PACMAN meeting: 09 July 2015 David Tshilumba D. Tshilumba, Delft, 09 July 2015

Available space Parallel kinematic Required: transmission with ratio 𝑛= 1 3 Desired output dofs: x , y, 𝜃 𝑥 , 𝜃 𝑦 Dofs to be constrained: z and 𝜃 𝑧 Actuators: linear motors Sensors: optical rulers Strategy: 2x2dofs stage y z x Parameters to be optimized: Sensors location Transmission volume Transmission stiffness Transmission dynamics Rolling angle minimized Xy translation D. Tshilumba, Delft, 09 July 2015

Actuators Linear piezo motor Stepper motor + piezo stack Resolution Linear piezo motor Vs Hybrid actuator Linear piezo motor Stepper motor + piezo stack Resolution 0.03nm > 0.15nm Range 20mm Max lateral force 20N 255N Max torque 0.5Nm 1.5Nm Option 1: 15kCHF Option 2: < 8kCH Piezo stepper have 2 modes (analog and stepping) Piezo stepper directly provides mm stroke without backlash when travel direction changes High resolution in stepping mode (10nm steps!) Piezo stepper has a high locking force at power off (through friction …N) Price (approx. …EUR) Steppermotor+ harmonic drive stepping resolution: (certainement moins bonne) Backlash when changing rotation direction of stepper motor in transmission (screw (differential? Or ball?)) Stepper motor has a high stall torque thanks to harmonic drive ratio(…Nm) - Less sensitive to external torques - Cheaper solution (approx. …EUR) D. Tshilumba, Delft, 09 July 2015 3

Compliant mechanism classification Distributed compliance mechanisms Lumped compliance mechanisms Stress distributed in overall structure Large range achievable Low on-axis stiffness Low input force required Stress concentrated in hinges Smaller rane achievable Higher on-axis stiffness Higher input force required CATIA figure of both versions Add figure with first 3 modes (unloaded) of these to show the difference on axis stiffness - For same thickness of hinge, k_rot of leaf and k_rot of notch is … larger => overall on axis stiffness ratio of … CCL for our application we are interested in large on axis stiffness first, and then a large stroke in the millimeter range => lumped compliance more appropriate Lumped compliance mechanism more appropriate for our application D. Tshilumba, Delft, 09 July 2015 4

Compliant joints Elementary joints Leaf spring Notch hinge b r Hyp: 𝑏>10ℎ , 𝐿>10ℎ Hyp: 𝑟 𝑒 >5 𝐾 𝛼𝑀 = 𝐸𝐼 𝐿 𝛼 𝑎𝑑𝑚 = 2 𝜎 𝑎𝑑𝑚 𝐿 𝐸ℎ 𝐾 𝛼𝑀 = 2𝐸𝑏 𝑒 2.5 9𝜋 𝑟 e b 𝐾 𝑡𝑟𝑎𝑐 = 𝐸𝐴 𝐿 𝛿 𝑡𝑟𝑎𝑐,𝑎𝑑𝑚 = 𝜎 𝑎𝑑𝑚 𝐿 𝐸 𝐾 𝑡𝑟𝑎𝑐 =0.353 𝐸𝑏 𝑒 𝑟 𝛿 𝑠ℎ𝑒𝑎𝑟,𝑎𝑑𝑚 = 𝜎 𝑎𝑑𝑚 𝐿 2 3𝐸ℎ h 𝐾 𝑠ℎ𝑒𝑎𝑟 = 12𝐸𝐼 𝐿 3 𝐾 𝑠ℎ𝑒𝑎𝑟 =0.218 𝐸𝑏 𝑒 1.5 𝑟 1.5 L Elastic deflection of flexible member Small deflection hypothesis D. Tshilumba, Delft, 09 July 2015 5

Compliant joints Materials: Aluminium (green) Titanium (magenta) Steel (blue) Notch stiffer in rotation: 𝐾 𝛼𝑀,𝑁𝑜𝑡𝑐ℎ 𝐾 𝛼𝑀,𝑙𝑒𝑎𝑓 = 24 9𝜋 2 𝐿 ℎ 0.5 Small achievable deflection with usual material Notch: < 5°; leaf: < 15° D. Tshilumba, Delft, 09 July 2015 6

Inverted lever mechanism Parameters to consider: Coupling stiffness Pivot stiffness Intrinsic flexure overall stiffness Example of 1dof lever mechnism Lumped mass representation 𝑥 𝑜𝑢𝑡 𝑥 𝑖𝑛 = −1 𝑛 𝑖𝑑 𝑘 3 𝑘 2 1+ 1 𝑛 𝑖𝑑 2 + 1 𝑛 𝑖𝑑 2 + 𝑘 3 𝑘 1 Ratios 𝑘 3 𝑘 2 and 𝑘 3 𝑘 1 must be minimized! Assumptions: lever arm is rigid angular stiffness of the springs ignored This could be done by applied via such a design. Of course this first Issues to tackle for long range displacement: excessive stress in hinges internal eigen modes D. Tshilumba, Delft, 09 July 2015

Compliant amplification mechanisms Example : 1dof double lever 2x2dofs Double levers (n1=2/3, n2=1/2) Actuators : casing in dark grey and driving axis in orange Hinges dimensions: Notch (b x r x e) : 50mm x 10mm x 1.5mm Leaf spring (b x L x h): 50mm x 20mm x 1.5mm Comparison of static performance between notch-based and leaf spring-based configuration D. Tshilumba, Delft, 09 July 2015 8

Compliant amplification mechanisms Example : 1dof double lever 280MPa 0.5Nm Small deviation in ratio compared with ideal case Significance of Link flexibility with notches Stress distribution in leafsprings Piezo motor input displacement limited CCL: Not possible to achieve stroke of 1mm with lumped compliace notch-based flexure bearing only! Excessive torque on the ceramics for y>… 9

Stress release concept: “floating” base of flexure bearing MR fluid + flexure bearing MR fluid composition: Carrier fluid Carbonyl Iron particles additives Parameter Value Particle size 0.05 up to 10 m Operating temperature -40°C up to 150°C No field viscosity 0.1 up to 1Pa.s Maximum field ~250 kA/m If magnetic field activated: Alignment of magnetic particles Rapid change of viscosity (~ 10 5 within ms) Possible to become solid D. Tshilumba, Delft, 09 July 2015

Stress release concept: “floating” base of flexure bearing Base point clamped through high friction with MR fluid Actuator imposes displacement MR fluid + flexure bearing Stress build up in flexure no Yield stress reached MR fluid variable locking in series with flexure bearing Control of fluid viscosity to block and modulate friction between base points of the flexure and the fluid Could provide damping during the progressive transmission of the elastic energy of the flexure to the fluid yes Modification of MR fluid properties (viscosity, state) Stress release through elastic energy transmission from the flexure to the MR fluid Base points slide because of low friction with MR fluid Requirements: Detection of yield stress Progressive transmission of the elasitc energy of the flexure D. Tshilumba, Delft, 09 July 2015

Control technique Control techniques for positioning: Input filtering Open loop Perfect step (broad spectrum) Filtering input step (low pass, notch) D. Tshilumba, Delft, 09 July 2015

Control technique Control techniques for positioning: Input shaping Open loop Step decomposition D. Tshilumba, Delft, 09 July 2015

Control technique Control techniques for positioning: Input shaping Step decomposition Constraint equations: Residual vibrations 𝑉 ω,ξ = 𝐴 ∑ 𝐴 ↑ = 𝑒 −𝜉𝜔 𝑡 𝑛 𝐶(𝜔,𝜉) 2 + 𝑆(𝜔,𝜉) 2 𝐶 𝜔,𝜉 = 𝑖=1 𝑛 𝐴 𝑖 𝑒 𝜉𝜔 𝑡 𝑖 cos⁡(𝜔 1− 𝜉 2 𝑡 𝑖 ) S 𝜔,𝜉 = 𝑖=1 𝑛 𝐴 𝑖 𝑒 𝜉𝜔 𝑡 𝑖 sin⁡(𝜔 1− 𝜉 2 𝑡 𝑖 ) Amplitude 𝐴 𝑖 >0 and 𝑖=1 𝑛 𝐴 𝑖 =1 Time optimality Min ( 𝑡 𝑛 ) Robustness 𝑠𝑙𝑜𝑝𝑒= 𝑑 𝑑𝜔 𝑉(𝜔,𝜉) Minimization of residual vibration in total response by destructive interference. This interference patern is created by applying substeps with specif amplitudes Ai applied at specific time ti. These parameters are determined by solving … with the 5 Constraint equations. The zv shaper is calculated based on the assumption that we have a perfect knowledge of the freq and damping of the dominant eigen mode. Therefore, alternative shapers have been developed in order to gain some robustness. Of course there are trade-offs! D. Tshilumba, Delft, 09 July 2015

Control technique Control techniques for positioning: Input shaping Perfect knowledge Error of 30% Basic shaper: ZV shaper Sensitive to mode parameters alternative shapers (ZVD, SI) Trade off: shaper duration 𝑡 𝑛 Multi mode shapers achievable Several shapers exist. The most basic one is the Zero vibration shaper. The amplitudes Ai and times ti are calculated by constraining V=0. The curve here shows the sensitivity of the shaper to an error in the knowledge of the frequency of the dominant mode of the system. As you can see, the ZV shaper is quite sensitive to an error in the resonance freq(the percentage of residual vibration is higher than 5% for very small deviation in freq) Becaus we never really know perfectly the resonance freq, it is usefull to develop alternative shapers. The ZVD shaper is obtained by forcing the slope=0 at the modeling freq (i,e, at wm). The insensitivity zone (i.e. where V<5%) is increased. It is also possible to design shapers with specified insensivity (i.e. ) The price to pay to get more and more robust shaper is an increased risetime. So D. Tshilumba, Delft, 09 July 2015

Control technique Active feedback PID controller: 𝜔 𝑛 = 𝑘 𝑚 𝜔 𝑛 = 𝑘 𝑚 𝑡 𝑟 = 1.5 𝜔 𝑛 𝑡 𝑠 = 4.6 𝜉 𝜔 𝑛 𝐶 𝑠 = 𝑘 𝑝 + 𝑘 𝑑 𝑠+ 𝑘 𝑖 𝑠 PID controller: P-term: increases the speed D-term: increases the damping I-term: removes the steady state error P-term decreases the rise time since it increases the resonance freq: applying a control force proportional to the absolute displacement of the mass is equivalent to adding a spring connecting the mass to a fix point in space. => the total stiffness connecting the mass to the environment is increased. Since w=sqrt(k/m),w is increased. And Next steps: Compare theses techniques on the type 1 prototype D. Tshilumba, Delft, 09 July 2015

Setup PI Piezo stacks P225 Heidenhain LIP 281 linear encoders NI PXI MK II spectrum analyser D. Tshilumba, Delft, 09 July 2015

Setup Linear encoders: Manual Phase unwraping Direct interpolation in post processing from measured signals I1 and I2 sub-nanometric measurement resolution (cfr slide 29) D. Tshilumba, Delft, 09 July 2015

Positioning_test_Type1: Calibration Horizontal encoder readout calibration 1 Ref signal: 2.048m horizontal, 5Hz Sampling rate:4096 Hz PZT controllers: ON Transport pin: unlocked Horizontal encoder readout too small 𝑘 𝑐𝑜𝑟𝑟 = 1 1−𝑑 𝑘 𝑐𝐻 𝑑=0,170𝑚𝑚, 𝑘 𝑐𝐻 =5.05𝑟𝑎𝑑/𝑚 D. Tshilumba, Delft, 09 July 2015

Positioning_test_Type1: Calibration Horizontal encoder readout calibration 2 - Measured coupling between vertical and roll of magnet. Possible explanation: torque applied on the magnet by the shear pins because of initial non zero roll angle Ref signal: 2.048m vertical, 5Hz Sampling rate:4096 Hz PZT controllers: ON Transport pin: unlocked Parasitic readout in horizonatal encoder 𝑑=0,170𝑚𝑚, 𝑘 𝑐𝑉 =1.1765𝑟𝑎𝑑/𝑚 D. Tshilumba, Delft, 09 July 2015

Positioning_test_Type1_Encoders noise No input signal Sampling rate:4096 Hz PZT controllers: OFF Transport pin: locked Significant drift in lateral encoder Resolution: ~0.6𝑛𝑚 in vertical encoder ~0.4𝑛𝑚 in lateral encoder known drift source (temperature) D. Tshilumba, Delft, 09 July 2015

Positioning_test_Type1_PZTnoise No input signal Sampling rate:4096 Hz PZT controllers: ON Transport pin: unlocked Significant drift in lateral encoder Resolution: ~1𝑛𝑚 in vertical encoder ~1𝑛𝑚 in lateral encoder known drift source (temperature) D. Tshilumba, Delft, 09 July 2015

Positioning_test_Type1measurements Horizontal line Ref signal: 2.048m horizontal, 5Hz Sampling rate:4096 Hz PZT controllers: ON Transport pin: unlocked D. Tshilumba, Delft, 09 July 2015

Positioning_test_Type1measurements Vertical line Ref signal: 2.048m vertical, 5Hz Sampling rate:4096 Hz PZT controllers: ON Transport pin: unlocked 15% offset in amplitude Lateral deviation of 65nm D. Tshilumba, Delft, 09 July 2015

Positioning_test_Type1measurements Oblic line Ref signal: 512nm oblic, 5Hz Sampling rate:4096 Hz PZT controllers: ON Transport pin: unlocked 15% offset in amplitude (vertical) 18% offset in amplitude (lateral) Lateral deviation of 15nm D. Tshilumba, Delft, 09 July 2015

Positioning_test_Type1measurements Circle Ref signal: 100nm circle, 5Hz Sampling rate:4096 Hz PZT controllers: ON Transport pin: unlocked D. Tshilumba, Delft, 09 July 2015

Next steps Perform positioning measurement on type 1 with 2D trajectories (August 2015) Asses performance of input shaping on type 1 prototype in 1dof (vertical) (september 2015) Choose amplification mechanism topology (july 2015) 3D geometry of amplification mechanism (2dofs) (october 2015) Model stress release concept (november 2015) D. Tshilumba, Delft, 09 July 2015 28

The END D. Tshilumba, Delft, 09 July 2015 29

Piezo stepper Specs Linear Piezoelectric motor   Load and dimensional specifications Value Stiffness in motion direction 150 N/μm Holding force capacity (passive) 800 N Drive force capacity 600 N Maximum lateral force allowed 20 N Maximum torque allowed in the direction of the driving rod 0.5 Nm Maximum torque allowed generated by lateral force Fully extended maximum length 125 mm Maximum lateral dimension 80 mm Maximum mass of the actuator 1.5 kg Motion specifications   Value Nominal elongation without external force or restraint (closed loop) 20 mm Closed loop resolution 5 nm Open loop resolution 0.03 nm Minimal incremental displacement (in step mode) 10 nm up to 10um Travel in analog mode 3um Maximum velocity (in nano stepping mode) 0.4 mm/s D. Tshilumba, CERN, 03 February 2015

Compliant amplification mechanisms Topologies Lever structure 4-bar structure Double Symmetric 4-bar structure Trianglular structure Parasitic lateral displacement Stiffness decrease at output Parasitic lateral displacement Higher Stiffness at output No parasitic lateral displacement Non linear amplification ratio No torque applied on piezo (symmetry) Low eigen frequency compact - Buckling force for clamped clamped boundary conditions D. Tshilumba, Delft, 09 July 2015 35

Positioning_test_Type1measurements Vertical line Ref signal: 2.048m vertical, 5Hz Sampling rate:4096 Hz PZT controllers: ON Transport pin: unlocked D. Tshilumba, Delft, 09 July 2015