19_1216 Differential-Algebraic Relaxation and Set-Valued Algebraic Loop for Realization of Sliding Mode Systems

Presented at the International Workshop on Nonsmooth Systems and Their Applications in Control and Robotics.

https://personal.utdallas.edu/~oxm130230/sw19/

Transcript

1. 1 Differential-Algebraic Relaxation Differential-Algebraic Relaxation and Set-Valued Algebraic Loop and Set-Valued Algebraic Loop for Realization of for Realization of Sliding Mode Systems Sliding Mode Systems Department of Mechanical Systems Engineering Hiroshima University, Japan Ryo Kikuuwe https://home.hiroshima-u.ac.jp/kikuuwe/ https://www.youtube.com/user/kikuuwe/
2. 3 Today's Talk Today's Talk  [1] Implicit Integration for Simulation of Coulomb Friction  [2] Differential-Algebraic Relaxation for Simulation of Coulomb Friction  [3] Differential-Algebraic Relaxation for Sliding Mode Control  [4] Set-valued Algebraic Loop for Force Control of robots  [5] Concluding Remarks
3. 4 [1] [1] Implicit Integration Implicit Integration (How I arrived in this idea.) (How I arrived in this idea.) [1] [1] Implicit Integration Implicit Integration (How I arrived in this idea.) (How I arrived in this idea.)
4. 5 My motivation: from automobile industry My motivation: from automobile industry  Needs: robots for assisting manual fine positioning task.  Theme: What kind of resistive forces are suited? f v  Idea: Coulomb friction force would be helpful!!  Problem: Discontinuity of Coulomb friction causes problem in digital control.  My interest moved into: Discontinuities and Nonsmooth Systems.
5. 6 Coulomb Friction Coulomb Friction where  Static friction state = “sliding mode"  Control to realize Coulomb friction- like response = sliding mode control with sliding surface v = 0.  Main problem: chattering due to zero-velocity crossing.
6. 7 Why Chattering Happens? Why Chattering Happens?  “Natural” friction  No chattering  “Artificial” friction  Chattering happens  Difference: latency in the closed loop. f v f e device device f v computer f e actuator sensor
7. 8 What if there is no latency? What if there is no latency?  Let's consider this simplistic case:  When “sgn” is closed with a delayless feedback loop, it is just a saturation. [Kikuuwe et al., 2006: IEEE-TRO] [Kikuuwe et al., 2006: ICRA] where
8. 10 It is not new! It is not new! Normal cone of K at y Projection on K from u K = [-1,1]  Not new at all, but this simple expression led me to various techniques for control and simulation. y O Convex set K u  It's just a special case of a long-known relation between the normal cone and the projection.
9. 10 Algorithm Algorithm analytical solution using implicit (backward) Euler discretization No Latency = Implicit Discretization No Latency = Implicit Discretization [Kikuuwe et al.: IEEE TRO, 2006] f v f e Differential Inclusion (continuous time) Differential Inclusion (continuous time) Algebraic Inclusion (discrete time) Algebraic Inclusion (discrete time)
10. 11 Application to Robot Control Application to Robot Control [Kikuuwe et al., IROS 2006] robot robot velocity command force virtual mass virtual mass velocity human operator human operator Implicit integration is used here  Admittance control to realize friction-like response
11. 13 This is how I was cited This is how I was cited  … An interesting contribution is in Kikuuwe et al. (2005) where the implicit discretization of Coulomb friction is rediscovered and extended to more sophisticated multivalued nonsmooth models (equations (14a)-(14c) in Kikuuwe et al., 2005 exactly correspond to the procedure described in Fig. 1.17) ...  Acary & Brogliato: “Numerical Methods for Nonsmooth Dynamical Systems”, Springer, 2008.  (At the time of 2005, I wasn't aware of Acary & Brogliato's works.) V. Acary B. Brogliato
12. 14 Limitation Limitation  If the set-valuedness is closed with an algebraic loop, it can be converted into “saturation.” + ¡  To utilize this principle, the innermost loop must be closed in the computer. G G device device G G computer computer device device computer  Good J  Bad L  Good J
13. 15 [2] [2] Differential Algebraic Relaxation Differential Algebraic Relaxation [2] [2] Differential Algebraic Relaxation Differential Algebraic Relaxation
14. 16 friction force f velocity v Cases Implicit Method Cannot Handle Cases Implicit Method Cannot Handle  When the loop is passing through the real world  When the loop is too complicated, e.g., including another signum... robot
15. 17 Solution: “Diff. Alg. Relaxation” Solution: “Diff. Alg. Relaxation”  Put a high-gain feedback through signum. robot f v e _ ¡ +  It “relaxes” the input-output constraint. K and B should be set as high as possible.  Now we have a Differential Algebraic Inclusion (DAI).
16. 18 Physical Interpretation Physical Interpretation [Kikuuwe et al.: IEEE-TRO, 2006] disp. e  “Relaxed” system velocity friction force  “Original” system  Friction acts on a massless “proxy” object.  The “proxy” and the device are connected through a stiff spring-damper element.  Friction acts on the device friction force f velocity v friction force f velocity v ¡ e _ velocity friction force
17. 19 input u output f K B e Derivation of Algorithm Derivation of Algorithm  Continuous time (diff. alg. inclusion)  Discrete time (algorithm to solve above)  Discrete time (algebraic inclusion) [Kikuuwe et al.: IEEE Trans. on Robotics, 2006] output f input u ¡ +
18. 20 Examples Examples  sliding & rolling of a sphere  Injection simulator  bipedal robot simulator  use of many point-wise frictional contact handlers
19. 21 Summary Summary  Implicit integration is not convenient in the case where the innermost loop surrounding “sgn” passes through a complicated system.  The “Differential-Algebraic Relaxation” is a solution for such a case. robot f v e _ ¡ + disp. e friction force f velocity v ¡ e _
20. 22 [3] [3] Differential Algebraic Relaxation Differential Algebraic Relaxation for for Sliding Mode Control Sliding Mode Control [3] [3] Differential Algebraic Relaxation Differential Algebraic Relaxation for for Sliding Mode Control Sliding Mode Control
21. 23  Consider a simple second order system:  Let's say we want to realize p → p d .  Set the sliding surface as and let's set the controller as  Then, we have  As long as |this part| < HF/M, we have ¾ → 0 in finite time and also p → p d asymptotically. Simple Sliding Mode Controller Simple Sliding Mode Controller
22. 24 Let's implement it Let's implement it  “Explicit” implementation results in chattering  Simple smoothing makes it just a saturated PD.  Not good because velocity measurement noise is magnified by the high D gain. T T device device ¿ p p d computer
23. 25 Solution: Diff. Alg. Implementation Solution: Diff. Alg. Implementation ¿ ¿ ¿ [Kikuuwe & Fujimoto: ICRA2006] [Kikuuwe et al., 2010, TRO ] PID: SMC: ¿ ¿ ¿ ¿  Determine ¿ and to satisfy the following differential algebraic inclusion. T T computer device device ¿ p p d q  We named it: “Proxy-based Sliding Mode Control (PSMC)”
24. 26 f f f f Derivation of the Controller Derivation of the Controller analytical solution implicit Euler discretization Algebraic Inclusion (continuous time) Algebraic Inclusion (continuous time) Differential Algebraic Inclusion (continuous time) Differential Algebraic Inclusion (continuous time) Controller Controller
25. 27 It behaves like this It behaves like this  Sinusoidal desired trajectory and disturbances  Step response PID: SMC:
26. 28 Practical Advantage of PSMC Practical Advantage of PSMC  PID Control  PSMC  It is as accurate as but safer than PID.  In normal operation, its accuracy is the same as PID control.  After torque saturation, its resuming motion is slow and smooth, and there is no overshoots. fast slow position velocity
27. 29 Advantage (continued) Advantage (continued)  It is as accurate as but safer than PID.  Fast resuming from small errors (accuracy; PID)  Slow resuming from large errors (safety; SMC) fast slow position velocity time position slow fast PID: SMC: ¿ ¿ ¿ ¿  Robustness of SMC is not preserved, but the robustness of PID is usually enough for manipulators.  SMC is rather good for saturation management. (Quite a non- standard purpose!! Not to be confused!!)
28. 30 Voice of a User Voice of a User [Van Damme et al., ICRA2007]  PSMC was applied to pneumatic robots  Intended for rehabilitation robots [Van Damme et al., 2009 IJRR] @ Vrije Univ. Brussel, Belgium [Beyl et al., 2011, Advanced Robotics]
29. 31 Users of PSMC Users of PSMC [Van Damme et al., 2007]  Rehabilitation robots,  Assistive robots  Piezzoactuators  Master-slave system  Tactile sensing system  Motion platform  Fuel cells  Microswimmers, etc. [Kashiri et al. 2016] [Jin et al. 2016] [Gu et al. 2015] [Tanaka et al., 2010] [Yoshimoto et al. 2015] [Prieto et al. 2013] [Hasturk et al. 2011] [Chen et al., 2016] [Liao et al. 2015] a [Nishi & Katsura, 2015]
30. 32 Application: Unilateral Teleoperator Application: Unilateral Teleoperator  Modified PSMC with joint-torque limit + task-space velocity limit  Quick and natural response  Safe even under rough command  Compliant to external force
31. 33 Even a 4-Year-Old Can Use It Even a 4-Year-Old Can Use It
32. 34 Equivalence Equivalence  PID controller with torque bound + an anti windup realized by a “set-valued algebraic loop” position ＋ ＋ ＋ － position ＋ ＋  SMC with differential-algebraic relaxation ¿ p pd ＋ ¿ p pd ＋ － ＋
33. 35 Summary Summary  I have applied the “differential-algebraic relaxation” to a simple sliding-mode position controller.  The resultant controller has been found quite useful.
34. 36 [4] [4] Force Control with Force Control with Set-valued Algebraic Loop Set-valued Algebraic Loop [4] [4] Force Control with Force Control with Set-valued Algebraic Loop Set-valued Algebraic Loop
35. 37 environment environment position contact force position command position controller position controller torque command Proxy Proxy external force ＋ ＋ ＋ Admittance Control Admittance Control  AKA: “Position-based Impedance Control”  Realizes specified mechanical impedance on the end-effector. f Mv Bv  Requires force sensor.  If Robot tracks a Proxy accurately, robot’s dynamics becomes close to the “proxy” dynamics  Robot’s dynamics is suppressed by position controller.  Suited for robots with high-ratio gear boxes & large inertia.
36. 38 Conventional Admittance Controller Conventional Admittance Controller  It consists of a “Proxy Dynamics” and “Position Controller”  Robot position qs does not influence Proxy position qx . → External force may pull Robot apart from Proxy. environment environment position contact force position command position controller position controller torque command Proxy Proxy external force ＋ ＋ ＋ ↓ position controller ←proxy dynamics =
37. 39 Flaw of admittance control Flaw of admittance control  Responds only to force sensor inputs.  Does not react to out-of- sensor contacts. (Position controller resists to it.)  It may cause damage or injury. environment environment position contact force position command position controller position controller torque command Proxy Proxy external force ＋ ＋ ＋
38. 40 What if actuator torques are limited? What if actuator torques are limited?  Still unsafe!  Proxy does not respond to the out-of-sensor contact Once ⇒ the force is removed, Robot snaps back to Proxy.  Holding Robot and pushing the force sensor results in unpredictable behavior environment environment position contact force position command position controller position controller torque command Proxy Proxy external force ＋ ＋ ＋
39. 41 A new method A new method  Includes a normal-cone feedback.  Now it is Diff. Alg. Inclusion with unknowns qx and ¿.  It’s an algebraic loop without latency ←proxy .. ↓ position controller environment environment position contact force position command position controller position controller torque command Proxy Proxy external force ＋ ＋ ＋ － [Kikuuwe, 2019, IEEE-TRO]
40. 42  (1) does not permit ¿ outside [{F, F ].  As far as , it’s equivalent to normal admittance control.  When , the proxy acceleration is determined so that (2) with holds true. (2) (1) Property of this DAI Property of this DAI  When the torque is saturated, the normal-cone part acts to pull the robot to the proxy, holding the proxy near the robot.
41. 43 Implementation for discrete time Implementation for discrete time  Discretize it with implicit Euler method ( e.g., 　　　　　　　　　　　　 )  Solve analytically  We get an ordinary algorithm without set-valuedness
42. 44 Algorithm Algorithm  Constants:  No iterative computation. Computationally cheap.
43. 45 Implementation Implementation  New controller with the torque limits 1.5Nm to 6Nm plus gravity+friction compensators  Friction compensator developed by Iwatani & Kikuuwe (2017, SICE- JCMSI)
44. 46  Pushing the link  [C] Robot departs from proxy, and snaps back later  [N] Proxy follows robot Experiments & Results Experiments & Results  [C] Conventional Method  [N] New method  Pushing the link with holding force sensor  [C] Proxy moves away and Robot tries to chase it.  [N] Proxy stays with Robot.
45. 47 Equivalence Equivalence  Admittance controller with torque bound realized by set-valued algebraic feedback position force f external force ＋ ＋ ＋ env env － position force f external force ＋ ＋ ＋ env env －  Sliding-mode-like impedance controller with differential- algebraic relaxation
46. 48 Summary Summary  A torque-bounded admittance controller is realized with a set-valued algebraic loop thorough a normal cone.  It has a similar structure to the differential-algebraic relaxation.  This approach may be useful for bounding the control inputs and states.
47. 49 [5] [5] Concluding Remarks Concluding Remarks [5] [5] Concluding Remarks Concluding Remarks
48. 50 Concluding Remarks Concluding Remarks  Implicit Integration  One way to realize sliding modes  Differential-Algebraic Relaxation  Another way to realize sliding modes, e.g., simulation of Coulomb friction and SMC of robots  Set-valued Algebraic Loop  May be useful for bounding control inputs and states Set-valued Algebraic loop Differential- algebraic relaxation Equivalent!