Newton Euler equations in general coordinates

Transcription

1 Newton Euler equations in general coorinates y ertol ongar an Frank Kirchner Robotics Innoation Center, DFKI GmbH, remen, Germany Abstract For the computation of rigi boy ynamics, the Newton Euler equations represent a crucial relation unifying the laws of motion by Newton an Euler using the language of instantaneous screws Typically, Newton Euler equations are state in spatial or in boy coorinates, respectiely In this paper, a general formulation of Newton Euler equations is proie for arbitrary reference systems In particular, the general form unifies the known equations in spatial an boy coorinates To the best of the authors knowlege, this relation between the spatial an the boy form has not been reporte in literature The noel formulation is base on the concept of time ifferentiation with respect to moing reference systems Introuction Newton s secon law of motion states that the force f which acts upon a translating particle, a zero-imensional boy, is proportional to the time eriatie of the particle s linear momentum h = m with mass m an elocity, all measure in an inertial system In Euler s formulation lieira, 2007, this is epresse in the ifferential equation f = h = M = M = M a Here, a = enote the the acceleration ector an the mass matri M = m I with ientity matri I = I 3 is introuce for sake of formal consistency with Euler s secon law of motion, below Euler s secon law of motion states that the torque τ which acts upon a rotating boy is proportional to the time eriatie of the boy s angular momentum λ = M 3 ω with inertia matri M 3 an pseuoector ω enoting the boy s angular elocity, all measure in an inertial system This is epresse in the ifferential equation τ = λ = M 3 ω = M 3 ω + M 3 ω 2 The angular acceleration is enote by α = ω The eriatie M 3 is gien within the appeni While the mass in Newton s law is assume to be a uniersal quantity Arema, 2005, the inertia of the moing boy changes with respect to the inertial system, so that the chain rule is applie in the ifferentiation of Euler s law In this paper, the Newton Euler equations that subsume Equations an 2 by means of twists an wrenches are formulate with respect to an arbitrary reference system The remainer of the paper is structure as follows: necessary quantities an their notation are introuce in the net section In Section 3, the generalize formulation of the Newton Euler equations is gien an interrelations to corresponing formulations in spatial an in boy coorinates are iscusse In Section 4, a conclusion is rawn Technical computations of time eriaties are compile in Appeni A

2 2 rerequisites Newton Euler equations in general coorinates 2 2 oses an isplacements A pose matri R 4 4 has the form = with attitue E S3 an E p 0 position p R 3 A trajectory is a sorte set of poses t with t > 0 A isplacement matri D R 4 4 has the shape D = with rotation R S3 an translation R t 0 t R 3 The matri D = D Q = Q escribes the passie isplacement between two poses SE3 an Q SE3 The ajoint representation D A = AD R of D has the shape D A = R 0 t R R 22 Screws, twists, an wrenches Geometric screws A screw ξ R is enote in lücker coorinates as ξ = ν µ subsuming the irection ν an the moment µ The matri representation of a screw ξ is enote by ξ R ν3 +ν an has the shape ξ 2 = with ν = The ν µ 0 0 +ν 3 0 ν ν 2 +ν 0 ajoint representation of a screw ξ is enote by ξ a = aξ R an has the shape ξ a = ν 0 For a screw ξ = ν µ ν µ, the screw ector fiel ξ : R 3 R 3 is ealuate at any point with coorinates p to the moment µ = ξ p gien as ξ : p µ + ν p The ector fiel ξ is characterize by the constitutie equation Minguzzi, 203 ξ p ξ q = ν p q, 2 for arbitrary points an Q with coorinates p an q The interchange operator R is efine as = accoring to Hunt, 2003, an McCarthy an Soh, I I 0 hysical Screws Twists an wrenches are geometric screws which are equippe with physical interpretations an units as screws from the force an an from the elocity omain, respectiely oth are enote in ray-coorinates in accorance with hillips, 2007, Tsai, 999, Mason, 200, Daison et al, 2004, an McCarthy an Soh, 200 A wrench is enote by W = f τ, subsuming force f an torque τ, an a twist is enote by V = ω, subsuming angular an linear elocity, ω an For the ector fiel V of a twist V = ω, Equation 2 is caste into the elocityifference equation Uicker et al, 2003, gien, with = V p an Q = V q, as Q = ω p q 22 Analogously, the constitutie equation of a wrench W = f τ, is the torque-ifference equation of the ector fiel W, gien, with τ = W p an τ Q = W q, as τ τ Q = f p q Attributing quantities For the remainer of the ocument, a notation is introuce to specify the reference system an, if necessary, the ealuation point of a quantity The reference system of a quantity is enote as ] accoring to Sathaye, 20 In aition, the ealuation The matri representation is also calle a all ector, see Uicker et al, 203 The elocity-ifference equation is also name the funamental formula of the rigi boy in Minguzzi, 203

3 REREQUISITES 3 point of a quantity is optionally enote as ] Together, a quantity is specifie by an ealuation point an a reference system as ] Reference systems For arbitrary reference systems, SE3 an Q SE3, the homogenize coorinates R 4 of a point are characterize with the funamental ientity by Sathaye, 20 as = ] = I ] = ] = Q ] Q 24 Similarly, the ientity is state for the representing matrices of a pose SE3 as = ] = I ] = ] = Q ] Q 25 From the preious ientities, the transformation rules for a coorinate change are euce; for a pose matri, for eample, as ] = Q ] Q = D Q ] Q, an analogously, for the homogenize coorinates of a point Ealuation points The ealuation of the screw ector fiels ξ, V, an W, which are specifie by the ifference Equations 2, 22, an 23, rea with the introuce notation as µ] = µ = ξ p, ] = = V p, an τ ] = τ = W p The funamental ientities for a screw, corresponing to Equations 24 an 25, are gien for its matri representation ξ, its lücker ector ξ = ν µ, an its ajoint representation ξ a as ξ = ξ ] = I ξ ] I = ξ ] = Q ξ ] Q Q Q, 2 ξ = ξ ] = I A ξ ] = A ξ ] = ξ ] Q, 27 QA Q ξ a = ξ a] = A ξ a] A = Q A ξ a] Q Q Q A 28 From these ientities, transformation rules for a change of reference system an ealuation point can be euce As an eample, the matri twist ξ ] with respect to the reference, is obtaine from the matri twist ξ ] Q with respect to the reference Q Q as ξ ] = D Q ξ ] Q Q D Q With the conention D] Q Q = D Q, the notation of the transformation rule can be refine for two specific cases For coinciing ealuation points, the matri D = R 0 0 is enote as D]QQ Q = D, so that the transformation reas ξ ] Q = D] QQ Q ξ ] Q Q D] QQ Q Similarly, for coinciing attitues, the isplacement matri D = 0 I t Q is enote as D] QQ = D, so that the transformation reas ξ ] Q = D] Q QQ ξ ] Q Q D] QQ Q Motion laws y means of the notation ] at ref, Newton s law of motion from Equation, is rephrase as f] = M ] where enotes the position of the particle Euler s law of motion from Equation 2 is rephrase as τ ] C = M 3] C ω +M 3 ] C ω where C enotes the center of mass an is chosen coincient to C Thus, the notation proes helpful to emphasize the feasibility constraints that unerlie these two motion laws 24 Mass an inertia The coorinates of the center of mass C of a boy, specifie by mass m an mass ensity function ρ, are compute as c = m ρ V The rotatie inertia matri V M 3 ] C C R 3 3 is compute as ] C = M3 ρ C C 2 V V If the ealuation point coincies with the origin of the reference system, it can be omitte

4 Newton Euler equations in general coorinates 4 Figure Sketch of a moing boy, together with a moing reference system an the inertial system For simplification, the orientations E t an E t E are assume to be constant oer time t The absolute elocities, of boy an of obserer, are inicate by re an green arrows The relatie elocity is compute as the ifference = + an is inicate by blue arrows For an arbitrary ealuation point, the rotatie inertia matri M 3 ] C is etermine by the Huygens Steiner theorem, also known as the parallel ais theorem Selig, 2005 With respect to an arbitrary reference system E S3, the rotatie inertia matri M 3 ] E is compute as M 3 ] E = R EC M 3 ] C R EC The affine inertia matri M R subsumes the rotatie inertia M 3 R 3 3 an the translatie mass M = m I At the center of mass C, the affine matri M ] C C has the shape ] C M = M3] C C 0 With respect to an arbitrary reference system SE3, C 0 M the spatial inertia matri M ] is compute ia the skew ais theorem Selig, 2005 M ] = DA C T M ] C C DA C, 29 a generalize ersion of the Huygens Steiner theorem As in Section 23, refine statements can be mae in case of coinciing ealuation points or reference attitues In case that D only changes the ealuation point, D = 0 I t, or the reference system D = R 0 0, the formulation can be refine by enoting D as D = D] CC, C or as D = D] C, respectiely y means of the screw swap operator the first factor D A C T in Equation 29 is alternately epresse as D A C T = D C A The funamental ientity for the affine inertia matrices reas M = M ] = A T M ] A = Q A T M ] Q Q QA 20 As aboe, the ientities can be aopte for pure rotations with coinciing ealuation points an for pure translations with coinciing reference attitues 25 Time eriaties with respect to a mobile reference The concept of time eriaties with respect to mobile reference systems is, for eample, introuce by Featherstone, 2008 The time eriatie of a moing point with homogeneous coorinates = t is etermine with respect to a moing reference system with matri = t as ] = ] = ] = ] = ] + ] ] ] 2 ] = + V In a physical interpretation, the three terms ], ], an V ] escribe the relatie elocity of with respect to, the absolute elocity of with respect to, an the absolute elocity of with respect to an ealuate at ;

5 A GENERAL FRMULATIN 5 all epresse in coorinates with respect to the reference system y introucing ] = ], ] = ] = ], an ] = ] = V ], Equation 2 is rephrase as ] = ] + ] ] an = ] ], 22 resembling the forms absolute elocity = relatie elocity + obserer elocity an relatie elocity = absolute elocity obserer elocity In Figure, a sketch for an eample setup is proie where the point of the boy is name b Similar to Equation 24, a ifferential, funamental ientity can be state as = ] = I ] = ] = Q ] 23 Q A term ] escribes the same global eriatie as = ], in coorinates with respect to reference system In contrast, ] escribes a local eriatie The corresponing eriaties of poses, screws, an inertias are proie in Appeni A2 3 A general formulation 3 Newton Euler equations in general coorinates The Newton Euler equations in general coorinates state that the wrench W which acts upon a moing boy at a point fie in the lamina of the non-inertial reference system is proportional to the time eriatie of the boy s momentum M V, both epresse with respect to the reference This is epresse by the ifferential equation W ] = The equation is state in more etail as ] W M V = M V ] + V a ] ] 3 ] M V, 32 by applying the transformations ] ] M V = A M V = A A M V ] = A A M V ] + A M V ] = M V ] + A V a A M V ] = M V ] + ] V a M V ], using Equation 27, the chain rule, epansion, sorting, the ientity A = V a A from Equation A 2, an Equation Ealuation Spatial coorinates With respect to the static reference frame, the secon summan of Equation 32 anishes, so that Equation 32 simplifies to W ] = M V ] Applying the chain rule, the equation is further ealuate to ] ] ] W = M ω ] + ] ω M ] α ω a T ] ] ω 33 = M a] + M ] Here, the time eriatie of the spatial inertia matri, = ω a T M M ] ] + ω a M from Equation A 3 in the appeni is use together with the ientity

6 Newton Euler equations in general coorinates ω a ω = 0 For the case that the ealuation point coincies with the center of mass C, Equation 33 coers ] C ] τ C W = = f M3] C 0 0 M α C a] + ω] C M3]C ω]c 0 0 ω] C M ]C The equations are state in this form, for eample, by Featherstone, 2008 oy coorinates With respect to a reference system moing together with the boy, the first summan M V ] of Equation 32 is simplifie by using the time inariance of the boy inertia, M ] ω = ] M ω ] = ] M αa ] Thus, Equation 32 simplifies to ] ] α ω a T ] ] ] ω W = M a] + M 34 For the case that the ealuation point coincies with the center of mass C, Equation 34 coers ] C ] τ C W = = M3] C 0 α C f 0 M a] + ω] C M3]C ω]c 0 0 ω] C M ]C The Newton Euler equations are state in this form, for eample, by Murray et al, 994, an by Fossen, 20 General coorinates For the case of an arbitrarily moing reference frame, the general Newton Euler equations of Equation 32 rea in components ] f ] τ = ω a ] ] ω ω M + M 35 The twist V ] in last factor M V ] is etermine accoring to the ientity in Equation 27, the inertia matri M ] accoring to the ientity Equation 20 The first summan M V ] is epane to M ] ] V ] + M ] V The twist eriatie V ] is etermine ia Equation A 4, the inertia eriatie M ] is etermine ia Equation A 5 Similar to the physical interpretation of Equation 22, the terms f τ ], M ω ], an ω a M ω ] in Equation 35 can be characterize by the form absolute wrench = relatie wrench + obserer wrench In the language of screws, the sum resembles the ifference between real an apparent forces which are istinguishe for non-inertial reference systems 4 Conclusions This paper proies a general formulation of the Newton Euler equations obtaine by ifferentiation with respect to non-inertial reference systems an by using a suitable notation scheme The general equations unify the known formulations in spatial an in boy coorinates an clarify their interrelation The erie connection can simplify communicating mechanics of rigi boies Aitionally, the noel formulation of the Newton Euler equations might proe applicable to questions arising in computational rigi boy mechanics with regar to moing reference frames In robotics, for eample, mobile reference systems are of interest when the interaction of a robot s en-effector link with the robot s enironment is require to be escribe from the point of iew of another moing link of the mechanical eice Acknowlegments The work presente in this paper was fune by the Feeral Ministry of Eucation an Research MF within the project Recupera Grant 0-IM-400A

7 Appeni A Time eriaties AENDI 7 A Spatial time eriaties Attitues an poses The time eriaties with respect to the inertial reference system relate to an attitue matri E = E t S3 are E = E T = ω E = ω T E T E T = A E = E T ω T = E ω The time eriaties relatie to the inertial reference system relate to the ajoint representation A = A of a pose matri = t SE3 are A = V a A A = A V a A 2 A T = A T V a T A T = V a T A T The first equation is etene to A A = A V a A = A V a, a formula which generalizes the rotational ientity E T E = ω ET E = E T ω Twists The time eriatie of a twist V with respect to the inertial reference system is enote by A = α a = ω = V Inertia matrices The time eriatie of spatial inertia matri M with respect to the inertial reference system is M ] = = M M ] = V a T M ] ] M V a A 3 The equation is obtaine by the transformations using Equation 20, the chain rule, an ientities from Equation A 2 M ] = = M M ] = D A T M ] DA = D A T M ] DA D A T M ] D A = V a T D A T M ] DA + D A T M ] DA V a = V a T M ] M ] V a A2 General time eriaties oses The time eriatie of a pose matri = t with respect to a mobile reference system with matrices = t is obtaine in analogy to Equation 2 as ] = ] = ] ] = = ] + ] ] = + ] V ] In the first line, the funamental ientity Equation 25 is applie twice In aition, the inariance of the inertial system is use Net, the chain rule is applie In the thir line, the epression is epane, simplifie by using = ] V, an sorte The computation principle remains in the following eriations Screws The time eriatie of a screw matri ξ = ξ t with respect to a mobile reference system with matrices = t is compute as ξ ] = ξ ] = ] ξ = ] ξ = ] ξ + ξ ] + ξ ] = ] ξ V + ] ξ ] ξ ] V ]

8 Newton Euler equations in general coorinates 8 The funamental ientity of Equation 2 is applie in the first line For the simplification, the ientity = = = ] V is use In lücker coorinates, the corresponing epression reas ξ] = ] ξ + V a ] ξ ] A 4 Inertia matrices The time eriatie of an inertia matri M = M t with respect to a mobile reference system with matrices = t is M ] = ] M V a M ] ] T ] V ] a M A 5 The equation is obtaine with the ientity Equation 20 as M ] = ] A T M A = A T A T M ] A A ] = + A M T A T M ] + ] M A A ] = M V a M ] ] T ] V ] a M In the last transformation, the simplifications, with inerse to, so that ξ = ξ, A A = A V a A = V a = ] V a A T A T = A T V a T A T = A V a A T are use They are erie by means of the ientities gien in Equation A 2 = V a ] T REFERENCES Arema, M 2005 Newton-Euler Dynamics Springer Daison, J K, Hunt, K H, an ennock, G R 2004 Robots an Screw Theory: Applications of Kinematics an Statics to Robotics for Uniersity ress Featherstone, R 2008 Rigi oy Dynamics Algorithms Springer Fossen, T I 20 Hanbook of Marine Craft Hyroynamics an Motion Control John Wiley & Sons Hunt, K H 2003 Don t Cross-Threa the Screw! Journal of Robotic Systems, 207 Mason, M T 200 Mechanics of Robotic Manipulation MIT ress McCarthy, J M an Soh, G S 200 Geometric Design of Linkages Springer Minguzzi, E 203 A geometrical introuction to screw theory European Journal of hysics, 343 Murray, R M, Li, Z, an Sastry, S S 994 A Mathematical Introuction to Robotic Manipulation CRC ress lieira, A R E 2007 Euler s Contribution to Classical Mechanics In 2th IFToMM Worl Congress hillips, J 2007 Freeom in Machinery: Volume 984 an Volume Cambrige Uniersity ress Sathaye, A 20 Notes on Similarity Lecture Notes, MA322 Selig, J M 2005 Geometric Funamentals of Robotics Springer Tsai, L-W 999 Robot Analysis: The Mechanics of Serial an arallel Manipulators John Wiley & Sons Uicker, J J, ennock, G R, an Shigley, J E 2003 Theory of Machines an Mechanisms for Uniersity ress Uicker, J J, Raani,, an Sheth, N 203 Matri Methos in the Design Analysis of Mechanisms an Multiboy Systems Cambrige Uniersity ress