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IMA Conference on Mathematics of 9 – 11 September 2015, St Anne’s College, University of Oxford

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Newton–Euler equations in general coordinates

By Bertold Bongardt and Frank Kirchner Robotics Innovation Center, DFKI GmbH, Bremen, Germany

Abstract

For the computation of , the –Euler equations represent a crucial relation unifying the laws of by Newton and Euler using the language of instantaneous screws. Typically, Newton–Euler equations are stated in spatial or in body coordinates, respectively. In this paper, a general formulation of Newton–Euler equations is provided for arbitrary reference systems. In particular, the general form uniﬁes the known equations in spatial and body coordinates. To the best of the authors’ knowledge, this relation between the spatial and the body form has not been reported in literature. The novel formulation is based on the concept of diﬀerentiation with respect to moving reference systems.

1. Introduction Newton’s law of motion states that the f which acts upon a translating , a zero-dimensional body, is proportional to the time of the particle’s linear h = m · v with m and v, all measured in an inertial sys- tem. In Euler’s formulation (Oliveira, 2007), this is expressed in the diﬀerential equation d d d f = dt (h) = dt (M · v) = M · dt (v) = M · a . (1.1) d Here, a = dt (v) denote the the vector and the mass M = m · I with identity matrix I = I 3 is introduced for sake of formal consistency with Euler’s second law of motion, below. Euler’s second law of motion states that the τ which acts upon a rotating body is proportional to the time derivative of the body’s

λ = M3 · ω with matrix M3 and pseudovector ω denoting the body’s , all measured in an inertial system. This is expressed in the diﬀerential equation d d d d τ = dt (λ) = dt (M3 · ω) = dt (M3) · ω + M3 · dt (ω) . (1.2) d d The is denoted by α = dt (ω). The derivative dt (M3) is given within the appendix. While the mass in Newton’s law is assumed to be a ‘universal’ quantity (Ardema, 2005), the inertia of the moving body changes with respect to the inertial system, so that the chain rule is applied in the diﬀerentiation of Euler’s law. In this paper, the Newton–Euler equations – that subsume Equations 1.1 and 1.2 by means of twists and wrenches – are formulated with respect to an arbitrary reference system. The remainder of the paper is structured as follows: necessary quantities and their notation are introduced in the next section. In Section 3, the generalized formulation of the Newton–Euler equations is given and interrelations to corresponding formulations in spatial and in body coordinates are discussed. In Section 4, a conclusion is drawn. Technical computations of time are compiled in Appendix A. IMA Conference on Mathematics of Robotics 9 – 11 September 2015, St Anne’s College, University of Oxford

Newton–Euler equations in general coordinates 2 2. Prerequisites 2.1. Poses and displacements

4×4 EP p A pose matrix P ∈ R has the form P = 0 1 with attitude EP ∈ SO(3) and 3 (t) p ∈ R . A is a sorted set of poses P with t > 0. A 4×4 R t matrix D ∈ R has the shape D = 0 1 with R ∈ SO(3) and translation 3 −1 t ∈ R . The matrix D = DPQ = P · Q describes the passive displacement between two Ad 6×6 poses P ∈ SE(3) and Q ∈ SE(3). The adjoint representation D = Ad(D) ∈ R of Ad  R 0  D has the shape D = t × RR .

2.2. Screws, twists, and wrenches 6 ν  Geometric screws. A screw ξ ∈ R is denoted in Pl¨ucker coordinates as ξ = µ sub- suming the direction ν and the µ. The matrix representation† of a screw ξ is ⊗ 0 −ν +ν ⊗ 4×4 ⊗ ν µ  3 2  ⊗ +ν 0 −ν denoted by ξ ∈ R and has the shape ξ = 0 0 with ν = 3 1 . The −ν2 +ν1 0 ad 6×6 adjoint representation of a screw ξ is denoted by ξ = ad(ξ) ∈ R and has the shape  ⊗  ad ν 0 ν  ~ 3 3 ξ = µ⊗ ν⊗ . For a screw ξ = µ , the screw vector ﬁeld ξ : R → R is evaluated ~  at any point P with coordinates p to the moment µP = ξ p given as ξ~ : p 7→ µ + ν × p . The vector ﬁeld ξ~ is characterized by the constitutive equation (Minguzzi, 2013) ξ~p − ξ~q = ν × (p − q) , (2.1) for arbitrary points P and Q with coordinates p and q. The interchange 4 ∈ 6×6 0 I R is deﬁned as 4 = I 0 according to Hunt, 2003, and McCarthy and Soh, 2010. Physical Screws. Twists and wrenches are geometric screws which are equipped with physical interpretations (and units) as screws from the force and and from the velocity domain, respectively. Both are denoted in ray-coordinates in accordance with (Phillips, 2007), (Tsai, 1999), (Mason, 2001), (Davidson et al., 2004), and (McCarthy and Soh, f  2010). A wrench is denoted by W = τ , subsuming force f and torque τ , and a twist ω  is denoted by V = v , subsuming angular and linear velocity, ω and v. ~ ω  For the vector ﬁeld V of a twist V = v , Equation 2.1 is casted into the ‘velocity- ~  ~  diﬀerence equation’‡ (Uicker et al., 2003), given, with vP =V p and vQ =V q , as

vP − vQ = ω × (p − q) . (2.2)

f  Analogously, the constitutive equation of a wrench W = τ , is the ‘torque-diﬀerence ~ ~  ~  equation’ of the vector ﬁeld W , given, with τP = W p and τQ = W q , as

τP − τQ = f × (p − q) . (2.3)

2.3. Attributing quantities For the remainder of the document, a notation is introduced to specify the reference system P and, if necessary, the evaluation point X of a quantity. The reference system

of a quantity • is denoted as [•]P according to Sathaye, 2011. In , the evaluation † The matrix representation is also called a ‘Ball vector’, see Uicker et al., 2013. ‡ The velocity-diﬀerence equation is also named the ‘fundamental formula of the rigid body’ in Minguzzi, 2013. IMA Conference on Mathematics of Robotics 9 – 11 September 2015, St Anne’s College, University of Oxford

2.4. Mass and inertia The coordinates of the center of mass C of a body B, speciﬁed by mass m and mass 1 R function ρ, are computed as c = m · V ρ(v) · v dV . The rotative inertia matrix C 3×3 [M3]C ∈ R is computed as Z  C  ⊗ 2 M3 = ρC v · −(v ) dV . C V † If the evaluation point coincides with the origin of the reference system, it can be omitted. IMA Conference on Mathematics of Robotics 9 – 11 September 2015, St Anne’s College, University of Oxford

Newton–Euler equations in general coordinates 4

Figure 1. Sketch of a moving body B, together with a moving reference system P and the (t) (t) inertial system O. For simpliﬁcation, the orientations EP and EE are assumed to be constant over time t. The absolute , vOB of body B and vOP of observer P , are indicate by red

and green arrows. The vPB is computed as the diﬀerence vPB = −vOP + vOB and is indicated by blue arrows.

X For an arbitrary evaluation point X, the rotative inertia matrix [M3]C is determined by the –Steiner theorem, also known as the parallel axis theorem (Selig, 2005). With respect to an arbitrary reference system E ∈ SO(3), the rotative inertia matrix X X X −1 [M3]E is computed as [M3]E = REC · [M3]C · REC . 6×6 3×3 The aﬃne inertia matrix M6 ∈ R subsumes the rotative inertia M3 ∈ R and C the translative mass M = m·I. At the center of mass C, the aﬃne matrix [M6]C has the C [M ]C 0  shape M  = 3 C . With respect to an arbitrary reference system P ∈ SE(3), 6 C 0 M P the spatial inertia matrix [M6]P is computed via the skew axis theorem (Selig, 2005)

 P Ad −T  C Ad −1 M = (D ) · M6 · (D ) , (2.9) 6 P PC C PC a generalized version of the Huygens–Steiner theorem. As in Section 2.3, reﬁned state- ments can be made in case of coinciding evaluation points or reference attitudes. In case I t R 0 that D only changes the evaluation point, D = ( 0 1 ), or the reference system D = ( 0 1 ), PC PP the formulation can be reﬁned by denoting D as D = [D]CC , or as D = [D]PC , respec- Ad −T tively. By means of the screw swap operator 4 the ﬁrst factor (DPC ) in Equation 2.9 Ad −T Ad is alternately expressed as (DPC ) = 4 · (DPC ) · 4. The fundamental identity for the aﬃne inertia matrices reads

 O Ad −T  P Ad −1 Ad −T  Q Ad −1 M6 = M6 = (P ) · M6 · (P ) = (Q ) · M6 · (Q ) . (2.10) O P Q As above, the identities can be adopted for pure (with coinciding evaluation points) and for pure translations (with coinciding reference attitudes).

2.5. Time derivatives with respect to a mobile reference The concept of time derivatives with respect to mobile reference systems is, for exam- ple, introduced by Featherstone, 2008. The time derivative of a moving point X with x  x (t) homogeneous coordinates 1 = 1 is determined with respect to a moving reference system P with matrix P = P (t) as  d x  = P −1 ·  d x  = P −1 · d  x   = P −1 · d P ·  x   dt 1 P dt 1 O dt 1 O dt 1 P = P −1 · d (P ) ·  x  + P · d  x   (2.11) dt 1 P dt 1 P = d  x  + V ⊗ ·  x  . dt 1 P P P 1 P

d x   d x  ⊗ x  In a physical interpretation, the three terms dt [ 1 ]P ,[ dt 1 ]P , and [V P · 1 ]P describe the relative velocity of X with respect to P , the absolute velocity of X with respect to O, and the absolute velocity of P with respect to O and evaluated at X; IMA Conference on Mathematics of Robotics 9 – 11 September 2015, St Anne’s College, University of Oxford

A GENERAL FORMULATION 5 all expressed in coordinates with respect to the reference system P . By introducing X ∼ d x  X X ∼ d x  X X ∼ ⊗ x  [vPX ]P = dt [ 1 ]P ,[vOX ]P = [vX ]P = [ dt 1 ]P , and [vOP ]P = [vP ]P = [V P · 1 ]P , Equation 2.11 is rephrased as  X  X  X  X  X  X vOX = vPX + vOP and vPX = vOX − vOP , (2.12) P P P P P P resembling the forms ‘absolute velocity = relative velocity + observer velocity’ and ‘relative velocity = absolute velocity − observer velocity’. In Figure 1, a sketch for an example setup is provided (where the point x of the body B is named b). Similar to Equation 2.4, a diﬀerential, fundamental identity can be stated as d x  =  d x  = I ·  d x  = P ·  d x  = Q ·  d x  . (2.13) dt 1 dt 1 O dt 1 O dt 1 P dt 1 Q d d d A term [ dt (•)]P describes the same ‘global’ derivative as dt (•) = dt [(•)]O , in coordinates d with respect to reference system P . In contrast, dt [•]P describes a ‘local’ derivative. The corresponding derivatives of poses, screws, and are provided in Appendix A.2.

3. A general formulation 3.1. Newton–Euler equations in general coordinates The Newton–Euler equations in general coordinates state that the wrench W which acts upon a moving body at a point X ﬁxed in the lamina of the non-inertial reference

system P is proportional to the time derivative of the body’s momentum 4·M6 ·V , both expressed with respect to the reference P . This is expressed by the diﬀerential equation

X X h i h d  i W = dt 4 · M6 · V . (3.1) P P The equation is stated in more detail as

X X X X h i d h i h adi h i W = dt 4 · M6 · V + V P · 4 · M6 · V , (3.2) P P P P by applying the transformations

X X h d  i −1 Ad d h i −1 Ad d  Ad  X  4 · M6 · V = (P ) · 4 · M6 · V = (P ) · P · 4 · M6 · V dt dt dt P P O

−1 Ad  d Ad  X Ad d  X  = (P ) · P · 4 · M6 · V + P · 4 · M6 · V dt P dt P

d  X −1 Ad ad Ad  X = 4 · M6 · V + (P ) · (V · P ) · 4 · M6 · V dt P P P d  X  adX  X = 4 · M6 · V + V · 4 · M6 · V , dt P P P P d Ad ad Ad using Equation 2.7, the chain rule, expansion, sorting, the identity dt P = V · P from Equation A 2, and Equation 2.8. 3.2. Evaluation Spatial coordinates. With respect to the static reference frame O, the second summand  X d  X of Equation 3.2 vanishes, so that Equation 3.2 simpliﬁes to W = 4 · M6 · V . P dt P Applying the chain rule, the equation is further evaluated to h iX h iX h ωiX h iX h ωiX W = 4 · M · d + d 4 · M · 6 dt v dt 6 v O O O O O (3.3) h iX h αiX   ωadT h iX h ωiX = 4 · M6 · + − · 4 · M6 · . O a O v O v O Here, the time derivative of the spatial inertia matrix, d M X = (− ω ad)T · M X + dt 6 O v 6 O M X · − ω ad from Equation A 3 in the appendix is used together with the identity 6 O v IMA Conference on Mathematics of Robotics 9 – 11 September 2015, St Anne’s College, University of Oxford

Newton–Euler equations in general coordinates 6 ω ad ω  v · v = 0. For the case that the evaluation point X coincides with the center of mass C, Equation 3.3 covers

h iC h iC  C  h iC  C C C  τ [M3]O 0 α [ω]O × [M3]O · [ω]O 0 4 · W = = 0 M · + 0 [ω]C × M · [v]C . O f O a O O O The equations are stated in this form, for example, by Featherstone, 2008. Body coordinates. With respect to a reference system B moving together with the body, the ﬁrst summand d 4 · M · V X of Equation 3.2 is simpliﬁed by using the time dt 6 P d  ω X  X d  ω X  X invariance of the body inertia, 4 · M6 · = 4 · M6 · = 4 · M6 · dt v B B dt v B B  α X . Thus, Equation 3.2 simpliﬁes to a B

X X X ad T X X X h i h i h αi h  ωB   i h i h ωi W = 4 · M6 · + − · 4 · M6 · . (3.4) B B a B vB B B v B For the case that the evaluation point X coincides with the center of mass C, Equation 3.4 covers h iC h iC  C  h iC  C C C  τ [M3]B 0 α [ω]B × [M3]B · [ω]B 0 4 · W = = 0 M · + 0 [ω]C × M · [v]C . B f B a B B B The Newton–Euler equations are stated in this form, for example, by Murray et al., 1994, and by Fossen, 2011. General coordinates. For the case of an arbitrarily moving reference frame P , the general Newton–Euler equations of Equation 3.2 read in components

X X ad X X h f i d h  ωi h ωP  i h  ωi = dt 4 · M6 · + · 4 · M6 · . (3.5) τ P v P vP P v P

X X The twist [V ]P in last factor [4 · M6 · V ]P is determined according to the identity in X Equation 2.7, the inertia matrix [M6]P according to the identity Equation 2.10. The ﬁrst summand d [4 · M · V ]X is expanded to 4 · ( d [M ]X · [V ]X + [M ]X · d V X ). The dt 6 P dt 6 P P 6 P dt P d X d X twist derivative dt [V ]P is determined via Equation A 4, the inertia derivative dt [M6]P is determined via Equation A 5. f  X d Similar to the physical interpretation of Equation 2.12, the terms [ τ ]P , dt [4 · M6 · ad ω  X ωP  ω  X v ]P , and [ vP · 4 · M6 · v ]P in Equation 3.5 can be characterized by the form ‘absolute wrench = relative wrench + observer wrench’. In the language of screws, the sum resembles the diﬀerence between real and apparent which are distinguished for non-inertial reference systems.

4. Conclusions This paper provides a general formulation of the Newton–Euler equations obtained by diﬀerentiation with respect to non-inertial reference systems and by using a suitable notation scheme. The general equations unify the known formulations in spatial and in body coordinates and clarify their interrelation. The derived connection can simplify communicating of rigid bodies. Additionally, the novel formulation of the Newton–Euler equations might prove applicable to questions arising in computational rigid body mechanics with regard to moving reference frames. In robotics, for example, mobile reference systems are of interest when the interaction of a robot’s end-eﬀector link with the robot’s environment is required to be described from the point of view of another moving link of the mechanical device.

Acknowledgments. The presented in this paper was funded by the Federal Ministry of Education and Research (BMBF) within the project Recupera (Grant 01-IM-14006A). IMA Conference on Mathematics of Robotics 9 – 11 September 2015, St Anne’s College, University of Oxford

Ad −1 d Ad Ad −1 ad Ad Ad −1 ad The ﬁrst equation is extended to (P ) · dt P = (P ) ·V ·P = (P ) ·V , T d T ⊗ T ⊗ a formula which generalizes the ‘rotational’ identity E · dt (E) = E ·ω ·E = (E ·ω) .

Twists. The time derivative of a twist V with respect to the inertial reference system O α  d ω  d is denoted by A = a = dt v = dt (V ).

Inertia matrices. The time derivative of spatial inertia matrix M6 with respect to the inertial reference system O is

d d  O  d O ad T  O  O ad (M6) = M6 = M6 = −(V ) · M6 − M6 · V . (A 3) dt dt O dt O O O The equation is obtained by the transformations (using Equation 2.10, the chain rule, and identities from Equation A 2)