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CLEMSON U N I V E R S I T Y

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.

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  1. A Nonlinear Tracking Controller for a Haptic Interface Steer-by-Wire SystemsP. Setlur, D. Dawson, J. Chen, and J. WagnerDepartments of Mechanical and Electrical/Computer EngineeringConference on Decision and Control, December 2002, Las Vegas CLEMSON U N I V E R S I T Y

  2. 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. q1, t1 Driver input torque qa, t1 T1 Feedback Motor Drive Motor T2 qa, t2 Tire/Road interface forces q2, t2 System Description Steer-by-wire system with haptic interface Conventional system Primary Subsystem Secondary Subsystem

  4. 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. 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. 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. q1, t1 T1 Feedback Motor Drive Motor , - Scaling factors (gear ratios) T2 q2, t2 System Model Primary Subsystem I1,I2- Lumped inertia of Primary and Secondary subsystems Damping and Friction effects Secondary Subsystem

  8. Reference Model - Concept User feels no difference between these two cases “Impedance Control Technique”

  9. q1, t1 qd, t1 T1 qd, t2 Reference Model Primary Subsystem Target Conventional system • 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).

  10. Adaptive Control • 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 Driver Experience Tracking error Locked Tracking error Parameter Update Laws

  11. Adaptive Control • 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 Parameter estimation errors

  12. Elimination of Torque Measurements • 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, T1 and T2 are designed as Torque Observers (to be designed)

  13. . . . . . . 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 e1 = e2 = 0 then t1 = t1 and t2 = t2 (Identification of tire road forces). • The filtered tracking errors are redefined for this problem as ^ ^ . s1 0 e1, e1, e10 Analysis will be presented only for the Primary System. The analysis for the secondary system is based on similar arguments

  14. Unmeasurable Disturbance 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 Add and subtract (s1(t) is NOT measurable) Standard Signum function (sign function in matlab) Robust control like term Feedback term

  15. Elimination of Torque Measurements • A non-negative function Va1(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. q1, t1 I1 = 6.8 X 10-2 Kg-m2 B1 = 1 X 10-5 Kg-m2/s K1 = 1 X 10-7 N-m = 1 t1 = 5t exp(-0.005t) . Nx(.) = Bxqx + Kxqx T1 T2 I2 = 54.2 Kg-m2 B2 = 1 X 10-2 Kg-m2/s K2 = 1 X 10-4 N-m = 1 t2 = -200 tanh(q2) q2, t2 Simulation Results • Simulated system was assumed to have the following parameters

  17. qd, t1 qd, t2 Simulation Results • The target dynamics were generated using • Further to evaluate performance, a conventional system was simulated IT = 2 Kg-m2 BT = 1 Kg-m2/s KT = 1 N-m aT1 = 1 aT2 = 0.1 Ia = I1 + I2 = 54.268 Kg-m2 Ba = B1 + B2 = 1.001 X 10-2 Kg-m2/s Ka = K1 + K2 = 1.001 X 10-4 N-m a1 = 1 a2 = 1

  18. (N-m) 1 t Simulation Results - Adaptive Control 40 20 0 0.4 q d1 0.3 0.2 Angular Displacement (rad) 0.1 q a 0 -0.05 0 50 100 150 200 time (s)

  19. -3 x 10 Simulation Results - Adaptive Control 6 4 e 2 0 e 1 -4 Tracking Error (rad) -8 -12 -14 70 60 T 2 40 Control Torques (N-m) 20 T 1 0 -10 0 50 100 150 200 time (s)

  20. Simulation Results - EMK Extension 70 60 40 T 2 Control Torques (N-m) 20 T 1 0 -10 0.08 ~ t 2 0.04 ~ t Torque Observation Errors (N-m) 1 0 -0.04 -0.06 0 50 100 150 200 time (s)

  21. Experimental Results - EMK Extension Drive Motor Steering Wheel Torque Sensors Feedback Motor Rack LVDT Hydraulic Damper Current Sensors Preamplifiers

  22. 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 I1 = 0.0725 Kg-m2 B1 = 0.3 Kg-m2/s K1 = 0 N-m I2 = 2.5 X 10-3 Kg-m2 B2 = 2 X 10-3 Kg-m2/s K2 = 0 N-m IT = 2 Kg-m2 BT = 0.3 Kg-m2/s KT = 0 N-m aT1 = 10 aT2 = 1 b1 = 500 Ks = 700 r1 = 1 r2 = 10

  23. 2 0.4 1 0.2 qd ,q1, q2 (rad) 0 0 e1, e2 (rad) -0.2 -1 -0.4 -2 0 10 20 30 40 50 0 10 20 30 40 50 time (s) time (s) 3 2 1 T1 , T2 0 -1 -2 -3 0 10 20 30 40 50 time (s) Experimental Results - EMK Extension

  24. 4 3 2 1 t1 ,t1(N-m) ^ 0 5 -1 4 -2 3 -3 2 -4 1 t2 ,t2(N-m) ^ 0 -1 -2 -3 Experimental Results - EMK Extension 0 10 20 30 40 50 time (s)

  25. 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 a1 and a2 • Torque capacity of the Feedback motor • Repeatability of driver input - Choice of r • larger value control torques have to change quickly (motors are inductive systems)

  26. Concluding Remarks • PresentedVehicle Steering System Model for the Steer-by-wire configuration. • Presented the Adaptive tracking control algorithmto 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

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