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PREPRINT 1
On the Dynamics of Floating-base Robots:
Linking the Recursive Formulation to the
Reduced Euler-Lagrange Equations
Hrishik Mishra, Gianluca Garofalo, Alessandro M. Giordano, Christian Ott
Abstract —For rigid Floating-base Robotic Systems
(FRS), the unreduced equations of motion are often de-
scribed by the inertia-coupled dynamics of both, its shape
and the FRS-base. Matrix transformations of these equa-
tions revealed a block-diagonal inertia in an alternative
dynamics description in past works. However, the structure
of the transformed matrix of Coriolis/Centrifugal (CC) terms
was not examined, and is the primary subject of this paper.
Unlike past works, we adhere to the Lagrangian approach,
which leads to the inertia-decoupled dynamics of shape
and locked velocities. Firstly, we propose a novel two-
part structure of the CC matrix with self-evident velocity
dependencies. In particular, we prove that the first part de-
pends only on the shape velocity and exhibits the passivity
property. We also prove that the second part depends only
on the locked velocity and is skew-symmetric. Secondly, in
the off-diagonal partition of the latter CC matrix, we isolate
the commutative part. Thirdly, using the separation of the
velocity dependencies in the shape dynamics, we extract
the curvature term, which is a key quantity in locomotion
methods. Fourthly, through the ensuing simplifications, the
complete CC matrix is obtained using the closed-form com-
putation of its partitioned matrices. Finally, we demonstrate
the practical benefit of the proposed CC matrix structure for
FRS through controller design and locomotion analysis.
Index Terms—Manipulator dynamics, Mobile robots, Or-
bital robotics, Robot control, Robot motion.
I. INT ROD UC TIO N
Floating-base Robotic Systems (FRS) have emerged as a
key feature in domains such as On-Orbit Servicing (OOS)
[1], [2], humanoids [3], [4] and, aerial [5] and underwater
[6] robotics, see Fig. 1. The dynamics of a FRS are defined
by a system of Hamel’s equations [7], [8], which consist of
an Euler-Poincar´e equation and an Euler-Lagrange equation
corresponding to the motion of the FRS-base and the shape
(joints), respectively. Hamel’s equations are particularly at-
tractive due to the computational efficiency of its recursive
form [9, §9.4]. This dynamics description also possesses the
passivity (or skew-symmetric) property, which is common in
mechanical systems, and is useful for stability analyses in
motion control of FRS [2]. For an underactuated FRS, i.e.
without external forces, shape dynamics are obtained through a
Hrishik Mishra, Gianluca Garofalo, Alessandro M. Giordano and
Christian Ott are with the Institute of Robotics and Mechatron-
ics, German Aerospace Center (DLR), Wessling, Germany (e-mail:
hrishik.mishra@dlr.de)
g1k
g1k
g1k
g1
g1
g1
qk
qk
qk{1}
{1}
{1}
{O}
Fig. 1. Floating-base Robotic Systems (FRS) with configuration
(g1, q), where g1∈SE(3) is the pose of the FRS-base frame, {1},
relative to the inertial frame {O}, and q∈Rnare the n-joint positions.
Lagrangian reduction of the Hamel’s form, which yields the re-
duced (free-floating [1]) shape dynamics on a momentum map
level-set. This simplification is an outcome of the commonly-
known FRS property of momentum conservation. In geometric
mechanics, this property is modeled as a Pfaffian-like velocity
constraint through a quantity called the mechanical connection
(see [10, Ex. 4]). For a FRS, this qunatity is distinctly non-
flat (curvature) [7], i.e. the constraint is not preserved over a
closed path (gait) in shape-space. In other words, a gait causes
a net displacement of the FRS-base, see Fig. 2. In fact, a key
feature of locomotion approaches from geometric mechanics
is to approximate the displacement per gait with area integrals
of the curvature [11], [12], [13], [14].
Initial condition final condition
gait
q1
q2
t0tf
q1(0)
q1(0)
q2(0)
q2(0)
Fig. 2. Net displacement of the FRS-base due to a gait. Left: FRS initial
(t0) configuration; Center: gait with starting point (green) and direction
(arrow); Right: FRS final (tf) configuration.
For this class of mechanical systems, the Reduced Euler
Lagrange (REL) equations [15, §5.3][8, pp. 141] alternatively
describe the dynamics. In particular, the REL equations consist
2 PREPRINT
of locked and shape variations as an Euler-Poincar´e equation
and an Euler-Lagrange equation, respectively. The structure
of the REL equations provide a useful insight into the FRS
dynamics through a block-diagonal inertia, the separation of
velocity dependencies in the Coriolis/Centrifugal (CC) terms
and the apparentness of the curvature form. A first step in this
direction by the robotics community was a matrix transforma-
tion of the Hamel’s equations [4]. This transformation revealed
the useful block-diagonal inertia property. It is important to
acknowledge that although an explicit reference to the REL
equations was not made, the resulting dynamics [4] were its
spatial-momentum equivalent [16, eq. 8]. A similar result was
obtained in the domain of multi-agent systems [17].
However, a direct link between kinematic chain recursive
dynamics and REL equations has not been established before.
A negative consequence was that only the block-diagonal
structure of inertia was exploited [4], [18], whereas the struc-
tural properties of the CC matrix were not examined. In fact,
the use of matrix transformations resulted in a placeholder CC
matrix, which concealed its precise structure. This resulted in
a CC matrix, which (a) had a superficial coupling between
the shape dynamics and the group variable, (b) lacked a
commutative property (like fixed-base robots, see [19]), and
(c) did not reveal the computed curvature. We point out that
the curvature computation in FRS locomotion analyses [11],
[12], [13], [14] is restricted to symbolic methods, which limits
their applicability to simple FRS systems [7], like in Fig.2.
To this end, the contributions of this paper are the following.
For the REL equations of a FRS, firstly, we propose a CC
matrix, which is obtained through a novel closed-form com-
putation of its partitioned matrices with self-evident velocity
dependencies. In particular, the computation is decomposed
into two CC matrices, first of which depends on the shape
velocity, while the second depends on the locked velocity. We
prove that the former CC matrix of block-diagonal terms sat-
isfies the passivity property, while the second satisfies a skew-
symmetric property. Secondly, the off-diagonal partition of
the latter CC matrix is further separated into its commutative
and non-commutative parts. This property enables exchange
of velocity arguments in the computation of the quadratic
CC forces for simplification in specific cases, e.g. velocity
observer design, like in [19]. Thirdly, from the proposed CC
matrix, we derive a recursive closed-form expression of the
curvature. Finally, the merit of the proposed structure is shown
through the resulting novelty in two FRS applications, namely
a Lyapunov-based control method and locomotion analyses.
The paper is organized as follows. In Sec. II, a summary
of useful concepts on SE(3) motion is given. In Sec. III, the
FRS dynamics descriptions used in robotics and geometric
mechanics are unified, which leads to the REL equations.
The main result is stated and derived in Sec. IV. Herein, the
novel properties of the CC matrix are proved. In Sec. V, the
closed-form expression of curvature is derived. In Sec. VI,
applications that benefit from the proposed CC matrix are
provided. Concluding remarks are given in Sec. VII.
II. MO DEL IN G MOTIO N ON SE(3) GROU P
In this section, relevant details about motion on SE(3) group
are provided. The reader is referred to App. .1 for the matrix
descriptions of introduced quantities. The pose of a rigid body
is a matrix group representation of SE(3), which is written
as g= (R, p), where R∈SO(3) is the rotation matrix and
p∈R3is the position. The identity of the SE(3) group is I4,4,
where Ik,k is an identity matrix of dimension k. The tangent
space at I4,4is the se(3) algebra, which is referenced in body
and spatial frames. Analogously, the cotangent space at I4,4is
the dual space of wrenches, denoted as se(3)∗. The se(3) alge-
bra and its dual se(3)∗are isomorphic to the space of velocity
twists and wrenches on R6using (•)∧:R6→se(3),se(3)∗
and (•)∨:se(3),se(3)∗→R6, e.g. given a twist, V ∈ R6,
V∧∈se(3). A pose between two frames is subscripted with
symbols of both frames, e.g. in Fig. 1, the pose of {k}relative
to {1}is g1k. Poses and velocities that are subscripted once
are referenced relative to the inertial frame, {O}, e.g. (g1, V1)
are the pose and the velocity of the FRS-base, {1}, relative
to {O}. The Adjoint action, Ad : se(3) →s e(3), of a pose g,
transforms elements of se(3) algebra between spatial and body
frames as Vs= AdgV, see [20]. In the following text, for
simplicity of notation, we indicate the pose argument in Ad by
its ordered frames. e.g. Adg1k= Ad1k. The adjoint map of the
se(3) algebra onto itself is ad :se(3) →se(3). This is denoted
as adVand its coadjoint map is ad⊤
V:se(3)∗→se(3)∗.
In the following, given A∈Rn×n,A≻0denotes positive-
definiteness. 0k,l is a k×lmatrix of zeros, and for l= 1,
index is omitted. Given X∈Rl, Y ∈Rm, and Z∈Rl×m,
hX, Y iZ=X⊤Z Y .
Kinematics/Dynamics of a Rigid body: Given a rigid body
(e.g. kth link in Fig. 1) with pose gk∈SE(3), the SE(3)
reconstruction formula is,
Kinematicsn˙gk=gkV∧
k, gk≡(Rk, pk)(1)
where V∧
k∈se(3) is the body-referenced Lie algebra. The
Euler-Poincar´
eequation is written using se(3) ∼
=R6as,
DynamicsnMk˙
Vk−ad⊤
VkMkVk=Fk,(2)
where Fk∈R6∼
=se(3)∗is the body wrench, Mk∈R6×6
is the inertia matrix (see App. .1) depending on mk>0
and Ik∈R3×3≻0, which denote the body’s mass and in-
ertia tensor respectively. The operator ad⊤
VkMkin (2) en-
capsulates SE(3) structural coefficients and is a key ma-
trix that will also appear later in the context of FRS
dynamics. Firstly, note that it is not skew-symmetric.
This property is useful for Lyapunov-based analysis in
robotics. Secondly, given V, W ∈R6∼
=se(3), the bivariate
map (ad⊤
VMk)Wdoes not exhibit a commutative prop-
erty, i.e. (ad⊤
VMk)W6= (ad⊤
WMk)V, which is due to the
non-abelian nature of SE(3). To this end, the operator,
ad∼
•:se(3) →se(3)∗, is alternatively defined [21] such that,
(ad⊤
VMk)W= (ad∼
MkW)V, (see App. .1) (3)
Therefore, (3) provides a bilinear map using the ad∼
(•)
operator, which is skew-symmetric and also serves to exchange
velocity arguments for purposes of analysis.
H. MISHRA et al.: ON THE DYNAMICS OF FLOATING-BASE ROBOTS 3
III. DYN AM IC S OF T HE FRS
In this section, we state the FRS dynamics descriptions,
which are prevalent in both, robotics [4], [22] and geometric
mechanics [8], [15], [16]. In the text, the shape dependencies
(q) and velocity-dependencies of the dynamic quantities are
provided in declaration and may be omitted later for brevity.
Def. 1: A FRS is a multibody system of n+ 1 rigid
links (see Fig. 1), which comprises of the FRS-base and n
holonomic joints. The configuration space of the FRS, as seen
in Fig. 1, is Q ≡ SE(3) ×Rnwith coordinates r= (g1, q)∈
Q, where g1∈SE(3) is the pose of the FRS-base and q∈Rn
denotes the nshape positions of the robotic manipulator.
A. Hamel’s equations
Considering potential forces (e.g. gravity) as external, the
presence of the group coordinate g1yields a reduced La-
grangian [15, §5] with inertia tensor, M∈R(6+n)×(6+n), as
l(q, V ) = 1
2hV, V iM(q), where V=V⊤
1˙q⊤⊤is the con-
figuration space velocity, V∧
1=g−1
1˙g1∈se(3) for the FRS-
base pose. The dynamics that result from lare the Hamel’s
equations [7, eq. 4,5], and are written in vector notation as,
Mb(q)Mbq(q)
Mbq(q)⊤Mq(q)
|{z }
M(q)
˙
V1
¨q+C(q, V )V1
˙q=F1
τ
|{z}
F
(4)
where Mb, Mbq , Mqare the locked, coupling and manipulator
inertias, respectively, C∈R(6+n)×(6+n)is the CC matrix, and
F1∈R6∼
=se(3)∗and τ∈Rnare the forces acting on the
FRS-base and joints, respectively. The main advantage of (4)
is its computation using recursive methods [3], [23].
A brief outline of recursive computations of the matrices in
(4) is given below. Note that the link velocity, Vk=Tk(q)V,
Tk=Ad−1
1k(q)Jk(q), with a link Jacobian Jkrelative to
the FRS-base. The recursive expressions of the matrices can
be obtained using Tk, e.g. as shown by [3]. In particular,
M=X
k
T⊤
kMkTk,(5)
C=X
k
T⊤
k−ad⊤
VkMkTk+Mk˙
Tk(6)
The partitioning in Tkcan be used to further add detail to the
dynamic matrices, e.g. in M=MbMbq
M⊤
bq Mq, we get,
Mb=X
k
Ad−⊤
1kMkAd−1
1k, Mbq =X
k
Ad−⊤
1kMkJk
Mq=X
k
J⊤
kMkJk,(7)
B. REL equations: The motivation
The dynamic description in Sec. III-A and its properties
only describe the FRS as a kinematic chain. However, the
FRS also belongs to the class of mechanical systems, which
are uniquely characterized by a conservation property of the
momentum map (defined below).
Def. 2: The momentum map (or generalized momentum)
[7, App.] is a mapping, J:TQ → se(3)∗, which can be
defined as J= Ad−⊤
1MbV1+Mbq ˙q, and physically rep-
resents the total spatial momentum of the FRS.
Specifically, the invariance of l(q, V )to g1, implies a SE(3)
symmetry in the FRS, which leads to a conservation law on
Jaccording to Noether’s theorem [15, §4.1]. An alternative
dynamics description, which explicitly shows this property,
is the Reduced Euler-Lagrange (REL) equations [15, §5.3].
In prior works, [4, eq. 15], the Routhian quantity (J,˙q)[8]
was used to describe the REL equations, which were obtained
using matrix transformations. The reported benefits were a
block-diagonal inertia and the vanishing of the CC terms
in the momentum equation. Although, it was not reported,
we note that the equations proposed in [4] were the spatial
form of the REL equations [16]. The main problem with
this representation, however, is that it introduces an apparent
coupling between g1and the shape dynamics [16, eq. 8]. Thus,
in this paper, we develop a body formulation of the REL
equations, i.e. using body velocity quantities. To this end, we
first define the locked velocity for the FRS as follows.
Def. 3: Locked velocity is a configuration-dependent ve-
locity shift [8, pp. 141] of the FRS-base velocity, V1, as
µ=V1+Al(q) ˙q, where Al=M−1
bMbq is the dynamic-
coupling factor [1].
It has the physical interpretation of being the velocity of
the instantaneous equivalent rigid FRS (locked shape) and is
the body velocity corresponding to the momentum map, i.e.
µ=M−1
bAd⊤
1J. It is a non-integrable quantity.
Using µ, a new system velocity is defined as
ξ=µ⊤˙q⊤⊤∈R6+n, which is related to the configuration
velocity, V, through a linear transformation as V=L(q)ξ,
where L=I6,6−Al
0n,6In,n . By applying the method of
dynamic congruent transformations from [4], the REL
equations are obtained as,
Mb(q) 06,n
0n,6Λq(q)
|{z }
Λ(q)
˙
ξ+Γb(q, V ) Γbq (q, V )
−Γbq(q, V )⊤Γq(q, V )
|{z }
Γ(q,V )
ξ=F(8)
where Λ,Γ∈R(6+n)×(6+n)are the transformed matrices of
inertia and CC terms, and F=F⊤
1(τ− A⊤
lF1)⊤⊤is the
transformed covector of all forces acting on the FRS.
In (8), Γis a placeholder CC matrix, which is obtained
through a transformation and does not provide a special struc-
ture. In particular, the velocity dependencies are not clearly
separated. This limits specific applications that require the par-
titioned CC terms, preferably with closed-form computation in
terms of individual velocity dependencies, i.e. µ, ˙q. Moreover,
while prior works [4] exploited the block-diagonal inertia of
the REL equations, the CC matrix has not been investigated.
It is worth appreciating that the symbolic structure of the
REL equations has been analytically revealed by the geometric
mechanics community [8], [15], [16]. In this structure, notable
key aspects of the CC terms were velocity dependency sepa-
ration and the apparentness of the curvature term, which will
be summarized next.
4 PREPRINT
Geometric Mechanics: In geometric mechanics, the conser-
vation of Jis written as a Pfaffian-like velocity constraint.
To describe the constraint, a geometric quantity called the
mechanical connection is defined as below.
Def. 4: [10, §3.2] [24, def. 3.1] The mechanical connection
is a map A:TQ → se(3) which physically quantifies the
generalized velocity corresponding to J, and is written as
A= (Ms
b)−1J, where Ms
b= Ad⊤
g−1
1MbAdg−1
1is simply the
spatial locked inertia [15, eq. 5.3.1]. Using Def. 2,
(Ad−⊤
1ΛbAd−1
1)−1J= Ad1I6,6Al(q)V1
˙q
|{z }
A(q,V )
(9)
and Al=M−1
bMbq is alternatively called the local mechanical
connection.
The Pfaffian-like constraint for the FRS is obtained
by setting, A= 06, which results in the zero momen-
tum (J= 06) velocity, i.e. Vh=(−Al˙q)⊤˙q⊤⊤. This
is the horizontal velocity and defines the zero momentum
shape Lagrangian, ˆ
l˙q=1
2hVh, VhiM=1
2h˙q, ˙qiΛq. Its orthog-
onal complement, i.e. the vertical velocity, is obtained as
Vv=V−Vh=µ⊤0⊤
n⊤. This defines the non-zero mo-
mentum Lagrangian as ˆ
lµ=1
2hVv, VviM=1
2hµ, µiMb.
Lemma 1: For the unforced FRS, a decoupled reduced
Lagrangian, ˆ
l(q, ξ) = ˆ
l˙q+ˆ
lµ, resuts in the REL equations [15,
§5.3], which, in matrix-based notation [16], [25] is,
Mb˙µ+dMb
dt µ=ad⊤
µMbµ−ad⊤
Al˙qMbµ(10)
Λq¨q+dΛq
dt ˙q−∂
2∂q h˙q, ˙qiΛq=˜
N(q, µ, ˙q)(11)
˜
N=−DAl( ˙q)⊤Mbµ+∂hµ, µiMb
2∂q − A⊤
lad⊤
µMbµ(12)
where DAl(q, ˙q)•=dAl( ˙q)• −[(Al˙q)∧,(Al•)∧]is the local
curvature1,2 of the mechanical connection. The first term in
DAlmeasures the intrinsic change in Al˙qacross the shape-
space; and the second measures the extrinsic change in it as
the space of allowable velocities rotates with the body frame,
{1}, due to the non-commutativity of SE(3) [12].
Proof: The Euler-Poincare part is (10), and is derived
in App. .2. In (11), the L.H.S is the standard Euler-Lagrange
part of the shape dynamics, while ˜
Ncollects the µ-dependent
terms. Its expanded form is proved in App. .3.
Accordingly with new system velocity, ξ, the FRS-base
pose, g1, is reconstructed alternatively as,
˙g1=g1µ− Al(q) ˙q∧(13)
We stated the unforced case of the full FRS dynamics in
(10) and (11) to highlight the following aspects. Firstly, a
key observation is that (10) and (11) correspond to the body
momentum and the shape (reduced) dynamics, respectively.
1The curvature is a measure of the net displacement of the FRS-base over
a gait. A formal treatment of curvature is provided later in Sec. VI.
2The sign convention of curvature lacks consistency in literature. In [15,
§5.3], [16], [25], the curvature appears on the R.H.S of the shape dynamics
with a negative sign and we adopt this convention.
Secondly, the CC terms have an elegant separation of velocity-
dependent CC couplings in terms of ( ˙q, ˙q)on L.H.S and
( ˙q, µ),(µ, µ)on R.H.S. Thirdly, note that by exploiting the
apparentness of the curvature, it can be extracted from (12) of
the REL equations. Fourthly, (10) and (11) do not provide a
skew-symmetric property for the full FRS dynamics. Hence, in
this paper, we start by simplifying the recursive expressions of
forced dynamics from previous works in [3], [4] and separate
the CC couplings as above. Instead of the exact form of the
REL equations, we derive a reformulation in the next section,
which provides a skew-symmetric property.
IV. PRO PO SED FO RM OF THE RE DUC ED
EUL ER -LAGR AN GE (REL) EQUATI ONS
In this section, we state and prove the main result of this
paper. Firstly, we introduce fundamental matrices [26], i.e.
matrices that are used to write the CC terms, which appear
in the Lagrangian form of (10) and (11), as linear operations.
A key characteristic of these matrices is that they are directly
related to the locked inertia, Mb.
Def. 5: Locked inertia velocity (LIV) matrix: A symmetric
matrix, P(q, x), given an arbitrary shape velocity, x∈Rn,
which is interpreted as P( ˙q) = dMb
dt when x= ˙q(see (10)).
Def. 6: Locked inertia derivative (LID) matrix: A matrix,
S(q, x)⊤, given an arbitrary locked velocity x∈R6∼
=se(3),
which arises in the partial derivative of the locked kinetic
energy relative to shape in (11), and can be written as
∂hµ,µiMb
∂q =S(q, µ)⊤µ, when x=µ.
In both, (10) and (11), a common matrix is of the form
ad⊤
Al˙qMb. In the spirit of our arguments in Sec. II, we use (3)
as ad⊤
Al˙qMbµ=ad∼
MbµAl˙q, to define the following matrix.
Def. 7: Interaction (IM) matrix: A matrix ad∼
MbyAl, given
an arbitrary locked velocity y∈R6∼
=se(3), which captures
the product of the structure coefficients of the local mechanical
connection,Al, and the SE(3) group.
Theorem 1: Main result: For the FRS in Def. 1, the pro-
posed form of Reduced Euler-Lagrange equations are written
using matrix-based recursive forms as,
Mb(q) 06,n
0n,6Λq(q)
|{z }
Λ(q)
˙µ
¨q+1
2P( ˙q) 06,n
0n,6˜
Γ′
q( ˙q)
|{z }
D˙q(q,ξ)
µ
˙q
|{z}
ξ
=ad∼
Mbµ−1
2S(µ)−ad∼
MbµAl
1
2S(µ)⊤− A⊤
lad∼
Mbµ−˜
B(q, µ)
|{z }
Dµ(q,µ)
µ
˙q
+F1
τ− A⊤
lF1
|{z }
F
(14)
where Λis the block-diagonal inertia tensor, and D˙q,Dµare
the two proposed CC matrices which depend only on ˙qand
µ, respectively. The CC matrices are written in terms of the
fundamental matrices defined in Definitions 5-7. Additionally,
˜
Bis directly related to the curvature of the mechanical
connection and ˜
Γ′is the CC matrix that results in the ( ˙q, ˙q)-
coupling in the shape dynamics. Fcontains external forces
acting on the FRS.
H. MISHRA et al.: ON THE DYNAMICS OF FLOATING-BASE ROBOTS 5
In particular,
•Λq(q) = X
k
J⊤
kMkJk− A⊤
lMbAl(15a)
•P( ˙q) = −X
k
Ad−⊤
1kad⊤
Jk˙qMk+MkadJk˙qAd−1
1k(15b)
•S(µ)⊤=X
k=b
J⊤
kad⊤
Ad−1
1kµMk+ad∼
MkAd−1
1kµAd−1
1k(15c)
•˜
Γ′
q( ˙q) = X
k
˜
J⊤
kad∼
Mk˜
Jk˙q˜
Jk+Mk˙
˜
Jk,(15d)
•˜
B(µ) = −A⊤
lad∼
MbµAl+S(µ)TAl− A⊤
lS(µ)
−X
k
(J⊤
kMk∇(Ad−1
1kµ)Jk)(15e)
where, given a velocity X∈R6∼
=se(3),
∇X=M−1
k(ad⊤
XMk+ad∼
MkX−MkadX)
The closed-form expressions for all matrices in (14) are
provided in the next two subsections.
Proof: We recall from Sec. III-B that, instead of using
(q, V ), we use (q , ξ)to define the dynamics. A consequence
of this is that, in contrast to Sec. III-A, the link velocity is
determined as, Vk=˜
Tk(q)ξ, such that,
˜
Tk(q) = Ad−1
1k(q)˜
Jk(q)(16)
where, ˜
Jk=Jk−Ad−1
1kAlis the generalized Jacobian [1]
for the kth link. For the proof of Theorem 1, firstly, we obtain
Λ,Γin (8) alternatively3by replacing Tk(q)with ˜
Tk(q)in the
recursive computations given in (5) and (6). This computation
provides a block-diagonal inertia as in (8). To determine the
forces on the R.H.S of (14), using V=Lξ in the power
equality we get,
ξ⊤F=V⊤F⇒ F =L⊤F(17)
which yields the forces on R.H.S of (14). These observations
have previously been made and are not new [4].
The main contribution of this paper starts with the closed-
form expressions for the matrices of inertia velocity, P( ˙q),
and inertia derivative, S(µ), in part 1. In part 2, we separate
the velocity dependencies in the recursive expressions and in
part 3, we prove and reveal the form in the main result in (14).
Throughout this section, the following identities will be used
for simplifying the expressions.
X
k
Ad−⊤
1kMk˜
Jk= 06,n (18a)
X
k
Ad−⊤
1kMk˙
˜
Jk=X
k
Ad−⊤
1kad⊤
Jk˙qMk˜
Jk(18b)
X
k
Ad−⊤
1kad∼
Mk˜
Jk˙qAd−1
1k= 06,n (18c)
X
k
Ad−⊤
1kad⊤
Ad−1
1kµMk˜
Jk= 06,n (18d)
X
k
Ad−⊤
1kad⊤
Jk˙q−ad⊤
˜
Jk˙qMk˜
Jk= 06,n (18e)
3In contrast to the approach in [4, eq. 15], which used matrix transfor-
mations, a recursive method is used here to ease the proof of Theorem 1.
To prove (18a), in the recursion of (7), the expression of
˜
Jkis used instead of Jk, while all the other identities are
a direct consequence of (3), (18a) and, (65a) and (65b) from
the Prop. 4 in the Appendix.
Proof part 1: (LIV/LID matrices): For the matrices in
Def. 5 and Def. 6, the recursion for Mbin (7) is used. The
closed-form expression of the matrix in Def. 5, P(q, ˙q), is
obtained by computing d
dt Mb(q), and using the time-derivative
of the Ad operator as,
P( ˙q) = −X
k
Ad−⊤
1kad⊤
Jk˙qMk+MkadJk˙qAd−1
1k(19)
For the derivation of the closed-form expression of the LID
matrix in Def. 6, the result is presented as the following
corollary of Lemma 4, which is given in the Appendix.
Corollary 1: A corollary of Lemma 4 is that, given x∈R6,
∂
∂q hµ, µiMb=S(µ)⊤µ, S(µ)⊤= 2 X
k=b
Πk(x)⊤(20)
where the closed-form of Πk(µ)⊤=J⊤
kad⊤
Ad−1
1kµMkAd−1
1kis
derived in (72) of Lemma 4 (Appendix).
Proof: Using the result in Lemma 4 and invoking the
velocity exchange property in (3), the result follows.
Proof part 2: (Separation of velocity dependencies): For
this part, the partitions of the CC matrix, Γ, in (??) are
expanded. The main idea of this proof is to start from these
expressions and separate the terms according to their velocity
dependencies for each of the four blocks in (??). In fact, the
key feature of the structure in (14) is the isolation of different
terms according to the dependency on shape ( ˙q) and locked
(µ) velocities. This step will reveal the fundamental matrices
defined above. For the CC matrix, Γ, using (18b) yields
Γb=X
k
Ad−⊤
1k−ad⊤
VkMk−MkadJk˙qAd−1
1k(21a)
Γbq =X
k
Ad−⊤
1k−ad⊤
VkMk˜
Jk+Mk˙
˜
Jk
=X
k
Ad−⊤
1k−ad⊤
VkMk+ad⊤
Jk˙qMk˜
Jk
(21b)
Γqb =X
k
˜
J⊤
k−ad⊤
VkMk−MkadJk˙qAd−1
1k(21c)
Γq=X
k
˜
J⊤
k−ad⊤
VkMk˜
Jk+Mk˙
˜
Jk.(21d)
To derive the form in (14), Γmust also be factorized into
( ˙q, ˙q),( ˙q , µ)and (µ, µ)couplings. The key idea, is to use (16)
to write Vk=Ad−1
1kµ+˜
Jk˙qand split ad⊤
VkMkas the sum of
contributions depending on ˙qand µ, for example,
ad⊤
VkMk= (ad⊤
Ad−1
1kµ+ad⊤
˜
Jk˙q)Mk(22)
Firstly, we simplify Γbby splitting ad⊤
VkMkand using the
expansion for P( ˙q)from (19) as,
Γb=X
k
Ad−⊤
1k−ad⊤
(Ad−1
1kµ−Al˙q)+ad⊤
Jk˙qMk
−MkadJk˙qAd−1
1k
=−ad⊤
µMb+ad⊤
Al˙qMb+P( ˙q)
(23)
6 PREPRINT
where Prop. 5 (Appendix) is used to eliminate the summation.
Note that the last matrix in (23) is the LIV matrix in Def. 5.
Next, for Γbq , we also use Prop. 5 (Appendix) after expand-
ing ˜
Jkand using identity (18e), to get
Γbq =X
k
Ad−⊤
1kad⊤
(−Ad−1
1kµ−˜
Jk˙q+Jk˙q)Mk˜
Jk
=−X
k
Ad−⊤
1kad⊤
Ad−1
1kµ+ad⊤
(˜
Jk˙q−Jk˙q)Mk˜
Jk
=6,n
(24)
where (18d) and (18e) are straightforwardly applied.
Remark 1: Observe that the simplifications in (23) and (24)
in top row of (8) results in the momentum equation of (11).
Next, for Γqb, using (22) and (16) to expand the terms in
(21c), we get,
Γqb =X
k
˜
J⊤
k−ad⊤
(Ad−1
1kµ+˜
Jk˙q)Mk−MkadJk˙qAd−1
1k
=X
k−J⊤
kad⊤
Ad−1
1kµ+A⊤
lAd−⊤
1kad⊤
Ad−1
1kµ
−˜
J⊤
kad ˜
Jk˙qMk−˜
J⊤
kMkadJk˙qAd−1
1k
(25)
Upon expanding, followed by invoking Prop. 5 (Appendix) for
the second term, the matrix A⊤
lad⊤
µMbis obtained. Therefore,
(25) is rewritten as,
Γqb =A⊤
lad⊤
µMb−X
k
J⊤
kad⊤
Ad−1
1kµMkAd−1
1k
|{z }
˜
S(q,µ)⊤
−X
k˜
J⊤
kad⊤
˜
Jk˙qMk+˜
J⊤
kMkadJk˙qAd−1
1k
|{z }
B1(q, ˙q)
(26)
The last term in Γto consider is Γq, which is expanded
using (22) as,
Γq=−X
k
˜
J⊤
kad⊤
Ad−1
1kµMk˜
Jk
|{z }
B2(q,µ)
+X
k
˜
J⊤
k−ad⊤
˜
Jk˙qMk˜
Jk+Mk˙
˜
Jk
|{z }
Γ′
q(q, ˙q)
,(27)
This concludes the simplification obtained through separa-
tion of dependencies in the recursive formulation.
Proof part 3: (Matching the proposed REL equations): In
L.H.S of (14), we see that the Dµmatrix only has µdepen-
dencies whereas Γqb in Γhas ˙qdependencies too. Also, note
that ˙
Λq6= Γ′
q+ Γ′⊤
q. So, to simplify the structure further, we
invoke the the following identities,
B1( ˙q)µ=X
k
˜
J⊤
kad∼
MkAd−1
1kµ˜
Jk−MkadAd−1
1kµJk˙q
=B3(q, µ) ˙q
(28)
Γ′
q( ˙q) ˙q=X
k
˜
J⊤
k−ad∼
Mk˜
Jk˙q˜
Jk+Mk˙
˜
Jk˙q
=˜
Γ′
q(q, ˙q) ˙q
(29)
˜
S(µ)⊤µ=1
2X
k
(Πk+˜
Πk)µ=1
2S(µ)⊤µ(30)
These three equations follow straight-forwardly by using prop-
erty (3) and adXY=−adYXfor rearrangement.
By using the identities in (28), (29) and (30), we get,
Γqbµ+ Γq˙q=−1
2S(µ)⊤µ+A⊤
lad⊤
Al˙qMbµ
+˜
B(q, µ) ˙q+˜
Γ′
q( ˙q) ˙q
(31)
where ˜
B=−(B2+B3). Expanding ˜
B, using ˜
Jk, followed by
invocation of (18d) to cancel terms results in (15e).
Remark 2: With the above observations, note that (31)
provides the velocity dependency separation which match the
shape dynamics in (11). Also, ˜
Γ′
q( ˙q) ˙q=dΛq( ˙q)
dt −∂h˙q, ˙qiΛq
2∂q ˙q.
From remarks 1 and 2, the recursive expressions for the
REL equations in (10) and (11) have therefore been derived.
However, these two equations do not provide a system-wide
skew-symmetric property, which is ubiquitous in robotics. To
that end, we propose the following commutative property
which leads to a reformulation of the REL equations in (14).
Property 1: Given locked velocities, x, y ∈R6, and shape
velocity, z∈Rn, the following commutativity properties for
the fundamental matrices from Def. 5 and Def. 6 hold.
S(q, x)⊤y=S(q, y )⊤x, P (q, z )y=S(q, y)z(32)
which extends the result for generalized coordinates in [26,
Prop. 5,6] to the locked-shape velocity formulation here.
Proof: For the first, the expansion of S⊤in Corollary 1
is used followed by property (3). For the second, also using
property (3) and adXY=−adYXleads to the result.
The Prop. 1 leads to the following corollary, which is used
to obtain the result in Theorem 1.
Corollary 2: Given locked velocity µ∈R6∼
=se(3)∗and
shape velocity ˙q∈Rn,P( ˙q)µ=1
2P( ˙q)µ+1
2S(µ) ˙q.
Using Corollary 2 on (23), followed by using property (3)
for both ad⊤terms, we obtain the first row in (14). Using
the same property for the A⊤
lad⊤
µMbµterm in (31) provides
the second row in (14). Rearranging as ( ˙q, ˙q)on L.H.S and
all ( ˙q, µ),(µ, µ)on R.H.S, we obtain the proposed D˙q,Dµ
matrices in (14).
With Theorem 1, a novel formulation of the CC matrix
as two CC matrices, namely D˙q,Dµwas obtained. We point
out, that the Γmatrix in (8) is analogous to the result in [4,
eq. 18]. However, the block matrices therein were functionally
dependent on Vand not the chosen velocity corresponding to
the decoupling transformation. In contrast, by further simplifi-
cation of Γin this paper, block matrices in (14) are explicitly
H. MISHRA et al.: ON THE DYNAMICS OF FLOATING-BASE ROBOTS 7
defined in terms of the system velocity ξ. For applications, the
following useful properties are established.
Property 2: Given x∈R6, y ∈Rn, and z=x⊤y⊤⊤,
the skew-symmetric property, z⊤(˙
Λ−2Γ)z= 0, can be
viewed in (14), as satisfying the following,
x⊤(dMb
dt −P( ˙q))x= 0, y⊤(dΛq
dt −2˜
Γ′
q)y= 0
xTad∼
Mbµx= 0, y⊤˜
B(q, µ)y= 0, z⊤Dµz= 0
(33)
Proof: The first follows straightforwardly. The second
follows by using the corresponding matrix expansions in (15a)
and (15d). For the third, fourth and fifth, ad∼
Mbµ,˜
Bin (15e)
and Dµin (14), respectively, are skew-symmetric. Note that,
in (14), the the CC matrix of block-diagonal terms, D˙q,
satisfying the passivity property depends only on ˙qwhile the
CC matrix of off-diagonal terms, Dµsatisfying the skew-
symmetric property depends only on µ.
In many FRS applications, the momentum dynamics ((14),
top row) are preferred in an inertial frame located at the CoM,
called {C}. The pose of this frame, gc∈SE(3), is simply a
right SE(3) translation of g1, i.e gc=g1g1c. Correspondingly,
this results in a change of basis for µ, which is an Adjoint
transformation of the SE(3) element, g1c, i.e. µc= Ad−1
1cµ.
For such a case, the following property is useful.
Property 3: Given gc∈SE(3), the right SE(3) translation
corresponds to the following.
1) Dynamic transformations:
•¯
Λb= Ad⊤
1cMbAd1c•¯
P= Ad⊤
1cPAd1c
•ad∼
¯
Λbµc= Ad⊤
1cad∼
MbµAd1c•¯
Al= Ad−1
1cAl
•¯
S= Ad⊤
1cS(Ad1cµc)•˜
B(µc)≡˜
B(Ad1cµc)
•ad∼
¯
Λbµc
¯
Al= Ad⊤
1cad∼
ΛbµAl•¯
Fb= Ad⊤
1cFb
2) The shape matrices Λq,˜
Γ′remain invariant.
Proof: The proof follows from using equivariant trans-
formations and is summarized in Lemma 5 (Appendix).
Thus, given (q, ˙q, g1c, µc), the L.H.S entries of Prop. 3, can
be used to rewrite (14) in the {C}frame. Two key observations
are made here as a corollary.
Corollary 3: The skew-symmetric Prop. 2 is preserved after
the SE(3) translation in Prop. 3. The shape-dynamics (bottom
row (14)) is invariant4to the right SE(3) translation.
Proof: The first can be follows straightforwardly from
Prop. 3. For the second, note item 2 in Prop. 3 and also that,
S(Ad⊤
1cµc) Ad1cµc=S(µ)⊤µ, ˜
B(Ad1cµc) = ˜
B(µ)
¯
A⊤
lad∼
¯
Λbµcµc=A⊤
lad∼
Λbµµ(34)
V. CU RVATURE IN THE FRS
In this section, we introduce the concept of the curvature
of the mechanical connection, and use (14) to provide a novel
closed-form expression for it. Firstly, we observe that a FRS
is uniquely characterized by a mechanical connection, which
is not preserved over closed-loops (gait) in shape-space. In
4Representation in prior works [4] prevented revealing this invariance.
other words, the FRS-base undergoes a net displacement due
to the gait, which modifies the basis for representing the local
mechanical connection, see Fig. 2. This phenomenon is called
the falling-cat problem [8].
Let us assume J= 06⇒µ= 06for simplicity of exposi-
tion, which reduces (13) to
˙g1=g1(−Al˙q)∧(35)
with initial condition is g1(0) = I4,4. Under an Abelian group
assumption for g1(it is not), the solution for (35) [13] is,
g1(1) = exp Ztf
t0−Al˙qdt= exp Z∂C−Aldq(36)
where exp(X)≡exp(X∧), X ∈R6is the SE(3) exponential
[20]. Note that the time-integral in first is replaced for a path-
integral over a path ∂C. For a closed path, i.e. gait, Stokes’
theorem [13] is applied to convert the path integral in (36) to
an area integral over the area Cover shape-space Rnas,
g1(1) = exp −Z ZC
DAl( ˙q)dA(37)
where, DAl(X)Y=dAl(X)Y−[AlX, AlY], X, Y ∈Rnis
a map DAl:Rn×Rn→se(3) and is the curvature of the
mechanical connection, i.e. the covariant derivative of the
local connection form, −AlXalong shape trajectories given
by Y. The curvature is akin to the curl of the vector field
−AlX[11, §5.2]. The integral of dAl(q, X )Yin (37) is
referred to as the nonconservative contribution and captures
the change in mechanical connection due to change in shape
positions. Likewise, the integral of the Lie bracket term
−[(AlX)∧,(AlY)∧]is the the primary non-commutative con-
tribution and captures the change in mechanical connection
due to non-commutativity of SE(3) [12].
Since SE(3) is non-Abelian, (37) cannot be directly used.
In fact, there is no exact solution for (35). However, in [12],
the corrected Body Velocity Integral (cBVI), i.e.
ζ=−Ztf
t0Al(q) ˙qdt =−Z ZC
DAl(q)dA (38)
was shown as an accurate approximation in an optimally
computed minimum perturbation coordinates.
Plotting the integrand DAlcomponent-wise over the gait’s
domain produces the Constraint Curvature Function (CCF)
surfaces, see Fig. 3. By computing the CCF surface volume
under the gait area, an approximate ¯g1= exp(ζ)per gait cycle
was obtained in [12]. We elaborate the main idea for a 2-joint
FRS (see Fig. 2). In Fig. 3, the CCF for the kth component5,
(DAl)kis plotted on the left as a surface with an overlay
of a circular gait (red). The gait is defined by basis vectors
q1, q2∈R2. On the right of Fig. 3, the volumetric mesh of the
CCF surface under the gait area (red) is shown. The positive
sense (arrows) is given by the direction in which the gait
is executed. By computing the volume of the CCF surfaces
for each component (k), we obtain the cBVI component, ζk.
Finally, ¯g1= exp(ζ)is the approximate net displacement of
the FRS-base due to the chosen gait.
5In se(3) ∼
=R6,k∈[1,6], where the first three basis are translational
and the last three are rotational.
8 PREPRINT
4
2
0
-2
-4
-5
0
-0.2
-0.4
0.2
0.4
0
5
1
0.5
0
-0.5
0
0.5
0
0.05
0.1
0.15
0.2
0.25
q1
q1q2
q2
(DAl)k
gk
Fig. 3. CCF surface for the kth basis corresponding to a gait (red
circle) using 2-joints. Mesh volume (right) under the gait area provides
an approximate estimate of FRS-base displacement per gait.
In [14, eq. 20], an alternative approximation of (13) was
used. In any case, the critical quantity in FRS-locomotion
methods from geometric mechanics [12], [13], [14] is DAl(q).
Thus far, curvature has been done symbolically computed,
which limits applicability to simple FRS. To the best of our
knowledge, there is no recursive method for this purpose. To
this end, we report the following result.
Theorem 2: In recursive form, the local curvature [15,
Def. 3.5], DAl(q, X )Y=dAl(X)−adAlXAlY, is written
using the result from Theorem 1 as,
DAl(q, X )Y=Mb(q)−1B⊤(q, X)Y , X, Y ∈Rn(39)
where,
B(q, X ) = X
kJ⊤
k(Mk∇JkX+ 2MkadJkXAd−1
1k
−S(AlX)T+A⊤
lP(X) + A⊤
lad⊤
AlXMb
(40)
Proof: The recursive form of the curvature is obtained
by matching (11) and bottom row of (14). Following Remark
2, by elimination, the only remaining terms are −˜
B(µ) ˙qin
(14) and −(DAl(q, ˙q))⊤Mbµin (11), which are equal6.
Considering a generalized velocity Xinstead of ˙q, (15e)
is reformulated as −˜
B(µ)X=B(q, X )µto obtain the new
matrix operator in (40). This reformulation of −˜
B(µ)X
is performed by applying the properties adXY=−adYX,
(3), and the proposed Prop. 1. Hence, by equating,
−˜
B(µ)X=B(X)µ=−(DAl(q, X ))⊤Mbµ, we get the result
in Theorem 2. In particular, we further isolate the non-
conservative part as,
dAl(X)Y=−M−1
b−S(AlX) + P(X)Al
+X
kJ⊤
k(Mk∇JkX+ 2MkadJkXAd−1
1k⊤(41)
In the next section, we will demonstrate how the curvature
in Theorem 2 is exploited for gaits using more than 2-DoF.
VI. AP PL ICATION OF PROPO SE D DYN AM IC S
In this section, we emphasize the merit of Theorems 1 and
2 through practical FRS applications. In Sec. VI-A, an output
feedback stabilization problem of an FRS is analyzed using
6This observation has also been remarked in [25, §4]
a Lyapunov-based approach. This highlights the benefit of
the proposed CC matrix structure from Theorem 1 and its
reported properties. In Sec. VI-B, we demonstrate the utility
of Theorem 2 for curvature computation, which is required in
geometric nonholonomic locomotion methods.
A. Output Feedback Stabilization with Non-zero
Momentum
We consider the shape (joints) control of an orbital robot,
i.e. gravity-free FRS, without spacecraft (FRS-base) actuation
(F1= 06) for the non-zero momentum case (J 6= 06),
see Fig. 4. For the full state-feedback case, this problem has
received limited attention [27], [28] in literature7. The key
idea in these works was to isolate the disturbing CC terms in
shape dynamics and correspondingly compensate for them in
the control law. In both works, the momentum equation was
written in the basis of the spatial momentum, J. The reported
disadvantage of the results was that the shape dynamics CC
terms were coupled with the SE(3) variable, g1, through R1,
as discussed in Sec. III-B. As a result, the controllers required
attitude measurements. However, we remark that this was an
apparent coupling because in the proposed Prop. 3, we proved
that the shape dynamics is completely independent of g1.
{1}
{O}
J 6= 0
Fig. 4. Diagram of an underactuated (F1= 06) orbital robot (FRS)
with non-zero momentum, i.e. J 6= 06. A controller is designed to
regulate the shape (joints) while considering an unmeasured spacecraft
velocity.
In [27], a matrix transformation of the recursively-computed
CC matrix in (4) was used to obtain the compensation torques.
Note that although such a transformation reveals a useful
block-diagonal inertia, it does not naturally provide separated
velocity dependencies in the CC matrix. This resulted in
compensation torques, which were a function of both (J,˙q)
and cancelled out power-preserving terms, as will be shown in
the following analysis. In contrast, [28] employed a Routhian
approach, which revealed compensation torques that only
depended on J. However, the CC terms were reported as
partial derivatives and lacked recursive formulations, which
limits applicability for systems with high DoF.
To motivate the problem of output feedback stabilization,
we note that the orbital robot is a FRS, which is composed
of a high-bandwidth robotic subsystem and low-bandwidth
7In [27], an end-effector task was addressed, but the presented theory is
equally applicable for joint task like in [28].
H. MISHRA et al.: ON THE DYNAMICS OF FLOATING-BASE ROBOTS 9
spacecraft’s Guidance-Navigation-Control (GNC) system. The
aforestated approaches [27], [28] are contingent on the estima-
tion of the spacecraft states [29] by the GNC system. However,
this premise ignores the negative effects [29, pp. 12] of low
bandwidth of the GNC bus, which limits the GNC estimation
rate. Hence, despite the high bandwidth of the robotic sys-
tem (≈1[kHz]), feedback control is not. Alternatively, while
confining completely to the robotic system, an exteroceptive
pose sensor (e.g. camera) may be used at a marginally higher
rate ≈10[Hz] [30, §6.1]. In this approach, the spacecraft
velocity is the unmeasurable state (due to the lack of a
sensor) and it needs to be estimated. Therefore, we consider
the aforestated control problem for an orbital robot with an
unmeasured spacecraft velocity. The overarching goal is to
exploit the measurements, which are readily available on the
robotic system, so that the proposed controller is implemented
in situ. Hence, we propose the controller as an interconnection
of a spacecraft velocity observer and a shape (joints) control
law. In particular, the observer consists of kinematics and
dynamics equations with error injection, while the control law
is the sum of feed-forward (to compensate J 6= 06) and task-
based torques.
The estimation problem of the spacecraft states, namely
(g1, V1), is formalized next. Let ˆr= (ˆg1, q)∈ Q denote the es-
timated configuration, where ˆg1is the spacecraft pose estimate
of a virtual frame {ˆ
1}, see Fig. 5. The corresponding velocity
of this configuration is ˆ
V=ˆ
V⊤
1˙q⊤⊤. The configuration
state estimation error, η∈SE(3) between ˆg1(t)and g1, is
defined8as η(g1,ˆg1) = ˆg−1
1g1. Using the log map, we obtain
an error term as log(η) = ǫ∧∈se(3) and ǫ=ψ⊤
1r⊤
1⊤,
where ψ1and r1are the orientation and translational errors,
respectively (see [21]).
g1
ˆg1
qk
qk
{1}
{ˆ
1}
{O}
Fig. 5. An orbital robot (right) with configuration (g1, q), where
g1∈SE(3) is the spacecraft frame’s pose, {1}, relative to the inertial
frame, {O}. For its estimate (left), the observer problem is to use the
available measurements, (q, ˙q), and a slow-sampled g1(dashed) to
estimate V1.
The key idea here is to ensure that the state ˆµ(t)→µ(t)
as ˆg1(t)→g1(t), which results in ˆ
V1(t)→V1(t)and is our
observer design goal. Next, the proposed observer and the
control law will be described as a part of the controller through
the main equations. The design is divided into three parts.
1) SE(3)Group Observer:The kinematic part of the ob-
server is chosen with the same geometric structure as (13)
8ηis a left-invariant error [31, eq. 6], i.e η(gg1, g ˆg1) = η,g∈SE(3).
with an error injection term as follows,
˙
ˆg1= ˆg1ˆµ−AdηAl˙q
|{z }
ˆ
V1
+ AdηKǫ∧
(42)
where K:R6→R6is the observer kinematic gain which is
determined through stability analysis in Sec. VI-A.4.
Error kinematics: The observer error kinematics are derived
by taking the time derivative of the pose error, η= ˆg−1
1g1, as,
˙η=−ˆg−1
1˙
ˆg1ˆg−1
1g1+ ˆg−1
1˙g1
⇒˙η=η(µ−Ad(η−1)ˆµ−Kǫ)∧
⇒(η−1˙η)∨=µ−Ad(η−1)ˆµ
|{z }
µe
−Kǫ (43)
where µeis the observer velocity error. Using the differential
of exponential from [21, Lemma 2] on (43), we obtain,
˙ǫ=Br(ǫ)(η−1˙η)∨=Br(ǫ)(µe−Kǫ)(44)
where Br(•)is the SE(3) Jacobian [21, Th. 2].
2) Euler-Poincar ´
e Observer:Before describing the equa-
tions of the dynamics observer, we need to obtain a vector
comparison between the locked velocity (µ) and the observer
velocity (ˆµ). The Adη−1term acts as the transport oper-
ator, which helps in defining the correct velocity error as
µe=µ−ˆµo, where ˆµo= Adη−1ˆµ, as is evident in (43).
Following the discussion above, we compute the velocity
error dynamics by taking the time-derivative of µeand using
(43), as follows,
d
dtµe=d
dt (µ−Ad(η−1)ˆµ)
= ˙µ−Ad(η−1)˙
ˆµ−ad(µe−Kǫ)ˆµo(45)
Substituting for ˆµo=µ−µeand using the properties,
adXX= 06and adXY=−adYXin (45), we get,
d
dtµe= ˙µ−Ad(η−1)˙
ˆµ+adµµe+adKǫ ˆµo(46)
Therefore, the observer goal is to determine Ad(η−1)˙
ˆµ.
To this end, the observer for the Euler-Poincar ´e equation is
proposed with a geometric structure similar to the first row of
(14) with locked velocity ˆµo,
MbAd−1
η˙
ˆµ=−1
2P( ˙q)−ad⊤
ˆµoMbˆµo
−1
2S(ˆµo) + ad∼
MbˆµoAl˙q+Fo−MbadK ǫ ˆµo
(47)
where Fo∈R6∼
=se(3)∗is a design input force which is
determined by stability analysis in Sec. VI-A.4, and the last
term is simply meant to cancel out the corresponding term in
(46).
Velocity error dynamics: Substituting (47) into (46), and
using actual locked dynamics ( ˙µ) from (14), we get,
d
dtµe=Mb−1−1
2P( ˙q)µe−1
2S(µe) + ad∼
MbµeAl˙q
+ (ad⊤
µMbµ−ad⊤
ˆµoMbˆµo−Mbadµµe)− Fo(48)
where µ−ˆµo=µehas been used for simplification.
The bracketed ad⊤terms can be further simplified
10 PREPRINT
in terms of ˜
Coperator using Lemma 6 in Appendix.
Denoting, Γ1( ˙q, µ, µe) = −1
2P( ˙q) + ˜
C(µ, µe)and
˜
S(µe) = −(1
2S(µe) + ad∼
MbµeAl), (48) is rewritten as,
dµe
dt =Mb−1Γ1( ˙q, µ, µe)µe+˜
S(µe) ˙q− Fo(49)
Note that the velocity error dynamics in (49) is a function
of the controlled quantities µ, ˙q. Thus a separation principle
between observer and controller design is not feasible since
both,µand ˙qmust be additionally stabilized. In the considered
scenario, the spacecraft has no external actuation, while the
robotic system acts as a payload to perform the manipulation
task. Hence, a task-based control law is proposed next to
stabilize ˙q, while µis treated as an external input.
3) Shape-space Controller:Therefore, the controller con-
sists of the aforementioned observer equations and addition-
ally, a shape (joints) control law. We propose the control law
with two types of decoupling compensation torques, ˆτi∈Rn,
which generalizes the results in [27], [28] using the same
dynamic description from Theorem 1 as,
τ=τc+ ˆτi, i = 1,2
ˆτ1=−(1
2S(ˆµo)⊤− A⊤
lad∼
Mbˆµo)ˆµo+˜
B(ˆµo) ˙q
ˆτ2=−(1
2S(ˆµo)⊤− A⊤
lad∼
Mbˆµo)ˆµo
(50)
where τc∈Rnis a task controller, which is required to
perform a manipulator task, and is determined through stability
analysis in Sec. VI-A.4.
Shape-space velocity dynamics: Denoting Γq1=˜
Γ′
q( ˙q)and
Γq2=˜
Γ′
q( ˙q) + ˜
B(µ)corresponding to iin (50) for ease of no-
tation, the shape velocity dynamics are written by substituting
(50) in second row of (14) as,
¨q=Λ−1
q−Γqi ˙q+1
2(S(µ)⊤µ−S(ˆµo)⊤ˆµo)
− A⊤
lad∼
Mbµµ−ad∼
Mbˆµoˆµo+τc(51)
The quadratic terms in (51) are simplified by, firstly, substitut-
ing µe=µ−ˆµoand then applying the commutative Prop. 1
for S⊤to get
S(µ)⊤µ−S(ˆµo)⊤ˆµo=2S(µ)⊤−S(µe)⊤µe(52)
Secondly, using (3), followed by Lemma 6 (in Appendix)
for the Coperator, we get,
A⊤
l(ad∼
Mbµµ−ad∼
Mbˆµoˆµo)
=A⊤
l(ad⊤
µMbµ−ad⊤
ˆµoMbˆµo) = A⊤
lC(µ, µe)µe
(53)
Hence, substituting (52) and (53) in (51), and adding and
subtracting A⊤
lad∼
Mbµeµeto get ˜
S⊤from S⊤(see (49)), we
obtain the shape velocity dynamics as,
¨q= Λ−1
qτc−Γqi ˙q+˜
S(µe)⊤+ Γ2(µ, µe)µe
Γ2(µ, µe) = S(µ)⊤− A⊤
l(C(µ, µe) + ad∼
Mbµe)
(54)
Fig. 6 is the block diagram of the proposed controller (blue)
with the corresponding equations for an orbital robot (orange).
Proposed
Observer
Control law
+
Controller
Orbital Robot
(Plant)
g1
(ˆg1,ˆ
V1,ˆµ)
(q, ˙q)τc
τ
ˆτi
g1(k)(42),(47)
(50)
(4)
Fig. 6. A block-diagram of the output feedback stabilization controller
(blue) for an orbital robot with partial state measurements, (g1, q, ˙q).
4) Stability analysis:Here, we prove the stability of the
proposed method’s error dynamics, i.e. (44), (49) and (54). For
the task-oriented control input, τc, in (50), given a task setpoint
qd∈Rn, a positive potential function Φq(q, qd)is defined with
bounds as Φq<Φq<Φq. Using its differential dΦq, the time-
derivative is ˙
Φq=dΦ⊤
q˙q. We denote ∆q=q−qdas the task
error, and thus Φq≡Φq(∆q). Therefore, using (44), (49) and
(54), the error dynamics is compactly written as,
˙x=A(x, µ)x+G(q)u(55)
where x=ǫ⊤µ⊤
e˙q⊤∆q⊤⊤,u=F⊤
oτ⊤
c⊤and
G⊤=06,6−Mb−10n,60n,6
06,n 06,n Λ−1
q0n,n
A=
−BrKBr06,n 06,n
06,6Mb−1Γ1Mb−1˜
S06,n
0n,6Λ−1
q(˜
S⊤+ Γ2)−Λ−1
qΓqi 0n,n
0n,60n,6In,n 0n,n
Hence, the control problem for the orbital robot is reduced
to choosing usuch that the closed-loop dynamics of (55) ex-
hibits a stability property. Note that (55) is a non-autonomous
system due a to dependency on the external input µ.
Lemma 2: For an orbital robot, defined by (4) or (14), an
output feedback stabilization controller consisting of
1) SE(3) group observer defined as (42) with an error
function η= ˆg−1
1g1such that tr(η(0)) 6=−1,
2) an Euler-Poincar ´e observer as (47),
3) a shape-space control law as (50),
such that the input variable uin (55) is chosen using,
Fo=kB⊤
rǫ+P( ˙q)ˆµo+ad⊤
Al˙qMbˆµo+MbadAl˙qˆµo, with the
observer parameter k > 0, and τc=−dΦq(q, qd)−Dq˙qfor
the task with potential Φq(q, qd)and damping gain, Dq≻0,
then the closed-loop error dynamics in (55) ensures uniform
asymptotic stability of the state, x.
Proof: To use Lyapunov’s direct method, the Lyapunov
candidate is chosen as,
W=1
2hǫ, ǫiK
| {z }
W1
+1
2hµe, µeiMb
| {z }
W2
+1
2h˙q, ˙qiΛq
|{z }
W3
+Φq(∆q)(56)
such that K≻0. We consider Wfor an open connected region
x(0) ∈Ψ⊂R12+2n, and there exist bounds as,
α(||x||)≤W(x)≤α(||x||)(57)
H. MISHRA et al.: ON THE DYNAMICS OF FLOATING-BASE ROBOTS 11
Taking time-derivative along trajectories and using error dy-
namics in (55), we get,
˙
W=x⊤Rx+ ˙q⊤(τc+dΦq)−µ⊤
eFo(58)
where R=
−KBrK KBr06,n 06,n
06,6(Γ1+1
2˙
Mb)˜
S06,n
0n,6(˜
S⊤+ Γ2) (−Γqi +1
2˙
Λq) 0n,n
0n,60n,60n,n 0n,n
We choose K=kI6,6as a scalar to apply the property
Br(ǫ)ǫ=ǫ[21, Lemma 3] to the block matrix (1,1) position.
The block matrix position (2,2) is eliminated due to, 1st in
skew-symmetric Prop. 2 and, also, skew-symmetric ˜
Cfrom
Lemma 6 contained in Γ1. We remark that, to eliminate the
block matrix (3,3) position, if i= 1, the 2nd skew-symmetric
Prop. 2 is used, whereas for i= 2, the 4th of Prop. 2 is also
invoked. Hence, we get,
˙
W=−k2||ǫ||2+kǫ⊤Brµe+ ˙q⊤(Γ2+ 2 ˜
S⊤)µe
+ ˙q⊤(τc+dΦq) + µ⊤
eFo
(59)
In (59), the term ˙q⊤(Γ2+ 2 ˜
S⊤)µeis simplified by applying
Lemmas 7 and 8 in Appendix. This leads to,
˙
W=µ⊤
ekB⊤
rǫ−P+ 2ad⊤
Al˙qMbµe
+P( ˙q) + MbadAl˙q+ad⊤
Al˙qMb)µ− Fo
+ ˙q⊤(τc+dΦq)−k2||ǫ||2
(60)
Finally, choosing Fo, τcfrom Lemma 2, we get,
˙
W=−k2||ǫ||2−˙q⊤Dq˙q+µ⊤
e(MbadAl˙q−ad⊤
Al˙qMb)
|{z }
skew-symmetric
µe
⇒˙
W≤ −k2||ǫ||2−σ(Dq)||˙q||2≤0(61)
which proves the uniform stability of the state xabout origin.
In the following, we use positive constants, ci>0and to prove
asymptotic stability, the following steps are needed. Firstly,
note that that ||µ(t)|| < c1due to the boundedness of || ˙q||,
which results from (61), and that ||µ(0)|| is also bounded.
This is verified by evaluating the top row of (14). Using
this observation and (61), we conclude that |¨
W|< c2, which
allows us to invoke Barbalat’s Lemma and prove that ˙
W→0,
which means ||ǫ||,|| ˙q|| → 0.
For asymptotic stability, we invoke Matrosov’s theorem [32]
for non-autonomous systems with two nested auxiliary func-
tions. Firstly, choosing W1=W1+W2−µ⊤
eMbǫ, as in [21],
we get lim ˙
W→0˙
W1=−hµe, µeiMb, which is sign-definite
and non-zero. Satisfying conditions of Matrosov theorem,
we conclude ||µe|| → 0. Secondly, choosing W2=hdΦq,˙qi,
we get lim(˙
W ,||µe||)→0˙
W2=−hdΦq, dΦqiΛ−1
q, which is
also sign-definite and non-zero. Hence, we conclude that
||dΦq(∆q)||,||∆q|| → 0. Using these arguments, uniform
asymptotic stability of xfollows.
Thus, the structure in Theorem 1 resulted in two novel
control laws, each of which which bears a dependency sim-
ilarity to those reported in both, [27] and [28]. However,
we additionally treat the output feedback stabilization case.
In particular, upon observer convergence, the compensation
torques, ˆτ1, in (50) achieved complete decoupling from µ,
but this resulted in a (µ, ˙q)dependency. We point out that
although an analogous dependency, (J,˙q), was found in
[27], the control law did not achieve complete decoupling.
This is because matrix transformations of the CC matrix do
not lead to separation of velocity dependencies. Through ˆτ2,
we showed that complete decoupling is not necessary for
asymptotic convergence, and that the compensation of only
the µ-quadratic terms is sufficient. This observation was made
in [28], but for J-quadratic terms, and was a consequence of
the Lagrangian dynamics description. Indeed, by linking the
recursive dynamics with the REL equations, we inherited its
structural advantages while finding applicability to FRS with
higher DoF. It is worth noting that the control law in Lemma
2, in contrast to [27], [28], was shown to be independent of
the group variable, g1. Most importantly, the commutativity in
Prop. 1 and the skew-symmetric Prop. 2 were pivotal to the
stability analysis that led to the result in Lemma 2. To the best
of the author’s knowledge, such an analysis would not have
been possible with existing FRS dynamics descriptions.
B. Nonholonomic locomotion
Consolidating the ideas in Sec. V for µ= 06, the question
that we pose in this section is: Given a closed path in shape-
space, q(t)∈Rn, can the net displacement in FRS-base
be estimated without explicitly solving the integral of (35).
To this end, we rely on the established results from geo-
metric mechanics [11], [12], [13] for 2-DoF gaits. However,
we generalize these 2-DoF gait-approaches to higher DoF
shape variables by separating the gait-space from shape-space.
Firstly, let us consider the case of executing gaits using m-
joints, where 2≤m≤n. Secondly, let r=r1r2⊤be the
gait coordinates which are related to the shape-space through
a linear map as,
r= Ψ(q−qc)⇒˙r= Ψ ˙q(62)
where qc∈Rnis the origin of the gait coordinate system
and Ψ∈R2×mmaps the basis of shape to gait, like a
transformation matrix. Note that Ψis not invertible if m > 2.
The main concept is illustrated in Fig. 7, where a circular gait
(on a hyperplane) in an anti-clockwise sense is executed using
m= 3 shape variables.
For this concept, we exploit the proposed method from The-
orem 2 in the following algorithm to answer the introductory
question of this subsection.
1) Select mjoints out of nin shape-space which are used
to design the gait.
2) Create a discrete set of points in the domain of the gait,
i.e, (r1, r2)∈ D, where ϑr≡(−π, π ]×(−π, π].
3) Given Ψ−1, for all r∈ D, compute discrete set of points
in domain of shape, q=qc+ Ψ−1r∈ϑq.
4) Express the normalized basis vectors of rin terms of
normalized basis of qas e1, e2∈Rn.
5) Choose a minimum perturbation coordinate frame,
{V}, and use the Adjoint action of the pose
12 PREPRINT
q1
q2
q3
r1
r2
qc
q(0)
R
R3
R2
Fig. 7. A2-DoF gait-space on a hyperplane with gait variables (r1, r2)
and its origin at qcin a surrounding 3-DoF shape space with shape
variables (q1, q2, q3), which execute the gait starting at q(0) in an anti-
clockwise sense.
gv1∈SE(3) to map the curvature9to the new frame
as DAv
l= Adv1DAl.
6) For each q∈ϑq, compute the CCF using the expression
DAv
l(q, e1)e2using (39) from Theorem 2 and item 5.
7) For a given closed path, q(t)∈Rn, compute the the
volume of DAv
l(q, e1)e2under the trajectory to obtain
the cBVI component-wise, i.e. ζi.
8) Compute exp(ζ)to approximately estimate the net dis-
placement in the frame {V}.
9) Using gv1(0) = gv1(1), the net displacement of the FRS-
base is computed as g1(1) = exp(ζ)gv1(1).
As an example, we used a LWR-4+ robot with
n= 7 joints, whose dynamic/kinematic parameters were
reported in [2]. The FRS-base was chosen with a
mass, mb= 3.5[Kg] and principal moments of inertia,
Ib≡(0.12,0.14,0.12)[Kg.m2]. The gait-shape map was cho-
sen as ψ−1=0 1/√2 1/√2 0 03
0 0 0 1 03⊤
for a circular
gait with qc=q(0) = 07to be executed using joints numbered
2,3,4,m= 3. The gait was commanded to the FRS as
˙r=π
2sin(πt) cos(πt). The basis vectors were chosen
as ei=ψ−1(i), i = 1,2, where ψ−1(i)is its ith column. In
accordance to [33], a coordinate system located at the CoM of
the FRS and oriented along its instantaneous principal axes, i.e.
virtual chassis frame, was chosen as the minimal perturbation
coordinate frame {V}. In this frame, the locked inertia is
completely diagonalized. With this step, the translation com-
ponents k= 1,2,3can be ignored since the CoM of the
FRS does not change due to shape motion, i.e. the curvature
components are zero. This was also verified in simulation.
The above steps were performed with a discrete grid size
of 30 ×30 for ϑr. The specific CCF surfaces are shown in
Fig. 8 with the gait overlaid in white. By visual inspection,
it can be seen that the encased volumes in k= 4,5bases
are small and the maximal displacement is expected along
the negative k= 5 basis (enlarged). Indeed, upon integrating
(35), the final orientation after the gait was found to be
(−0.4228,0.1682,−13.73)[°] in XYZ sense. By using the ap-
proximation, it was found to be (−0.9623,0.9873,−13.25)[°].
Using the pose estimate ¯g1= exp(ζ∧)and ground truth g1, the
9Equivariance property of the curvature form [15, eq. 3.3.1]
error was ||log(g−1
1¯g1)|| = 0.017. The approximation error is
comparable to the results obtained in [11].
1
0
-1
1
0
-0.2
0
0.2
-1
1
0
-1
1
0
-0.5
0
0.5
-1
1
0.5
0
-0.5
-1
1
0
-1
1
-1.5
-1
-0.5
0
0.5
1.5
(DAv
l)4
(DAv
l)5
(DAv
l)6
r1
r1
r1
r2
r2
r2
Fig. 8. CCF surface plots for the rotational bases k= 4,5,6of the
virtual chassis frame, {V}, for a 2-DoF circular gait (white) using 3-
joints.
The key point here is that the recursive method in Theorem
2 computes the exact curvature quantity without introducing
any errors of its own. Additionally, the main advantage of the
recursive computation over the symbolic method is the ease of
adding/removing joints seamlessly to the gait analysis using
the same FRS model. In fact, this flexibility motivated us to
analyze gait and shape as separate spaces, so that the existing
2-DoF gait approaches can be extended to the case of higher
shape DoF, for which we proposed an algorithm.
VII. CONC LU SIO N
Hence, in this paper, we established a congruence between
REL equations from geometric mechanics and the recursive
forms from FRS literature as the overarching objective. In
this pursuit, we derived a novel closed-form computation of
the CC matrix, which clearly shows the separation of locked
and shape-space velocity dependencies. From the proposed
dynamics in (14), we obtained three new results. Firstly, we re-
vealed a more elaborate form of the skew-symmetric/passivity
property in Prop. 2, which is not only a reformulation, but
also has practical significance in control design. Secondly,
we established commutativity properties between two sub-
matrices of the CC matrix in Prop. 32 and demonstrated its
utility in aiding Lyapunov-based observer design. Finally, we
also derived a closed-form expression for the curvature form
of the FRS in Lem. 2, which is a key quantity in locomotion.
APPEN DI X
1) Matrix Representation:Given g≡(R, p)∈SE(3) with
body velocity twist, V=ω⊤v⊤⊤(declared in Sec. II),
the following quantities are detailed,
g=R p
01,31,V∧=ω×v
01,30
Adg=R03,3
p×R R ,adV=ω×03,3
v×ω×(63)
where (•)×is a skew-symmetric matrix for the vector and, ω
(v) is the angular (linear, respectively) velocity. For kth-link
(see Sec. II) with momentum hk=hT
ωhT
vT=MkVk,
Mk=Ik03,3
03,3mkI3,3,ad∼
MkVk=hω×hv×
hv×03,3(64)
H. MISHRA et al.: ON THE DYNAMICS OF FLOATING-BASE ROBOTS 13
Property 4: Corresponding to a group action of g∈SE(3),
the following properties hold,
•AdgadVAd−1
g=adAdgV(65a)
•Ad⊤
gad∼
MVAdg=ad∼
Ad⊤
gMV(65b)
Property 5: Given the FRS (see Fig. 1) defined in Def. 1,
for V ∈ R6∼
=se(3),
X
k
Ad−⊤
1kad⊤
Ad−1
1kVMkAd−1
1k=ad⊤
VMb(66)
Proof: Using (65a) in Prop. 4 for simplification of L.H.S,
followed by substitution of the recursive expansion of Mbfrom
(7), we get the result.
2) Proof of Lem. 1 (10):The momentum equation [16, eq. 6]
is,
hdp
dt , ηi=hp, [(−Al˙q+ Λ−1
bp)∧, η∧]∨i, η ∈R6(67)
Moving ηto the left on L.H.S and R.H.S after using SE(3) Lie-
bracket isomorphism, [X∧,•]∨=adX, X ∈R6, substituting
p=Mbµand, eliminating ηyields the result.
3) Proof of Lem. 1 (11):To obtain closed-form expressions
for ˜
Nin (11), we recall [24, eq. 3.11.19]10, which provides a
scalar product form in terms of body momentum, p=Mbµ,
as,
h˜
N , δqi=−Dp, dAl(q, ˙q , δq)−[(Al˙q)∧,(Alδq)∧]∨
+1
2
∂(Mb−1p)
∂q δq + [(Mb−1p)∧,(Alδq)∧]∨E
(68)
In (68), the isomorphism, [X∧,•]∨=adXas in Lem. .2 is
used. Also, in vector notation, dAl(q, ˙q , δq) = dAl(q, ˙q)δq.
And, ∂(Mb
−1p)
∂q =−Mb−1∂ Mb(Mb
−1p)
∂q . Moving all δq terms
towards the left in both L.H.S and R.H.S of (68), and substi-
tuting p, we get,
hδq, ˜
Ni=Dδq, −(dAl(q, ˙q)⊤− A⊤
lad⊤
Al˙q)Mb(q)µ
+1
2
∂µ⊤Mb(q)µ
∂q − A⊤
lad⊤
µMb(q)µE(69)
Removing δq variations, we get (12).
4) Proof in Def. 5:The following Lemma, which is an
application of [3, Prop. 4], is key to the proof and is stated in
the context of (74).
Lemma 3: Given a column-wise detail of Jacobian as,
Jk=J1
k... Jn
kfor the kth link, using [3, Prop. 4] for
the jth joint and a velocity X∈R6∼
=se(3), we have,
∂Ad−1
1kX
qj
=−adJj
kAd−1
1kX
⇒∂Ad−1
1kX
∂q =h−adJ1
kAd−1
1kX ...i(70)
10In [24, eq. 3.11.19], the µ-dependent terms are on the L.H.S.
Lemma 4: The partial derivative of the scalar form
hx, yiMb, x, y ∈R6with respect to qis computed as,
∂
∂q hx, yiMb=X
k=bΠk(x)⊤+˜
Πk(x)y(71)
X
k=b
Πk(x)⊤=X
k=b
J⊤
kad⊤
Ad−1
1kxMkAd−1
1k(72)
X
k=b
˜
Πk(x) = X
k=b
J⊤
kad∼
MkAd−1
1kxAd−1
1k(73)
Proof: Firstly note that,
∂
∂q hx, yiMb=∂
∂q X
k=b
x⊤Ad−⊤
1k(q)MkAd−1
1k(q)y
=X
k=b
Πk(x)⊤y+x⊤Πk(y)
(74)
where Πk(x) = Ad−⊤
1kMk
∂Ad−1
1kx
∂q . Now, we invoke
the result of Lemma 3, and apply the property,
adJj
kAd−1
1kx=−adAd−1
1kxJj
kin it. Isolating, Jj
kto obtain
Jk, we first obtain ∂Ad−1
1kx
∂q =adAd−1
1kxJk. Substituting this in
Πkyields (72).
The result in second of (74) can be conveniently rewritten as
a linear operator form as (71) such that xTΠk(y) = ˜
Πk(x)y.
This velocity exchange property appears similar to the one in
(3) and is actually a consequence of it. By simply exploiting
the property in (3) in x⊤Πk(y), we obtain (73).
Lemma 5: (Proof of Prop. 3): The first row is simply a
tensor Adjoint transformation due to g1ctranslation. In the
second row, the former is an outcome of (65b), while the latter
is simply an Adjoint transformation of the local mechanical
connection. The third row is also an Adjoint transformation
of the corresponding matrices. The last one follows from
Adjoint transformation of the local mechanical connection and
application of (65b).
Since we exploited body Jacobians ˜
Jk, which are invariant
to a right translation, invariance of Λq,˜
Γ′follows.
Lemma 6: For body velocities, V1,V2,Ve∈R6∼
=se(3),
such that, Ve=V1−V2, given inertia Λ, the following holds,
ad⊤
V1ΛV1−ad⊤
V2ΛV2=C(V1, Ve)Ve(75)
where C(V1, Ve) = (ad⊤
V1Λ + ad∼
ΛV1−ad⊤
VeΛ).
Proof: see [21, Lemma 6].
A useful skew-symmetric operator is defined using Cas,
˜
C(V1, Ve) = C(V1, Ve)−ΛadV1,˜
C⊤=−˜
C(76)
Lemma 7: (For (59)) Using commutative Prop. 32 for S, P ,
h˙q, µeiΓ2=hµe,˙qiΓ⊤
2
=µ⊤
eS(µ)−(Mbadµ−ad∼
Mbµ)Al˙q
=µ⊤
eP( ˙q) + (MbadAl˙q+ad⊤
Al˙qMbµ)µ
(77)
Lemma 8: (For (59)) Using commutative Prop. 32 for S, P ,
h˙q, µei2˜
S⊤=hµe,˙qi2˜
S=µ⊤
e−P−2ad⊤
Al˙qMbµe(78)
14 PREPRINT
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