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NON-LINEAR OBSERVER FOR SLIP PARAMETER ESTIMATION OF UNMANNED WHEELED VEHICLES
NON-LINEAR OBSERVER FOR SLIP PARAMETER ESTIMATION OF UNMANNED WHEELED VEHICLES Zibin Song, Yahya Zweiri, Lakmal D Seneviratne and Kaspar Althoefer
{zibin.song; yahya.zweiri, k.althoefer and lakmal.seneviratne} @kcl.ac.uk Department of Mechanical Engineering, King’s College London, U.K Abstract: This paper presents a non-linear sliding mode observer (SMO) to estimate slip parameters based on the vehicle kinematic model and sensor feedback. Lyapunov stability theory is used to establish the stability conditions. It is shown that the observer will converge in a finite time, provided the observer gains satisfy constraints. Specially designed linear and 2-D test rigs are used to validate the proposed observer. Experimental results show that the SMO can estimate the slip parameters to a high accuracy. Thus the proposed observer has the potential to be used on Unmanned Ground Vehicles equipped with GPS or Inertial sensors. Copyright © 2007 IFAC Keywords: Sliding mode observer, Extended Kalman Filter, Slip Parameters, Wheeled Vehicles, Test rigs 1. INTRODUCTION Wheeled vehicle slip parameters (linear slip and slip angle) are required for traction control and trajectory tracking purposes. However, the accurate estimation of the slip parameters is a complex problem. It has been recognized that it is difficult to directly measure slip parameters (Bevly, et al., 2000). (Lee, et al., 2004) measured wheel slips directly using a fifth wheel mounted to the rear bumper. (Bevly, et al., 2000). proposed a method for measuring wheel slip angle using GPS velocity measurements. However, noise imposes limitations on the slip angle measurement. Karl Iagnemma et. al (Iagnemma, et al., 2002) developed a test bed to measure wheel slips. (Angelova, et al., 2006) presented a method to predict slip, using stereo images, assuming that terrain appearance and geometry is known in advance. In (Ray, 1995), an Extended Kalman Filter (EKF) was employed to estimate wheel slip and slip angle based on a simplified dynamic model of a wheeled vehicle. The use of an EKF to estimate slip parameters of a tracked vehicle with no knowledge of soil properties was examined in (Le, 1997). (Scheding, et al., 1999) presented an Extended Kalman Filter for the estimation of slips of a wheeled loader. (Ste´phant, J. and Charara, 2004) proposed a sliding mode observer to estimate the vehicle slip angle based on a simplified dynamic model. (Song, et al., 2006) proposed an Utkin observer for the slip estimation of a tracked vehicle. The robustness and superior performance of the sliding mode observer
over an Extended Kalman Filter was demonstrated using both the simulations and experimental results. This paper proposes a sliding mode observer (SMO) for the estimation of wheel slip and slip angle based on a full kinematic model of a wheeled vehicle and simple sensor feedback. The SMO is validated by comparing the measured slip parameters with the estimated slip parameters, showing very good agreement. Using Lyapunov theory, it is shown that the error dynamics converges after a finite time, provided the gains are selected to satisfy certain stability constraints. The paper is structured as follows: In the Section 2, the kinematic equations of a wheeled vehicle are presented. In Section 3, the SMO is developed and the stability proof is given. Section 4 describes the experimental set up. In Section 5, the SMO is validated by comparing the slip estimates against the measured data and compared to an Extended Kalman Filter and conclusions are drawn in Section 6. 2. WHEELED KINEMATIC MODEL WITH SLIP CONSTRAINTS There are several papers in the literature on the kinematic modelling of wheeled vehicles (Muir and Neuman, 1987; Alexander and Maddocks, 1989; Shim, et al., 1995). In general, the wheel and terrain contact is assumed to be a point contact, and the kinematic equations are derived assuming pure rolling, considering no slip. The kinematic modelling of wheeled vehicles with slip constraints is seldom
studied in the literature. Here, we are concerned with one single wheel as shown in Fig. 1. The wheel can rotate about the vertical axis and can steer a certain angle. There are two control inputs, steering angle, δ and wheel angular velocity, ω .
∂ ( δ)
φ
Fig. 1. Free Body Diagram of a single wheel vehicle e av Tr
y
y&
ω
o
n tio V
c ire ld
x
∂
φ x&
θ
x&
measurements. Since sensor measurements are prone to random noise, the observers have to be robust against noise and disturbances. Sliding mode observers are increasingly used in different applications (Ste´phant and Charara, 2004; Song, et al., 2004; ). Since sensors contain noise, the sliding mode observer (SMO) is a popular approach to reconstruct states in non-linear systems since it can deal with noise and uncertainty in the dynamic system (Utkin and Shi, 1999; Edwards and Spurgeon, 1998). This paper presents a non-linear sliding mode observer for the slip parameter estimation based on the kinematic model of a wheeled vehicle and simple sensor feedback. The SMO minimizes the error between the predicted and measured vehicle trajectory, converging the estimated states to the real states, and then the slip parameters can be estimated. With a sliding mode observer, the control action switches from one value to another in finite time, and this may cause chattering; to avoid this effect, a low pass filter is employed. The sliding mode observer designed in this study supplies a supplementary system with a discontinuous switching component, and then intentionally creates a sliding motion in the supplementary system (Zweiri, et al., 2006).
r
Fig. 2.Wheel Velocities The linear slip of a wheel with respect to the terrain is defined as follows, Fig. 2: V x x ω r − x (1) i = 1− = 1− = =− s ωr ωr ωr ωr where, x& , ω and Vs are longitudinal, wheel angular and slip velocities. r is the radius of the rigid wheel and (x, y) represents a local frame attached to the vehicle. Slip angle is given by, Fig. 2. y& ∂ = tan −1 (2) x& During the experimentation, a single wheel kinematic model is used, in which arbitrary steering angle is equivalent to slip angle in this case. One wheel kinematic modeling is the focus of this paper as single wheel is characterised by wheel slip and sip angle where a vehicle exhibits two characterises. Using velocity relationship, kinematic equations of a single wheel are written as follows (3) X& = x&[cos φ − tan ∂ sin φ ] & (4) Y = x&[sin φ + tan ∂ cos φ ]
x& (5) r (1 − i ) where, x& = ωr(1 − i) , θ is the angular displacement of the single wheel.
θ& = ω =
The methodology used here is based on the sliding mode design principles presented in (Song, et al., 2006; Utkin and Shi, 1999). The basic principle is extended hence to estimate the slip parameters ( i, ∂ ) using Equations (3-5). It is noted that X, Y are linearly dependent variables. The sliding mode observer of rotational motion of a single wheel takes the form of: (6) Xˆ = x cos φ + L sign( X − Xˆ ) + L ( X − Xˆ ) 1
2
& (7) θˆ = L3 sign(θ − θˆ) + L2 (θ − θˆ) Subtracting Equation (3) from Equation (6) and Equation (5) from Equation (7) yields error dynamics X& = − x& sin φ tan ∂ − L1 sign( X − Xˆ ) − L2 ( X − Xˆ ) (8)
θ& =
x& − L3 sign(θ − θˆ) − L2 (θ − θˆ) r (1 − i)
When the sliding mode is reached, X ≈ 0 and θ ≈ 0 i.e. X → Xˆ and θ → θˆ , then the Equations (8) and (9) can be rewritten as follows: − x& sin φ tan ∂ − L sign( X − Xˆ ) − L ( X − Xˆ ) ≈ 0 (10) 1
In a system where not all state variables are easily measured, observers allow the estimation of those state variables based on readily available
2
x& − L3 sign(θ − θˆ) − L2 (θ − θˆ) ≈ 0 (11) r (1 − i) Given the longitudinal velocity of the wheel, x& , sliding gains, L1 , L2 , L3 and the vehicle trajectory X , θ , then the slip parameters ( i, ∂ ) can be calculated from Equations (10) and (11): ∂ = ± arccos
x& 2 sin 2 φ ( L sign( X − Xˆ )) 2 + x& 2 sin 2 φ 1
3. SLIDING MODE OBSERVER
(9)
i = 1−
eq
x& ( L3 sign(θ − θˆ))eq r (12)
where (⋅)eq represents a lowpass filter (Utkin and Shi, 1999).
-are operated at different speeds to generate slips. By varying the wheel and the carriage motors speeds, various wheel slip values can be generated.
Lyapunov Stability Analysis Let the switching functions be: s1 = X − Xˆ s2 = θ − θˆ If the sliding conditions, s1 s&1 < 0 and s2 s&2 < 0 are satisfied, the stability can be enforced and the error dynamic will converge in a finite time (Utkin and Shi, 1999). Applying this approach: A Lyapunov function is defined as 1 V1 = s12 2 Taking the time derivative of the Lyapunov function gives V&1 = s1 s&1 = − x& sin φ tan(∂ ) s1 − L1 sign( s1 ) s1 − L2 s12
Fig. 3. Linear Test rig
≤ x& sin φ tan ∂ s1 − L1 s1 < 0 From this follows that if L1 >| x& sin φ tan ∂ |+
+
, where, ⋅ represents upper
bound then V&1 < 0 , 1 2 s2 , 2 taking the time derivative of the Lyapunov function gives x& V&2 = s2 s&2 = s2 − L3 sign( s2 ) s2 − L2 s22 r (1 − i)
Similarly let the 2nd Lyapunov function be V2 =
x& ≤ s2 − L3 s2 r (1 − i ) From this follows that if L3 >
x& r (1 − i )
+
then V&2 < 0 4. EXPERIMENTAL SETUP Photographs of the test rigs used in this research are shown in Figs. 3 and 4. The algorithm is tested in two stages: on specially designed linear and 2-D test rigs where the slip parameters can be accurately measured and controlled. Fig. 3 shows the picture of a linear test rig with a single wheel used in this research. The linear test rig is equipped with electric motors and measurement sensors, and serves as an experimental platform for algorithm validation purposes. The test rig is mainly comprised of a wheel assembly, a carriage assembly, a controller, sensors, a data acquisition system, and a soil box. The wheel assembly is not vertically constrained and can move up and down freely. The wheel in the experiments is 0.198 m in diameter and 0.098 m in width. The experimental soils are contained in a soil box 0.1 × 0.85 × 0.8 m3. During experiments using the linear test rig, two motors -one driving the wheel and another driving the chain attached to the carriage
Fig. 4 2-D Test rig The 2-D Test rig consists of a wheel assembly, a carriage assembly, an arm assembly, a controller, sensors, a data acquisition system and a cylindrical soil container, see Fig. 4. It is noted that the carriage imposes no restriction on vertical motion of the wheel assembly and hence the wheel assembly can move up and down freely. A vertical beam can move freely inside the carriage holder. The carriage above the arm can move towards and away from the rotation centre of the arm. The experimental soils are contained a cylindrical container with a diameter of 0.85 m and height of 0.15 m. During experimentation, one motor driving the wheel was used, dragging the arm along. The Sliding Mode Observer and the Extended Kalman Filter require measurements of x& , X , φ and θ as inputs. Three sensors including an arm rotary encoder, a pillar rotary encoder, and linear position transducer, assist in measuring the longitudinal velocity of the centre of gravity of the wheel assembly. θ is measured by wheel encoder. φ is measured by the arm rotational and pillar encoders, see Fig. 4. φ = θ1 + θ 2 , see the definition of θ1 and θ 2 in Fig. 5 X is measured by a position transducer and the arm encoder, see Fig. 4.
the EKF is less accurate. The mean RMS errors for the SMO and EKF are 10.42% and 19.56% respectively. Noise, with a magnitude up to ±5% of measurements, used to simulate unknown noise is superimposed on the observer inputs in order to test the robustness of the proposed observer. The mean RMS errors in slip estimate using the SMO and EKF are 10.8% and 31.58%. It can be seen that the sliding mode observer is very robust against noise.
Y
θ2 P
∂
C
y
θ1
x
o
X
0.4
Q 0.35
0.3
Fig.5 Loop velocity vector Diagram
0.25
1
1 1
1
1 1
where, δ c is the angle between the velocity of vehicle and wheel plane.
δ c = tan
−1
r& sin θ1 + r cos θ1θ&1 r& cos θ − r sin θ θ& 1
1 1
where, θ1 , θ&1 -measured using the arm encoder, see Fig. 6, θ , θ& -measured using the pillar encoder, 2
2
see Fig. 4. The slip and slip angle measurement approach is described in detail in Appendix A. In order to implement the SMO algorithm in Unmanned Ground Vehicles, sensors to measure the vehicle longitudinal velocity and vehicle trajectory are required. Of currently available sensing system, sensors such as GPS, DGPS, Inertial Sensors (particularly for high acceleration), Visual Odometry based sensor, or a combination of above sensors, could provide the necessary observer inputs to implement the proposed algorithm (Non-contact speed and distance measurement; Helmick, et al., 2004).
Slip
0.2
X = r cos θ1 x& is measured by the position transducer, the arm encoder. x& = (r& cos θ − r sin θ θ& )2 + (r& sin θ + r cos θ θ& ) 2 cos δ
0.15
0.1
0.05
c
0
-0.05
5
10
15
20
25 Time (s)
30
35
40
45
Fig. 6. Comparison between measured slip (***) and estimated slip using the SMO (---) and EKF (-.-.-.) Table 1 Garside 14/25
Slip Ranges 0% - 10% 10% -20% 20% - 30% 30% - 40% 40% -50% 50% -60% 60% - 70% 70% - 90%
Mean RMS error in slip using SMO 10.42% 10.28% 8.57% 9.98% 3.13% 2.77% 2.29% 1.39%
Mean RMS error in slip using EKF 19.56% 32.66% 15.76% 34.77% 6.64% 5.73% 5.46% 5.02%
Table 1 shows the RMS errors for slip estimate using the SMO and the EKF. It can be seen that the estimated slip parameters are very close to measured slip parameters.
5. EXPERIMENTAL RESULTS The sliding mode observer and Extended Kalman Filter require measurements of x& , X , φ and θ as inputs. Experiments were carried out on Garside 14/25 sand, Garside 60 sand and Garside Iron sand provided by Aggregate Industries UK. generate The slip can be accurately measured and controlled in this manner. In all the experiments, the depth of the soils is 80 mm. 5.1 Linear Test rig In test I, Garside 14/25 sand was used. The sliding gains, L2 and L3 are 0.06 and 1.60 s −1 . Fig. 6 shows the comparison between the measured slip, and the estimates of the slip using the SMO and EKF. The estimated slip using the SMO has a very good agreement with the measured slip, but the slip using
Similarly, tests II and III are carried out on Garside 60 Sand (Soil 2) and Garside Iron Sand (Soil 3) respectively. The slip covers a range within 0% 90%, with 10% interval between 0%-50%, 20% interval between 50% - 90%. Fig. 7 shows that the Mean RMS errors in estimated slips using the SMO and EKF on two type of soils. The Mean RMS errors in estimated slips using the SMO is less than 15%. 5.2 2D Test rig In the first test, Garside 14/25 Sand was used. The sliding gains, L1 , L2 , L3 are 0.01, 0.06 and 1.4 s −1 respectively. The Figs. 8 and 9 shows the measured slip parameters, and the estimated slip parameters using the SMO and EKF.
35 SMO on Soil 2 EKF on Soil 2 SMO on Soil 3 EKF on Soil 3
30
designed to accurately control and measure wheel slip and the slip angle which is hard to measure in practice. A single wheel was commanded to move on Garside 14/25 sand, Garside 60 sand and Garside Iron sand to study the effect of different soil 1.4
1.2
1
Slip angle in rad
The estimated slip parameters using the SMO have very good agreement with the measured slip parameters, but the estimated slip parameters using the EKF are less accurate. The mean RMS errors in slip using the SMO and EKF are 3.04% and 13.36% respectively. The mean RMS errors in slip angle using the SMO and EKF are 4.07% and 886.47% respectively. Table 2 shows the estimated slip parameters using the SMO and EKF. It can be seen that the SMO is more accurate than the EKF.
0.8
0.6
0.4
Mean RMS error in %
25
0.2 20
0
2
4
6
8
15
10 Time (s)
12
14
16
18
10
Fig. 9 Comparison between measured slip angle (***) and estimated slip angle using the SMO (---) and EKF (-.-.-.)
5
0
0
0.1
0.2
0.3
0.4
0.5 Slip range
0.6
0.7
0.8
0.9
1
140
Fig. 7 Mean RMS errors over slip range 0% - 90% using the SMO and EKF
Slip Slip Slip Slip
120
using SMO using EKF angle using SMO angle using EKF
Mean RMS error in %
100
1
80
60
40
0.5
Slip
20
0 0
-0.5
2
4
6
8
10 Time (s)
12
14
16
18
This paper presents a non-linear sliding mode observer to estimate the slip parameters based on a kinematic model of one single wheel and on- board sensor measurements. Lyapunov theory was used to prove the convergence of error dynamics. The sliding gains are chosen according to stability analysis and shown to be bounded. To validate the observer proposed, linear and 2-D test rigs are specially
0.2
0.3
0.4 Slip range
0.5
0.6
0.7
0.8
150 Slip Slip Slip Slip
using SMO using EKF angle using SMO angle using EKF
Mean RMS error in %
100
50
0
6. CONCLUSIONS
0.1
Fig. 10 mean RMS errors in estimated slip parameters using the SMO and EKF on Garside 60 Sand
Fig. 8 Comparison between measured slip (***) and estimated slip using the SMO (---) and EKF (-.-.-.) Similarly, tests II and III are carried out on Garside 60 Sand and Garside Iron Sand respectively. The slip covers a range within 0% - 90%, with 10% interval between 0%-50%, 20% interval between 50% - 90%. Figs. 10 and 11 show that the Mean RMS errors in estimated slip parameters using the SMO and EKF on Garside 60 Sand and Garside Iron Sand respectively. The Mean RMS errors in estimated slip parameters using the SMO is less than 15%.
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 11 mean RMS errors in estimated slip parameters using the SMO and EKF on Garside Iron Sand types on the slip parameters. Experimental results show that the estimated slip parameters using the sliding mode observer have very good agreement with measured ones as shown in Tables 1-2. The
sliding mode observer proves to be more robust against noise. Table 2 Garside 14/25 mean RMS error in
0% - 10%
mean RMS error in
∂
i
Slip Ranges SMO 18.47%
EKF 98.73%
SMO 10.37%
EKF 141.44%
10% -20%
8.04%
45.62%
23.76%
74.89%
20% - 30%
3.8%
21.62%
8.44%
39.4%
30% - 50%
3.39%
19.72%
6.7%
34.88%
50% -70%
1.3%
6.5%
3.78%
15.11%
REFERENCES Bevly, D.M., J.C. Gerdes, C. Wilson and G. Zhang (2000), The Use of GPS based Velocity Measurements for Improved Vehicles State Estimation, Proceeding of the American Control Conference Chicago, Illinois, pp. 2538-2542, June. Lee, C., K. Hedrick and K. Yi (2004), Real-Time Slip-Based Estimation of Maximum Tire-Road Friction Coefficient, IEEE/ASME Transactions on Mechatronics., Vol. 9, No. 2, June. Iganemma, Karl, Hassan Shibly and Steven Dubowsky (2002), On-Line Terrain Parameter Estimation for Planetary Rovers, Proceedings of the 2002 IEEE International Conference on Robotics and Automation, Washington, DC May, 2002, pp. 3142-3147. Angelova, A., L. Matthies, D. Helmick, G. Sibley and P. Peroma (2006), Learning to Predict Slip for Ground Robots, Proceedings of the 2006 IEEE International Conference on Robotics and Automation, Orlando, Florida-May, pp. 33243331. Ray, L.R. (1995), Nonlinear State and Tire Force Estimation for Advanced Vehicle Control, IEEE Transaction on Control Systems Technology, Vol.3, No.1. March. Le, A.Tuan (1997), Estimation of Track-soil Interactions for Autonomous Tracked Vehicles, In proceedings of the 1997 IEEE International Conference on Robotics and Automation, Vol. 2, pp. 1388-1393, April. Scheding, S., G. Dissanayake, E. Mario Nebot, and H. Durrant-Whyte (1999), An Experiment in Autonomous Navigation of an Underground Mining Vehicle, IEEE Transaction on Robotics and Automation, Vol. 15, No. 1, pp. 85-95, February. Ste´phant, J. and A. Charara (2004), Virtual Sensor: Application to Vehicle Sideslip Angle and Transerval Forces, IEEE Transaction on Industrial Electronics, Vol. 51, No. 2, pp. 278289, April. Song, Z.B., Yahya H Zweiri, Lakmal D Seneviratne and Kaspar Althoefer (2006), Non-linear Observer for Slip Estimation of Skid-steering Vehicles, Proceedings of the 2006 IEEE International Conference on Robotics and
Automation, Orlando, Florida-May, pp 14991504. Muir, P.F., C.P. Neuman (1987), “Kinematic modeling of wheeled vehicles,” International journal of Robotics research, Vol. 4, No. 2, pp. 281-329. Alexander, J.C., J.H. Maddocks (1989), On the kinematics of wheeled mobile robots, International journal of Robotics research, Vol. 8, No. 5, pp. 15-27. Shim, H.S., J.H. Kim and K. Koh (1995), Variable Structure Control Nonholonomic Wheeled Mobiel Robot, Proceedings of the 1995 IEEE International Conference on Robotics and Automation, pp. 1694-1700. Murray, R.M. and S.S. Sastry (1993), “Nonholonomic Motion Planning: Steering Using Sinusoids,” IEEE Transactions on Automatic Control, Vol. 38, No. 5, pp. 700-716, May. Utkin, V., J. Guldner, and J. Shi (1999), Sliding Mode Control in Electromechanical Systems, Taylor & Francis Inc., 325 Chesnut Street, Philadelphia, PA 19106. Edwards, C. and S.K. Spurgeon, "Sliding mode control : theory and applications," London; Taylor & Francis, 1998. Zweiri, Y. H., J. F. Whidborne, and L. D. Seneviratne (2006), Diesel engine indicated and load torque estimation using a non-linear observer, Proc. IMechE Vol. 220 Part D: J. Automobile Engineering, pp. 775-785. Non-contact speed and distance measurement, [online] available Racelogic http://www.racelogic.co.uk/?show=VBOXAccuracy. Helmick, D.M., Y. Cheng, D.S. Clouse, and L.H. Matthies (2004), Pathing following using visual odometry for a mars rover in high-slip environments. pp. 772-789. IEEE Aerospace Conference Proceedings. Appendix A 2-D test rig measurement method: ω -measured by the wheel encoder, see Fig. 4 Outputs ∂ -measured using the arm encoder, the position transducer and the pillar encoder, see Figs 4 and 5. ∂ = π − (θ1 + θ 2 ) + δ c i -measured using the arm encoder, the position transducer, the pillar encoder and wheel encoder, see Figs. 4 and 5.
{zibin.song; yahya.zweiri, k.althoefer and lakmal.seneviratne} @kcl.ac.uk Department of Mechanical Engineering, King’s College London, U.K Abstract: This paper presents a non-linear sliding mode observer (SMO) to estimate slip parameters based on the vehicle kinematic model and sensor feedback. Lyapunov stability theory is used to establish the stability conditions. It is shown that the observer will converge in a finite time, provided the observer gains satisfy constraints. Specially designed linear and 2-D test rigs are used to validate the proposed observer. Experimental results show that the SMO can estimate the slip parameters to a high accuracy. Thus the proposed observer has the potential to be used on Unmanned Ground Vehicles equipped with GPS or Inertial sensors. Copyright © 2007 IFAC Keywords: Sliding mode observer, Extended Kalman Filter, Slip Parameters, Wheeled Vehicles, Test rigs 1. INTRODUCTION Wheeled vehicle slip parameters (linear slip and slip angle) are required for traction control and trajectory tracking purposes. However, the accurate estimation of the slip parameters is a complex problem. It has been recognized that it is difficult to directly measure slip parameters (Bevly, et al., 2000). (Lee, et al., 2004) measured wheel slips directly using a fifth wheel mounted to the rear bumper. (Bevly, et al., 2000). proposed a method for measuring wheel slip angle using GPS velocity measurements. However, noise imposes limitations on the slip angle measurement. Karl Iagnemma et. al (Iagnemma, et al., 2002) developed a test bed to measure wheel slips. (Angelova, et al., 2006) presented a method to predict slip, using stereo images, assuming that terrain appearance and geometry is known in advance. In (Ray, 1995), an Extended Kalman Filter (EKF) was employed to estimate wheel slip and slip angle based on a simplified dynamic model of a wheeled vehicle. The use of an EKF to estimate slip parameters of a tracked vehicle with no knowledge of soil properties was examined in (Le, 1997). (Scheding, et al., 1999) presented an Extended Kalman Filter for the estimation of slips of a wheeled loader. (Ste´phant, J. and Charara, 2004) proposed a sliding mode observer to estimate the vehicle slip angle based on a simplified dynamic model. (Song, et al., 2006) proposed an Utkin observer for the slip estimation of a tracked vehicle. The robustness and superior performance of the sliding mode observer
over an Extended Kalman Filter was demonstrated using both the simulations and experimental results. This paper proposes a sliding mode observer (SMO) for the estimation of wheel slip and slip angle based on a full kinematic model of a wheeled vehicle and simple sensor feedback. The SMO is validated by comparing the measured slip parameters with the estimated slip parameters, showing very good agreement. Using Lyapunov theory, it is shown that the error dynamics converges after a finite time, provided the gains are selected to satisfy certain stability constraints. The paper is structured as follows: In the Section 2, the kinematic equations of a wheeled vehicle are presented. In Section 3, the SMO is developed and the stability proof is given. Section 4 describes the experimental set up. In Section 5, the SMO is validated by comparing the slip estimates against the measured data and compared to an Extended Kalman Filter and conclusions are drawn in Section 6. 2. WHEELED KINEMATIC MODEL WITH SLIP CONSTRAINTS There are several papers in the literature on the kinematic modelling of wheeled vehicles (Muir and Neuman, 1987; Alexander and Maddocks, 1989; Shim, et al., 1995). In general, the wheel and terrain contact is assumed to be a point contact, and the kinematic equations are derived assuming pure rolling, considering no slip. The kinematic modelling of wheeled vehicles with slip constraints is seldom
studied in the literature. Here, we are concerned with one single wheel as shown in Fig. 1. The wheel can rotate about the vertical axis and can steer a certain angle. There are two control inputs, steering angle, δ and wheel angular velocity, ω .
∂ ( δ)
φ
Fig. 1. Free Body Diagram of a single wheel vehicle e av Tr
y
y&
ω
o
n tio V
c ire ld
x
∂
φ x&
θ
x&
measurements. Since sensor measurements are prone to random noise, the observers have to be robust against noise and disturbances. Sliding mode observers are increasingly used in different applications (Ste´phant and Charara, 2004; Song, et al., 2004; ). Since sensors contain noise, the sliding mode observer (SMO) is a popular approach to reconstruct states in non-linear systems since it can deal with noise and uncertainty in the dynamic system (Utkin and Shi, 1999; Edwards and Spurgeon, 1998). This paper presents a non-linear sliding mode observer for the slip parameter estimation based on the kinematic model of a wheeled vehicle and simple sensor feedback. The SMO minimizes the error between the predicted and measured vehicle trajectory, converging the estimated states to the real states, and then the slip parameters can be estimated. With a sliding mode observer, the control action switches from one value to another in finite time, and this may cause chattering; to avoid this effect, a low pass filter is employed. The sliding mode observer designed in this study supplies a supplementary system with a discontinuous switching component, and then intentionally creates a sliding motion in the supplementary system (Zweiri, et al., 2006).
r
Fig. 2.Wheel Velocities The linear slip of a wheel with respect to the terrain is defined as follows, Fig. 2: V x x ω r − x (1) i = 1− = 1− = =− s ωr ωr ωr ωr where, x& , ω and Vs are longitudinal, wheel angular and slip velocities. r is the radius of the rigid wheel and (x, y) represents a local frame attached to the vehicle. Slip angle is given by, Fig. 2. y& ∂ = tan −1 (2) x& During the experimentation, a single wheel kinematic model is used, in which arbitrary steering angle is equivalent to slip angle in this case. One wheel kinematic modeling is the focus of this paper as single wheel is characterised by wheel slip and sip angle where a vehicle exhibits two characterises. Using velocity relationship, kinematic equations of a single wheel are written as follows (3) X& = x&[cos φ − tan ∂ sin φ ] & (4) Y = x&[sin φ + tan ∂ cos φ ]
x& (5) r (1 − i ) where, x& = ωr(1 − i) , θ is the angular displacement of the single wheel.
θ& = ω =
The methodology used here is based on the sliding mode design principles presented in (Song, et al., 2006; Utkin and Shi, 1999). The basic principle is extended hence to estimate the slip parameters ( i, ∂ ) using Equations (3-5). It is noted that X, Y are linearly dependent variables. The sliding mode observer of rotational motion of a single wheel takes the form of: (6) Xˆ = x cos φ + L sign( X − Xˆ ) + L ( X − Xˆ ) 1
2
& (7) θˆ = L3 sign(θ − θˆ) + L2 (θ − θˆ) Subtracting Equation (3) from Equation (6) and Equation (5) from Equation (7) yields error dynamics X& = − x& sin φ tan ∂ − L1 sign( X − Xˆ ) − L2 ( X − Xˆ ) (8)
θ& =
x& − L3 sign(θ − θˆ) − L2 (θ − θˆ) r (1 − i)
When the sliding mode is reached, X ≈ 0 and θ ≈ 0 i.e. X → Xˆ and θ → θˆ , then the Equations (8) and (9) can be rewritten as follows: − x& sin φ tan ∂ − L sign( X − Xˆ ) − L ( X − Xˆ ) ≈ 0 (10) 1
In a system where not all state variables are easily measured, observers allow the estimation of those state variables based on readily available
2
x& − L3 sign(θ − θˆ) − L2 (θ − θˆ) ≈ 0 (11) r (1 − i) Given the longitudinal velocity of the wheel, x& , sliding gains, L1 , L2 , L3 and the vehicle trajectory X , θ , then the slip parameters ( i, ∂ ) can be calculated from Equations (10) and (11): ∂ = ± arccos
x& 2 sin 2 φ ( L sign( X − Xˆ )) 2 + x& 2 sin 2 φ 1
3. SLIDING MODE OBSERVER
(9)
i = 1−
eq
x& ( L3 sign(θ − θˆ))eq r (12)
where (⋅)eq represents a lowpass filter (Utkin and Shi, 1999).
-are operated at different speeds to generate slips. By varying the wheel and the carriage motors speeds, various wheel slip values can be generated.
Lyapunov Stability Analysis Let the switching functions be: s1 = X − Xˆ s2 = θ − θˆ If the sliding conditions, s1 s&1 < 0 and s2 s&2 < 0 are satisfied, the stability can be enforced and the error dynamic will converge in a finite time (Utkin and Shi, 1999). Applying this approach: A Lyapunov function is defined as 1 V1 = s12 2 Taking the time derivative of the Lyapunov function gives V&1 = s1 s&1 = − x& sin φ tan(∂ ) s1 − L1 sign( s1 ) s1 − L2 s12
Fig. 3. Linear Test rig
≤ x& sin φ tan ∂ s1 − L1 s1 < 0 From this follows that if L1 >| x& sin φ tan ∂ |+
+
, where, ⋅ represents upper
bound then V&1 < 0 , 1 2 s2 , 2 taking the time derivative of the Lyapunov function gives x& V&2 = s2 s&2 = s2 − L3 sign( s2 ) s2 − L2 s22 r (1 − i)
Similarly let the 2nd Lyapunov function be V2 =
x& ≤ s2 − L3 s2 r (1 − i ) From this follows that if L3 >
x& r (1 − i )
+
then V&2 < 0 4. EXPERIMENTAL SETUP Photographs of the test rigs used in this research are shown in Figs. 3 and 4. The algorithm is tested in two stages: on specially designed linear and 2-D test rigs where the slip parameters can be accurately measured and controlled. Fig. 3 shows the picture of a linear test rig with a single wheel used in this research. The linear test rig is equipped with electric motors and measurement sensors, and serves as an experimental platform for algorithm validation purposes. The test rig is mainly comprised of a wheel assembly, a carriage assembly, a controller, sensors, a data acquisition system, and a soil box. The wheel assembly is not vertically constrained and can move up and down freely. The wheel in the experiments is 0.198 m in diameter and 0.098 m in width. The experimental soils are contained in a soil box 0.1 × 0.85 × 0.8 m3. During experiments using the linear test rig, two motors -one driving the wheel and another driving the chain attached to the carriage
Fig. 4 2-D Test rig The 2-D Test rig consists of a wheel assembly, a carriage assembly, an arm assembly, a controller, sensors, a data acquisition system and a cylindrical soil container, see Fig. 4. It is noted that the carriage imposes no restriction on vertical motion of the wheel assembly and hence the wheel assembly can move up and down freely. A vertical beam can move freely inside the carriage holder. The carriage above the arm can move towards and away from the rotation centre of the arm. The experimental soils are contained a cylindrical container with a diameter of 0.85 m and height of 0.15 m. During experimentation, one motor driving the wheel was used, dragging the arm along. The Sliding Mode Observer and the Extended Kalman Filter require measurements of x& , X , φ and θ as inputs. Three sensors including an arm rotary encoder, a pillar rotary encoder, and linear position transducer, assist in measuring the longitudinal velocity of the centre of gravity of the wheel assembly. θ is measured by wheel encoder. φ is measured by the arm rotational and pillar encoders, see Fig. 4. φ = θ1 + θ 2 , see the definition of θ1 and θ 2 in Fig. 5 X is measured by a position transducer and the arm encoder, see Fig. 4.
the EKF is less accurate. The mean RMS errors for the SMO and EKF are 10.42% and 19.56% respectively. Noise, with a magnitude up to ±5% of measurements, used to simulate unknown noise is superimposed on the observer inputs in order to test the robustness of the proposed observer. The mean RMS errors in slip estimate using the SMO and EKF are 10.8% and 31.58%. It can be seen that the sliding mode observer is very robust against noise.
Y
θ2 P
∂
C
y
θ1
x
o
X
0.4
Q 0.35
0.3
Fig.5 Loop velocity vector Diagram
0.25
1
1 1
1
1 1
where, δ c is the angle between the velocity of vehicle and wheel plane.
δ c = tan
−1
r& sin θ1 + r cos θ1θ&1 r& cos θ − r sin θ θ& 1
1 1
where, θ1 , θ&1 -measured using the arm encoder, see Fig. 6, θ , θ& -measured using the pillar encoder, 2
2
see Fig. 4. The slip and slip angle measurement approach is described in detail in Appendix A. In order to implement the SMO algorithm in Unmanned Ground Vehicles, sensors to measure the vehicle longitudinal velocity and vehicle trajectory are required. Of currently available sensing system, sensors such as GPS, DGPS, Inertial Sensors (particularly for high acceleration), Visual Odometry based sensor, or a combination of above sensors, could provide the necessary observer inputs to implement the proposed algorithm (Non-contact speed and distance measurement; Helmick, et al., 2004).
Slip
0.2
X = r cos θ1 x& is measured by the position transducer, the arm encoder. x& = (r& cos θ − r sin θ θ& )2 + (r& sin θ + r cos θ θ& ) 2 cos δ
0.15
0.1
0.05
c
0
-0.05
5
10
15
20
25 Time (s)
30
35
40
45
Fig. 6. Comparison between measured slip (***) and estimated slip using the SMO (---) and EKF (-.-.-.) Table 1 Garside 14/25
Slip Ranges 0% - 10% 10% -20% 20% - 30% 30% - 40% 40% -50% 50% -60% 60% - 70% 70% - 90%
Mean RMS error in slip using SMO 10.42% 10.28% 8.57% 9.98% 3.13% 2.77% 2.29% 1.39%
Mean RMS error in slip using EKF 19.56% 32.66% 15.76% 34.77% 6.64% 5.73% 5.46% 5.02%
Table 1 shows the RMS errors for slip estimate using the SMO and the EKF. It can be seen that the estimated slip parameters are very close to measured slip parameters.
5. EXPERIMENTAL RESULTS The sliding mode observer and Extended Kalman Filter require measurements of x& , X , φ and θ as inputs. Experiments were carried out on Garside 14/25 sand, Garside 60 sand and Garside Iron sand provided by Aggregate Industries UK. generate The slip can be accurately measured and controlled in this manner. In all the experiments, the depth of the soils is 80 mm. 5.1 Linear Test rig In test I, Garside 14/25 sand was used. The sliding gains, L2 and L3 are 0.06 and 1.60 s −1 . Fig. 6 shows the comparison between the measured slip, and the estimates of the slip using the SMO and EKF. The estimated slip using the SMO has a very good agreement with the measured slip, but the slip using
Similarly, tests II and III are carried out on Garside 60 Sand (Soil 2) and Garside Iron Sand (Soil 3) respectively. The slip covers a range within 0% 90%, with 10% interval between 0%-50%, 20% interval between 50% - 90%. Fig. 7 shows that the Mean RMS errors in estimated slips using the SMO and EKF on two type of soils. The Mean RMS errors in estimated slips using the SMO is less than 15%. 5.2 2D Test rig In the first test, Garside 14/25 Sand was used. The sliding gains, L1 , L2 , L3 are 0.01, 0.06 and 1.4 s −1 respectively. The Figs. 8 and 9 shows the measured slip parameters, and the estimated slip parameters using the SMO and EKF.
35 SMO on Soil 2 EKF on Soil 2 SMO on Soil 3 EKF on Soil 3
30
designed to accurately control and measure wheel slip and the slip angle which is hard to measure in practice. A single wheel was commanded to move on Garside 14/25 sand, Garside 60 sand and Garside Iron sand to study the effect of different soil 1.4
1.2
1
Slip angle in rad
The estimated slip parameters using the SMO have very good agreement with the measured slip parameters, but the estimated slip parameters using the EKF are less accurate. The mean RMS errors in slip using the SMO and EKF are 3.04% and 13.36% respectively. The mean RMS errors in slip angle using the SMO and EKF are 4.07% and 886.47% respectively. Table 2 shows the estimated slip parameters using the SMO and EKF. It can be seen that the SMO is more accurate than the EKF.
0.8
0.6
0.4
Mean RMS error in %
25
0.2 20
0
2
4
6
8
15
10 Time (s)
12
14
16
18
10
Fig. 9 Comparison between measured slip angle (***) and estimated slip angle using the SMO (---) and EKF (-.-.-.)
5
0
0
0.1
0.2
0.3
0.4
0.5 Slip range
0.6
0.7
0.8
0.9
1
140
Fig. 7 Mean RMS errors over slip range 0% - 90% using the SMO and EKF
Slip Slip Slip Slip
120
using SMO using EKF angle using SMO angle using EKF
Mean RMS error in %
100
1
80
60
40
0.5
Slip
20
0 0
-0.5
2
4
6
8
10 Time (s)
12
14
16
18
This paper presents a non-linear sliding mode observer to estimate the slip parameters based on a kinematic model of one single wheel and on- board sensor measurements. Lyapunov theory was used to prove the convergence of error dynamics. The sliding gains are chosen according to stability analysis and shown to be bounded. To validate the observer proposed, linear and 2-D test rigs are specially
0.2
0.3
0.4 Slip range
0.5
0.6
0.7
0.8
150 Slip Slip Slip Slip
using SMO using EKF angle using SMO angle using EKF
Mean RMS error in %
100
50
0
6. CONCLUSIONS
0.1
Fig. 10 mean RMS errors in estimated slip parameters using the SMO and EKF on Garside 60 Sand
Fig. 8 Comparison between measured slip (***) and estimated slip using the SMO (---) and EKF (-.-.-.) Similarly, tests II and III are carried out on Garside 60 Sand and Garside Iron Sand respectively. The slip covers a range within 0% - 90%, with 10% interval between 0%-50%, 20% interval between 50% - 90%. Figs. 10 and 11 show that the Mean RMS errors in estimated slip parameters using the SMO and EKF on Garside 60 Sand and Garside Iron Sand respectively. The Mean RMS errors in estimated slip parameters using the SMO is less than 15%.
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 11 mean RMS errors in estimated slip parameters using the SMO and EKF on Garside Iron Sand types on the slip parameters. Experimental results show that the estimated slip parameters using the sliding mode observer have very good agreement with measured ones as shown in Tables 1-2. The
sliding mode observer proves to be more robust against noise. Table 2 Garside 14/25 mean RMS error in
0% - 10%
mean RMS error in
∂
i
Slip Ranges SMO 18.47%
EKF 98.73%
SMO 10.37%
EKF 141.44%
10% -20%
8.04%
45.62%
23.76%
74.89%
20% - 30%
3.8%
21.62%
8.44%
39.4%
30% - 50%
3.39%
19.72%
6.7%
34.88%
50% -70%
1.3%
6.5%
3.78%
15.11%
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Automation, Orlando, Florida-May, pp 14991504. Muir, P.F., C.P. Neuman (1987), “Kinematic modeling of wheeled vehicles,” International journal of Robotics research, Vol. 4, No. 2, pp. 281-329. Alexander, J.C., J.H. Maddocks (1989), On the kinematics of wheeled mobile robots, International journal of Robotics research, Vol. 8, No. 5, pp. 15-27. Shim, H.S., J.H. Kim and K. Koh (1995), Variable Structure Control Nonholonomic Wheeled Mobiel Robot, Proceedings of the 1995 IEEE International Conference on Robotics and Automation, pp. 1694-1700. Murray, R.M. and S.S. Sastry (1993), “Nonholonomic Motion Planning: Steering Using Sinusoids,” IEEE Transactions on Automatic Control, Vol. 38, No. 5, pp. 700-716, May. Utkin, V., J. Guldner, and J. Shi (1999), Sliding Mode Control in Electromechanical Systems, Taylor & Francis Inc., 325 Chesnut Street, Philadelphia, PA 19106. Edwards, C. and S.K. Spurgeon, "Sliding mode control : theory and applications," London; Taylor & Francis, 1998. Zweiri, Y. H., J. F. Whidborne, and L. D. Seneviratne (2006), Diesel engine indicated and load torque estimation using a non-linear observer, Proc. IMechE Vol. 220 Part D: J. Automobile Engineering, pp. 775-785. Non-contact speed and distance measurement, [online] available Racelogic http://www.racelogic.co.uk/?show=VBOXAccuracy. Helmick, D.M., Y. Cheng, D.S. Clouse, and L.H. Matthies (2004), Pathing following using visual odometry for a mars rover in high-slip environments. pp. 772-789. IEEE Aerospace Conference Proceedings. Appendix A 2-D test rig measurement method: ω -measured by the wheel encoder, see Fig. 4 Outputs ∂ -measured using the arm encoder, the position transducer and the pillar encoder, see Figs 4 and 5. ∂ = π − (θ1 + θ 2 ) + δ c i -measured using the arm encoder, the position transducer, the pillar encoder and wheel encoder, see Figs. 4 and 5.