Material. Lecture 8 Backlash and Quantization. Linear and Angular Backlash. Example: Parallel Kinematic Robot. Backlash.

Transcription

1 Lectre 8 Backlash and Qantization Material Toda s Goal: To know models and compensation methods for backlash Lectre slides Be able to analze the effect of qantization errors Note: We are sing analsis methods from previos lectres (describing fnctions, small gain theorem etc.), and these have references to the corse book(s). Qantizer / F in Linear and Anglar Backlash x ot F ot Example: Parallel Kinematic Robot Gantr-Ta robot: Need backlash-free gearboxes for high precision T in T ot θ in θ ot EU-project: SMErobot TM Backlash ead-zone Model Backlash (glapp) is present in most mechanical and hdralic sstems increasing with wear bad for control performance ma case oscillations Note: A gear box withot an backlash will not work if temperatre rises Force (Torqe) x ot (θ in θ ot ) Often easier to se model of the form ( ) x ot ( ) Uses implicit assmption: F ot = F in, T ot = T in. Can be wrong, especiall when no contact. The Standard Model Servo motor with Backlash Assme instead ẋ ot = ẋ in when in contact ẋ ot = when no contact No model of forces or torqes needed/sed x ot θ ot P-control of servo motor θ ref b θin + K +st s θ in θ ot θ in How does the vales of K and affect the behavior?

2 Effects of Backlash escribing Fnction for a Backlash.5.5 No backlash K =.5,, Backlash K =.5,, Oscillations for K = bt not for K =.5 or K =. Wh? Limit ccle becomes smaller if is made smaller, bt it never disappears d If A > d then d v N(A) = b + ia A ( π arcsin b = A π with Im N(A) ր Re 8 8 a = d ( ) d π A ( d A ( ) d A + ) d A and d A ) else N(A) =. minte exercise escribing Fnction Analsis 3 Nqist iagrams.8.. Inpt and otpt of backlash Std the plot for the describing fnction for the backlash on the previos slide. Where does the fnction end for A and wh? N(A) Imaginar Axis 3 K = /N(A) K = K =.5 Real Axis For K =, =.: intersection between G(jω) and /N(A) occrs for A =.33, ω =. Simlation: A =.33, ω = π/5. =. escribing fnction predicts oscillation well! Limit ccles? Backlash Limit Ccles The describing fnction method is onl approximate. Can one determine conditions that garantee stabilit? Rewrite the sstem as θ in θ ot θ in BL θot + K b θin θ in θ ot +st s G(s) Note that the block BL satisfies G(s) Note: θ in and θ ot will not converge to zero θ ot = { θin in contact otherwise Idea: Consider instead θ in and θ ot Analsis b small gain theorem Analsis b circle criterion Backlash block has gain (from θ in to θ ot ) Hence closed loop is BIBO stable provided that G(s) is asmptoticall stable and G(iω) < for all ω Backlash map from θ in to θ ot is in the sector [, ]. /k = and /k = Hence closed loop is stable if Re G(iω) > for all ω. (For or motor example this proves stabilit when K < )

3 Backlash compensation Linear Controller esign Introdce phase lead to avoid the /N(A) crve: Mechanical soltions Instead of onl a P-controller we choose K(s) = k +st +st ead-zone Linear controller design + k +st +st θin θ in θ b ot +st s Backlash inverse Backlash Inverse Controller K(s) = k +st +st Simlation with T =.5, T =. x ot.5 8 Nqist iagrams.5, with/withot filter 5 5 Imaginar Axis with filter x ot, with/withot filter withot filter 5 5 Real Axis No limit ccle, oscillation removed! Idea: Let jmp ± when ẋ ot shold change sign. Works if the backlash is directl on the sstem inpt! Backlash Inverse Example Perfect compensation If + if (t) > (t ) (t) = if (t) < (t ) (t ) otherwise = then xot (t) = (t) (perfect compensation) < : Under-compensation (decreased backlash) > : Over-compensation, often gives oscillations Motor with backlash on inpt, P-controller Example Under compensation Example Over compensation

4 Backlash More advanced models Example: Parallel Kinematic Robot Gantr-Ta robot: Need backlash-free gearboxes for ver high precision Warning: More detailed models needed sometimes Model what happens when contact is attained Model external forces that inflence the backlash Model mass/moment of inertia of the backlash. EU-project: SMErobot TM Rotational to Linear motion Backlash in gearbox and rails Gear box Rack-and-pinion (Swe. kggstng ) Motor connects here Remed: Use two motors in opposite directions: One motor can act as spring and brake to redce backlash. Need measrements on both motor and arm-side. Backlash compensation Qantization Qantizer / How accrate shold the converters be? (8- bits?) What precision is needed in comptations? (8- bits?) From master thesis b B. Brochier, Control of a Gantr-Ta Strctre, LTH, See also master theses b j. Schiffer and L. Halt, 9. Qantization in A/ and /A converters Qantization of parameters Rondoff, overflow, nderflow in operations NOTE: Compare with (different) limits for qantizer/dead-zone rela in Lectre. Linear Model of Qantization escribing Fnction for eadzone Rela Model the qantization error as a stochastic signal e independent of with rectanglar distribtion over the qantization size. Works if qantization level is small compared to the variations in e Q + e.8... e Rectanglar noise distribtion over [, ] has the variance V ar(e) = + / e f e de = e de = / Lectre N(A) = πa /A for A > and zero otherwise

5 escribing Fnction for Qantizer escribing Fnction for Qantizer Qantizer / N ( A ) A/δ A < ( ) N(A) = n k πa A n < A < n+ (See exercise) k= The maximm vale is /π.7 for A.7. Predicts limit ccle if Nqist crve intersects negative real axis to the left of π/.79. Shold design for gain margin > /.79=.7! Note that redcing onl redces the size of the limit oscillation, the oscillation does not vanish completel. 5 minte exercise Example Motor with P-controller. N ( A ) Rondoff at inpt, =.. Nqist crve intersects at.5k. Hence stable for K < withot qantization. Stable oscillation predicted for K > /.7 =.57. (a) A/δ How does the shape of the describing fnction relate to the nmber of possible limit ccles and their stabilit. What if the Nqist plot Otpt (b) Otpt (c) 5 K =.8 K =. 5 intersects the negative real axis at.8? intersects the negative real axis at? intersects the negative real axis at? Otpt K =. 5 Time Example oble integrator with nd order controller Nqist crve Qantization at A/ converter oble integrator with nd order controller, =. A/ controller /A process Inpt Otpt Otpt Time escribing fnction: A / =., period T = 39 Simlation: A =. and T = 8 Qantization at /A converter Qantization Compensation oble integrator with nd order controller, =. Inpt Unqantized Otpt Time Use improved converters, (smaller qantization errors/larger word length) Linear design, avoid nstable controller, ensre.3 gain margin Use the tracking idea from anti-windp to improve /A converter controller igital /A + Analog escribing fnction: A / =.5, period T = 39 Simlation: A =.5 and T = 39 Better prediction, since more sinsoidal signals Use analog dither, oversampling and digital low-pass filter to improve accrac of A/ converter + A/ filter decim. 5

6 Toda s Goal To know models and compensation methods for backlash Be able to analze the effect of qantization errors No More Lectre This Week! Qantizer /