the nonlinear tire behaviors are not excited in normal vehicle maneuvers.

The prediction accuracy of two scenarios suggests that modeling error of the unsprung mass system may greatly affect the accuracy of the parameter identification and thus the dynamics predictions. Therefore, a detail modeling and/or real-time system identification of the unsprung mass system may be needed to improve the feasibility of this approach. Besides, this research also shows that the vehicle parameter identification is challenging because the system has a low degree-of-observability. Therefore, increasing the SNR of the sensor systems and careful designs of the weighting matrix of the identification algorithm are recommended.

A schematic plot a vehicle and four coordinate systems (global frame, road frame, vehicle frame and auxiliary frame).

Block diagram of the vehicle dynamics prediction system.

The driving maneuvers for the illustrative simulation. The upper plot is the steering wheel angle and the lower plot is the wheel torques applying on four tires. The frequency is 1 Hz.

Comparisons of the vehicle dynamics from the simulated vehicle dynamics, the sensor outputs, and the sensor fusion system outputs. The vehicle dynamics are presented in the global frame. The error standard deviations are calculated from the 5th second to the 10th second.

Comparisons of the vehicle dynamics from the simulated vehicle dynamics and the sensor fusion system outputs. The vehicle dynamics are presented in the aux-frame.

The identification of the vehicle mass and moment of inertia. The mean values are calculated from the 15th second to the 10th second.

The identification of the tire tracking stiffness and cornering stiffness. The mean values are calculated from the 15th second to the 10th second.

Predictions of the vehicle dynamics in a left-hand turn event. The prediction inaccuracy is 0.51% on average, calculated from the 10.25th second to the 15th second.

Predictions of the vehicle dynamics in a rollover event. The prediction system successfully predicts the rollover incident. The prediction inaccuracy is 27.3% on average, calculated from the 10.25th second to the 11.5th second.

The relations between the slip ratio and the longitudinal tire adhesive force.

The relations between the slip angle and the lateral tire adhesive force.

Sensor output rates and noise characteristics.

Output |
Noise | |
---|---|---|

| ||

Standard deviation | ||

GPS | 5 Hz | 0.4° |

(attitude measurement) | ||

GPS | 5 Hz | horizontal: 1 m |

(position measurement) | vertical: 3 m | |

Suspension | 1 Hz | 0.001 m |

displacement sensor | ||

Accelerometer | 1 kHz | 0.02 m/s^{2} |

Gyroscope | 1 kHz | 0.08°/s |

Tachometer | 1 kHz | 2°/s |

The measurement bias is not considered.

The relations between the estimation error of sensor fusion system and the relative inaccuracy of the parameter identification.

relative inaccuracy | _{x} |
_{y} |
_{z} | |
---|---|---|---|---|

infinite | 1.94% | 2.68% | 0.33% | |

SNR | 30 dB | 2.50% | 14.62% | 1.52% |

20 dB | 14.91% | 60.28% | 10.41% | |

10 dB | 69.05% | 93.63% | 54.11% |

Different weighting matrices result in different convergence rate.

convergence rate |
_{x} |
_{y} |
_{z} | |
---|---|---|---|---|

diag{0.625, 0.625, 0.625} | 0.08 s | 1.63 s | 0.24 s | |

Q_{4} |
diag{0.125, 0.125, 0.125} | 6.19 s | 18.2 s | 3.32 s |

diag{0.025, 0.025, 0.025} | 36 s | Nan |
25 s | |

diag{0.005, 0.005, 0.005} | Nan |
Nan |
Nan |

The convergent rate is defined at the time when estimated value reaches 90% of the real value.

_{4} and _{4} are both designed as a diagonal matrix. _{4} varies in each case, while _{4} is kept the same as diag{1, 1, 1}.

The value “Nan” means that the convergence time is too long to calculate.