Kinematic transformation of mechanical behavior Neville Hogan

Transcription

1 inematic transfrmatin f mechanical behavir Neville Hgan Generalized crdinates are fundamental If we assume that a linkage may accurately be described as a cllectin f linked rigid bdies, their generalized crdinates are a fundamental requirement fr any mdel f mechanical behavir. Hwever, t describe functinal behavir it will ften be necessary t epress mechanical behavir as it appears in a different frame f reference, fr eample, the Cartesian crdinates f the end-pint. nwledge f the gemetry relating the tw frames is sufficient t transfrm mechanical behavir, but care is required. Transfrmatin t end-pint crdinates Epress the kinematic equatins relating end-pint crdinates t generalized crdinates. L The relatins between incremental displacements, velcities, frces and mmenta are btained by differentiating and using pwer cntinuity. L d d J d v Jω t τ J f t η J p Nte that transfrmatin f mtin variables (displacement, velcity) is always welldefined frm generalized crdinates t any ther crdinates. Cnversely, the transfrmatin f frce variables (frce, mmentum) is always well-defined t generalized crdinates frm any ther crdinates. In general, the inverse f these transfrmatins may nt be well-defined. Inertia Inertia relates mmentum and velcity. η Iω If every rigid bdy in the linkage has nn-zer mass, the inertia tensr is psitive-definite and its inverse eists. T define stred kinetic energy, the inverse f this relatin is required, the causally-preferred frm fr an inertia. 1 η ω I Transfrmatin t end-pint crdinates is a straightfrward matter f substitutin. 1 v M p 1 v J I J t p page 1

2 M 1 J I J t 1 The inverse inertia at the end-pint can always be defined. Due t linkage gemetry it varies with end-pint psitin and linkage cnfiguratin. In generalized crdinates, inertia (and hence inverse inertia) is psitive definite. In end-pint crdinates, inverse inertia is nly psitive semi-definite (strictly nn-negative); in sme cnfiguratins it may lse rank. In thse cases the end-pint inertia appraches infinity frce may be applied but n mtin results. This is an argument fr cnsidering an inertial mechanism fundamentally t be an admittance, nt an impedance. Frictin Energy dissipatin (frictin) is characterized by a relatin between frce and velcity. In end-pint crdinates a nnlinear frm is as fllws. f Π ( v) 1 where Π ( ) dentes a functin with f t v 0 and f 0 at v 0. The crrespnding frcevelcity relatin in generalized crdinates is always well-defined and is btained by substitutin. t τ J Π J ω Π, ω Nte that the linkage kinematics intrduces a dependence n cnfiguratin. Damping Damping is defined as the gradient f frce with respect t velcity. In end-pint crdinates: Π B v In generalized crdinates: Π B ω t B J B J Frm this we see that it is always pssible t find the generalized-crdinate damping crrespnding t a specified end-pint damping. Hwever, this generalized-crdinate damping may nt have full rank and its inverse may nt be defined. Fr eample, if the dimensin f cnfiguratin space (the number f generalized crdinates) eceeds the dimensin f end-pint space (the number f end-pint crdinates) the generalizedcrdinate damping crrespnding t any end-pint damping has rank less than the dimensin f cnfiguratin space. In the limit f very large damping, the end-pint is effectively immbilized but the linkage, which has mre degrees f freedm than the end-pint, remains free t mve 2. 1 This assumes a sign cnventin with pwer psitive int any passive element. page 2

3 Generalized-crdinate damping in end-pint crdinates T transfrm a full-rank generalized-crdinate damping t end-pint crdinates we epress it in admittance frm. 1 ω B τ The crrespnding end-pint damping is btained by substitutin. t v J B 1 J B f J 1 B J t 1 Again we see that even if the generalized-crdinate damping is independent f cnfiguratin, the end-pint damping will, in general, vary with cnfiguratin. Static impedance Damping and inertia (tw f an infinite set f admittance parameters) transfrm between crdinates in the same way. Hwever, static impedance r admittance parameters transfrm differently. A general frm fr static impedance (i.e., frce-psitin behavir) in generalized crdinates is τ Φ, where Φ( ) dentes a functin and dentes a zer f the functin, a cnfiguratin at which the (generalized) frce is zer. Causal arguments 3 indicate this impedance frm must always eist thugh it may nt always have an inverse (admittance) frm. Cnsider the differential f this functin. dτ d + d In the fllwing, we assume the zer cnfiguratin des nt change: d 0. Stiffness is defined as the gradient f frce with respect t displacement. Nte that stiffness is (in general) a functin f cnfiguratin. As the static impedance frm always eists, stiffness may always be defined thugh it may vanish r lse rank at sme cnfiguratins. When the stiffness has full rank, cmpliance is defined as the inverse f stiffness. C 1 inematic stiffness Differentiate the frce transfrmatin t find the relatin between stiffness in end-pint and generalized crdinates. 3 Hgan, N. (1985) Bil. Cyb. 52: page 3

4 t t J dτ J df + fd This reveals the fundamental difficulty: quite aside frm any elastic behavir, any nnzer frce acts thrugh cnfiguratin-dependent mment arms 4 t prduce an apparent stiffness. Dente the apparent kinematic stiffness by Γ. t J Γ f Nte that the kinematic stiffness vanishes fr zer frce. Hwever, if nn-zer it may be psitive r negative (stabilizing r destabilizing) depending n the rientatin f the frce. End-pint stiffness in generalized crdinates Static impedance in end-pint crdinates may be epressed as fllws f Φ (, ) where dentes a psitin at which frce is zer. The crrespnding static impedance in generalized crdinates is always well-defined and is btained by substitutin. t t τ J f J Φ L, L Φ, ( ) End-pint stiffness: Stiffness in generalized crdinates: t J t L f + J t Γ + J J If the stiffness is evaluated at zer net end-pint frce, the kinematic stiffness vanishes. t f J J 0 As with damping, we see that it is always pssible t find the generalized-crdinate stiffness crrespnding t a specified end-pint stiffness. Hwever, this generalizedcrdinate stiffness may nt have full rank and its inverse (generalized-crdinate cmpliance) may nt be defined. If the dimensin f cnfiguratin space eceeds the dimensin f end-pint space the generalized-crdinate stiffness crrespnding t any end-pint stiffness has rank less than the dimensin f cnfiguratin space. In the limit f very large end-pint stiffness, the end-pint is effectively immbilized but the linkage, which has mre degrees f freedm than the end-pint, remains free t mve. 4 These mment arms are defined by the clumns f the Jacbian. page 4

5 Generalized-crdinate cmpliance in end-pint crdinates T transfrm a full-rank generalized-crdinate stiffness t end-pint crdinates we epress it in cmpliance frm. t d C dτ C J df + Γd t ( 1 C Γ) d C J df t ( 1 C Γ) d C J df Γ d J t d ( ) f Assuming the inverse eists, this epressin may be slved fr d and d. 1 t ( Γ) J f d d d Jd J 1 t d C df C 1 J ( Γ) J df t ( Γ) J If the cmpliance is evaluated at zer end-pint frce, the kinematic stiffness vanishes and the epressin simplifies. t C f JC J 0 If the Jacbian may be inverted these epressins are equivalent t thse fund abve. t t J C J Γ hence Γ + J J 1 1 ( ) In general the inverse Jacbian may nt eist but, prvided the difference between generalized-crdinate stiffness and kinematic stiffness can be inverted, end-pint cmpliance can always be defined. Hwever, it may nt always have full rank, and hence it may nt always be pssible t define end-pint stiffness. Remark Nte the prfund influence f linkage kinematics n all cmpnents f impedance. Fr eample, at singular cnfiguratins (at which the Jacbian lses rank) all f the admittance parameters (cmpliance, inverse damping, inverse inertia, etc.) lse rank and all crrespnding impedance parameters (stiffness, damping, inertia, etc.) becme undefined (i.e., apprach infinity in at least ne directin). Thus pse is ne f the mst imprtant ways t mdulate interactive behavir. page 5