Design and Analysis of Series Elasticity in Closed-loop Actuator Force Control by David William Robinson B.S., Mechanical Engineering Brigham Young University April 1994 S.M., Mechanical Engineering Massachusetts Institute of Technology June 1996 Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2000 c Massachusetts Institute of Technology 2000. All rights reserved. Author............................................................................ Department of Mechanical Engineering May 11, 2000 Certifiedby........................................................................ Gill A. Pratt Associate Professor of Electrical Engineering and Computer Science Thesis Supervisor Certifiedby........................................................................ David Trumper Associate Professor of Mechanical Engineering Thesis Committee Chair Accepted by . ...................................................................... Ain A. Sonin Chairman, Department Committee on Graduate Students Design and Analysis of Series Elasticity in Closed-loop Actuator Force Control by David William Robinson Submitted to the Department of Mechanical Engineering on May 11, 2000, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering Abstract Series elastic actuators have a spring intentionally placed at the actuator output. Measuring the spring strain gives an accurate measurement for closed-loop actuator force control. The low spring stiffness allows for high control gain while maintaining actuator stability. This gives series elastic actuators many desirable properties including high bandwidth at moderate force amplitudes, low output impedance, large dynamic range, internal error rejection and tolerance to shock loading. However, as a consequence of the elasticity, the large force bandwidth capabilities of the actuator are reduced when operating at power saturation limits. Series elasticity is examined with three models. First, it is generalized by using a minimal actuator model. This mathematical model consists of an ideal velocity source actuator, linear spring and proportional controller. Series elasticity is then demonstrated in two case studies of physical actuator systems. The first is a linear hydraulic piston with a servo valve and the second is an electric motor with a geared linear transmission. Both case studies have a linear spring and low complexity control systems. The case studies are analyzed mathematically and verified with physical hardware. A series elastic actuator under simple closed-loop control is physically equivalent to a second order system. This means that an equivalent mass defined by the control system and physical parameters, is effectively in series with the physical spring connected to the actuator load. Non- dimensional analysis of the dynamics clarifies important parametric relationships into a few key dimensionless groups and aids understanding when trying to scale the actuators. The physical equivalent abstractions and non-dimensional dynamic equations help in the development of guidelines for choosing a proper spring stiffness given required force, speed and power requirements for the actuator. Thesis Supervisor: Gill A. Pratt Title: Associate Professor of Electrical Engineering and Computer Science 2 For Sarah 3 Acknowledgments MIT has given me an incredible education the last four years. It has stretched me academically and personally. I am very grateful to the people that have been with me to share this whole experience. I want to thank my advisor Professor Gill Pratt for his advice, guidance, help, time, and enthu- siasm. I appreciate all that I have learned from him as an advisor, teacher, mentor and especially as a friend. I also want to thank the other members of my doctoral thesis committee: Professor David Trumper, Professor Haruhiko Asada, and Dr. J. Kenneth Salisbury. They have all made significant contributions to this thesis. Their direction and encouragement as a group and as individuals has been tremendous. The Leglab is a great place to work because of the people. Dan Paluska, Ben Krupp, Jerry Pratt, Chris Morse, Andreas Hofmann, Greg Huang, Robert Ringrose, Mike Wessler, Allen Parseghian, Hugh Herr, Bruce Deffenbaugh, Olaf Bleck, Ari Wilkenfeld, Terri Iuzzolino, Joanna Bryson, Jianjuen Hu, Chee-Meng Chew, and Peter Dilworth have been great friends and colleagues. They have filled this experience with wonderful camaraderie in both work and fun. I especially appreciate my good friends Brandon Rohrer, Sean Warnick, and Rick Nelson with whom I have shared the MIT engineering Ph.D. road. The time I spent working and in counsel with these men has been choice. They have each been marvelous examples to me in the ways that they balance family, service, work, school, and fun. I thank my parents, brothers, sisters, and their families for their constant support, encourage- ment, and words of confidence. Mom and Dad are the ones who made it possible to start this journey and the whole family have always been there to see me through. I particularly appreciate the support of my two daughters. Hannah’s smiles, hugs and clever wit have kept me going. MaryAnn’s recent arrival gave me the desire and motivation finish up quickly. They have both helped me to keep life in its proper perspective. Finally, I give deep appreciation to my eternal companion Sarah. Our experience together has been full to overflowing. I am thankful for her faith, consistency, gentleness, kindness, and love. When you are with Sarah, how can your experience be anything but great! 143, always. This research was supported in part by the Defense Advanced Research Projects Agency under contract number N39998-00-C-0656 and the National Science Foundation under contract numbers IBN-9873478 and IIS-9733740. 4 Contents 1 Introduction 17 1.1Thesis............................................ 17 1.2Motivation......................................... 18 1.2.1 ActuationandForceControl........................... 18 1.2.2 SeriesElasticActuators.............................. 19 1.3Highlightsofthesisresults................................. 20 1.3.1 Actuators...................................... 20 1.3.2 Bandwidth..................................... 21 1.3.3 OutputImpedance................................. 23 1.3.4 LoadMotion.................................... 25 1.4ThesisContributions.................................... 26 1.5ThesisContents....................................... 26 1.6Noteonthesisdata..................................... 27 2 Background and Related Work 29 2.1ForceControl........................................ 29 2.1.1 PassiveCompliance................................ 30 2.1.2 ActiveControl................................... 31 2.2ApplicationsofForceControl............................... 32 2.3 Robot Actuators and Active Force Control . 33 2.3.1 Electro-Magnetic.................................. 34 2.3.2 Hydraulic...................................... 35 2.3.3 Pneumatic..................................... 36 2.3.4 Others........................................ 36 2.4IntentionallyCompliantRobotActuators........................ 37 2.4.1 SeriesElasticActuators.............................. 37 2.4.2 OtherElectro-mechanicalCompliance...................... 38 2.4.3 HydraulicCompliance............................... 41 2.5Summary.......................................... 42 3 Linear Series Elastic Actuators 43 3.1GeneralModel....................................... 43 3.1.1 Elasticity...................................... 44 3.1.2 ControlSystem................................... 44 3.1.3 Motor........................................ 45 3.1.4 System Inputs . 46 3.2MinimalLinearModelDerivation............................. 47 3.2.1 GeneralPowerDomainOpen-LoopModel.................... 47 3.2.2 GeneralClosed-LoopModel............................ 48 3.3MinimalModelAnalysis.................................. 48 3.3.1 Case1:FixedLoad–Closed-loopBandwidth.................. 49 3.3.2 Case 1: Fixed Load – Large force bandwidth . 52 5 3.3.3 Case2:ForcedLoadMotion–OutputImpedance............... 54 3.3.4 ImpactTolerance.................................. 57 3.4MassLoad.......................................... 58 3.4.1 Forcesontheload................................. 58 3.5DimensionalAnalysis.................................... 61 3.6GeneralModelSummary................................. 63 4 Hydro-Elastic Case Study 65 4.1ModelDerivation...................................... 65 4.1.1 ModelDefinition.................................. 65 4.1.2 PowerDomainModel............................... 68 4.1.3 Closed-loopModel................................. 68 4.1.4 Two input cases . 69 4.2ModelAnalysis....................................... 69 4.2.1 Saturation and Large Force Bandwidth . 70 4.2.2 Case1:Closed-loopBandwidth.......................... 72 4.2.3 Impedance:Case2................................. 74 4.2.4 ProportionalControl................................ 75 4.3EffectofLoadMass.................................... 76 4.3.1 LoadForces..................................... 76 4.3.2 LoadMotion.................................... 78 4.4PhysicalActuator..................................... 78 4.4.1 ComponentSelection................................ 78 4.4.2 ChoosingtheSpringConstant.......................... 79 4.4.3 Physical Actuator Characteristics . 80 4.5Hydro-ElasticSummary.................................