1. Automotive Research Center Robotics and Mechatronics A Nonlinear Tracking Controller for a Haptic Interface Steer-by-Wire Systems A Nonlinear Tracking Controller for a Haptic Interface Steer-by-Wire Systems P. Setlur, D. Dawson, J. Chen, and J. Wagner Departments of Mechanical and Electrical/Computer Engineering Conference on Decision and Control, December 2002, Las Vegas CLEMSON U N I V E R S I T Y

2. Automotive Research Center Robotics and Mechatronics Presentation Outline Introduction –System Description and Problem Statement –Problem Motivation –Past Research Model Development –System model –Reference model concepts Adaptive Control Design –Error Definitions –Control Design –Stability Proof Extension to Eliminate Torque Measurements Numerical Simulation Results Experimental Results –Setup –Preliminary Results Conclusion

3. Automotive Research Center Robotics and Mechatronics System Description Steer-by-wire system with haptic interface Conventional system Primary Subsystem     T1T1 Feedback Motor Secondary Subsystem

4. Automotive Research Center Robotics and Mechatronics Problem Motivation Advent of Hybrid Vehicles is due to scarcity in fossil fuel and environmental concerns –engine may be cycled on/off : Hydraulic steering systems not feasible –power limitations : mandate efficient technologies Steer-by-wire systems provide –improved vehicle response ( electrical systems are faster) –ability to use additional driver input devices ( joystick) Varied preferences in amount of feedback and feel –most important feedback to the driver, after vision Flexibility in vehicle design

5. Automotive Research Center Robotics and Mechatronics Haptic Interface - Goals Accurate reproduction of driver commands at the wheel Provide force feedback to the driver –Use feedback motor in steer-by-wire systems –Ability to scale inputs Displacement of the driver input device should be governed by a set of target dynamics –Tunable dynamics that permit various choices of “road feel” –Adaptive techniques to compensate for unknown system parameters Elimination of force measurement –Identification of tire/road interface forces

6. Automotive Research Center Robotics and Mechatronics Past Research Liu et al. - worked on estimating the effect of force feedback in a driving simulator (1995) Gillespie et al. - proposed use of force reflecting joysticks to cancel “feedthrough” dynamics in aircrafts (1999) Qu et al. - showed how a “dynamic robust-learning control” scheme can compensate for disturbances that are bounded and sufficiently smooth (2002) Lewis et al. - detailed description of the “impedance control” technique (1993) Setlur et al. - controller to achieve trajectory tracking for steer-by-wire systems (2002) Mills et al. - developed detailed models for steer-by-wire systems (2001)

7. Automotive Research Center Robotics and Mechatronics System Model Secondary Subsystem Primary Subsystem I 1, I 2 - Lumped inertia of Primary and Secondary subsystems Damping and Friction effects, - Scaling factors (gear ratios)     T1T1 Feedback Motor

8. Automotive Research Center Robotics and Mechatronics Reference Model - Concept User feels no difference between these two cases “Impedance Control Technique”

9. Automotive Research Center Robotics and Mechatronics If follows, then the driver feels as if he were driving a conventional vehicle with inertia, damping and friction function. Target system parameters are chosen so that the reference trajectories remain bounded at all times (reference system dynamics are BIBO stable). Reference Model Target Conventional system     T1T1 Primary Subsystem

10. Automotive Research Center Robotics and Mechatronics To quantify the control objective, the following error signals are defined After taking the time derivatives of the filtered tracking errors, the open-loop error system can be rewritten as To achieve the control objectives outlined, the control torques are designed as Filtered Tracking Errors Adaptive Control Driver Experience Tracking error Locked Tracking error Parameter Update Laws

11. Automotive Research Center Robotics and Mechatronics After substituting the control in the open-loop error system, the closed-loop error system can be written as A non-negative function is defined as After differentiating the above function with respect to time, and substituting the closed-loop error systems, we obtain Adaptive Control Parameter estimation errors

12. Automotive Research Center Robotics and Mechatronics For this extension, all system parameters are assumed to be known. The target dynamics are generated using estimated torques. The tracking error signals are defined as before After taking second derivative with respect to time and using the system and reference dynamics, we obtain the open-loop error system The control torques, T 1 and T 2 are designed as Elimination of Torque Measurements Torque Observers (to be designed)

13. Automotive Research Center Robotics and Mechatronics Elimination of Torque Measurements After substituting the control design in the open-loop error system, the closed-loop error system can be written as Clearly, if e 1 = e 2 = 0 then  1 =  1 and  2 =  2 (Identification of tire road forces). The filtered tracking errors are redefined for this problem as ^^ s 1 0 e 1, e 1, e 1 0... Analysis will be presented only for the Primary System. The analysis for the secondary system is based on similar arguments....

14. Automotive Research Center Robotics and Mechatronics Elimination of Torque Measurements After taking the first time derivative and using the system and reference dynamics, we obtain the open-loop error system Based on the above structure, the torque observer is designed as After substituting the observer in the open-loop error system, the closed-loop error system can be written as Standard Signum function (sign function in matlab) Feedback term Unmeasurable Disturbance Robust control like term Add and subtract (s 1 (t) is NOT measurable)

15. Automotive Research Center Robotics and Mechatronics Elimination of Torque Measurements A non-negative function V a1 (t) is defined as After differentiating the above function with respect to time, and substituting the closed-loop error system, we obtain After integrating both sides and performing some manipulations, we obtain So,. Similarly, we can show. From Babalat’s Lemma, and

16. Automotive Research Center Robotics and Mechatronics Simulation Results Simulated system was assumed to have the following parameters I 1 = 6.8 X 10 -2 Kg-m 2 B 1 = 1 X 10 -5 Kg-m 2 /s K 1 = 1 X 10 -7 N-m = 1  1 = 5t exp(-0.005t)     T1T1 I 2 = 54.2 Kg-m 2 B 2 = 1 X 10 -2 Kg-m 2 /s K 2 = 1 X 10 -4 N-m = 1  2 = -200 tanh(  2 ) N x (.) = B x q x + K x q x.

17. Automotive Research Center Robotics and Mechatronics The target dynamics were generated using Further to evaluate performance, a conventional system was simulated Simulation Results I T = 2 Kg-m 2 B T = 1 Kg-m 2 /s K T = 1 N-m  T1 = 1  T2 = 0.1 I a = I 1 + I 2 = 54.268 Kg-m 2 B a = B 1 + B 2 = 1.001 X 10 -2 Kg-m 2 /s K a = K 1 + K 2 = 1.001 X 10 -4 N-m  1 = 1  2 = 1

18. Automotive Research Center Robotics and Mechatronics Simulation Results - Adaptive Control 050100150200 0 20 40 time (s)  1 (N-m) -0.05 0 0.1 0.2 0.3 0.4 Angular Displacement (rad)  d1  a

19. Automotive Research Center Robotics and Mechatronics -14 -12 -8 -4 0 4 6 x 10 -3 Tracking Error (rad) e 1 e 2 -10 0 20 40 60 70 Control Torques (N-m) T 2 T 1 050100150200 time (s) Simulation Results - Adaptive Control

20. Automotive Research Center Robotics and Mechatronics -10 0 20 40 60 70 Control Torques (N-m) 050100150200 time (s) T 1 T 2 -0.06 -0.04 0 0.04 0.08 Torque Observation Errors (N-m)  2   1  Simulation Results - EMK Extension

21. Automotive Research Center Robotics and Mechatronics Experimental Results - EMK Extension Steering Wheel Hydraulic Damper LVDT Drive Motor Feedback Motor Rack Torque Sensors Preamplifiers Current Sensors

22. Automotive Research Center Robotics and Mechatronics Experimental Results - EMK Extension Tests were performed to identify the parameters of the system. The following results were obtained The target system was chosen to have the following parameters The control gains were chosen to be I 1 = 0.0725 Kg-m 2 B 1 = 0.3 Kg-m 2 /s K 1 = 0 N-m I 2 = 2.5 X 10 -3 Kg-m 2 B 2 = 2 X 10 -3 Kg-m 2 /s K 2 = 0 N-m I T = 2 Kg-m 2 B T = 0.3 Kg-m 2 /s K T = 0 N-m  T1 = 10  T2 = 1  1 = 500 K s = 700  1 = 1  2 = 10

23. Automotive Research Center Robotics and Mechatronics Experimental Results - EMK Extension -0.4 -0.2 0 0.2 0.4 e 1, e 2 (rad) 01020304050 time (s) 01020304050 -2 0 1 2 time (s)  d,  1,  2 (rad) -3 -2 0 1 2 3 T 1, T 2 01020304050 time (s)

24. Automotive Research Center Robotics and Mechatronics -4 -3 -2 0 1 2 3 4  1,  1 (N-m) ^ 01020304050 time (s) -3 -2 0 1 2 3 4 5  2,  2 (N-m) ^ Experimental Results - EMK Extension

25. Automotive Research Center Robotics and Mechatronics Experimental Results - EMK Extension Torque sensor measurements –Noisy –Drift –Low resolution Target system dynamics involves twice integrating the torque signals for Adaptive control Gearing factor  1 and  2 Torque capacity of the Feedback motor Repeatability of driver input - Choice of  –larger value control torques have to change quickly (motors are inductive systems)

26. Automotive Research Center Robotics and Mechatronics Concluding Remarks Presented Vehicle Steering System Model for the Steer-by-wire configuration. Presented the Adaptive tracking control algorithm to ensure that –vehicle follows driver commands –driver is provided a haptic feedback Proposed an EMK extension that eliminates the need for torque sensor measurements –identified tire/road interface forces Simulation Results verify the efficacy of the proposed control laws Preliminary Experimental Results were presented to discuss practical issues Future work would involve –Control algorithm to compensation of parametric uncertainties without measurement of torque –Incorporation of visual feedback for driver-in-loop tests