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Redundant configuration coordinates and nonholonomic velocity coordinates in analytical mechanics
Proceedings of the 9th Vienna International Conference on Proceedings of the 9th Vienna International Conference on Mathematical Proceedings ofModelling the 9th Vienna International Conference on Mathematical Proceedings ofModelling the 9th Vienna International Conference on Vienna, Austria, February 21-23, 2018 Mathematical Modelling Vienna, Austria, February 21-23, 2018 Available online at www.sciencedirect.com Mathematical Modelling Vienna, Austria, February 21-23, 2018 Vienna, Austria, February 21-23, 2018
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IFAC PapersOnLine 51-2 (2018) 409–414
Redundant configuration coordinates and Redundant configuration coordinates and Redundant configuration coordinates and Redundant configuration coordinates and nonholonomic velocity coordinates nonholonomic velocity coordinates nonholonomic velocity coordinates nonholonomic velocity coordinates in analytical mechanics in analytical mechanics in analytical mechanics in analytical mechanics Matthias Matthias Konz, Konz, Joachim Joachim Rudolph Rudolph Matthias Konz, Joachim Rudolph Matthias Konz, Joachim Rudolph Chair Chair of of Systems Systems Theory Theory and and Control Control Engineering, Engineering, Saarland University, 66123 Saarbr¨ u Germany, Chair Theory Control Engineering, Saarland University, 66123and Saarbr¨ ucken, cken, Germany, Chair of of Systems Systems Theory and Control Engineering, (e-mail: {m.konz, j.rudolph}@lsr.uni-saarland.de) Saarland University, 66123 Saarbr¨ u cken, (e-mail: {m.konz, j.rudolph}@lsr.uni-saarland.de) Saarland University, 66123 Saarbr¨ ucken, Germany, Germany, (e-mail: {m.konz, j.rudolph}@lsr.uni-saarland.de) (e-mail: {m.konz, j.rudolph}@lsr.uni-saarland.de)
Abstract: Abstract: Several Several established established concepts concepts of of analytical analytical mechanics mechanics are are reviewed reviewed and and extended extended to to Abstract: Severalconfiguration established concepts of analytical mechanicsvelocity are reviewed and extended to include redundant coordinates and nonholonomic coordinates. The include redundant coordinates and nonholonomic coordinates. The main main Abstract: Severalconfiguration established concepts of analytical mechanicsvelocity are reviewed and extended to include redundant configuration coordinates and nonholonomic velocity coordinates. The main motivation for coordinates is the resulting be whereas motivation for redundant redundant coordinates is that that and the nonholonomic resulting formulations formulations can be global, global,The whereas include redundant configuration coordinates velocity can coordinates. main minimal coordinates might necessarily be local depending on the topology of the configuration motivation for redundant coordinates is that the resulting formulations can be global, whereas minimal coordinates mightcoordinates necessarilyisbethat local the topology configuration motivation for redundant thedepending resulting on formulations canofbethe global, whereas space for The formulations for equations, minimal necessarily be on the configuration space e.g. e.g.coordinates for SO(3). SO(3).might The resulting resulting formulations for Lagrange’s Lagrange’s equations,of the Gibbs-Appell minimal coordinates might necessarily be local local depending depending on the the topology topology of the the Gibbs-Appell configuration space e.g. for the SO(3). The resulting formulations for Lagrange’s equations, the Gibbs-Appell equations coefficients of are These are then equations and coefficients of the the Levi-Civita Levi-Civita connection are derived. derived. Thesethe areGibbs-Appell then applied applied space e.g. and for the SO(3). The resulting formulationsconnection for Lagrange’s equations, to the example of the free rigid body to derive the well-known Newton-Euler equations. equations and the coefficients of the Levi-Civita connection are derived. These are then to the example of the free rigid body to derive the well-known Newton-Euler equations. equations and the coefficients of the Levi-Civita connection are derived. These are then applied applied to the of rigid to derive well-known Newton-Euler equations. to the example example of the the free freeFederation rigid body body derive the the well-known © 2018, IFAC (International of to Automatic Control) Hosting Newton-Euler by Elsevier Ltd. equations. All rights reserved. Keywords: mechanics, first-principles, energy methods, redundant Keywords: mechanics, first-principles, energy methods, redundant coordinates, coordinates, rigid rigid body body Keywords: Keywords: mechanics, mechanics, first-principles, first-principles, energy energy methods, methods, redundant redundant coordinates, coordinates, rigid rigid body body 1. components 1. INTRODUCTION INTRODUCTION components of of the the angular angular velocity velocity ω ω are are not not derivatives derivatives 1. INTRODUCTION components of the angular velocity ω are not derivatives of any coordinates. of any coordinates. 1. INTRODUCTION components of the angular velocity ω are not derivatives of any coordinates. Methods Methods from from analytical analytical mechanics mechanics are are widely widely used used by by of any coordinates. engineers for modeling and as inspiration for control design Methods from analytical mechanics are widely used by engineers for modeling and mechanics as inspiration control design Methods from analytical arefor widely used by 1.2 1.2 Outline Outline and One key therein is use Outline engineers for inspiration for and optimization. optimization. Oneand keyas aspect therein is nthe the design use of of 1.2 engineers for modeling modeling and asaspect inspiration for control control design 1.2 Outline where n (independent) generalized coordinates q ∈ R and optimization. One key aspect therein is the use of n A possible n (independent) generalized q ∈ isRnnthewhere and optimization. One keycoordinates aspect therein use of possible extension extension of of the the Lagrangian Lagrangian formalism formalism to to ininwhere n A (independent) generalized coordinates q ∈The Rn following is the degree of freedom of the system. A possible extension of the Lagrangian formalism to include redundant coordinates was proposed in Konz is the degree of freedom of the system. where n A (independent) generalized coordinates q ∈The R following clude redundant coordinates was proposed in Konztoand and possible extension of the Lagrangian formalism inbrief example should motivate that this restriction can be is the degree of freedom of the system. The following redundant coordinates was proposed in derivation Konz and Rudolph (2015). This approach allows aa direct brief motivate thatsystem. this restriction can be clude is theexample degree should of freedom of the The following Rudolph (2015). This approach allows direct derivation clude redundant coordinates was proposed in Konz and brief example should motivate that this restriction can be of the Euler impractical for certain systems. (2015). This approach allows a direct derivation equations (2). impractical forshould certainmotivate systems.that this restriction can be Rudolph brief example of the Euler equations (2). Rudolph (2015). This approach allows a direct derivation impractical for certain systems. of the Euler equations (2). impractical for certain systems. of thepresent Euler equations (2). The article extends these The present article extends these concepts concepts further further by by 1.1 1.1 Motivating Motivating example example The present article extends these concepts further meby formulating other established concepts of analytical 1.1 Motivating example formulating other established concepts of analytical meThe present article extends these concepts further by 1.1 Motivating example formulating other ofestablished concepts of analytical mechanics in terms redundant configuration coordinates chanics in terms redundantconcepts configuration coordinates formulating other ofestablished of analytical meA popular approach for the derivation of equations of in terms of redundant configuration coordinates A popular approach for the derivation of equations of chanics and nonholonomic velocity coordinates. The and minimal minimal nonholonomic velocity coordinates. The funfunin terms of redundant configuration coordinates A popular approach for the derivation of equations of chanics motion (EOM) is formalism: Choose minimal nonholonomic velocity coordinates. The funmotion (EOM) is the the Lagrangian Lagrangian formalism: Choose gengenA popular approach for nthe derivation of equations of and damental axiom for the discussions is the Langrangedamental axiom for the discussions is the Langrangeand minimal nonholonomic velocity coordinates. The funmotion (EOM) is theq Lagrangian formalism: Choose genand formulate the Lagrangian eralized coordinates ∈ R axiom for (in thecontrast discussions is the LangrangetheChoose Lagrangian eralized coordinates ∈ Rnnn and formulate motion (EOM) is theq Lagrangian formalism: gen- damental d’Alembert principle to principle d’Alembertaxiom principle to Hamilton’s Hamilton’s principle damental for (in thecontrast discussions is the Langrangeformulateqthe Lagrangian eralized coordinates q∈ Rn and L(q, q) ˙ in terms of the chosen coordinates and their time L(q, q) ˙ incoordinates terms of the chosen coordinates andLagrangian their time for formulateqthe eralized q∈ R and the article). principle (in for the previous previous article). d’Alembert principle (in contrast contrast to to Hamilton’s Hamilton’s principle principle L(q, q) ˙ in terms of the chosen coordinates q and their time d’Alembert derivatives q. ˙˙ The EOM can evaluated the previous article). derivatives q. The EOM can be be evaluated qfrom from L(q, q) ˙ in terms of the chosen coordinates and their time for for the previous article). derivatives q. ˙ dThe EOM can be evaluated from The first section of the present work deals with geometry, ∂L ∂L The first section of the present work deals with geometry, derivatives q. ˙ dThe can 0, be evaluated from ∂L EOM − ∂L (1) firstshape section of the present workspace dealsand withhow geometry, i.e. of the it d ∂∂L ∂Li = i − ∂q = 0, ii = = 1, 1, .. .. .. ,, n. n. (1) The i.e. the the of of thetheconfiguration configuration it can can The firstshape section present workspace dealsand withhow geometry, dt q ˙ i i d ∂L ∂L − = 0, i = 1, . . . , n. (1) dt ∂ q˙ii − ∂q ii = 0, i = 1, . . . , n. i.e. the shape of the configuration space and how it can be parameterized, and with kinematics, i.e. how config(1) dt ∂ q ˙ be parameterized, and with kinematics, i.e. how configi.e. the shape of the configuration space and how it can ∂q However, this ∂ q˙i systems ∂q i parameterized, and with kinematics, i.e. how configuration and velocity coordinates are related. The second However, for fordtsome some systems this approach approach may may lead lead to to be uration and velocity coordinates are related. The second be parameterized, and with kinematics, i.e. how configHowever, for some systems this approach may lead to awkward and velocity coordinates are related. The second section with and the concept of awkward expressions. expressions. However, for some systems this approach may lead to uration section deals deals with kinetics kinetics and establishes establishes theThe concept of uration and velocity coordinates are related. second awkward expressions. section deals with kinetics and establishes the concept of a generalized force. We will motivate three formulations awkward expressions. Consider for example a rigid body fixed at one point: A a generalized force. We will motivate three formulations section deals with kinetics and establishes the concept of Consider for example a rigid body fixed at one point: A aforgeneralized force. We will motivate three formulations deriving the generalized force of inertia: Lagrange’s Consider for example a rigid body fixed at one point: A popular choice of generalized coordinates are the Cardan for deriving the generalized force of inertia: Lagrange’s a generalized force. We will motivate three formulations popular choice of generalized areone thepoint: Cardan Consider for example a rigid coordinates body fixed at A equations deriving the on generalized force of inertia: Lagrange’s based the energy, the popular choice of generalized coordinates are the cumberCardan for angles roll, yaw, will quite equations based the kinetic kinetic energy, the Gibbs-Appell Gibbs-Appell deriving the on generalized force of inertia: Lagrange’s angles for forchoice roll, pitch, pitch, yaw, which which will lead lead to toare quite popular of generalized coordinates the cumberCardan for equations on the kinetic energy, the Gibbs-Appell based on the acceleration energy, and the angles for roll, pitch, yaw, which will lead toconfigurations. quite cumber- equations based on some that are for particular energy, the LeviLevion the the acceleration kinetic energy, the and Gibbs-Appell some EOM EOM that are singular singular for will particular angles for roll, pitch, yaw, which lead toconfigurations. quite cumber- equations based on the acceleration energy, and the LeviCivita connection for the inertia metric. Finally, we will some EOM that are singular for particular configurations. Alternatively, the EOM for particular system Civita connection inertia metric. Finally, weLeviwill based onfor thethe acceleration energy, and the Alternatively, theare EOM for this this particular configurations. system can can be be equations some EOM that singular for particular Civita connection for the inertia metric. Finally, we will apply these formulations to the example of a free rigid Alternatively, the EOM for this particular system can be given directly by the Euler equations apply these formulations to the example of a free rigid Civita connection for the inertia metric. Finally, we will given directly by Euler Alternatively, thethe EOM for equations this particular system can be apply these formulations to the example of a free rigid body. given directly by˙ the Euler equations body. apply these formulations to the example of a free rigid R = R ω , Θ ω ˙ + ω Θω = 0. (2) given directly by the Euler equations R˙˙ = R ω , Θω˙ + ω Θω = 0. (2) body. R˙ = R ωmatrix , Θω˙ + ω Θω = 0. the angular (2) body. with the orientation R ∈ SO(3), R = R ω , Θ ω ˙ + ω Θω = 0. (2) 2. with the orientation matrix R ∈ SO(3), the angular 3 2. CONFIGURATION CONFIGURATION AND AND KINEMATICS KINEMATICS with the orientation matrix R ∈ SO(3), the angular velocity ω ∈ R · operator 3 , the inertia matrix Θ, and the 2. CONFIGURATION AND KINEMATICS · operator velocity ∈ R33 , the inertia and the with theω orientation matrixmatrix R ∈ Θ,SO(3), the angular 3 2. CONFIGURATION AND KINEMATICS velocity ω ∈b = R3a, × theb, inertia matrix Θ, and the · operator such that a a, b ∈ R . 3 such thatω a∈b = a, b ∈ Rmatrix . velocity R a, × theb, inertia Θ, and the · operator 2.1 2.1 Redundant Redundant configuration configuration coordinates coordinates such that ab = a × b, a, b ∈ R333 . Redundant configuration coordinates such that known ab = a × b, a,the b ∈ R . equations (2) cannot be 2.1 It It is is well well known that that the Euler Euler equations (2) cannot be 2.1 Redundant configuration coordinates It is wellfrom known that theequations Euler equations (2) cannot be The introductory example should demonstrate that it may derived Lagrange’s (1) directly since the derived Lagrange’s (1) directly since the It is wellfrom known that theequations Euler equations (2) cannot be The introductory example should demonstrate that it may derived from Lagrange’s equations (1)generalized directly since the The introductory example should demonstrate that degree it may entries the matrix R not coordibe to coordinates than entries of offrom the rotation rotation matrix R are are not generalized coordibe useful useful to use use more more coordinates than the the actual actual derived Lagrange’s equations (1) directly since the The introductory example should demonstrate that degree it may not generalized coordientries of theare rotation matrixby RR are be useful ton use more coordinates than the actual degree nates (they constrained of freedom of the system, i.e. redundant coordinates. We R = I3 ). Moreover, the nates (they constrained R = I ). Moreover, the of freedom n of the system, i.e. redundant coordinates. We entries of theare rotation matrixby RR are not generalized coordibe useful to use more coordinates than the actual degree 3 nates (they are constrained by R ). Moreover, the of freedom n of the system, i.e. redundant coordinates. We R = I3 3 nates (they are constrained by R R = I3 ). Moreover, the of freedom n of the system, i.e. redundant coordinates. We
Copyright © 2018, 2018 IFAC 1 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright 2018 IFAC 1 Copyright ©under 2018 responsibility IFAC 1 Control. Peer review of International Federation of Automatic Copyright © 2018 IFAC 1 10.1016/j.ifacol.2018.03.070
Proceedings of the 9th MATHMOD 410 Vienna, Austria, February 21-23, 2018
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will denote these by x ∈ Rν , ν ≥ n and their collection of constraint equations as φ(x) = 0. The set X of mutually admissible coordinate values is the configuration space ν
X = {x ∈ R | φ(x) = 0},
n = dim X = ν −rank
∂φ ∂x
2.3 Numerical stabilization In practice, problems may arise due to bad initial conditions φ(x0 ) = 0 or numerical integration errors. To counter this, we can add a stabilization term −Λφ to the kinematic equation to get ξ d x˙ = [A Ψ ] ⇒ (8) dt φ = −Λφ. −Λφ
(3)
and may also be regarded as an embedded manifold. A parameterization by minimal coordinates q ∈ Rn , on the other hand, corresponds to a chart of the configuration manifold. This chart cannot be global unless the configuration space is isomorphic to Rn (which is not the case for the previous example X ∼ = SO(3) R3 ). Consequently one has to use several charts (e.g. at least 4 charts to cover SO(3)) to have a global parameterization for X. On the other hand, the Whitney embedding theorem (Lee, 2003, Th. 6.13) states that any smooth manifold admits a global, smooth embedding for sufficiently large ν.
i.e. for an appropriate Λ ∈ R(ν−n)×(ν−n) , the error in the geometric constraint converges exponentially. However, since numerical issues are beyond the scope of this article, we will assume in the following that the constraint is always fulfilled. Also this problem is irrelevant for instance when using the model for control design.
This is the main motivation for the use of redundant coordinates: The ability to formulate global EOM.
2.4 Directional derivatives and their commutation On the embedded configuration manifold it does not make much sense to take derivatives in the coordinate directions ∂/∂xα . What does make sense are derivatives in the directions tangent to the manifold. By construction (5), the columns of A form a basis for the tangent space of X. For the derivative in these tangent directions we introduce the notation 1 ∂ ∂i = Aα , i = 1, . . . , n. (9) i ∂xα Since the matrix A(x) varies over the configuration space, the directional derivatives do not commute in general. Instead we have the following commutation relation: ∂i (∂j f ) − ∂j (∂i f ) ∂ β ∂f β ∂ α ∂f = Aα A − A A i i j ∂xα j ∂xβ ∂xβ ∂xα β β ∂Aj ∂f α ∂Ai − A = Aα i j ∂xα ∂xα ∂xβ 2 ∂2f ∂ f β + Aα A − . (10) i j ∂xα ∂xβ ∂xβ ∂xα
2.2 Minimal velocity coordinates First note that the geometric constraint is equivalent to its time derivative supplemented by the corresponding initial condition, i.e. ∂φ ˙ φ(x) = 0 ⇔ φ(x, x) ˙ = (x) x˙ = 0, φ(x 0 ) = 0. (4) ∂x
Since the Jacobian ∂φ/∂x(x) might not have full rank, let Φ(x) ∈ R(ν−n)×ν be a matrix with the same kernel ker Φ = ker ∂φ/∂x but with full rank.
Due to the nonlinearity of the geometric constraint φ(x) = 0 it might be impossible to eliminate some of the redundant coordinates x without introducing singularities. After differentiation of the nonlinear geometric constraint, we have a linear kinematic constraint Φ(x)x˙ = 0. Here, there is no harm in choosing a set of minimal velocity coordinates ξ(t) ∈ Rn for x. ˙
One way of constructing such a set is as follows: Choose Y (x) ∈ Rn×ν such that the combination with Φ(x) is regular Y (x) ξ x˙ = , rank Y (x) = ν Φ(x) 0 Y (x)
0
This is not very useful yet, since it contains ∂f /∂xα alone, α instead of Aα i (∂f /∂x ) = ∂i f . To fix this, insert the identity (6b) and use the derivatives of the identities (6a) to obtain ∂i (∂j f ) − ∂j (∂i f ) ∂f = ∂i Aβj − ∂j Aβi Aσk Yβk + Ψκσ Φκβ ∂xσ
−1 Y (x)
ξ x˙ = [A(x) Ψ (x)] = A(x) ξ. 0
⇒
(5)
For future reference, note the resulting identities Y A = In , ΦΨ = Iν−n , Y Ψ = 0, ΦA = 0, (6a) (6b) AY + Ψ Φ = Iν . The crucial consequence of the kinematic equation (5) is d dt φ
=
∂φ ˙ ∂x x
=
∂φ ∂x A
ξ = 0,
σ = δβ
∂f = ∂i Aβj − ∂j Aβi Yβk Aσk σ ∂x k k
(7)
γij =−γji
i.e. the geometric constraint is preserved. All we need to care about is a valid initial condition x 0 ∈ X as already pointed out in (4).
∂k f
∂Φκβ ∂Φκα ∂f α β + A i Aj − Ψκσ σ . β α ∂x ∂x ∂x
0
(11)
0
The second term vanishes because the difference in the parenthesis is essentially a reordering of ∂ 2 φκ /∂xα ∂xβ −
Velocity variables that are not necessarily derivatives of coordinates, here ξ, are sometimes called nonholonomic velocity coordinates in Hamel (1949) or quasi-velocities in Lurie (2002).
1 Here, and in the following, the summation convention over double indices i, j, k = 1, . . . , n and α, β = 1, . . . , ν is used.
2
Proceedings of the 9th MATHMOD Vienna, Austria, February 21-23, 2018
Matthias Konz et al. / IFAC PapersOnLine 51-2 (2018) 409–414
∂ 2 φκ /∂xβ ∂xα = 0. Finally, since (11) holds for an arbitrary function f (x), we can summarize in operator form k k ∂i ∂j − ∂j ∂i = γij ∂k , γij = ∂i Aβj − ∂j Aβi Yβk (12)
construction (5) of velocity coordinates ξ, we define the (minimal) displacement coordinates hi = Yαi δxα and get i the relation δxα = Aα i h . Then we can express velocity and displacement as δrp = ∂i rp hi (16) r˙ p = ∂i rp ξ i , and (14) as mp¨rp − FA (17) hi p , ∂i rp = 0. p
k and call γij the commutation coefficients.
The right hand side of (12) appears in the context of Lagrange’s equation in Boltzmann (1902) and Hamel (1904). In the contemporary literature on this context these quank tities γij are sometimes called the Boltzmann three-index symbols in Lurie (2002) or Hamel coefficients in Bremer (2008). In Misner et al. (1973) and Bloch et al. (2009) they appear as the commutators of basis vectors.
Since the displacement coordinates h1 , . . . , hn are free and arbitrary, the terms in the brackets have to vanish. The resulting equation m ¨ r , ∂ r = FA i = 1, . . . , n (18) p p i p p , ∂ i rp , p p
However, all these sources assume minimal configuration coordinates X = Rn and consequently deal with square matrices A = Y −1 . So we can regard the definition (12) as a generalization for embedded manifolds. It can also be shown that Y can be replaced by the (Moore–Penrose) pseudoinverse of A + , giving even closer analogy to the established formulas.
fiM
3.2 Inertia force and inertia matrix Expressing the accelerations ¨rp in terms of the coordinates ¨rp = ∂j rp ξ˙j + ∂k ∂j rp ξ j ξ k (19) and substituting into (18) yields m ∂ r , ∂ r ξ˙j + m ∂ r , ∂ ∂ r ξ j ξ k . fiM = p p i p j p p p i p k j p Mij Γijk (20)
Using the derivative of the identities (6a), the commutation coefficients can also be expressed as ∂Yβk ∂Yαk β k γij = − (13) Aα i Aj . ∂xβ ∂xα Now the following becomes obvious: The commutation i vanish if the corresponding velocity coorcoefficients γjk i dinates ξ are integrable, i.e. ⇒
Yαi =
fiA
can be regarded as the balance of the generalized inertia force f M and the generalized applied force f A .
2.5 Nonholonomy of the velocity coordinates
∃ π i : π˙ i = ξ i = Yαi x˙ α
411
Here we introduced the inertia matrix M (x) = M (x) ∈ Rn×n and the quantities Γ (x) ∈ Rn×n×n that will be investigated in the following.
∂π i ∂xα
3.3 Connection coefficients First, note the identities (21a) ∂k Mij = Γijk + Γjik , s γkj Msi = Γijk − Γikj (21b) and combine them while permuting the indices to yield Γijk = ∂k Mij − Γjik s = ∂k Mij + γik Msj − Γjki s Msj − ∂i Mkj + Γkji = ∂k Mij + γik s s = ∂k Mij + γik Msj − ∂i Mkj + γij Msk + Γkij s s = ∂k Mij + γik Msj − ∂i Mkj + γij Msk + ∂j Mik − Γikj s s = ∂k Mij + γik Msj − ∂i Mkj + γij Msk + ∂j Mik s − γkj Msi − Γijk . (22) With some reordering we find that Γijk = 21 ∂k Mij + ∂j Mik − ∂i Mjk s s s + γij Msk + γik Msj − γjk Msi , (23) i.e. the quantities Γijk can be computed from the inertia matrix M and the commutation coefficients γ , which in turn only depend on the choice of coordinates.
∂Yβi ∂Yαi ∂ 2 πi i = = ⇒ γjk = 0, i = 1, . . . , n ∂xβ ∂xβ ∂xα ∂xα which is not the case in general. ⇒
3. LAGRANGE-D’ALEMBERT’S PRINCIPLE Consider a system of N particles with positions rp ∈ R3 , p = 1 . . . N w.r.t. to an inertial frame and particle masses mp ∈ R > 0. The applied forces FA p collect all forces acting on a particle (gravitation, control forces, . . . ) that are not reaction forces. The Lagrange-d’Alembert principle states: N FA (14) rp , δrp = 0 p − mp¨ p=1
where δrp is called a virtual displacement of a particle and ·, · is the standard scalar product. Virtual displacements must be compatible with the constraints in the sense that they are tangent vectors to a possible motion. 3.1 Introduce coordinates
We express the particle positions rp = rp (x) in terms of the (possibly redundant) configuration coordinates x. Then the velocity r˙ p and the virtual displacement δrp can be expressed by ∂rp α ∂rp α x˙ , δrp = δx . (15) r˙ p = ∂xα ∂xα If the coordinates x are redundant, the displacements δx are constrained by (∂φ/∂x)δx = 0. Analogous to the
Regarding Mij as the coefficients of a Riemannian metric k and γij as the coefficients of the Lie-bracket w.r.t. the chosen basis vectors ∂i , then Γijk are the corresponding coefficients of the Levi-Civita connection. To the best of the authors’ knowledge, the only source that states this formula explicitly (still restricted to minimal configuration coordinates) is (Misner et al., 1973, eq. 8.19a). For the 3
Proceedings of the 9th MATHMOD 412 Vienna, Austria, February 21-23, 2018
Matthias Konz et al. / IFAC PapersOnLine 51-2 (2018) 409–414
special case of minimal configuration coordinates q and s the velocity coordinates ξ = q, ˙ so ∂i = ∂q∂ i and γij = 0, definition (23) reduces to the Christoffel symbols (of the first kind).
potential V(r1 , . . . , rN ) and remaining external forces FE p. For the generalized applied force this yields ∂V fiA = FE , ∂ i rp p − p ∂rp = FE (28) p , ∂i rp − ∂i V(r1 (x), . . . , rN (x)) . p
3.4 Inertia force derived from kinetic energy
V (x)
fiE
Rearrange the inertial force (18) using the product rule, the commutation relation (12) and ∂i rp = ∂ r˙ p /∂ξ i : d j ˙ r m , ∂ r − ˙ r , ∂ ∂ r ξ fiM = p p i p p j i p p dt d k j ˙ r m , ∂ r − ˙ r , (∂ ∂ r − γ ∂ r )ξ = p p i p p i j p k p ij p dt d ∂ r˙ p ∂ r˙ p k j ˙ = ˙ r m , + γ ξ ˙ r , − ˙ r , ∂ r p p p p i p ij p dt ∂ξ i ∂ξ k d ∂T ∂T k j = + γij ξ − ∂i T (24) dt ∂ξ i ∂ξ k with the kinetic energy 2 1 1 T = (25) m ||˙ r || = m ∂ r , ∂ r ξ i ξ j . p p 2 2 p p p i p j p
3.7 Equations of motion
The proposed EOM for a system of particles decompose into kinematics and kinetics: The kinematic relation is i (29) x˙ α = Aα i ξ , x(t0 ) ∈ X. The kinetics can be derived from either of the three equivalent approaches d ∂L k j ∂L + γij ξ − ∂i L = fiE , (30a) dt ∂ξ i ∂ξ k ∂S ⇔ + ∂i V = fiE , (30b) ∂ ξ˙i ⇔ Mij ξ˙j + Γijk ξ j ξ k + ∂i V = fiE , ξ(t0 ) ∈ Rn (30c)
Mij
In (24) we found Lagrange’s differential operator for the kinetic energy in the case of the redundant coordinates x and nonholonomic velocity coordinates ξ. A similar expression has been derived in Hamel (1904) with the slight but crucial difference that it requires minimal configuration coordinates. The same result can be derived from Hamilton’s principle, see Konz and Rudolph (2015).
where (30a) with L = T − V is Lagrange’s equation, (30b) is commonly called Gibbs-Appell’s equations and (30c) are sometimes called the explicit EOM. These formulations have the same structure as the established ones, but with generalized definitions of the directional derivative ∂i , the k commutation coefficients γij , and the connection coefficients Γijk in order to incorporate redundant configuration coordinates.
3.5 Inertia force derived from acceleration energy
It should be stressed again that geometric constraints only apply to the initial condition φ(x(t0 )) = 0, whereas the EOM are ν + n ordinary, first-order differential equations.
Rearrange the inertial force using ∂i rp = ∂¨rp /∂ ξ˙i : ∂¨rp ∂ 1 fiM = m ¨r , = mp ||¨rp ||2 p p p i ˙ ∂ξ ∂ ξ˙i p 2
(26)
4. A SINGLE FREE RIGID BODY
S
For the present purpose, a rigid body can be considered as a finite number of particles that have a constant distance dpq = const. ∈ R to each other. The corresponding configuration space is (31) X = {(r1 , . . . , rN ) ∈ R3N | ||rp − rq || = dpq }. A discussion on the configuration space of a rigid body can be found in any textbook on classical mechanics e.g. (Goldstein, 1951, sec. 4-1). It is crucial here to note that while dim X = 6, X R6 .
˙ is commonly called the where the scalar function S(x, ξ, ξ) acceleration energy. This formulation was first proposed by Gibbs (1879) and by Appell (1900) with a focus on nonholonomic systems. The acceleration energy expressed in terms of the coordinates x and ξ is j k ˙j+ m ∂ r , ∂ r m ∂ r , ∂ ∂ r ξ ξ ξ S = ξ˙i 21 p i p j p p i p k j p p p Mij
+
1 2
Γijk
p
mp ∂l ∂i rp , ∂k ∂j rp ξ i ξ j ξ k ξ l ,
4.1 Configuration coordinates
(27)
S0
where we recovered the inertia matrix M and the connection coefficients Γ previously defined in (20). The term S0 is independent of the acceleration ξ˙ and consequently ˙ vanishes in the derivative ∂S/∂ ξ.
R ¯r1 r
m1 r1
3.6 Conservative forces V E It is common to split the applied forces FA p = Fp + F p V into forces Fp = −∂V/∂rp that can be derived from a
Fig. 1. A rigid body composed of particles and the body fixed frame. 4
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Consider the position r and orientation R of a body fixed frame as illustrated in Fig. 1: 1 1 1 1 R1 R2 R3 r r = r 2 ∈ R3 , R = R12 R22 R32 ∈ SO(3). (32) R13 R23 R33 r3
x˙
x = [r 1 , r 2 , r 3 , R11 , R12 , R13 , R21 , R22 , R23 , R31 , R32 , R33 ] ∈ R12 .
Since R has to be a rotation matrix, the coordinates x are geometrically constrained by φ(x) = 0.
R2 0 0 0
R3 0 0 0
1 v 2 0 0 0 v3 0 −R3 R2 v 1 , R3 0 −R1 ω2 ω −R2 R1 0 ω3 A
(37)
ξ
which can also be written in the more familiar form r˙ = R v, R˙ = R ω . (38) These kinematic relations can be found in many textbooks on the subject.However, Some authors, use the combination [ r˙ , ω ] as velocity coordinates. This would suit the formalism of (5) as well: simply replace the rotation matrix in the upper left corner of (36) and (37) with the identity I3 . Notice that there is no physical motivation for any particular choice of coordinates x and ξ. The point is rather that the particular choice made here will lead to convenient mathematical expressions.
We use the coefficients of the position r and the rotation matrix R as configuration coordinates for the rigid body
⇔
r˙ R1 R˙ 1 0 = R˙ 2 0 ˙ 0 R3
Then we can express the position rp of each particle as rp = r + R¯rp , ¯rp = const., p = 1, . . . , N (33) where ¯rp is the constant particle position w.r.t. the body fixed frame.
R R = I3 , det R = +1
413
(34)
The orthonormality condition R R = I3 yields only 6 independent constraint equations due to symmetry and already implies det R = ±1. Since the determinant is a continuous function, we can conclude that the orthonormality condition produces two disjoint subspaces. The additional constraint det R = +1 simply picks one of them without affecting the dimension of the configuration space. See (Frankel, 1997, sec. 1.1d) for a more thorough discussion. It should be stressed that this (redundant) parameterization by r and R is global, whereas any parameterization by minimal coordinates can only be local.
Commutation coefficients. Plug the matrices Y and A from (35) and (37) into the definition (12) of the comk mutation coefficients γij . Recalling the skew symmetry k k condition γij = −γji , the nonzero coefficients are 4 5 6 2 3 3 1 1 2 = γ56 = γ64 = γ45 = 1. = γ61 = γ15 = γ42 = γ53 = γ34 γ26 (39) They can be combined with the velocity ξ to define the matrix k j k=1...6 ω 0 γij ξ i=1...6 = = −ad (40) ξ . v ω
4.2 Velocity coordinates and kinematics
4.3 Inertia
A popular choice of velocity coordinates for a free rigid body is the translational velocity v expressed in the body fixed frame and the angular velocity ω defined as ˙ ∨ ∈ R3 v = R r˙ ∈ R3 , ω = (R R) (35)
Kinetic energy and inertia matrix. The velocity r˙ p of a particle in terms of the chosen velocity coordinates ξ = [v ω ] is ˙ rp = R(v − ¯rp ω). (41) r˙ p = r˙ + R¯ Therefore, we can write the kinetic energy of the free rigid body as ¯rp I3 v 1 T = 2 [v , ω ] (42) m p p ω ¯rp ¯rp ¯rp s mI3 m m s Θ
We can rearrange these expressions as a matrix vector structure and add the independent eqs. from ∂φ/∂x = 0 to match the kinematic relations proposed in (5): 1 0 0 0 R1 v 0 0 0 v 2 R2 3 R3 0 0 0 v 1 1 0 0 − 12 R2 2 R3 ω 1 2 0 − 1 R 0 r˙ ω 2 3 2 R1 3 0 1 R − 1 R 0 R˙ 1 ω 2 2 2 1 (36) = R 0 0 2R ˙ 2 0 0 1 R˙ 0 0 0 0 2R2 3 0 0 0 0 2R3 x˙ 0 0 0 R R2 3 0 0 R3 0 R1 0 0 R2 R1 0 ξ Y = Y 0 Φ
M
with the inertia parameters: 1 m= m , s=m m ¯r , p p p p p
Θ=
p
¯rp ¯rp , (43) mp
namely the total mass m ∈ R > 0, the center of mass s ∈ R3 , and the moment of inertia Θ = Θ ∈ R3×3 > 0 w.r.t. the chosen body fixed frame.
where R1 , R2 , and R3 are the columns of the rotation matrix R, defined only for the sake of readability. Note that det Y = 8(det R)4 = 8, so the system (36) is globally invertible: 5
Connection coefficients. Since the inertia matrix M here is constant, the connection coefficients (23) consist only of the terms with the commutation coefficients γ previously stated in (39), i.e.
s s s Γijk = 12 γij Msk + γik Msj − γjk Msi = −Γjik . (44)
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5. CONCLUSIONS
Taking into account this skew symmetry, the non-zero coefficients are (45a) Γ324 = Γ135 = Γ216 = m, 1 (45b) Γ254 = Γ364 = Γ515 = Γ616 = ms , 2 Γ424 = Γ145 = Γ365 = Γ626 = ms , (45c) (45d) Γ434 = Γ535 = Γ146 = Γ256 = ms3 , Γ564 = 12 Θ11 − Θ22 − Θ33 , Γ644 = Γ565 = Θ12 , (45e) Γ645 = 12 Θ22 − Θ33 − Θ11 , Γ454 = Γ566 = Θ13 , (45f) Γ456 = 12 Θ33 − Θ11 − Θ22 , Γ455 = Γ646 = Θ23 . (45g)
We reviewed some well-established concepts of analytical mechanics using redundant configuration coordinates x and minimal velocity coordinates ξ in contrast to the commonly used parameterization in q and q. ˙ This established additional geometric ingredients A and γ and extended the definition of the connection coefficients Γ . The formulations are covariant, so for the special choice of coordinates x = q and ξ = q˙ (which imply A = In and γ = 0). We obtain the familiar expressions such as Lagrange’s equations and the Christoffel symbols. The use of redundant configuration coordinates x ∈ X vs. minimal coordinates q ∈ Rn is like the use of an embedding vs. a chart for the configuration space. If the configuration space is not isomorphic to Rn every chart is necessarily local and so are the EOM in terms of minimal coordinates. With an embedding we have the chance to achieve a global parameterization as shown in the example for the rigid body. Notice that even though the motivations come from differential geometry the derivations and their application only require elementary calculus.
Acceleration energy. Differentiate (41) once more to get the acceleration of a particle ¯rp ω˙ + ω ¯rp ω)). ¨rp = R( v˙ − (v − (46) Substituting this expression into the definition (26) of the acceleration energy S yields ω 0 mI3 m s v˙ s v 1 mI3 m ˙ + S=ξ 2 m v ω ω˙ ω s Θ m s Θ ξ˙
M
+
1 2
−ad ξ
ξ
M
¯rp ω)) ω ¯rp ω) . m (ω (v − (v − p p
The proposed formulations can also give valuable insight and inspiration for the design of global control laws for mechanical systems, as demonstrated in Konz and Rudolph (2016).
(47)
S0 (ξ )
The term S0 collects the terms that are independent of the acceleration coefficients ξ˙ that consequently do not enter the equations of motion.
REFERENCES Appell, P.E. (1900). Sur une forme g´en´erale des ´equations de la dynamique. J. Reine Angew. Math., 121, 310–319. Bloch, A.M., Marsden, J.E., and Zenkov, D.V. (2009). Quasivelocities and Stabilization of Relative Equilibria of Underactuated Nonholonomic Systems. In Proc. 48th IEEE Conf. on Decision and Control. ¨ Boltzmann, L. (1902). Uber die Form der Lagrangeschen Gleichungen f¨ ur nichtholonome, generalisierte Koordinaten. In Wissenschaftliche Abhandlungen, volume 3, 682–692. Johann Ambrosius Barth. Bremer, H. (2008). Elastic Multibody Dynamics: A Direct Ritz Approach. Springer. Frankel, T. (1997). The Geometry of Physics. Cambridge University Press. Gibbs, J.W. (1879). On the fundamental formulae of dynamics. Am. J. Math., 2(1), 49–64. Goldstein, H. (1951). Classical Mechanics. AddisonWesley. Hamel, G. (1904). Die Lagrange-Eulerschen Gleichungen der Mechanik. Z. f¨ ur Mathematik u. Physik, 50, 1–57. Hamel, G. (1949). Theoretische Mechanik. Springer. Konz, M. and Rudolph, J. (2015). Equations of Motion with Redundant Coordinates for Mechanical Systems on Manifolds. In Proc. 8th Vienna Int. Conf. on Mathematical Modelling, 681–682. Konz, M. and Rudolph, J. (2016). Beispiele f¨ ur einen direkten Zugang zu einer globalen, energiebasierten Modellbildung und Regelung von Starrk¨orpersystemen. at Automatisierungstechnik, 64(2), 96–109. Lee, J.M. (2003). Introduction to Smooth Manifolds. Springer. Lurie, A.I. (2002). Analytical Mechanics. Springer. Misner, C.W., Thorne, K.S., and Wheeler, J.A. (1973). Gravitation. W. H. Freeman and Company.
4.4 Gravity Assuming that the body is moving close to the surface of the Earth, the particles are subject to the gravitational force Fp = mp g, where g ∈ R3 are the coefficients of the gravitational acceleration w.r.t. the inertial frame. The resulting generalized force and potential energy are fiV = mp g, ∂i rp = ∂i mp g, rp . (48) p p V
In terms of the configuration coordinates x and with the previously defined parameters (43) this is R (mg) V , V = mg, r + Rs . (49) f = s R (mg)
4.5 Equation of motion
Evaluation of either of the eqs. (30) yields the kinetics ω 0 mI3 m mI3 m s v˙ s v + ω˙ ω v ω m s Θ m s Θ M
ξ˙
−ad ξ
fM
+
M
ξ
E R (mg) F = E . τ s R (mg)
fV
(50)
fE
Combining this with the kinematics (37) yields EOM for a rigid body subject the gravitation and the external forces f E = (F E , τ E ). These equations are commonly called the Newton-Euler equations. 6
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Redundant configuration coordinates and Redundant configuration coordinates and Redundant configuration coordinates and Redundant configuration coordinates and nonholonomic velocity coordinates nonholonomic velocity coordinates nonholonomic velocity coordinates nonholonomic velocity coordinates in analytical mechanics in analytical mechanics in analytical mechanics in analytical mechanics Matthias Matthias Konz, Konz, Joachim Joachim Rudolph Rudolph Matthias Konz, Joachim Rudolph Matthias Konz, Joachim Rudolph Chair Chair of of Systems Systems Theory Theory and and Control Control Engineering, Engineering, Saarland University, 66123 Saarbr¨ u Germany, Chair Theory Control Engineering, Saarland University, 66123and Saarbr¨ ucken, cken, Germany, Chair of of Systems Systems Theory and Control Engineering, (e-mail: {m.konz, j.rudolph}@lsr.uni-saarland.de) Saarland University, 66123 Saarbr¨ u cken, (e-mail: {m.konz, j.rudolph}@lsr.uni-saarland.de) Saarland University, 66123 Saarbr¨ ucken, Germany, Germany, (e-mail: {m.konz, j.rudolph}@lsr.uni-saarland.de) (e-mail: {m.konz, j.rudolph}@lsr.uni-saarland.de)
Abstract: Abstract: Several Several established established concepts concepts of of analytical analytical mechanics mechanics are are reviewed reviewed and and extended extended to to Abstract: Severalconfiguration established concepts of analytical mechanicsvelocity are reviewed and extended to include redundant coordinates and nonholonomic coordinates. The include redundant coordinates and nonholonomic coordinates. The main main Abstract: Severalconfiguration established concepts of analytical mechanicsvelocity are reviewed and extended to include redundant configuration coordinates and nonholonomic velocity coordinates. The main motivation for coordinates is the resulting be whereas motivation for redundant redundant coordinates is that that and the nonholonomic resulting formulations formulations can be global, global,The whereas include redundant configuration coordinates velocity can coordinates. main minimal coordinates might necessarily be local depending on the topology of the configuration motivation for redundant coordinates is that the resulting formulations can be global, whereas minimal coordinates mightcoordinates necessarilyisbethat local the topology configuration motivation for redundant thedepending resulting on formulations canofbethe global, whereas space for The formulations for equations, minimal necessarily be on the configuration space e.g. e.g.coordinates for SO(3). SO(3).might The resulting resulting formulations for Lagrange’s Lagrange’s equations,of the Gibbs-Appell minimal coordinates might necessarily be local local depending depending on the the topology topology of the the Gibbs-Appell configuration space e.g. for the SO(3). The resulting formulations for Lagrange’s equations, the Gibbs-Appell equations coefficients of are These are then equations and coefficients of the the Levi-Civita Levi-Civita connection are derived. derived. Thesethe areGibbs-Appell then applied applied space e.g. and for the SO(3). The resulting formulationsconnection for Lagrange’s equations, to the example of the free rigid body to derive the well-known Newton-Euler equations. equations and the coefficients of the Levi-Civita connection are derived. These are then to the example of the free rigid body to derive the well-known Newton-Euler equations. equations and the coefficients of the Levi-Civita connection are derived. These are then applied applied to the of rigid to derive well-known Newton-Euler equations. to the example example of the the free freeFederation rigid body body derive the the well-known © 2018, IFAC (International of to Automatic Control) Hosting Newton-Euler by Elsevier Ltd. equations. All rights reserved. Keywords: mechanics, first-principles, energy methods, redundant Keywords: mechanics, first-principles, energy methods, redundant coordinates, coordinates, rigid rigid body body Keywords: Keywords: mechanics, mechanics, first-principles, first-principles, energy energy methods, methods, redundant redundant coordinates, coordinates, rigid rigid body body 1. components 1. INTRODUCTION INTRODUCTION components of of the the angular angular velocity velocity ω ω are are not not derivatives derivatives 1. INTRODUCTION components of the angular velocity ω are not derivatives of any coordinates. of any coordinates. 1. INTRODUCTION components of the angular velocity ω are not derivatives of any coordinates. Methods Methods from from analytical analytical mechanics mechanics are are widely widely used used by by of any coordinates. engineers for modeling and as inspiration for control design Methods from analytical mechanics are widely used by engineers for modeling and mechanics as inspiration control design Methods from analytical arefor widely used by 1.2 1.2 Outline Outline and One key therein is use Outline engineers for inspiration for and optimization. optimization. Oneand keyas aspect therein is nthe the design use of of 1.2 engineers for modeling modeling and asaspect inspiration for control control design 1.2 Outline where n (independent) generalized coordinates q ∈ R and optimization. One key aspect therein is the use of n A possible n (independent) generalized q ∈ isRnnthewhere and optimization. One keycoordinates aspect therein use of possible extension extension of of the the Lagrangian Lagrangian formalism formalism to to ininwhere n A (independent) generalized coordinates q ∈The Rn following is the degree of freedom of the system. A possible extension of the Lagrangian formalism to include redundant coordinates was proposed in Konz is the degree of freedom of the system. where n A (independent) generalized coordinates q ∈The R following clude redundant coordinates was proposed in Konztoand and possible extension of the Lagrangian formalism inbrief example should motivate that this restriction can be is the degree of freedom of the system. The following redundant coordinates was proposed in derivation Konz and Rudolph (2015). This approach allows aa direct brief motivate thatsystem. this restriction can be clude is theexample degree should of freedom of the The following Rudolph (2015). This approach allows direct derivation clude redundant coordinates was proposed in Konz and brief example should motivate that this restriction can be of the Euler impractical for certain systems. (2015). This approach allows a direct derivation equations (2). impractical forshould certainmotivate systems.that this restriction can be Rudolph brief example of the Euler equations (2). Rudolph (2015). This approach allows a direct derivation impractical for certain systems. of the Euler equations (2). impractical for certain systems. of thepresent Euler equations (2). The article extends these The present article extends these concepts concepts further further by by 1.1 1.1 Motivating Motivating example example The present article extends these concepts further meby formulating other established concepts of analytical 1.1 Motivating example formulating other established concepts of analytical meThe present article extends these concepts further by 1.1 Motivating example formulating other ofestablished concepts of analytical mechanics in terms redundant configuration coordinates chanics in terms redundantconcepts configuration coordinates formulating other ofestablished of analytical meA popular approach for the derivation of equations of in terms of redundant configuration coordinates A popular approach for the derivation of equations of chanics and nonholonomic velocity coordinates. The and minimal minimal nonholonomic velocity coordinates. The funfunin terms of redundant configuration coordinates A popular approach for the derivation of equations of chanics motion (EOM) is formalism: Choose minimal nonholonomic velocity coordinates. The funmotion (EOM) is the the Lagrangian Lagrangian formalism: Choose gengenA popular approach for nthe derivation of equations of and damental axiom for the discussions is the Langrangedamental axiom for the discussions is the Langrangeand minimal nonholonomic velocity coordinates. The funmotion (EOM) is theq Lagrangian formalism: Choose genand formulate the Lagrangian eralized coordinates ∈ R axiom for (in thecontrast discussions is the LangrangetheChoose Lagrangian eralized coordinates ∈ Rnnn and formulate motion (EOM) is theq Lagrangian formalism: gen- damental d’Alembert principle to principle d’Alembertaxiom principle to Hamilton’s Hamilton’s principle damental for (in thecontrast discussions is the Langrangeformulateqthe Lagrangian eralized coordinates q∈ Rn and L(q, q) ˙ in terms of the chosen coordinates and their time L(q, q) ˙ incoordinates terms of the chosen coordinates andLagrangian their time for formulateqthe eralized q∈ R and the article). principle (in for the previous previous article). d’Alembert principle (in contrast contrast to to Hamilton’s Hamilton’s principle principle L(q, q) ˙ in terms of the chosen coordinates q and their time d’Alembert derivatives q. ˙˙ The EOM can evaluated the previous article). derivatives q. The EOM can be be evaluated qfrom from L(q, q) ˙ in terms of the chosen coordinates and their time for for the previous article). derivatives q. ˙ dThe EOM can be evaluated from The first section of the present work deals with geometry, ∂L ∂L The first section of the present work deals with geometry, derivatives q. ˙ dThe can 0, be evaluated from ∂L EOM − ∂L (1) firstshape section of the present workspace dealsand withhow geometry, i.e. of the it d ∂∂L ∂Li = i − ∂q = 0, ii = = 1, 1, .. .. .. ,, n. n. (1) The i.e. the the of of thetheconfiguration configuration it can can The firstshape section present workspace dealsand withhow geometry, dt q ˙ i i d ∂L ∂L − = 0, i = 1, . . . , n. (1) dt ∂ q˙ii − ∂q ii = 0, i = 1, . . . , n. i.e. the shape of the configuration space and how it can be parameterized, and with kinematics, i.e. how config(1) dt ∂ q ˙ be parameterized, and with kinematics, i.e. how configi.e. the shape of the configuration space and how it can ∂q However, this ∂ q˙i systems ∂q i parameterized, and with kinematics, i.e. how configuration and velocity coordinates are related. The second However, for fordtsome some systems this approach approach may may lead lead to to be uration and velocity coordinates are related. The second be parameterized, and with kinematics, i.e. how configHowever, for some systems this approach may lead to awkward and velocity coordinates are related. The second section with and the concept of awkward expressions. expressions. However, for some systems this approach may lead to uration section deals deals with kinetics kinetics and establishes establishes theThe concept of uration and velocity coordinates are related. second awkward expressions. section deals with kinetics and establishes the concept of a generalized force. We will motivate three formulations awkward expressions. Consider for example a rigid body fixed at one point: A a generalized force. We will motivate three formulations section deals with kinetics and establishes the concept of Consider for example a rigid body fixed at one point: A aforgeneralized force. We will motivate three formulations deriving the generalized force of inertia: Lagrange’s Consider for example a rigid body fixed at one point: A popular choice of generalized coordinates are the Cardan for deriving the generalized force of inertia: Lagrange’s a generalized force. We will motivate three formulations popular choice of generalized areone thepoint: Cardan Consider for example a rigid coordinates body fixed at A equations deriving the on generalized force of inertia: Lagrange’s based the energy, the popular choice of generalized coordinates are the cumberCardan for angles roll, yaw, will quite equations based the kinetic kinetic energy, the Gibbs-Appell Gibbs-Appell deriving the on generalized force of inertia: Lagrange’s angles for forchoice roll, pitch, pitch, yaw, which which will lead lead to toare quite popular of generalized coordinates the cumberCardan for equations on the kinetic energy, the Gibbs-Appell based on the acceleration energy, and the angles for roll, pitch, yaw, which will lead toconfigurations. quite cumber- equations based on some that are for particular energy, the LeviLevion the the acceleration kinetic energy, the and Gibbs-Appell some EOM EOM that are singular singular for will particular angles for roll, pitch, yaw, which lead toconfigurations. quite cumber- equations based on the acceleration energy, and the LeviCivita connection for the inertia metric. Finally, we will some EOM that are singular for particular configurations. Alternatively, the EOM for particular system Civita connection inertia metric. Finally, weLeviwill based onfor thethe acceleration energy, and the Alternatively, theare EOM for this this particular configurations. system can can be be equations some EOM that singular for particular Civita connection for the inertia metric. Finally, we will apply these formulations to the example of a free rigid Alternatively, the EOM for this particular system can be given directly by the Euler equations apply these formulations to the example of a free rigid Civita connection for the inertia metric. Finally, we will given directly by Euler Alternatively, thethe EOM for equations this particular system can be apply these formulations to the example of a free rigid body. given directly by˙ the Euler equations body. apply these formulations to the example of a free rigid R = R ω , Θ ω ˙ + ω Θω = 0. (2) given directly by the Euler equations R˙˙ = R ω , Θω˙ + ω Θω = 0. (2) body. R˙ = R ωmatrix , Θω˙ + ω Θω = 0. the angular (2) body. with the orientation R ∈ SO(3), R = R ω , Θ ω ˙ + ω Θω = 0. (2) 2. with the orientation matrix R ∈ SO(3), the angular 3 2. CONFIGURATION CONFIGURATION AND AND KINEMATICS KINEMATICS with the orientation matrix R ∈ SO(3), the angular velocity ω ∈ R · operator 3 , the inertia matrix Θ, and the 2. CONFIGURATION AND KINEMATICS · operator velocity ∈ R33 , the inertia and the with theω orientation matrixmatrix R ∈ Θ,SO(3), the angular 3 2. CONFIGURATION AND KINEMATICS velocity ω ∈b = R3a, × theb, inertia matrix Θ, and the · operator such that a a, b ∈ R . 3 such thatω a∈b = a, b ∈ Rmatrix . velocity R a, × theb, inertia Θ, and the · operator 2.1 2.1 Redundant Redundant configuration configuration coordinates coordinates such that ab = a × b, a, b ∈ R333 . Redundant configuration coordinates such that known ab = a × b, a,the b ∈ R . equations (2) cannot be 2.1 It It is is well well known that that the Euler Euler equations (2) cannot be 2.1 Redundant configuration coordinates It is wellfrom known that theequations Euler equations (2) cannot be The introductory example should demonstrate that it may derived Lagrange’s (1) directly since the derived Lagrange’s (1) directly since the It is wellfrom known that theequations Euler equations (2) cannot be The introductory example should demonstrate that it may derived from Lagrange’s equations (1)generalized directly since the The introductory example should demonstrate that degree it may entries the matrix R not coordibe to coordinates than entries of offrom the rotation rotation matrix R are are not generalized coordibe useful useful to use use more more coordinates than the the actual actual derived Lagrange’s equations (1) directly since the The introductory example should demonstrate that degree it may not generalized coordientries of theare rotation matrixby RR are be useful ton use more coordinates than the actual degree nates (they constrained of freedom of the system, i.e. redundant coordinates. We R = I3 ). Moreover, the nates (they constrained R = I ). Moreover, the of freedom n of the system, i.e. redundant coordinates. We entries of theare rotation matrixby RR are not generalized coordibe useful to use more coordinates than the actual degree 3 nates (they are constrained by R ). Moreover, the of freedom n of the system, i.e. redundant coordinates. We R = I3 3 nates (they are constrained by R R = I3 ). Moreover, the of freedom n of the system, i.e. redundant coordinates. We
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will denote these by x ∈ Rν , ν ≥ n and their collection of constraint equations as φ(x) = 0. The set X of mutually admissible coordinate values is the configuration space ν
X = {x ∈ R | φ(x) = 0},
n = dim X = ν −rank
∂φ ∂x
2.3 Numerical stabilization In practice, problems may arise due to bad initial conditions φ(x0 ) = 0 or numerical integration errors. To counter this, we can add a stabilization term −Λφ to the kinematic equation to get ξ d x˙ = [A Ψ ] ⇒ (8) dt φ = −Λφ. −Λφ
(3)
and may also be regarded as an embedded manifold. A parameterization by minimal coordinates q ∈ Rn , on the other hand, corresponds to a chart of the configuration manifold. This chart cannot be global unless the configuration space is isomorphic to Rn (which is not the case for the previous example X ∼ = SO(3) R3 ). Consequently one has to use several charts (e.g. at least 4 charts to cover SO(3)) to have a global parameterization for X. On the other hand, the Whitney embedding theorem (Lee, 2003, Th. 6.13) states that any smooth manifold admits a global, smooth embedding for sufficiently large ν.
i.e. for an appropriate Λ ∈ R(ν−n)×(ν−n) , the error in the geometric constraint converges exponentially. However, since numerical issues are beyond the scope of this article, we will assume in the following that the constraint is always fulfilled. Also this problem is irrelevant for instance when using the model for control design.
This is the main motivation for the use of redundant coordinates: The ability to formulate global EOM.
2.4 Directional derivatives and their commutation On the embedded configuration manifold it does not make much sense to take derivatives in the coordinate directions ∂/∂xα . What does make sense are derivatives in the directions tangent to the manifold. By construction (5), the columns of A form a basis for the tangent space of X. For the derivative in these tangent directions we introduce the notation 1 ∂ ∂i = Aα , i = 1, . . . , n. (9) i ∂xα Since the matrix A(x) varies over the configuration space, the directional derivatives do not commute in general. Instead we have the following commutation relation: ∂i (∂j f ) − ∂j (∂i f ) ∂ β ∂f β ∂ α ∂f = Aα A − A A i i j ∂xα j ∂xβ ∂xβ ∂xα β β ∂Aj ∂f α ∂Ai − A = Aα i j ∂xα ∂xα ∂xβ 2 ∂2f ∂ f β + Aα A − . (10) i j ∂xα ∂xβ ∂xβ ∂xα
2.2 Minimal velocity coordinates First note that the geometric constraint is equivalent to its time derivative supplemented by the corresponding initial condition, i.e. ∂φ ˙ φ(x) = 0 ⇔ φ(x, x) ˙ = (x) x˙ = 0, φ(x 0 ) = 0. (4) ∂x
Since the Jacobian ∂φ/∂x(x) might not have full rank, let Φ(x) ∈ R(ν−n)×ν be a matrix with the same kernel ker Φ = ker ∂φ/∂x but with full rank.
Due to the nonlinearity of the geometric constraint φ(x) = 0 it might be impossible to eliminate some of the redundant coordinates x without introducing singularities. After differentiation of the nonlinear geometric constraint, we have a linear kinematic constraint Φ(x)x˙ = 0. Here, there is no harm in choosing a set of minimal velocity coordinates ξ(t) ∈ Rn for x. ˙
One way of constructing such a set is as follows: Choose Y (x) ∈ Rn×ν such that the combination with Φ(x) is regular Y (x) ξ x˙ = , rank Y (x) = ν Φ(x) 0 Y (x)
0
This is not very useful yet, since it contains ∂f /∂xα alone, α instead of Aα i (∂f /∂x ) = ∂i f . To fix this, insert the identity (6b) and use the derivatives of the identities (6a) to obtain ∂i (∂j f ) − ∂j (∂i f ) ∂f = ∂i Aβj − ∂j Aβi Aσk Yβk + Ψκσ Φκβ ∂xσ
−1 Y (x)
ξ x˙ = [A(x) Ψ (x)] = A(x) ξ. 0
⇒
(5)
For future reference, note the resulting identities Y A = In , ΦΨ = Iν−n , Y Ψ = 0, ΦA = 0, (6a) (6b) AY + Ψ Φ = Iν . The crucial consequence of the kinematic equation (5) is d dt φ
=
∂φ ˙ ∂x x
=
∂φ ∂x A
ξ = 0,
σ = δβ
∂f = ∂i Aβj − ∂j Aβi Yβk Aσk σ ∂x k k
(7)
γij =−γji
i.e. the geometric constraint is preserved. All we need to care about is a valid initial condition x 0 ∈ X as already pointed out in (4).
∂k f
∂Φκβ ∂Φκα ∂f α β + A i Aj − Ψκσ σ . β α ∂x ∂x ∂x
0
(11)
0
The second term vanishes because the difference in the parenthesis is essentially a reordering of ∂ 2 φκ /∂xα ∂xβ −
Velocity variables that are not necessarily derivatives of coordinates, here ξ, are sometimes called nonholonomic velocity coordinates in Hamel (1949) or quasi-velocities in Lurie (2002).
1 Here, and in the following, the summation convention over double indices i, j, k = 1, . . . , n and α, β = 1, . . . , ν is used.
2
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∂ 2 φκ /∂xβ ∂xα = 0. Finally, since (11) holds for an arbitrary function f (x), we can summarize in operator form k k ∂i ∂j − ∂j ∂i = γij ∂k , γij = ∂i Aβj − ∂j Aβi Yβk (12)
construction (5) of velocity coordinates ξ, we define the (minimal) displacement coordinates hi = Yαi δxα and get i the relation δxα = Aα i h . Then we can express velocity and displacement as δrp = ∂i rp hi (16) r˙ p = ∂i rp ξ i , and (14) as mp¨rp − FA (17) hi p , ∂i rp = 0. p
k and call γij the commutation coefficients.
The right hand side of (12) appears in the context of Lagrange’s equation in Boltzmann (1902) and Hamel (1904). In the contemporary literature on this context these quank tities γij are sometimes called the Boltzmann three-index symbols in Lurie (2002) or Hamel coefficients in Bremer (2008). In Misner et al. (1973) and Bloch et al. (2009) they appear as the commutators of basis vectors.
Since the displacement coordinates h1 , . . . , hn are free and arbitrary, the terms in the brackets have to vanish. The resulting equation m ¨ r , ∂ r = FA i = 1, . . . , n (18) p p i p p , ∂ i rp , p p
However, all these sources assume minimal configuration coordinates X = Rn and consequently deal with square matrices A = Y −1 . So we can regard the definition (12) as a generalization for embedded manifolds. It can also be shown that Y can be replaced by the (Moore–Penrose) pseudoinverse of A + , giving even closer analogy to the established formulas.
fiM
3.2 Inertia force and inertia matrix Expressing the accelerations ¨rp in terms of the coordinates ¨rp = ∂j rp ξ˙j + ∂k ∂j rp ξ j ξ k (19) and substituting into (18) yields m ∂ r , ∂ r ξ˙j + m ∂ r , ∂ ∂ r ξ j ξ k . fiM = p p i p j p p p i p k j p Mij Γijk (20)
Using the derivative of the identities (6a), the commutation coefficients can also be expressed as ∂Yβk ∂Yαk β k γij = − (13) Aα i Aj . ∂xβ ∂xα Now the following becomes obvious: The commutation i vanish if the corresponding velocity coorcoefficients γjk i dinates ξ are integrable, i.e. ⇒
Yαi =
fiA
can be regarded as the balance of the generalized inertia force f M and the generalized applied force f A .
2.5 Nonholonomy of the velocity coordinates
∃ π i : π˙ i = ξ i = Yαi x˙ α
411
Here we introduced the inertia matrix M (x) = M (x) ∈ Rn×n and the quantities Γ (x) ∈ Rn×n×n that will be investigated in the following.
∂π i ∂xα
3.3 Connection coefficients First, note the identities (21a) ∂k Mij = Γijk + Γjik , s γkj Msi = Γijk − Γikj (21b) and combine them while permuting the indices to yield Γijk = ∂k Mij − Γjik s = ∂k Mij + γik Msj − Γjki s Msj − ∂i Mkj + Γkji = ∂k Mij + γik s s = ∂k Mij + γik Msj − ∂i Mkj + γij Msk + Γkij s s = ∂k Mij + γik Msj − ∂i Mkj + γij Msk + ∂j Mik − Γikj s s = ∂k Mij + γik Msj − ∂i Mkj + γij Msk + ∂j Mik s − γkj Msi − Γijk . (22) With some reordering we find that Γijk = 21 ∂k Mij + ∂j Mik − ∂i Mjk s s s + γij Msk + γik Msj − γjk Msi , (23) i.e. the quantities Γijk can be computed from the inertia matrix M and the commutation coefficients γ , which in turn only depend on the choice of coordinates.
∂Yβi ∂Yαi ∂ 2 πi i = = ⇒ γjk = 0, i = 1, . . . , n ∂xβ ∂xβ ∂xα ∂xα which is not the case in general. ⇒
3. LAGRANGE-D’ALEMBERT’S PRINCIPLE Consider a system of N particles with positions rp ∈ R3 , p = 1 . . . N w.r.t. to an inertial frame and particle masses mp ∈ R > 0. The applied forces FA p collect all forces acting on a particle (gravitation, control forces, . . . ) that are not reaction forces. The Lagrange-d’Alembert principle states: N FA (14) rp , δrp = 0 p − mp¨ p=1
where δrp is called a virtual displacement of a particle and ·, · is the standard scalar product. Virtual displacements must be compatible with the constraints in the sense that they are tangent vectors to a possible motion. 3.1 Introduce coordinates
We express the particle positions rp = rp (x) in terms of the (possibly redundant) configuration coordinates x. Then the velocity r˙ p and the virtual displacement δrp can be expressed by ∂rp α ∂rp α x˙ , δrp = δx . (15) r˙ p = ∂xα ∂xα If the coordinates x are redundant, the displacements δx are constrained by (∂φ/∂x)δx = 0. Analogous to the
Regarding Mij as the coefficients of a Riemannian metric k and γij as the coefficients of the Lie-bracket w.r.t. the chosen basis vectors ∂i , then Γijk are the corresponding coefficients of the Levi-Civita connection. To the best of the authors’ knowledge, the only source that states this formula explicitly (still restricted to minimal configuration coordinates) is (Misner et al., 1973, eq. 8.19a). For the 3
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special case of minimal configuration coordinates q and s the velocity coordinates ξ = q, ˙ so ∂i = ∂q∂ i and γij = 0, definition (23) reduces to the Christoffel symbols (of the first kind).
potential V(r1 , . . . , rN ) and remaining external forces FE p. For the generalized applied force this yields ∂V fiA = FE , ∂ i rp p − p ∂rp = FE (28) p , ∂i rp − ∂i V(r1 (x), . . . , rN (x)) . p
3.4 Inertia force derived from kinetic energy
V (x)
fiE
Rearrange the inertial force (18) using the product rule, the commutation relation (12) and ∂i rp = ∂ r˙ p /∂ξ i : d j ˙ r m , ∂ r − ˙ r , ∂ ∂ r ξ fiM = p p i p p j i p p dt d k j ˙ r m , ∂ r − ˙ r , (∂ ∂ r − γ ∂ r )ξ = p p i p p i j p k p ij p dt d ∂ r˙ p ∂ r˙ p k j ˙ = ˙ r m , + γ ξ ˙ r , − ˙ r , ∂ r p p p p i p ij p dt ∂ξ i ∂ξ k d ∂T ∂T k j = + γij ξ − ∂i T (24) dt ∂ξ i ∂ξ k with the kinetic energy 2 1 1 T = (25) m ||˙ r || = m ∂ r , ∂ r ξ i ξ j . p p 2 2 p p p i p j p
3.7 Equations of motion
The proposed EOM for a system of particles decompose into kinematics and kinetics: The kinematic relation is i (29) x˙ α = Aα i ξ , x(t0 ) ∈ X. The kinetics can be derived from either of the three equivalent approaches d ∂L k j ∂L + γij ξ − ∂i L = fiE , (30a) dt ∂ξ i ∂ξ k ∂S ⇔ + ∂i V = fiE , (30b) ∂ ξ˙i ⇔ Mij ξ˙j + Γijk ξ j ξ k + ∂i V = fiE , ξ(t0 ) ∈ Rn (30c)
Mij
In (24) we found Lagrange’s differential operator for the kinetic energy in the case of the redundant coordinates x and nonholonomic velocity coordinates ξ. A similar expression has been derived in Hamel (1904) with the slight but crucial difference that it requires minimal configuration coordinates. The same result can be derived from Hamilton’s principle, see Konz and Rudolph (2015).
where (30a) with L = T − V is Lagrange’s equation, (30b) is commonly called Gibbs-Appell’s equations and (30c) are sometimes called the explicit EOM. These formulations have the same structure as the established ones, but with generalized definitions of the directional derivative ∂i , the k commutation coefficients γij , and the connection coefficients Γijk in order to incorporate redundant configuration coordinates.
3.5 Inertia force derived from acceleration energy
It should be stressed again that geometric constraints only apply to the initial condition φ(x(t0 )) = 0, whereas the EOM are ν + n ordinary, first-order differential equations.
Rearrange the inertial force using ∂i rp = ∂¨rp /∂ ξ˙i : ∂¨rp ∂ 1 fiM = m ¨r , = mp ||¨rp ||2 p p p i ˙ ∂ξ ∂ ξ˙i p 2
(26)
4. A SINGLE FREE RIGID BODY
S
For the present purpose, a rigid body can be considered as a finite number of particles that have a constant distance dpq = const. ∈ R to each other. The corresponding configuration space is (31) X = {(r1 , . . . , rN ) ∈ R3N | ||rp − rq || = dpq }. A discussion on the configuration space of a rigid body can be found in any textbook on classical mechanics e.g. (Goldstein, 1951, sec. 4-1). It is crucial here to note that while dim X = 6, X R6 .
˙ is commonly called the where the scalar function S(x, ξ, ξ) acceleration energy. This formulation was first proposed by Gibbs (1879) and by Appell (1900) with a focus on nonholonomic systems. The acceleration energy expressed in terms of the coordinates x and ξ is j k ˙j+ m ∂ r , ∂ r m ∂ r , ∂ ∂ r ξ ξ ξ S = ξ˙i 21 p i p j p p i p k j p p p Mij
+
1 2
Γijk
p
mp ∂l ∂i rp , ∂k ∂j rp ξ i ξ j ξ k ξ l ,
4.1 Configuration coordinates
(27)
S0
where we recovered the inertia matrix M and the connection coefficients Γ previously defined in (20). The term S0 is independent of the acceleration ξ˙ and consequently ˙ vanishes in the derivative ∂S/∂ ξ.
R ¯r1 r
m1 r1
3.6 Conservative forces V E It is common to split the applied forces FA p = Fp + F p V into forces Fp = −∂V/∂rp that can be derived from a
Fig. 1. A rigid body composed of particles and the body fixed frame. 4
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Consider the position r and orientation R of a body fixed frame as illustrated in Fig. 1: 1 1 1 1 R1 R2 R3 r r = r 2 ∈ R3 , R = R12 R22 R32 ∈ SO(3). (32) R13 R23 R33 r3
x˙
x = [r 1 , r 2 , r 3 , R11 , R12 , R13 , R21 , R22 , R23 , R31 , R32 , R33 ] ∈ R12 .
Since R has to be a rotation matrix, the coordinates x are geometrically constrained by φ(x) = 0.
R2 0 0 0
R3 0 0 0
1 v 2 0 0 0 v3 0 −R3 R2 v 1 , R3 0 −R1 ω2 ω −R2 R1 0 ω3 A
(37)
ξ
which can also be written in the more familiar form r˙ = R v, R˙ = R ω . (38) These kinematic relations can be found in many textbooks on the subject.However, Some authors, use the combination [ r˙ , ω ] as velocity coordinates. This would suit the formalism of (5) as well: simply replace the rotation matrix in the upper left corner of (36) and (37) with the identity I3 . Notice that there is no physical motivation for any particular choice of coordinates x and ξ. The point is rather that the particular choice made here will lead to convenient mathematical expressions.
We use the coefficients of the position r and the rotation matrix R as configuration coordinates for the rigid body
⇔
r˙ R1 R˙ 1 0 = R˙ 2 0 ˙ 0 R3
Then we can express the position rp of each particle as rp = r + R¯rp , ¯rp = const., p = 1, . . . , N (33) where ¯rp is the constant particle position w.r.t. the body fixed frame.
R R = I3 , det R = +1
413
(34)
The orthonormality condition R R = I3 yields only 6 independent constraint equations due to symmetry and already implies det R = ±1. Since the determinant is a continuous function, we can conclude that the orthonormality condition produces two disjoint subspaces. The additional constraint det R = +1 simply picks one of them without affecting the dimension of the configuration space. See (Frankel, 1997, sec. 1.1d) for a more thorough discussion. It should be stressed that this (redundant) parameterization by r and R is global, whereas any parameterization by minimal coordinates can only be local.
Commutation coefficients. Plug the matrices Y and A from (35) and (37) into the definition (12) of the comk mutation coefficients γij . Recalling the skew symmetry k k condition γij = −γji , the nonzero coefficients are 4 5 6 2 3 3 1 1 2 = γ56 = γ64 = γ45 = 1. = γ61 = γ15 = γ42 = γ53 = γ34 γ26 (39) They can be combined with the velocity ξ to define the matrix k j k=1...6 ω 0 γij ξ i=1...6 = = −ad (40) ξ . v ω
4.2 Velocity coordinates and kinematics
4.3 Inertia
A popular choice of velocity coordinates for a free rigid body is the translational velocity v expressed in the body fixed frame and the angular velocity ω defined as ˙ ∨ ∈ R3 v = R r˙ ∈ R3 , ω = (R R) (35)
Kinetic energy and inertia matrix. The velocity r˙ p of a particle in terms of the chosen velocity coordinates ξ = [v ω ] is ˙ rp = R(v − ¯rp ω). (41) r˙ p = r˙ + R¯ Therefore, we can write the kinetic energy of the free rigid body as ¯rp I3 v 1 T = 2 [v , ω ] (42) m p p ω ¯rp ¯rp ¯rp s mI3 m m s Θ
We can rearrange these expressions as a matrix vector structure and add the independent eqs. from ∂φ/∂x = 0 to match the kinematic relations proposed in (5): 1 0 0 0 R1 v 0 0 0 v 2 R2 3 R3 0 0 0 v 1 1 0 0 − 12 R2 2 R3 ω 1 2 0 − 1 R 0 r˙ ω 2 3 2 R1 3 0 1 R − 1 R 0 R˙ 1 ω 2 2 2 1 (36) = R 0 0 2R ˙ 2 0 0 1 R˙ 0 0 0 0 2R2 3 0 0 0 0 2R3 x˙ 0 0 0 R R2 3 0 0 R3 0 R1 0 0 R2 R1 0 ξ Y = Y 0 Φ
M
with the inertia parameters: 1 m= m , s=m m ¯r , p p p p p
Θ=
p
¯rp ¯rp , (43) mp
namely the total mass m ∈ R > 0, the center of mass s ∈ R3 , and the moment of inertia Θ = Θ ∈ R3×3 > 0 w.r.t. the chosen body fixed frame.
where R1 , R2 , and R3 are the columns of the rotation matrix R, defined only for the sake of readability. Note that det Y = 8(det R)4 = 8, so the system (36) is globally invertible: 5
Connection coefficients. Since the inertia matrix M here is constant, the connection coefficients (23) consist only of the terms with the commutation coefficients γ previously stated in (39), i.e.
s s s Γijk = 12 γij Msk + γik Msj − γjk Msi = −Γjik . (44)
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5. CONCLUSIONS
Taking into account this skew symmetry, the non-zero coefficients are (45a) Γ324 = Γ135 = Γ216 = m, 1 (45b) Γ254 = Γ364 = Γ515 = Γ616 = ms , 2 Γ424 = Γ145 = Γ365 = Γ626 = ms , (45c) (45d) Γ434 = Γ535 = Γ146 = Γ256 = ms3 , Γ564 = 12 Θ11 − Θ22 − Θ33 , Γ644 = Γ565 = Θ12 , (45e) Γ645 = 12 Θ22 − Θ33 − Θ11 , Γ454 = Γ566 = Θ13 , (45f) Γ456 = 12 Θ33 − Θ11 − Θ22 , Γ455 = Γ646 = Θ23 . (45g)
We reviewed some well-established concepts of analytical mechanics using redundant configuration coordinates x and minimal velocity coordinates ξ in contrast to the commonly used parameterization in q and q. ˙ This established additional geometric ingredients A and γ and extended the definition of the connection coefficients Γ . The formulations are covariant, so for the special choice of coordinates x = q and ξ = q˙ (which imply A = In and γ = 0). We obtain the familiar expressions such as Lagrange’s equations and the Christoffel symbols. The use of redundant configuration coordinates x ∈ X vs. minimal coordinates q ∈ Rn is like the use of an embedding vs. a chart for the configuration space. If the configuration space is not isomorphic to Rn every chart is necessarily local and so are the EOM in terms of minimal coordinates. With an embedding we have the chance to achieve a global parameterization as shown in the example for the rigid body. Notice that even though the motivations come from differential geometry the derivations and their application only require elementary calculus.
Acceleration energy. Differentiate (41) once more to get the acceleration of a particle ¯rp ω˙ + ω ¯rp ω)). ¨rp = R( v˙ − (v − (46) Substituting this expression into the definition (26) of the acceleration energy S yields ω 0 mI3 m s v˙ s v 1 mI3 m ˙ + S=ξ 2 m v ω ω˙ ω s Θ m s Θ ξ˙
M
+
1 2
−ad ξ
ξ
M
¯rp ω)) ω ¯rp ω) . m (ω (v − (v − p p
The proposed formulations can also give valuable insight and inspiration for the design of global control laws for mechanical systems, as demonstrated in Konz and Rudolph (2016).
(47)
S0 (ξ )
The term S0 collects the terms that are independent of the acceleration coefficients ξ˙ that consequently do not enter the equations of motion.
REFERENCES Appell, P.E. (1900). Sur une forme g´en´erale des ´equations de la dynamique. J. Reine Angew. Math., 121, 310–319. Bloch, A.M., Marsden, J.E., and Zenkov, D.V. (2009). Quasivelocities and Stabilization of Relative Equilibria of Underactuated Nonholonomic Systems. In Proc. 48th IEEE Conf. on Decision and Control. ¨ Boltzmann, L. (1902). Uber die Form der Lagrangeschen Gleichungen f¨ ur nichtholonome, generalisierte Koordinaten. In Wissenschaftliche Abhandlungen, volume 3, 682–692. Johann Ambrosius Barth. Bremer, H. (2008). Elastic Multibody Dynamics: A Direct Ritz Approach. Springer. Frankel, T. (1997). The Geometry of Physics. Cambridge University Press. Gibbs, J.W. (1879). On the fundamental formulae of dynamics. Am. J. Math., 2(1), 49–64. Goldstein, H. (1951). Classical Mechanics. AddisonWesley. Hamel, G. (1904). Die Lagrange-Eulerschen Gleichungen der Mechanik. Z. f¨ ur Mathematik u. Physik, 50, 1–57. Hamel, G. (1949). Theoretische Mechanik. Springer. Konz, M. and Rudolph, J. (2015). Equations of Motion with Redundant Coordinates for Mechanical Systems on Manifolds. In Proc. 8th Vienna Int. Conf. on Mathematical Modelling, 681–682. Konz, M. and Rudolph, J. (2016). Beispiele f¨ ur einen direkten Zugang zu einer globalen, energiebasierten Modellbildung und Regelung von Starrk¨orpersystemen. at Automatisierungstechnik, 64(2), 96–109. Lee, J.M. (2003). Introduction to Smooth Manifolds. Springer. Lurie, A.I. (2002). Analytical Mechanics. Springer. Misner, C.W., Thorne, K.S., and Wheeler, J.A. (1973). Gravitation. W. H. Freeman and Company.
4.4 Gravity Assuming that the body is moving close to the surface of the Earth, the particles are subject to the gravitational force Fp = mp g, where g ∈ R3 are the coefficients of the gravitational acceleration w.r.t. the inertial frame. The resulting generalized force and potential energy are fiV = mp g, ∂i rp = ∂i mp g, rp . (48) p p V
In terms of the configuration coordinates x and with the previously defined parameters (43) this is R (mg) V , V = mg, r + Rs . (49) f = s R (mg)
4.5 Equation of motion
Evaluation of either of the eqs. (30) yields the kinetics ω 0 mI3 m mI3 m s v˙ s v + ω˙ ω v ω m s Θ m s Θ M
ξ˙
−ad ξ
fM
+
M
ξ
E R (mg) F = E . τ s R (mg)
fV
(50)
fE
Combining this with the kinematics (37) yields EOM for a rigid body subject the gravitation and the external forces f E = (F E , τ E ). These equations are commonly called the Newton-Euler equations. 6