Adaptive Observer for PTWV
8.3 Observer Evaluation and Simulation
8.3 Observer Evaluation and Simulation
In this section, the proposed interconnected observer for the combined longitudinal and lateral dynamics of PTWV is evaluated by co-simulation withBikeSimc software. A PTWV model is chosen from the dataset Big Sport Baseline8 bodies and default parameters. The simulations are carried out in two maneuvers:
Maneuver 1: Urban scenic road includes acceleration and braking scenarios with a high friction coefficient µ=0.9.
Maneuver 2: Handling road course with variable speed.
The motorcycle behavior, including longitudinal and lateral dynamics requires three inputs: the rider’s steering torque applied on the handlebars and the two braking torques applied on both front and rear wheels to reduce the longitudinal velocity.
In the following, if the actual state vy,vx andFyf,Fyr andFxf,Fxr are unknown, the state estimation can be validated from lateral and longitudinal accelerations as follow:
Eq1: ˆay= (Fˆyfm+Fˆyr), Eq2 : ˆay=vˆ˙y+vxψˆ˙
Eq3: ˆax= (Fˆxfm+Fˆxr), Eq4 : ˆax=vˆ˙x−vyψˆ˙+Cd/m.vˆx2 (8.43) The first simulation is carried out from FHWA (Federal Highway Administration), this maneuver demon-strates an acceleration and braking test includes three phases.
0 10 20 30
The first phase (0 < t <10(s)) is an acceleration phase, where only the drive torque T is applied on the rear wheel. In the second phase (10< t <20(s)), no braking or engine torque is applied and the main body is subject to lateral motion in response to the generated tire forces whereas the front body is subject to steering motion as imposed by the applied rider’s steering torqueτ on the motorcycle handlebar. The third phase (20< t <30(s)), is the braking phase where a braking torqueBf andBr are applied to both front and rear wheels to reduce the forward speed of the PTWV from 35m/sto 10m/s.
0 10 20 30
Figure 8.3: Maneuver 1: BikeSimmeasured states.
The inputs of the lateral and longitudinal models are the braking torques at the front and rear wheels, the drive engine torque and the steering torque depicted in figure8.2. The measured state used in the observer design are given in figure8.3. The lateral and forward accelerations (axanday) are also used in the observer design as well as to validate the estimation of the unmeasured states from equation (8.43).
148 Chapter 8. Interconnected Observers for PTWV
Figure 8.4: Maneuver 1: Actual states (in blue) compared to estimated states (dashed red).
Estimation results of this scenario are depicted on figure8.4, which are the lateral and longitudinal speeds, the front and rear longitudinal forces, the front and rear cornering forces. From equation (8.43), the unmeasured state (vx,vy,Fxi,Fyi) are validated and depicted in figure8.5.
Figure 8.5: Maneuver 1: Validation of the estimated states.
We can see on figure (8.4) the non-measured states (the lateral and longitudinal speeds, the front and rear longitudinal forces, the front and rear cornering forces) compared to the actual data acquired from BikeSim sensors. These results show the ability of the designed observer to well recover simultaneously the interconnected longitudinal and lateral states of the motorcycle motion. Also we can remark a rapid transient phase of the observer. Indeed, these plots show some differences, in particular in the lateral speed and the lateral forces. This means that the lateral model is the most affected by the modeling errors between theBikeSim model and the sharp model used in the observer design. In fact, the two body sharp model is a pure lateral dynamic model valid for a various constant forward speeds. Indeed, the lateral model is slightly
8.3. Observer Evaluation and Simulation 149 affected by the longitudinal motion because it does not take into account speed variation when accelerating or braking. Despite modeling errors and the speed variation, the state estimation error still have ISpS performance and the interconnected observer still provide good estimation.
In the second maneuver, the motorcycle undergoes an oncoming traffic in road course with variable speed.
The figures 8.6show the input signals in the longitudinal and lateral models whereas figures8.7depict the measured states along the track. The unmeasured state are estimated, depicted in figures8.8, and validated in figure8.9.
Figure 8.7: Maneuver 2: Measured states.
150 Chapter 8. Interconnected Observers for PTWV
Figure 8.8: Maneuver 2: Actual states (in blue) compared to estimated states (dashed red).
0 20 40 60 80
time [s]
-4 -2 0 2 4 6
ax[m.s2 ]
Longitudinal acceleration
actual (Fxf+Fxr)/M
˙
vx−ψv˙ y+ (Cd/M)vx2
0 20 40 60 80
time [s]
-15 -10 -5 0 5 10 15
ay[m.s2 ]
Lateral acceleration
actual (Fyf+Fyr)/M vy+ ˙ψvx
Figure 8.9: Maneuver 2: Validation of the estimated states.
This track test includes a speed variations, in which the rider alternate between acceleration and braking in turns, this maneuver is very common in riding situations. It clearly demonstrates all the capabilities of the interconnected observer of estimating the unknown states of both longitudinal and lateral motorcycle dynamics. Moreover, the forward speed is varying between 15 and 40m/ssimultaneously with the braking and engine torques and with lateral rider action in order to test the estimation performances independently of the longitudinal velocity variations. As for the first test, this scenario illustrate that the observer rapidly and accurately estimates the state of the interconnected model with minimal error even for extreme riding situations. Despite some small estimation errors owing to modeling uncertainties, one can conclude that the interconnected observer provides satisfactory results.
8.4. Final Remarks 151
8.4 Final Remarks
The main contribution of this work is to extend the existing works on the estimation of two-wheeled vehicle’s lateral dynamics by the estimation of the longitudinal motion. The dependencies between these two motions interfere on the observability of the estimators. In this scope, this chapter dealt with the estimation of the quasi LPV out-of-plane and in-plane motorcycle motion. The interconnected observer formulation of the estimation problem is presented and evaluated throughout co-simulation with a high-end motorcycle simulation. This method is based on the decomposition of motorcycle model into two quasi LPV subsystems, then each quasi LPV subsystems model of the vehicle is transformed into Takagi-Sugeno (TS), the result is formalized using Lyapunov theory and the Input to State Practical Stability (ISpS) formulated as an optimization problem under Linear Matrix Inequalities (LMI) aiming to minimize the error estimation bound.
The observer allows the reconstruction of relevant non-measurable states of the PTWV: the forward speed and the longitudinal tire forces from the first sub-observer and lateral speed, roll angle and the cornering forces from the second sub-observer.